Taylor–Proudman theorem

Derivation

The Navier–Stokes equations for steady flow, with zero viscosity and a body force corresponding to the Coriolis force, are

${\displaystyle \rho (\mathbf{u}\cdot \mathrm{\nabla })\mathbf{u}=\mathbf{F}-\mathrm{\nabla }p,}$

where ${\displaystyle \mathbf{u}}$ is the fluid velocity, ${\displaystyle \rho }$ is the fluid density, and ${\displaystyle p}$ the pressure. If we assume that ${\displaystyle F=\mathrm{\nabla }\mathrm{\Phi }=-2\rho \mathbf{\Omega }\times \mathbf{u}}$ is a scalar potential and the advective term on the left may be neglected (reasonable if the Rossby number is much less than unity) and that the flow is incompressible (density is constant), the equations become:

${\displaystyle 2\rho \mathbf{\Omega }\times \mathbf{u}=-\mathrm{\nabla }p,}$

where ${\displaystyle \mathrm{\Omega }}$ is the angular velocity vector. If the curl of this equation is taken, the result is the Taylor–Proudman theorem:

${\displaystyle (\mathbf{\Omega }\cdot \mathrm{\nabla })\mathbf{u}=\mathbf{0}.}$

To derive this, one needs the vector identities

${\displaystyle \mathrm{\nabla }\times (A\times B)=A(\mathrm{\nabla }\cdot B)-(A\cdot \mathrm{\nabla })B+(B\cdot \mathrm{\nabla })A-B(\mathrm{\nabla }\cdot A)}$

and

${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }p)=0}$

and

${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }\mathrm{\Phi })=0}$

(because the curl of the gradient is always equal to zero). Note that ${\displaystyle \mathrm{\nabla }\cdot \mathbf{\Omega }=0}$ is also needed (angular velocity is divergence-free).

The vector form of the Taylor–Proudman theorem is perhaps better understood by expanding the dot product:

${\displaystyle {\mathrm{\Omega }}_x\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }x}+{\mathrm{\Omega }}_y\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }y}+{\mathrm{\Omega }}_z\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }z}=0.}$

In coordinates for which ${\displaystyle {\mathrm{\Omega }}_x={\mathrm{\Omega }}_y=0}$, the equations reduce to

${\displaystyle \frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }z}=0,}$

if ${\displaystyle {\mathrm{\Omega }}_z\ne 0}$. Thus, all three components of the velocity vector are uniform along any line parallel to the z-axis.