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University of Southern California Dissertations and Theses
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Generation and degeneration of long internal waves in lakes
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Generation and degeneration of long internal waves in lakes
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GENERATION AND DEGENERATION OF LONG INTERNAL WAVES IN LAKES by Takahiro Sakai A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (AEROSPACE ENGINEERING) December 2008 Copyright 2008 Takahiro Sakai Dedication To Chieko, Emma and Luca. ii Acknowledgments I thank my advisor Prof. Larry Redekopp so much. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vi List of Figures viii Abstract xiv Chapter 1: Introduction 1 Chapter 2: Two-layer model 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Evolution model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 E®ects of environmental parameters . . . . . . . . . . . . . . . . . . . . . 14 2.4.1 E®ect of spatial wind stress distribution . . . . . . . . . . . . . . . 15 2.4.2 E®ect of Wedderburn number and depth ratio. . . . . . . . . . . . 17 2.4.3 E®ect of wind stress duration . . . . . . . . . . . . . . . . . . . . . 18 2.4.4 E®ect of layer aspect ratio . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.5 E®ectoftemporalwindstressvariationsandinternalwaveresonance 23 2.5 Down scaling of energy spectra . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 E®ects of variable topography and width . . . . . . . . . . . . . . . . . . . 28 2.7 Modelingofre°ectionanddissipationduringshoalingonslopingboundaries 32 2.7.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Chapter 3: Multi-modal model 41 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Derivation of a nonlinear, multi-modal system . . . . . . . . . . . . . . . . 43 3.3 The two-mode evolution model . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Wind forced response of two-mode model . . . . . . . . . . . . . . . . . . 57 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 iv 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Chapter 4: Large-lake model 71 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Linear hydrostatic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.5 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5.1 Evolution on uniform depth . . . . . . . . . . . . . . . . . . . . . . 87 4.5.2 Evolution on variable depth . . . . . . . . . . . . . . . . . . . . . . 100 4.6 Wind-forced evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Chapter 5: Numerical method 117 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Spectral representation of vector function . . . . . . . . . . . . . . . . . . 120 5.3 Linear Evolution Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3.1 Spectral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.3.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.4 Nonlinear Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.4.1 Model formulation and algorithmic approach . . . . . . . . . . . . 135 5.4.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Chapter 6: Concluding summary 144 Appendix A 152 Appendix B 154 Appendix C 157 v List of Tables 2.1 Parameters of the experimental runs. The density di®erence was set as ¢½¼ 20 kg m ¡3 for all runs. The maximum initial surface displacement ³ 0 is measured at the vertical end wall with a plus(+) or a minus(-) sign to distinguish ³ 0 is either above(+) or below(-) the equilibrium interface surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1 Coe±cientsofselectednonlineartermsfordi®erentmetalimnionthickness h 2 . The thickness and the strati¯cation of the hypolimnion are ¯xed as h 3 = 3 and N 2 h = 0, respectively. The smoothing parameter is chosen as ± =0:1 for h 2 =1 & 2, ± =0:05 for h 2 =0:5, and ± =0:025 for h 2 =0:25. 54 3.2 Coe±cients of selected nonlinear terms for di®erent strati¯cation N 2 h in thehypolimnion. Thethicknessesofthemetalimnionandthehypolimnion are ¯xed as h 2 =1 and h 3 =1, respectively. The smoothing parameter is ¯xed as ± =0:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Coe±cientsofselectednonlineartermsfordi®erenthypolimnionthickness h 3 . Thethicknessofthemetalimnion,thestrati¯cationofthehypolimnion and the smoothing paramter are ¯xed as h 2 =1, N 2 h =0 and ± =0:1. . . 55 3.4 ThemodalenergyE tr transferredfromverticalmode-1tomode-2andthe energy transfer coe±cients ® 1 & ® 2 for various h 2 , N 2 h and h 3 . (W =1:5) 66 4.1 Modalamplitudesbasedonthelineartheoryfordi®erentBurgernumbers B and wind forcing durations t 0 =T i . Amplitudes are normalized by the amplitude of the lowest wave mode (M1R1+). . . . . . . . . . . . . . . . . 109 5.1 Instantaneous error of isopycnal amplitude Z at t = 5:475 sampled at di®erent radial locations for various truncation limit N for time integra- tion with M3R3- wave mode. Values are normalized by the initial wave amplitude. Values in parentheses in r =0:35 column are obtained with a halved time step ¢t=0:0025. . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.2 Energy E computed at di®erent time for various truncation limits N for time integration with M1R1- wave mode. Values are normalized by their initial values at t=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 vi 5.3 Energy E computed at di®erent time for various truncation limits N for timeintegrationwithM3R13-wavemode. Valuesarenormalizedbytheir initial values at t=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.4 Energy E at t = 5 obtained at di®erent truncation limits N for various time step sizes ¢t for time integration with M1R1- wave mode. Values are normalized by their initial values at t = 0. No value (-) implies that the time integration is unstable due to excessive time step size. . . . . . . 135 vii List of Figures 2.1 Two-layer density strati¯ed lake model. . . . . . . . . . . . . . . . . . . . 9 2.2 (a) Topography is even-extendedin the computational domain. The wind stress ¿ s is halved and mirrored about x=L. Direction of the wind stress in the left and the right half domain are opposite each other. (b) Uni- formandsinusoidalwindstressdistributionsde¯nedinthecomputational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Time series of the conserved quantities E 1 , E 2 and E 3 . . . . . . . . . . . . 14 2.4 Snap shots of the interface displacement f p for uniform (left column) and sinusoidal (right column) wind stress distributions. Wind stress blowing to the right withW =1 is applied for 0·t·1=4. . . . . . . . . . . . . . 16 2.5 Internal wave energy E 2 for uniform and sinusoidal wind stress distri- butions as a function of time. Energies are scaled by the energy of the uniform stress case at t=1=4.. . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 (a) Energy calculated for di®erent Wedderburn numbersW as a function of layer thickness ratio h 1 =h 2 . (b) Energy calculated for di®erent layer thickness ratio h 1 =h 2 as a function of inverse Wedderburn numberW ¡1 . . 18 2.7 Energy input to the internal wave ¯eld as a function of the wind forcing duration. Energy is scaled by its value at t=1=4. . . . . . . . . . . . . . 19 2.8 Internal wave evolution under sustained, uniform wind blowing to the right. 19 2.9 (a) Energy calculated for di®erent wind forcing durations t 0 with a ¯xed Wedderburn number W = 1 as a function of the layer thickness ratio h 1 =h 2 . (b) Energy calculated for di®erent wind forcing durations with a ¯xed layer thickness ratio h 1 =h 2 = 1=5 as a function of Wedderburn number. Energies are scaled by the value at t 0 =1=4. . . . . . . . . . . . 20 2.10 Shockformationtime(t s ¡t 0 )asafunctionofinverseWedderburnnumber W ¡1 and the layer depth ratio h 1 =h 2 . Bold lines and thin lines represent the forcing duration t 0 = 1=4 and t 0 = 3=8, respectively. On break lines, no shock is formed during four seiche periods after the wind is turned o®. 21 viii 2.11 (a) Energy computed at t 0 = 1=4 as a function of the upper layer aspect ratio h 1 =L. Energies are scaled by the value at h 1 =L=0:002. (b) Energy versuswindforcingdurationt 0 forh 1 =L=0:002and0.0002. Energiesare scaled by the value at t 0 =1=4. . . . . . . . . . . . . . . . . . . . . . . . . 22 2.12 (a) Modeled periodic, pulse wind stress input function T(t). Time series of ¯eld energies for: (b) di®erent pulse duration T 0 with a ¯xed forcing period T w = T s and (c) di®erent forcing period T w with a ¯xed forcing pulse duration T 0 =1=4. Energies in graph (b) are scaled by the value for T 0 = 1=4 at t = 1=4, and energies in graph (c) are scaled by the value at t=1=4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.13 Energy gain as a function of forcing period T w for T 0 =1=4. . . . . . . . . 25 2.14 Wave number spectra as a function of time for Wedderburn numbers (a) W ¡1 =0:5, (b)W ¡1 =1 and (c)W ¡1 =1:5. The wave length ¸ is scaled by L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.15 Timeseriesof(a)totalspectralenergy,(b)energycontainedin0·¸ ¡1 · 2 and (c) energy contained in 10 · ¸ ¡1 · 40 for di®erent Wedderburn numbersW. Energies are scaled by the value forW =1 at t=1=4.. . . . 27 2.16 Sloping topography set up for the numerical simulation. The wind stress is applied in either (a) up-slope or (b) down-slope direction. . . . . . . . . 28 2.17 Internal wave evolution over the sloping topography with h b = 3h 1 for up-slope wind (left column), and down-slope wind (right column). . . . . 29 2.18 Shock formation time (t s ¡t 0 ) as a function of the slope hight h b .. . . . . 29 2.19 Variable width set up for numerical simulation. Depth is constant (h 2 = 5h 1 ). Windstressisappliedineither(a)narrowingor(b)wideningdirection. 30 2.20 InternalwaveevolutionthroughvariablechannelwidthwithW r =W l =1=3 for (a) narrowing-wind and (b) widening-wind. . . . . . . . . . . . . . . . 30 2.21 Shock formation time (t s ¡t 0 ) as a function of the contraction ratio W r =W l . 31 2.22 Comparison of temporal signal of the interface displacement f p sampled at x = 1=4 and x = 3=4 for (a) sloping topography case with h b = 3h 1 and (b) variable width case with W r =W l =1=3. . . . . . . . . . . . . . . . 32 2.23 (a) Modi¯cation of the sloping boundary and (b) schematic of the eddy viscosity function º s de¯ned along the modi¯ed slope. . . . . . . . . . . . 33 ix 2.24 (a) Dimension of the laboratory tank. The tank is 30 cm wide (constant). Threewavegages(WGA,WGB,WGC)areinstalledtomeasuretheinter- face surface displacement. (b) Initial interface surface tilt with upwelling at the slope. (c) Initial interface surface tilt with downwelling at slope. All these ¯gures are not to scale. . . . . . . . . . . . . . . . . . . . . . . . 34 2.25 Time series of the isopycnal surface displacement ³ p measured at WGB for Run 2, Run 12, Run 20 and Run 28. Solid lines are obtained from the numerical simulations. Dot-dash lines are obtained from the labo- ratory experiments by Boegman et al. (2005a). The wave signals under the left and the right arrows are incoming wave packet on to the slope and the re°ected wave packet from the slope, respectively. The re°ection coe±cients (see Figure 2.26) are computed for each pair of incoming and re°ected wave packet labeled by a number (1 or 2) under each arrow. . . . 37 2.26 Comparison of re°ection coe±cients E r =E i obtained from the laboratory experiments and the numerical model. Dash number in each label indi- cates the number of re°ection as indicated in Figure 2.25. Dash lines are §0:15 deviation from the equal line. . . . . . . . . . . . . . . . . . . . . . 38 3.1 Basin con¯guration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Pro¯le of buoyancy frequency N 2 , vertical mode-1 eigenfunction Á 1 and vertical mode-2 eigenfunction Á 2 for di®erent (a) metalimnion thickness h 2 , (b) hypolimnion strati¯cation N 2 h , and (c) hypolimnion thickness h 3 . . 53 3.3 Model stability limit in inverse Wedderburn number W ¡1 as a function of the metalimnion thickness h 2 for a ¯xed total depth (h 1 +h 2 +h 3 = 5). (h s =1:0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 Penetration of the wind stress into the metalimnion. . . . . . . . . . . . . 57 3.5 Evolution of the isopycnal amplitude for (a) mode-1 and (b) mode-2 with the wind stress penetration depth h s =1. . . . . . . . . . . . . . . . . . . 58 3.6 Evolution of the isopycnal amplitude for (a) mode-1 and (b) mode-2 with the wind stress penetration depth h s =1:5. . . . . . . . . . . . . . . . . . 60 3.7 (a) Modal energies and (b) modal forcing coe±cients as a function of the wind stress penetration depth h s for the metalimnion thickness h 2 = 1 (solid line) and h 2 =2 (dash line). (h 3 =3, W =1:5) . . . . . . . . . . . 62 3.8 Modal energies (upper row) and corresponding energy transfer E tr (lower row) as a function of time for (a) h 2 = 1 and (b) h 2 = 2. (W = 1:5, h s =1, h 1 +h 2 +h 3 =5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 x 3.9 Theisopycnalamplitude(upperrow),kineticenergydensity(middlerow) and potential energy density (bottom row) at t=3:25 over the right half domain for (a) mode-1 and (b) mode-2. Energies are normalized by the total system energy. (h 2 =1, h 3 =3, W =1:5, h s =1:5) . . . . . . . . . . 65 3.10 Modal energy transfer E tr (from mode-1 to mode-2) as a function of inverseWedderburnnumberW ¡1 fordi®erentstrati¯cationpro¯les. (h 3 = 3, h s =1:5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.11 Sloping depth con¯guration. . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.12 Isopycnal amplitudes of (a) mode-1 and (b) mode-2 over sloping topogra- phy at di®erent time. (W =1:5, h s =1:5).. . . . . . . . . . . . . . . . . . 68 3.13 Isopycnal amplitudes of (a) mode-1 and (b) mode-2 over uniform topog- raphy (h 3 =5) at di®erent time. (W =1:5, h s =1:5) . . . . . . . . . . . . 69 4.1 Eigenfrequency! asafunctionofscaledBurgernumberB=cforthelowest three azimuthal (M) and radial (R) modes. . . . . . . . . . . . . . . . . . 72 4.2 Variable depth model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 Evolution of the Kelvin wave (M1R1+). Snap shots of the isopycnal amplitude Z (upper row) and the magnitude of the velocity amplitude p U 2 +V 2 (lower row) are taken at (a) t = 0, (b) t = 8, (c) t = 16 and (d) t=24. Contour level step is 0.05 for all the plots. The lowest contour level shown in the velocity contour plot is 0.05. . . . . . . . . . . . . . . 88 4.4 Developed view of the Kelvin wave front (B = 8, A 0 =¡0:3, ¤ = 0:025, t=18). The front position and its propagating direction are indicated by an arrow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Azimuthal location of the minimum value of the isopycnal amplitude Z (the leading Kelvin wave trough) as a function of time. . . . . . . . . . . . 90 4.6 Timeseriesofthemaximumazimuthalgradientoftheisopycnalamplitude Z at r = 1 for: (a) di®erent amplitudes A 0 with ¯xed Burger number B = 4 and aspect ratio ¤ = 0:025; (b) di®erent Burger numbers with ¯xed jA 0 j = 0:3 and ¤ = 0:025; and (c) di®erent aspect ratios with ¯xed B =4 andjA 0 j=0:3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.7 Comparison of the isopycnal amplitude Z for di®erent Burger numbers (a)B =2, (b)B =4and(c)B =8att=0(upperrow)andt=16(lower row). Contour level step is 0.05.. . . . . . . . . . . . . . . . . . . . . . . . 92 4.8 Comparison of the isopycnal amplitude Z for di®erent aspect ratios (a) ¤ = 0:04, (b) ¤ = 0:03 and (c) ¤ = 0:02 at t = 17:5. Contour level step is 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 xi 4.9 ComparisonofpotentialenergyspectrumoftheKelvinwave(M1R1+)at t=12 for di®erent initial amplitudes (a) A 0 =-0.2, (b) A 0 =¡0:3 and (c) A 0 =¡0:4 along with the corresponding isopycnal amplitude Z. . . . . . 94 4.10 (a) Azimuthal modal energies of the Kelvin wave (M1R1+) at di®erent times. Energies in lower modes are magni¯ed in (b). . . . . . . . . . . . . 95 4.11 Azimuthal modal energies of the Kelvin wave (M1R1+) at t=20 for: (a) di®erent initial amplitudes A 0 with ¯xed Burger number B = 4; and (b) di®erent Burger numbers with ¯xed amplitude A 0 =¡0:3. . . . . . . . . . 96 4.12 Evolution of the Poincar¶ e wave (M1R1-). Snap shots of the isopycnal amplitude Z (upper row) and the magnitude of the velocity amplitude p U 2 +V 2 (lower row) are taken at (a) t=0, (b) t=0:7, (c) t=1:4 and (d) t=24. Contour level step for the isopycnal amplitude is 0.05. . . . . . 97 4.13 Time series of the minimum and the maximum values of the isopycnal amplitude Z for Poincar¶ e waves (M1R1-) of di®erent amplitudes. . . . . . 98 4.14 Comparison of potential energy spectrum of the Poincar¶ e wave (M1R1-) for di®erent initial amplitudes (a) A 0 =-0.2 and (b) A 0 =¡0:4 at t = 4, and (c) A 0 = ¡0:3 at later time t = 20 along with the corresponding isopycnal amplitude Z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.15 PseudorecurrenceofaPoincar¶ e-typewave(M1R1+withtheBurgernum- ber B = 1=2). Snap shots of the isopycnal amplitude Z are taken at (a) t = 0, (b) t = 8(1:97T), (c) t = 16(3:93T), (d) t = 24(5:9T) and (e) t=32(7:86T). Contour level step is 0.05. . . . . . . . . . . . . . . . . . . 99 4.16 Snap shots of the isopycnal amplitudes Z of the Kelvin waves at t = 8 (upper contour panels) and at t = 16 (lower contour panels) for di®erent depth pro¯les. The depth pro¯les is depicted on the top of each column. . 101 4.17 Timeseriesofthemaximumazimuthalgradientoftheisopycnalamplitude Z at r =1 for (a) slant depth pro¯le and (b) parabolic depth pro¯le. . . . 102 4.18 Comparison of the isopycnal amplitudes for the uniform depth case (Z 1 ) and the slant depth case (Z 2 ) at (a) t = 2:7(1:96T), (b) t = 3:4(2:47T) and (c) t = 4:1(2:98T). The contour level step for these plots are set to be 0.05. On the most right column the di®erence of Z 1 and Z 2 scaled by the initial amplitudejA 0 j is plotted with a contour level step 0.02. . . . . 103 4.19 Time series of the maximum and the minimum values of the isopycnal amplitudeZ ofthePoincar¶ ewaveovertheslantdepthpro¯lewith¢h=1. For comparison purpose the time series for the linear hydrostatic model and for di®erent depth cases are also included in the ¯gure. . . . . . . . . 104 xii 4.20 (a) Time series of the total energy for di®erent Wedderburn numbers W and (b) time series of kinetic (KE) and potential (PE) energies forW =2. 106 4.21 Total energy as a function of the Wedderburn numberW. . . . . . . . . . 106 4.22 (a)EnergyratioasafunctionofBurgernumberB fordi®erentwindstress pro¯les and (b) energy ratio as a function of the Wedderburn number W for di®erent Burger numbers. . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.23 FrequencypowerspectrumoftheisopycnalamplitudeZ sampledat(r;µ)= (1;0) for (a) nonlinear, non-hydrostatic model and (b) linear hydrostatic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.24 (a)Estimatedamplituderatioand(b)correspondingenergyratiobetween the Kelvin wave and the Poincar¶ e wave modes as a function of the Burger numberB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.1 Exact solution of the isopycnal amplitude Z for M1R1- wave mode at t = 0 (a), numerical solution at t = 13:75 (b), and corresponding vector ¯eld (c) obtained with the spectral truncation N =5. Contour level step is 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2 Exact solution of the isopycnal amplitude Z for M3R3- wave mode at t = 0 (a), numerical solution at t = 5:475 (b), and corresponding vector ¯eld(c)obtainedwiththespectraltruncation N =15. Contourlevelstep is 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.3 Evolution of a Kelvin wave (M1R1+) initial condition of amplitude A 0 = ¡0:3 in the weakly-nonlinear and weakly-dispersive model. Snap shots of isopycnal surface Z are taken at (a) t = 0, (b) t = 8, (c) t = 16 and (d) t=24. Contour level step is 0.05. . . . . . . . . . . . . . . . . . . . . . . . 140 5.4 Azimuthal modal energy spectrum at various time for the evolution ¯eld of the Kelvin wave initial condition (M1R1+). . . . . . . . . . . . . . . . . 141 5.5 EvolutionofaPoincar¶ ewave(M1R1-)initialconditionofamplitudeA 0 = ¡0:3 in the weakly-nonlinear and weakly-dispersive model. Snap shots of isopycnal surface Z are taken at (a) t = 0, (b) t = 3, (c) t = 3:4 and (d) t=24. Contour level step is 0.05. . . . . . . . . . . . . . . . . . . . . . . . 142 5.6 local minima and maxima of the isopycnal surface Z for di®erent initial ampliutdes (Poincar¶ e M1R1- mode) as a function of time. . . . . . . . . . 142 xiii Abstract Thenonlinearevolution,generationanddegenerationofwind-driven,basin-scaleinternal waves in lakes are investigated employing weakly-nonlinear, weakly-dispersive evolution models. The models studied are based on rational, asymptotic approximations of the hydrodynamicequationsofmotion,andincludeatwo-layermodel,amulti-modalmodel, and a large-lake model with the e®ect of earth's rotation. It is found that nonlinearity, in conjunction with the dispersive nature of the °uid medium, plays a principle role in (i)theearlystageofdegenerationofbasin-scalewavesthroughnonlinearsteepeningand subsequentgenerationofoscillatorywaves;and(ii)thetransferofenergyamongmultiple vertical modes in the internal ¯eld. Strong dependence of these nonlinear processes on the background strati¯cation, the lake geometry, the horizontal extent of a lake, and the spatio-temporal wind stress function are demonstrated and quanti¯ed through a series of numerical simulations of the di®erent models. xiv Chapter 1 Introduction Hydrodynamicmotionintheinterioroflakesiscrucialtoissuessuchaswaterqualityand ecological sustainability. The interior dynamics is energized primarily by wind action, with the deposition of energy into basin-scale motions through a transfer of the wind stress across the free surface boundary layer. In typical stably-strati¯ed lakes the action of a surface wind stress drives not only high-frequency surface waves, whose region of in°uenceismostlycon¯nedtotheupper-mixedlayer, butalsoforcesthroughvolumetric transport in the closed basin a signi¯cant tilting of the internal density ¯eld, inducing a basin-scale pressure gradient. Then, when the wind stress subsequently diminishes, the unbalanced, large-scale horizontal pressure gradient relaxes through a dynamic process leadingtotheappearanceofaspectrumofinternalwaveswhosestateatanysubsequent time determines, and dominates, the internal weather of a closed basin. The consequent internal wave ¯eld contains, in general, energy in scales ranging from basin-scale to those on the order of the mixed-layer depth, and in frequencies ranging from the seiche period up to the buoyancy frequency (Saggio & Imberger 1998). Motions encompassing this wide range of scales are generally propagative, possess three dimensional features, and can e®ectively transfer momentum and kinetic energy throughout the basin and to boundary domains where the energy is principally dissipated (Imberger 1994). Hence, internal waves serve as energy distributors and assert a pivotal role in driving transport and mixing processes in lakes, and they de¯ne the °ux path of biological and chemical particleswithinabasin(°uxpathproblem: Imberger1998). Understandingtheprincipal mechanismswherebybio-geochemicaltransportoccursinabasiniscrucialtoestablishing management procedures and remediation policies for vulnerable and fragile fresh water resources. 1 Hydrodynamicsofalakeismoreorlessa®ectedbyCoriolisaccelerationintroducedby the earth's rotation, and Coriolis acceleration asserts a determining role in lake motions when the horizontal scale is su±ciently large. In a lake that is small enough to neglect the e®ect of earth's rotation, a typical, internal response initiated by wind stress forcing over the lake surface is followed by the internal seiche, a basin-scale standing wave. It hasbeen100yearssincethetheoreticalfoundationofthewind-generated,internalseiche wasestablishedbyWedderburn(1907,1909,1912and1915). Inalargelake,ontheother hand,atypicalresponseisdominatedbyKelvinandPoincar¶ ewaves. Earlierdevelopment ofthestudyofthesebasin-scalewaveswasachievedquiteawhilelaterbyMortimer(1963 & 1968). A Kelvin wave is a shore-trapped wave having a large alongshore current, and it propagates in a cyclonic direction along the shore. The Kelvin waves having large amplitude of the order of 10 meters are often observed in large lakes. A Poincar¶ e wave is an o®shore type of wave having its largest perturbation current located near the basin center, rotating anti-cyclonically. These waves and even higher wave modes can be simultaneously excited during a wind forcing event, and they propagate, interacting with each other and also with the lake bathymetry, generating complex responses in the internal ¯eld (Csanady 1975). The energy deposited into such wind-generated, basin- scale, internal motion is eventually transformed through a down-scale cascade of energy across the spectrum of internal waves into small scale turbulence and other dissipative motions over the full extent of the basin boundaries. This internal energy transfer is a nonlinear process, and it has long been one of the most important problems in physical limnology. An accumulating body of ¯eld observations has clearly revealed the existence of the internalseicheanditsfrequentdegenerationintoasteepnonlinearwavefront(Hornetal. 2001; Stashchuk et al. 2005; Farmer 1978; Hunkins & Fliegel 1973, etc.). Furthermore, thisbodyofobservationshasprovided¯rmevidenceofpacketsofpropagatingoscillatory waves. Generation of nonlinear, longwave packets following the early steepening of the initial basin-scale tilt is one of the important pathways for energy down-scaling. Such 2 sub-basin-scale waves shoal at sloping boundaries, spatially con¯ned regions where they further steepen and break due to strong nonlinear advection, dissipating much of their energy through bottom friction and turbulent mixing (Michallet & Ivey 1999; Helfrich 1992; Vlasenko & Hutter 2002; Boegman et al. 2005a). Nonlinear steepening and gener- ation of high frequency wave packets are, however, strongly dependent on the strength and spatio-temporal distribution of the applied wind stress, and also on the background environment (strati¯cation and bathymetry). The functional dependence of the energy deposition into the internal wave spectrum, and the evolution of the spectrum, on the multiple parameters of the problem is not well understood, deserving further study. For modeling of basin-scale internal waves, a simple two-layer strati¯cation model has been preferably used since it was ¯rst applied by Watson (1904) for quantifying the internal seiche frequency observed in Loch Ness. The two-layer model is particularly useful for theoretical studies, o®ering signi¯cant reduction of parameters to be consid- ered. The strati¯cation is generally continuous, and its structure varies seasonally, with consequent seasonal e®ects on the evolution of internal waves (Antenucci et al. 2000). In a continuously strati¯ed °uid, as is well-known from linear analysis, the vertical dis- tribution of °uid velocities and displacement of isopyicnal surfaces possess multi-modal structures,andthehigherverticalmodesprovideshorterverticallengthscales. Thetwo- layer model accounts only for the ¯rst baroclinic mode, and it implicitly truncates all the other baroclinic modes in the ¯eld. Many ¯eld observations have been made which capture the multi-modal nature (e.g., Mortimer 1952; Wiegand & Chamberlain 1987; MÄ unnich et al. 1992; Roget & Zamboni 1997; Boehrer 2000, etc.). The multi-modal feature is not an isolated phenomenon in lakes, but has also been reported in oceanog- raphy (e.g., Bogucki et al. 2005; Garkema 2003). Energy transfer among vertical modes is an important nonlinear phenomenon which has drawn much attention in recent years (Garkema 2003; HÄ uttemann & Hutter 2001; Vlasenko & Hutter 2001). 3 In this thesis we explore the nonlinear e®ects in basin-scale °uid motions through an asymptotic modeling approach. In Chapter 2 we propose a two-layer, variable- environmental, weakly-nonlinear model that allows rapid simulation to reasonably explore a considerable range of parameter settings for various wind forcing scenarios (Sakai&Redekopp2008b). Wequantifythedown-scaleenergyprocessofinternalwaves from the basin-scale to scales which are of order several times the upper-mixed-layer depth. In Chapter 3 we derive a multi-modal, weakly-nonlinear, wind-forced evolution model (Sakai & Redekopp 2008d). The multi-modal approach yields an evolution equa- tion for each vertical mode with inter-modal interactions through nonlinear terms. For fundamentalstudy,welimittheverticalmodesinthemodeltothe¯rsttwomodeswhich are energetically dominant in many cases. The model is numerically simulated, and we study the modal energetics for various parameters of the modeled strati¯cation and the wind forcing. In Chapter 4, we derive a weakly-nonlinear model for a large, circular lake of variable depth (Sakai & Redekopp 2008c). With this model we explore the nonlinear evolution and the energetics of Kelvin and Poincar¶ e waves under parametric e®ects of windforcing, waveamplitudes, Coriolisaccelerations, lakedimensionsandtopographies. For the simulation of the model, we devised a new numerical method as described in Chapter 5 (Sakai & Redekopp 2008a). 4 Chapter 2 Two-layer model 2.1 Introduction Forsmalltomediumsizedlakes, particularlynarrowlakeswherethee®ectoftheearth's rotation is negligible, a wind blowing over the surface of stably strati¯ed lake induces a downwind transport in the epilimnion. The result in a closed basin is an accumulation of epilimnion water near the leeward shore with a consequent excess of hypolimnion water at the windward end due to volume conservation. There is an attendant, but very slight, up-slope tilt of the free surface, but the down-slope tilt in the metalimnion in the wind direction is much more pronounced (Mortimer 1952; Monismith 1986; Stevens & Imberger1996). Ifthewindissustainedforasigni¯cantfractionoftheinternalpendulum (alt.,seiche)period,andthenrelaxes,theunbalancedbaroclinicpressuregradientcauses the tilted thermocline to tend to return to equilibrium. This return to equilibrium occurs, initially at least, through formation of a basin-scale internal seiche. Although a barotropic seiche also emerges over the free surface, the basin-scale internal seiche is far more energetic (cf. Heaps 1984). The energy deposited into wind-generated, basin-scale, internal motion is eventually transformed through a down-scale cascade of energy. This energy transfer occurs across the spectrum of internal waves into small scale turbulence andotherdissipativemotionsoverthefullextentofthebasinboundaries. Thedominant part of the energy is dissipated by turbulent mixing in the bottom boundary under long internalwavesandbythebreakingandinteractionofinternalwavesatsloping,shore-line boundaries (Michallet & Ivey 1999). A much smaller fraction of the energy is dissipated by turbulent mixing which is driven by wave-induced shear instability in the interior of the basin. These mixing and dissipation processes, as well as the turbulent mixing 5 process in the upper surface layer, are an important and active area of research (see review by WÄ uest & Lorke 2003). Field observations in Lake Biwa by Saggio & Imberger (1998) suggest that the decay of internal wave energy is much faster than predicted by estimates based on any typical internal dissipation rate. They also suggest that the shorter-scale, higher-frequency wavestrappedinthemetalimnionseemtodistributetheenergyandreshapetheinternal energy spectrum. Generation and dissipation of such high frequency waves are yet to be identi¯ed. Thorpe et al. (1996) discussed potential sources of the high frequency waves, listing such possibilities as: i) nonlinear steepening of the basin scale waves leading to the appearance of a packet of internal solitary waves (ISWs); ii) the thermocline jump propagating around local irregular and rough bathymetry generates local disturbances which radiate internal waves; and iii) a region of strong shear with low Richardson number generated across the thermocline as long waves propagate, generating locally- unstable patches with radiating internal waves. An accumulating body of ¯eld observations has clearly revealed the existence of the internal seiche and its frequent degeneration into a steep nonlinear wave front. Further- more, this body of observations has provided ¯rm evidence of packets of propagating ISWs wherein a higher frequency dynamics is present (e.g., Thorpe & Hall 1972; Farmer 1978; Hunkins & Fliegel 1973; Wiegand & Carmack 1986). Laboratory experiments (Horn et al. 2001) have elucidated the steepening of the initial seiche into a nonlinear front, along with its subsequent evolution into packets of bidirectionally-propagating long waves. These longwave packets seem to exhibit a dominant balance between non- linearity and non-hydrostatic (dispersive) e®ects, the balance underlying the existence of permanent features such as solitary waves (Grimshaw 2002). Laboratory experiments have also elucidated processes associated with the shoaling and re°ecting of ISWs at a sloping boundary. Shoaling ISWs are found to break and dissipate a substantial frac- tion of their energy due to turbulent mixing, but a non-trivial re°ection in terms of a transformed packet occurs (Boegman et al. 2005a; Michallet & Ivey 1999; Helfrich 6 1992). Generation of nonlinear, longwave packets (nominal ISWs) following the early steepening of the initial basin-scale tilt is, therefore, one of the very important pathways for energy down-scaling. Nonlinear steepening and generation of ISWs are, however, stronglydependentonthestrengthandspatio-temporaldistributionoftheappliedwind stress, and also on the background environment (strati¯cation and bathymetry). The functional dependence of the energy deposition into the internal wave spectrum, and the evolution of the spectrum, on the multiple parameters of the problem is not well understood. Horn et al. (2001) studied the degeneration of a basin scale seiche into higher-frequency solitary waves for a range of wind-driven tilts and background environments by using a laboratory model. Boegman et al. (2005b) described the ener- getics of a basin-scale seiche, its consequent nonlinear surge, and its later evolution in frequency space by using the same laboratory model. They also conducted experiments with a sloping boundary and characterized the energy loss of ISW packets in terms of re°ection coe±cients and the frequency spectrum (Boegman et al. 2005a). Several ¯eld observationswerealsoexaminedbytheseauthors. Fromanumericalpointofview, how- ever, there is a dearth of models that admit a means for rapid exploration of parametric e®ects on the energetics and dynamics of the internal weather in closed basins. There are two principal reasons for this state of a®airs in lake hydrodynamics: i) most numer- ical models are based on the hydrostatic approximation and therefore cannot capture the non-hydrostatic e®ects essential to front evolution and energy down-scaling; and ii) ISWs in numerical models of many lakes lie in subgrid scales, and practical methods for modeling and parameterizations of generation and dissipation of such subgrid scale waves is yet to be developed (e.g., Boegman et al. 2004). Hence, further understanding of the generation, evolution and dissipation of longwave features in closed basins (e.g., ISWs), especially the understanding of parametric e®ects via a rapid-simulation tool, can go a long way toward facilitating the development of reliable and useful numerical models of the broad range of scales associated with wind-driven lake hydrodynamics. 