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A.7 Gravity waves in a vertical shear flow

Here we give a simple derivation of the dispersion relation for internal gravity waves in the presence of a vertical differential rotation in order to illustrate the phenomenon of critical layers (Section 8.4). For further elaboration see, e.g., Andrews et al. (1987); Fritts et al. (1998); Staquet and Sommeria (2002Jump To The Next Citation Point); MacGregor (2003).

We begin with the anelastic equations of Appendix A.2, neglecting rotation, magnetic fields, and dissipation. Furthermore, we consider only 2D flows in the equatorial plane, setting vh and its latitudinal derivatives equal to zero. We then linearize the anelastic equations about a zonal flow which may vary with radius: ^ U0(r)f. Equations (40View Equation) and (41View Equation) then become

1--@-( 2 ) --1---@mf- r2@r r mr + rsinh @f = 0, (102)
-- 2 @mr-+ U-@mr- - rU-0 = - @P--- rg, (103) @t r @f r2 @r
@m U @m @P ---f + -----f + S0mr = - ---, (104) @t r @f @f
-- @S U @S dS --- + ----- = - vr---, (105) @t r @f dr
where -- mr = rvr, -- mf = rvf, and S is the vertical shear:
1-@-- S0 = r @r (rU0) . (106)
The equation of state, Equation (43View Equation), remains unchanged. The final term on the left-hand-side is the centrifugal force associated with the imposed zonal flow and may be eliminated by redefining the reference state pressure in order to balance it. The next step is to make the WKB approximation, assuming the wavelength is much less than the radius of the Sun and much less than the pressure scale height. We define local coordinates x and z such that dx = rdf and dz = - dr, and we expand all variables in terms of Fourier modes oc exp [i(kxx + kzz - st)]. Substitution into Equations (102View Equation)-(105View Equation) then yields
kzmz + kxmx = 0, (107)
-- i(s - kxU0) mz = kzP + -rg S, (108) CP
i(s - kxU0) mx + mzS0 = kxP, (109)
-- -i (s - kxU0)S = vzdS/dr. (110)
Combining these three equations into a single dispersion relation yields
( ) (s - kxU0)2 k2x + k2z - N 2k2x = i(s - kxU0) S0kxkz, (111)
where -- N 2 = (g/CP )dS/dr. If the Doppler-shifted frequency, s- kxU0, is nonzero, and if the vertical wavelength is much smaller than the scale of the shear, then we may neglect the right-hand-side of Equation (112View Equation). The result may then be expressed as
s - kxU0 = N cos y, (112)
where y is the angle that the wavevector makes with the vertical:
( ) k2x 1/2 cosy = ± -2----2- . (113) kx + kz
Rearranging Equation (112View Equation) yields an expression for the vertical wavenumber:
( ) 2 2 ----N-2----- kz = kx 2 - 1 . (114) (s- kxU0)
Often it is assumed that k2z» k2x in which case the last term in Equation (114View Equation) may be neglected, giving
k2 N 2 -z2-~ ----------2. (115) kx (s - kxU0)
As the wave approaches a critical layer where s - kxU0 --> 0, the vertical wavenumber increases without bound. In the presence of a toroidal magnetic field, Equation (114View Equation) becomes
( 2 ) k2z = k2x --------N----------- 1 , (116) (s - kxU0)2 - v2Ak2x
where vA is the Alfvén speed (Barnes et al., 1998).

The phase velocity and group velocity implied by Equation (112View Equation) are given by

( ) sk- |kx| k-- cp = k2 = kxU0 + N k k2 (117)
( ) -@s- @s-- N-|kx|- k2z- cg = @k ^x + @k ^z = U0^x + k3 k ^x - kz^z , (118) x z x
where k = kx^x + kzz^ and k = (k2x + k2z)1/2. In a stationary medium (U0 = 0), the phase velocity and group velocity are perpendicular: cp.cg = 0. Furthermore, the vertical components of cp and cg are of the opposite sign. The energy propagation and the ray path of the wave are along the direction defined by the group velocity (e.g., Staquet and Sommeria, 2002).

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