The Effect of Stratification and Bathymetry on Internal Seiche Dynamics by Paul David Pricker B.Sc. in Chemistry, Dalhousie University (1989) M.Sc. in Physics, University of Toronto (1991) Submitted to the Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil and Environmental Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2000 @ Massachusetts Institute of Technology 2000. All rights reserved. Signature of Author ............. .......... ............. .. ............................ Department of Civil and Environmental Engineering 26 July 2000 Certified by ..................... ......... /Heidi M. Nepf Associate Professor, Civil and Environmen al Engineering -77 ;hesis Supervisor Accepted by ..... ..................... ....................................... Daniele Veneziano Chairman, Department Committee on Graduate Students MASSACHUSETTS INSTITUTE OF TECHNOLOGY SEP 1 5 2000 LIBRARIES The Effect of Stratification and Bathymetry on Internal Seiche Dynamics by Paul David Fricker Submitted to the Department of Civil and Environmental Engineering on 26 July 2000, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil and Environmental Engineering Abstract Internal seiches are basin-scale standing waves which oscillate within the body of a stratified lake or bay. In thermally stratified mid-latitude lakes, for which the surface-to-bed density difference is typically 0 (1 - 3kg/m 3) during summer, buoyancy within the water column supports rela- tively large amplitude waves, with horizontal and vertical fluid displacements as large as ~10% of the lake dimensions, depending on the strength of wind forcing. In strongly-stratified lakes, seiching is often the only dynamic process occuring in the hypolimnion, since direct wind-driven motions are constrained to the epilimnion, and small-scale (i.e. progressive) internal waves can- not propagate outside the pycnocline. Internal seiches therefore provide the principal conduit for converting wind energy into boundary, hypolimnetic, and effective diapycnal mixing. We begin by investigating the dependence of internal seiche structure (i.e., the velocity field) on lake bathymetry and stratification, using a two-dimensional linear, inviscid model to compute numerical seiche solutions for a series of idealized configurations. This is followed by an analysis of the fundamental seiche (V1H1) in the Upper Mystic Lake (UML; Winchester, Massachusetts), including a comparison of model results to field observations (thermistor chain temperature time series), and an assessment of the seasonal evolution of bed velocity distribution. We next evaluate the viscous damping of internal seiches by modifying the inviscid formulation with the addition of a benthic boundary layer flow. A generalized expression for the decay rate (a) is derived through a perturbation analysis using the solvability condition on the combined inviscid/first-perturbation-order system. The resulting a is equivalent in form to the integral of seiche kinetic energy at the bed (as for surface waves), weighted by an additional coefficient which accounts for effects of buoyancy and bathymetry. Comparison to other, physically-based derivation methods reveals that a can be interpreted equivalently as the rate of stress working by the seiche on the bed boundary. Finally, using numerical solutions for the three dominant seiches in the UML, we find that buoyancy effects generate roughly an order of magnitude increase in a for each mode. The estimated relative damping rates account for the apparent rapid decay of the fundamental (V1H1) seiche, and are consistent with the observed persistence of the dominant higher mode (V3H1). Buoyancy effects therefore appear to be an important factor governing seiche climate in the UML. Thesis Supervisor: Heidi M. Nepf Title: Associate Professor, Civil and Environmental Engineering Acknowledgments First I must thank my supervisor, Professor Heidi M. Nepf, for allowing me to freely explore new ideas and avenues of research, even though I couldn't always explain where they were leading. Heidi endured my enthusiasm for equations, and constantly reminded me that a clear, well-developed presentation of information is essential for communicating scientific research. I would also like to thank the members of my committee, Professor Harold F. Hemond (who was also my supervisor during my first year at MIT) and Professor Chiang C. Mei, for their input and support. I am grateful for the help and friendship of my fellow students at the Parsons Lab; thanks go especially to Hrund Andrad6ttir, Chin-Hsien Wu, Laura DePaoli, Marco Ghisalberti, Brian White, Enrique Vivoni, David Senn, Daniel Pedersen, Susan Brown, Gordon Ruggaber, Frederic