7 In this report we quantify the down-scale energy process of internal waves from the basin-scale to scales which are of order 5-10 times the controlling °uid depth (typically the upper-mixed-layer depth). Our approach is to use a simpli¯ed theoretical model which includes variable environmental and forcing parameters. The vertical structure is taken in a most simpli¯ed form, a two-layer model, and the e®ect of the earth's rotation is neglected as the model is restricted to high-aspect ratio basins (i.e., the length-to-width ratio is large). These simpli¯cations are advantageous to construction of a simulation model that captures leading-order nonlinear and dispersive e®ects, and yet allows rapid simulation to reasonably explore a considerable range of parameter set- tings for various wind forcing scenarios. In x2.2 we describe a driven-damped, variable- environmental, higher-order Korteweg-de Vries (KdV) model applicable to describing bi-directional propagation of internal waves which are long relative to the controlling depth (i.e., the mixed-layer depth) in a con¯ned basin. In x2.4 we apply the model to a rectangular lake for di®erent wind stress distributions, demonstrate internal wave resonance under periodic wind forcing, and apply the model to various background envi- ronments and wind forcing strengths. Inx2.5 we discuss the down-scale energy transfer in lakes by use of a spatio-temporal energy spectrum obtained from model simulations. Inx2.6themodelisappliedtolakeswithvariabletopographyandwidth. Inx2.7wealso attempttoextendthemodeltothecaseofaslopingendwallandcalibratethesimulation model against results from laboratory experiments by Boegman et al. (2005a). 2.2 Evolution model WeconsideralakeoflengthLhavingastable,two-layerdensitystrati¯cationasdepicted in Figure 2.1. The length of the lake, and particularly its width, is assumed to be su±ciently small so that the e®ect of the earth's rotation is negligible. The upper layer with density ½ 1 and thickness h 1 overlies the heavier, lower layer with density ½ 2 and variable thickness h 2 (x). If the metalimnion (thermocline) of a strati¯ed lake is 8 h 1 h 2 z = ζ(x,t) z=0 ρ 1 ρ 2 x z x=0 x=L τ s Figure 2.1: Two-layer density strati¯ed lake model. su±ciently thin, a two-layer model can be used to rationally approximate the lowest mode dynamics. Such a model, of course, precludes any leakage of energy into higher verticalmodes,ane®ectthatalmostsurelyisanimportantelementintheenergytransfer to shorter scales in a closed basin, especially for waves propagating through horizontal contractions and undergoing `re°ections' from sloping end walls. Such e®ects will be addressed in a subsequent study where coupled-mode evolution equations are derived and simulated (Sakai & Redekopp 2008d). The internal wave motion in this two-layer, interfacial model is marked by an inter- facial displacement z = ³(x;t) from the equilibrium level. The lake is assumed to have a variable width (i.e., variable layer widths W 1 (x) and W 2 (x) representing the average widths of the respective layers), but its average measure is always assumed to be narrow compared to the length L. In this limit a lateral averaging process can be employed to obtain a laterally-averaged value for the dependent variables such as ³(x;t). As the density ratio ½ 1 =½ 1 across the free surface is large compared to the internal density ratio ½ 1 =½ 2 , the upper surface is assumed to be °at, a condition that ¯lters out any surface wave motion. We do allow for the existence of a surface stress ¿ s (x;t), however, to capture the e®ect of an applied wind stress and its potential for forcing internal wave motions. Since the propagation space (x-direction) is bounded, it is essential to allow a full bi-directionalityinanyasymptoticapproximationoftheforced,long-wavedynamics. To this end, a long-wave evolution equation of Boussinesq form is required. However, as shownbyHornet al.(2002), theleading-ordercontributionofbi-directionalpropagation 9 canbeaccountedforbyuseofanextended-folded-domainrepresentationoftheunbiased, second-orderintimeBoussinesqmodelresultingina¯rst-orderintimeKorteweg-deVries model. As the asymptotic methodology underlying the rational derivation of such a model equation is quite standard, and relevant details already appear in Horn et al. (2002), with extensions to variable width and multiple wave modes by Redekopp (2008), we choose to simply present our evolution model here: ³ t +c 0 ³ x + c 0 2 ½ dlnc 0 dx + dlnW dx ¾ ³ + 3 2 ®c 0 ³³ x +® 2 c 0 ³ 2 ³ x + ¯ 2 c 0 ³ xxx =¡ k b c 0 2h 2 2 C f ³j³(x;t)¡³(2L¡x;t)j¡ k s u 2 ¤0 4c 0 ©(x;t); (2.1) where coe±cients are de¯ned by the relations c 2 0 = (½ 2 ¡½ 1 )gh 1 h 2 ½ 1 h 2 +½ 2 h 1 ; ®= ½ 2 h 2 1 ¡½ 1 h 2 2 (½ 1 h 2 +½ 2 h 1 )h 1 h 2 ; ® 2 = 3 h 2 1 h 2 2 " 7 8 µ ½ 2 h 2 1 ¡½ 1 h 2 2 ½ 1 h 2 +½ 2 h 1 ¶ 2 ¡ ½ 2 h 3 1 +½ 1 h 3 2 ½ 1 h 2 +½ 2 h 1 # ; ¯ = h 1 h 2 3 µ ½ 1 h 1 +½ 2 h 2 ½ 1 h 2 +½ 2 h 1 ¶ ; k s = ½ 1 h 2 ½ 1 h 2 +½ 2 h 1 ; k b = ½ 2 h 1 ½ 1 h 2 +½ 2 h 1 : 9 > > > > > > > > > > > > = > > > > > > > > > > > > ; (2.2) In this extended, forced-dissipative (KdV) model, the e®ect of variable depth h 2 (x) is accounted for through spatially-varying coe±cients, especially the variable long-wave phase speed c 0 (x) and its derivative. The e®ect of variable lake width is contained in the term containing a single depth-averaged width function W(x). The terms on the right-hand side, respectively, represent the e®ect of bottom friction through use of a dimensionless friction coe±cient C f and the e®ect of a varying surface wind stress described by the function ©(x;t). AselucidatedinHornet al.(2002),equation(2.1)isemployedinaperiodiccomputa- tional domain [0;2L] de¯ned via an even extension of the original physical domain [0;L] aboutx=L(seeFigure2.2a). Then,thephysicalsolution³ p isobtainedbyeven-folding 10 π/2 1 -1 0 L 2L x X(x) x=0 x=L x=2L h 2 (x) h 1 τ s /2 τ s /2 (a) (b) Figure 2.2: (a) Topography is even-extended in the computational domain. The wind stress¿ s is halved andmirrored about x=L. Direction of thewind stress in the leftand the right half domain are opposite each other. (b) Uniform and sinusoidal wind stress distributions de¯ned in the computational domain. of the solution about x=L; that is, ³ p =³(x;t)+³(2L¡x;t): (2.3) In this way equation (2.1) serves as a ¯rst-order model capturing the dominant physical e®ects of the wind-generated wave ¯eld in narrow lakes. The reduction to a ¯rst-order intimedynamicalmodelisaverysigni¯cantstep. Particularlyinthatitformsthebasis for a rapid-integration model useful for exploring the role of di®erent parametric e®ects in de¯ning the internal wave ¯eld set up by wind forcing. Themodelde¯nedin(2.1)ispresentedindimensionalform. Inwhatfollowsitisuse- ful to re-cast the equation in dimensionless form. To this end the interface displacement is scaled with h 1 , the epilimnion depth; the propagation coordinate x is scaled with the basin length L; and time is scaled with the nominal internal seiche period 2L=c 00 , where c 00 is the long-wave phase speed computed using a base value of the lower layer depth 11 h 20 . Furthermore,thewindstressfunction©(x;t)isscaledwiththesquareofareference friction velocity u 2 ¤0 . Using these scales, equation (2.1) takes the non-dimensional form f t +2c 0 f x +3®c 0 ff x +2® 2 c 0 f 2 f x + µ h 1 L ¶ 2 ¯c 0 f xxx +c 0 ½ dlnc 0 dx + dlnW dx ¾ f =¡k b L h 1 µ h 1 h 2 ¶ 2 C f fjf(x;t)¡f(2¡x;t)j¡ 1 2 k s W ¡1 X(x)T(t): (2.4) The coe±cients c 0 , ®, ® 2 , ¯, k b and k s are now recast in dimensionless form, the depen- dent variable f(x;t) is simply ³=h 1 , and (x;t) are dimensionless space-time coordinates scaledwiththeirappropriatereferencevalues. Thecorrespondingphysicalsolutiongiven intherelation(2.3)isnowdenotedasf p (=³ p =h 1 ). Thewindstressdistributionfunction ©(x;t) has been separated into a dimensionless spatial part X(x) and a dimensionless temporal part T(t), and the use of non-dimensional variables has introduced the Wed- derburn numberW. It is de¯ned by W = c 2 00 h 1 u 2 ¤0 L : (2.5) TheWedderburnnumberisusefulforparameterizingthestrengthofthewindstress,and measures the magnitude of the baroclinic pressure gradient (c 0 f x »c 00 h 1 =L) relative to the vertical gradient of the wind stress (»u 2 ¤0 =c 0 ). 2.3 Numerical method To simulate the model described by (2.4) we employ the pseudo-spectral method similar to that in Fornberg & Whitham (1978). Since the spatio-temporal model equation is integratedin thespatially-periodic computationaldomain, thespatialderivativescan be computed accurately and e±ciently by using the fast Fourier transform. The 3rd-order multi-step scheme is used for time integration. The computer program MKDV, in which these numerical methods are implemented, was originally developed by Horn et al. for 12 simulating the initial value problem for a two-layer model for a rectangular lake. It has been shown in their work that the simulation results using MKDV agree qualitatively with their laboratory experiments (Horn et al. 2002). The program was modi¯ed to accommodatetheintroductionofappliedwindforcingandabenthic(turbulent)friction in order to meet the objectives of this study. The simulation results presented here used a total of N x =1024 spatial mesh points in the computational domain [0;2] and a time step of ¢t=5£10 ¡6 unless as otherwise noted. In the specialized limit of a °at bottom and constant width (i.e., homogeneous coef- ¯cients), and if the wind forcing and boundary layer friction terms are absent, the KdV equation possesses in¯nitely many conserved densities for either a periodic or an unbounded domain(e.g., Drazin & Johnson 1989). In order to validate our modi¯ed program, we evaluate the leading three conserved quantities E 1 = Z 2 0 fdx; E 2 = Z 2 0 f 2 dx; E 3 = Z 2 0 ½ 3 2 ®c 0 f 3 + 1 2 ® 2 c 0 f 4 ¡ 3 2 ¯c 0 f 2 x ¾ dx; 9 > > > > > > > = > > > > > > > ; (2.6) from simulation results. f(x;t) is a solution to (2.4) in the computational domain [0;2]. Sincetheresultedcomputationaldomainis¯niteandperiodic,theseintegrationsmustbe evaluated over the computational domain. E 1 represents conservation of mass and E 2 is that of energy. The second term in E 3 is obtained by including the cubic nonlinear term in (2.4). These conserved quantities are evaluated for several simulations using di®erent sets of parameters. Figure 2.3 shows time series of the conserved quantities for one of thetypicalruns. Thephysicalparameterswerechosenas h 1 =h 2 =1=5andh 1 =L=0:002 in the simulation. A uniform wind stress with Wedderburn number W = 1:0 is applied for 0 · t · 1=4 starting from an initial state with a °at interface at rest (i.e., E 1 is zero theoretically). The calculated value of E 1 is O(10 ¡7 ), which is the order of machine precision(theprogramwasruninsingleprecisionarithmetic). ThemagnitudesofE 2 and 13 0 1 2 3 4 0 2 4 x 10 −7 E 1 0 1 2 3 4 0 0.01 E 2 0 1 2 3 4 −1 0 x 10 −4 t E 3 Figure 2.3: Time series of the conserved quantities E 1 , E 2 and E 3 . E 3 increase until t = 1=4 whereupon the wind forcing is discontinued. Thereafter, the latter two conserved densities remain virtually constant, con¯rming a very respectable simulation ¯delity. 2.4 E®ects of environmental parameters In this section we consider the role of various parametric e®ects on the wind-generated wave ¯eld in a lake devoid of geometric inhomogeneities; that is, in a \box" lake having constant depth and constant width. Referring to the model (2.4), and after invoking the Boussinesq approximation (½ 1 ¼ ½ 2 ), the set of basic control parameters are reduced to h 1 =L, h 1 =h 2 and W with a ¯xed friction coe±cient C f = 0:0025. We study here the e®ects of these parameters on the evolution of the nonlinear wave ¯eld driven by wind forcing, and seek to clarify the transfer of internal wave energy from the basin scale to progressively shorter internal waves. 14 2.4.1 E®ect of spatial wind stress distribution ThespatialwindstressdistributionfunctionX(x)isprescribedarbitrarilyprovidedonly that it is antisymmetric about x=1 in the extended computational domain [0;2]. This antisymmetry requirement may cause the prescribed distribution X(x) to be discontin- uous at both the physical and computational boundaries. Such spatial discontinuities in the forcing tend to produce small-scale numerical oscillations every time step, and they soon pollute the numerical solution. To alleviate these spurious high wave num- ber oscillations, one can introduce an arti¯cial smoothing of the distribution X(x) at the discontinuities. Alternatively, one can apply a spectral (low pass) ¯ltering of the numerical solution after a speci¯ed number of time steps while the wind forcing term is active. Both treatments are e®ective, but we mainly used the spectral ¯ltering, ¯ltering out the upper 7/8 of the total Fourier modes during the time the wind forcing is active. If the forcing continues for an extensive time (e.g., seex2.4.3 &x2.4.5), or if the forcing is relatively strong so that nonlinearity becomes important and ISWs emerge during the time wind forcing is active, the use of spatial stress smoothing is preferred over post spectral ¯ltering. In this study, we limit the choices for X(x) to either a uniform or a sinusoidal distribution, and examine the in°uence of di®erent stress densities. The integral of the stressoverthephysicalsurfaceineithercaseistakentobeequalinanycomparisonoflake responses (see Figure2.2b). Figure 2.4showsthe simulatedwaveevolutionfor these two di®erent stress distributions with the physical parameters h 1 =h 2 =5 and h 1 =L=0:002. A uniform wind stress with W = 1 is applied for 0· t· 1=4, after which it is turned o®. The interface, which is uniformly °at initially, is progressively depressed at the leeward end and elevated at the windward end. The depression propagates windward and the elevation propagates leeward. The propagation speed of these interfacial pulses is essentially equal to the linear, long-wave phase speed. At t = 1=4 the tilted surface closely approximates a tilted straight line when the stress distribution is uniform, and is smoothly curved when the stress distribution is sinusoidal. After the wind is turned 15 −0.5 0 0.5 t=0 Uniform stress t=0.13 t=0.25 t=1 0 1 t=2 −0.5 0 0.5 Sinusoidal stress 0 1 Figure 2.4: Snap shots of the interface displacement f p for uniform (left column) and sinusoidal (right column) wind stress distributions. Wind stress blowing to the right withW =1 is applied for 0·t·1=4. o®, the interface begins to oscillate as a standing wave (internal seiche). However, a nonlinear steepening occurs forming a front on the background seiche. This front is gradually steepened (see t = 1) and, at some later time, nonlinear and dispersive e®ects become closely balanced allowing the formation of a train of oscillatory waves (ISWs) which gradually disperse throughout the domain (see t = 2). The wave train re°ects at boundaries, propagating back and forth in the physical domain. As evident in Figure 2.4, the peak amplitude of waves forced by the sinusoidal stress distribution are relatively larger than those forced by the uniform stress distribution. Figure 2.5 shows the time series of the integral E 2 , which is a direct measure of the total energy of the internal wave ¯eld forced by the wind (see Appendix A). A wind stress duration of t = 1=4 is used as a reference value as it corresponds to the time when the interface is deformed from its equilibrium position everywhere except at the lake center. The model includestheboundarylossterm,butadecayintheenergyisalmostimperceptibleforthe total integration time of the present simulation and for the selected friction coe±cient. 16 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t E 2 sinusoidal stress uniform stress Figure 2.5: Internal wave energy E 2 for uniform and sinusoidal wind stress distributions as a function of time. Energies are scaled by the energy of the uniform stress case at t=1=4. Observing Figure 2.5, even though the integrated stress distributions are identical, the sinusoidaldistributiondepositsgreaterenergyintheinternalwave¯eldthantheuniform stress case. As depicted in Figure 2.2b, the locally higher stress density in the middle of the physical domain for the sinusoidal stress case induces a locally larger pressure gradient and, therefore, a locally steeper interface (see Figure 2.4 at t=1=4). 2.4.2 E®ect of Wedderburn number and depth ratio Figure 2.6a depicts the energy deposited in the internal wave ¯eld as a function of the layer depth ratio for ¯xed Wedderburn number. A uniform wind stress was applied for 0· t· 1=4, and h 1 =L was chosen as 0.002. As the lower layer becomes more shallow with respect to the upper layer, the energy input for ¯xed wind stress diminishes. The multiplying constant k s of the forcing term in (2.4) is reduced for smaller lower layer depth (see (2.2)). Hence the lower layer depth limits the energy gained from the wind. Figure 2.6b shows the energy input for various W ¡1 for several layer depth ratios. As expected, larger values of W ¡1 yield higher values of the energy input to the internal wave ¯eld. 17 (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 h 1 /h 2 (E 2 ) 1/4 W −1 = 0.5 W −1 = 1 W −1 = 2 (b) 0 0.5 1 1.5 0 0.01 0.02 0.03 0.04 W −1 /h 1 h 2 =0.2 h 1 /h 2 =0.5 h 1 /h 2 =1 (E 2 ) 1/4 Figure 2.6: (a) Energy calculated for di®erent Wedderburn numbers W as a function of layer thickness ratio h 1 =h 2 . (b) Energy calculated for di®erent layer thickness ratio h 1 =h 2 as a function of inverse Wedderburn numberW ¡1 . 2.4.3 E®ect of wind stress duration The previous results correspond to a wind forcing that is sustained for a ¯xed time of t 0 = 1=4. Increasing the duration of wind forcing is nominally expected to increase the energy in the wave ¯eld. However, the internal response of the lake, including nonlinear e®ects, can impose a competing e®ect counteracting the continued steepening of the interface. Figure 2.7 shows the maximum value of E 2 for various duration (t 0 ) of a uniform wind stress withW =1. The energy input is normalized by that received after a forcing time of t 0 = 1=4. The energy gain increases with the forcing duration until a time of t 0 < 1=2. At this time the energy deposited into the internal wave ¯eld is maximum, and the internal lake response decreases as the stress is continually applied. The corresponding internal dynamics under sustained wind are shown in Figure 2.8, where selected snap shots of the interface surface displacement for sustained rightward wind forcing are presented. The surface is tilted down leeward for 0 < t < 1=2, but it returns toward the equilibrium line for 1=2<t<1. The return of the surface is caused by the natural free mode seiche. This seesaw-like motion repeats approximately with a seiche period. As exhibited in Figure 2.8, a nonlinear front gradually steepens, and 18 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 E 2 /(E 2 ) 1/4 t 0 Figure 2.7: Energy input to the internal wave ¯eld as a function of the wind forcing duration. Energy is scaled by its value at t=1=4. −0.5 0 0.5 t=0 t=0.5 t=1 0 1 t=1.5 −0.5 0 0.5 t=2 t=2.5 t=3 0 1 t=3.5 Figure 2.8: Internal wave evolution under sustained, uniform wind blowing to the right. eventually evolves into a train of solitary waves (ISWs). The ISW wave train spreads, and goes back and forth, in the domain without signi¯cant gain in wave energy (see Figure 2.8 at t = 2 through t = 3). After t = 3:5 the wave ¯eld is ¯lled by the shorter wave length scales of ISWs, and the wave ¯eld becomes signi¯cantly more complex as the di®erentwavecomponentscannot separate in theclosed domain. Asimulationwith sustained forcing for 0·t· 20 revealed that the seiching is damped in the presence of an unsteady wave ¯eld that persists throughout the domain. 19 (a) (b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h 1 /h 2 0 0.5 1 1.5 2 2.5 (E 2 ) 1/4 E / 2 t 0 =1/8 t 0 =3/8 t 0 =1/4 t 0 =1/2 0 0.5 1 1.5 W −1 (E 2 ) 1/4 E / 2 t 0 =1/8 t 0 =3/8 t 0 =1/4 t 0 =1/2 0 0.5 1 1.5 2 2.5 Figure 2.9: (a) Energy calculated for di®erent wind forcing durations t 0 with a ¯xed Wedderburn numberW =1 as a function of the layer thickness ratio h 1 =h 2 . (b) Energy calculated for di®erent wind forcing durations with a ¯xed layer thickness ratio h 1 =h 2 = 1=5 as a function of Wedderburn number. Energies are scaled by the value at t 0 =1=4. The results shown in Figure 2.7 provide a very useful reference de¯ning the energy input to the internal wave ¯eld for an isolated wind event. Figure 2.9a shows the energy input from wind for di®erent forcing durations from 1=8 to 1=2 when the energies are normalized by the value at t 0 = 1=4 for each h 1 =h 2 . The energy input normalized in this way is independent of the depth ratio, varying with duration of forcing as shown in Figure 2.7. Figure 2.9b is similar to 2.9a except varying W ¡1 with ¯xed h 1 =h 2 = 1=5. Normalized energies again are essentially constant for di®erent W ¡1 . These results suggest that the energy input can be characterized by the pro¯le shown Figure 2.7 together with the data in Figure 2.6a and Figure 2.6b. In order to characterize the signi¯cance of the nonlinearity, we de¯ne a shock forma- tion time t s which satis¯es the relation r = ¯ ¯ ¯ ¯ @f p @x (x;t s ) ¯ ¯ ¯ ¯ max ¯ ¯ ¯ ¯ @f p @x (x;t 0 ) ¯ ¯ ¯ ¯ max : (2.7) 20 0.5 0.5 1 1 1 2 2 2 2 3 3 3 3 h 1 /h 2 W −1 0.5 0.5 1 1 1 1 2 2 2 2 3 3 3 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 2.10: Shock formation time (t s ¡t 0 ) as a function of inverse Wedderburn number W ¡1 and the layer depth ratio h 1 =h 2 . Bold lines and thin lines represent the forcing duration t 0 =1=4 and t 0 =3=8, respectively. On break lines, no shock is formed during four seiche periods after the wind is turned o®. The prede¯ned parameter r is a ratio of maximum interfacial surface gradient between t = t s and t = t 0 (that present at the end of wind forcing). As time increases, the gradient ratio r increases due to the nonlinear steepening of the wave front. When the gradient ratio is su±ciently large, the wave front starts to generate a packet of ISWs. We refer to the evolved wave front as a `shock' for convenience, and quite arbitrarily take r =10 for convention. Figure 2.10 shows the shock formation time for variousW ¡1 and layer depth ratios with a ¯xed h 1 =L = 0:002. As expected from above discussions, the shock is formed earlier for larger W ¡1 and deeper lower layers. Also, a longer wind forcing duration generates the shock earlier, so long as t 0 ·1=2. If h 1 ¼h 2 , the leading nonlinearcoe±cient®becomesclosetozeroandtheshockisnotformedunlessthecubic nonlinear term is included in the model. Figure 2.10 shows a ¯nite shock formation time for h 1 ¼h 2 due to the e®ect of the cubic nonlinearity in the model (2.4). 21 (a) log 10 (h 1 /L) −4 −3.5 −3 −2.5 0.99 1 1.01 (E 2 ) 1/4 (b) 0 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 t h 1 /L=0.0002 h /L=0.002 1 E / 2 (E 2 ) 1/4 Figure 2.11: (a) Energy computed at t 0 = 1=4 as a function of the upper layer aspect ratio h 1 =L. Energies are scaled by the value at h 1 =L = 0:002. (b) Energy versus wind forcing duration t 0 for h 1 =L = 0:002 and 0.0002. Energies are scaled by the value at t 0 =1=4. 2.4.4 E®ect of layer aspect ratio Theupperlayeraspectratioh 1 =Lscalesonlythedispersiveandbottomfrictiontermsin the non-dimensional version of the modi¯ed KdV model (2.4). As the length of the lake increases, the strength of dispersive e®ects diminish. Here the aspect ratio determines the length scale of (solitary) waves with respect to the lake length. Our numerical simulation showed that the actual wave lengths for di®erent aspect ratios with the same layer thicknesses were approximately the same, an expected result since the controlling dimension in long wave theory is the smallest layer depth. The longer the length of the lake, the greater the area over which bottom friction acts, although the magnitude of the frictional damping is usually small. In Figure 2.11, we compared the ¯eld energies E 2 for di®erent h 1 =L with a layer depth ratio h 1 =h 2 = 1=5 and a uniform wind stress withW =1. The energy increased slightly for shorter lake length, but the increment is less than 0.3% for change of h 1 =L by factor of 10 (Figure 2.11a). This slight change in energy is caused by the bottom friction that is scaled by (h 1 =L) ¡1 . The scaled energies for various forcing duration are approximately the same as shown in Figure 2.11b. If the upper layer thickness is the same, the energy input from wind is proportional to the 22 length of the lake for ¯xed h 1 =h 2 . With the observations made in the previous section, the energy input pro¯le (Figure 2.7) is essentially universal for h 1 =h 2 , h 1 =L andW. 2.4.5 E®ect of temporal wind stress variations and internal wave res- onance Westudyinthissectiontheroleofdi®erenttemporalvariationsofthewindstressonthe energydepositiontotheinternalwave¯eldinaclosedbasin. Sincethemodelpossessesa distinct free-mode natural frequency, it is expected that the wave ¯eld possibly becomes resonantiftheperiodofthewindforcingisclosetothenaturalfrequency. Thorpe(1974) noted that the large amplitude internal wave observed in Loch Ness is possibly caused by resonance to periodic wind forcing. To study such internal wave resonance, we set up a periodic, unit-pulse wind stress input function as shown in Figure 2.12a. Each pulse is positive and one-sided (wind blows from left to right) with bandwidth T 0 , and the period of the pulse is T w . To demonstrate internal resonance, we let T w be equal to a seiche period T s , and internal wave evolutions are simulated for T 0 = 1=4, 1=2, 3=4 and 1 with a spatially uniform stress, and ¯eld energies are calculated for these cases as shown in Figure 2.12b. The physical parameters were chosen as h 1 =h 2 = 1=5, h 1 =L = 0:002 and W = 1. For T 0 =1=4, 1=2 and 3=4, the energy increases after every pulse input. In the present case thebottomboundarylossissosmallthattheincreasedenergyisnotdampede®ectively. Although it is not shown in this report, the interfacial surface displacement for these input functions shows that the amplitudes of the waves become larger after each wind event. The ¯eld energy becomes the highest with T 0 = 1=2, and further increase in the pulse duration decreases the input energy because of the reaction of the free mode seiche. For continuous stress input (T 0 =1), the energy rather oscillates with the seiche period, and amplitudes of the oscillations decrease for large t. For large t, the internal seiche is damped, and the unsteady, short waves overwhelm the domain. Although the 23 0 1 2 3 4 2 4 6 8 10 12 14 16 18 t E 2 /(E 2 ) 1/4 T 0 =1/4 T 0 =1/2 T 0 =3/4 T 0 =1 E 2 /(E 2 ) 1/4 0 1 2 3 4 0 1 2 3 4 5 t T w =0.5 T w =0.8 T w =1 T w =1.2 T w =1.5 2 T 0 t 1 T(t) 3 T w 0 s [T ] 1 (a) (b) (c) Figure 2.12: (a) Modeled periodic, pulse wind stress input function T(t). Time series of ¯eld energies for: (b) di®erent pulse duration T 0 with a ¯xed forcing period T w =T s and (c) di®erent forcing period T w with a ¯xed forcing pulse duration T 0 =1=4. Energies in graph (b) are scaled by the value for T 0 =1=4 at t=1=4, and energies in graph (c) are scaled by the value at t=1=4. amplitudesdiminishintimeduetothedampingoftheinternalseiche,theaverageenergy for each seiche period is approximately the same. The ¯eld energies increase when the wind forcing and internal seiche are in-phase, and vice versa when the forcing and seiche are out-of-phase. Here, `in-phase' is when the interfacial surface is tilting down on the leeward side, and `out-of-phase' is when the surface is tilting up on the leeward side during wind forcing. If the wind forcing and seicheareinphaseperiodically,theinternalwave¯eldbecomesresonantasdemonstrated above. Ifin-phaseandout-of-phaseforcingarerepeatedoneafteranother,thewave¯eld is expected to be neutral energetically (one particular case is the sustained wind forcing discussed above). Figure 2.12c shows time series of internal wave energies for di®erent 24 0.4 0.6 0.8 1 1.2 1.4 1.6 0 1 2 3 4 T w E (2) /E (1) Figure 2.13: Energy gain as a function of forcing period T w for T 0 =1=4. forcingperiodsT w . ForT w =1=2and3=2,energiesoscillateintimebecausetheseforcing periods repeat in-phase and out-of-phase forcing periodically, while the energy rapidly increases for the resonant case (T w =1). For the intermediate cases (T w =0:8 and 1:2), the energies do not continuously increase as the resonant cases, but energies increase for the ¯rst three consecutive wind forcing events. Here, we denote the initial energy level after the ¯rst wind forcing as E (1) , and so for the next energy level after the second forcing as E (2) . Figure 2.13 shows the energy gain E (2) =E (1) for various forcing periods. The energy gain has a maximum value about 4 at T w =1 and minima at T w =1=2 and 3=2. From Figure 2.13, if the wind forcing period is approximately 0:7 < T w < 1:3, the energy grows after two consecutive wind pulses (E (2) =E (1) >1). 2.5 Down scaling of energy spectra In order to quantify the energy down scaling from the basin scale to smaller scales, the time evolution of the wave spectra was computed. Figure 2.14 shows spectral evolu- tions obtained from numerical simulations with di®erentW ¡1 . The physical parameters were chosen as h 1 =h 2 = 1=5 and h 1 =L = 0:002, and a uniform wind stress was applied for a quarter seiche period. Low wave-number, basin-scale waves dominate the energy 25 λ −1 0 1 2 3 4 0 10 20 30 40 50 −3 −2 −1 0 log (energy) 10 (a) or less or less λ −1 0 1 2 3 4 0 10 20 30 40 50 −3 −2 −1 0 log (energy) 10 (b) or less λ −1 t 0 1 2 3 4 0 10 20 30 40 50 −3 −2 −1 0 log (energy) 10 (c) or less Figure 2.14: Wave number spectra as a function of time for Wedderburn numbers (a) W ¡1 =0:5, (b)W ¡1 =1 and (c)W ¡1 =1:5. The wave length ¸ is scaled by L. spectrum for the early stages after a wind event. Energy °ow into higher wave-number scales becomes apparent approximately at the same time that the nonlinearity, in the presence of very weak dispersion, causes a shock front to form. Subsequently, as ISWs evolve from this front, shorter-scale waves spread over the basin due to the in°uence of dispersion. The in°uence of dispersion in the wave spectra is manifested as a gradual decreaseofenergyinthehigherwavenumberbandassociatedwithshocksandISWs. In Figure 2.15, we computed a) the total wave energy by integrating the spectra over entire wave number range; b) the energy contained in the basin scale seiche by integrating over the wave number band 0 · ¸ ¡1 · 2; and c) the energy contained in shocks and ISWs which is apportioned to the wave number band about 10 · ¸ ¡1 · 40, where ¸ is the 26 0 1 2 3 4 0 1 2 3 4 5 total energy W −1 =0.5 W −1 =1.0 W −1 =1.5 0 1 2 3 4 0 1 2 3 4 5 energy in 0 ≤ λ −1 ≤ 2 0 1 2 3 4 0 1 2 3 4 5 energy in 10 ≤ λ −1 ≤ 40 t (a) (b) (c) Figure2.15: Timeseriesof(a)totalspectralenergy,(b)energycontainedin0·¸ ¡1 ·2 and (c) energy contained in 10 · ¸ ¡1 · 40 for di®erent Wedderburn numbers W. Energies are scaled by the value forW =1 at t=1=4. wavelengthscaledbyL. Thetotalenergyemployedhereisdi®erentfromE 2 inprevious sections (i.e., E 2 is proportional to the sum of kinetic and potential energy of the °ow ¯eld as shown in Appendix A, but the total energy computed here is proportional only to potential energy). The total energy oscillates in the early stage due to seiching. As ISWs evolve, high frequency oscillations emerge due to formation of short-scale standing waves arising during re°ection of ISWs from the end walls. The larger the imposed wind stress, the greater the total energy deposited into the low wave number band, which then °ows into the nonlinear wave energy band. Again as shown in Figure 2.15, basin scale motion contains most of the total energy during the early stage, but it decreases 27 hb 5h1 h1 0.25L 0.5L 0.25L Wind (a) (b) Figure 2.16: Sloping topography set up for the numerical simulation. The wind stress is applied in either (a) up-slope or (b) down-slope direction. gradually as nonlinear waves appear in the evolution process following front formation and subsequent action of dispersion. 2.6 E®ects of variable topography and width In this section, we ¯rst apply the modi¯ed KdV model as described by (2.1) for variable topographies. In order to study the e®ects of the variable topographies, we set up a simple sloping topography model as sketched in Figure 2.16. The basin is the deepest at the left end wall with h 2 = 5h 1 . The topography possesses a long slope with a hight h b in the middle of the domain, and it is connected to °at surfaces. The slope and °at level surfaces are connected by a smooth curve in order to avoid unphysical numerical noises which are usually caused by topographical discontinuities. The basin width is set to constant, and the upper layer aspect ratio is ¯xed h 1 =L = 0:002. The wind stress distribution is ¯xed uniform. Since the topography is asymmetric, the dynamical responses associated with wind forcing directions are expected to be di®erent. Figure 2.17 shows the internal wave evolutions for up-slope and down-slope wind directions with W = 1. It can be observed that a packet of ISWs appears earlier for the down- slope wind than for the up-slope wind. Figure 2.18 shows the shock formation time for various h b . From Figure 2.18, the di®erence in the shock formation time increases as h b increases. When the wind blows in the up-slope direction, the wave front is initiated at the shallower end, where the surface depression (or gradient of the wave front) is 28 down-slope wind −0.4 0 0.4 t=0.25 t=1.5 t=2.5 −0.4 0 0.4 up-slope wind t=3 0 1 0 1 Figure 2.17: Internal wave evolution over the sloping topography with h b = 3h 1 for up-slope wind (left column), and down-slope wind (right column). 