In
fluid mechanics, the Taylor–Proudman theorem (after
Geoffrey Ingram Taylor and
Joseph Proudman) states that when a solid body[clarification needed] 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.
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 (reasonable if the
Rossby number is much less than unity) and that the
flow is incompressible (density is constant), the equations become:
where is the
angular velocity vector. If the
curl of this equation is taken, the result is the Taylor–Proudman theorem:
The
Taylor column is an imaginary cylinder projected above and below a real cylinder that has been placed parallel to the rotation axis (anywhere in the flow, not necessarily in the center). 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 (1870-1923), a mathematician at Cambridge University, in 1897.[1]: 506 [2] Proudman published another derivation in 1916 and Taylor in 1917, then the effect was demonstrated experimentally by Taylor in 1923.[3]: 648 [4]: 245 [5][6]