Critical balance in a rapidly rotating fluid ($k_{\perp }U\sim 2\Omega |k_{\parallel }|/k_{\perp }$)

Balance the inertial wave rate against horizontal nonlinear advection. The anisotropy $ϵ=|k_{\parallel }|/k_{\perp }$ is then of the same order as the Rossby number $k_{\perp }U/(2\Omega )$. Horizontal geostrophic flow gives kinematic pressure amplitude $\delta p\sim 2\Omega U/k_{\perp }\sim U^2/ϵ$. The first ageostrophic flow is $O(ϵU)$ and supplies the small horizontal divergence balancing $-{\partial }_z{u}_{\parallel }$. Rapid inertial oscillations are excluded by this slow-time ordering.