0 1 2 3 4 0.5 1 1.5 2 h b t s −t 0 up-slope wind down-slope wind W = 1 -1 W = 1.3 -1 Figure 2.18: Shock formation time (t s ¡t 0 ) as a function of the slope hight h b . restricted by the shallow water e®ect (see Figure 2.6). The amplitude of the front is being diminished as the front advances in the down-slope direction. On the contrary, when the wind blows in the down-slope direction, the wave front is initiated at the deeper end, and the front advances in up-slope direction, where the front is ampli¯ed. Consequently, as the slope becomes high, the nonlinear e®ects (shock, ISWs) appear earlier for the down-slope wind. 29 Wr Wl 0.25L 0.5L 0.25L (a) (b) Figure 2.19: Variable width set up for numerical simulation. Depth is constant (h 2 = 5h 1 ). Wind stress is applied in either (a) narrowing or (b) widening direction. narrowing wind −0.5 0 0.5 t=0.25 t=1 t=1.5 −0.5 0 0.5 widening wind t=2 0 1 0 1 Figure 2.20: Internal wave evolution through variable channel width with W r =W l =1=3 for (a) narrowing-wind and (b) widening-wind. In contrast to the variable topography case, all the coe±cients (except width-e®ect term) in the modi¯ed KdV equation (2.1) are independent of variable width. To study thee®ectsofvariablewidthwesetupasimplegeometryassketchedinFigure2.19. The basin has a greater width W l on the left section and a narrower width W r on the right section, and these sections are connected by a smoothly blended straight taper section. We ¯x h 1 =h 2 = 1=5, h 1 =L = 0:002, and use a uniform wind stress distribution. Similar to the variable topography case, we apply the wind stress in narrowing and widening directions. Internal wave evolutions for each wind direction are compared in Figure 2.20. Shock formation and emergence of a train of ISWs appear earlier for wind blowing towardthewidersectionthanforwindblowingtowardthenarrowersection. Di®erences 30 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 W r /W l t s − t 0 narrowing-wind widening-wind W =0.7 -1 W =1 -1 Figure2.21: Shockformationtime(t s ¡t 0 )asafunctionofthecontractionratioW r =W l . in the shock formation time increase for higher contraction ratios as shown in Figure 2.21. The reason for the di®erence is similar to that for the variable topography case. If the wind blows in a widening direction, the wave front is initiated at the wider end, anditadvancesin thenarrowingdirectionwherethewavefrontisampli¯ed. Ifthe wind blows in the narrowing direction, the wave front is initiated at the narrower end, and it advances in the widening direction where the amplitude of the front is diminished. As the result, the nonlinear shock emerges earlier for a widening wind than for a narrowing wind. Figure 2.22 shows temporal signal of the interface surface displacement at x = 1=4 and x = 3=4 during the time range the wave packet propagating: (a) in the up-slope direction over a variable topography case with h b =3h 1 and (b) in the narrowing direc- tion in a variable width case with W r =W l = 1=3. These sample points corresponds to the end points of the sloping bottom or the variable width. It is observed that the wave packetisampli¯edforbothcases. Itcanbeshownthatthewaveamplitudesare,inturn, diminished when the wave packet is propagating in the opposite direction. From the ¯g- ure the wave ampli¯cation ratio is about 1.1 for the variable topography case and about 2.0 for the variable width case. Although contraction ratios of the basin section area are of similar magnitude, the wave ampli¯cation of the variable width case is considerably 31 2.2 2.3 2.4 2.5 2.6 2.7 2.8 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 t x=0.25 x=0.75 f p 1.2 1.3 1.4 1.5 1.6 1.7 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 t x=0.25 x=0.75 f p (a) (b) Figure 2.22: Comparison of temporal signal of the interface displacement f p sampled at x = 1=4 and x = 3=4 for (a) sloping topography case with h b = 3h 1 and (b) variable width case with W r =W l =1=3. largerthanthatofthevariabletopographycase. Thisimpliesthattheamplitudesofthe wave packet are e®ectively ampli¯ed (or diminished) through a channel having variable width rather than having variable depth. 2.7 Modeling of re°ection and dissipation during shoaling on sloping boundaries In many strati¯ed lakes the thermocline intersects the sloping boundary at the ends of a basin. Fieldobservationsandlaboratoryexperimentshavesuggestedthat,whenthewind generated internal wave front re°ects from the sloping boundary, a signi¯cant fraction of its energy is dissipated due to strong mixing at the boundary. This mixing process has an important implication for biological productivity in aquatic systems (Ostrovsky et al. 1996). In this section we attempt to apply the KdV-type model to narrow basins with a sloping end wall. First, we propose a geometrical modi¯cation to the sloping boundary and, second, introduce an additional dissipation which accounts for the strong dissipationontheslope. Thismodi¯edmodelisthencomparedwiththeresultsobtained from experiments. 32 L (original lake length) δ L h 1 b e h 1 x O x s modified (a) x s O ν so εν so ν s (x) x (b) Figure 2.23: (a) Modi¯cation of the sloping boundary and (b) schematic of the eddy viscosity function º s de¯ned along the modi¯ed slope. 2.7.1 Modeling The theoretical model we used in previous sections is only applicable to lakes with vertical end walls, and waves are assumed to undergo complete re°ection at the walls. Since the lower layer depth approaches zero at the sloping boundary, the wave phase speed also approaches zero. Then the wave is \choked" at the boundary, invalidating a basic premise in the derivation of the KdV model. To prevent the wave choking and yet allow for inclusion of rudimentary aspects of wave shoaling, we introduce a modi¯ed geometry to represent a fully sloping boundary. In Figure 2.23a we introduce a shelf region with depth b e h 1 , where b e is a free parameter. A vertical wall is placed at the end of the shelf by a distance ±L from the point of intersection between an equilibrium interface surface and the slope. ± is a second free parameter, and it is intended to adjust the wave phase (seiche frequency). Since the modi¯ed domain has vertical end walls, we can extend the modi¯ed domain to the computational domain (Figure 2.2). In order to account for the dissipation and mixing at the sloping boundary, we add an additional second-order dissipation term in the model, i.e., ³ t +c(³)³ x + ¯ 2 c 0 ³ xxx + 1 2 dc 0 dx ³ =F¡D+º s ³ xx : (2.8) 33 (b) ζ 0 (c) 600 cm 29 cm 307 cm WGB rigid lid (a) WGA WGC Figure 2.24: (a) Dimension of the laboratory tank. The tank is 30 cm wide (constant). Three wave gages (WGA, WGB, WGC) are installed to measure the interface surface displacement. (b) Initial interface surface tilt with upwelling at the slope. (c) Initial interface surface tilt with downwelling at slope. All these ¯gures are not to scale. The nonlinear terms are now represented by a nonlinear phase speed c(³) function, and º s is an eddy viscosity which may depend on space. F andD are the wind forcing term and bottom boundary loss term, respectively, de¯ned in (2.1). The functional form of º s needs to be designed. Figure 2.23b shows one conceptual sketch of the distribution of º s over the modi¯ed domain, assuming its maximum value at the end wall and decreasing to zero over the shoaling region. 2.7.2 Application Boegman et al. (2005a) reported experimental results of longwave packets generated in a closed basin with a sloping boundary. In this section we apply our proposed model to their laboratory tank, and compare our simulation with their experimental results. The laboratory tank is 6m long, 29cm(= H) deep and 30cm wide (Figure 2.24a). One end wall is inclined with a slope of either S = 1=10 or S = 3=20. The other end wall is vertical. The tank is ¯lled with fresh water overlaying saline water, and it is closed by a rigid lid. Before performing an experiment, the tank is rotated to obtain a desired initial tilt of the interface surface. Then at t = 0 the tank is rapidly rotated to the horizontal position so that the interface is initially inclined at the original tilt angle of the tank. The initial interface surface is either upwelling or downwelling on the slope (Figure2.24bandFigure2.24c, respectively). Threewavegagesareinstalledtomeasure the displacement of interface surface. 34 Since this experiment corresponds to an initial value problem, wind forcing term is turned o® in our simulations. The initial interface surface tilt is halved and even-folded onto the computational domain. For direct comparison to experiment, we include the dissipativee®ectcorrespondingtoalaminarboundarylayeralongthesurfaceofthetank (cf. Keulegan 1948; Miles 1976). The model equation employed here is written ³ t +c(³)³ x + ¯ 2 c 0 ³ xxx + 1 2 dc 0 dx ³ =º s (x)³ xx + 1 4¼ r c 0 º 2 h 1 h 2 ½ 1 h 2 +½ 2 h 1 £ ½ ½ 2 h 2 2 µ 1+ 2h 2 b ¶ + ½ 1 h 2 1 µ 1+ 2h 1 b ¶¾Z 1 ¡1 jkj 1 2 (¡1+isgnk) b ³(k;t)e ikx dk; (2.9) where º, b and b ³ are kinetic viscosity, width of the tank and Fourier transform of ³, respectively. Sincethewaveamplitudeinthisstudyisrelativelylarge, weemployafully nonlinear phase velocity: c(³)=c 0 · 1+3 (h 1 ¡h 2 )(h 1 ¡h 2 ¡2³) (h 1 +h 2 ) 2 £ 0 @ s (h 1 ¡³)(h 2 +³) h 1 h 2 ¡ h 2 ¡h 1 +2³ h 2 ¡h 1 1 A 3 5 : (2.10) AccordingtoOstrovsky&Stepanyants(2005),(2.10)wasfoundbySlunyaevetal.(2003) fromthefullynonlinear,non-dispersivelongwaveequation. Althoughexpressionsforthe nonlinear, higher order dispersive term are also available (see Ostrovsky & Grue 2003; Koop & Butler 1981), we used only the weakly dispersive term in (2.9). The spatial distribution of the eddy viscosity º s is modeled by a Gaussian function of the form º s (x)=º s0 exp " ¡ µ x x s p jln²j ¶ 2 # ; (2.11) where ² is a small fraction of the eddy viscosity at the end of the slope x = x s (Figure 2.23). We¯xed²=0:01forallsimulationsandº s0 wasadjusted toyieldcorrespondence 35 Run S h 1 =H ³ 0 =h 1 ± º s0 [m 2 s ¡1 ] 2 3/20 0.18 -0.86 0.0 0.04 12 3/20 0.20 +0.85 0.07 0.04 20 1/10 0.20 -0.77 0.0 0.002 28 1/10 0.20 +0.58 0.02 0.002 Table 2.1: Parameters of the experimental runs. The density di®erence was set as ¢½¼20 kg m ¡3 for all runs. The maximum initial surface displacement ³ 0 is measured at the vertical end wall with a plus(+) or a minus(-) sign to distinguish ³ 0 is either above(+) or below(-) the equilibrium interface surface. with experimental results; i.e., the interface displacement signal obtained at wave gage B (WGB) so that the re°ecting wave packet is su±ciently dissipated after re°ection. We chose four experimental runs for comparison. Parameters for each run are sum- marized in Table 2.1. It can be shown from equation (2.10) that, if the amplitude is large, the nonlinear phase velocity may cross zero and become negative for given h 1 =h 2 . In such a case the wave can be \stagnated" near the shelf region. The critical amplitude is smaller as the shelf becomes shallower. To avoid this stagnation, the shelf needs to be su±ciently deep. In this study we simply ¯xed b e =1:5 for all runs. The shelf extension parameter ± was adjusted and determined so that the phase of the wave packet mea- sured at WGB matches the experimental result. The value of ± for each run is listed on Table 2.1. Since the domain length is modi¯ed in the simulation, the initial condition is also modi¯ed. In the modi¯ed domain, the initial surface slope was kept the same and the position of the interface was adjusted so that the positive and negative volumes are equal. The computational domain was discretized into 512 points, and a time step ¢t=0:0005 [s] was used for all runs. InterfacedisplacementsmeasuredatWGBarecomparedinFigure2.25. Forallruns, thephaseofthewavefrontmatchesqualitativelyforthe¯rstandsecondsetofincoming and outgoing packets. For Run2 and Run20 (both are upwelling initial conditions at slope), the amplitude of the front during initial re°ection is about 20% smaller in the 36 −0.03 0 0.03 ζ [m] Run 2 1 2 1 2 −0.03 0 0.03 ζ [m] Run 20 1 1 2 2 −0.03 0 0.03 ζ [m] Run 12 1 2 1 2 0 50 100 150 200 250 300 350 400 −0.03 0 0.03 ζ [m] time [s] Run 28 1 1 2 2 Figure 2.25: Time series of the isopycnal surface displacement ³ p measured at WGB for Run 2, Run 12, Run 20 and Run 28. Solid lines are obtained from the numerical simu- lations. Dot-dash lines are obtained from the laboratory experiments by Boegman et al. (2005a). The wave signals under the left and the right arrows are incoming wave packet on to the slope and the re°ected wave packet from the slope, respectively. The re°ection coe±cients (see Figure 2.26) are computed for each pair of incoming and re°ected wave packet labeled by a number (1 or 2) under each arrow. simulation than in the experiment. Since the eddy viscosity was assumed stationary, this was caused by excessive dissipation of the initial wave front at the slope. In the ¯rst incoming packet, about three to ¯ve solitons are well predicted by the numerical model. For Run12, the ¯rst incoming packet possesses larger amplitude oscillations, and the wave amplitudes signi¯cantly deviate from the experiment. The packet is initiated at the vertical end wall where the eddy viscosity is zero. Since the initial amplitude is large (³ 0 =h 1 = 0:85), some additional eddy dissipation may need to be considered. For Run28 where ³ 0 =h 1 = 0:58, no such high amplitude oscillation was observed in the numerical simulation. For Run12 and Run28, by taking account of the resolution of the 37 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 experiment model Run2−1 Run2−2 Run12−1 Run12−2 Run20−1 Run20−2 Run28−1 Run28−2 Figure 2.26: Comparison of re°ection coe±cients E r =E i obtained from the laboratory experiments and the numerical model. Dash number in each label indicates the number of re°ection as indicated in Figure 2.25. Dash lines are §0:15 deviation from the equal line. experimental measurement, which is §2 mm, the ¯rst two or three major solitons are well predicted by the numerical model (see 200·t·230 [s]). We de¯ne here a re°ection coe±cient E r =E i as a ratio of the energy of the re°ected wave packet over that of the incoming wave packet: E r E i = 1 t 3 ¡t 2 Z t 3 t 2 ¢½³ 2 dt 1 t 2 ¡t 1 Z t 2 t 1 ¢½³ 2 dt : (2.12) The interval [t 1 ;t 2 ] and [t 2 ;t 3 ] are intervals of incoming and re°ected packets, respec- tively. There°ectioncoe±cientsforexperimentalandnumericalrunswerecomputedand compared in Figure 2.26. Five out of eight re°ection coe±cients fall into §0:15 devi- ation range. All re°ection coe±cients obtained from Run12 numerical simulation are about 50% smaller than those of the experiment, primarily because the high amplitude oscillations are not damped at the vertical end wall in the numerical model. Although 38 the ¯rst re°ection coe±cient for Run2 is well predicted by numerical model, the sec- ond re°ection coe±cient is about 50% smaller than that of experiment. From Figure 2.26, the numerical model is over damping for the second re°ected packet, but at this stage, wave amplitudes are quite small and experimental resolution is comparable to the wave amplitudes. Measured re°ection coe±cients can vary signi¯cantly by such an error. From Table 2.1, eddy viscosities seem to depend primarily on slope S. A change in slope by a factor of 1.5 results in a change in the eddy viscosity by a factor of 100. For large slope º s can increase, but the region of slope becomes shorter in the domain. Hence, total dissipation e®ects are expected to be weaker for larger slopes. The domain extension parameter ± had to be adjusted for Run12 and Run28 cases in which initial conditionsarebothdownwellingatslope. Duetotheverylimitednumberofexperiments for comparison, the functional dependence of ± is left unknown. 2.8 Conclusions A variable environmental, driven-damped KdV-type model was applied to the problem of wind-forced, internal wave generation in narrow lakes, and the model was found to be useful for obtaining ¯rst-order physical information for various sets of parameters with minimal computational e®ort. The spatial distribution of the wind stress over the lake surface is an important factor to determine the total energy input to the internal wave ¯eld. Even if the integral of the stress distribution is the same, the total energy inputs are quite di®erent as demonstrated by using sinusoidal and uniform wind stress distributions. We were able to also demonstrate an internal wave resonance phenomena. If the wind blowing over a lake surface is near its seiche frequency, it is possible that the wave is ampli¯ed and the ¯eld energy continues to grow. Duration of wind forcing is an important factor in predicting the total ¯eld energy. Maximum energy is obtained when the wind blows for half a seiche period. Therefore, the wind forcing frequency and duration of each forcing event are important parameters 39 forstudyingthedynamicalresponseoflakes. Suchdynamicalbehaviorcanbeeasilysim- ulated by using the present model. We studied e®ects of key environmental parameters: the layer depth ratio h 1 =h 2 , the upper layer aspect ratio h 1 =L and Wedderburn number W. Energyinputwithrespecttothewindforcingdurationappearstobeuniversal. The form of wind energy input curve does not depend on these parameters. Wehavequanti¯edthee®ectofnonlinearitybyde¯ningashockformationtime. The shockformationtimeiswellparameterizedbyusingW ¡1 andh 1 =h 2 . Wehavequanti¯ed the energy down scale processes by computing time evolution of wave spectra. It was shown (in Figure 2.15) that the basin scale energy is continually transferred to shorter length scales (solitary waves). Such shorter length scale waves are generated by the interactions between nonlinear and nonhydrostatic e®ects in the equations of motion. The numerical model was applied to variable topography and variable width lake models. Wave ampli¯cation e®ect along slope or contraction was well produced by the numericalmodel. Similartoresultsforrectangularlakes,basinscaleenergyiscontinually transferred to shorter length scales. Although variable topography introduces spatially- varying coe±cients in each term in the numerical model, these coe±cients are constant forvariablewidthwithconstanttopography. Asdemonstratedinx2.6,thevariablewidth is a more e®ective wave ampli¯er than the variable depth. Our numerical model was applied to shoaling along a sloping boundary by introduc- ing a shelf region and by introducing a second order dissipation term multiplied by an eddy viscosity. It was found to be possible to produce results that match experiments qualitatively. We attempted to reduce the number of free parameters to the maximum eddy viscosity º 0 and a shelf extension parameter ±. º 0 appears to be dependent primar- ily on the slope. Parameterizations of ± is left unknown. Since this is our ¯rst attempt to apply a simple numerical model to a sloping boundary, further study of the modeling and parametrization are left for future work. 40 Chapter 3 Multi-modal model 3.1 Introduction Thermally strati¯ed lakes are often subject to wind stress forcing, generating basin scale internalwavesthataretheprimaryenergysourcefordrivingmaterialtransportinlakes. Formodelingof suchlonginternalwavesinlakes, a simple two-layerstrati¯cationmodel has been preferably used since its establishment in early 20-th century. The model is a reasonable approximation as long as the lake is strongly strati¯ed (e.g., during summer)andthedensitystrati¯cationiscon¯nedtoathinlayerbetweenahomogeneous epilimnion and a homogeneous hypolimnion. The strati¯cation is generally continuous, and its structure varies seasonally, with consequent seasonal e®ects on the evolution of internal waves (Antenucci et al. 2000). In a continuously strati¯ed °uid, as is well- known from linear analysis, the vertical distribution of °uid velocities and displacement ofisopyicnalsurfacespossessmulti-modalstructures. Thetwo-layermodelaccountsonly for the ¯rst baroclinic mode, and it implicitly neglects all the other baroclinic modes in the ¯eld. In the ¯rst vertical mode, the isopycnal displacement is only pronounced near the thermocline, and the entire °uid column moves vertically in the same direction. In the second mode, the isopycnal displacement is pronounced at the upper and lower parts of the thermocline, and the vertical °uid motions there take place in opposite directions, stretching and compressing the thermoclinic layer. The second vertical velocity mode has a single vertical node at the thermocline. There are increasing numbers of nodes for higher vertical modes, and their locations are spread increasingly toward the upper and bottom surfaces, providing even shorter vertical length scales. Many ¯eld observations 41 have been made which capture the multi-modal nature. Early discovery of multi-modal response was achieved by Mortimer (1952). He identi¯ed the second vertical mode from a vertical temperature record in Windermere, and applied a model having three homogeneous layers to calculate the frequency of the second mode seiche. Years later, ¯eld observations in several lakes revealed internal responses dominated by the ¯rst and second modes (Wiegand & Chamberlain 1987; MÄ unnich et al. 1992; Roget & Zamboni 1997; Boehrer 2000). In small lakes, when the frequency of the second vertical mode is near that of diurnal wind forcing, a resonant response of the second mode may occur. Observations have been reported of apparent resonance in actual lakes (Wiegand & Chamberlain 1987; MÄ unnich et al. 1992). Responses higher than the second mode have been also reported in small lakes (seventh to tenth mode dominated response in Frains lake by LaZerte 1980; third mode dominated response in lake Banyoles by Roget & Zamboni 1997). The multi-modal feature is not an isolated phenomenon in lakes, but has also been reported in oceanography (e.g., Bogucki et al. 2005; Garkema 2003). Itisroutinetosolvetheverticalnormalmodeequationforagivenstrati¯cationpro¯le to obtain eigenfrequencies and eigenfunctions for particular modes of interest. However, such normal mode equations are based on linear theory, and the solutions therefore are linearlyindependent, yieldingnoinformationaboutmodalinteractions. Garkema(2001, 2003) adopted a multi-modal approach, and formulated a weakly-non1inear and weakly- dispersive multi-modal evolution system that was successfully applied to the oceanogra- phy problem. Utilizing a strati¯cation pro¯le in the Bay of Biscay in which the third verticalmode isdominant, hedemonstratedthatenergymayleak(notdissipation) from the third mode to lower modes. Also, the amount of energy leakage was shown to be highly dependent on the strength of the strati¯cation. HÄ uttemann & Hutter (2001) observed emergence of solitary waves of the second vertical mode when a mode-one soli- ton ran over a sill in a long laboratory channel. At the same time, Vlasenko & Hutter (2001) simulated the two-dimensional Navier-Stokes equations in the same con¯guration 42 as HÄ uttemann's experiment, and detailed structure of the °ow ¯eld was obtained, con- ¯rming that both the ¯rst and the second mode solitons are very close to those obtained by Korteweg-de Vries theory. To understand the full basin energetics in lakes, it is essential to determine the modal energy distribution among dominant vertical modes in a system allowing full bi- directional propagation of the linearly independent modes. Nonlinear models are the essential for capturing inter-mode energy transfer. In this paper, we derive a weakly- nonlinear,wind-forcedevolutionmodelbyapplyingthemulti-modalapproachthatyields an evolution equation for each vertical mode with inter-modal interactions through non- linear terms. For fundamental study, we limit the vertical modes in the model to the ¯rst two modes which are energetically dominant in many cases. The model is numeri- cally solved, and we study the modal energetics for various parameters of the modeled strati¯cation and the wind forcing. 3.2 Derivation of a nonlinear, multi-modal system Weconsideraclosedbasincontainingcontinuouslystrati¯ed°uid. Toisolatethephysics from the e®ect of the earth's rotation, we neglect Coriolis acceleration of °uid elements. Assuming width of the lake is su±ciently narrow and uniform, we start with the two- dimensional,incompressible,Boussinesqapproximated,inviscidequationsofmotionthat are perturbed from the basic state of hydrostatic equilibrium: u x +w z =0; u t +uu x +wu z =¡p x +¿ z ; w t +uw x +ww z =¡p z ¡¾; ¾ t +u¾ x +w¾ z =N 2 w: 9 > > > > > > > > = > > > > > > > > ; (3.1) In these equations, subscripts denote partial derivatives, p is the density normalized pressure, ¿ is the density normalized horizontal stress to account for wind forcing and 43 h b x = 0 x = L z = 0 Figure 3.1: Basin con¯guration. bottom friction along the horizontal (x) axis, ¾ is the perturbed buoyancy (¾ =½g=½ 0 ), andN isbuoyancyfrequencyde¯nedbyusingareferencedensity ½ 0 , staticdensity½ s (z) and gravity g as N 2 (z)=¡ g ½ 0 d½ s (z) dz : (3.2) We assume the ¯eld domain that is enclosed by a non-deformable upper surface, vertical end walls and nonuniform (variable depth) bottom surface (Figure 3.1). Solutions of (3.1) are dictated by slip-free, impermeable and non-deformable boundary conditions: w =0 at z =0; w =¡u dh b dx at z =¡h b ; u=0 at x=0 and x=L: 9 > > > > > = > > > > > ; (3.3) The length of the lake is denoted by L, and h b is the variable depth of the °uid. We introduce a long wave scaling to horizontal space and time coordinates, and also de¯ne a slow space coordinate », (X;T)=¹(x;t); » =¹ 3 x; (3.4) where ¹ » h=l ¿ 1 is the long wave scaling parameter with h a depth scale (e.g., thickness of surface mixed layer) and l is typical (long) wave length scale. We assume 44 that the topography varies slowly in space (i.e., h b is a function of » only). We then expand the dependent variables in an asymptotic series of the form (u;p;¾)=²(u (1) ;p (1) ;¾ (1) )+² 2 (u (2) ;¢¢¢)+¢¢¢ ; w =¹²(w (1) +²w (2) +¢¢¢); ¿ =¹² 2 (¿ (1) +²¿ (2) +¢¢¢); 9 > > > > > = > > > > > ; (3.5) where ² » a=h ¿ 1 is the amplitude parameter, a representing a typical amplitude of long internal waves. Scaling of the stress ¿ is intentionally taken to be second-order in the amplitude parameter so that wind forcing or benthic frictional dissipation appears ¯rst in the second order balance (i.e., weak forcing and weak damping). We introduce the familiar Korteweg-de Vries (KdV) scaling (i.e., ² = ¹ 2 ), in order to balance weak nonlinearity and leading-order non-hydrostatic correction at the same level of approxi- mation. Transformingindependentvariablesusing(3.4), andsubstituting(3.5)into(3.1), the leading-order balance gives the linear equation set u (1) X +w (1) z =0; u (1) T +p (1) X =0; p (1) z +¾ (1) =0; ¾ (1) T ¡N 2 w (1) =0: 9 > > > > > > > > = > > > > > > > > ; (3.6) From (3.6) one can obtain a single equation in favor of w (1) : w (1) zzTT +N 2 w (1) XX =0: (3.7) One can seek a vertical normal mode solution in the form w (1) =W(X;»;T)Á(»;z); (3.8) 45 where Á = 0 at z = 0 (upper surface) and z =¡h b (bottom surface) to satisfy leading order boundary conditions. Substituting (3.8) into (3.7) one can construct a standard boundary value problem along the vertical line for every »: Á 00 n + N 2 (z) c 2 n (») Á n =0; Á n j z=0 =Á n j z=¡h b =0; n=1;2;¢¢¢ ; (3.9) where c n is an eigenvalue, Á n is the corresponding eigenfunction, and primes denote partial derivatives with respect to z. The corresponding orthogonality relation is Z 0 ¡h b Á 0 m Á 0 n dz = I n c 2 n ± mn ; Z 0 ¡h b N 2 Á m Á n dz =I n ± mn ; I n = Z 0 ¡h b N 2 Á 2 n dz; (3.10) where ± mn is Kroneker's delta. All dependent variables are now expanded using the consistency implied by (3.6): u (1) = X n U (1) n (X;»;T)Á 0 n (»;z); w (1) = X n W (1) n (X;»;T)Á n (»;z); p (1) = X n P (1) n (X;»;T)Á 0 n (»;z); ¾ (1) = X n Z (1) n (X;»;T)N 2 (z)Á n (»;z): 9 > > > > > > > > > > > > > = > > > > > > > > > > > > > ; (3.11) Substituting (3.11) into (3.6), employing the orthogonality relation (3.10), and eliminat- ing W (1) n and P (1) n , we obtain the coupled pair of evolution equations U (1) nT +c 2 n Z (1) nX =0; Z (1) nT +U (1) nX =0: 9 > = > ; (3.12) Asevidentin(3.11)Z n isthemodalisopycnalamplitudeand U n isthemodalamplitude for the horizontal velocity. Equation (3.12) de¯nes a linear, bi-directional, simple wave equation with phase speed c n . 46 Proceeding to the next order balance using the expansion (3.5) leads to the inhomo- geneous set u (2) X +w (2) z =¡u (1) » ; u (2) T +p (2) X =¡fu (1) u (1) X +w (1) u (1) z g+¿ (1) z ; p (2) z +¾ (2) =¡w (1) T ; ¾ (2) T ¡N 2 w (2) =¡f(u (1) ¾ (1) ) X +(w (1) ¾ (1) ) z g: 9 > > > > > > > > = > > > > > > > > ; (3.13) Note that the leading-order stress term ¿ (1) appears and that the boundary condition in the vertical direction gives w (2) =0 at z =0; w (2) =¡u (1) @h b @» at z =¡h b : 9 > = > ; (3.14) The term u (1) » in the ¯rst equation in (3.13) is the leading-order e®ect of slowly varying depth; the bracketed terms in the second equation (x-momentum) contain the leading- order nonlinear acceleration; the term w (1) T in the third equation (z-momentum) is the leading-order non-hydrostatic correction; and the bracketed terms in the last equation (continuity) de¯ne the leading-order buoyancy °ux correction. We expand the second-order variables u (2) , p (2) and ¾ (2) in the same manner as in (3.11), albeit w (2) can not be expanded by Á n because Á n does not satisfy the boundary condition (3.14). It is not necessary to expand w (2) to de¯ne the evolution at this level of approximation. Substituting the expansion of the dependent variables into the conservation of mass equation (the ¯rst equation in (3.13)), multiplying by Á 0 n , then integrating over the physical z domain and using orthogonality (3.10), one obtains I n c 2 n U (2) nX + Z 0 ¡h b Á 0 n w (2) z dz =¡ I n c 2 n U (1) n» ¡ X i U (1) i Z 0 ¡h b Á 0 n @Á 0 i @» dz: (3.15) 47 Using the boundary condition (3.14), the integral term on the left side of (3.15) can be evaluated via integration by parts Z 0 ¡h b Á 0 n w (2) z dz = X i [Á 0 n Á 0 i ] z=¡h b @h b @» U (1) i ¡ Z 0 ¡h b Á 00 n w (2) dz; (3.16) and w (2) can be eliminated by using the last equation in (3.13). Through this proce- dure (3.15) yields an evolution equation for Z n . The evolution equation for U n can be obtained by substituting the expansions into momentum equations in (3.13) and using the orthogonality relation (3.10). Some algebraic manipulation, the evolution of the leading-order ¯eld variables is obtained in the form: U (2) nT +c 2 n Z (2) nX =¡ X i X j n a (u) nij U (1) i U (1) jX +b (u) nij U (1) iX U (1) j o + X i d ni U (1) iXXT ¡ @c 2 n @» Z (1) n ¡c 2 n Z (1) n» ¡ X i ¿ Á 0 n @Á 0 i @» À c 2 i Z (1) i +k sn ¿ (1) s ¡k bn ¿ (1) b + 1 c 2 n D N 2 Á n ¿ (1) E ; Z (2) nT +U (2) nX =¡ X i X j n a (¾) nij (U (1) i Z (1) j ) X +b (¾) nij U (1) iX Z (1) j o ¡U (1) n» ¡ X i ½ c 2 n I n [Á 0 n Á i ] z=¡h b @h b @» + ¿ Á 0 n Á 0 i @» À U (1) i ¾ : 9 > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > ; (3.17) We point out that the ¯rst-order stress ¿ (1) has been divided into two parts: ¿ (1) s is the wind stress at the lake surface, and ¿ (1) b is the bottom shear stress. The coe±cients appearing in (3.17) are de¯ned by the following set of relations: a (u) nij = Á 0 n Á 0 i Á 0 j ® ; b (u) nij = * N 2 c 2 j Á 0 n Á i Á j + ; a (¾) nij = ¿ N 2 c 2 n Á n Á 0 i Á j À ; b (¾) nij = ¿ N 2 c 2 n Á 0 n Á i Á j À ; d nj =hÁ n Á j i; k sn = c 2 n I n Á 0 n j z=0 ; k bn = c 2 n I n Á 0 n j z=¡h b ; where h¢¢¢i´ c 2 n I n Z 0 ¡h (¢¢¢)dz: 9 > > > > > > > > > > > > > = > > > > > > > > > > > > > ; (3.18) 48 The term U (1) iX Z (1) j appearing in the second equation of (3.17) (i.e., continuity equa- tion) derives from the vertical buoyancy °ux term, and one observes that the equation can not be integrated with respect to X because of the presence of this term. Hence the velocity and isopycnal amplitudes can not be decoupled into a single, second-order- in-time wave equation. Combining (3.12) and (3.17), and transforming the independent variables back to their non-scaled form, we obtain the weakly-nonlinear evolution equa- tion set U nt +(c 2 n Z n ) x =¡ X i X j n a (u) nij U i U jx +b (u) nij U ix U j o + X i d ni U ixxt ¡ X i r ni c 2 i Z i +k sn ¿ s ¡k bn ¿ b + 1 c 2 n N 2 Á n ¿ ® ; Z nt +U nx =¡ X i X j n a (¾) nij (U i Z j ) x +b (¾) nij U ix Z j o ¡ X i s ni U i ; 9 > > > > > > > > > = > > > > > > > > > ; (3.19) where r ni = ¿ Á 0 n @Á 0 i @x À ; s ni = c 2 n I n [Á 0 n Á 0 i ] z=¡h b @h b @x +r ni : (3.20) The r ni coe±cient, containing e®ects of variable depth (alt., spatially varying eigen- value), can be further evaluated by using eigenfunction equation (3.9) and orthogonality condition (3.10). The ¯nal expression is given here without derivation: r ni = 8 > > > < > > > : (c n =c i ) 2 1¡(c i =c n ) 2 · Á 0 n Á 0 i ¸ z=¡h b I i I n dlnc 2 i dx ; if i6=n; 1 2 d dx ln µ I n c 4 n ¶ ; if i=n: (3.21) The wind stress ¿ s can be expressed in terms of a friction velocity u ¤0 and a prescribed, dimensionless stress distribution function F(x;t). ¿ s =u 2 ¤0 F(x;t): (3.22) 49 The vertical distribution of the horizontal stress induced by the wind, varying from its surfacevalue¿ s , isalsoneeded todetermine thecoe±cienthN 2 Á n ¿i in(3.19). Assuming only the wind stress contributes to the integral, we model ¿ using a static, vertical stress distribution function ~ ¿(z) ¿ =u 2 ¤0 F(x;t)~ ¿(z)=¿ s ~ ¿(z); (3.23) where ~ ¿ is dimensionless, and ~ ¿(0)=1. The bottom stress ¿ b can be modeled assuming that the boundary layer is turbulent and using a friction coe±cient C f : ¿ b =C f u b ju b j=C f X i U i (x;t)Á 0 i (z =¡h b ) ¯ ¯ ¯ ¯ ¯ ¯ X j U j (x;t)Á 0 j (z =¡h b ) ¯ ¯ ¯ ¯ ¯ ¯ ; (3.24) The value of C f is taken as 0.0025 in this study, and u b is the inviscid, wave-induced velocityatthebottomsurface. Wetakethecontributionofbottomfrictiontoaparticular mode in the form ¿ b ´¿ bn =C f U n Á 0 n (z =¡h b ) ¯ ¯ ¯ ¯ ¯ ¯ X j U j Á 0 j (z =¡h b ) ¯ ¯ ¯ ¯ ¯ ¯ : (3.25) 3.3 The two-mode evolution model In this study we limit the