Geostrophic adjustment and wave-vortex interaction in rotating shallow water

Nonlinear, rotational, and boundary effects in the geostrophic adjustment process are examined using a rotating f-plane shallow water model started from dam break initial conditions. Nonlinearity is shown to lead to the spontaneous generation of shocks, but their subsequent decay is inevitable and increases with nonlinearity and/or rotation. Long-time numerical integrations demonstrate the approach to the geostrophically-balanced end state and further show that the time required for adjustment to complete is in general, in this problem, long. When fluid is located initially only on one side of the dam and confined to a finite width channel, it adjusts by forming a wedge-like nose, along one channel wall, that travels at a speed in excess of the nonrotating dam break counterpart. Steady states are calculated; the steady mass transport in the channel decreases as rotation increases.; As a direct application, geostrophic adjustment is shown to be a dynamical mechanism to create cyclone/anticyclone asymmetry. It is shown that symmetric pairs of unbalanced initial conditions will adjust and eventually lose their symmetry at the balanced end state. The asymmetry increases with the initial imbalance and reaches a relative maximum for initial conditions on the scale of the deformation radius. 2-D numerical time-integrations confirm theoretically predicted end states, establish a relatively fast time scale for adjustment to complete, and elucidate adjustment scenarios not amenable to theoretical analysis (non-axisymmetric cases). Irreversible potential vorticity change is shown to be a byproduct of adjustment; the fast inertio-gravity wave component of the flow is conjectured to be responsible for the change.; This conjecture is investigated by considering a “wave-vortex” interaction problem: plane inertio-gravity waves impinging on a geostrophically balanced flow. A small amplitude asymptotic analysis in both one and two spatial dimensions clearly demonstrates a time-dependent, nonlinear, wave-vortex interaction. Numerical integrations extend this result to larger wave and vortex amplitude regimes. But at long times, after the waves have propagated away, it is shown that the vortex remains essentially unchanged from its initial state. Thought experiments show, however, that it would be too simplistic to conclude from this last result that there is no wave-vortex interaction.