The primary aim of this thesis is to understand the dynamics of oceanic mesoscale eddies induced
by a localised ‘plunger’ forcing in the rotating shallow-water model. Intended to be an elementary
representation of transient eddy forcing in the ocean, the properties of this forcing are motivated
by three-layer shallow-water double-gyre dynamics. The forcing is imposed on a zonal background
flow and we consider the linear flow response and its ‘quasi-nonlinear’ self-interaction. Aside from
the aim of developing the theory of shallow-water Rossby waves, we are motivated by the need to
be prepared for the development of stochastic parameterisations of oceanic mesoscale eddies.
The plunger forcing induces a rich spectrum of Rossby waves that redistribute potential vorticity
(PV) and zonal momentum, the properties of which have a strong and significant dependence
on the background flow. For a uniform background flow, the plunger drives northward PV flux
convergence, equivalent to a convergence of zonal momentum fluxes at the forced latitudes. This
is due to outward meridional propagation of Rossby waves which transport westward momentum
away from their source. The amplitude of the PV/zonal momentum redistribution has a robust
dependence on the direction and speed of the background flow, with, for example, maximal net-
northward PV flux convergence for a weak westward background flow. The amplitude of PV/zonal
momentum redistribution can be attributed to two factors: (1) resonance between the plunger and
the available Rossby wave spectrum, and (2) efficiency of nonlinear self-interaction.
For a Gaussian-jet background flow the flow response and quasi-nonlinear self-interaction are more
complicated due to the non-trivial background PV gradient. In this case we concentrate on the role
of plunger latitude, and show that results can be explained by considering Rossby wave refractive
index theory, critical layers and turning latitudes.