number of active vertical modes to the lowest two (V1: mode- one; V2: mode-two). This restriction is made in order to reduce the complexity of the evolution model while retaining the energetically-dominant modes in the system. To reduce the number of free parameters in the evolution equation, we introduce non- dimensional variables by use of the following scales: x by L; Z n and z by the surface mixing layer thickness h 1 ; t by 2L=c 0 , where c 0 is a reference phase speed taken as a spatial average of V1 phase speed; c n (or c 0 ) by N 0 h 1 , where N 0 is the maximum 50 buoyancy frequency; and U n by N 0 h 2 1 . After recasting the evolution equation (3.19) in a dimensionless form, we have an evolution equation set for V1 in the form: U t + 2 c 0 (c 2 1 Y) x = 2 c 0 f¹ 111 UU x +¹ 112 UV x +¹ 121 VU x +¹ 122 VV x g +S 2 fd 11 U xxt +d 12 V xxt g¡ 2 c 0 f· 11 Y +· 12 Zg + 2c 0 W ~ k s1 F(x;t)¡ 2 c 0 S k b1 C f UjUÁ 0 1 +VÁ 0 2 j z=¡h b ; Y t + 2 c 0 U x = 2 c 0 f¡¾ 111 (UY) x ¡¾ 112 (UZ) x ¡¾ 121 (VY) x ¡¾ 122 (VZ) x +º 111 YU x +º 112 YV x +º 121 ZU x +º 122 ZV x g ¡ 2 c 0 f¸ 11 U +¸ 12 Vg; 9 > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > ; (3.26) and corresponding set for V2 in the form: V t + 2 c 0 (c 2 2 Z) x = 2 c 0 f¹ 211 UU x +¹ 212 UV x +¹ 221 VU x +¹ 222 VV x g +S 2 fd 21 U xxt +d 22 V xxt g¡ 2 c 0 f· 21 Y +· 22 Zg + 2c 0 W ~ k s2 F(x;t)¡ 2 c 0 S k b2 C f VjUÁ 0 1 +VÁ 0 2 j z=¡h b ; Z t + 2 c 0 V x = 2 c 0 f¡¾ 211 (UY) x ¡¾ 212 (UZ) x ¡¾ 221 (VY) x ¡¾ 222 (VZ) x +º 211 YU x +º 212 YV x +º 221 ZU x +º 222 ZV x g ¡ 2 c 0 f¸ 21 U +¸ 22 Vg: 9 > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > ; (3.27) The varialbes U & Y in these equations are the velocity amplitude and the isopycnal amplitude of V1, respectively, and V & Z are the V2 counterparts. The quantity ~ k sn in (3.26) and (3.27) is the modi¯ed wind forcing factor de¯ned as ~ k sn =k sn + 1 c 2 n hN 2 Á n ~ ¿(z)i: (3.28) 51 The parameter W appearing in (3.26) and (3.27) is the Wedderburn number, which is inversely proportional to the magnitude of the wind stress, W = c 2 0 h 1 u 2 ¤0 L : (3.29) Also, S in (3.26) and (3.27) is the shallow water parameter, de¯ned as S =h 1 =L, which quadratically scales the dispersive term and inversely scales the bottom friction term. Withthisevolutionmodelinhand,wesetuptheverticalstructurewhichqualitatively represents typical strati¯cation pro¯les in lakes. We adopt a three-layer, continuously varying density structure comprising a well-mixed layer (epilimnion) of thickness h 1 , a thermoclinic layer (metalimnion) of thickness h 2 having uniform density gradient, and a weakly-strati¯ed deep water (hypolimnion) of thickness h 3 . The square of the corresponding buoyancy frequency is expressed by a simple formula in a dimensionless form: N 2 (z)= 1 2 ½ 1¡tanh µ z+1 ± ¶ +(1¡N 2 h ) · tanh µ z+1+h 2 ± ¶ ¡1 ¸¾ ; (3.30) where ± is a smoothing parameter across the interface between the metalimnion and either the epilimnion or hypolimnion, and N 2 h is the buoyancy frequency in the hypolimnion relative to the value in the metalimnion. We present in Figure 3.2 selected pro¯les of N 2 (z) for di®erent h 2 , N 2 h and h 3 , and their corresponding eigenfunctions of V1 (Á 1 ) and V2 (Á 2 ). The eigenfunctions are nor- malizedbytheirmaximumvalues. Ifthereisnostrati¯cationinthehypolimnion(Figure 3.2a&3.2c),Á 1 attainsthemaximumvaluewithinthemetalimnion,andÁ 2 hasextremal valuesnearthetopandthebottomportionsofthemetalimnion,implyingthatisopycnal displacements of V1 and V2 are both pronounced in the metalimnion, which is stretched and squeezed by V2. A slight increase in the strati¯cation of the hypolimnion leaves the shape of Á 1 nearly unchanged, but Á 2 is signi¯cantly altered having its maximum value shifted downward into the hypolimnion (Figure 3.2b). Strati¯cation in the hypolimnion 52 0 1 −5 −4 −3 −2 −1 0 N 2 z h 2 =0.5 h 2 =1 h 2 =2 0 1 φ 1 −1 0 1 φ 2 0 1 −5 −4 −3 −2 −1 0 z N h 2 =0 N h 2 =0.1 N h 2 =0.5 0 1 −1 0 1 N 2 φ 1 φ 2 0 1 −5 −4 −3 −2 −1 0 z h 3 =1 h 3 =2 h 3 =3 0 1 −1 0 1 N 2 φ 1 φ 2 (a) (b) (c) Figure 3.2: Pro¯le of buoyancy frequency N 2 , vertical mode-1 eigenfunction Á 1 and ver- ticalmode-2eigenfunctionÁ 2 fordi®erent(a)metalimnionthicknessh 2 ,(b)hypolimnion strati¯cation N 2 h , and (c) hypolimnion thickness h 3 . enhances vertical displacements in the hypolimnion via V2, and also enhances the hori- zontal motions at the lake bottom where the gradient of Á 2 is maximum. It is evident, therefore, that weak strati¯cation in the hypolimnion can signi¯cantly enhance benthic stimulation from wind-forced V2 internal waves. The vertical structure of N 2 (z) determines all the coe±cients in the evolution equa- tions (3.26) and (3.27). The coe±cients are computed by numerical integration for 53 h 2 0.25 0.5 1 2 ¹ 111 0.943 0.792 0.554 0.222 ¾ 111 -0.314 -0.264 -0.185 -0.074 º 111 0.314 0.264 0.185 0.074 ¹ 122 12.166 6.460 3.329 1.234 ¾ 122 -0.089 -0.087 -0.078 -0.049 º 122 0.175 0.169 0.148 0.090 ¹ 222 14.834 7.169 3.312 1.383 ¾ 222 -4.945 -2.390 -1.104 -0.461 º 222 4.945 2.390 1.104 0.461 ¹ 211 0.413 0.336 0.237 0.130 ¾ 211 0.467 0.426 0.373 0.331 º 211 15.195 7.200 3.345 1.522 Table3.1: Coe±cientsofselectednonlineartermsfordi®erentmetalimnionthicknessh 2 . The thickness and the strati¯cation of the hypolimnion are ¯xed as h 3 =3 and N 2 h =0, respectively. The smoothing parameter is chosen as ± =0:1 for h 2 =1 & 2, ± =0:05 for h 2 =0:5, and ± =0:025 for h 2 =0:25. several di®erent values of h 2 (Table 3.1), N 2 h (Table 3.2) and h 3 (Table 3.3), where we provide only coe±cients of self-nonlinear (¹ 111 , ¹ 222 ;¢¢¢) and coupling-nonlinear terms (¹ 122 , ¹ 211 ;¢¢¢) for the sake of later discussions. The most notable result is the sensitivity of the coe±cients with respect to the thickness of the metalimnion (see Table 3.1). Some of the coe±cients (¹ 222 , º 211 , ¹ 122 ) become very large as the metalimnion thickness decreases. Self-nonlinear coe±cients of V2 are larger than their V1 counter- parts by roughly an order of magnitude. This is due to the fact that the gradient of Á 2 in the middle of the metalimnion becomes larger for thinner metalimnion (see Figure 3.2a). Depending on the wave amplitudes, the appearance of these large coe±cients can cause the corresponding nonlinear terms to be larger than linear terms. The asymptotic assumptionthatwasusedtoderivetheevolutionmodelthenbecomesdisordered, neces- sitating that the model be restricted to wind-forcings that yield smaller amplitudes. In fact, when the evolution model was simulated (numerical method and run con¯guration 54 N 2 h 0 0.1 0.2 0.5 ¹ 111 0.554 0.495 0.412 0.155 ¾ 111 -0.185 -0.165 -0.137 -0.052 º 111 0.185 0.165 0.137 0.052 ¹ 122 3.329 -0.097 -0.563 -0.534 ¾ 122 -0.078 0.003 0.026 0.034 º 122 0.148 -0.006 -0.048 -0.059 ¹ 222 3.312 0.592 0.189 0.218 ¾ 222 -1.104 -0.197 -0.063 -0.073 º 222 1.104 0.197 0.063 0.073 ¹ 211 0.237 0.247 0.122 0.018 ¾ 211 0.373 0.490 0.423 0.358 º 211 3.345 2.969 1.532 0.792 Table 3.2: Coe±cients of selected nonlinear terms for di®erent strati¯cation N 2 h in the hypolimnion. The thicknesses of the metalimnion and the hypolimnion are ¯xed as h 2 =1 and h 3 =1, respectively. The smoothing parameter is ¯xed as ± =0:1. h 3 1 2 3 4 ¹ 111 0.000 0.389 0.554 0.642 ¾ 111 0.000 -0.130 -0.185 -0.214 º 111 0.000 0.130 0.185 0.214 ¹ 122 0.000 2.195 3.329 4.048 ¾ 122 0.000 -0.058 -0.078 -0.088 º 122 0.000 0.109 0.148 0.168 ¹ 222 3.113 3.275 3.312 3.325 ¾ 222 -1.038 -1.092 -1.104 -1.108 º 222 1.038 1.092 1.104 1.108 ¹ 211 0.271 0.254 0.237 0.225 ¾ 211 0.548 0.429 0.373 0.341 º 211 3.248 3.345 3.345 3.328 Table 3.3: Coe±cients of selected nonlinear terms for di®erent hypolimnion thickness h 3 . The thickness of the metalimnion, the strati¯cation of the hypolimnion and the smoothing paramter are ¯xed as h 2 =1, N 2 h =0 and ± =0:1. 55 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 h 2 W −1 stable (t < 2) unstable (t > 2) Figure 3.3: Model stability limit in inverse Wedderburn number W ¡1 as a function of the metalimnion thickness h 2 for a ¯xed total depth (h 1 +h 2 +h 3 =5). (h s =1:0) are brie°y described in x3.4), numerical instability was encountered as the strength of the wind stress forcing was increased. In Figure 3.3, for example, the threshold Wed- derburn number for achieving stable numerical integration up to two V1 seiche periods as a function of the metalimnion thickness for a ¯xed total depth is plotted. For thin- ner metalimnion, our numerical code is not capable of performing long-time integration for strong wind forcing. We also found during numerical testing that the instability is pronounced for higher numerical resolution. When all the nonlinear coupling terms between the two modes are turned o®, numerical integration becomes stable even for strong wind forcing. The precise mechanism of how the instability is triggered is not straight forward. We conjecture that the large nonlinear coe±cients enhance excessive energy transfer between V1 and V2 (i.e., widely disparate length scales), and also gen- erate excessive energy levels at high wave numbers. Adding strati¯cation to the hypolimnion decreases the magnitude of the nonlinear couplingcoe±cients(seeTable3.2),makingthemodellessnonlinearand,hence,allowing larger wind energy input to the system. Reducing the thickness of the hypolimnion decreases the magnitude of V1 nonlinear coe±cients, while V2 counterparts change only 56 N 2 (z) h1 h2 h s τ s τ(z) Figure 3.4: Penetration of the wind stress into the metalimnion. slightly. It is well known from KdV theory that the weak self-nonlinearity is identically zero for a completely symmetric vertical structure. This is realized in Table 3.3 for h 3 = 1 (h 1 = h 2 = 1), albeit values of the cross nonlinear terms (¹ 112 , ¹ 121 ;¢¢¢), which are not included in the table, are non-zero for this con¯guration. 3.4 Wind forced response of two-mode model When the wind stress is applied over the lake surface, a horizontal shear stress progres- sivelypenetratesacrosstheepilimnion. Heaps&Ramsbottom(1966)foundananalytical solution for the shear stress distribution by solving the two-dimensional, steady, linear hydrostatic equations. In this case, the wind induced shear stress decreases to zero lin- early across the epilimnion. For a continuously strati¯ed ¯eld, however, the shear stress distribution is not well known. The most commonly assumed form takes the shear stress diminishing linearly to a zero value at the base of the epilimnion, and the stress is zero beyond (e.g., see Monismith 1987). With this assumption, the stress term hN 2 Á n ~ ¿i in the wind stress factor ~ k s is identically zero. In this study, we adopt the linear stress function, but allow the stress to penetrate the metalimnion (see Figure 3.4). The stress 57 t = 0.25 t = 0.75 t = 1.25 t = 1.75 t = 2.25 t = 2.5 t = 2.75 0.4 -0.4 (a) (b) 0 Figure 3.5: Evolution of the isopycnal amplitude for (a) mode-1 and (b) mode-2 with the wind stress penetration depth h s =1. function is expressed by introducing a stress penetration depth h s as a parameter: ~ ¿(z)= 8 > < > : z+h s h s ; ¡h s <z·0; 0; otherwise. (3.31) The evolution equations (3.26) and (3.27) are solved numerically using the 4th-order compact ¯nite di®erence scheme (Lele 1992) for spatial discretization, and a forward-in- time 3rd-order Adams-Bashforth scheme. The 4th-order compact ¯lter (Lele 1992; Slinn & Riley 1998) is applied every 10 time steps for dealiasing and stabilization. Numerical resolution that follow was chosen using 1025 points for spatial domain and a time step ¢t = 5£ 10 ¡5 . This high resolution con¯guration, in conjunction with the spatial descretization scheme with spectral-like resolution, su±ciently resolves steep nonlinear fronts and oscillatory waves. Figure 3.5 shows the evolution of isopycnal amplitudes (Y and Z) at several selected times in a lake of uniform depth with h 2 = 1, h 3 = 3. In what follows, the smoothing 58 parameter ± = 0:1 and the shallow water parameter S = 1=500 are used except as otherwise noted. Uniform, rightward wind stress of W = 1:5 is applied for the ¯rst quarter V1 seiche period (t = 1=4), and the wind is turned o® thereafter. The wind stress penetration depth is chosen as h s = 1 (no penetration to the metalimnion). The model was integrated from an initial condition at rest. Looking at V1, at the end of the wind setup time (t = 1=4), the isopycnal surface is tilted almost linearly across the domain. After the wind setup, the surface tends to return dynamically to its initial equilibrium state, forming a basin scale seiche. After one-seiche period, a nonlinear wave front develops (as indicated by arrow in the ¯gure), and the front steepens as it propagates. Whenthefrontbecomessteeper(aboutt=1:75),anoscillatorywavepacket forms behind the front, owing to the realization of an approximate balance between the weak nonlinearity and the leading non-hydrostatic e®ect. The oscillatory waves spread as the degenerating front moves back and forth in the domain. The initial response of V2 appears near the end walls. The negative volume on the left side is steepened, and it evolves into a high wave number oscillatory wave packet as it propagates toward the basin interior. The wave phase speed of V1 is about one-third of V2 (c 1 = 0:939; c 2 =0:284). A distinct wave packet appears at t¼2 for V1, and t¼0:75 for V2. Since theselfnonlinearityofV2ismuchstrongerthanthatofV1(seeTable3.1: ¹ 111 =0:554; ¹ 222 = 3:312), wave lengths in the V2 wave packet are much shorter than their V1 counterparts. More interesting, when V1 waves re°ect from the end walls, footprints of V1 waves are evident in the V2 domain just during the V1 wave re°ection process (e.g., see V2 panels at t = 1:75, 2.25, 2.75 in Figure 3.5), which implies that energy is transferred from V1 to V2 during V1 wave re°ections. Footprints of V2 waves can also be seen in the V1 domain, but their amplitudes are very small and not signi¯cant energetically. Thenumericalresultdiscussedabove,however,shouldbelookedatwithcaution. The ¯eld response for both modes depends quite sensitively on the wind stress penetration depth h s . Figure 3.6 shows several corresponding ¯elds for evolution when the wind 59 t = 0.25 t = 1.75 t = 2.25 t = 2.75 (a) (b) 0.4 -0.4 0 Figure 3.6: Evolution of the isopycnal amplitude for (a) mode-1 and (b) mode-2 with the wind stress penetration depth h s =1:5. stress penetrates down to the mid-level of the metalimnion (h s = 1:5). The response of V1 is qualitatively similar to that in Figure 3.5, but the response of V2 is quite di®erent, exhibiting a substantially diminished wind energy input to V2. Hence, there is no generation of a V2 nonlinear front. Energy transfer from V1 to V2 during wave re°ection is still clearly observed. It is instructive to de¯ne the modal energy and to quantify aspects of the modal energetics. From the Euler and continuity equations in the Boussinesq limit, one can show (e.g., see Garkema 2003) that the conserved energy density is dE = 1 2 (u 2 +w 2 )+ 1 X n=0 B n (n+2)! ¾ n+2 N 2 ; (3.32) where B 0 =1; B n+1 =¡ ½ B n N 2 (z) ¾ z : (3.33) The energy density given in (3.32) has the familiar structure as a sum of kinetic energy and potential energy. However, the potential energy is expressed by an in¯nite series. 60 For a reasonable calculation of the energy, we de¯ne the total energy in the system with a variable buoyancy frequency by the integral relation E = Z x Z z ½ 1 2 ¡ u 2 +w 2 ¢ + 1 2 ¾ 2 N 2 + 1 6 (N 2 ) 0 ¾ 3 N 6 ¾ dzdx; (3.34) where only a leading order correction of the potential energy for non-uniform N 2 (z) is included. Field variables are expressed by using two vertical modes: u=UÁ 0 1 +VÁ 0 2 ; w =U x Á 1 +V x Á 2 ; ¾ =N 2 (YÁ 1 +ZÁ 2 ): 9 > > > > > = > > > > > ; (3.35) Substituting (3.35) into (3.34), and evaluating the integral assuming the lake depth is uniform, one obtains: E = ½ 1 2 I 1 c 2 1 £ hU 2 i+d 11 hU 2 x i+d 12 hU x V x i ¤ + 1 2 £ I 1 (hY 2 i¡c 2 1 ¾ 111 hY 3 i)+I 2 (º 211 ¡2¾ 211 )hZY 2 i ¤ ¾ V1 + ½ 1 2 I 2 c 2 2 £ hV 2 i+d 22 hV 2 x i+d 21 hU x V x i ¤ + 1 2 £ I 2 (hZ 2 i¡c 2 2 ¾ 222 hZ 3 i)+I 1 (º 122 ¡2¾ 122 )hYZ 2 i ¤ ¾ V2 ; (3.36) whereh¢¢¢idenotesintegrationoverthehorizontaldomain,andallcoe±cientsarerelated to coe±cients in the evolution equations (3.26) & (3.27). Terms in (3.36) are selectively grouped into V1 or V2. In each group, the ¯rst bracketed term represents the kinetic energy and the last bracketed term represents the potential energy. Using (3.36), modal energies were calculated at the end of wind forcing (t=1=4) as a function of h s , and results are exhibited in Figure 3.7. Energies are normalized by the total energy for evolution with a stress penetration depth corresponding to h s = 0. EnergyinputtoV2dramaticallydecreasesasthewindstresspenetratesthemetalimnion. 61 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 h s energy mode−1 mode−2 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 h s |k s | mode−1 mode−2 ~ (a) (b) Figure 3.7: (a) Modal energies and (b) modal forcing coe±cients as a function of the wind stress penetration depth h s for the metalimnion thickness h 2 = 1 (solid line) and h 2 =2 (dash line). (h 3 =3, W =1:5) The V1 energy also decreases as the stress penetrates down, but the change in energy is much less than that in V2. In the same ¯gure, values of the wind forcing factor ~ k sn are shown for the same range of h s . The forcing factor and modal energies exhibit the same trend. The forcing factor of V2 decreases to zero as the wind stress penetrates down to the half depth of the metalimnion. In this case, the wind energy is not injected to V2 directly. Energy of V2 does not vanish completely, however, because energy is still transferredfromV1toV2duringthewindforcingthroughnonlinearcoupling. InFigure 3.7,correspondingresultsforthickmetalimnion(h s =2)arealsopresented. Theirtrends arethesameasthoseofformercase,butthedecreaseinmodalenergiesash s increasesis slower. Althoughitisnotshowninthe¯gure,theforcingfactorj ~ k s jvanishesasthewind stress penetration increases to near the half-depth of the metalimnion. Modal energy partition becomes more sensitive as the metalimnion becomes thinner. In the modal energy de¯nition (3.36), the most interesting terms are hZY 2 i in V1 and hYZ 2 i in V2. These terms are the correlation between the potential energy of one mode and isopycnal amplitude of another mode. We call these terms the energy transfer terms. Of course, energy transfer is processed through `all' dependent variables which 62 are governed by the evolution equation set, but the energy transfer terms solely provide explicit modal energy exchange among all the other energy terms in (3.36). Another type of the modal interaction hU x V x i can be equally distributed to both modes, hence there is no explicit energy transfer through this term. Here we de¯ne, for convenient quanti¯cation purposes, the amount of explicit modal energy transfer E tr as a di®erence of the energy transfer terms E tr = 1 2 ® 1 hZY 2 i¡ 1 2 ® 2 hYZ 2 i; (3.37) where ® 1 and ® 2 are abbreviated representations of the energy transfer coe±cients ® 1 =I 2 (º 211 ¡2¾ 211 ); ® 2 =I 1 (º 122 ¡2¾ 122 ): 9 > = > ; (3.38) We interpret that if E tr > 0, the amount of energy jE tr j is transferred from V2 to V1, and vice versa for E tr <0. Figure 3.8 shows modal energies and E tr as a function of time for h 2 =1 and h 2 =2 with h s =1. Energies are normalized by the total energy at the end of forcing (t=1=4). The total energy is not necessarily conserved after the wind forcing, because (3.36) is stillanapproximation,andtheevolutionmodelincludesbottomfrictiondamping. Fluc- tuation amplitudes of the total energy for h 2 = 1 is larger than that for h 2 = 2. This quite probably occurs because the nonlinearity of the evolution model for a thinner met- alimnion is larger, requiring a higher-order correction in the energy expression. Looking attheh 2 =2case, energydampingduetothebottomfrictionisnegligible. Totalenergy fortheh 2 =1caseseemsslightlydampedduetothenumerical¯lteringtosuppresshigh wave number noises that arise from larger nonlinearity of V2. In all cases, the modal energies oscillate in time, and they are out of phase. The amount of energy transfer also oscillates in every half V1 seiche period. Furthermore, the energy is transferred from V1 to V2 for most of the time (E tr < 0). Comparing with Figure 3.5, the energy 63 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 t energy total mode−1 mode−2 0 0.5 1 1.5 2 2.5 3 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 t E tr mode−1 mode−2 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 t energy total mode−1 mode−2 0 0.5 1 1.5 2 2.5 3 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 t E tr mode−1 mode−2 (a) (b) Figure 3.8: Modal energies (upper row) and corresponding energy transfer E tr (lower row)asafunctionoftimefor(a)h 2 =1and(b)h 2 =2. (W =1:5,h s =1,h 1 +h 2 +h 3 = 5). transfer occurs when V1 waves re°ect against the end walls, leaving their footprints in V2 domain. When V1 waves leave the wall after the re°ection, the energy transferred into V2 during re°ection is returned to V1, with no permanent energy transfer between the modes. Figure 3.9 shows vertically integrated potential and kinetic energy densities of each mode during V1 wave re°ection at the right end wall (x=1). We chose h s =1:5 with h 2 = 1 to focus more particularly on the energy transfer from V1 to V2 during re°ection by suppressing initial energy input to V2. In the V1 packet, the potential energy is larger than the kinetic energy by more than a factor of two. In the V2 packet, the potential energy is much larger than the kinetic energy which is almost negligible. During V1 wave re°ection, the potential energy of V1 dominates near the end wall, because the isopycnal amplitude increases due to superposition of incident and re°ected waves and, also, the horizontal velocity (kinetic energy) approaches zero at the end wall. The dominance of potential energy in V1 is preferably transferred to V2. 64 −0.4 −0.2 0 Y −0.4 −0.2 0 Z 0 1 2 3 (KE) V1 0 1 2 3 (KE) V2 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 (PE) V1 x 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 (PE) V2 x (a) (b) Figure 3.9: The isopycnal amplitude (upper row), kinetic energy density (middle row) and potential energy density (bottom row) at t=3:25 over the right half domain for (a) mode-1 and (b) mode-2. Energies are normalized by the total system energy. (h 2 = 1, h 3 =3, W =1:5, h s =1:5) In Table 3.4, the maximum amount of the energy transfer from V1 to V2 and values of the modal energy transfer coe±cients (® 1 & ® 2 ) are tabulated for various sets of h 2 , N 2 h , h 3 and h s . The energy transfer becomes larger for smaller h 2 or N 2 h , as indicated by the fact that ® 1 becomes much larger than ® 2 . Values of ® 1 are much larger than ® 2 for all choices of h 3 , indicating a dominance of energy transfer from V1 to V2. The coe±cientsº 211 and¾ 211 in® 1 arisefromtheverticalintegralofthebuoyancy°uxterms (¾u & ¾w) of V1, which are coupled in the V2 evolution equation (3.27). º 211 arises from the vertical buoyancy °ux (¾w), and ¾ 211 arises form the horizontal buoyancy °ux (¾u). As seen in Table 3.1-3.3, º 211 is much larger than ¾ 211 . This implies that the vertical buoyancy °ux of V1 plays an important role in transferring energy from V1 to V2. Especially, a thinner metalimnion corresponds to larger º 211 , resulting in more pronouncedenergytransfertoV2. ItcanalsobeobservedfromTable3.4thatwindstress penetrationintothemetalimniongiveslittlechangeintheamountofenergytransfer. In Figure 3.10 we show the amount of energy transfer from V1 to V2 as a function of the Wedderburn number for selected pro¯les of buoyancy frequency. The fractional amount 65 h 2 N 2 h h 3 ® 1 ® 2 E tr (h s =1) E tr (h s =1:5) 1 0 3 1.387 0.262 0.120 0.113 1.5 0 3 1.206 0.318 0.041 0.037 2 0 3 0.999 0.302 0.017 0.016 3 0 3 0.506 0.000 0.004 0.004 1 0.1 3 0.582 -0.012 0.049 0.035 1 0.2 3 0.293 -0.108 0.016 0.011 1 0.5 3 0.084 -0.195 0.002 0.002 1 0 1 1.210 0.000 0.058 0.045 1 0 2 1.351 0.196 0.095 0.083 1 0 4 1.396 0.294 0.138 0.136 Table 3.4: The modal energy E tr transferred from vertical mode-1 to mode-2 and the energy transfer coe±cients ® 1 & ® 2 for various h 2 , N 2 h and h 3 . (W =1:5) of energy transfer increases quadratically with respect to the inverse of the Wedderburn number for all cases. 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 W −1 E tr h 2 =1, N h 2 =0 h 2 =1, N h 2 =0.1 h 2 =2, N h 2 =0 Figure3.10: Modalenergytransfer E tr (frommode-1tomode-2)asafunctionofinverse Wedderburn number W ¡1 for di®erent strati¯cation pro¯les. (h 3 =3, h s =1:5) 66 3.5 Discussion Examination of our evolution model shows that the modal energy partition to V1 and V2 depends greatly on the penetration depth of the wind stress into the metalimnion. Monismith (1987) observed in his laboratory experiment the dominant response of V1 without appearance of V2 for strong wind stress forcing. He also observed that V1 and V2 co-exist in response to weak wind stress forcing. This suggests that the wind stress penetration depth may depend on the strength of the wind stress (Wedderburn number),andsuchverticalpenetrationofthewindstressinconjunctionwiththevertical strati¯cation pro¯le determines the initial energy input to each vertical mode. Recently, Stashchuk et al. (2005) conducted numerical experiments to simulate the fully-nonlinear, baroclinic response of a near two-layer °uid in a rectangular, long, nar- row tank, an experimental con¯guration similar to that employed by Horn et al. (2001). Stashchuk et al. discovered that the metalimnion (interfacial) layer thickens behind the nonlinear wave front (see Figure 6 & 7 in their paper). They concluded that the layer thickening is attributed to the interaction of the wave front with the vertical side walls (see Figure 8 in their paper). They did not conclude that the widening is attributed to V2, due to the fact that the ¯eld velocity pro¯le did not comply with typical V2 eigenpro¯les. However, V1 and V2, or even higher vertical modes, can co-exist in the ¯eld, and they are superimposed, at least in a linear sense. This requires a decomposi- tion of the disturbance ¯eld into vertical modes, which indeed the present multi-modal approach does, to determine the modal energy spectrum. Their results exhibited that the widening part of the density layer persistently follows the tail of the nonlinear wave train. IfthislayerwideningisattributedtoV2, thisphenomenonimpliesthepermanent energytransferfromV1toV2, whichisnotobservedinthemodelsimulationspresented here. Their density layer is very thin (about 0:2h 1 ), and the initial V1 wave amplitude is large (0:9h 1 ). This con¯guration, however, is outside the operational range of the present weakly-nonlinear model. 67 0.5 L 0.3 L 2 5 1 h 1 h 1 h 1 Figure 3.11: Sloping depth con¯guration. t = 2.75 t = 3 t = 3.25 t = 3.5 0.5 -0.5 (a) (b) 0 Figure3.12: Isopycnalamplitudesof(a)mode-1and(b)mode-2overslopingtopography at di®erent time. (W =1:5, h s =1:5). Althoughthevariabledepthtermsintheevolutionmodeladmitmodalenergytrans- fer, our model simulations exhibited that inter-mode transfer is negligible in variable depth cases. For example, Figure 3.12 shows evolution of isopycnal amplitudes in a lake having a sloping topography as depicted in Figure 3.11. Wind stress penetration depth was chosen to be h s =1:5 to suppress energy input to V2 in order to see modal interac- tionduringV1wavespropagatingovertheslopingbottom. Att=3,V1wavesareabout to pass over the slope, but at this time no signi¯cant interaction is observed in the V2 packet. Modal interaction during V1 wave re°ection is rather outstanding as observed at t = 2:75 and t = 3:25. In Figure 3.13, corresponding pictures for the uniform depth con¯guration are shown. Wave amplitudes are larger than the former case because the wind forcing factor k sn and nonlinearity are larger in the uniform depth case. It can be observed that there are a series of small footprints of the V1 oscillatory waves in the V2 domain, but the V1 wave re°ection contributes a much larger footprint to the V2 68 t = 2.75 t = 3 t = 3.25 t = 3.5 0.5 -0.5 (a) (b) 0 Figure3.13: Isopycnalamplitudesof(a)mode-1and(b)mode-2overuniformtopography (h 3 =5) at di®erent time. (W =1:5, h s =1:5) domain. Qualitativestructuresofthemodalcomponentsofthewave¯eldinthevariable depth case and the uniform counterpart are still very similar. 3.6 Conclusions Amulti-modal,weakly-nonlinearmodelforthewind-forcedresponseofbasin-scaleinter- nalwavesinaninhomogeneousenvironmentwasderived. Thetwo-vertical-modeinterac- tion was investigated by numerically simulating the evolution model for several modeled Brunt-VÄ aisÄ alÄ apro¯lesandseveralwindforcingfunctions. Energydistributionamongthe modes was studied by de¯ning modal energy and energy transfer functions in truncated form. Initial modal energy partition right after uniform wind forcing of speci¯ed duration strongly depends on vertical structures of both the strati¯cation and the wind stress. Penetration of the wind stress into the metalimnion can signi¯cantly change the modal energy input, especially for mode-2 and for a thin metalimnion. Determining the initial modal energy partition following wind setup is very important because the subsequent evolution (especially mode-2) is strongly dependent on the modal energy input. Modal energy exchange via nonlinear processes is captured by the evolution model derivedhere. Modalenergytransferoccurspredominantlyduringmode-1wavere°ection 69 against vertical end walls where the potential energy of mode-1 is much larger than the kineticenergy. Theenergytransferfrommode-1tomode-2duringmode-1wavere°ection isnotapermanentprocess. Duringthere°ection, mode-1energyistransferredtomode- 2, and the transferred energy into mode-2 is returned to mode-1 after the re°ection. The amount of energy transfer between modes is a function of the Wedderburn number, and it strongly depends on the strati¯cation pro¯le. Vertical buoyancy °ux of mode-1 plays an important role in energy transfer to mode-2. Thin metalimnion con¯gurations increase the vertical buoyancy °ux of mode-2 that is induced by mode-1 via nonlinear modal coupling, enhancing the energy transfer from mode-1 to mode-2. 70 Chapter 4 Large-lake model 4.1 Introduction Hydrodynamics of a lake is more or less a®ected by Coriolis acceleration introduced by the earth's rotation, and Coriolis acceleration asserts a determining role in lake motions when the horizontal scale is su±ciently large. A useful measure of this scale is the Rossby radius R 0 =c s =f, where c s is a characteristic velocity scale and f is the inertial frequency. Since typical characteristic phase speeds of internal waves are smaller than that of surface waves by a few orders of magnitude, the e®ect of earth's rotation, as well as the amplitude of °uid motion, are signi¯cantly greater in the case of internal waves as opposed to barotropic motions (Csanady 1975). As a rule of thumb, if the width of a lake is greater than the Rossby radius, the e®ect of earth rotation needs to be taken into account. In such a large lake, a typical response excited by wind stress forcing over the lake surface is dominated by Kelvin and Poincar¶ e waves of vertical mode-one. A Kelvin waveisashore-trappedwavehavingalargealongshorecurrent,andtheamplitudedecays exponentially away from the shore. It propagates in a cyclonic direction along the shore with frequency less than the inertial frequency. A Poincar¶ e wave is an o®shore type of wave having its largest perturbation current located near the basin center, rotating anti-cyclonically with frequency larger than the inertial frequency. The theoretical foundations based on the linear hydrostatic assumption applied to the hydrodynamic equations of motion were developed by Lamb (1932) and Csanady (1967) for a circular lake of uniform depth. One of the beautiful results in this theory is a dispersion relation as shown in Figure 4.1, providing a discrete horizontal modal fre- quencyspace,witheithercyclonicoranti-cyclonicpolarity. Thedimensionlessfrequency 71 0 1 2 3 4 5 6 7 8 9 10 −5 −4 −3 −2 −1 0 1 2 3 4 5 B/c ω M1 M2 M3 R1 R2 R3 R1 R2 R3 Figure 4.1: Eigenfrequency ! as a function of scaled Burger number B=c for the lowest three azimuthal (M) and radial (R) modes. !=f is exhibited as a function of the earth's rotation, which is parameterized as a ratio between the lake radius r 0 relative to the Rossby radius R 0 (i.e., the Burger number B). Since the phase speed of internal waves is dependent on the strati¯cation, which is in general seasonally varying, the e®ect of the earth's rotation also varies seasonally (Antenucci et al. 2000). Real lakes are con¯ned within complex boundaries, and they are subject to tempo- rallyandspatiallyirregularforcing,resultingina\zoo"ofcomplexresponsescomprising internalwaves,topographicgyresandcoastaljets(Csanady1975). Topredictsuchphys- ical responses, three-dimensional hydrodynamic models have been developed and their use has expanded in recent years (e.g., Beletsky & O'Connor 1997; Wang & Hutter 1998; Hodges et al. 2000; Rueda et al. 2003). Nevertheless, the linear hydrostatic model is still a most useful theoretical tool because of its simplicity, and also because of its use for numerical or experimental validation. Application of the dispersion relation for a °at-bottom, circular basin is widely used for comparing the wave frequency spectrum 72 obtained from ¯eld, laboratory, or numerical experiments. Speci¯cally, gross energetics de¯nedbythemodelwereappliedtoLakeKinnerettoestimatethedissipationtimescale (Antenucci & Imberger 2001). Also, the model was recently applied to study horizontal transport of °uid particles for various wind forcing parameters, and chaotic advection of particles was explored (Stocker & Imberger 2003). Thelinearmodel,however,isvalidonlyiftheamplitudeof°uidmotionissu±ciently small relative to the controlling vertical length scale. Nonlinear e®ects in internal °uid responses have been well studied in a non-rotating frame, by which the physics is iso- lated from the e®ect of earth's rotation. Such studies allow one to eliminate the lateral horizontalcoordinatefromconsideration,greatlyfacilitatingthedevelopmentofreduced models amenable to rapid simulation (viz., Sakai & Redekopp 2008bc). It has been well recognized that a wave of ¯nite amplitude steepens as it propagates, and the front sub- sequently generates a train of oscillatory waves following its own tail (Horn et al. 2001; Stashchuk et al. 2005; Farmer 1978; Hunkins & Fliegel 1973, etc.). Such sub-basin-scale waves shoal at sloping boundaries, spatially con¯ned regions where they further steepen and break due to strong nonlinear advection, dissipating much of their energy through bottom friction and turbulent mixing (Michallet & Ivey 1999; Helfrich 1992; Vlasenko & Hutter 2002; Boegman et al. 2005a). Energy transfer among vertical modes is also an important nonlinear process which has drawn much attention in recent years (Garkema 2003; HÄ uttemann & Hutter 2001; Vlasenko & Hutter 2001). These nonlinear e®ects provideimportantimplicationstothelakeecologybydrivingtransportofbiologicaland chemical particles. Basin-scale waves having large amplitude (i.e., of the order of 10 meters) are often observed in large lakes. In this study we intend to explore the nonlinear e®ects in basin- scale °uid motions in a rotating frame through an asymptotic modeling approach. In x4.2 we formulate a weakly-nonlinear, weakly-dispersive, wind forced evolution model for a circular lake of variable depth. A circular lake boundary is chosen for numerical 73 simplicity, and to isolate the essential, intrinsic physics of rotating lakes from the geo- metric e®ect of radial variation of the shore boundary. In x4.3 we present a summary of the linear hydrostatic model to provide preliminaries for the rest of study. The non- linear evolution model developed here can be solved only numerically, and the method is described brie°y in x4.4. In x4.5 we apply the model to initial-value problems for KelvinandPoincar¶ ewavesseparately, exploringtheirevolutioncharacteristicsfordi®er- ent Burger numbers, amplitudes, lake dimensions, and topographies. In x4.6 we apply the model to wind-forced problems to determine the dominant wave modes emerging under natural forcing, and then also estimate their energy partition. We summarize the study inx4.7, o®ering some general perspectives based on results obtained in this study. 