Note that APE is independent of depth H, because all the APE is associated with the interfacial slope; only fluid motion in the vicinity of the interface changes potential energy, as is obvious by inspection of Fig. 8.9.
It is interesting to compare the APE associated with the tilted interface in Fig. 8.9 with the kinetic energy associated with the thermal wind current, which is in balance with it. Making use of the Margules formula, Eq. 7-21, the geostrophic flow of the upper layer is given by u2 = g'y/2Q. if the lower layer is at rest. Thus the kinetic energy of the flow per unit length is, on evaluation:
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