## The Taylorproudman Theorem

A remarkable property of geostrophic motion is that if the fluid is homogeneous (p uniform) then, as we shall see, the geostrophic flow is two dimensional and does not vary in the direction of the rotation vector, Q. Known as the Taylor-Proudman theorem, it is responsible for the glorious patterns observed in our dye shirring experiment, GFD Lab 0, shown again in Fig. 7.7. We discuss the theorem here and make much subsequent use of it, particularly in Chapters 10 and 11, to discuss the constraints of rotation on the motion of the atmosphere and ocean.

For the simplest derivation of the theorem, let us begin with the geostrophic relation written out in component form, Eq. 7-4. If p and f are constant, then taking the vertical derivative of the geostrophic flow components and using hydrostatic balance, we see that (dug/3z, dvg/3z) = 0; therefore the geostrophic flow does not vary in the direction of fz.

A slightly more general statement of this result can be obtained if we go right back to the pristine form of the momentum equation (Eq. 6-29) in rotating coordinates. If the flow is sufficiently slow and steady (Ro << 1) and F is negligible, it reduces to:

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