Reduced equations for an anisotropic rotating fluid ($D_{\perp }{\nabla }_{\perp }^2\Phi =2\Omega {\partial }_{z}w,D_{\perp }w=-2\Omega {\partial }_z\Phi$)

Under critical balance in a rapidly rotating fluid, define ${\mathbf{u}}_{\perp }^{(0)}=\hat{\mathbf{z}}\times {\nabla }_{\perp }\Phi$ and $D_{\perp }={\partial }_t+\{\Phi ,\cdot \}$. Leading horizontal incompressibility gives zero divergence, but at the next order ${\nabla }_{\perp }\cdot {\mathbf{u}}_{\perp }^{(1)}=-{\partial }_{z}w$. Its Coriolis acceleration contribution survives in the vertical vorticity equation, producing the displayed coupled system. The vertical momentum equation uses the leading pressure $p_0=2\Omega \Phi$. A flow that stays purely horizontal is independent of $z$ in this reduced system; the assertion is not an exact theorem for arbitrary fast solutions of the full rotating Euler equations.