Taylor–Proudman theorem


In fluid mechanics, the Taylor–Proudman theorem states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity, the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.

Derivation

The Navier–Stokes equations for steady flow, with zero viscosity and a body force corresponding to the Coriolis force, are
where is the fluid velocity, is the fluid density, and the pressure. If we assume that is a scalar potential and the advective term on the left may be neglected and that the flow is incompressible, the equations become:
where is the angular velocity vector. If the curl of this equation is taken, the result is the Taylor–Proudman theorem:
To derive this, one needs the vector identities
and
and
.
Note that is also needed.
The vector form of the Taylor–Proudman theorem is perhaps better understood by expanding the dot product:
In coordinates for which, the equations reduce to
if. Thus, all three components of the velocity vector are uniform along any line parallel to the z-axis.

Taylor column

The Taylor column is an imaginary cylinder projected above and below a real cylinder that has been placed parallel to the rotation axis. The flow will curve around the imaginary cylinders just like the real due to the Taylor–Proudman theorem, which states that the flow in a rotating, homogeneous, inviscid fluid are 2-dimensional in the plane orthogonal to the rotation axis and thus there is no variation in the flow along the axis, often taken to be the axis.
The Taylor column is a simplified, experimentally observed effect of what transpires in the Earth's atmospheres and oceans.

History

The result known as the Taylor-Proudman theorem was first derived by Sydney Samuel Hough, a mathematician at Cambridge University, in 1897. Proudman published another derivation in 1916 and Taylor in 1917, then the effect was demonstrated experimentally by Taylor in 1923.