In fluid mechanics, the Taylor–Proudman theorem (after G. I. 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 , 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 now make the assumption that is scalar potential and the advective term may be neglected (reasonable if the Rossby number is much less than unity) and that the flow is incompressible (density is constant) then the equations become:
To derive this, one needs the vector identities
(because the curl of the gradient is always equal to zero). Note that 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:
Now choose coordinates in which and then the equations reduce to
if . Note that the implication is that all three components of the velocity vector are uniform along any line parallel to the z-axis.
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.
- The Taylor-Proudman theorem was first derived by Sydney Samuel Hough (1870-1923), a mathematician at Cambridge University. See: Hough, S.S. (January 1, 1897). "On the application of harmonic analysis to the dynamical theory of the tides. Part I. On Laplace's "oscillations of the first species," and on the dynamics of ocean currents". Phil. Trans. R. Soc. Lond. A. 189: 201–257. Bibcode:1897RSPTA.189..201H. doi:10.1098/rsta.1897.0009.
- Taylor, G.I. (March 1, 1917). "Motion of solids in fluids when the flow is not irrotational". Proc. R. Soc. Lond. A. 93: 92–113. Bibcode:1917RSPSA..93...99T. doi:10.1098/rspa.1917.0007.
- Proudman, J. (July 1, 1916). "On the motion of solids in a liquid possessing vorticity". Proc. R. Soc. Lond. A. 92: 408–424. Bibcode:1916RSPSA..92..408P. doi:10.1098/rspa.1916.0026.