= Reduced equations for an anisotropic rotating fluid
{title2=$D_\perp\nabla_\perp^2\Phi=2\Omega\partial_z w,\quad D_\perp w=-2\Omega\partial_z\Phi$}
Under <critical balance in a rapidly rotating fluid>, define $\mathbf u_\perp^{(0)}=\widehat{\mathbf z}\times\nabla_\perp\Phi$ 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.
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