= Strict geostrophic balance
{title2=$2\boldsymbol\Omega\times\mathbf u=-\nabla\pi$}
For a homogeneous <incompressible flow>, strict <geostrophic balance> means that the <Coriolis acceleration> is exactly balanced by the <pressure gradient>, after conservative body-force potentials have been absorbed into the pressure per unit density $\pi$. It neglects <material acceleration> rather than merely assuming that its size is small. With constant vertical rotation, $\pi_z=0$ and taking the <curl> gives $\mathbf u_z=0$, the <Taylor–Proudman theorem>. Nonzero cross-contour motion over sloping rigid boundaries requires small departures from this strict limit.
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