4.2 Model formulation We consider a stable, continuously strati¯ed water body con¯ned in a cylindrical lake of variable depth. Wind stress is applied over the lake surface where we impose a slip-free, rigid-lid boundary condition to eliminate the barotropic wave mode from consideration. For the primitive equations used in this study, we choose the Boussinesq approximated, inviscid equations of motion for the hydrodynamic ¯eld perturbed from the hydrostatic state. We write the equations in the form r¢u=0; @u @t +f£u+u¢ru=¡rp+ @~ ¿ h @z ¡¾e z ; @¾ @t +u¢r¾ =N 2 w; 9 > > > > > = > > > > > ; (4.1) where the velocity ¯eld is u = ue r + ve µ + we z , the Coriolis vector is f = fe z , the density-divided pressure is p, the density-divided horizontal stress is ~ ¿ h = ¿ r e r +¿ µ e µ , the perturbed buoyancy is ¾ =½g=½ 0 , and the Brunt-VÄ aisÄ alÄ a frequencyN is de¯ned as N 2 (z)=¡ g ½ 0 d½ s (z) dz : (4.2) 74 h b b N 2 z = 0 Figure 4.2: Variable depth model. Solutions to (4.1) are dictated by free-slip, impermeable boundary conditions at all the basin boundary surfaces; namely w =0 at z =0; u¢r[z¡b(r;µ)]=0 at z =¡h b (r;µ); u=0 at r =r 0 ; 9 > > > > > = > > > > > ; (4.3) wherer 0 istheradiusofthelake, h b isthetotaldepthfunction,andbistheheightofthe topography measured form a reference horizontal surface (see Figure 4.2). We assume that h b (or b) varies slowly in space relative to the average depth. We apply a long-wave scaling to the radial coordinate, time and the Coriolis param- eter, de¯ning the scaled quantities (R;T)=¹(r;t); f =¹F; (4.4) where ¹(¿1) is the long-wave scaling parameter. The scaling of f is introduced so that the Coriolis force a®ects the dynamics at the leading-order approximation. Assuming further that the depth variation is slow on the scale of the typical wave length, we introduce a slow space radial coordinate » » =¹ 3 r: (4.5) 75 Afterintroducingthesescaledvariables,weexpandthedependentvariablesinanasymp- totic series by use of the amplitude parameter ² (²¿1); (u;v;p;¾)=²(u (1) ;v (1) ;p (1) ;¾ (1) )+² 2 (u (2) ;¢¢¢)+¢¢¢ ; w =¹²(w (1) +²w (2) +¢¢¢); ¿ r;µ =¹ 3 ²(¿ (1) r;µ +²¿ (2) r;µ +¢¢¢): 9 > > > > > = > > > > > ; (4.6) Further, we introduce the Korteweg-de Vries (KdV) scaling, ² = ¹ 2 , in order for weak nonlinearity and the leading non-hydrostatic correction to balance in the same order. The leading-order balance gives a linear, hydrostatic model: @u (1) @R + u (1) R + 1 R @v (1) @µ + @w (1) @z =0; @u (1) @T ¡Fv (1) =¡ @p (1) @R ; @v (1) @T +Fu (1) =¡ 1 R @p (1) @µ ; @p (1) @z ¡¾ (1) =0; @¾ (1) @T ¡N 2 (z)w (1) =0: 9 > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > ; (4.7) From (4.7), combining the horizontal divergence and curl of the horizontal momentum equations, and eliminating p (1) and ¾ (1) using the vertical momentum and continuity equations, one obtains the linear, long internal wave equation @ 2 @T 2 à @ 2 w (1) @z 2 ! +r 2 h fN 2 (z)w (1) g+F 2 @ 2 w (1) @z 2 =0; (4.8) wherer 2 h denotesthehorizontalLaplacian(i.e.,r 2 h ´@ 2 =@R 2 +R ¡1 @=@R+R ¡2 @ 2 =@µ 2 ). We note that w (1) vanishes at both the upper and the lower boundaries. Customarily, one can seek a solution for w (1) in terms of a slowly-varying vertical eigenfunction: w (1) »W(R;»;µ;T)Á(»;µ;z): (4.9) 76 Substituting(4.9)into(4.8), andafterenforcingperiodicityonN 2 Áintheazimuthal(µ) direction, we obtain the eigenvalue problem Á 00 l + N 2 (z) c 2 l (»;µ) Á l =0; Á l j z=0 =Á l j z=¡h b =0; l =1;2;¢¢¢ : (4.10) Thequantitiesc l andÁ l aretheeigenvalue(verticalmodalphasespeed)andcorrespond- ingeigenfunction,respectively,andtheprimedenotesaverticalderivative(Á 0 ´@Á=@z). Equation (4.10) is an ODE along a vertical line for a ¯xed horizontal coordinate, and it possesses the orthogonality relation Z 0 ¡h b Á 0 k Á 0 l dz = I l c 2 l ± kl ; Z 0 ¡h b N 2 Á k Á l dz =I l ± kl ; I l = Z 0 ¡h b N 2 Á 2 l dz; (4.11) where ± kl is Kroneker's delta. It is convenient to expand all dependent variables in (4.7) by using the eigenfunction in the form: u (1) = X l U (1) l (R;»;µ;T)Á 0 l (»;µ;z); v (1) = X l V (1) l Á 0 l ; w (1) = X l W (1) l Á l ; p (1) = X l P (1) l Á 0 l ; ¾ (1) = X l Z (1) l N 2 (z)Á l : 9 > > > = > > > ; (4.12) Aftersubstituting(4.12)into(4.7), andusingtheorthogonalityrelation(4.11), andthen eliminatingP l andW l , weobtain a setof evolutionequationsfor thevelocity amplitudes (U l and V l ) and the isopycnal amplitude (Z l ): @U (1) l @T ¡FV (1) l +c 2 l @Z (1) l @R =0; @V (1) l @T +FU (1) l + c 2 l R @Z (1) l @µ =¡ X l 0 ½ ± ll 0 @c 2 l @µ +® (µ) ll 0 c 2 l ¾ Z (1) l 0 R ; @Z (1) l @T + @U (1) l @R + U (1) l R + 1 R @V (1) l @µ =¡ X l 0 ® (µ) ll 0 V (1) l 0 R ; 9 > > > > > > > > > = > > > > > > > > > ; (4.13) 77 where ® (µ) ll 0 is de¯ned ® (µ) ll 0 = ¿ Á 0 l @Á 0 l 0 @µ À ´ c 2 l I l Z 0 ¡h b Á 0 l @Á 0 l 0 @µ dz: (4.14) In (4.13), summation terms are modal coupling terms resulting from variable depth in the azimuthal direction. Proceeding with next order balance in (4.1) gives @u (2) @R + u (2) R + 1 R @v (2) @µ + @w (2) @z =¡ @u (1) @» ; @u (2) @T ¡Fv (2) + @p (2) @R =¡ @p (1) @» ¡(u (1) ¢ru (1) )¢e r + @¿ (1) r @z ; @v (2) @T +Fu (2) + 1 R @p (2) @µ =¡(u (1) ¢ru (1) )¢e µ + @¿ (1) µ @z ; @p (2) @z +¾ (2) =¡ @w (1) @T ; @¾ (2) @T ¡N 2 w (2) =¡u (1) ¢r¾ (1) : 9 > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > ; (4.15) The second-order boundary conditions for w (2) are w (2) =0 at z =0; w (2) =u (1) @b @» + v (1) » @b @µ at z =¡h b : 9 > > = > > ; (4.16) Both weakly nonlinear terms and the leading non-hydrostatic term (@w (1) =@T) appear in (4.15) owing to use of the KdV scaling. The leading order e®ect of variable depth now appears in the boundary condition (4.16) explicitly. Without argument, we simply expand u (2) , v (2) , p (2) and ¾ (2) in a similar way in terms of the leading order variables as shown in (4.12). However, w (2) in the ¯rst equation (mass conservation) in (4.15) cannot be expanded by Á l , because Á l does not satisfy the boundary condition (4.16) at z = ¡h b . Thereby, when applying the orthogonality relation to the mass conservation equation,weintegratethe@w (2) =@z termviaintegrationbypartsapplyingtheboundary condition(4.16). Theremainingintegralcanbeeasilyevaluatedaftereliminatingw (2) by using the last equation in (4.15). Except for the handling of the w (2) term in such a way, 78 we follow the same procedure as in derivation of the leading order evolution equations (4.13). After some algebra, we ¯nd the counter part as follows: @U (2) l @T ¡FV (2) l +c 2 l @Z (2) l @R =¡c 2 l @Z (1) l @» ¡ X l 0 ½ ± ll 0 @c 2 l @» +® (r) ll 0 c 2 l 0 ¾ Z (1) l 0 ¡ X l 0 X l 00 ( ¯ (u;v) ll 0 l 00 à U (1) l 0 @U (1) l 00 @R + V (1) l 0 R @U (1) l 00 @µ ¡ V (1) l 0 V (1) l 00 R ! +¯ (w) ll 0 l 00 D (1) l 0 U (1) l 00 +¯ (µ) ll 0 l 00 V (1) l 0 U (1) l 00 R +¯ (w) ll 0 l 00 X l 000 ® (µ) l 0 l 000 V (1) l 000 U (1) l 00 R ) + X l 0 @ @T ( ° (w) ll 0 @D (1) l 0 @R +° (® 1 ) ll 0 à 1 R @V (1) l 0 @R ¡ V (1) l 0 R 2 !) +(c 2 l =I l )fÁ 0 l ¿ (1) r ¯ ¯ ¯ z=0 ¡ Á 0 l ¿ (1) r ¯ ¯ ¯ z=¡h b g+(1=c 2 l )hN 2 Á l ¿ (1) r i; 9 > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > ; (4.17) @V (2) l @T +FU (2) l + c 2 l R @Z (2) l @µ =¡ X l 0 ½ ± ll 0 @c 2 l @µ +® (µ) ll 0 c 2 l 0 ¾ Z (2) l 0 R ¡ X l 0 X l 00 ( ¯ (u;v) ll 0 l 00 à U (1) l 0 @V (1) l 00 @R + V (1) l 0 R @V (1) l 00 @µ + U (1) l 0 V (1) l 00 R ! +¯ (w) ll 0 l 00 D (1) l 0 V (1) l 00 +¯ (µ) ll 0 l 00 V (1) l 0 V (1) l 00 R +¯ (w) ll 0 l 00 X l 000 ® (µ) l 0 l 000 V (1) l 000 V (1) l 00 R ) + X l 0 @ @T ( ° (w) ll 0 1 R @D (1) l 0 @µ +° (® 2 ) ll 0 D (1) l 0 R +° (® 1 ) ll 0 1 R 2 @V (1) l 0 @µ +° (® 3 ) ll 0 V (1) l 0 R 2 ) +(c 2 l =I l )fÁ 0 l ¿ (1) µ j z=0 ¡Á 0 l ¿ (1) µ j z=¡h b g+(1=c 2 l )hN 2 Á l ¿ (1) µ i; 9 > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > ; (4.18) @Z (2) l @T + @U (2) l @R + U (2) l R + 1 R @V (2) l @µ = X l 0 ( c 2 l I l · Á 0 l Á 0 l 0 @b @» ¸ z=¡h b ¡® (r) ll 0 ) U (1) ll 0 ¡ @U (1) l @» + X l 0 c 2 l I l · Á 0 l Á 0 l 0 @b @µ ¸ z=¡h b V (1) l 0 » ¡ X l 0 ® (µ) ll 0 V (2) l 0 R ¡ X l 0 X l 00 ( ¾ (u;v) ll 0 l 00 à U (1) l 0 @Z (1) l 00 @R + V (1) l 0 R @Z (1) l 00 @µ ! ¡¾ (w) ll 0 l 00 D (1) l 0 Z (1) l 00 +¾ (µ) ll 0 l 00 V (1) l 0 Z (1) l 00 R ¡¾ (w) ll 0 l 00 X l 000 ® (µ) l 0 l 000 V (1) l 000 Z (1) l 00 R ) ; 9 > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > ; (4.19) 79 where, D (1) l = @U (1) l @R + U (1) l R + 1 R @V (1) l @µ ; ® (r) ll 0 = ¿ Á 0 l @Á 0 l 0 @» À ; ¯ (u;v) ll 0 l 00 =hÁ 0 l Á 0 l 0Á 0 l 00i; ¯ (w) ll 0 l 00 = ¿ Á 0 l Á l 0 N 2 c 2 l 00 Á l 00 À ; ¯ (µ) ll 0 l 00 = ¿ Á 0 l Á 0 l 0 @Á l 00 @µ À ; ¾ (u;v) ll 0 l 00 = ¿ Á l c 2 l Á 0 l 0N 2 Á l 00 À ; ¾ (w) ll 0 l 00 = ¿ Á l c 2 l Á l 0(N 2 Á l 00) 0 À ; ¾ (µ) ll 0 l 00 = ¿ Á l c 2 l Á l 0N 2 @Á l 00 @µ À ; ° (w) ll 0 =hÁ l Á l 0i; ° (® 1 ) ll 0 = X l 00 ° (w) ll 00 ® (µ) l 00 l 0 ; ° (® 2 ) ll 0 = @° (w) ll 0 @µ +° (® 1 ) ll 0 ; ° (® 3 ) ll 0 = @° (® 1 ) ll 0 @µ + X l 00 ® (µ) ll 00 ° (® 1 ) l 00 l 0 : 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > > > > > > > ; (4.20) Evolution equations (4.17), (4.18) and (4.19) contain weak nonlinearity, weak non- hydrostatic (dispersive) e®ect, slowly varying depth e®ect, wind stress and bottom fric- tion in the same order. These terms are all coupled, implying energy transfer among all vertical modes. Modal coupling of nonlinear terms is expressed by double and triple sums. With spatially variable coe±cients, behavior of these nonlinear terms is undoubt- edly expected to be quite complicated. We attempted ¯rst to numerically simulate a constant depth version of (4.17), (4.18) and (4.19) with multiple vertical modes. However, the model exhibited instability even with moderate amplitudes, which prevented integration for su±ciently long time. The instability is pronounced for the simulation with higher resolution, but it does not occur when the nonlinear coupling terms are all turned o®. We conjecture that such an insta- bilityproblemiscausedbysomewavenumberdependent, complexnonlinearinteraction inverticalmodalspace,whichisinherentintheevolutionequations. Althoughonemight be tempted to suppress the instability by adding hyper-viscosity terms in the evolution 80 equations, we rather chose to limit the present study to only an energetically dominant mode, namely the lowest vertical mode, for stable integration. Of course, restricting consideration to a single mode signi¯cantly reduces the com- plexity of the model. To further reduce the number of free parameters in the evolution model, we scale r by the lake radius r 0 , z and Z by the epilimnion (mixing layer) depth h 1 , N by its maximum value N 0 , t by r 0 =N 0 h 1 , c by N 0 h 1 , U and V by N 0 h 2 1 , and ¿ r;µ j z=0 by u 2 ¤0 where u ¤0 is a surface friction velocity induced by the wind stress. We neglect the bottom friction term (¿ r;µ j z=¡h b = 0), and assume also that the wind stress decreases to zero at the base of the epilimnion (i.e., hN 2 Á¿ r i = hN 2 Á¿ µ i = 0). After combining O(1) equations (4.13) and O(²) equations f(4.17), (4.18) and (4.19)g using (4.6), and transforming scaled coordinates back to the original coordinates using (4.4) and (4.5), we recast the evolution equations in a dimensionless form that is correct to O(²). @U @t ¡BV + @ @r (c 2 Z)=¡c 2 ® (r) Z ¡ ½ ¯ (u;v) µ U @U @r + V r @U @µ ¡ V 2 r ¶ +¯ (w) DU +¯ (µ;w) VU r ¾ +¤ 2 @ @t ½ ° (w) @D @r +° (® 1 ) µ 1 r @V @r ¡ V r 2 ¶¾ + 1 W k s ¿ r ; 9 > > > > > > > > > > > = > > > > > > > > > > > ; (4.21) @V @t +BU + 1 r @ @µ (c 2 Z)=¡c 2 ® (µ) Z r ¡ ½ ¯ (u;v) µ U @V @r + V r @V @µ + UV r ¶ +¯ (w) DV +¯ (µ;w) V 2 r ¾ +¤ 2 @ @t ½ ° (w) 1 r @D @µ +° (® 2 ) D r +° (® 1 ) 1 r 2 @V @µ +° (® 3 ) V r 2 ¾ + 1 W k s ¿ µ ; 9 > > > > > > > > > > > = > > > > > > > > > > > ; (4.22) @Z @t + @U @r + U r + 1 r @V @µ =· (r) U +· (µ) V r ¡ ½ ¾ (u;v) µ U @Z @r + V r @Z @µ ¶ ¡¾ (w) DZ +¾ (µ;w) VZ r ¾ ; 9 > > = > > ; (4.23) 81 where all vertical modal indices are dropped, and additional coe±cients are de¯ned ¯ (µ;w) =¯ (µ) +¯ (w) ® (µ) ; ¾ (µ;w) =¾ (µ) ¡¾ (w) ® (µ) ; k s = c 2 I Á 0 j z=0 ; · (r) = 1 I ¡ Á 0 ) 2 ¯ ¯ z=¡h b ¢ @b @r ¡® (r) ; · (µ) = 1 I (Á 0 ) 2 ¯ ¯ z=¡h b ¢ @b @µ ¡® (µ) : 9 > = > ; (4.24) There are two important dimensionless parameters in the horizontal momentum equa- tions (4.21) and (4.22); the Burger number B and the Wedderburn number W. The Burger numberB used in this study is de¯ned as a ratio between the lake radius r 0 and internal Rossby radius of deformation R 0 =N 0 h 1 =f: B = fr 0 N 0 h 1 = r 0 R 0 : (4.25) The Wedderburn number is de¯ned as a ratio between baroclinic pressure gradient and the wind stress: W = N 2 0 h 3 1 u 2 ¤0 r 0 : (4.26) Also, theparameter¤in(4.21)and(4.22)isanaspectratiode¯nedas¤= h 1 =r 0 , which quadratically scales the dispersive terms. The vertical structure of the strati¯cation and the nonuniform depth are implicitly included in the linear phase speed c and in each integral coe±cient de¯ned in (4.20) and (4.24). Since we neglect the in°uence of bottom friction, the model should represent the wind-generated response for the order of several days, which is a typical time scale for forced internal waves in large lakes (Csanady 1968ab; Antenucci & Imberger 2001). 4.3 Linear hydrostatic model Before proceeding to a simulation of the forced, nonlinear response, we summarize the set of exact solutions to the linear hydrostatic problem for uniform depth. Although the model has been well documented (Lamb 1932; Csanady 1967), we include the results in this section since they provide essential preliminaries for the rest of this report. The 82 modelisthesameastheleadingorderequation(4.13),whichwewriteinadimensionless form: @U @t ¡BV +c 2 @Z @r =0; @V @t +BU +c 2 1 r @Z @µ =0; @Z @t + @U @r + U r + 1 r @V @µ =0: 9 > > > > > = > > > > > ; (4.27) After constructing an equation in favor of Z, a wave-like solution is found by assuming Z » R(r)expi(mµ¡B!t), where ! is a dimensionless frequency scaled by f. The fundamental solution is written in the form: U =¡ A 0 c 2 B(! 2 ¡1) ½ m R(r) r ¡!R 0 (r) ¾ sin(mµ¡B!t¡± 0 ); V = A 0 c 2 B(! 2 ¡1) ½ m! R(r) r ¡R 0 (r) ¾ cos(mµ¡B!t¡± 0 ); Z = A 0 R(r)cos(mµ¡B!t¡± 0 ); 9 > > > > > > = > > > > > > ; (4.28) where A 0 is a wave amplitude, ± 0 is an initial phase of the wave, and R(r) is a radial eigenfunction normalized by its maximum value, i.e., R(r) = R ¤ (r)=jR ¤ (r)j max , where R ¤ (r) is written in terms of the Bessel(J) or the modi¯ed-Bessel(I) function R ¤ (r)= 8 > > < > > : I m (B ¤ r) if ! 2 <1; J m (B ¤ r) if ! 2 >1; and B ¤ = B c p j1¡! 2 j: 9 > > > > > > = > > > > > > ; (4.29) I m (B ¤ r) is the Kelvin wave (subinertial) solution, and J m (B ¤ r) is the Poincar¶ e wave (superinertial) solution. The radial wave number is implicitly contained in B ¤ through !,whichisaneigenfrequencydeterminedbyadispersionrelation; namely,fortheKelvin wave mode we have B ¤ I m¡1 (B ¤ )¡m µ 1+ 1 ! ¶ I m (B ¤ )=0: (4.30) 83 Since I m (r) is a positive, increasing function, (4.30) implies that the Kelvin wave can possess only a single radial mode of positive frequency. We distinguish the wave trav- eling direction by calling `cyclonic'(counter clockwise) for positive frequency, and `anti- cyclonic'(clockwise) for negative frequency. For ! 2 > 1, I m is replaced by J m in (4.30). The dispersion relation then gives increasing frequencies in discrete radial modes for both cyclonic and anticyclonic waves for given azimuthal mode (m) and Burger number B. Dispersion relations for the ¯rst three azimuthal and radial modes are plotted in Figure 4.1 as a function of the Burger number. For convention, we label the wave mode by a format `M(azimuthal mode)R(radial mode)'. The dimensionless frequency ! =1 is a critical case between Kelvin and Poincar¶ e wave modes, and the corresponding solution is easily found by assuming (U;V;Z)»exp(imµ¡Bt) in (4.27), yielding U =¡ A 0 B c 2(m+1) (r m+1 ¡r m¡1 )sin(mµ¡B c t¡± 0 ); V = A 0 B c 2(m+1) (r m+1 +r m¡1 )cos(mµ¡B c t¡± 0 ); Z = A 0 r m cos(mµ¡B c t¡± 0 ); 9 > > > > > > = > > > > > > ; (4.31) whereB c is a critical Burger number B c =c p m(m+1); (4.32) that is determined by requiring U = 0 at r = 1 when deriving (4.31). The Kelvin wave solution exists only if B > B c . It is easy to show that ! = ¡1 is physically irrelevant, since corresponding solution yields Z»r ¡m , which is singular at r =0. 4.4 Numerical method Since our evolution model derived in x4.2 is described in terms of polar coordinates, a numerical singularity is inevitable at r =0, even though the solution is regular (C 1 ) in 84 the domain (Boyd 2001). We ¯rst attempted to integrate the model equations (4.21), (4.22) and (4.23) through a ¯nite di®erence approach, but it failed to continue the integration for su±ciently long time due to emergence of numerical instability on the computational grid adjacent to r = 0. The numerical instability, in general, is caused by the inability of such a low order method to satisfy all necessary regularity conditions as r!0. To remedy this di±culty, we implemented the spectral method described here brie°y. Uponformulationofthespectralmodel,itisconvenient(butnotalwaysnecessary)to useradial°uxvariablesviathetransformationr(U;V)7¡!( ~ U; ~ V)insteadoftheoriginal variables. We expand the dependent variables by a Fourier exponential in the azimuthal directionandone-sidedJacobipolynomials(Verkley1997a;Matsushima&Marcus1995) in the radial direction. After projecting the variables onto a ¯nite dimensional subspace, sayf ~ U; ~ V;Zg¡ ¡ ! P N f ~ U N ; ~ V N ;Z N g, we have that ~ U N = N X m=¡N N+2 X n=jmj n¸1 ^ u mn fQ mn (r)¡± m0 (¡1) n 2 Q mn (r)ge imµ ; ~ V N = N X m=¡N N+2 X n=jmj n¸1 ^ v mn fQ mn (r)¡± m0 (¡1) n 2 Q mn (r)ge imµ ; Z N = N X m=¡N N X n=jmj ^ z mn Q mn (r)e imµ ; 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; (4.33) where Q mn (r) is the one-sided Jacobi polynomial Q mn (r)=r jmj P (0;jmj) k (2r 2 ¡1); n=jmj+2k; k =0;1;2:::; (4.34) and P (0;jmj) k (x) is the Jacobi polynomial of order k. Q mn (r) is an orthogonal polynomial of degree n with weight function r, namely Z 1 0 Q mn (r)Q mn 0(r)rdr = ± nn 0 2(n+1) : (4.35) 85 The one-sided Jacobi polynomial, by its nature, implicitly satis¯es the regularity con- dition at r = 0 for a scalar function (e.g., Z). However, the polynomial by itself is not satisfactory to guarantee the regularity at r = 0 for vector functions. Vector functions ( ~ U; ~ V) that are regular at r = 0 dictates the use of a kinematic constraint at r = 0, which is written here without derivation (see Sakai & Redekopp 2008a for details): N+2 X n=jmj (¡1) k (jmj+k)! k!jmj! f^ u mn +isgn(m)^ v mn g=0 for jmj¸1: (4.36) Also, the boundary condition of vanishing normal velocity ( ~ U =0 at r =1) is expressed in the spectral space as N+2 X n=jmj n¸1 n 1¡± m0 (¡1) n 2 o ^ u mn =0: (4.37) Substituting the expansion (4.33) into the evolution equations (4.21), (4.22) and (4.23),andafterusingorthogonalityrelation(4.35),plusitscounterpartintheazimuthal direction, the equations are transformed to a system of the ¯rst order ODEs in time in the spectral space. Nonlinear terms in the evolution equations are ¯rst evaluated in physical space at Gauss-Legendre type collocation points in the radial direction, and uniform collocation points in the azimuthal direction. Then, these physical values are projected to the spectral space via inverse transforms (Gaussian quadrature in radial and fast Fourier transform in azimuthal directions). Resolution of the collocation grid was chosen so that the spectral coe±cients are evaluated exactly without aliasing for a given truncation N. The system of ODEs, after combined with the kinematic constraint (4.36) and the boundary condition (4.37), were integrated simultaneously in time by usingthe4-thorderRunge-Kuttaschemeof¯xedtimestepsize. Thenumericalcodewas validatedbycomparingsimulationsagainsttheexactsolutionoftheinitialvalueproblem for the linear hydrostatic model described in x4.3. The numerical code was originally developedtosimulatemulti-vertical-modalequationsofarbitrarynumberf(4.17), (4.18) 86 and (4.19)g, but because of stability issues discussed in x4.2, we ran the code with only a single vertical mode to obtain the results discussed below. 4.5 Initial value problem 4.5.1 Evolution on uniform depth We ¯rst investigate the evolution of initial values for a circular lake of uniform depth. Vertical structure used in this study comprises an epilimnion of depth h 1 (mixing layer), a metalimnion of depth h 2 having constant density gradient (thermoclinic layer with constant Brunt-VÄ aisÄ alÄ a frequencyN 0 ), and a hypolimnion of uniform density, which we express by a formula, N 2 (z)= 1 2 ½ tanh µ z+1+h 2 ± ¶ ¡tanh µ z+1 ± ¶¾ ; (4.38) after scaling all vertical lengths by h 1 and N by N 0 . The parameter ± controls the thicknessoftransitionregionsattheupperandlowerendsofthemetalimnion. Wechoose h 2 = 1(= h 1 ), ± = 1=10 and the depth of hypolimnion h 3 = 3 (hence the total depth H =5). The vertical mode-one phase speed (c) is 0.9395 for this vertical structure after numerically solving the boundary value problem (4.10) using (4.38). The corresponding vertical eigenfunction is normalized by its maximum value. Exact solutions to the linear hydrostatic model described in x4.3 are used as the initial conditions. Of course, such exact solutions are not the solution of the evolution model f(4.21), (4.22) and (4.23)g. However, at the initial stage of evolution where nonlinearity is not signi¯cant, these solutions can serve as a reasonable approximation to the model solutions. In Figure 4.3, we show a typical picture of the evolution of an azimuthal mode-one Kelvin wave (M1R1+), where we chose B = 4, ¤ = 0:025 and initial wave amplitude A 0 =¡0:3, and we set the spectral resolution N =70 and used an integration time step ¢t = 0:0025. The negative side of the isopycnal amplitude (Z) propagates faster than 87 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0 0 0 0 0 0 0 0 0 0 + + + + 0 0 0 0 + + 0 0 + 0 0 + (a) (b) (c) (d) Figure 4.3: Evolution of the Kelvin wave (M1R1+). Snap shots of the isopycnal ampli- tude Z (upper row) and the magnitude of the velocity amplitude p U 2 +V 2 (lower row) are taken at (a) t = 0, (b) t = 8, (c) t = 16 and (d) t = 24. Contour level step is 0.05 for all the plots. The lowest contour level shown in the velocity contour plot is 0.05. its positive side, creating a front which gradually steepens as it travels in the cyclonic direction (t=8¼1:4T, where T is the linear wave period). The steep front evolves and it generates a train of shorter waves (t = 16¼ 2:7T), the leading front passes through its own oscillatory tail, and then the ¯eld becomes more complicated generating even shorter length-scale waves (t = 24¼ 4:1T). A width of a linear Kelvin wave l K can be de¯ned in a similar way to one used in the case of an in¯nite channel model, l K = R ¤ (r) R 0¤ (r) ¯ ¯ ¯ ¯ r=1 =B ¤¡1 I m (B ¤ ) I 0 m (B ¤ ) : (4.39) Since I m (x)=I 0 m (x)! 1 as x!1, then l K »B ¤¡1 for largeB ¤ (=B=c p 1¡! 2 »B=c, after noting ! ¿ 1 for large B from Figure 4.1). At the present Burger number, the along-shorehorizontal°uidmotion(seethesecondrowofFigure4.3)iscon¯nednearthe shore(l K =0:27),anditpersistsevenaftertheoscillatorywavesdevelop. Thesteepening 88 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.7 0.8 0.9 1 θ/π r 0 0 0 0 0 0 0 0 0 0 0 0 0.1 0.1 0.2 -0.1 -0.2 -0.3 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 Figure 4.4: Developed view of the Kelvin wave front (B = 8, A 0 = ¡0:3, ¤ = 0:025, t=18). The front position and its propagating direction are indicated by an arrow. of the Kelvin wave is not a new result by itself. It has been demonstrated by several authorsbyutilizingnonlinearmodels ineither semi-in¯niteorin¯nite domains. Bennett (1973) ¯rst demonstrated the steepening of Kelvin waves by analyzing a transversely geostrophic shallow water model. Tomasson & Melville (1990) demonstrated a triad resonance of a periodic Kelvin wave train by using a weakly nonlinear, dispersive model, andatthesametimetheirnumericalsimulationshowedsteepeningofKelvinwavefronts. Fedorov & Melville (1995) demonstrated steepening and subsequent breaking of Kelvin wave front by using a shallow water model. Also, the steepening of Kelvin waves on the equatorialthermoclinehasbeenstudiedbymanyauthors(e.g.,Fedorov&Melville2000; Boyd 1998). Regarding the steepening of Kelvin waves, Maxworthy (1983) produced a strongly nonlinear, solitary internal Kelvin wave of vertical mode-two in his laboratory tank, where it was clearly shown that the crest of the wave was curved backward as it traveled along a the side wall. Melville et al. (1989) later demonstrated the backward curvature of solitary Kelvin wave by numerically simulating a weakly nonlinear, dispersive model in a channel. Similar results are also found in our model simulation as shown in Figure 4.4, whereZ is developed over the r-µ plane forB =8 att=18. Lines of constant phase (trough) curves backward relative to their traveling direction. The trough of the leading wave is less curved due to the presence of the steep front, but troughs of following waves are clearly curved backward. This curvature owes to the nonlinear e®ect on the wave 89 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8 9 t/T θ/π A 0 =-0.2 A 0 =-0.4 Linear Figure 4.5: Azimuthal location of the minimum value of the isopycnal amplitude Z (the leading Kelvin wave trough) as a function of time. phasespeed(i.e.,wavesoflargeramplitudetravelfasterthanthoseofsmalleramplitude). In Figure 4.5 the azimuthal location of the minimum value of Z is plotted as a function of time for di®erent wave amplitudes, showing that the wave of larger amplitude travels faster. Small jumps at t¼ T for A 0 =¡0:4 and at t¼ 1:5T for A 0 =¡0:2, indicated by arrows in the ¯gure, implicitly indicate the emergence of oscillatory waves, where the minimum point of Z quickly shifts from the middle of the wave trough to the trough of the leading oscillatory wave following the steepened front. TherateofsteepeningofaKelvinwaveisastrongfunctionofamplitudeandenviron- mentalparameters(i.e., theverticalstructure,B and¤). Toquantifythis, wecalculated the maximum azimuthal gradient of the Z ¯eld along the basin perimeter for various sets of parameters. Figure 4.6a shows the gradient of the wave front as a function of time for di®erent initial amplitudes for ¯xedB =4 and ¤=0:025. Calculated gradients are normalized by their initial values. The ¯gure exhibits that waves of larger ampli- tude steepen quickly, reaching a larger maximum gradient, where the gradient does not increase any more owing to the balance between nonlinear and dispersive e®ects. Figure 4.6b showsthe gradientof the wavefrontas a functionof time for variousvalues ofB for 90 0 1 2 3 4 0 10 20 30 40 t/T [(dZ/dθ) r=1 ] max |A 0 |=0.4 |A 0 |=0.3 |A 0 |=0.2 0 4 8 12 16 20 24 0 10 20 30 t [(dZ/dθ) r=1 ] max B=0.5 B=B c B=2 B=4 B=6 B=8 0 1 2 3 4 0 10 20 30 40 50 60 70 0.02 0.04 Λ = 0.01 0.03 t/T [(dZ/dθ) r=1 ] max (a) (b) (c) Figure 4.6: Time series of the maximum azimuthal gradient of the isopycnal amplitude Z at r =1 for: (a) di®erent amplitudes A 0 with ¯xed Burger numberB =4 and aspect ratio ¤ = 0:025; (b) di®erent Burger numbers with ¯xed jA 0 j = 0:3 and ¤ = 0:025; and (c) di®erent aspect ratios with ¯xedB =4 andjA 0 j=0:3. ¯xed A 0 = ¡0:3 and ¤ = 0:025. The rate of steepening is signi¯cantly modi¯ed when B is relatively small (B < 2), resulting in a slower steepening rate for smaller B. But the wave is not steepened any more if B < B c , where the wave becomes super-inertial (Poincar¶ etype),althoughthepropagationdirectionofthewaveisstillcyclonic,thesame as that of Kelvin wave. 91 0 0 + (a) (b) (c) 0 0 0 0 + + + 0 0 0 0 0 + + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + + Figure 4.7: Comparison of the isopycnal amplitude Z for di®erent Burger numbers (a) B = 2, (b) B = 4 and (c) B = 8 at t = 0 (upper row) and t = 16 (lower row). Contour level step is 0.05. For larger B(> 4), the steepening rate is very similar. In Figure 4.7, we show snap shots of the Z ¯eld at ¯xed time (t=16) for di®erent values ofB along with their initial conditions. At t = 16, the wave front reached a near maximum slope, and the front already generated oscillatory waves on its tail. We also observe the ¯gure that wave length of the oscillatory tail becomes shorter for largerB, and the radial width of these waves becomes shorter for larger B (l K » c=B). Figure 4.6c shows the gradient of the wave front as a function of time for di®erent ¤ for ¯xed A 0 =¡0:3 andB = 4. As seen in (4.21) and (4.22), ¤ 2 scales the dispersive terms. For small ¤, the dispersive e®ect becomes very small and the evolution model becomes more like a non-dispersive shallow water model where the wave front evolves to a shock. The rate of steepening becomes greater, and it reaches a larger maximum for smaller ¤. For ¤ = 0:01, the gradient became so steep for the present spectral resolution that the numerical integration was terminated at t ¼ 2 due to emergence of Gibb's phenomenon. In Figure 4.8 we show 92 (a) (b) (c) + + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + + 0 0 0 0 + + 0 Figure 4.8: Comparison of the isopycnal amplitude Z for di®erent aspect ratios (a) ¤=0:04, (b) ¤=0:03 and (c) ¤=0:02 at t=17:5. Contour level step is 0.05. snapshots of the Z ¯eld at t = 17:5(¼ 3T) for di®erent values of ¤. It can be seen that the smaller ¤ (shallower lake) generates an oscillatory tail of shorter wave length. Steepeningofthewavefrontanditssubsequentgenerationofoscillatorywavesimplies nonlinear energy transfer from basin scale to smaller scales of °uid motion. A spectral method is very advantageous to quantifying the energy contained in each of the discrete wave numbers. In Figure 4.9 a horizontal wave number spectrum is shown of the verti- cally integrated potential energy at t=12 (¼2T) for di®erent amplitudes forB =4 and ¤=0:025, along with their corresponding snap shots of the Z ¯eld. The spectral energy densityisnormalizedbythetotalpotentialenergy, andthehorizontalwavenumber¯eld is a discrete set of Fourier azimuthal modes (m) and radial one-sided Jacobi polynomial modes (n) within the spectral truncation N =70. For clarity, the de¯nition of the ¯eld energy density is E´ 1 2 fu 2 +v 2 +w 2 g+ 1 2 ¾ 2 N 2 ; (4.40) where the ¯rst bracketed term corresponds to kinetic energy and the second corresponds topotentialenergy. Althoughthepotentialenergytermisexpressedbyanin¯niteseries of higher order terms (e.g, see Garkema 2003), they are less signi¯cant and hence we 93 azimuthal mode radial mode 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 70 azimuthal mode radial mode 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 or less log ( PE / total PE ) 10 (a) (b) (c) 70 azimuthal mode radial mode 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 + 0 0 + 0 0 0 0 0 0 0 0 + Figure 4.9: Comparison of potential energy spectrum of the Kelvin wave (M1R1+) at t = 12 for di®erent initial amplitudes (a) A 0 =-0.2, (b) A 0 = ¡0:3 and (c) A 0 = ¡0:4 along with the corresponding isopycnal amplitude Z. truncated them in this study. For a single vertical mode and uniform depth, (4.40) is expressed after scaling as E´ 1 2 ( (U 2 +V 2 )Á 02 (z)+¤ 2 µ @U @r + U r + 1 r @V @µ ¶ 2 Á 2 (z) ) + 1 2 Z 2 Á 2 (z): (4.41) The kinetic energy of the vertical component of velocity can be also neglected noting w=v » ¤ ¿ 1, but we retained it here. Initially, all the energy is concentrated in azimuthalmode-one. Inthelinearmodel,theenergyiscontainedinazimuthalmode-one at all times, which was con¯rmed in our numerical code by turning o® all the nonlinear terms. ForA 0 =¡0:2,thefrontisnotquitesteepenedatt=12andmostoftheenergyis stillcontainedinm=1. Forlargeramplitudes, thewavefrontbecomessteeperentailing oscillatory waves, and then more energy is found in higher modes. In Figure 4.10, we present a total energy spectrum as a function of azimuthal mode for several di®erent times for B = 4, ¤ = 0:025 and A 0 = ¡0:3. Spectral convergence is readily observed 94 2 energy 10 0 10 1 10 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 azimuthal mode 70 (truncation limit) (t = 0) t=4 t=8 t=12 t=16 t=20 azimuthal mode 10 0 10 1 10 −4 10 −3 10 −2 10 −1 10 0 energy (t = 0) t=4 t=8 t=12 t=16 t=20 (a) (b) Figure 4.10: (a) Azimuthal modal energies of the Kelvin wave (M1R1+) at di®erent times. Energies in lower modes are magni¯ed in (b). from the whole energy spectrum (Figure 4.10a). Looking at the enlarged plot (Figure 4.10b), the energy in higher modes increases in time, and the energy in m=1 decreases in exchange. At t = 16 (¼ 2:7T), a spectral energy peak emerges at m = 8, which corresponds to the azimuthal wave number of oscillatory waves. At t=20 (¼3:4T), the oscillatory wave gains more energy, being slightly stretched due to dispersion. Figure 4.11a shows a total energy spectrum at t = 20 for di®erent initial amplitudes. Again, more energy is found in higher modes at the expense of energy in m = 1 for larger amplitude. For A 0 =¡0:2 at t=20, the oscillatory tail is still premature, but for larger amplitudes the oscillatory tail is already developed and a spectral energy peak appears at m = 7. Figure 4.11b shows the similar azimuthal energy spectrum at t = 20 for di®erent values of B for A 0 =¡0:3. With this amplitude, the oscillatory tail is already developedatt=20. The¯gureindicatesthatthespectralenergypeakappearsinhigher azimuthal modes for larger B, which is qualitatively consistent with the observation of Figure 4.7. Recently,theinternalwaveevolutioninacircularbasinofuniformdepthwasstudied in laboratory experiments (Wake et al. 2004, 2005). However, the nonlinear steepening 95 (a) (b) energy azimuthal mode 10 0 10 1 10 −4 10 −3 10 −2 10 −1 10 0 (t = 0 ) |A 0 |=0.2 |A 0 |=0.3 |A 0 |=0.4 azimuthal mode energy 10 0 10 1 10 −4 10 −3 10 −2 10 −1 10 0 B=2 B=4 B=8 Figure 4.11: Azimuthal modal energies of the Kelvin wave (M1R1+) at t = 20 for: (a) di®erent initial amplitudes A 0 with ¯xed Burger numberB =4; and (b) di®erent Burger numbers with ¯xed amplitude A 0 =¡0:3. was not observed in their experiments. The vertical structure was a two-layer con¯g- uration, and for most of the experimental runs the upper and lower layer depths were set equal. In such a case, the coe±cient of the leading nonlinear term ¯ (u;v) is identi- cally zero, which can greatly reduce the chance to observe nonlinear e®ects. Also, the larger dissipation common in laboratory tanks possibly damps the °uid motion before the nonlinear steepening becomes signi¯cant. Forthenextinitialvalueproblem, wechooseaPoincar¶ ewaveofazimuthalmode-one andradialmode-one(M1R1-). AtypicalportraitofPoincar¶ ewaveevolutionobtainedby numerical simulation is shown in Figure 4.12 forB =4, ¤=0:025 and A 0 =¡0:3. The initial condition is symmetric, but the ¯eld quickly loses its symmetry as it evolves. The asymmetric¯eldsubsequentlytendstoreturntoanearsymmetric¯eld,andthisrepeats aperiodically. If the model is linear hydrostatic, the ¯eld is symmetric permanently, rotating in an anti-cyclonic direction with constant speed without changing shape. On the contrary to the Kelvin wave, the °uid velocity is the largest at r = 0 for Poincar¶ e waves. LookingatthelowerrowofFigure4.12,contoursofthemagnitudeofthevelocity aresymmetrichavingtheirmaximumatr =0initially,butthevelocitycontoursbecome 96 (a) (b) (c) (d) 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9 Figure 4.12: Evolution of the Poincar¶ e wave (M1R1-). Snap shots of the isopycnal amplitude Z (upper row) and the magnitude of the velocity amplitude p U 2 +V 2 (lower row) are taken at (a) t = 0, (b) t = 0:7, (c) t = 1:4 and (d) t = 24. Contour level step for the isopycnal amplitude is 0.05. eccentric having their maximum o® the basin center as the wave evolves. It can be also observed from the ¯gure that the isopycnal amplitude is larger on the side where the maximum velocity is found. This asymmetric modulation of wave amplitude persists for the rest of evolution (see t = 24¼ 17:5T in Figure 4.12), and there is no symptom of wave front steepening which was observed in Kelvin wave evolutions studied earlier in this section. Although the result is not shown in here, we simulated this case with much smaller ¤ (=0.001), but both results were indistinguishable. This is because the non-hydrostatic term is e®ective only when the ¯eld gradient is large, which is not the case for the Poincar¶ e wave evolution. In Figure 4.13, we plotted the maximum and minimumvaluesofZ asfunctionsoftimefordi®erentinitialamplitudes. Theamplitude modulationisaperiodicandbiasedasymmetricallyonthenegativeside, andtheamount of the modulation becomes larger for a larger initial amplitude. 