An Introduction to Internal Waves
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An introduction to internal waves T. Gerkema J.T.F. Zimmerman Lecture notes, Royal NIOZ, Texel, 2008 Por qu¶eme preguntan las olas lo mismo que yo les pregunto? Qui¶enpuede convencer al mar para que sea razonable? Pablo Neruda, El libro de las preguntas. Contents 1 Introduction 9 1.1 The ocean's inner unrest . 9 1.2 Restoring forces . 12 1.2.1 The ocean's strati¯cation . 13 1.2.2 The Earth's diurnal rotation . 14 1.3 Origins of internal waves . 16 1.4 Dissipation and mixing . 19 1.5 Overview . 20 Further reading . 21 2 The equations of motion 23 2.1 Introduction . 23 2.2 Fluid mechanics . 24 2.3 A brief introduction to thermodynamics . 26 2.3.1 Fundamentals . 26 2.3.2 Open systems . 28 2.3.3 Some de¯nitions . 29 2.3.4 The Gibbs potential and the equation of state . 30 2.4 Complete set of governing equations . 31 2.5 Internal-wave dynamics . 34 Further reading . 35 Appendix A: a closer look at the Coriolis Force . 36 Appendix B: sound waves . 37 3 Local static stability 39 3.1 The buoyancy frequency . 39 3.2 N in terms of density . 41 3.3 N in terms of temperature and salinity . 42 3.4 A practical example . 43 3.5 Special types of strati¯cation . 44 3.5.1 A ﬂuid in thermodynamic equilibrium . 45 3.5.2 A turbulently mixed ﬂuid . 46 3.6 Potential density . 47 3.6.1 Generalized potential density . 47 5 3.6.2 The common concept of potential density . 48 3.7 Limitations of the concept of local stability . 49 4 Approximations 51 4.1 Introductory remarks . 51 4.2 Overview . 52 4.3 Quasi-incompressibility . 53 4.3.1 Momentum equations . 53 4.3.2 Mass and energy equations . 55 4.3.3 Resulting equations . 57 4.4 The `Traditional Approximation' . 58 4.5 Linearization . 59 4.5.1 Resulting equations . 60 4.5.2 Energy equation . 61 4.6 Rigid-lid approximation . 62 4.7 Preview: methods of solution . 64 5 Internal-wave propagation I: method of vertical modes 67 5.1 General formulation . 67 5.1.1 Oscillatory versus exponential behaviour . 68 5.1.2 Orthogonality . 69 5.1.3 Hydrostatic approximation . 70 5.2 Uniform strati¯cation . 70 5.2.1 Dispersion relation . 71 5.2.2 Modal structure . 74 5.2.3 Superposition of modes . 77 5.3 Varying N: two layers . 78 5.3.1 Refraction and internal reﬂection . 80 5.3.2 Trapping of high-frequency waves (! > jfj) . 82 5.3.3 Trapping of low-frequency waves (! < jfj) . 82 5.4 A simple model for the ocean's strati¯cation . 84 5.4.1 Dispersion relation and vertical modes . 85 5.4.2 Scattering at the thermocline . 87 5.4.3 Interfacial waves . 88 5.5 Linearly varying N 2: Airy functions . 91 5.6 Non-traditional e®ects . 93 5.6.1 Frequency range . 95 5.6.2 Dispersion relation for constant N . 96 5.6.3 Expressions of other ¯elds . 97 5.6.4 Superposition of modes . 98 Appendix: the delta-distribution . 99 6 Internal wave-propagation II: method of characteristics 101 6.1 Basic properties of internal waves . 101 6.1.1 Dispersion relation and corollaries . 101 6.1.2 General solution . 104 6.1.3 Kinetic and potential energy . 105 6.2 Reﬂection from a sloping bottom . 106 6.3 Propagation between two horizontal boundaries . 108 6.4 Three-dimensional reﬂection . 109 6.5 Non-uniform strati¯cation . 111 6.5.1 WKB approximation . 112 6.5.2 Characteristic coordinates . 114 6.6 Non-traditional e®ects . 114 7 Internal tides 119 7.1 Barotropic tides . 119 7.2 Boundary vs. body forcing . 124 7.3 Generation over a step-topography . 126 7.4 Solutions for in¯nitesimal topography . 130 7.4.1 Uniform strati¯cation . 132 7.4.2 Three-layer model . 134 7.4.3 Interfacial tides . 137 7.5 Energetics and conversion rates . 