Rossby Solitary Wave in the Presence of a Critical Layer 
P. Caillol & R. H. Grimshaw 
Department of Mathematical Sciences, Loughborough University, UK 
Abstract 
This study considers the evolution of long Rossby waves in a horizontally sheared zonal current. 
We restrict attention to a stable flow so that the nonlinear time scale is long. These assumptions 
will enable the flow to organize itself into a large-scale coherent structure in the r/’egime where a 
competition sets in between weak nonlinearity and weak dispersion. This balance is often described 
by a Korteweg-de-Vries equation. The traditional assumption of a weak amplitude breaks down 
when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of 
a singularity in the leading order equation, which strongly modifies the flow in a critical layer. Here 
nonlinear eects are invoked to resolve this singularity, since the relevant geophysical flows are of high 
Reynolds number flows. Viscosity is also introduced in order to render the nonlinear critical layer 
solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic 
expansions, this inner flow will be matched at the edges of the critical layer with the outer flow. 
We will show that the critical-layer induced flow leads to a strong rearrangement of the related 
streamlines and consequently of the potential vorticity contours, particularly in the neighbourhood 
of the separatrices bounding the open and closed streamlines. The symmetry of the critical layer vis- 
`a-vis the critical level is also broken. This theory is thus relevant for the phenomenon of Rossby wave 
breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described 
by a Korteweg-de-Vries equation which is strongly modified by functionals of the amplitude and its 
temporal derivative which depend on the critical-layer shape, that is depression or elevation waves. 
These additional terms are made necessary at a certain order of the asymptotic expansion while 
matching the inner flow on the dividing streamlines. The new evolution equation is not in general 
integrable, but supports a family of solitary waves.