Inertia-gravity-waves in geostrophic turbulence

Abstract

The dynamics of the atmosphere and ocean can be decomposed into a slow, large-scale component, in near geostrophic and hydrostatic balance, and a fast component consisting mostly of inertia–gravity waves (IGWs). These waves exist because of restoring forces provided by a combination of gravity and density stratification, and the Coriolis effect due to rotation of the Earth. Their frequencies are high, from a few hours to minutes, while their wavelengths span a broad range of scales, from say 500 km in the atmosphere and 100 km in the ocean, down to metre scales.

IGWs are important for a number of processes since they induce a transport of mass and momentum, interact with the slow component of the motion, and ultimately affect global-scale circulation in the atmosphere and ocean. It is therefore important to be able to predict and understand their evolution to model weather, ocean dynamics and climate. This evolution is complicated by the fact that IGWs do not propagate in tranquil, homogeneous environments but in the complex field of eddies and currents that characterises the nonlinear dynamics of the slow motion and is described as geostrophic turbulence.

This thesis develops new mathematical models of the propagation of IGWs in geostrophic turbulence. The new models represent the turbulent background flow using random fields and capture its impact, through advection and refraction, in a statistically-averaged sense.

In the simplest case, wave energy can be mathematically represented by a single point in wavenumber space, corresponding to a plane wave with a well defined wavelength and direction of propagation. The presence of a background flow causes IGWs to scatter however, giving rise to changes in length-scales and directions of propagation. This corresponds to transfers of wave energy across wavenumber space. In this thesis we derive equations governing the statistics of these energy transfers. The derivations rely on the Wigner transform to define a wavenumber-resolving energy density, and on multiscale asymptotics to derive the equation it satisfies, a kinetic equation in which the scattering effect is represented through an integral in wavenumber space. In the limit of IGWs short compared with the turbulence scales, the WKB approximation applies and the scattering integral reduces to a diffusion operator in wavenumber space.

We examine scattering in a series of increasingly general scenarios. First, for scattering by flows with no vertical dependence, the governing equations for IGWs can be reduced to an equivalent shallow-water system. This setting is appropriate for studying low-mode internal tides propagating through a turbulent field of eddies which typically share similar length-scales. The kinetic equation shows that energy transfers are restricted to waves with the same frequency and identical vertical structure, and that they ultimately lead to an isotropic wave field when the turbulent flow is itself isotropic. The equation enables us to estimate characteristic time and length scales for scattering and isotropisation. We carry out simulations of internal tides generated by a planar wavemaker for the linearised shallow-water model to confirm the pertinence of these scales. A comparison with the numerical solution of the kinetic equation demonstrates the validity of the latter and illustrates how the interplay between wave scattering and transport shapes the wave statistics.

Second, we consider geostrophic turbulence with a non-trivial vertical structure consistent with Charney’s theory. This leads to radically different dynamics, with a cascade of energy towards small scale in both the horizontal and vertical. We explore this using the diffusion approximation to the kinetic equation that arises in the WKB limit. We derive explicit solutions for both initial-value and forced steady-state scenarios. In the forced case, diffusion leads to a k-2 wave energy spectrum, consistent with as-yet-unexplained features of observed oceanic and atmospheric spectra.

Third, we go beyond the WKB approximation to consider the full kinetic equation that applies in the absence of spatial-scale separation between IGWs and geostrophic flow. We demonstrate how the kinetic equation recovers the diffusion equation in the appropriate limit and show that it captures new dynamical features, negligible in the WKB limit but significant in practice including the reflection of upward- to downward-propagating waves. We compare the predictions of the kinetic equation and those of its diffusion approximation to high-resolution numerical simulations of the three-dimensional Boussinesq equations.

Taken together, the results provide a detailed description of the impact of geostrophic turbulence on IGWs, quantifying the spectral transfers that take IGW energy from the forcing scales to the dissipation scales and highlighting the manner in which turbulence shapes the distribution of IGW energy in the atmosphere and ocean.