The inertial instability of equatorial shear flows is studied, with a view to understanding observed phenomena in the Earth's stratosphere and mesosphere. The basic state
is a zonal flow of stratified fluid on an equatorial β-plane, with latitudinal shear. The
simplest self-consistent model of the instability is used, so that the basic state and
the disturbances are zonally symmetric, and a vertical diffusivity provides the scale
selection. We study the interaction between the inertial instability, which takes the
form of periodically varying disturbances in the vertical, and the mean flow, where
‘mean’ is a vertical mean.
The weakly nonlinear regime is investigated analytically, for flows with an arbitrary
dependence on latitude. An amplitude equation of the form dA/dt = A−k2A∫[mid ]A[mid ]2dt
is derived for the disturbances, and the evolving stability properties of the mean
flow are discussed. In the final steady state, the disturbances vanish, but there is a
persistent mean flow change that stabilizes the flow. However, the magnitude of the
mean flow change depends strongly on the initial conditions, so that the system has a
long memory. The analysis is extended to include the effects of Rayleigh friction and
Newtonian cooling, destroying the long-memory property.
A more strongly nonlinear regime is investigated with the help of numerical
simulations, extending the results up to the point where the instability leads to
density contour overturning. The instability is shown to lead to a homogenization of
fQ¯ around the initially unstable region, where f is the Coriolis parameter, and Q¯ is
the vertical mean of the potential vorticity. As the instability evolves, the line of zero
Q¯ moves polewards, rather than equatorwards as might be expected from a simple
self-neutralization argument.