Taylor–Proudman theorem
In fluid mechanics, the Taylor–Proudman theorem (after Geoffrey Ingram Taylor and Joseph Proudman) states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity formula_1, the fluid velocity will be uniform along any line parallel to the axis of rotation. formula_1 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 formula_4 is the fluid velocity, formula_5 is the fluid density, and formula_6 the pressure. If we assume that formula_7 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 formula_1 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 (because the curl of the gradient is always equal to zero). Note  that formula_14 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: In coordinates for which formula_16, the equations reduce to if formula_18. 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 (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 formula_19 axis, often taken to be the formula_20 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. Proudman published another derivation in 1916  and Taylor in 1917, then the effect was demonstrated experimentally by Taylor in 1923.