97 0 1 2 3 4 5 6 t/T −1 −1.1 −1.2 −1.3 −1.4 −1.5 1 0.9 0.8 0.7 Z Z min /|A | 0 max /|A | 0 |A 0 |=0.2 |A 0 |=0.3 |A 0 |=0.4 Figure 4.13: Time series of the minimum and the maximum values of the isopycnal amplitude Z for Poincar¶ e waves (M1R1-) of di®erent amplitudes. The Figures 4.14ab show the normalized potential energy spectrum of the ¯eld at t = 4 ¼ 3T for di®erent amplitudes, along with snapshots of the Z ¯eld. Although the actual spectral resolution is N = 70, the energy spectrum is truncated to N = 30 because of insigni¯cant energy in higher modes. Energy is concentrated in m = 1, and it spreads over higher radial modes rather than higher azimuthal modes, implying faster spectral convergence in azimuthal modes. More energy spreads over higher radial and azimuthal modes for larger initial amplitude (A 0 =¡0:4), and the ¯eld becomes more asymmetric. In Figure 4.14c, we also show the case at t=20¼14:5T for A 0 =¡0:3 to indicate that a similar modal energy structure persists even after a long time. The amplitude modulation is also observed for cyclonic, super-inertial wave modes (Poincar¶ e type). Figure 4.15 shows evolution of the isopycnal amplitude starting with the M1R1+ initial condition forB =1=2(<B c ). The ¯eld is symmetric initially, but the amplitude in the negative side increases and the amplitude in the positive side decreases (t = 8 ¼ 2T) as the wave evolves. Later, the asymmetry returns very nearly to the symmetryoftheinitialcondition(t=16¼4T),andthisisrepeatedapproximatelyevery 4T. The recurrence of the initial condition has been found in the integrable KdV model 98 azimuthal mode radial mode 0 10 20 30 0 10 20 30 azimuthal mode radial mode 0 10 20 30 0 10 20 30 azimuthal mode radial mode 0 10 20 30 0 10 20 30 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 or less log ( PE / total PE ) 10 (a) (b) (c) + 0 0 0 0 + 0 0 0 0 0 + Figure4.14: ComparisonofpotentialenergyspectrumofthePoincar¶ ewave(M1R1-)for di®erent initial amplitudes (a) A 0 =-0.2 and (b) A 0 =¡0:4 at t = 4, and (c) A 0 =¡0:3 at later time t=20 along with the corresponding isopycnal amplitude Z. (a) 0 0 0 + (b) + 0 0 0 (c) + 0 0 0 (d) + 0 0 0 (e) + 0 0 0 Figure4.15: PseudorecurrenceofaPoincar¶ e-typewave(M1R1+withtheBurgernumber B = 1=2). Snap shots of the isopycnal amplitude Z are taken at (a) t = 0, (b) t = 8(1:97T), (c) t=16(3:93T), (d) t=24(5:9T) and (e) t=32(7:86T). Contour level step is 0.05. 99 under periodic boundary conditions (Zabusky & Kruskal 1965). The KdV evolution involves steepening and generation of solitary waves which have amplitude-dependent phase speeds. The recurrence is achieved when all the soliton's phases become very close. However, the pseudo recurrence character found in the Poincar¶ e wave presented above is of a di®erent type because of the absence of front steepening and subsequent generation of solitary-like waves. 4.5.2 Evolution on variable depth We now examine brie°y e®ects of variable depth. We ¯x the metalimnion depth as the value used in the previous section (h 2 =1), and perturb the depth of hypolimnion from the base depth h 3 =3. We consider two simple, model topography pro¯les: i) a slanted bottom surface h b =1+h 2 +h 3 +¢hrcosµ; (4.42) and ii) a symmetric parabolic bottom surface h b =1+h 2 +h 3 ¡¢hr 2 ; (4.43) where ¢h is in each case a depth perturbation scaled by h 1 . These depth functions are regular at r =0. Vertical structure pro¯le (4.38), which is independent of h 3 , holds over the whole domain. We ¯rst consider evolution of Kelvin waves in an environment described by (4.42) and (4.43). In Figure 4.16, snapshots of isopycnal amplitude at two di®erent times are tabulatedforaslantbottomwith¢h=1andaparabolicbottomwith¢h=1. Included in Figure 4.16 are results for constant depth cases h 3 = 3 and h 3 = 2 for comparison. The Burger number and the aspect ratio were set atB =4 and ¤=0:025, respectively, and resolution of the numerical integration was set with N = 70 and ¢t = 0:0025. For all the cases in the ¯gure, the system was integrated from the common initial condition, which is a linear, hydrostatic, Kelvin wave solution for h 3 =3 with the initial amplitude 100 6 4 5 4 4 5 4 0 0 0 0 + + 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 + 0 0 0 + 0 0 0 0 0 0 + + Figure 4.16: Snap shots of the isopycnal amplitudes Z of the Kelvin waves at t = 8 (upper contour panels) and at t=16 (lower contour panels) for di®erent depth pro¯les. The depth pro¯les is depicted on the top of each column. A 0 = ¡0:3 (see Figure 4.3 at t = 0). The ¯eld evolution for the slant basin appears nearly the same as the original case of constant depth. However, the parabolic basin exhibits a slower phase speed and slower steepening than the original °at basin. The evolution in the parabolic basin rather resembles the one for the constant depth case h 3 = 2, the same depth at the basin perimeter in the parabolic basin. The slow phase speedprimarilyresultedfromthefactthatthelinearphasespeedissmallerforshallower depth (i.e., c = 0:9395 at h 3 = 3 and c = 0:8763 at h 3 = 2, and note that the wave frequency is scaled by c for givenB as implied in Figure 4.1). Figure 4.17 shows the maximum azimuthal slope of the isopycnal amplitude at the basin perimeter as a function of time for di®erent values of depth perturbation in both the slant and parabolic basin con¯gurations. For the slant basin, the slope °uctuates as the wave travels over the variable depth cyclonically. The °uctuation becomes greater for larger depth perturbation, but it is not relatively signi¯cant for the present amount 101 (a) (b) [(dZ/dθ) r=1 ] max 0 1 2 3 4 0 10 20 30 t/T const. depth (h =5h 1 ) parabola Δ h=0.5 parabola Δ h=1 const. depth (h =4h 1 ) 5 Δh b b 0 1 2 3 4 0 10 20 30 t/T [(dZ/dθ) r=1 ] max const. depth (h =5h ) taper Δ h=0.5 taper Δ h=1 Δh 5 1 b Figure 4.17: Time series of the maximum azimuthal gradient of the isopycnal amplitude Z at r =1 for (a) slant depth pro¯le and (b) parabolic depth pro¯le. of perturbation. On the contrary, the rate of steepening is signi¯cantly a®ected by the depthatthebasinperimeterintheparabolicbasin. Inthesame¯gure,itcanbeseenthat therateofsteepeningfor¢h=1andtheuniformdepthofh b =4fortheparabolicbasin are very similar. Similar to the phase speed argument above, the coe±cient of primary nonlineartermsin(4.21)and(4.22)are¯ (u;v) =¡0:3691ath 3 =3and¯ (u;v) =¡0:2594 at h 2 =2, implying less nonlinearity for smaller depth, and consequent slower growth of the wave front near the basin perimeter where much of ¯eld kinetic energy is con¯ned. Forthe slantbasin, the environmentalcoe±cients varysinusoidally centeredatvaluesat average depth along with the perimeter. This is why similar evolution pictures to those of the original °at basin appear in the case of the slant basin. Field evolution of the lowest Poincar¶ e waves (M1R1-) in basins of variable depth established above is considered for the rest of this section. Both physical and numerical parameter con¯gurations are the same as those for the Kelvin wave case noted above, although the Poincar¶ e wave simulation requires less numerical resolution due to lack of progressive front steepening. Figure 4.18 shows snap shots of the isopycnal amplitude at di®erent times for the slant basin of ¢h = 1, along with those for the original °at basin. These snap shots are selected in such a way that either the negative side of Z 102 6 4 5 Z - Z 2 1 |A | 0 (a) (c) (b) 0 + 0 0 0 + 0 0 0 + 0 + 0 0 0 0 0 + 0 0 0 + 0 0 0 + 0 + 0 0 0 0 0 + 0 0 0 + 0 0 0 + 0 + 0 0 0 0 0 Figure4.18: Comparisonoftheisopycnalamplitudesfortheuniformdepthcase(Z 1 )and theslantdepthcase(Z 2 )at(a)t=2:7(1:96T),(b)t=3:4(2:47T)and(c)t=4:1(2:98T). The contour level step for these plots are set to be 0.05. On the most right column the di®erence of Z 1 and Z 2 scaled by the initial amplitude jA 0 j is plotted with a contour level step 0.02. shoals (t=2:7 andt=4:1) or the positive side of Z shoals (t=3:4). Although the wave phases are nearly the same for the slant and °at basin, their amplitudes are di®erent. WealsopresentonthethirdcolumninFigure4.18thenormalizeddi®erenceofisopycnal amplitudes between those of the slant (Z 2 ) and °at basin (Z 1 ), i.e., (Z 2 ¡Z 1 )=jA 0 j. It can be seen from the ¯gure that the amplitude of the wave increases as the wave shoals, and the amplitude decreases as the wave travels into the deep side of the basin. 103 0 1 2 3 4 5 6 t/T −1 −1.1 −1.2 −1.3 −1.4 −1.5 −1 1 0.9 0.8 0.7 1 Z min Z max linear model linear model h b =5h 1 (const. depth) parabola Δh=1 /|A | 0 /|A | 0 Figure 4.19: Time series of the maximum and the minimum values of the isopycnal amplitude Z of the Poincar¶ e wave over the slant depth pro¯le with ¢h = 1. For com- parison purpose the time series for the linear hydrostatic model and for di®erent depth cases are also included in the ¯gure. We measured the maximum and minimum values of Z were measured as functions of time for the slant and °at basin and plotted in Figure 4.19. The ¯gure implies that the nonlinear amplitude modulation is dominant over the modulation induced by the variable depth. The modulation of amplitude for variable depth is about 10% of the original amplitude (see also the case for the linear, variable depth model in Figure 4.19), but the nonlinear modulation corresponds to 35%. In contrast to Kelvin waves, the kineticenergydensityisconcentratedatthebasincenterinPoincar¶ ewave¯elds. Hence, itisnaturallyexpectedthatthedepthchangecon¯nedtowardthebasinperimetera®ects Poincar¶ e ¯eld evolution only slightly. To demonstrate this, a time series of the extremal values of the isopycnal amplitude for the parabolic basin with ¢h=1 are also included in Figure 4.19. It is evident from the ¯gure that the extremal values obtained in the parabolic basin are close to those obtained in the °at basin. 104 4.6 Wind-forced evolution We apply our evolution model to the wind-forced problem, where multiple wave modes can be excited and they co-exist with, or may even generate, other modes via nonlinear interaction. Welimitourstudytoalakeofuniformdepthwhichissubjecttowindforcing of ¯nite duration. Uniform wind stress over the basin surface is ideal as a fundamental model, but it is not suited to the spectral model due to the fact that such a stress function is discontinuous at the basin perimeter, which can immediately destruct the spectral convergence of numerical solutions. Instead, we use a radially symmetric stress directed along the x(horizontal) axis written by a simple formula ¡ ! ¿ h = n s +1 n s (1¡r 2ns )f1¡u s (t¡t 0 )ge x ; (4.44) whereu s (t)istheHeavisidestepfunction,t 0 iswindforcingduration,ande x =cosµe r ¡ sinµe µ . This stress starts from a maximum value at the basin center and decreases to zero toward the perimeter. The average of the stress over the surface is unity, and the magnitude of the stress is controlled by the Wedderburn numberW. The parameter n s in (4.44) is an integer parameter to control the stress shape: n s = 1 gives a parabolic distribution, and increasing n s decreases the stress at the basin center while increasing the stress near the perimeter making the distribution close to uniform (j ¡ ! ¿ h j ! 1 as n s !1). Itisclearfrom(4.44)thatthewindenergyisinjectedonlythroughazimuthal mode-one. It is also important to note that (4.44) is a regular function at r = 0. The wind forcing duration is parameterized by a fraction · w of the inertial period T i ; i.e. t 0 =· w T i =· w 2¼ B : (4.45) Physical and numerical parameters are set to h b =5 (h 2 =h 1 =1 & h 3 =3), ¤=0:001, N =31 and ¢t=0:005, unless otherwise noted. 105 0 5 10 15 0 0.5 1 1.5 2 W=2.5 W=2 W=1.5 t total energy 0 5 10 15 0 0.2 0.4 0.6 0.8 1 KE PE t KE, PE (a) (b) Figure 4.20: (a) Time series of the total energy for di®erent Wedderburn numbers W and (b) time series of kinetic (KE) and potential (PE) energies for W =2. 10 −1 10 0 10 1 10 −3 10 −2 10 −1 10 0 B=1/2 W total energy B=1 B=2 B=4 B=8 empty dots : n = 1 filled dots : n = 8 s s Figure 4.21: Total energy as a function of the Wedderburn numberW. We computed the volume integral of the total energy density as a function of time for di®erent Wedderburn numbers for B = 4, n s = 1 and t 0 =T i =3, and show results in Figure4.20a. The¯eldenergyincreasesuntilthewindisturnedo®, andtheenergythen stays approximately constant for the rest of the evolution due to the absence of friction damping in the model. The total energy is not necessarily an exact constant, however, because we used the truncated expression (4.41). Figure 4.21 shows the total energy at t = t 0 as a function of the Wedderburn number for di®erent values B and n s . The 106 (a) (b) 0 2 4 6 8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 W <PE>/<KE> B=4 B=2 linear hydrostatic 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 W=4 W=1 open dots:n =1 filled dots: n = 8 B <PE>/<KE> M1R1+ M1R1− s s Figure 4.22: (a) Energy ratio as a function of Burger numberB for di®erent wind stress pro¯les and (b) energy ratio as a function of the Wedderburn number W for di®erent Burger numbers. total energy is proportional to W ¡2 regardless of the stress shape and Burger number. We also show in Figure 4.20b the volume integrated kinetic energy (KE) and potential energy (PE) as functions of time for W = 2. Oscillatory energy exchange between PE and KE is clearly observed, which readily suggests that multiple wave modes co-exist in the wind excited ¯eld. For a speci¯c wave mode in the linear hydrostatic model, one can easilyshowfrom(4.28)thatPEandKEareindependentoftimeforagivenamplitude(a wave of permanent form and frequency rotates around a basin center). If multiple wave modes co-exist in the wave ¯eld, an energy exchange between PE and KE can occur, primarily because frequencies of all the wave modes are distinct. Hence, in a real wave ¯eld, computing energy partitioning between PE and KE is meaningful only in a time average sense. InFigure4.22a, wecomputedtheenergypartitionbetweenPEandKEasafunction ofBurgernumberfort 0 =T i =3. Eachenergyisaveragedinanintervalofthelongestwave period for the given Burger number (i.e. Kelvin wave period), after the wind is turned o®. Thecomputedenergypartitionrangeswithin§10%comparedtothevalueaveraged in three Kelvin wave periods. We also included in the ¯gure the energy ratio for linear 107 hydrostaticKelvin(M1R1+)andthegravestPoincar¶ e(M1R1-)waves. Energypartition shows a preference in kinetic energy for the Poincar¶ e wave mode, especially for larger B. In Figure 4.22b, we also show the energy partition as a function of Wedderburn number for di®erent Burger numbers, along with the value obtained from the linear hydrostatic model. It can be seen from the ¯gure that the dependence of Wedderburn number on the energy partition is very weak. This suggests that the linear hydrostatic model can serve as a good tool for estimating energetics, at least in the initial stage of free evolution. Wind forced response of a linear hydrostatic, circular basin has been well documented (Csanady 1972; 1968ab; Stocker & Imberger 2003). Referring to these works, an analytical solution to the linear hydrostatic model subject to uniform wind stress can be obtained by a Laplace transform approach, and the ¯nal result is written in terms of our scaled variables as: U = k s W 1 X n=1 A n B(1¡! 2 1n ) ½ ! 1n R 0 1n (r)¡ R 1n (r) r ¾ cos(µ¡B! 1n t+¢ 0 ); V = k s W 1 X n=1 A n B(1¡! 2 1n ) ½ R 0 1n (r)¡! 1n R 1n (r) r ¾ sin(µ¡B! 1n t+¢ 0 ); Z = k s c 2 W 1 X n=1 A n R 1n (r)sin(µ¡B! 1n t+¢ 0 ); 9 > > > > > > > > > > = > > > > > > > > > > ; (4.46) where R 1n is a normalized modal eigenfunction of azimuthal mode-one and n-th radial mode with corresponding eigenfrequency ! 1n , phase shift ¢ 0 = B! 1n t 0 =2, and A n is a radial modal amplitude given by A n =2 ! 1n ¡1 1+! 1n ¡(B=c) 2 ! 3 1n sin µ B! 1n t 0 2 ¶ jR ¤ 1n (r)j max R ¤ 1n (1) : (4.47) Note that (4.46) is a solution for post wind forcing (t ¸ t 0 ), hence the geostrophic (steady) solution is canceled in the expression. From (4.46), all ¯eld variables are pro- portional to k s W ¡1 . This implies the total energy is proportional to W ¡2 , which is consistent with our numerical result presented in Figure 4.21. 108 (a) t 0 =T i =1=3 B R1+ R1- R2+ R2- R3+ R3- 1 1 0.372 0.072 0.097 0.038 0.047 2 1 0.481 0.029 0.049 0.049 0.080 4 1 0.368 0.064 0.259 0.043 0.108 (b) t 0 =T i =1=2 B R1+ R1- R2+ R2- R3+ R3- 1 1 0.238 0.106 0.149 0.006 0.011 2 1 0.263 0.054 0.115 0.007 0.014 4 1 0.262 0.027 0.098 0.014 0.040 (c) t 0 =T i =1 B R1+ R1- R2+ R2- R3+ R3- 1 1 0.375 0.020 0.056 0.010 0.018 2 1 0.302 0.047 0.083 0.012 0.024 4 1 0.064 0.023 0.090 0.015 0.040 Table 4.1: Modal amplitudes based on the linear theory for di®erent Burger numbers B and wind forcing durations t 0 =T i . Amplitudes are normalized by the amplitude of the lowest wave mode (M1R1+). Field response for the linear hydrostatic basin subject to uniform wind stress is constructed by an in¯nite family of radial modes for azimuthal mode-one waves. Table 4.1showsthemodalamplitudeforthe¯rstthreeradialmodesofcyclonicandanticyclonic wavescomputedby(4.47)fordi®erentBandt 0 . (4.47)impliesthatthemagnitudeofeach modalamplituderangesformzerotoitsmaximumvalueeveryT i =2! 1n . Unlessthewind blows too long (t 0 <T i ), the dominant wave modes are a Kelvin wave (M1R1+) having thelargestamplitudeandthegravestPoincar¶ ewave(M1R1-). Onceinitiatedbythewind forcing, these modal amplitudes are invariant for the rest of the response. In a nonlinear counterpart, although the wind energy is introduced only via azimuthal mode-one, the energy can spread over other azimuthal and radial modes via nonlinear interaction. In Figure 4.23a, we show a frequency power spectrum of the isopycnal amplitude sampled 109 1 10 −1 10 0 10 −4 10 −5 10 −3 10 −2 10 −1 10 0 10 frequency [1/T] spectral density M1R1+ M2R1+ M3R1+ M1R1- M2R1- M3R1- M1R2+/- M2R2+/- M3R2+/- 10 −5 10 −1 10 0 10 −4 10 −3 10 −2 10 −1 10 0 10 1 frequency [1/T] spectral density M1R1+ M1R1- M1R2- M1R3- (a) (b) Figure4.23: FrequencypowerspectrumoftheisopycnalamplitudeZ sampledat(r;µ)= (1;0) for (a) nonlinear, non-hydrostatic model and (b) linear hydrostatic model. at (r;µ) = (1;0) for B = 4 and W = 3:5. The sampling interval is 0 · t · 50, which corresponds to eight and one-half Kelvin wave periods. At this Wedderburn number, the isopycnal amplitude is relatively small (» 0:12), and the steepening of the Kelvin wave becomes noticeable only after about t = 40. We did not sample the signal beyond t=50 due to insu±cient numerical resolution to resolve the steepened front. In the ¯gure, frequencies of the ¯rst three azimuthal and the ¯rst two radial wave modes determined from the linear hydrostatic model are indicated. The spectral peaks agree with these frequencies very well. The Kelvin wave and the gravest Poincar¶ e modes are still dominant modes even in the long time interval. A similar trend was also found for di®erentBurgernumbersandsamplingpoints,althoughtheseresultsarenotshownhere. We also simulated the same case under the linear hydrostatic con¯guration having all nonlinearandnon-hydrostatictermsturnedo®inournumericalcode. Thecorresponding frequency spectrum is shown in Figure 4.23b. The spectral density has peaks at only azimuthal mode-one waves, which implies there is no energy leakage to other azimuthal modes in the numerical code. 110 Unlessamodelislinearhydrostatic,itisnotastraightforwardtasktodeterminethe energy contained in a given wave mode. Motivated by the above results that the Kelvin andgravestPoincar¶ emodesaredominant,weaddresstheenergypartitionbetweenthese dominant modes. In a linear hydrostatic sense, we try to estimate roughly the energy partition from our results obtained by numerical simulation. Assuming the post-wind forcing ¯eld is dominated by two wave modes, we write the ¯eld solution as a linear superposition of the modes U = 2 X n=1 A n P n (r)cos(µ¡B! n t¡± n ); V = 2 X n=1 A n Q n (r)sin(µ¡B! n t¡± n ); Z = 2 X n=1 A n R n (r)sin(µ¡B! n t¡± n ); 9 > > > > > > > > > > = > > > > > > > > > > ; (4.48) where index n = 1 and n = 2 represent Kelvin wave and Poincar¶ e wave mode, respec- tively. Radial structure functions of the velocities P n (r) and Q n (r) are de¯ned after (4.28) with m=1, so that P n (r)=¡ c 2 B(1¡! 2 n ) ½ R n (r) r ¡! n R 0 n (r) ¾ ; Q n (r)= c 2 B(1¡! 2 n ) ½ ! n R n (r) r ¡R 0 n (r) ¾ : 9 > > > = > > > ; (4.49) The volume integral of the potential energy is computed as PE = 1 2 Z 0 ¡H Z 2¼ 0 Z 1 0 N 2 (z)Á 2 (z)Z 2 (r;µ;t)rdrdµdz = ¼ 2 I ½ A 2 1 Z 1 0 R 2 1 rdr+A 2 2 Z 1 0 R 2 2 rdr +2A 1 A 2 cos(B(! 2 ¡! 1 )t+± 2 ¡± 1 ) Z 1 0 R 1 R 2 rdr ¾ : (4.50) 111 Thelasttermin(4.50)isatime°uctuatingpartofthePEowingtodistinctfrequenciesof the wave modes. Looking at Figure 4.20b, the primary period of the energy oscillation is measured to be 1.2. Using eigenfreqencies for Kelvin (! 1 = 0:268) and Poincar¶ e (! 2 =¡1:14) modes for B = 4, the oscillation period is computed to be 1.12, which is close to the measured value. Figure 4.20b shows a modulation of the amplitude of the energy oscillation, possibly owing to contributions from higher wave modes that are not accounted in this two mode truncation. Taking the time average of (4.50), the energy °uctuation term vanishes leaving hPEi= ¼ 2 I ½ A 2 1 Z 1 0 R 2 1 rdr+A 2 2 Z 1 0 R 2 2 rdr ¾ : (4.51) Carrying out the same procedure for the kinetic energy counterpart yields hKEi= ¼ 2 I c 2 ½ A 2 1 Z 1 0 (P 2 1 +Q 2 2 )rdr+A 2 2 Z 1 0 (P 2 2 +Q 2 2 )rdr ¾ : (4.52) From (4.51) and (4.52), solving for A 2 1 and A 2 2 yields the amplitude ratio A 2 A 1 = r ¡°¹ 1 +¾ 1 °¹ 2 ¡¾ 2 ; (4.53) where ° is a ratio of potential to kinetic energy, and ¾ n and ¹ n are the radial modal energy densities of potential and kinetic energy: ° = hPEi hKEi ; ¾ n = ¼ 2 I Z 1 0 R 2 n rdr and ¹ n = ¼ 2 I c 2 Z 1 0 (P 2 n +Q 2 n )rdr: (4.54) The quantity ° is readily estimated from numerical simulation, and ¾ n and ¹ n are also readily computed using the eigenfunctions. After computing the amplitude ratio by (4.53), the modal energy ratiohE 2 i=hE 1 i is computed hE 2 i hE 1 i = ¾ 2 +¹ 2 ¾ 1 +¹ 1 µ A 2 A 1 ¶ 2 : (4.55) 112 (a) (b) B 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 n = 1 s A 2 /A 1 n = 8 s LinearFP W=1 W=4 0 2 4 6 8 10 0 5 10 15 20 n = 1 s B <E 2 >/<E 1 > LinearFP W=1 W=4 n = 8 s Figure 4.24: (a) Estimated amplitude ratio and (b) corresponding energy ratio between the Kelvin wave and the Poincar¶ e wave modes as a function of the Burger number B. The amplitude and energy ratios of Kelvin and Poincar¶ e waves as functions of B, after using the potential to kinetic energy ratio estimated previously in Figure 4.22a, are shown in Figure 4.24. Corresponding results for the linear, forced problem (LFP) described by (4.46) and (4.47) are included in Figure 4.24. For larger n s (wind stress approaches a uniform distribution), the amplitude as well as energy ratios get close to those of LFP. However, the amplitude ratio deviates substantially from that of LFP for smallerB. Onthecontrarytotheuniformwind,thecurlofanonuniformstress(ourcase) does not vanish, and a non-zero curl of the stress generates relative vorticity during the wind forcing. This implies that the relative vorticity serves as an inhomogeneous term in the radial eigenfunction equation (Bessel's equation), modifying the eigenfunction by necessitating particular solutions in addition to the free modal (Bessel) solutions. The wind generated vorticity (i.e., particular solution) remains even after the wind is turned o® to conserve the total vorticity in the system. Recalling our de¯nition of the forcing duration t 0 in (4.45), t 0 increases in proportional to B ¡1 . With these issues in mind, we believe that the long-time, wind-generated relative vorticity ¯eld becomes signi¯cant compared to Kelvin or Poincar¶ e wave modes, which can consequently render (4.53) and (4.55)invalid, since(4.53)and(4.55)arebasedonasimpleassumptionthatadmitsonly 113 KelvinandPoincar¶ ewavemodesinthe¯eld. We¯rstdoubtedthattheratiodiscrepancy for small B might be caused by the nonlinearity, but we discarded the possibility after we obtained nearly the same results (e.g., see Figure 4.22b) by running the code for the restricted, linear hydrostatic con¯guration. Observing (4.53), requiring the inside of the square root to be positive, we have the constraints that ¾ 2 ¹ 2 <° < ¾ 1 ¹ 1 or ¾ 1 ¹ 1 <° < ¾ 2 ¹ 2 : (4.56) These constraints imply that the gross potential-to-kinetic energy ratio must be within a range between those of Kelvin and Poincar¶ e waves. But, as seen in Figure 4.22a, ° is outside the range for small B. This discrepancy most likely derives from the non-zero curl of the wind stress as discussed above. From Figure 4.24b, the Poincar¶ e wave energy is pronounced for larger B (> 2), althoughtheKelvinwavehasalargeramplitudeasseeninFigure4.24a. Thisisbecause, for large B, the frequency of the Poincar¶ e wave approaches -1, which causes horizontal velocities to become pronounced via the relation (U;V) / (1¡! 2 ) ¡1 as described in (4.28)and(4.46);andalsotheKelvinwavebecomesmorelocalizedtothebasinperimeter (l K »c=B). For n s =1 (parabolic stress shape), the stress is biased at the basin center, favoring energy input in Poincar¶ e waves over the Kelvin wave, whose energy is con¯ned near the basin perimeter. 4.7 Conclusions A weakly-nonlinear, weakly-dispersive, wind-forced, variable environmental evolution model is derived for a continuously strati¯ed circular basin. The model was numerically simulated with the vertical modes in the ¯eld restricted to include only the lowest ver- tical mode. We investigated ¯rst the ¯eld evolution starting from initial conditions cor- responding to hydrodynamically-balanced, linear Kelvin and Poincar¶ e waves in a basin of both uniform and perturbed depth. Then we investigated the wind-forced response, 114 and evaluated our simulation results employing the linear hydrostatic model with an emphasis on energetically-dominant Kelvin and Poincar¶ e waves. Althoughthelineartheorygivesasetofwavesolutionsofpermanentformandspeed, our simulation exhibited that those linear solutions are not preserved in the nonlinear evolution. The Kelvin wave steepens as it travels, and the steepened front generates oscillatory waves owing to a balance between weak nonlinearity and non-hydrostatic e®ects. Through this nonlinear e®ect, the ¯eld energy is transferred from basin scale to sub-basin scales. The rate of steepening is a strong function of the wave amplitude, Burger number, vertical structure, depth-to-horizontal scale ratio, and typical depth withinaRossbyradiusalongthebasinperimeter. The¯eldkineticenergyisstillcon¯ned nearthebasinperimeterevenafterthedevelopmentofoscillatorywaves. Energytransfer from Kelvin to Poincar¶ e wave modes is thus insigni¯cant. On the other hand, Poincar¶ e wave evolution does not exhibit such a hyperbolic character as observed in Kelvin wave evolution. The amplitude of a Poincar¶ e wave is modulated, losing its symmetry and later returning to \near" symmetry as it evolves, exhibitingapseudorecurrencecharacter. Theamplitudemodulationisastrongfunction oftheinitialwaveamplitude, andthedepthatthebasincenter. Theenergyiscontained primarily in azimuthal mode-one with preference on rather higher radial modes. This structureofthemodalenergyispersistentforlongerevolutiontimesfollowingtheinitial adjustment. The¯eldkineticenergyiso®-shorebiasedatalltimes,withminordeviation from symmetry. Similar to Kelvin wave evolution, there is no signi¯cant energy transfer from the Poincar¶ e to the Kelvin wave mode. UniformornearuniformwindstressexcitesKelvinandPoincar¶ ewavemodespredom- inantly. If the forcing is not too strong, the linear theory can still be an e®ective tool to estimate the initial energy resident in these dominant modes. Also, the frequency infor- mationobtainedformthelineardispersionrelationisquiteusefultodiagnosetheinternal wave spectrum in the frequency space. After a single wind forcing event, the amplitude of the Kelvin wave is greater than the Poincar¶ e counterpart. This is quite misleading, 115 however, becausethetotalenergyoftheKelvinwavebecomesmuchlessthanthatofthe Poincar¶ e wave, especially for large lakes, or perhaps for lakes with weak strati¯cation. Consideringthe¯eldevolutioncharacterofthesewaves,energydissipationisexpectedto be greatly dependent on the lake dimension and the strati¯cation (Burger number). In large lakes for example, the shore-con¯ned, post-steepened, oscillatory Kelvin waves can shoal and eventually break as it passes through three dimensional irregular bathymetry, enhancing mixing at the shore and radiating near-buoyancy-frequency, short internal waves o® the shore. Poincar¶ e waves instead, can lose energy via bottom friction around the basin center due to the large bottom current there. To estimate the gross energy dissipation rate in a basin, it is important to estimate the energy partition in these dominant modes at the starting point. Hydrodynamic models for large lakes are usually based on the hydrostatic assump- tion, one that has been rationalized by noting that the horizontal length scale is much largerthantheverticalscale,renderingverticaladvectionnegligibleattheleadingorder. However, if the horizontal ¯eld gradient becomes very large due to nonlinear steepening, verticaladvectioncannotremainnegligible,asdemonstratedbythenumericalsimulation presented in this study. The non-hydrostatic, oscillatory waves are usually subgrid scale in large lakes. Even a non-hydrostatic model is unable to capture the physics unless otherwise using su±ciently ¯ne ¯nite di®erence mesh. To improve the ¯delity of the hydrodynamic models, there will be a need to model the dissipation rate of such subgrid scale waves. Our study has focused on the evolution of predominant modes in a context of weak nonlinearity with weak topography perturbation in a circular basin. This simplicity is probably why the energy exchange between these dominant modes appeared very weak. Sloping shelf and irregular perimeter are expected to greatly enhance such modal energy transfer through bathymetric gyres and strong nonlinearity. It should be added that the higher vertical modes are also expected to play an important role in nonlinear energy transfer. These are important issues deserving future study. 