Much of the utility of arises from its conservation properties, which we will now derive following Müller (1995) (see also Pedlosky, 1987). The equations of motions for a compressible fluid may be expressed as follows:
With a little manipulation, the curl of Equation (86) yields the vorticity equation:
We emphasize that the general form of Ertel’s theorem, Equation (91) follows directly from the fundamental Equations (85)-(87) with no additional assumptions. It is, therefore, valid throughout the solar interior. However, the concept of potential vorticity is of little use in the convection zone where entropy contours are chaotic and convoluted and the dynamics are not strongly constrained by Equation (91). Its importance lies mainly in the lower tachocline where the entropy gradient is predominantly radial and is approximately proportional to the vertical vorticity.
To appreciate the implications of Equation (91), consider a layer in the lower tachocline bounded by two isentropic surfaces and . Without further approximation, Equation (91) may be written in flux form as (Müller, 1995)
If we neglect the Lorentz force and dissipation, then we have local as well as global potential vorticity conservation, as expressed by Equation (92). We now linearize this equation in order to illustrate some of the wave modes it supports. We consider the simplest case in which the density is constant across the layer bounded by and and the entropy gradient is also constant and entirely radial. In this case, the potential vorticity is proportional to the vertical vorticity and the linearized Equation (92) becomes
Another class of Rossby waves occurs when , and therefore the thickness of the layer between and , varies with latitude. Consider the motion of a vertical vortex column in this case. Again, we will assume constant density. If Equation (92) is satisfied, the absolute vorticity in the vortex column, , will be small where the layer thickness is small (large ), and large where the layer thickness is large (small ). In other words, conservation of potential vorticity implies that a vortex column will become taller and narrower when moving toward a thicker part of the layer, spinning up in the process as it tends to conserve its angular momentum. Conversely, when it moves toward a thinner part of the layer it will become shorter and wider and it will spin down. The induced vorticity can act as a restoring force which gives rise to longitudinally-propagating waves analogous to the classical shallow-water Rossby waves discussed in many geophysical textbooks (e.g., Pedlosky, 1987; Tritton, 1988; Vallis, 2005; see also McIntyre, 1998; McIntyre, 2003 for an insightful description of the Rossby wave mechanism). More generally, in a spherical shell, background potential vorticity gradients arising from latitudinal entropy and density variations can modify the Rossby-Haurwitz solutions discussed above to produce Rossby wave modes which possess horizontal divergence.
© Max Planck Society and the author(s)