Abstract
We examine the late-time evolution of an inviscid zonally symmetric shallow-water flow on the surface of a rotating spherical earth. An arbitrary initial condition radiates inertia–gravity waves that disperse across the spherical surface. The simpler problem of a uniformly rotating (f-plane) shallow-water flow on the plane radiates these waves to infinity, leaving behind a nontrivial steady flow in geostrophic balance (in which the Coriolis acceleration balances the horizontal hydrostatic pressure gradient). This is called “geostrophic adjustment.” On a sphere, the waves cannot propagate to infinity, and the flow can never become steady due to energy conservation (at least in the absence of shocks). Nonetheless, when energy is conserved a form of adjustment still takes place, in a time-averaged sense, and this flow satisfies an extended form of geostrophic balance dependent only on the conserved mass and angular momentum distributions of fluid particles, just as in the planar case. This study employs a conservative numerical scheme based on a Lagrangian form of the rotating shallow-water equations to substantiate the applicability of these general considerations on an idealized aqua-planet for an initial “dam” along the equator in a motionless ocean.
Highlights
Geostrophic adjustment theory describes the intricate manner in which an arbitrary initial rotating stratified flow transforms to a final, steady, and geostrophically balanced state
The simpler problem of a uniformly rotating (f-plane) shallow-water flow on the plane radiates these waves to infinity, leaving behind a nontrivial steady flow in geostrophic balance
When energy is conserved a form of adjustment still takes place, in a time-averaged sense, and this flow satisfies an extended form of geostrophic balance dependent only on the conserved mass and angular momentum distributions of fluid particles, just as in the planar case
Summary
Geostrophic adjustment theory describes the intricate manner in which an arbitrary initial rotating stratified flow transforms to a final, steady, and geostrophically balanced state. The Lagrangian formulation is useful numerically for studying the transient flow evolution, as conservation of mass, angular momentum, and potential vorticity on each fluid particle can be exactly satisfied at all times. The Lagrangian approach is unique in that no high-order viscous terms are required to stabilize the numerical solution This is of primary importance when studying the nonlinear rotating shallow-water equations in spherical coordinates that struggle with the coordinate singularity at the poles.[9,10]. The zonal symmetry means that there is an exact balanced state, a completely stationary state, entirely determined by the Lagrangian mass and angular momentum distributions These distributions determine the potential vorticity distribution for zonal flows.
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