= Stratification–rotation analogy
Uniformly stratified, nonrotating, two-dimensional ideal <Boussinesq equations> in coordinates $(x,z)$ are equivalent to unstratified rotating equations independent of $Y$, with $(X,Y,Z)=(z,y,-x)$, <rotation> $2\boldsymbol\Omega=N\mathbf e_Z$, <velocity> $(U,V,W)=(w,\sigma/N,-u)$, and the same reduced <pressure>. The transverse rotating <velocity> represents <buoyancy perturbation>, while the rotation-axis <velocity> represents the stratified horizontal <velocity>. Matching prescribed forces may have components in the invariant plane, but no transverse component. Physical <free surface> conditions are not preserved: the physical vertical direction and its normal <velocity> are different under the mapping.
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