Nonlinear and time-dependent equivalent-barotropic flows

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. The idea was proposed by Charney (J. Met., vol. 6 (6), 1949, pp. 371-385) for modelling a barotropic atmosphere. More recently, steady, linear formulations have been used to study oceanic flows. In this paper, 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 three problems in which the behaviour of homogeneous flows has been well established either experimentally, analytically or observationally 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 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 well-known 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, wind-stress and bottom friction effects. However, in the equivalent-barotropic simulations with 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. A third problem consists of performing simulations of the wind-driven flow over realistic bottom topography in the Gulf of Mexico. The formation of the so-called Campeche gyre is explored. It is found that such circulation may be consistent with the equivalent-barotropic dynamics.