The Rotation‐Modified Kadomtsev‐Petviashvili Equation: An Analytical and Numerical Study

Abstract

The rotation-modified Kadomtsev-Petviashvili equation, derived by Grimshaw in 1985, is studied both analytically and numerically to determine the structure of solutions which are initially localized. It is shown that solitary-like waves can be found, whose wavefronts are curved in a direction transverse to the propagation direction, which remain unsteady, and which are always accompanied by trailing Poincare waves. These effects are more noticeable as the effects of rotation are increased.(l.la)

Long-term evolution of strongly nonlinear internal solitary waves in a rotating channel

“…The wave suffered remarkable attenuation due to the radiation of secondary perturbances. Similar results were obtained by Grimshaw and Tang (1990) for continuously stratified fluids.…”

“…This remark concerns basically the formation of a slowly attenuated packet of Kelvin waves as an ultimate state of the evolution of a large-amplitude internal solitary wave. At the first stage of evolution, an initially twodimensional ISW starts to evolve according to the classical scenario described by the RMKP equation (e.g., Katsis and Akylas, 1987;Grimshaw and Tang, 1990). More specifically, it starts to transform into a Kelvin wave (with exponential decay of the wave amplitude across the channel) and its front curved backwards.…”

“…The detailed analysis of the evolution process of ISWs in a rotating channel was performed in a number of theoretical works (Grimshaw, 1985;Katsis and Akylas, 1987;Grimshaw, 1988;Melville et al, 1989;Grimshaw and Tang, 1990;Akylas, 1991). The analysis is normally based on the Rotation Modified Kadomtsev-Petviashvili (RMKP) equation for the wave amplitude A(x, y, t), where x, y and t are along-channel, across-channel coordinate and time, respectively.…”

“…Note that Eqs. (6) and (7) are coupled in the nonlinear terms, and, as pointed out by Grimshaw and Tang (1990), this is the ultimate cause that inhibits the existence of a stationary solution. To illustrate the mechanism, let us suppose the inexistence of Poincaré modes at t=0 (A i =0, i=1, .., n, for t=0).…”

“…Even though the wave profile consistent with the boundary condition (2) is taken as the initial wave, it also starts to evolve according to the scenario described by Katsis and Akylas (1987) and Grimshaw and Tang (1990). One of the conclusions of these papers was that the exponential decrease of the wave amplitude across the channel is not a crucial condition for the wave evolution.…”