Nonlinear and time-dependent equivalent-barotropic flows with topography

Some oceanic and atmospheric flows may be modelled as
equivalent-barotropic systems, in which the horizontal fluid velocity
varies in magnitude at different vertical levels while keeping the same
direction. The governing equations at a specific level are identical to
those of a homogeneous flow over an equivalent depth, determined by a
pre-defined vertical structure. Most oceanic studies using the
equivalent-barotropic approach are focused on steady, linear
formulations. In this work, the nonlinear, time-dependent model with
variable topography is examined. To include nonlinear terms, we assume
suitable approximations and evaluate the associated error in the
dynamical vorticity equation. The model is solved numerically to
investigate the equivalent-barotropic dynamics in comparison with a
purely barotropic flow. We consider two problems in which the behaviour
of homogeneous flows has been well-established either experimentally or
analytically in past studies. First, the nonlinear evolution of cyclonic
vortices around a topographic seamount is examined. It is found that the
vortex drift induced by the mountain is modified according to the
vertical structure of the flow. When the vertical structure is abrupt,
the model effectively isolates the surface flow from both inviscid and
viscous topographic effects (due to the shape of the solid bottom and
Ekman friction, respectively). Second, the wind-driven flow in a closed
basin with variable topography is studied (for a flat bottom this is the
so-called Stommel problem). For a zonally uniform, negative wind-stress
curl in the homogeneous case, a large-scale, anticyclonic gyre is formed
and displaced southward due to topographic effects at the western slope
of the basin. The flow reaches a steady state due to the balance between
topographic, β, wind-stress and bottom friction effects. However, in the
equivalent-barotropic simulations with an abrupt vertical structure,
such an equilibrium cannot be reached because the forcing effects at the
surface are enhanced, while bottom friction effects are reduced. As a
result, the unsteady flow is decomposed as a set of planetary waves.