On a Theory of Amplitude Vacillation in Baroclinic Waves: Some Numerical Solutions

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  • 1 Department of Mathematics, Monash University, Clayton, Australia 3168
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Abstract

In an earlier paper (Smith, 1977) it is shown that when viscous effects are important only on a time scale much longer than that for incipient wave growth, the amplitude evolution of a marginally unstable baroclinic wave in a two-layer, quasi-geostrophic zonal flow is governed by an infinite system of ordinary differential equations. These equations have a steady solution which under certain conditions is unstable with respect to small perturbations in wave amplitude. In the case where viscous effects are nonzero but are exceedingly small, the asymptotic analysis in Smith (1977) shows that a stable limit cycle solution is also possible and when the steady solution is unstable, an initially incipient wave evolves toward the limit cycle, which represents an amplitude vacillation of the wave.

In this paper, some numerical integrations of the amplitude equations are presented for the case of moderate viscosity. These are compared with solutions obtained from the amplitude equations derived by Pedlosky (1971) in a theory which omits a certain boundary condition an the mean zonal flow (Smith, 1974). Although the two sets of amplitude equations differ considerably, our results confirm the important prediction of Pedlosky that for sufficiently small viscosity and/or if the steady solution is unstable, an incipient wave evolves to a state in which its amplitude undergoes regular pulsations, or vacillations, described by a stable limit cycle solution. However, the parameter range for which the steady solution is unstable is widely different in the two analyses, except for vanishingly small viscosity.

Abstract

In an earlier paper (Smith, 1977) it is shown that when viscous effects are important only on a time scale much longer than that for incipient wave growth, the amplitude evolution of a marginally unstable baroclinic wave in a two-layer, quasi-geostrophic zonal flow is governed by an infinite system of ordinary differential equations. These equations have a steady solution which under certain conditions is unstable with respect to small perturbations in wave amplitude. In the case where viscous effects are nonzero but are exceedingly small, the asymptotic analysis in Smith (1977) shows that a stable limit cycle solution is also possible and when the steady solution is unstable, an initially incipient wave evolves toward the limit cycle, which represents an amplitude vacillation of the wave.

In this paper, some numerical integrations of the amplitude equations are presented for the case of moderate viscosity. These are compared with solutions obtained from the amplitude equations derived by Pedlosky (1971) in a theory which omits a certain boundary condition an the mean zonal flow (Smith, 1974). Although the two sets of amplitude equations differ considerably, our results confirm the important prediction of Pedlosky that for sufficiently small viscosity and/or if the steady solution is unstable, an incipient wave evolves to a state in which its amplitude undergoes regular pulsations, or vacillations, described by a stable limit cycle solution. However, the parameter range for which the steady solution is unstable is widely different in the two analyses, except for vanishingly small viscosity.

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