Inertia-Gravity Wave Generation: a WKB Approach
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Inertia-gravity wave generation: a WKB approach Jonathan Maclean Aspden Doctor of Philosophy University of Edinburgh 2010 Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Jonathan Maclean Aspden) iii iv Abstract The dynamics of the atmosphere and ocean are dominated by slowly evolving, large-scale motions. However, fast, small-scale motions in the form of inertia-gravity waves are ubiquitous. These waves are of great importance for the circulation of the atmosphere and oceans, mainly because of the momentum and energy they transport and because of the mixing they create upon breaking. So far the study of inertia-gravity waves has answered a number of questions about their propagation and dissipation, but many aspects of their generation remain poorly understood. The interactions that take place between the slow motion, termed balanced or vortical motion, and the fast inertia-gravity wave modes provide mechanisms for inertia-gravity wave generation. One of these is the instability of balanced flows to gravity-wave-like perturbations; another is the so-called spontaneous generation in which a slowly evolving solution has a small gravity-wave component intrinsically coupled to it. In this thesis, we derive and study a simple model of inertia-gravity wave generation which considers the evolution of a small-scale, small amplitude perturbation superimposed on a large-scale, possibly time-dependent flow. The assumed spatial-scale separation makes it possible to apply a WKB approach which models the perturbation to the flow as a wavepacket. The evolution of this wavepacket is governed by a set of ordinary di®erential equations for its position, wavevector and its three amplitudes. In the case of a uniform flow (and only in this case) the three amplitudes can be identi¯ed with the amplitudes of the vortical mode and the two inertia-gravity wave modes. The approach makes no assumption on the Rossby number, which measures the time-scale separation between the balanced motion and the inertia-gravity waves. v The model that we derive is ¯rst used to examine simple time-independent flows, then flows that are generated by point vortices, including a point-vortex dipole and more complicated flows generated by several point vortices. Particular attention is also paid to a flow with uniform vorticity and elliptical streamlines which is the standard model of elliptic instability. In this case, the amplitude of the perturbation obeys a Hill equation. We solve the corresponding Floquet problem asymptotically in the limit of small Rossby number and conclude that the inertia-gravity wave perturbation grows with a growth rate that is exponentially small in the Rossby number. Finally, we apply the WKB approach to a flow obtained in a baroclinic lifecycle simulation. The analysis highlights the importance of the Lagrangian time dependence for inertia-gravity wave generation: rapid changes in the strain ¯eld experienced along wavepacket trajectories (which coincide with fluid-particle trajectories in our model) are shown to lead to substantial wave generation. vi Contents Abstract vi List of ¯gures xii 1 Introduction 1 1.1 Geophysical fluid dynamics . 1 1.2 Outline of thesis . 3 2 Geophysical fluid dynamics 7 2.1 Introduction . 7 2.2 Rotation and strati¯cation . 7 2.2.1 Coriolis e®ect . 7 2.2.2 Rossby number . 12 2.2.3 The Brunt-VÄaisÄalÄafrequency . 12 2.3 Governing equations . 14 2.3.1 Introduction . 14 2.3.2 Boussinesq approximation . 15 2.3.3 The Boussinesq equations . 16 2.4 Balance relations . 17 2.5 Potential vorticity . 18 2.6 Conclusion . 20 3 Inertia-gravity waves 21 3.1 Introduction . 21 3.2 Time-scale separation . 22 3.3 Dispersion relation . 23 3.4 Generation mechanisms . 25 3.4.1 Spontaneous generation . 