system further while retaining the leading-order physics of the instability. First, we note that, in general, $\omega\ll\Omega$, as may be judged from the exact results of YG05 or equation (\[omegaR\]), for wavelengths that are not significantly shorter than the radial pressure scale $L$. Equation gives us $$u=(2i\Omega\omega/\kappa^2)v\sim (\omega/\Omega)v.$$ It is then easy to show that in equation the first term (the acceleration term $-i\omega u$) can be neglected. It follows that, to leading order, the $x$ and $y$ equations of motion of the centre of mass relax to a form of *geostrophic balance*, i.e. Coriolis forces effectively cancel out the pressure gradient and the buoyancy force. This is important, as it means that pressure perturbations do not effectively drive centre-of-mass motions in the orbital plane. So this crucial stabilising tendency is consigned to a subdominant role. It also means that the centre of mass executes modified epicycles in the orbital plane, to leading order.

Second, we neglect the $u$ perturbation in the two continuity equations and which we are permitted to do if we restrict ourselves to long radial wavelengths. From equations - we can obtain the following scalings $$u \sim \frac{\omega}{\Omega}\, v \sim 
\mathrm{max}(k_x\frac{\omega}{\Omega^2}\,h, \omega\frac{g_e}{\Omega^2}\,\delta).$$ Combining these with and 