116 Chapter 5 Numerical method 5.1 Introduction Di®erential equations in physics are often described in a polar coordinate system. Such equations comprise with coe±cients varying as 1=r or 1=r 2 , and these terms are clearly singular at r = 0. When the equations, especially the nonlinear ones, are to be solved numerically,carefulconsiderationmustbegiventothesesingularitiessothatthenumer- ics can yield physically feasible solutions as r!0. The spectral methods, provided that the solution is expanded by appropriate polynomials, can yield smooth and bounded solutions, allowing one to simulate the physics of the problem without anxieties about having unphysical solutions near r = 0 at any time. For comprehensive discussions of spectral methods for problems with coordinate singularities, we refer to Chapter 18 in Boyd (2001). Physically well-behaved solutions in polar coordinates must satisfy a regularity con- dition that is solely a consequence of the coordinate transformation, independent of governing equations. In particular, an arbitrary scalar function Á de¯ned in the polar coordinates (r;µ) is expanded in a Fourier series of the form Á(r;µ)= 1 X m=¡1 Á m (r)e imµ : (5.1) If Á(r;µ) is analytic (C 1 ) at r =0, then in some vicinity of r =0, Á m (r) is expanded in a Taylor series of the form Á m (r)= 1 X k=0 Á mn r jmj+2k ; (5.2) 117 where n is a cumulated radial modal index de¯ned n = jmj + 2k. This regularity condition is proved by writing the expansion (5.1) in terms of Cartesian coordinates (x;y) with Á m (r) expanded by arbitrary powers of r, and requiring that all the terms are polynomial in x and y (e.g., see Boyd 2001; Eisen et al. 1991). The best example of Á m (r) is the Bessel function of the ¯rst kind J m (¸r), the radial eigenfunction of a Laplacian operator r 2 Á = ¡¸ 2 Á. The J m (¸r), having m-th order zeros at r = 0 with the same parity as m (even or odd function about r = 0 depending on m being even or odd), is expanded exactly in the form of (5.2). Although such a Bessel function seems to be suitable for the expansion function, since the de¯ning di®erential equation (Bessel's equation)isnonsingularattheouterboundary, therateofconvergencecanbedegraded, and the Gibb's phenomenon possibly emerges near the boundary (pp.29-35 in Gottlieb & Orszag 1977). For the radial expansion function in a bounded domain, therefore, it is ideal that the function satis¯es not only the regularity condition at r = 0, but also the de¯ning di®erential equation is singular at the outer boundary (Matsushima & Marcus 1995). The one-sided Jacobi polynomial, proposed independently by Matsushima & Marcus (1995) and Verkley (1997a), satis¯es these requirements. In a scaled domain 0·r· 1, the one-sided Jacobi polynomial Q mn (r) is de¯ned Q mn (r)=r jmj P (®;¯) k (2r 2 ¡1); (5.3) where P (®;¯) k (x) is the Jacobi polynomial (see section 22 in Abramowitz & Stegun 1972) and ® and ¯ (both > ¡1) are arbitrary parameters. Since P (®;¯) k (2r 2 ¡1) is an even polynomial of degree 2k, Q mn (r) is a polynomial containing powers r jmj , r jmj+2 , ¢¢¢, r jmj+2k . This readily implies that Á m (r) admits a direct expansion in terms of Q mn (r). 118 Furthermore, noting the fact that P (®;¯) k (x) is an orthogonal polynomial in ¡1· x· 1 with respect to the weight (1¡x) ® (1+x) ¯ , the orthogonality relation is written as Z 1 0 Q mn (r)Q mn 0(r)w(r)dr = ± nn 0 2(2k+®+¯+1) ¡(k+®+1)¡(k+¯+1) k!¡(k+®+¯+1) ; (5.4) and the weight function w(r) is w(r)=(1¡r 2 ) ® r 2(¯¡jmj)+1 : (5.5) Since the Jacobian for polar coordinates is r, the obvious choice is ® = 0 and ¯ =jmj, theparametersetusedbyMatsushima&MarcusandVerkley. RecentlyLivermore et al. (2007) proposed the polynomial with the new set ® =¡1=2 and ¯ =jmj¡1=2, the so called Worland polynomial in their paper. This particular set of parameters gives the weightfunction1= p 1¡r 2 ,thesameasthatforChebyshevpolynomials. Theirnumerical experiments showed that both the original one-sided Jacobi and the Worland polyno- mials behaved similarly, but when approximating a boundary layer function, Worland polynomials achieved a smaller local error close to the boundary. Theone-sidedJacobipolynomialshavebeenproventoworksuccessfullyforproblems involving scalar functions. Matsushima & Marcus and Livermore et al. tested the one-sided Jacobi polynomials for Bessel's eigenvalue problem, and it was shown that a much faster convergence rate was achieved than when using Chebyshev polynomials for the expansion function. Verkley (1997b) successfully applied the one-sided Jacobi polynomialsforsolvinganincompressible°uiddynamicequationinadisc. Forproblems involving vector functions, however, only a few applications of the one-sided Jacobi polynomials exist. Since the regularity condition of the vector function is di®erent from the scalar counterpart, some modi¯cation of the one-sided Jacobi basis is necessary. Leonard&Wray(1982)usedmodi¯edone-sidedJacobipolynomials,multiplyingafactor (1¡r 2 ) to satisfy the boundary condition at r = 1, and constructed a divergence-free velocity ¯eld for viscous °ow in a pipe. Ishioka (2003a & 2003b) modi¯ed the one-sided 119 Jacobi polynomials in the same manner, and constructed Galerkin-type vector basis functions completely satisfying the regularity and boundary conditions for solving the shallow water equation in a disc. As shown in the next section, the expansion coe±cients of radial (r) and azimuthal (µ) components of an analytic vector function are coupled to each other. For this rea- son, the expansion functions for radial and azimuthal vector components are designed necessarilyinacoupledformaswell. Addingrequirementstosatisfythephysicalbound- ary conditions for the vector components, the resulting basis functions become compli- cated and, consequently, the Galerkin-type formulation and subsequent implementation becomeinevitablycomplicated. Inthispaperweproposeatau-methodthatallowsoneto keep using the original form of the one-sided Jacobi polynomial (5.3) for the radial basis functions. The regularity and physical boundary conditions are satis¯ed by adjusting the extra spectral (tau) coe±cients. Through this approach, the coupling of the vector basis function is avoided, and the drawback of using Galerkin-type basis functions can be alleviated. Applications of the proposed method to hydrodynamic model equations are described inx5.3 andx5.4. 5.2 Spectral representation of vector function Sincevectorsarenotinvariantunderthetransformationbetweentwocoordinatesystems, the regularity condition of the vector functions is essentially di®erent from that of the scaler functions. This can be derived in many ways (e.g., see Lewis & Bellan 1990; Ishioka 2003a; Orszag 1974). In what follows we summarize the results. An arbitrary vector function u´ue r +ve µ de¯ned in the polar coordinates (r;µ) is expanded in a Fourier series of the form u(r;µ)= 1 X m=¡1 u m (r)e imµ : (5.6) 120 If u(r;µ) is analytic at r = 0, then in some vicinity of r = 0, u m (r) is expanded in a Taylor series of the form u m (r)= 1 X k=0 n¸1 u mn¡1 r jmj+2k¡1 ; (5.7) where n=jmj+2k, and the leading coe±cient u mjmj¡1 satis¯es the relation u mjmj¡1 +isgn(m)v mjmj¡1 =0 forjmj¸1: (5.8) The radial expansion function u m (r) has a base factor r jmj¡1 , the opposite parity to the scalar counterpart (5.2). The relation (5.8) is a coupled constraint hidden in the analytic vector components, a so called the kinematic constraint (Orszag 1974). Only satisfying the parity condition (5.7) does not guarantee the regularity of the vector function. Satisfyingthekinematicconstraintprovidesanimportantphysicalimplication. For an instructive purpose, suppose that we compute the divergence of the vector ¯eld at r =0, r¢u= @u @r + u r + 1 r @v @µ : (5.9) Substituting the expansion (5.6) and (5.7) into above expression, we have r¢u= 1 X m=¡1 1 X k=0 n¸1 f(jmj+2k)u mn¡1 +imv mn¡1 gr jmj+2k¡2 e imµ : (5.10) Thelowestradialpowerisr ¡1 whenm=§1andk =0,anditscoe±cientisu §10 §iv §10 . These values are identically zeros provided that the kinematic constraint (5.8) holds. Satisfying the kinematic constraint at m =§1 guarantees that r¢u is ¯nite as r! 0. But this is not su±cient to guarantee the regularity of r¢u at r = 0. Noting the fact that the divergence is a scalar function, the radial components of r¢u must be expanded in the form (5.2). The above expansion (5.10), therefore, must have an m-th order zero. Since the lowest order term of (5.10) is r jmj¡2 when k = 0, its coe±cient 121 jmj(u mjmj¡1 + isgn(m)v mjmj¡1 ) must vanish. This is accomplished by satisfying the kinematic constraint (5.8). Field equations usually contain dependent variables of both scalar and vector forms. Since the powerseries in vector functions is one degree less than that in scalar functions, theradialexpansionfunctionsbecomedi®erentfromthescalarcounterparts. Inorderto achieve minimal degeneration of the radial expansion functions for the spectral method beingpresentedhereafter,weintroduceanewvectorfunction ~ u,aradial°uxofude¯ned as ~ u(r;µ)´ru= 1 X m=¡1 ~ u m (r)e imµ : (5.11) The corresponding radial expansion function ~ u m (r) is of the same parity as that of the scalar functions (5.7) ~ u m (r)= 1 X k=0 n¸1 ~ u mn r jmj+2k ; (5.12) where the radialindex of ~ u mn isincreased byone. Althoughthe scalar functions contain a constant term if m=0, ~ u m (r) does not contain such a constant term as a consequence of multiplication of r. The kinematic constraint is essentially the same as (5.8) with the radial indices increased by one, namely ~ u mjmj +isgn(m)~ v mjmj =0 forjmj¸1: (5.13) The expansion functions in the tau-method do not necessarily satisfy the boundary conditions or even the kinematic constraint. Therefore, we require the radial expansion functions to satisfy only the parity condition (5.12). Using the one-sided Jacobi polyno- mial Q mn (r) de¯ned by (5.3), ~ u can be approximated by a ¯nite series in a triangular truncation form ~ u N (r;µ)= N X m=¡N N+¿ X n=jmj n¸1 ~ u mn ½ Q mn (r)¡± m0 (¡1) k (¯+1) k k! Q 00 (r) ¾ e imµ ; (5.14) 122 where ± m0 is a Kronecker's delta, and (¢¢¢) k is a Pochhammer symbol de¯ned as (z) k = ¡(z +k)=¡(z). Note that the spectral coe±cient ~ u mn is di®erent from that de¯ned in (5.12). If m 6= 0, the radial expansion function is Q mn (r), identical with that for the scalar functions. For m = 0, since the parity condition (5.12) does not allow constant terms, a constant contribution (i.e., Q 0n (0) given by (17) in Appendix B) is subtracted fromtheoriginalpolynomial(notethefactorQ 00 (r)=1). Usingtheorthogonality(5.4), the spectral coe±cient ~ u mn is determined through an integral transformation ~ u mn = (2k+®+¯)k!¡(k+®+¯+1) ¼¡(k+®+1)¡(k+¯+1) Z 2¼ 0 Z 1 0 ~ u N (r;µ)Q mn (r)w(r)e ¡imµ drdµ; (5.15) where n = jmj+2k (=jmj, jmj+2, ¢¢¢, N). This transformation formula is identical with the scalar function counterpart. The radial expansion in (5.14) has extra ¿-coe±cients indexed n = N +2, N +4, ¢¢¢, N+¿. By adjusting these ¿-coe±cients, the kinematic constraint and the boundary conditions are satis¯ed. Although the kinematic constraint is imposed only on the coef- ¯cients of r jmj terms in ~ u m (r), in the polynomial expansion (5.14) all the polynomials Q mn (r) contain r jmj . Using the expansion coe±cient of r jmj in Q mn (r) obtained from the series expansion (7) given in Appendix B, the corresponding kinematic constraint is expressed in a coupled form over the radial indices N+¿ X n=jmj (¡1) k (¯+1) k k! f~ u mn +isign(m)~ v mn g=0 forjmj¸1: (5.16) The physical boundary conditions can be expressed similarly in terms of the spectral coe±cients. Forexample,supposethatwewanttoimposeaDirichletboundarycondition u = 0 at r = 1 (i.e., ~ u = 0 at r = 1). Employing a property of the one-sided Jacobi polynomial (15) given in Appendix B, we have that N+¿ X n=jmj n¸1 1 k! n (®+1) k ¡± m0 (¡1) k (¯+1) k o ~ u mn =0 for all m: (5.17) 123 The radial truncation limit (N +¿) depends on the number of physical boundary conditions. For example, if the boundary condition is imposed on either u or v, the radial truncation limit should be chosen at N +2, that is, the boundary condition is satis¯edbyadjusting ~ u mN+2 (or ~ v mN+2 ),andthekinematicconstraint(5.16)issatis¯ed by adjusting ~ v mN+2 (or ~ u mN+2 ). If the boundary condition is imposed on both u and v, the radial truncation limit should be chosen at N +4. With the expansion (5.14), the physical value u N , computed by dividing ~ u N by r, is in turn numerically singular at r = 0. Practically, however, this is not a problem. The vector u at r = 0 is multi-valued (in µ-direction) due to the nature of the polar coordinates,andsuchvaluesarenotusuallyneededinthespectralmethods(e.g.,Gauss- Radau or Gauss-Legendre type radial collocation grids exclude the point r = 0). It is possible to use the crude vector form u that can be expanded by using the one-sided Jacobi polynomials (e.g.,u m (r)»r jmj¡1 P (®;¯) k (2r 2 ¡1)form6=0 with use of the weight w(r) = r 3 for the inner product). However, employing the radial °ux form ~ u is rather moreadvantageouscomputationally. Asmentionedabove,theradialexpansionfunctions of ~ u given by (5.14) are the same as those of the scalar functions except only m = 0 and, also noting the fact that their radial derivatives are identical, one can simplify the computer program and save the computer memory by de¯ning common basis matrices. 5.3 Linear Evolution Model We¯rstapplythetau-methodtoalinear,hyperbolic-typeevolutionmodelthathasbeen frequently used to study the hydrodynamics of large lakes. The equation set is given by @U @t ¡BV +c 2 @Z @r =0; @V @t +BU +c 2 1 r @Z @µ =0; @Z @t + @U @r + U r + 1 r @V @µ =0: 9 > > > > > = > > > > > ; (5.18) 124 Here we denote the dependent variables: the amplitudes of the radial velocity U, the azimuthal velocity V, and the isopycnal surface Z. (U;V) T is a vector, and Z is a scalar ¯eld. c is a linear wave phase speed, and B is called the Burger number, a parameter de¯ning the e®ect of Coriolis acceleration. We solve the model equation in a unit disc under the slip-free boundary condition (U =0 at r =1). 5.3.1 Spectral formulation Before presenting results we describe the spectral formulation of the problem based on the tau-method. Introducing the radial °ux of the vector ¯eld de¯ned by the relation ( ~ U; ~ V) T =r(U;V) T , the equation (5.18) is written in the form: @ ~ U @t =R (u) ; @ ~ V @t =R (v) ; @Z @t =R (z) : 9 > > > > > > = > > > > > > ; (5.19) Theapproximatedsolutionsetf ~ U N ; ~ V N ;Z N gissoughtinatruncatedseriesoftheform: Z N = N X m=¡N N X n=jmj z mn X mn (r;µ); 0 @ ~ U N ~ V N 1 A = N X m=¡N N+2 X n=jmj n¸1 0 @ u mn v mn 1 A Y mn (r;µ); 9 > > > > > > > = > > > > > > > ; (5.20) where the expansion functions X mn and Y mn are de¯ned as X mn (r;µ)=Q mn (r)e imµ ; Y mn (r;µ)=fQ mn (r)¡± m0 (¡1) k ge imµ : 9 > = > ; (5.21) 125 Q mn (r) is a one-sided Jacobi polynomial with ®=0 and ¯ =jmj written explicitly here as Q mn (r)=r jmj P (0;jmj) k (2r 2 ¡1); (5.22) wheren=jmj+2kwithk =0;1;2;¢¢¢. N representsatruncationlimitfortheazimuthal mode. Truncation limit of the radial mode for Z is also N. The radial truncation limit for the vector ¯eld is chosen to be N +2 in order to satisfy the boundary condition and the kinematic constraint with the spectral coe±cients of the highest order. We de¯ne an inner product in the form hX;Yi´ 1 2¼ Z 2¼ 0 Z 1 0 XY ¤ rdrdµ; (5.23) where the upper script (*) denotes the complex conjugate. Orthogonality identities of X mn and Y mn are given by hX mn ;X m 0 n 0i= ± mn 0± nn 0 2(n+1) ; hY mn ;X m 0 n 0i=hX mn ;X m 0 n 0i if n6=0: 9 > > = > > ; (5.24) Substituting the expansion (5.20) into the equation (5.19), and taking the inner prod- uct with respect to X ¤ mn (r;µ), we obtain, after using the orthogonality (5.24), a set of ordinary di®erential equations (ODE) for u mn , v mn and z mn . For given m, we have equations to determine du mn =dt and dv mn =dt as following du mn dt =2(n+1)hR (u)N ;X mn i; dv mn dt =2(n+1)hR (v)N ;X mn i; 9 > = > ; (5.25) for n=jmj;jmj+2;¢¢¢N. In this expression, R (¢¢¢)N denotes an approximation of R (¢¢¢) calculated using the approximated solution (5.20). To describe how to satisfy both the physicalboundaryconditionandthekinematicconstraint,wesupposem6=0hereunless as otherwise noted. 126 Thephysicalboundarycondition( ~ U =0atr =1)issatis¯edbyadjustingthehighest coe±cient u mN+2 . Di®erentiating the expression of the boundary condition (5.17) with respect to t, we have N+2 X n=jmj n¸1 f1¡± m0 (¡1) k g du mn dt =0: (5.26) Since du mn =dt are explicitly determined from the ¯rst equation in (5.25) for n · N, du mN+2 =dt is explicitly determined from the boundary condition (5.26). Next, thekinematicconstraintissatis¯edbyadjustingthehighestcoe±cient v mN+2 . Di®erentiating the kinematic constraint (5.16) with respect to t, we have N+2 X n=jmj (¡1) k (jmj+1) k k! ½ du mn dt +isgn(m) dv mn dt ¾ =0 for m6=0: (5.27) Since values of du mn =dt are now all known, and values of dv mn =dt are determined by the second equation in (5.25) for n · N, dv mN+2 =dt is explicitly determined from the kinematic constraint (5.27). Form=0,wehavethesameequationset(5.25)for ~ U and ~ V. Similarlytheboundary condition is satis¯ed by the highest coe±cient in (5.26). The values of dv 0n =dt are explicitlydeterminedfrom(5.25)forallnand,therefore,theimpositionofthekinematic constraint is unnecessary. The values of dz mn =dt are determined explicitly for all m and n by the equation similar to the vector ¯eld dz mn dt =2(n+1)hR (z)N ;X mn i: (5.28) Thevaluesofdu mn =dt,dv mn =dtanddz mn =dtarenowalldetermined,andthesevalues can be integrated forward-in-time through an appropriate ODE integration scheme. In this study we used the ¯xed time step, fourth-order Runge-Kutta method. Since the present numerical method was originally developed for solving nonlinear equations being presented in the following section, the inner products in (5.25) and 127 (5.28) were numerically evaluated. Numerical evaluation of the inner products involving nonlinear terms is described in x5.4. At each time step, after integrating u mn , v mn and z mn , the solution in the physical space is computed by (5.20) through the basis matrix multiplication. The radial derivatives of the solution are computed by (5.20) with the basis replaced by its derivative that is computed by the formula given in Appendix B. The values R (u)N , R (v)N and R (z)N are then obtained at the physical collocation grid, and then their inner products are readily computed. The analytical solution to the evolution system (5.18) is described in pp.317-324 in Lamb (1932) and in Csanady (1967). Writing the solution in the form of the radial velocity °ux ¯eld (i.e., ( ~ U; ~ V) T =r(U;V) T ), ~ U =¡ A 0 c 2 B(! 2 ¡1) ~ U m (r)sin(mµ¡B!t); ~ V = A 0 c 2 B(! 2 ¡1) ~ V m (r)cos(mµ¡B!t); Z = A 0 Z m (r)cos(mµ¡B!t); 9 > > > > > > = > > > > > > ; (5.29) whereA 0 isawaveamplitude,m(>0)isanazimuthalwave-number,andradialfunctions ~ U m (r) and ~ V m (r) are written ~ U m (r)=mZ m (r)¡!rZ 0 m (r); ~ V m (r)=m!Z m (r)¡rZ 0 m (r): 9 > = > ; (5.30) Z m (r) is a radial eigenfunction normalized by its maximum value; that is, Z m (r) = Z ¤ m (r)=jZ ¤ m (r)j max , and Z ¤ m (r) is either the Bessel(J) or the modi¯ed-Bessel(I) function depending on the value of ! Z ¤ m (r)= 8 > > < > > : I m (¸r) if ! 2 <1; J m (¸r) if ! 2 >1; and ¸= B c p j1¡! 2 j: 9 > > > > > > = > > > > > > ; (5.31) 128 The solution corresponding to ! 2 < 1 is characterized as the Kelvin wave, and that corresponding to ! 2 > 1 is characterized as the Poincar¶ e wave. Radial wave-number is included in ¸ implicitly through !, which that is the eigenfrequency determined by the dispersion relation. For the Poincar¶ e wave mode, the eigenfrequency is given by the transcendental relation ¸J m¡1 (¸)¡m µ 1+ 1 ! ¶ J m (¸)=0: (5.32) This equation is readily solved by the half interval method. For the Kelvin wave mode (! 2 <1), the J-Bessel function is replaced with the I-Bessel function in (5.32). We distinguish the wave traveling direction by calling `cyclonic' (counter clockwise rotation) for positive frequency (! > 0) and `anti-cyclonic' for negative frequency (! < 0). Forconvention,welabelthefundamentalwavemodethroughaformat`M(azimuthal mode)R(radialmode)'withthesignplus(+)forcyclonicornegative(-)foranti-cyclonic wave mode. SincetheI-BesselfunctionpossessesthesameparityasthatoftheJ-Besselfunction, the radial eigenfunction Z m (r) fully satis¯es the regularity condition (5.2). Also, using the series expansions for Bessel functions and complex exponentials for trigonometric functions, it can be easily veri¯ed that the coe±cient of the leading term r m¡1 of the exact solution (U;V) T fully satis¯es both the parity condition (5.7) and the kinematic constraint (5.8). 129 The exact solution can be expanded by Q mn (r) analytically (see Appendix C). In particular, the radial functions of unscaled form f ~ U ¤ m ; ~ V ¤ m ;Z ¤ m g for the Poincar¶ e wave mode are expressed as ~ U ¤ m (r)= 2 ¸ 1 X k=0 (¡1) k (n+1)f[(1+!)m+2!(k+1)]J n+1 (¸) ¡¸!J n (¸)gQ mn (r); ~ V ¤ m (r)= 2 ¸ 1 X k=0 (¡1) k (n+1)f[(1+!)m+2(k+1)]J n+1 (¸) ¡¸J n (¸)gQ mn (r); ~ Z ¤ m (r)= 2 ¸ 1 X k=0 (¡1) k (n+1)J n+1 (¸)Q mn (r); 9 > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > ; (5.33) where n = m + 2k. The Kelvin wave counterpart has a similar form with J-Bessel function replaced with I-Bessel function and omitting the factor (¡1) k in the above expression. An asymptotic expression of J-Bessel function given by 9.3.1 in Abramowitz & Stegun is J n (x)» 1 p 2¼n ³ ex 2n ´ n for n!1 with x ¯xed: (5.34) Thisexpressionimpliesthattheexpansioncoe±cientsoftheradialfunctions(5.33)have the property of in¯nite order, providing the spectral convergence. This is also true for the Kelvin wave solutions because the identity I n (x) = (¡i) n J n (ix) yields the same asymptotic relation as (5.34). 5.3.2 Simulation results Thespectralmethoddescribedabovewasimplementedinacomputerprogramwrittenin FORTRAN90,andnumericalcomputationwasperformedindoubleprecisionarithmetic. Truncated form of the exact solution (5.33) was used as the initial value for the spectral evolution equations (5.25) and (5.28). Parameters chosen for the example runs are at B = 4 and c = 0:9394668213. An exact solution of the isopycnal amplitude Z for the 130 (a) (b) (c) 0 0 0 0 + - + - Figure 5.1: Exact solution of the isopycnal amplitude Z for M1R1- wave mode at t=0 (a), numericalsolutionatt=13:75(b), andcorrespondingvector¯eld(c)obtainedwith the spectral truncation N =5. Contour level step is 0.2. anti-cyclonic Poincar¶ e wave of azimuthal mode one and radial mode one (M1R1-) is presented in Figure 5.1a. Solutions presented in the ¯gure are all normalized by the (initial) wave amplitude A 0 . This initial solution ¯eld starts to rotate anti-cyclonically around r = 0 at a constant frequency without changing its spatial structure. In the same ¯gure, numerical solutions of both the isopycnal and the velocity amplitude ¯elds at t = 13:75 are presented. At this time the ¯eld rotated around the center ten times. The spectral truncation was chosen at N = 5 with the time step ¢t = 0:005, although much larger time step is permissible for stable time integration. The solution ¯eld at t = 13:75 is very close to the exact solution qualitatively. At this truncation only two one-sided Jacobi modes are contained in the spectral approximation of Z at m = 1. In Figure 5.2 a similar comparison is presented for the anti-cyclonic Poincar¶ e wave of azimuthalmodethreeandradialmodethree (M3R3-). The solution¯eldis comparedat t = 5:475 corresponding to approximately ten local oscillation periods (10£2¼=B!) or three and one-third wave rotations. The spectral truncation in this example was chosen at N =15 with ¢t=0:005. Although the spatial structure of the solution ¯eld is more complicated than the previous example, at this truncation only six one-sided Jacobi modesarepresentinthespectralapproximationof Z atm=3. Again, thesolution¯eld at t=5:475 is already very close to the exact solution qualitatively. 131 (a) (b) (c) 0 0 0 0 0 0 0 0 0 0 0 0 + - + - + - + - + - + - + - + - + - + - + + - + - + - + - + - + - + - - Figure 5.2: Exact solution of the isopycnal amplitude Z for M3R3- wave mode at t=0 (a), numericalsolutionatt=5:475(b), andcorrespondingvector¯eld(c)obtainedwith the spectral truncation N =15. Contour level step is 0.2. N r =0:35 r =0:7 r =1 10 7:24£10 ¡1 (7:24£10 ¡1 ) 5:49£10 ¡1 7:70£10 ¡2 15 9:48£10 ¡3 (9:72£10 ¡3 ) 2:19£10 ¡4 4:37£10 ¡3 20 6:44£10 ¡5 (6:65£10 ¡5 ) 2:45£10 ¡5 3:56£10 ¡5 25 3:96£10 ¡7 (3:46£10 ¡9 ) 2:79£10 ¡7 2:19£10 ¡7 30 4:11£10 ¡7 (1:73£10 ¡8 ) 2:78£10 ¡7 2:29£10 ¡7 Table5.1: InstantaneouserrorofisopycnalamplitudeZ att=5:475sampledatdi®erent radial locations for various truncation limit N for time integration with M3R3- wave mode. Values are normalized by the initial wave amplitude. Values in parentheses in r =0:35 column are obtained with a halved time step ¢t=0:0025. In order to quantify the fast convergence of the numerical solutions, local error of the isopycnal amplitude was sampled at di®erent radial locations on the horizontal axis (µ =0)ata¯xedtimeforvarioustruncationlimits. TheresultsfortheM3R3-wavemode at t = 5:475 are shown in Table 5.1. The sampling locations were chosen at r = 0:35, 0:7 and 1, and these locations are very close to either the local maxima or minima of the isopycnal amplitude ¯eld (see Figure 5.2). The error values decrease uniformly and exponentially fast up to the N = 25 (N25) truncation limit, and they converge to some values of certain order regardless of increasing the truncation limit. For large truncation 132 limitthegrossnumericalerrorcanbecomedominatedbythetimediscretizationerror. In fact,whenhalvingthetimestepsizeto¢t=0:0025,thenumericalerrorsatN25andN30 decrease as supplemented in Table 5.1. Reducing the time step size for small truncation limitdoesnothelpreducetheerroratall,implyingthatthespatialdescretizationerroris ratherdominantinthegrosserror. Hence,thenumericalerrorcandecreaseexponentially fastasafunctionofthetruncationlimit, providedthetimestepsizeissu±cientlysmall. For the next validation we examine the global error of the numerical solutions by calculating conserved quantities of the system (5.18). The spatial integrals of the isopy- cnal amplitude (M), the relative vorticity (W) and the total energy (E) are conserved quantities de¯ned as following M = Z 2¼ 0 Z 1 0 Zrdrdµ; W = Z 2¼ 0 Z 1 0 ½ @V @r + V r ¡ 1 r @U @µ ¾ rdrdµ; E = Z 2¼ 0 Z 1 0 1 2 © U 2 +V 2 +c 2 Z 2 ª rdrdµ: 9 > > > > > > > = > > > > > > > ; (5.35) The integral isopycnal amplitude M is always conserved in the method presented here. The isopycnal amplitude Z is time integral of the divergence of the velocity ¯eld (see (5.18)). According to the Gauss divergence theorem, and recalling the normal velocity vanishing boundary condition (U = 0 at r = 1), the surface integral of the divergence ¯eld is identically zero (alt. M =0). The present numerical method exactly satis¯es the boundary condition through the spectral tau-equation (5.26), guaranteeing M =0 at all times. The integral vorticity W and the energy E, however, are not conserved in the presentmethod. AlthoughtheazimuthalexpansionwithFouriermodeshelpstheintegral vorticity be conserved except at the zero-th azimuthal mode (m = 0), the energy, as a quadraticquantity, isnotconservedatallazimuthalmodes. The evolutionequations for the tau-coe±cients (5.26) and (5.27) are di®erent from the spectral evolution equations (5.25) projected from the original system equation (5.18). Hence, the present method 133 N t=1 t=3 t=6 5 1.0002782417 0.9995259713 1.0003938509 10 1.0000001149 1.0000000043 1.0000000093 15 0.9999999998 0.9999999988 0.9999999976 20 0.9999999998 0.9999999988 0.9999999976 Table 5.2: Energy E computed at di®erent time for various truncation limits N for time integration with M1R1- wave mode. Values are normalized by their initial values at t=0. N t=1 t=2 t=3 10 2.4267690168 3.0829407465 2.1702447582 15 0.9945572323 0.9945807190 0.9974414656 20 0.9999898056 0.9999872125 1.0000006896 25 0.9999999502 0.9999998006 0.9999997031 30 0.9999999506 0.9999998023 0.9999997034 Table 5.3: Energy E computed at di®erent time for various truncation limits N for time integration with M3R13- wave mode. Values are normalized by their initial values at t=0. seeks for the solutions to the modi¯ed equations, a nature of the tau-method (Gottlieb & Orszag 1977; Boyd 2001). Although the present method renders the numerical scheme non-conservative, the deviationoftheconservedquantitiesdiminishasfastasthenumericalsolutionconverges. To demonstrate this statement, the integral energy E was calculated at di®erent times forvarioustruncationlimitsforthetimeintegrationwithM1R1-(Table5.2)andM3R3- (Table 5.3) wave modes. The integration time step was ¯xed at ¢t = 0:005. All the energy values are normalized by the initial value at t=0. It is evident from these tables that the energies uniformly converge exponentially fast up to values of certain order. The convergence rate of the M1R1- wave mode is much faster than that of the more complicated M3R3- wave mode. Increasing the truncation limit does not continually diminish the energy deviations because the time descretization error dominates in the 134 ¢t N =5 N =10 N =20 0.1 0.9937544325 - - 0.05 0.9997005005 0.9998037882 - 0.01 0.9999750209 0.9999999218 0.9999999368052 0.005 0.9999752403 0.9999999763 0.9999999980248 0.001 0.9999752525 0.9999999863 0.9999999999994 Table 5.4: Energy E at t = 5 obtained at di®erent truncation limits N for various time step sizes ¢t for time integration with M1R1- wave mode. Values are normalized by their initial values at t = 0. No value (-) implies that the time integration is unstable due to excessive time step size. larger truncation limit as suggested above. A more detailed example of the e®ect of the time step size and the truncation limit is presented in Table 5.4, where the energy E of the M1R1- wave mode at t = 5 for the time integration in di®erent truncation limits are tabulated for several time step sizes. It is evident from the table that the spectral convergence to an exact value (unity) is achieved only if the time step size is su±ciently small (e.g., ¢t = 0:01 gives exponential convergence up to N10 truncation, but at reducing the time step to ¢t=0:001 increases the convergence limit to N20). 5.4 Nonlinear Evolution 5.4.1 Model formulation and algorithmic approach The next example we apply the method to is a weakly-nonlinear, weakly-dispersive evo- lution model that is an asymptotic derivation based on the nonlinear system whose lin- earized reduction yields the linear model presented in the previous section. The asymp- totic projection allows the elimination of the vertical structure, yielding a nonlinear 135 system for the evolution of the horizontal structure of a speci¯c vertical eigenmode. The equation set we study is given @U @t ¡BV +c 2 @Z @r =¡ ½ ® µ U @U @r + V r @U @µ ¡ V 2 r ¶ +¯DU ¾ +¹ 2 ° @ @t µ @D @r ¶ ; @V @t +BU + c 2 r @Z @µ =¡ ½ ® µ U @V @r + V r @V @µ + UV r ¶ +¯DV ¾ +¹ 2 ° @ @t µ 1 r @D @µ ¶ ; @Z @t + @U @r + U r + 1 r @V @µ =¡ ½ ¯ µ U @Z @r + V r @Z @µ ¶ +®DZ ¾ ; 9 > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > ; (5.36) where D is the horizontal divergence of the velocity ¯eld D = @U @r + U r + 1 r @V @µ ; (5.37) and®,¯,¹and° areconstantcoe±cients. ThelinearphasespeedcandtheBurgernum- ber B were kept at the same values as those in the previous section (c = 0:9394668213; B = 4), and the additional coe±cients were chosen as ® = ¡0:3690737980, ¯ = ¡0:1845368990, ° = 2:007821008 and ¹ = 1=40, which correspond to a speci¯c physical con¯guration. The solutions to the equation set are sought in a unit disc with the phys- ical boundary condition U =0 at r =1. Besides nonlinear terms, the velocity equations containdispersivetermsconsistingofthetimederivativeofthesecondspatialderivatives of velocity variables. Except handling of the dispersive terms, the numerical formulation is similar to that of the linear model discussed in the previous section. 136 Transforming the velocity ¯eld via the relation ( ~ U; ~ V) T = r(U;V) T , we write the equation in the form: @ ~ U @t =R (u) + @ @t fL (u) r ~ U +L (v) r ~ Vg; @ ~ V @t =R (v) + @ @t fL (u) µ ~ U +L (v) µ ~ Vg; @Z @t =R (z) ; 9 > > > > > > = > > > > > > ; (5.38) where L (¢¢¢) r and L (¢¢¢) µ are linear operators for the dispersive terms. Spectral evolution equations of the velocity ¯eld are now coupled in the radial modal space due to the dispersive terms. The velocity equations in the spectral space are written in the form: du mn dt ¡ d dt N+2 X n 0 =jmj n¸1 fa (u) mn;n 0 u mn 0 +a (v) mn;n 0 v mn 0g=2(n+1)hR (u)N ;X mn i; dv mn dt ¡ d dt N+2 X n 0 =jmj n¸1 fb (u) mn;n 0 u mn 0 +b (v) mn;n 0 v mn 0g=2(n+1)hR (v)N ;X mn i; 9 > > > > > > > > = > > > > > > > > ; (5.39) for n=jmj,jmj+2,¢¢¢, N. The constant a (¢¢¢) mn;n 0 is de¯ned by a (¢¢¢) mn;n 0 =2(n+1) Z 1 0 Q mn (r)L (¢¢¢) r;m fQ mn 0(r)¡± m0 (¡1) k grdr; (5.40) whereL (¢¢¢) r;m isalinearoperatorforazimuthalmodem. Anotherconstantb (¢¢¢) mn;n 0 isde¯ned in the same way with L (¢¢¢) r;m replaced with L (¢¢¢) µ;m . The integrals were evaluated exactly through the Gaussian quadrature. Combining the evolution equations (5.39), the boundary condition (5.26), and the kinematic constraint (5.27), one can construct a linear system for du mn =dt and dv mn =dt in the form: A m dv m dt =b m : (5.41) 137 Inthisequation,A m isasystemmatrix, thevectorv m consistsofallsetsof(u mn ;v mn ) T forgivenm,andthevectorb m consistsoftherighthandsideof(5.39),(5.26)and(5.27). The size of A m for arbitrary m(6= 0) is (2b(N ¡jmj)=2c+4)£(2b(N ¡jmj)=2c+4)), where b¢¢¢c denotes the °oor function (i.e., bxc gives the largest integer less than or equal to x). The inverse matrices A ¡1 m are computed for all m before starting the time integration, and they are multiplied to b m to determine dv m =dt at every time step. Innerproductsontherighthandsideoftheequations(5.39),nowinvolvingnonlinear terms, are numerically evaluated in accordance with the same philosophy described in Verkley (1997b). To be speci¯c to our problem, we write an inner product in the form hR (u)N (r;µ);X mn (r;µ)i= Z 1 0 R (u)N m Q mn (r)rdr: (5.42) Here R (u)N m (r) is an inner product of R (u)N with respect to e ¡imµ de¯ned as R (u)N m (r)= 1 2¼ Z 2¼ 0 R (u)N (r;µ)e ¡imµ dµ: (5.43) Since the maximum wave number of the integrand is 3N (i.e., 2N for R (u)N and N for e ¡imµ ), the integral (5.43) is calculated exactly through a discrete Fourier transform R (u)N m (r)= 1 K K X k=1 R (u)N (r;µ k )e ¡imµ k ; (5.44) where µ k =(2¼=K)(k¡1) and K¸3N+1. This transform can be computed e±ciently employingthefastFouriertransformwitha choiceof K atapowerof 2. Afterobtaining R (u)N m (r), the radial integral in (5.42) is evaluated through the Gauss-Legendre type quadrature. Since (5.42) is an integral with respect to the weight r, it is convenient to use zeros of a Jacobi polynomial of the form P (1;0) n (x) for the abscissas (Krylov 2006). The quadrature formula is written explicitly as Z 1 0 R (u)N m (r)Q mn (r)rdr = L X l=1 R (u)N m (r l )Q mn (r l )w l ; (5.45) 138 where r l is the l-th zero of P (1;0) L (1¡2r), and corresponding weight w l is given by w l = 1 4r 2 l [P (1;0) 0 L (1¡2r l )] 2 : (5.46) Since the maximum degree of the polynomial in the integrand is 3N +4 (i.e., 2N +2 for R (u)N m and N +2 for Q mn ), the quadrature formula (5.45) is exact, provided L ¸ (3N +5)=2. 