where now H is interpreted as the vertical scale of the motion. The horizontal length scale Eq. 7-23 is known as the Rossby radius of deformation. It is the scale at which the effects of rotation become comparable with those of stratification. More detailed analysis shows that on scales smaller than Lp, the pressure adjusts to the velocity field, whereas on scales much greater than Lp, the reverse is true and the velocity adjusts to the pressure.

For the values of g and Q appropriate to our cylinder collapse experiment, Lp ~ 7 cm if H = 10 cm. This is roughly in accord with the observed slumping scale of our salty cylinder in Fig. 7.16. We shall see in Chapters 8 and 9 that Lp ~ 1000 km in the atmospheric troposphere, and Lp ~ 30 km in the main thermocline of the ocean; the respective deformation radii set the horizontal scale of the ubiquitous eddies observed in the two fluids.

7.3.5. Thermal wind in pressure coordinates

Equation 7-16 pertains to an incompressible fluid, such as water or the ocean. The thermal wind relation appropriate to the atmosphere is untidy when expressed with height as a vertical coordinate (because of p variations). However, it becomes simple when expressed in pressure coordinates. To proceed in p coordinates, we write the hydrostatic relation:

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