An example of soliton behaviour in a rotating baroclinic fluid

Abstract

Pedlosky (1972 a ) has derived a set of envelope equations which describe the nonlinear evolution of wave-packets in a marginally stable or unstable baroclinic shear flow using a quasi-geostrophic two-layer model on a beta-plane. We show that these envelope equations can be transformed either into the sine-Gordon equation ( ϕ ξז = sin ϕ ) for real amplitude or the so-called self-induced transparency (s. i. t.) equations of nonlinear optics for complex amplitude. The initial value problem for both equations can be solved analytically by use of the inverse scattering method and it is well known that a variety of soliton solutions exist. This two-layer system is therefore a new example of a physical system exhibiting soliton behaviour whose evolution is predictable for all time. Solutions of the sine-Gordon equation on a spatially periodic domain are more relevant for comparison with annulus experiments. Numerical integrations suggest that initial data recurs almost periodically in time thereby giving qualitative agreement with results from rotating fluid annulus experiments. The significance of such solutions to meteorology and oceanography is discussed.