The process of nonlinear geostrophic adjustment in the presence of a boundary (i.e.
in a half-plane bounded by a rigid wall) is examined in the framework of a rotating
shallow water model, using an asymptotic multiple-time-scale theory based on the
assumed smallness of the Rossby number ε. The spatial scale is of the order of the
Rossby scale. Different initial states are considered: periodic, ‘step’-like, and localized.
In all cases the initial perturbation is split in a unique way into slow and fast
components evolving with characteristic time scales f−1 and (εf)−1, respectively. The
slow component is not influenced by the fast one, at least for times t [les ] (fε)−1, and
remains close to geostrophic balance. The fast component consists mainly of linear
inertia–gravity waves rapidly propagating outward from the initial disturbance and
Kelvin waves confined near the boundary.
The theory provides simple formulae allowing us to construct the initial profile of
the Kelvin wave, given arbitrary initial conditions. With increasing time, the Kelvin
wave profile gradually distorts due to nonlinear-wave self-interaction, the distortion
being described by the equation of a simple wave. The presence of Kelvin waves
does not prevent the fast–slow splitting, in spite of the fact that the frequency
gap between the Kelvin waves and slow motion is absent. The possibility of such
splitting is explained by the special structure of the Kelvin waves in each case
considered.
The slow motion on time scales t [les ] (εf)−1
is governed by the well-known quasigeostrophic potential vorticity equation for the elevation. The theory provides an
algorithm to determine initial slow and fast fields, and the boundary conditions to
any order in ε. For the periodic and step-like initial conditions, the slow component
behaves in the usual way, conserving mass, energy and enstrophy. In the case of a
localized initial disturbance the total mass of the lowest-order slow component is
not conserved, and conservation of the total mass is provided by the first-order slow
correction and the Kelvin wave.
On longer time scales t [les ] (ε2f)−1
the slow motion obeys the so-called modified
quasi-geostrophic potential vorticity (QGPV) equation. The theory provides initial
and boundary conditions for this equation. This modified equation coincides exactly
with the ‘improved’ QGPV equation, derived by Reznik, Zeitlin & Ben Jelloul (2001),
in the step-like and localized cases. In the periodic case this equation contains an
additional term due to the Kelvin-wave self-interaction, this term depending on the
initial Kelvin wave profile.