139 Appendix A: Integral expressions I . 143 Appendix B: Integral expressions II . 143 8 Internal solitons 145 8.1 Observations . 145 8.2 Korteweg-de Vries (KdV) equation . 147 8.2.1 E®ect of nonlinearity . 148 8.2.2 E®ect of dispersion . 151 8.2.3 KdV for interfacial waves . 152 8.2.4 A heuristic `derivation' . 152 8.2.5 Soliton solution . 153 8.3 Derivation of the KdV equation . 157 8.3.1 Basic equations . 157 8.3.2 Scaling and small parameters . 159 8.3.3 Lowest order . 161 8.3.4 Next order . 162 8.3.5 Final result . 164 8.4 Inverse-scattering theory . 164 8.4.1 The scattering problem . 165 8.4.2 The meaning of the discrete spectrum . 168 8.4.3 Calculation of the number of emerging solitons . 169 8.5 Internal tides and solitons . 171 8.5.1 Disintegration of an interfacial tide . 172 8.5.2 'Local generation' by internal-tide beams . 174 8.5.3 Decay and dissipation . 175 8.6 Limitations of KdV . 176 Further reading . 179 Appendix: Form-preserving solutions of the extended KdV equation . 180 9 Miscellaneous topics 183 9.1 Internal-wave attractors . 183 9.1.1 Vertical wall . 185 9.1.2 Slope . 186 9.1.3 Discussion . 186 9.2 Nonlinear e®ects { or the absence thereof: general remarks . 188 9.3 Generation of higher harmonics . 190 9.3.1 Formulation of the problem . 190 9.3.2 General solution . 192 9.3.3 Solutions for reﬂection from a uniform slope . 194 9.3.4 Examples . 195 9.4 Wave-wave interactions . 195 9.5 Internal-wave spectra . 197 Bibliography 201 8 Chapter 1 Introduction Waves are all around us, but it is actually hard to say what a wave is. This is be- cause it is an immaterial thing: a signal, a certain amount of energy propagating through a medium. The medium we consider is water, seawater in particular. Even though water waves as such are immaterial, they are supported by the oscillatory movement of the water parcels, and this indeed forms a way of de- tecting them. But it is always important not to confuse the water motion with the wave itself. The following analogies may help to clarify this point: \A bit of gossip starting in London reaches Edinburgh very quickly, even though not a single individual who takes part in spreading it travels be- tween these two cities. There are two quite di®erent motions involved, that of the rumour, London to Edinburgh, and that of the persons who spread the rumour. The wind, passing over a ¯eld of grain, sets up a wave which spreads out across the whole ¯eld. Here again we must distinguish between the motion of the wave and the motion of the separate plants, which undergo only small oscillations. We have all seen the waves that spread in wider and wider circles when a stone is thrown into a pool of water. The motion of the wave is very di®erent from that of the particles of water. The particles merely go up and down. The observed motion of the wave is that of a state of matter and not of matter itself. A cork ﬂoating on the wave shows this clearly, for it moves up and down in imitation of the actual motion of the water, instead of being carried along by the wave." [16, pp. 104-105] 1.1 The ocean's inner unrest Waves at the ocean's surface are a familiar sight. These lecture notes deal with waves that propagate beneath the surface; they are mostly hidden from eyesight. Occasionally, however, they produce a visible response at the ocean's surface. 9 An example is shown in Figure 1.1, a photograph taken from the Apollo-Soyuz spacecraft in 1975, when it passed over Andaman Sea, north of Sumatra. Fig. 1.1: A photograph from the Apollo-Soyuz spacecraft, made over Andaman Sea, showing stripes due to internal waves. The stripes stretch over 100 km or.