25 3.4.2 Generation through instabilities . 26 3.5 Conclusion . 28 4 WKB approach 29 4.1 Introduction . 29 4.2 Derivation of Equations . 30 4.2.1 Adding a perturbation . 30 4.2.2 WKB Theory . 31 vii Contents Contents 4.2.3 Applying the WKB theory . 32 4.2.4 Vorticity and divergence . 35 4.2.5 Potential vorticity . 36 4.2.6 Eliminating½ ^0 ............................ 37 4.2.7 Final equations . 39 4.2.8 Recovering the intrinsic frequency . 40 4.2.9 Solving the system . 41 4.3 Non-dimensionalising . 42 4.4 Removing the singularity at m =0..................... 42 4.5 Energy . 44 4.6 Conclusion . 45 5 Simple flows 47 5.1 Introduction . 47 5.2 No Flow . 47 5.3 Pure Strain Field . 48 5.4 Transverse shear . 52 5.5 Strain and Shear . 54 5.6 Frontogenesis flow . 58 5.7 Conclusion . 61 6 Point-vortex model 63 6.1 Introduction . 63 6.2 Point vortices . 64 6.3 Dipole . 66 6.3.1 Wavenumber and amplitude equations . 67 6.3.2 Non-dimensionalising . 69 6.4 Polarisation . 69 6.4.1 Eigensolution . 69 6.4.2 Finding Av and Ag§ ......................... 71 6.5 Initialisation . 73 6.6 Results . 74 6.7 Elliptical trajectories within a dipole . 76 6.8 Complex time dependent flows . 79 6.8.1 Introduction . 79 6.8.2 Initialisation . 82 6.8.3 Results . 83 6.9 Conclusion . 88 7 Elliptical instability 89 7.1 Introduction . 89 7.2 Formulation . 91 7.3 WKB analysis . 96 7.4 The Stokes phenomenon . 98 7.5 Calculating M ................................ 99 7.6 Using exponential asymptotics to calculate S . 102 7.7 Analysis of the ® and ¯ integrals . 107 7.7.1 The asymptotics of ® for small and large values of ¹ . 107 7.7.2 The asymptotics of ® for small and large values of à . 109 7.7.3 The e®ect of ¯ ............................110 7.8 Position and thickness of the instability bands . 110 viii CONTENTS CONTENTS 7.9 Comparison with numerical results . 111 7.10 Justifying the hydrostatic approximation . 114 7.11 Conclusion . 115 8 Baroclinic lifecycle 117 8.1 Introduction . 117 8.2 Baroclinic instability . 118 8.3 Model setup . 120 8.4 Modifying the data . 121 8.5 Interpolation . 122 8.5.1 Smoothing the data ¯elds . 123 8.6 Initialisation . 123 8.7 Results . 132 8.8 Conclusion . 138 9 Conclusion 139 A Change of coordinates 143 Bibliography 145 ix Contents Contents x List of Figures 2.1 The set up used in the derivation of the Coriolis force. 9 4.1 The form of a wavepacket. 31 5.1 Inertia-gravity-waves in the case of no flow. 49 5.2 The streamlines and velocity ¯eld of a pure strain ¯eld. 50 5.3 The energy of a wavepacket in a pure strain ¯eld. 53 5.4 The velocity ¯eld of a transverse shear flow. 54 6.1 The streamlines and velocity ¯eld of a point vortex induced dipole. 68 6.2 Trajectories in a flow generated by a quasi-geostrophic dipole. 75 6.3 The evolution of the wavenumbers as a wavepacket sweeps past a dipole. 76 6.4 Inertia-gravity waves generated as a wavepackets sweeps past a dipole. 77 6.5 The ¯nal amplitudes of the inertia-gravity wave mode. 78 6.6 The elliptical trajectory of a wavepacket in close proximity to a dipole. 79 6.7 The wavenumbers evolution on an elliptical trajectory in a dipole. 80 6.8 The amplitudes evolution on an elliptical trajectory in a dipole. 81 6.9 The trajectory, wavenumbers, amplitudes, and local Rossby number of a wavepacket in a random strain flow. 84 6.10 The positions of the wavepacket and point vortices when growth is observed. 85 6.11 The trajectory, wavenumbers, amplitudes, and local Rossby number of a wavepacket in a random strain flow. 87 7.1 The stream lines and velocity ¯eld of an anticyclonic elliptical flow. 92 7.2 The paths of the integrals used in the calculation of S. 105 7.3 Contours of the parameters ® and ¯ governing the maximum growth rates.109 7.4 Numerical estimates of the local maxima of the growth rates. 112 7.5 Growth rates in anticyclonic flows. 113 7.6 Growth rates in cyclonic flows. ..