Rotating fluid

A rotating fluid is described in a rotating reference frame, where the Coriolis acceleration and centrifugal acceleration supplement the physical forces.

For homogeneous, incompressible, inviscid flow with constant rotation vector and negligible inertia, taking the curl of $2\mathbf{\Omega }\times \mathbf{u}=-\nabla p$ gives $(\mathbf{\Omega }\cdot \nabla )\mathbf{u}=0$. 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.

In nearly columnar flow between rigid boundaries, a geostrophic contour is a contour of column height divided by the Coriolis parameter. If relative vorticity is negligible, conservation of shallow-water potential vorticity constrains leading geostrophic flow to these contours. Motion across them stretches or squashes columns, generating relative vorticity; for constant $f$, the contours are simply contours of $H$.

For a layer rotating at angular speed $\Omega$, of depth $d$ and kinematic viscosity $\nu$, the Taylor number compares the Coriolis force to viscous damping through the squared ratio $(2\Omega d^2/\nu {)}^2$. Definitions in other geometries may include different geometric factors; this is the plane-layer convention.

The traditional approximation retains only the locally vertical component of planetary rotation in the Coriolis force. It gives the horizontal acceleration $f\hat{\mathbf{z}}\times {\mathbf{u}}_h$ and neglects terms involving the horizontal component of the rotation vector and vertical velocity.

The Coriolis parameter is $f=2\Omega \sin \phi$ for planetary rotation rate $\Omega$ and latitude $\phi$. It controls the vertical component of the Coriolis acceleration acting on horizontal flow.

An f-plane approximates the Coriolis parameter by a constant $f_0$ over a limited horizontal region.

A beta plane retains the leading meridional variation $f=f_0+\beta y$ of the Coriolis parameter.

An inertial oscillation is the horizontal circular motion of an unforced parcel in a rotating fluid. On an f-plane, $\dot{u}-fv=0$ and $\dot{v}+fu=0$, so the speed is constant and the angular frequency is $|f|$.

On a beta plane, the turning rate of an inertial oscillation is larger on the poleward side of its orbit than on the equatorward side. For initial speed $V$ and $\beta V/f_0^2\ll 1$, this asymmetry produces westward mean drift of magnitude $\beta V^2/(2f_0^2)$.