Chagnon, and all the others I will wish I had added later. A special badge of honor goes to those who survived helping me with field work. I would also like to thank my MIT friends-at- large, Karuna Mohindra, Sally Stiffier, Edmund Carlevale, Janni Moselsky, and Row Selman, as well as those who helped me remember the world outside MIT, Sanjeev Seereeram and Cecilia Mercado, Darren and Teresa Farmer, Phil Trowbridge and Laura Bonk, and my friends at home, George Xidos, Chris White, Phil Gunn, Susan Walsh, Emmanuel and Matina Xidos, Strat and Pat Poulos, Cyril and Peggy White, and John and Cathy Arab. I would also like to acknowledge Russell J. Boyd at Dalhousie University, for being a good scientist, and for making my first efforts in research so enjoyable and rewarding. Above all, I thank my wife Kamla and my children Alexander and Kristina, for their love, and for the tremendous sacrifices they have made over the past few years. This thesis is truly a group effort. And finally, I thank my family, Aubrey, Joan and Mike Fricker, without whose love and support this work would not have been possible. 3 Contents 1 Internal seiches 12 1.1 Introduction ... .. ... .. ... ... .. ... .. ... ... .. ... .. ... 12 1.1.1 Thesis outline ... .. ... ... ... .. ... .. ... ... .. ... .. 13 1.2 Review of Literature and Methods . ... .. ... .. ... .. ... .. ... .. 14 1.2.1 Background .. ... .. ... ... ... .. ... ... .. ... ... .. 15 1.2.2 Summary of seiche models ... .. ... ... .. ... ... .. ... ... 20 1.3 Excitation of seiches: continuous stratification . ... ... ... ... ... .. .. 26 1.3.1 Amplitude evolution .. ... ... ... .. ... ... .. ... ... .. 26 1.3.2 Reformulation . .. ... ... .. ... ... .. ... ... .. ... ... 33 1.4 Summary . ... .... ... .... ... .... .... ... .... ... .... 37 2 Bathyrnetry, stratification, and internal seiche structure 43 2.1 Introduction .. ... .... ... ... ... ... ... .... ... ... ... .. 44 2.2 Numerical Method .. ... ... .... ... ... ... ... .. .... ... 46 2.3 Test case: the Upper Mystic Lake . .... ... .... ... .... .. .... 50 2.3.1 Site description . ... .... ... ... ... .... ... ... .... .. 50 2.3.2 Data collection and analysis ... ... ... ... .... ... ... .... 52 2.4 Results and Discussion .. ... ... .... ... ... ... ... ... ... ... 56 2.4.1 Comparison to the model ... ... .... ... ... ... .... ... 56 2.4.2 Stratification and bed velocities ... .... ... ... ... .... ... 58 2.4.3 Bathymetry and bed velocities . ... ... ... .... ... ... .... 65 2.4.4 Seasonal variation of bed velocities in the UML . .... ... .... ... 67 4 2.5 Conclusions . ... ... ... ... ... ... ... ... ... ... ... ... 69 2.6 Acknowledgments .. ... .... ... ... .... .. ... ... .... ... 70 3 Viscous damping of internal seiches 75 3.1 Introduction .. ... ... ... ... .. ... ... ... .. .. .. .. .. .. 7 6 3.2 Analytical formulation . ... ... ... ... .. ... .. .. .. .. .. .. .. 7 7 3.2.1 Outer flow .. ... ... ... ... ... .... .. .. .. .. .. .. .. 79 3.2.2 The boundaries . ... .. ... ... .. ... .. .. .. .. .. .. .. 8 0 3.2.3 Benthic boundary layer ... ... ... .... .. .. .. .. .. .. .. 82 3.3 Perturbation . ... ... ... ... .. ... ... ... .. .. .. .. .. .. 8 5 3.3.1 Boundary conditions . ... ... ... ... ... .. ... ... ... 8 5 3.3.2 Governing equation and frequency change ... .. .. ... ... .. ... 87 3.4 Physical interpretation .. ... .... ... ... .... .... .... .... 9 2 3.4.1 Bed stresses .... ... .... .... ... .... .. .. .. .. .. .. 9 2 3.4.2 Dissipation within the boundary-layer ... ... ... ... .. ... 93 3.4.3 Pressure working . ... .. ... ... .. ... .. ... ... .. ... 95 3.5 Applications 1: Simple systems .. ... ... .. ... .. ... ... .. ... .. 9 6 3.5.1 Cylindrical and rectangular basins (3D flows) . .. .. .. .. .. .. .. 9 7 3.5.2 Longitudinal seiches (modelled as two-dimensional flows) . .. ... ... 99 3.6 Applications 2: Numerical calculations . ... .. ... .. .. ... ... ... 100 3.6.1 Numerical method ... ... .... .... ... .. ... ... .. ... 101 3.6.2 Model configurations and parameters . .. ... .. ... ... .. ... 102 3.6.3 Numerical solutions .. .... .... ... .... ... ... .. ... .. 106 3.7 Results and discussion .. ... ... ... ... ... .. .. ... ... .. ... 108 3.7.1 Buoyancy and the benthic boundary-layer..... .. ... .. ... ... 108 3.7.2 Decay rates . .. .. .. .. .. .. .. .. .. .. .. ... ... ... 114 3.8 Conclusions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 115 3.9 Appendix: Solution for the boundary-layer flow .. ... .. .. .. .. .. 117 4 The effect of buoyancy and bathymetry on internal seiche decay 124 4.1 Introduction . .. .. .. .. .. .. .. .. .. .. .. ..... .. 126 5 4.2 Internal seiche decay ....... ....... ....... ... 127 4.2.1 Numerical model for the inviscid flow