## Wave radiation in simple geophysical models

##### Abstract

Wave radiation is an important process in many geophysical flows. In particular, it is by wave
radiation that flows may adjust to a state for which the dynamics is slow. Such a state is
described as “balanced”, meaning there is an approximate balance between the Coriolis force
and horizontal pressure gradients, and between buoyancy and vertical pressure gradients. In
this thesis, wave radiation processes relevant to these enormously complex flows are studied
through the use of some highly simplified models, and a parallel aim is to develop accurate
numerical techniques for doing so.
This thesis is divided into three main parts.
1. We consider accurate numerical boundary conditions for various equations which support
wave radiation to infinity. Particular attention is given to discretely non-reflecting
boundary conditions, which are derived directly from a discretised scheme. Such a boundary
condition is studied in the case of the 1-d Klein-Gordon equation. The limitations
concerning the practical implementation of this scheme are explored and some possible
improvements are suggested. A stability analysis is developed which yields a simple stability
criterion that is useful when tuning the boundary condition. The practical use of
higher-order boundary conditions for the 2-d shallow water equations is also explored; the
accuracy of such a method is assessed when combined with a particular interior scheme,
and an analysis based on matrix pseudospectra reveals something of the stability of such
a method.
2. Large-scale atmospheric and oceanic flows are examples of systems with a wide timescale
separation, determined by a small parameter. In addition they both undergo constant
random forcing. The five component Lorenz-Krishnamurthy system is a system with a
timescale separation controlled by a small parameter, and we employ it as a model of
the forced ocean by further adding a random forcing of the slow variables, and introduce
wave radiation to infinity by the addition of a dispersive PDE. The dynamics are reduced
by deriving balance relations, and numerical experiments are used to assess the effects of
energy radiation by fast waves.
3. We study quasimodes, which demonstrate the existence of associated Landau poles of a
system. In this thesis, we consider a simple model of wave radiation that exhibits quasimodes,
that allows us to derive some explicit analytical results, as opposed to physically
realistic geophysical fluid systems for which such results are often unavailable, necessitating
recourse to numerical techniques. The growth rates obtained for this system, which
is an extension of one considered by Lamb, are confirmed using numerical experiments.