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A.6 Potential vorticity and Rossby waves

Much insight into the nature of rotating, stratified flows can be obtained by considering the concept of potential vorticity. Although it may be unfamiliar to many astrophysicists, it has been used extensively in geophysical applications for some time (e.g., Pedlosky, 1987Jump To The Next Citation PointMüller, 1995Jump To The Next Citation PointMcIntyre, 1998Jump To The Next Citation Point). In a solar physics context, we may define the potential vorticity, TT, as follows:
w . \~/ S TT = --a-----, (84) r
where wa is the absolute vorticity relative to an inertial frame, wa = w + 2_O_0. For the purposes of this section, S and r should be regarded as the total specific entropy and density field, not perturbations relative to a reference state as in Appendix A.2.

Much of the utility of TT arises from its conservation properties, which we will now derive following Müller (1995Jump To The Next Citation Point) (see also Pedlosky, 1987Jump To The Next Citation Point). The equations of motions for a compressible fluid may be expressed as follows:

@r- @t + \~/ .(rv) = 0, (85)
Dv r ----= - \~/ P + r \~/ V - 2r_O_0× v + A, (86) Dt
DS-- Dt = Q, (87)
where the vector A includes the Lorentz force and viscous dissipation and the heating term Q includes thermal diffusion as well as viscous and Ohmic heating. The potential V may include both gravitational and centrifugal forces. The thermodynamic variables S, r, and P should again be regarded as the total specific entropy, density, and pressure. We may also reasonably assume an ideal gas equation of state so [cf. Equation (37View Equation)]
S 1 ( P ) ( r ) --- = --ln --- - ln --- . (88) CP g P0 r0

With a little manipulation, the curl of Equation (86View Equation) yields the vorticity equation:

D \~/ r × \~/ P ---wa = (wa.\ ~/ ) v - wa ( \~/ .v) +-----2----+ \~/ × A. (89) Dt r
If we divide by r and incorporate the continuity Equation (85View Equation) we may rewrite this as
( ) ( ) D-- wa- wa- \~/ r-×- \~/ P- -1 Dt r = r . \~/ v + r3 + r \~/ × A. (90)
We may now take the dot product of Equation (90View Equation) with the entropy gradient \~/ S, and with the help of a few more vector identities, the result may be expressed as:
DTT-= wa-. \~/ Q + r -1 \~/ S.(\ ~/ × A) . (91) Dt r
If we can neglect the Lorentz force and dissipation, then A = Q = 0 and we obtain
DTT ----= 0. (92) Dt
This is known as Ertel’s theorem and states that potential vorticity is conserved following a fluid element. For a thorough discussion of its applicability and implications, see Pedlosky (1987Jump To The Next Citation Point) and Müller (1995Jump To The Next Citation Point).

We emphasize that the general form of Ertel’s theorem, Equation (91View Equation) follows directly from the fundamental Equations (85View Equation)-(87View Equation) 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 (91View Equation). Its importance lies mainly in the lower tachocline where the entropy gradient is predominantly radial and TT is approximately proportional to the vertical vorticity.

To appreciate the implications of Equation (91View Equation), consider a layer in the lower tachocline bounded by two isentropic surfaces S1 and S2. Without further approximation, Equation (91View Equation) may be written in flux form as (Müller, 1995)

@ ---(r TT) = - \~/ .Fpv, (93) @t
where Fpv is the potential vorticity flux
Fpv = r TT v- A× \~/ S - waQ. (94)
The dot product of Equation (94View Equation) with \~/ S yields
@S- Fpv. \~/ S = r TT @t . (95)
Using this result and integrating Equation (93View Equation) over the volume V between surfaces S1 and S2, we find
integral -d r TT dV = 0. (96) dt V
Thus, the total density-weighted potential vorticity, r TT, contained between surfaces S1 and S2 is conserved even in the presence of the Lorentz force and viscous, thermal, and Ohmic dissipation. In the limit where S1 and S2 are infinitesimally close together, the volume element becomes
dS dV = -----dA, (97) | \~/ S |
where dA is an area element lying within the isentropic surface S. Equation (96View Equation) then becomes (McIntyre, 1998Jump To The Next Citation Point2003Jump To The Next Citation Point)
integral b TT dA = 0, (98) S
where b = r/| \~/ S |. Thus, the total potential vorticity on an isentropic surface is conserved with respect to the weighing factor b, which is related to the mass per unit area contained between S1 and S2: dm = b dS.

If we neglect the Lorentz force and dissipation, then we have local as well as global potential vorticity conservation, as expressed by Equation (92View Equation). 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 S1 and S2 and the entropy gradient is also constant and entirely radial. In this case, the potential vorticity TT is proportional to the vertical vorticity and the linearized Equation (92View Equation) becomes

@wr- 2_O_0- @t - r vsinh = 0. (99)
We now consider divergenceless horizontal motions so the velocity is given by a streamfunction Z as follows:
v = \~/ × (Z ^r) . (100)
If we then expand Z in terms of spherical harmonics Y lm and assume a time dependence oc exp(- ist), Equation (99View Equation) yields
s = - -2_O_m---. (101) l(l + 1)
This is the dispersion relation for Rossby-Haurwitz waves which propagate in a retrograde direction with a longitudinal phase speed proportional to the rotation rate _O_ and inversely proportional to l(l + 1), where l is the spherical harmonic degree.

Another class of Rossby waves occurs when @S/@r, and therefore the thickness of the layer between S1 and S2, varies with latitude. Consider the motion of a vertical vortex column in this case. Again, we will assume constant density. If Equation (92View Equation) is satisfied, the absolute vorticity in the vortex column, wa, will be small where the layer thickness is small (large \~/ S), and large where the layer thickness is large (small \~/ S). 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., PedloskyJump To The Next Citation Point, 1987; TrittonJump To The Next Citation Point, 1988; VallisJump To The Next Citation Point, 2005; see also McIntyreJump To The Next Citation Point, 1998; McIntyreJump To The Next Citation Point, 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.

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