Dynamical characteristics of decaying Lamb couples

Abstract

A multiple-scale adiabatic asymptotic theory is developed to describe the dissipation of the solitary Lamb couple or modon solutions of the two-dimensional Navier-Stokes equations. The transport equations describing the evolution of the Lamb couple are obtained as solvability conditions for a direct asymptotic expansion assuming a relatively large but finite Reynolds number and are equivalent to globally-integrated leading-order enstrophy and energy balances. The asymptotic theory predicts that the spectral or spatial characteristics of the decaying Lamb couple are temporally invariant and that there is a simple exponential decay in the amplitude and translation speed. We compare the predictions of the theory with a high-resolution numerical simulation. The global and local predictions of the theory and the results of the numerical simulation are in very good agreement. As well, we present a time-series of vorticity-stream function scatter diagrams as derived from the numerical simulation to show that thenon-analytic linear vorticity-stream function relationship is being continuously maintained during the perturbed evolution of the Lamb couple.