= Taylor–Proudman theorem
{c}
{title2=$\partial_{\parallel}\mathbf u=0$}
For homogeneous, incompressible, inviscid flow with constant rotation vector and negligible inertia, taking the <curl> of $2\boldsymbol\Omega\times\mathbf u=-\nabla p$ gives $(\boldsymbol\Omega\cdot\nabla)\mathbf u=0$. Thus a <geostrophic flow> is independent of distance along the rotation axis. Body forces modify the relation by their <curl>; it applies unchanged in unforced exterior regions. A nontrivial exterior column cannot both obey this leading balance and decay along the rotation axis.
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