5.4.2 Simulation results For numerical examples we consider the evolution of the initial value de¯ned by (5.30) in the previous section. Although the solution (5.30) now becomes only a leading order, asymptotic solution (at vanishing amplitude) to the present nonlinear model (5.36), it is still interesting to see whether the model is stably integrated for a long time yielding physically reasonable solutions through the proposed numerical method. In Figure 5.3 an evolution of the azimuthal mode one Kelvin wave (M1R1+) initial conditionofamplitudeA 0 =¡0:3isshownatseveraldi®erenttimes. Spectralresolution waschosenatN =70withtimestep¢t=0:0025. Withthisspatialresolutiontheradial collocation is at L = 108 points, and the azimuthal collocation is at K = 256 points. Only a solution of the isopycnal surface Z is shown in the ¯gure because it is the time integral of the divergence of the velocity ¯eld, containing important physical essence of the velocity ¯eld. Observing the ¯gure, the negative side of the isopycnal surface travels faster than the positive side, generating a front that gradually steepens as it travels in the cyclonic (counter clockwise) direction (see Figure 5.3b corresponding to t = 8 ¼ 1:4T, where T is a linear wave period). The front continually steepens as the wave ¯eld evolves and it starts to generate a train of oscillatory waves (see Figure 5.3c corresponding to t = 16 ¼ 2:7T). The leading front passes eventually through its own oscillatory tail, generating ripples (see Figure 5.3d corresponding to t = 24 ¼ 4:1T), and the ¯eld becomes more complicated. The steepening of the wave front is a familiar 139 (a) (b) (c) (d) + + 0 0 + + + + + + 0 0 0 0 0 0 0 0 0 0 + 0 0 0 0 0 Figure 5.3: Evolution of a Kelvin wave (M1R1+) initial condition of amplitude A 0 = ¡0:3 in the weakly-nonlinear and weakly-dispersive model. Snap shots of isopycnal surface Z are taken at (a) t=0, (b) t=8, (c) t=16 and (d) t=24. Contour level step is 0.05. e®ect of nonlinearity, and subsequent generation of oscillatory waves results from an approximate balance between the nonlinearity and the dispersive e®ect in the model. This solution behavior is very similar to that found in the well known Korteweg-de Vries equation, the fundamental model describing the weakly-nonlinear and the weakly- dispersivee®ect. Thetimeintegrationconsistsof28000timestepswasperformedstably. In Figure 5.4 the total energy density (the integrand of the energy expression E in (5.35)) was calculated as a function of azimuthal mode m for several di®erent times. Modalenergieswereallnormalizedbythetotalsystemenergy. Thespectralconvergence isreadilyobservedfromthe¯gure. Initiallytheenergyisconcentratedonlyinazimuthal mode m = 1. The initial energy is subsequently transferred to higher modes as the 140 0 10 20 30 40 50 60 70 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 m t=4 t=8 t=16 t=20 t=24 E m Figure 5.4: Azimuthal modal energy spectrum at various time for the evolution ¯eld of the Kelvin wave initial condition (M1R1+). nonlinear wave front steepens until the oscillatory waves are generated, and then the spectral convergence rate becomes almost saturated. For the next example, an evolution of the Poincar¶ e wave of azimuthal mode one and radial mode one (M1R1-) was examined and results are presented in Figure 5.5. The numerical run con¯guration is the same as that in the Kelvin wave example described above. The initial condition is symmetric, but the ¯eld quickly loses its symmetry as it rotatesintheanti-cyclonic(clockwise)direction(Figure5.5b). Theevolvingasymmetric ¯eldtendstoreturntoanearsymmetric¯eld(Figure5.5c),andthisrepeatsaperiodically for the rest of evolution. The numerical model was stably integrated, and it was run up to t = 24 which corresponds to about seventeen and half wave rotations around the center (Figure 5.5d). InFigure5.6themaximumandminimumvaluesoftheisopycnalsurfacefordi®erent initial wave amplitudes A 0 are presented as a function of time. Values are normalized by the initial wave amplitude. The modulation of the amplitudes becomes larger as the initial amplitude increases, implying that the modulation is a nonlinear phenomenon. 141 (a) (c) (d) (b) 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + Figure 5.5: Evolution of a Poincar¶ e wave (M1R1-) initial condition of amplitude A 0 = ¡0:3 in the weakly-nonlinear and weakly-dispersive model. Snap shots of isopycnal surface Z are taken at (a) t = 0, (b) t = 3, (c) t = 3:4 and (d) t = 24. Contour level step is 0.05. 0 1 2 3 4 5 6 7 8 t −1 −1.1 −1.2 −1.3 −1.4 −1.5 1 0.9 0.8 0.7 Z min /|A 0 |Z max /|A 0 | |A 0 |=0.2 |A 0 |=0.3 |A 0 |=0.4 Figure 5.6: local minima and maxima of the isopycnal surface Z for di®erent initial ampliutdes (Poincar¶ e M1R1- mode) as a function of time. 142 This pseudo recurrence of the Poincar¶ e wave ¯eld has not been recognized before, but is clearly revealed by simulation employing the present numerical method. 5.5 Conclusions In this paper we proposed a tau-method approach that can fully satisfy the regularity condition (kinematic constraint) and the boundary conditions of the solutions to vector ¯eld equations de¯ned in the polar coordinates. This approach avoids an explicit cou- pling of radial basis functions of the polar vector ¯eld variables, and it enables one to utilize the original, simple form of the one-sided Jacobi polynomials as the radial basis if the vector ¯eld variables are transformed into the radial °ux form. Also, the method provides °exibilities in specifying the boundary conditions without modi¯cation of the basis functions. Especially for the problem involving both vector and scalar functions, therefore, the implementation of the method is expected to be easier than that of the Galerkin method. The spectral convergence was demonstrated through the stable time integrationofevolutionequationscomprisingboththescalarandthevector¯elds. Since the tau-method seeks the solution to the modi¯ed equation, the conserved quantities of theoriginalequationmodelarenotgenerallyconserved. Butthevariationofthequantity can be diminished to some negligible level with su±cient spectral and time resolutions. 143 Chapter 6 Concluding summary In this thesis we investigated the nonlinear evolution, generation and degeneration of basin-scale internal waves generated by wind-stress forcing employing three types of weakly-nonlinear, weakly-dispersive evolution models. These models capture di®erent physical mechanisms and their role in transferring energy to di®erent modes of motion. In what follows we brie°y summarize the conclusions. A simple, two-layer, KdV type model was proposed as a rapid simulation tool to study nonlinear e®ects and energetics of the vertical mode-one, basin-scale waves over a wide range of environmental and wind forcing parameters. It was con¯rmed that the weak-nonlinearity and the weak-non-hydrostatics play a principal role in the initial degeneration of the basin-scale internal waves through the nonlinear steepening and subsequent generation of oscillatory waves. The generation and degeneration of the basin-scale waves depend signi¯cantly on the strati¯cation, the lake depth and the wind forcing scenarios. Energy transfer of the basin-scale waves among the lowest two, energetically domi- nant, vertical modes was studied employing a multi-modal evolution model. Energy of the basin-scale waves is transferred to other vertical modes predominantly through the nonlinear interaction. When an energetically dominant, vertical mode-one wave re°ects from the basin end wall, some fraction of the energy in the vertical mode-one is trans- ferred to the vertical mode-two, primarily through the vertical buoyancy °ux. This vertical energy transfer is a strong function of the strati¯cation. We also point out that the modal energy partition initiated by the wind stress forcing is also a strong function of the vertical pro¯le of the wind stress penetrating the mixing layer. 144 An evolution model of basin-scale waves under the e®ect of earth's rotation was derived for a circular basin of variable depth. A new numerical method was devised and successfully implemented to simulate the model. The Kelvin wave steepens and subsequently generates, a packet of oscillatory waves as it evolves, exhibiting a similar hyperbolic character as KdV type models. The Poincar¶ e wave, on the contrary, does not steepen. It was found that the amplitude is modulated, losing its symmetry and later returning to near symmetry as it evolves, exhibiting a pseudo recurrence character. These nonlinear processes, as well as the wind energy deposition to these wave modes, are strongly dependent on the Burger number. Presentstudiesexamineinparticulartheinitialstageofgenerationanddegeneration ofwind-forcedbasin-scaleinternalwaves,andcapturemainlythelowerendoftheinternal energy spectrum of a lake. Energy transfer to scales on the order of the lake depth and smaller is outside the domain of the present models. Nevertheless, the scales captured in the models studied here do allow us to bridge into subgrid scales of many full-domain simulations. The models were idealized to have a slowly varying depth con¯ned by the vertical side-boundaries. Sloping shelf and irregular bathymetry, which are common in most of lakes, are expected to enhance the nonlinear energy transfer across the whole energy spectrum and they comprise regions where a substantial fraction of internal wave energy is dissipated. Also, from a water quality point of view, it is important to study the material transport processes in lakes where the nonlinearity of the hydrodynamics is expected to play a crucial role. These are vital issues to be pursued in future studies. 145 Bibliography Abramowitz, M., & Stegun, I. A. 1972. Handbook of mathematical functions. Dover. Antenucci, J., & Imberger, J. 2001. Energetics of long intenral gravity waves in large lakes. Limnol. Oceanogr., 46(7), 1760{1773. Antenucci, J., Imberger, J., & Saggio, A. 2000. Seasonal evolution of the basin-scale internal wave ¯eld in a large strati¯ed lake. Limnol. Oceanogr., 45(7), 1621{1638. Beletsky, D., & O'Connor, W. P. 1997. Numerical simulation of internal Kelvin waves and coastal upwelling fronts. J. Phys. Oceanogr., 27, 1197{1215. Bennett, J. R. 1973. A theory of large-amplitude Kelvin waves. J. Phys. Oceanogr., 3, 57{60. Boegman, L., Ivey, G. N., & Imberger, J. 2004. An internal solitary wave parameteriza- tionforhydrodynamiclakemodels. Proceedingsof15thAustralasianFluidMechanics Conference. The University of Sydney, Sydny, Australia. Boegman, L., Ivey, G. N., & Imberger, J. 2005a. The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr., 50(5), 1620{1637. Boegman, L., Ivey, G. N., & Imberger, J. 2005b. The energetics of large-scale internal wave degeneration in lakes. J. Fluid Mech., 531, 159{180. Boehrer, B. 2000. Modal response of a deep strati¯ed lake: western Lake Constance. J. Goephys. Res., 105(C12), 28837{28845. Bogucki, D. J., Redekopp, L. G., & Barth, J. 2005. Internal solitary waves in the coastal mixing and optics 1999 experiment: multimodal structure and resuspension. J. Geophys. Res., 110(C2024), doi:10.1029/2003JC002253. Boyd, J.P.1998. Higherordermodelsforthenonlinearshallowwaterwaveequationson the equatorial bata-plane with application to Kelvin wave frontogenesis. Dyn. Atmos. Oceans, 28, 69{91. Boyd, J. P. 2001. Chebyshev and Fourier spectral methods. 2nd edn. Dover. Csanady, G. T. 1967. Large-scale motion in the Great Lakes. J. Geophys. Res.,72(16), 4151{4162. 146 Csanady, G. T. 1968. Wind-driven summer circulation in the Great Lakes. J. Geophys. Res., 73(8), 2579{2589. Csanady, G. T. 1972. Response of large strati¯ed lakes to wind. J. Phys. Oceanogr., 2(1), 3{13. Csanady, G. T. 1975. Hydrodynamics of large lakes. Ann. Rev. Fluid Mech.,7, 357{386. Drazin, P. G., & Johnson, R. S. 1989. Solitons: an introduction. Cambridge university press. Chap. 5, pages 90{95. Eisen, H., Heinrichs, W., & Witsch, K. 1991. Spectral collocation methods and polar coordinate singularities. J. Comput. Phys., 96, 241{257. Farmer, D. M. 1978. Observations of long nonlinear internal waves in lake. J. Phys. Oceanogr., 8, 63{73. Fedorov, A. V., & Melville, W. K. 1995. Propagation and breaking of nonlinear Kelvin waves. J. Phys. Oceanogr., 25, 2518{2531. Fedorov, A. V., & Melville, W. K. 2000. Kelvin fronts on the equatorial thermocline. J. Phys. Oceanogr., 30, 1692{1705. Fornberg, B., & Whitham, G. B. 1978. A numerical and theoretical study of certain nonliear wave phenomena. Phil. Trans. R. Soc. Lond., A289, 373{404. Garkema, T. 2001. Internal and interfacial tides: beam scattering and local generation of solitary waves. J. Mar. Res., 59, 227{255. Garkema,T.2003. Developmentofinternalsolitarywavesinvariousthermoclineregines - a multi-modal approach. Nonl. Proc. in Geophys., 10, 397{405. Gottlieb, D., & Orszag, S. 1977. Numerical analysis of spectral methods; theory and applications, CBMS-NSF regional conference series in applied mathemathics. SIAM, Philadelphia, PA. Grimshaw, R. 2002. Internal solitary waves. Kluwer Academic Publishers. (in Environ- mental strati¯ed °ows, ed. by Grimshaw, R.). Pages 1{27. Heaps, N. S. 1984. Vertical structure of current in homogeneous and strati¯ed waters. Springer Verlag Wien-New York. (in Lake hydrodynamics, ed. by Hutter, K.). Pages 153{207. Heaps, N. S., & Ramsbottom, A. E. 1966. Wind e®ects on the water in a narrow two- layered lake. Phil. Trans. R. Soc. Lond. A, 259, 391{430. Helfrich, K. R. 1992. Internal solitary wave breaking and run-up on a uniform slope. J. Fluid Mech., 243, 133{154. Hodges, B. R., Imberger, J., Saggio, A., & Winters, K. B. 2000. Modeling basin-scale internal waves in a strati¯ed lake. Limnol. Oceanogr, 45(7), 1603{1620. 147 Horn, D.A., Imberger, J., &Ivey, G.N.2001. Thedegenerationoflarge-scaleinterfacial gravity waves in lakes. J. Fluid Mech., 434, 181{207. Horn, D. A., Imberger, J., Ivey, G. N., & Redekopp, L. G. 2002. A weakly nonlinear model of long internal waves in closed basins. J. Fluid Mech., 467, 269{267. Hunkins, K., & Fliegel, M. 1973. Internal undular surges in Seneca lake: A natural occurrence of solitons. J. Geophys. Res., 78, 539{548. HÄ uttemann, H., & Hutter, K. 2001. Boroclinic solitary waves in a two-layer °uid system with di®usive interface. Experiments in °uids, 30, 317{326. Imberger, J. 1994. Transport processes in Lakes: A review. Elsevier Science B. V. (in Limnology Now: A Paradigm of Planetary Problems, ed. by Margalef, R.). Pages 99{193. Imberger, J. 1998. Flux paths in strati¯ed lake: A review. Physical processes in lakes and oceans. American Geophysical Union. (in Coastal and esturine studies, ed. by Imberger, J.). Pages 1{18. Ishioka, K. 2003a. Spectral model for shallow-water equation on a disk I. Basic formu- lation. J. Japan Soc. of Fluid Mech., 22, 345{358. Ishioka, K. 2003b. Spectral model for shallow-water equation on a disk II. Numerical examples. J. Japan Soc. of Fluid Mech., 22, 429{441. Keulegan, G. H. 1948. Gradual damping of solitary waves. J. Res. Natl Bur. Stand.,40, 487{498. Koop, C. G., & Butler, G. 1981. An investigation of internal solitary waves. J. Fluid Mech., 112, 225{251. Krylov, V. I. 2006. Approximated calculation of integrals. Dover. Lamb, Sir H. 1932. Hydrodynamics. 6th edn. Dover. LaZerte, B. D. 1980. The dominating higher order vertical modes of the internal seiche in a small lake. Limnol. Oceanogr., 25(5), 846{854. Lele,S.1992. Compact¯nitedi®erenceschemeswithspectral-likeresolution. J.Comput. Phys., 103, 16{42. Leonard, A., & Wray, A. 1982. A new numerical method for the simulation of three- dimensional °ow in a pipe. In: Krause, E. (ed), Proceedings, eighth international conference on numerical methods in °uid dynamics. Aachen, Germany: Springer- Verlag, Berlin. Lewis, H. R., & Bellan, P. M. 1990. Physical constraints on the coe±cients of Fourier expansions in cylindrical coordinates. J. Math. Phys., 31(11), 2592{2596. 148 Livermore, P. W., Jones, C. A., & Worland, S. J. 2007. Spectral radial basis functions for full sphere computations. J. Comput. Phys., 227, 1209{1224. Matsushima, T., & Marcus, P. S. 1995. A spectral method for polar coordinates. J. Comput. Phys., 120, 365{374. Maxworthy, T. 1983. Experiments on solitary intenral Kelvin waves. J. Fluid Mech., 129, 365{383. Melville, W. K., Tomasson, G. G., & Renouard, D. P. 1989. On the stability of Kelvin waves. J. Fluid Mech., 206, 1{23. Michallet, H., &Ivey, G.N.1999. Experimentsonmixingduetointernalsolitarywaves. J. Geophys. Res., 104(C6), 13467{13477. Miles, J. W. 1976. Damping of weakly nonlinear shallow-water waves. J. Fluid Mech., 76, 251{257. Monismith, S. 1986. An experimental study of the upwelling response of strati¯ed reser- voirs to surface shear stress. J. Fluid Mech., 171, 407{439. Monismith, S. 1987. Modal response of reservoirs to wind stress. J. Hydraul. Eng., 113(10), 1290{1306. Mortimer, C. H. 1952. Watermovementsin lakes during summer strati¯cation; evidence from the distribution of termperature in Windermere. Phil. Trans. Roy. Soc. London, B, 236, 355{404. Mortimer, C. H. 1963. Frontiers in physical limnology with particular reference to long waves in rotating basins. Proc. 5th Conf., no. 10. Great Lakes Res. Div., Univ. Michigan. Mortimer,C.H.1968. Internal waves and associated currents observed in Lake Michigan during the summer of 1963. Special Rep. 1. Ctr. For Great Lakes Studies, Univ. Wis. MÄ unnich, M., WÄ uest, A., & Imboden, D. M. 1992. Observations of the second vertical mode of the internal seiche in an alpine lake. Limnol. Oceanogr., 37(8), 1705{1719. Orszag, S. A. 1974. Fourier series on spheres. Mon. Weather Rev., 102, 56{75. Ostrovsky, I., Yacobi, Y. Z., Walline, P., & Kalikhman, I. 1996. Seiche-induced mixing: Its impact on lake productivity. Limnol. Oceanogr., 41(2), 323{332. Ostrovsky, L. A., & Grue, J. 2003. Evolution equations for strongly nonlinear internal waves. Phys. Fluids, 15, 2934{2948. Ostrovsky,L.A.,&Stepanyants,Y.A.2005. Internalsolitonsinlaboratoryexperiments: Comparison with theoretical models. Chaos, 15, 037111. 149 Roget, E., Salvad G., & Zamboni, F. 1997. Internal seiche climatology in a small lake where transversal and second vertical modes are usually observed. Limnol. Oceanogr., 42(4), 663{673. Rueda, J. R., Schladow, S. G., & P¶ almarsson, S. ¶ O. 2003. Basin-scale intenral wave dynamics during a winter cooling period in a large lake. J. Geophys. Res., 108(C3), 3907, doi:10.1029/2001JC000942. Saggio, A., & Imberger, J. 1998. Internal wave weather in a stra¯¯ed lake. Limnol. Oceanogr., 43, 1780{1795. Sakai, T., & Redekopp, L. G. 2008a. An application of one-sided Jacobi polynomial for spectral modeling of vector ¯eld in polar coordinates. To be submitted. Sakai, T., & Redekopp, L. G. 2008b. A parametric study of the generation and degener- ation of wind forced, long internal waves in narrow lakes. To be submitted. Sakai,T.,&Redekopp,L.G.2008c. Aweaklynonlinearevolutionmodelforlonginternal waves in a large lake. To be submitted. Sakai,T.,&Redekopp,L.G.2008d. A weakly nonlinear model for multi-modal evolution of wind-generated long internal waves in a closed basin. To be submitted. Slinn, D. N., & Riley, J. J. 1998. A model for the simulation of turbulent boundary layers in an incompressible strati¯ed °ow. J. Comput. Phys., 144, 550{602. Slunyaev, A. V., Pelinovsky, E. N., Poloukhina, O. E., & Gavrilyuk, S. L. 2003. The Gardner equation as the model for long internal waves. Pages 368{369 of: Topical problems of nonlinear wave physics. Proceedings of the international symposium. Stashchuk, N., Vlasenko, V., & Hutter, K. 2005. Numerical modelling of disintegration of basin-scale internal waves in a tank ¯lled with strati¯ed water. Nonl. Proc. in Geophys., 12, 955{964. Stevens, C., &Imberger, J.1996. Theinitialresponseofastrati¯edlaketosurfaceshear stress. J. Fluid Mech., 312, 39{66. Stocker, R., & Imberger, J. 2003. Energy partitioning and horizontal dispersion in a strati¯ed rotating lake. J. Phys. Oceanogr., 33, 512{529. Thorpe, S. A. 1974. Near-resonant forcing in a shallow two-layer °uid: a model for the internal surge in Loch Ness? J. Fluid Mech., 63, 509{527. Thorpe, S. A., & Hall, A. 1972. The internal surge in Loch Ness. Nature, 23, 96{98. Thorpe, S. A., Keen, J. M., Jiang, R., & Lemmin, U. 1996. High-frequency internal waves in Lake Geneva. Phil. Trans. R. Soc. Lond. A, 354, 237{257. Tomasson, G. G., & Melville, W. K. 1990. Nonlinear and dispersive e®ects in Kelvin waves. Phys. Fluids A, 2(2), 189{193. 150 Verkley, W.T.M.1997a. Aspectralmodelfortwo-dimensionalincompressible°uid°ow in a circular basin I. Mathematical foundation. J. Comput. Phys., 136, 100{114. Verkley,W.T.M.1997b. Aspectralmodelfortwo-dimensionalincompressible°uid°ow in a circular basin II. Numerical examples. J. Comput. Phys., 136, 115{131. Vlasenko, V., & Hutter, K. 2001. Generation of second mode solitary waves by the interaction of a ¯rst mode soliton with a sill. Nonl. Proc. in Geophys., 8, 223{239. Vlasenko, V., & Hutter, K. 2002. Numerical experiments on the breaking of solitary internal waves over a slope-shelf topography. J. Phys. Oceanogr., 32, 1779{1793. Wake, G. W., Gula, J., & Ivey, G. N. 2004. Periodic forcing of baroclinic basin-scale waves in a rotating stra¯¯ed basin. Proceedings of 15th Austrarian °uid mechanics conference. The University of Sydney, Sydney, Austraria. Wake, G. W., Ivey, G. N., & Imberger, J. 2005. The temporal evolution of baroclinic basin-scale waves in a rotating circular basin. J. Fluid Mech., 523, 367{392. Wang, Y., & Hutter, K. 1998. A semi-implicit semispectral primitive equation model for lake circulation dynamics and its stability performance. J. Comput. Phys., 139, 209{241. Watson, E.R.1904. MovementsofthewatersonLochNessasindicatedbytemperature observations. Goegr. J., 24, 430{437. Wedderburn, E. M. 1907. An experimental investigation of the temperature changes occurring in fresh-water lochs. Proc. Roy. Soc. Edinb., 29, 2{20. Wedderburn, E. M. 1909. Temperature observations in Loch Garry (Invernessshire). With notes on currents and seiches. Proc. Roy. Soc. Edinb., 29, 98{135. Wedderburn, E. M. 1912. Temperature observations in Loch Earn, with a further con- tribution to the hydrodynamical theory of the temperature seiche. Proc. Roy. Soc. Edinb., 48, 629{695. Wedderburn, E. M., & Young, A. W. 1915. Temperature observations in Loch Earn. Part II. Trans. Roy. Soc. Edinb., 50, 741{767. Wiegand, R. C., & Carmack, E. 1986. The climatology of internal waves in deep tem- perate lake. J. Geophys. Res., 91, 3951{3958. Wiegand, R. C., & Chamberlain, V. 1987. Internal waves of the second vertical mode in a strati¯ed lake. Limnol. Oceanogr., 32(1), 29{42. WÄ uest, A., & Lorke, A. 2003. Small-scale hydrodynamics in lakes. Annu. Rev. Fluid Mech., 35, 373{412. Zabusky,N.J.,&Kruskal,M.D.1965. Interactionof\Solitons"inacollisionlessplasma and the recurrence of initial states. Phys. Rev. Lett., 15, 240{243. 151 Appendix A Total energy of the °ow ¯eld ConsiderthelinearBoussinesqequationforaninterfacialwaveinamediumwithconstant depth: (h 1 u 1 ) t ¡c 2 0 ³ x =0; ³ t ¡(h 1 u 1 ) x =0: 9 > = > ; (1) From these equations, we can construct the conservation equation @ @t ½ h 2 1 u 2 1 +c 2 0 ³ 2 2 ¾ + @ @x f¡c 2 0 h 1 u 1 ³g=0: (2) The similar equation exists for the lower layer. Adding such conservation equations yields @ @t ½ h 2 1 u 2 1 +h 2 2 u 2 2 2 +c 2 0 ³ 2 ¾ + @ @x fc 2 0 (h 2 u 2 ¡h 1 u 1 )³g=0: (3) The ¯rst bracketed term is proportional to the total energy density, which contains hor- izontal kinetic energy of the °uid in the upper and lower layers and the potential energy resulting from de°ection of interfacial surface. The second bracketed term is propor- tional to the total horizontal momentum °ux of the upper and lower layers. Integrating (3)overthephysicaldomain[0;L], andusingthevelocityvanishingboundarycondition, we get @ @t Z L 0 ½ 1 2 (h 2 1 u 2 1 +h 2 2 u 2 2 )+c 2 0 ³ 2 ¾ dx=0: (4) 152 Using f and g to represent the right- and left-going displacement functions in the extended computational domain (seex2.2), u 1 , u 2 and ³ can be expressed u 1 =¡ c 0 h 1 (f¡g); u 2 = c 0 h 2 (f¡g); ³ =f +g: 9 > > > > > > = > > > > > > ; (5) Substituting (5) into (4), we have @ @t Z L 0 f 2 +g 2 dx= @ @t Z 2L 0 f 2 dx=0; (6) after using symmetry relation (2.3). Thus, the integral of square of the amplitude over the computational domain is a conserved quantity, which can be interpreted as the total energy of the °ow ¯eld. 153 Appendix B Some properties of the Jacobi polynomials We summarize some properties of the Jacobi polynomials referred in this paper. Although many useful properties are found in Verkley (1997a), Matsushima & Mar- cus (1995) and Ishioka (2003a), care must be taken in the application because their de¯nitions of the radial basis functions are slightly di®erent. In this paper, we follow the de¯nition established by Verkley (1997a). The one-sided Jacobi polynomials are de¯ned here as Q mn (r)=r jmj P (®;¯) k (x) where x = 2r 2 ¡1 and n =jmj+2k. The arbitrary parameters (®;¯) are kept as they are so as not to loose the generality. Series expansions Power series of the one-sided Jacobi polynomials is given by Q mn (r)=r jmj P (®;¯) k (x)=(¡1) k k X k 0 =0 (¡1) k 0(¯+1) k (¯+1+k+®) k 0 (¯+1) k 0k 0 !(k¡k 0 )! r jmj+2k 0 : (7) From this expansion the coe±cient of the leading power r jmj is used to obtain the kine- matic constraint (5.16). The inverse expansion of (7) is also given by r jmj+2k = k X k 0 =0 (2k 0 +®+¯+1)¡(¯+k+1)(k¡k 0 +1) k 0 (k+k 0 +®+¯+1)¡(¯+k 0 +1)(k 0 +®+¯+1) k Q mn 0(r): (8) 154 The derivations of above expansions for a particular parameter set (®;¯) = (0;jmj) are found in Verkley (1997a). Recurrence relations TherecurrencerelationtocalculateP (®;¯) k (x)isgivenby22.7.1inAbramowitz&Stegun: 2(n+1)(n+®+¯+1)(2n+®+¯)P (®;¯) k+1 (x) =f(2n+®+¯+1)(® 2 ¡¯ 2 )+(2n+®+¯) 3 xgP (®;¯) k (x) ¡2(n+®)(n+¯)(2n+®+¯+2)P (®;¯) k¡1 (x): (9) This recurrence is stably advanced from starting values P (®;¯) 0 (x) = 1 and P (®;¯) 1 (x) = ¡(¯ +1)+(¯ +®+2)r 2 . Then the value of Q mn (r) is obtained after multiplying the one-sided factor r jmj to P (®;¯) k (x). ForcalculationofthederivativesLivermoreetal.(2007)introducedageneralmethod that employs a direct formula d l dx l P (®;¯) k (x)= (®+¯+n+1) 2 l P (®+l;¯+l) k¡l (x): (10) Using this property the recurrence formula for the ¯rst derivative of Q mn (r) is expressed as r dQ mn (r) dr =r jmj fjmjP (®;¯) k (x)+2r 2 (®+¯+k+1)P (®+1;¯+1) k¡1 (x)g; (11) and the second derivative can be also expressed as r 2 d 2 dr 2 Q mn (r)=r jmj fjmj(jmj¡1)P (®;¯) k (x) +2(2jmj+1)(®+¯+k+1)r 2 P (®+1;¯+1) k¡1 (x) +4(®+¯+k+1) 2 r 4 P (®+2;¯+2) k¡2 (x)g: (12) 155 For k·1 (alt. jmj·n·jmj+1), the derivatives are computed by the following: r d dr r jmj P (®;¯) 0 (x)=jmjr jmj ; r 2 d 2 dr 2 r jmj P (®;¯) 0 (x)=jmj(jmj¡1)r jmj ; r 2 d 2 dr 2 r jmj P (®;¯) 1 (x)=f¡(jmj¡1) 2 (¯+1)+(jmj+1) 2 (®+¯+2)r 2 gr jmj : 9 > > > > > > = > > > > > > ; (13) Boundary values The standardization of the Jacobi polynomials is de¯ned by 22.2.1 in Abramowitz & Stegun as P (®;¯) k (1)= 0 @ k+® k 1 A = (®+1) k k! : (14) Hence, the boundary value of Q mn (r) at r =1 is given by Q mn (1)= r jmj P (®;¯) k (2r 2 ¡1) ¯ ¯ ¯ r=1 = (®+1) k k! : (15) Theboundaryvalueofthe¯rstderivativeofQ mn (r)canbededucedusing(14)and(11), given by the following dQ mn (r) dr ¯ ¯ ¯ ¯ r=1 = ½ jmj+ 2k(®+¯+k+1) ®+1 ¾ (®+1) k k! : (16) Using a symmetry property P (®;¯) k (¡x)=(¡1) k P (¯;®) k (x) givenby22.4.1 in Abramowitz & Stegun, Q 0n (0) is obtained as Q 0n (0)= P (®;¯) k (2r 2 ¡1) ¯ ¯ ¯ r=0 =P (®;¯) k (¡1)=(¡1) k P (¯;®) k (1)=(¡1) k (¯+1) k k! ; (17) where(14)isusedinthelaststep. Thispropertyisusedtoconstructtheradialexpansion function of the radial velocity °ux given by (5.14). Accordingly, the Dirichlet boundary condition of the radial velocity °ux (5.17) is obtained using the property (15). 156 Appendix C Derivation of expansion coe±cients We describe the derivation of the expansion coe±cients of the exact solution to the linear evolution model givenin (5.33). Here weconsider expanding the non-scaled radial function of radial velocity °ux ~ U ¤ m (r) in terms of the one-sided Jacobi polynomials with parameterset(®;¯)=(0;m), thatisQ mn =r m P (0;m) k (2r 2 ¡1)assumingm>0. Taking the Poincar¶ e wave mode J m (¸r) for Z m (r) in (5.30), and using the recurrence relation J 0 m (x)=J m¡1 (x)¡mJ m (x)=x, ~ U ¤ m (r) is written ~ U ¤ m (r)=(1+!)mJ m (¸r)¡¸!rJ m¡1 (¸r): (18) Expansion of J m (¸r) in terms of Q mn (r) is already derived by Verkley (1997b) as fol- lowing J m (¸r)= 2 ¸ 1 X k=0 (¡1) k (n+1)J n+1 (¸)Q mn (r); and n=m+2k: (19) ThetermrJ m¡1 (¸r)in(18)canbeexpandedintermsofQ mn (r)byfollowingthesimilar proceduretoderiveaboveexpansiondescribedinVerkley(1997b). Usingthepowerseries expansion of the Bessel function, rJ m¡1 (¸r) is written rJ m¡1 (¸r)= µ ¸ 2 ¶ m¡1 1 X k=0 (¡1) k k!(m¡1+k)! µ ¸ 2 ¶ 2k r m+2k : (20) 157 Expansion of polynomial power r m+2k in terms of Q mn (r) is given by (8) in Appendix A with (®;¯)=(0;m) r m+2k =r n = 1 X k 0 =0 (m+2+2k 0 )(k¡k 0 +1) k 0 (m+2+k+k 0 )(m+2+k) k 0 Q mn 0(r): (21) Substituting (21) into (20), and converting terms with Pochhammer symbol to the fac- torial form, we get rJ m¡1 (¸r)= µ ¸ 2 ¶ m¡1 1 X k=0 k X k 0 =0 (¡1) k (m+k)(m+1+2k 0 ) (k¡k 0 )!(m+1+k+k 0 )! µ ¸ 2 ¶ 2k Q mn 0(r): (22) Changing the order of summations we write this expression as rJ m¡1 (¸r)= 1 X k 0 =0 (m+1+2k 0 ) £ ( µ ¸ 2 ¶ m¡1 1 X k=k 0 (¡1) k (m+k) (k¡k 0 )!(m+1+k+k 0 )! µ ¸ 2 ¶ 2k ) : (23) The bracketed sumf¢¢¢g is decomposed into two sums f¢¢¢g= µ ¸ 2 ¶ m¡1 1 X k=k 0 (¡1) k (k¡k 0 )!(m+k+k 0 )! µ ¸ 2 ¶ 2k ¡(k 0 +1) µ ¸ 2 ¶ m¡1 1 X k=k 0 (¡1) k (k¡k 0 )!(m+1+k+k 0 )! µ ¸ 2 ¶ 2k : (24) Letting k¡k 0 =l, this is written in a following form f¢¢¢g=(¡1) k 0 µ 2 ¸ ¶ " µ ¸ 2 ¶ m+2k 0 1 X l=0 (¡1) l l!(m+2k 0 +l)! µ ¸ 2 ¶ 2l # ¡(¡1) k 0 (k 0 +1) µ 2 ¸ ¶ 2 " µ ¸ 2 ¶ m+1+2k 0 1 X l=0 (¡1) l l!(m+1+2k 0 +l)! µ ¸ 2 ¶ 2l # : (25) 158 Recalling the series expansion of the Bessel function, the ¯rst square bracketed term is J m+2k 0(¸), and the second bracketed one is J m+2k 0 +1 (¸). Writing the expression explicitly, we have that f¢¢¢g=(¡1) k 0 µ 2 ¸ ¶ J m+2k 0(¸)+(¡1) k 0 (k 0 +1) µ 2 ¸ ¶ 2 J m+2k 0 +1 (¸): (26) Substituting (26) into (23) and using m+2k 0 =n 0 , rJ m¡1 (¸r) is expanded in the form rJ m¡1 (¸r)= 2 ¸ 1 X k 0 =0 (¡1) k 0 (n 0 +1) ½ J n 0(¸)¡ 2 ¸ (k 0 +1)J n 0 +1 (¸) ¾ Q mn 0(r): (27) Substituting (27) and (19) into (18), we obtain the expansion coe±cients of ~ U ¤ m (r) given in (5.33). The expansion of ~ V ¤ m (r) is obtained through a procedure similar to that described above. Corresponding expansion coe±cients for the Kelvin wave mode I m (¸r) are easily deduced by replacing ¸ with i¸, and using the identity J n (ix) = (i) n I n (x) in (5.33). Itturnsoutthatthe J-Besselfunctionsarereplacedwith I-Besselfunctions, and alternating signs (¡1) k are eliminated in (5.33). 159
Abstract (if available)
Abstract
The nonlinear evolution, generation and degeneration of wind-driven, basin-scale internal waves in lakes are investigated employing weakly-nonlinear, weakly-dispersive evolution models. The models studied are based on rational, asymptotic approximations of the hydrodynamic equations of motion, and include a two-layer model, a multi-modal model, and a large-lake model with the effect of earth's rotation. It is found that nonlinearity, in conjunction with the dispersive nature of the fluid medium, plays a principle role in (i) the early stage of degeneration of basin-scale waves through nonlinear steepening and subsequent generation of oscillatory waves
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Wave induced hydrodynamic complexity and transport in the nearshore
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Creator
Sakai, Takahiro
(author)
Core Title
Generation and degeneration of long internal waves in lakes
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Aerospace Engineering
Publication Date
12/01/2008
Defense Date
10/20/2008
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
earth rotation,internal waves,Kelvin wave,lake hydrodynamics,multi-modal model,non-hydrostatic model,nonlinear waves,numerical simulation,OAI-PMH Harvest,Poincare wave,two-layer model,weakly nonlinear model
Language
English
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Electronically uploaded by the author
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Advisor
Redekopp, Larry G. (
committee chair
), Domaradzki, Julian A. (
committee member
), Hammond, Douglas E. (
committee member
), Maxworthy, Tony (
committee member
), Ziane, Mohammed (
committee member
)
Creator Email
taksakai@hotmail.com,tsakai@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m1843
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UC1194823
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etd-Sakai-2524 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-132533 (legacy record id),usctheses-m1843 (legacy record id)
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etd-Sakai-2524.pdf
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132533
Document Type
Dissertation
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Sakai, Takahiro
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texts
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University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
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Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
earth rotation
internal waves
Kelvin wave
lake hydrodynamics
multi-modal model
non-hydrostatic model
nonlinear waves
numerical simulation
Poincare wave
two-layer model
weakly nonlinear model