Inertial instability in two-layer flows

Inertial instability occurs in rotating fluid systems when the absolute vorticity takes the oppositesigntotheCoriolisparameterf. Ithasbeenobservedinatmospheresandoceans, particularly near the equator where the latitudinal shear of the zonal winds can exceed f. Most previous studies of inertial instability adopt a continuously stratified fluid, for which the instability takes the form of overturning cells in the meridional plane. Here we instead study the instability using the zonally symmetric two-layer shallow water equations. We use a momentum-conserving interfacial friction, and show that the linear instability problem is then directly analogous to that of the continuously stratified system inthelimitofinfinitePrandtlnumber. Solutionsforlinearinstabilitiesforauniformshear flow on the equatorial beta plane are given in detail.
We then study frictionless nonlinear instabilities, using both weakly nonlinear theory and numerical solutions. On the equatorial β-plane, a third-order system of amplitude equations is derived, and their behaviour is verified and then extended into a moderately nonlinear regime numerically. On the f-plane, the nonlinear instability of a hyperbolic tangent shear flow is studied. Here the weakly nonlinear analysis requires a different scaling to the equatorial case, and the resulting system of amplitude equations is also different. The properties of this system are studied in depth, and the periodic oscillations that result are interpreted in terms of the evolving linear stability of the mean flow. The results are extended into a moderately nonlinear regime numerically.