The limiting form of zonally asymmetric inertial instability

Atmospheric flows are prone to inertial instability when the potential vorticity, q, takes the opposite sign to the Coriolis parameter, f. In the equatorial atmosphere, this configuration typically occurs due to the latitudinal shear of a zonal flow. Corresponding instabilities, which play an important role in moderating cross-equatorial shear, have been observed in the upper stratosphere near solstice, although they are likely to occur elsewhere too. With this in mind, we study the linear normal-mode instability of a zonal flow for which fq is somewhere negative, but with otherwise arbitrary latitudinal shear. A perturbation analysis is performed in the limit of large vertical wavenumber, yielding asymptotic series for the growth rate and structure of the modes at finite zonal wavenumber k.

When k is zero, the analysis describes the well-known zonally symmetric inertial instability as modes which are highly localised around the minimum of fq. However, the zonally asymmetric modes are more complicated: although they are still highly localised, the latitudinal structure of the most unstable mode is oscillatory, in contrast to the zonally symmetric case. For small k the growth rate always decreases from its zonally symmetric value, whilst for larger k a condition is given for these localised modes to exist. We argue that these modes may justifiably be described as a zonally asymmetric inertial instability.

The results are used to predict the growth rate of inertial instability for some simple shear flows. By comparing with results obtained numerically from the unapproximated linear equations, it is shown that the growth rate can be well predicted by using the first two or three terms of the perturbation expansion, even when the vertical wavenumber is relatively small. For the case of a uniform shear flow on an equatorial β-plane, the corresponding expressions for the growth rate and phase speed are given explicitly for k ≠ 0.

The same kind of ideas can be applied to study the inertial instability of a class of zonally asymmetric basic flows, now describing instabilities that are localised in the cross-stream direction. Time permitting, we will discuss the possible instability of Rossby waves at the equator by such a mechanism, and how this can provide a link between waves of an extratropical origin and the zonal momentum balance.