Topographic wave radiation and modon decay

Abstract

The effect of topographic wave radiation on isolated eddies is modelled by an f-plane modon propagating parallel to an infinitely long escarpment. It is assumed that the lengthscale of the modon is much smaller than the lengthscale on which the topographic waves evolve. This enables the linearised equations of motion to be solved and the asymptotic (far-field, large-time) bchaviour of the topographic wave field is subsequently described. A steady wave-like response is found when the modon moves within the range of possible topographic wave phase speeds. An anomalous case exists when the modon speed is the same as the long topographic wave group speed. This implies that energy cannot escape from the vicinity of the modon and consequently the response eventually becomes nonlinear. It is shown that the appropriate equation to describe the evolution of topographic waves in this case is a forced KdV equation.In the cases of the steady linear response and the nonlinear response (for positive forcing), the energy flux carried by the topographic waves is calculated. Conservation of energy and Lagrangian conservation of vorticity extrema, along with a slowly varying assumption concerning the evolution of the modon, are used to show that the radius and speed of the modon decrease exponentially. The e-folding time associated with the decay is shown to be of possible significance when compared to typical ocean eddy lifetimes.

The motion of a singular vortex near an escarpment

“…Condition (3.9) is the requirement that the fluid be at rest far from the escarpment, while (3.10) and (3.11) are obtained by integrating the governing equations across the escarpment. Similar linear initial-boundary value problems for the escarpment topology have previously been considered by, among others, Johnson (1985), Johnson & Davey (1990) and McDonald (1992McDonald ( , 1996, where details of the derivation of the boundary conditions and the solution procedure can be found. The solution to this problem is obtained through standard Fourier transform methods.…”

The motion of an intense vortex near topography

“…As demonstrated in McDonald (1996) the large time response is dominated by waves of zero phase velocity, i.e. near the poles of the steady part of (33).…”

“…It is well known that the linearized version of (1) with escarpment topography has free topographic wave solutions with phase velocities c p in the range − 6 c p 6 0 (see e.g. Johnson & Davey 1989;McDonald 1996). An immediate consequence is that, whatever the initial value of L, at large times the vortex must resonate with the topographic wave field, i.e.…”

“…As in Johnson & Davey (1989) and McDonald (1996), the solution (33) consists of a steady, localized disturbance and an unsteady term representing propagating topographic wave modes. It should be noted that for y = 0 the solution (32) satisfies ∇ 2 η − η = 0 which is further justification for ignoring the Jacobian terms in (29).…”

“…it moves with a velocity which matches a possible topographic wave phase speed. Such waves have group velocity c g = c p (McDonald 1996) and so energy must be lost through topographic wave radiation. The response of the vortex to this energy loss is the subject of the next section.…”