Abstract
We use the tools of exponential-asymptotic analysis to describe the solutions of
the three-dimensional Boussinesq equations and of Lorenz's five-component model
that arise in geophysical fluid dynamics. In particular we use both of these models
to study the generation of high-frequency motions known as inertia-gravity waves
from slow balanced motion via the Stokes phenomenon. To start with we give an
example demonstrating the Stokes phenomenon and to give an idea of the general
approach of these techniques.
We derive a full asymptotic expansion of the solutions of the Scorer equation
which is an inhomogeneous Airy equation. The Stokes phenomenon causes the
particular integral to switch on solutions of the Airy equation as a Stokes line is
crossed. We identify the Stokes lines and the anti-Stokes lines, and most importantly calculate the Stokes multipliers. We do this by studying the Borel-Laplace
transform of the solutions of the homogeneous and inhomogeneous part together
with examining the behaviour of late-terms in the asymptotic expansion of the
particular integral.
We study homoclinic solutions of Lorenz's five-component model. We linearise
the solution and hence we can use methods analogous to the ones mentioned
above. Here the slow motion is represented by the first function in the linear
expansion of the solution. It switches on the consequent term in the expansion
representing the fast motion as a Stokes line is crossed.
By considering solutions to the Boussnesq equations that consist of a sheared
flow and a perturbation we narrow our problem to a differential equation that
governs the amplitude of the inertia-gravity waves. Analytically the particular
integral represents the slow motion that switches on solutions of the homogeneous
equations which represent the fast oscillation. It follows from this that we can
derive a power series expansion of the Stokes multipliers where we derive an
expression for the first terms. The other terms we calculate numerically.
We use the Fourier transform to reduce the Boussinesq equations to an or¬
dinary differential equation. When we then carry out the inverse transform we
use our previous result to replace the the integrand with its dominant behaviour.
As the saddle point of the exponent governing the integrand collides with the
endpoints for certain values we use Bleistein's method together with numerical
integration to estimate the inverse.