INTERNAL GRAVITY WAVES

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1 INTERNAL GRAVITY WAVES B. R. Sutherland Departments of Physics and of Earth&Atmospheric Sciences University of Alberta

2 Contents Preface List of Tables vii xi 1 Stratified Fluids and Waves Introduction Stratified fluids Stratified liquids and the ocean Stratified gases and the atmosphere Internal gravity waves Co-ordinate systems Lagrangian and Eulerian frames of reference Equations of state Equations of state for air Equations of state for sea water Conservation of internal energy Thermodynamics of dry air Potential temperature of a gas Thermodynamics of sea water Conservation of mass 19 i

3 ii INTERNAL GRAVITY WAVES 1.9 Incompressibility Conservation of momentum The buoyancy force The pressure gradient force The Coriolis force The viscous damping force Hydrostatic balance The Boussinesq and Anelastic Approximations The speed of sound Boussinesq liquids Non-Boussinesq liquids Anelastic gases Boussinesq gases Conservation of angular momentum Vorticity in uniform-density fluid Vorticity in stratified fluid Potential vorticity Conservation of mechanical energy Kinetic and potential energy Available potential energy The Bernoulli equation Wave theory Representation of plane waves The dispersion relation Phase velocity Group velocity Representation of wavepackets Plane wave and wavepacket evolution equations Wave modes Cross-correlations 67 2 Interfacial waves Introduction Surface waves Finite-amplitude equations Small-amplitude approximation Dispersion relation, phase and group speeds Polarisation relations Fluid parcel motion 84

4 Bruce R. Sutherland iii Energy transport Momentum deposition The Stokes drift Momentum transport Shallow water equations Interfacial waves in a two-layer fluid Equations of motion Interfacial waves in infinite-depth fluid Interfacial waves in finite-depth fluid Shallow interfacial waves Interfacial waves in multi-layer fluids Laterally bounded interfacial waves Shear flows Derivation of equations Rayleigh waves Shear layer instability in uniform-density fluid Shear instability of interfacial waves Interfacial waves influenced by rotation Small-amplitude wave equations Inertial waves Energetics of inertial waves Internal waves in uniformly stratified fluid Introduction The buoyancy frequency Vertical oscillations of a liquid Vertical oscillations of an ideal gas Stable, neutral and uniform stratification Diagonal oscillations of a liquid Boussinesq internal waves Equations of motion in a liquid Equations of motion in a Boussinesq gas Dispersion relation Phase and group velocity Polarization relations Hydrostatic and nonhydrostatic waves Evanescent disturbances Transport by Boussinesq internal waves Zero mass/internal energy transport 166

5 iv INTERNAL GRAVITY WAVES Mechanical energy transport Wave action and pseudoenergy Momentum transport Wave-induced mean flow Bounded internal waves Modes in rectangular box Modes in non-uniform stratification Internal waves near slopes Non-uniformly stratified shear flows Equations of motion Singularities Stability criteria Piecewise-linear theory Waves in uniform flow with uniform stratification Trapped internal gravity waves Shear instability in stratified fluid Non-Boussinesq internal waves Non-Boussinesq waves in a liquid Anelastic waves in a gas Internal waves influenced by rotation Nonlinear considerations Introduction Weakly nonlinear theory The nonlinear pendulum Weakly nonlinear theory for partial differential equations The nonlinear Schrödinger equation Modulational stability theory Weakly nonlinear interfacial waves Theory for interfacial waves in infinitely deep fluid Deep water waves Deep interfacial plane waves Deep interfacial wavepackets Interfacial waves in finite-depth fluid Solitary waves Nonlinear shallow water equations Moderately long weakly nonlinear interfacial waves Weakly nonlinear internal waves Perturbation Theory 241

6 Bruce R. Sutherland v Modulational stability Breakdown of internal waves into turbulence Parametric subharmonic instability Overturning instabilities Shear instabilities Overturning driven by self-acceleration Generation mechanisms Introduction Oscillating bodies Oscillating cylinder in inviscid fluid Oscillating cylinder in viscous fluid Oscillating sphere One- and two-layer flow over topography Equations of motion for a one-layer fluid One-layer small-amplitude hydrostatic flow One layer small-amplitude nonhydrostatic flow Two-layer flow Steady stratified flow over topography Froude and Long numbers Flow over small-amplitude periodic hills Internal waves over small-amplitude localized hills Internal waves over large-amplitude localized hills Internal waves over localized two-dimensional hills Downslope windstorms Tidal flow over topography Barotropic tides Inertia gravity wave beams Wave propagation and spectra Introduction Extrinsic and intrinsic frequencies Ray theory General theory Ray theory for waves in two dimensions Ray Theory for Interfacial Waves Surface waves approaching a beach Interfacial waves approaching a slope Ray Theory for Internal Waves 328

7 vi INTERNAL GRAVITY WAVES Internal waves in two-dimensions Critical levels Reflection levels Caustics Eckart resonance and tunnelling Eckart resonance Transmission and reflection of incident plane waves Internal Wave Spectra Oceanic internal waves Atmospheric internal waves 355 A Suggestions for Further Reading 360 A.1 Textbooks 360 A.2 Review Articles 361 A.3 Journal Articles 361 A.3.1 Anelastic and other non-boussinesq equations 361 A.3.2 Momentum and energy transport 362 A.3.3 Stratified shear flow stability 362 A.3.4 Internal waves at sloping boundaries and attractors 362 A.3.5 Evolution of finite amplitude internal gravity waves 363 A.3.6 Evolution of internal solitary waves 363 A.3.7 Parametric subharmonic instability of internal gravity waves 364 A.3.8 Internal wave interaction with its induced mean flow 364 A.3.9 Interaction between internal wave beams 364 A.3.10 Internal wave breakdown 365 A.3.11 Internal wave generation by oscillating bodies 365 A.3.12 Internal wave generation by steady flow over topography 366 A.3.13 Internal solitary wave generation by flow over topography 366 A.3.14 Inertia gravity wave generation by tidal flow 367 A.3.15 Internal waves at critical and reflection levels 367 A.3.16 Internal wave ducting and tunnelling 368 A.3.17 Internal waves in transient and veering flows 368 A.3.18 Oceanic internal wave spectra 369 A.3.19 Atmospheric internal wave spectra 370 A.3.20 Drag parameterization in general circulation models 370 Index 373

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