Due to its buoyant nature, penetrative convection is particularly efficient at exciting gravity waves. These are, in general, influenced by the Coriolis force (i.e., they are inertial gravity waves) but if their period is close to the buoyancy period () of a few hours then rotation may be neglected. For illustration, we consider a Cartesian domain defined such that and are the local longitude and latitude coordinates and is the height (antiparallel to ). Of particular interest in a tachocline context is the interaction of gravity waves with a vertical shear. The dispersion relation for smallwavelength internal gravity waves in a verticallysheared zonal flow, is
where is the frequency, is the component of the wave vector in the direction of the shear and is the angle it makes with the horizontal (see Appendix A.7). The direction of phase propagation is given by the angle but in a stationary medium (), the group velocity is perpendicular to the phase velocity (Appendix A.7). The highestfrequency waves have and have a horizontal phase velocity ( or ).The intrinsic frequency of the wave, is set by the wave generation process, for example the timescale which characterizes penetrative convection. As the wave propagates vertically, this frequency is Doppler shifted by the background flow, . For illustration, we will assume . If the zonal phase speed of the wave is parallel to the mean flow (), the wave may encounter a critical layer where the Dopplershifted frequency approaches zero. The resulting dynamics are illustrated in panel a of Figure 27. In a solar context, the vertical coordinate may be regarded as increasing downward, with at the base of the convection zone.

Similar dynamics can also occur in the presence of a horizontal shear, as illustrated in panel b of Figure 27. In this case we have a zonal flow which depends on , the local latitudinal coordinate, . If a wave propagates horizontally against the mean flow, it may encounter a trapping plane at where the Dopplershifted frequency approaches the BruntVäisälä frequency, . The horizontal group velocity again approaches the mean flow speed, and the latitudinal wavenumber, increases without limit according to WKB theory. The wave will again break or dissipate by thermal or viscous diffusion before is reached, inducing a net momentum flux from the source region of the waves to the vicinity of the trapping plane. The nonlinear breaking of internal gravity waves near a trapping plane and the associated mass and momentum transport has recently been modeled numerically by Staquet and Huerre (2002).
In the Sun, waves are unlikely to dissipate solely by critical layer absorption (or the analogous process near a trapping plane). Rather, they dissipate mainly by radiative diffusion. Still, the processes discussed above give some insight into the resulting momentum transport. In the presence of a prograde zonal flow with vertical shear, a prograde wave will have a lower vertical group velocity and a higher vertical wave number than a retrograde wave. Thus, the prograde wave will be more readily dissipated by thermal diffusion even if it does not encounter a critical layer. The net result is a convergence of prograde momentum which acts to accelerate the mean flow. As the zonal velocity increases, Doppler shifts are amplified and waves travel shorter distances before they are dissipated. The region of convergence moves upward (toward lower ) while lower layers (higher ) decelerate again as a result of the reduced wave flux. In this way, oscillating zonal flows can be established which are analogous to the QuasiBiennial Oscillation (QBO) in the Earth’s stratosphere (Baldwin et al., 2001).
Wavedriven flows such as these in the solar tachocline have been studied by several authors (Fritts et al., 1998; Kumar et al., 1999; Kim and MacGregor, 2001, 2003; Talon et al., 2002). Kim and MacGregor (2001, 2003), in particular, considered a simple 1D model for a zonal flow with vertical shear , in which momentum transport by radiativelydamped gravity waves is offset by viscous diffusion. Two waves were included in the model, prograde and retrograde, with horizontal velocities parallel and antiparallel to the mean flow, respectively. As the turbulent viscosity was decreased, the temporal response of the resulting zonal flow underwent a transition from stationary to periodic, to quasiperiodic, and eventually to chaotic. A periodic solution is illustrated in Figure 28. When only a single wave was included in the presence of a background shear, the solutions were stationary and tended to produce countergradient angular momentum transport, accelerating the mean flow.

Waves which are not filtered out by shear or other processes in the tachocline will propagate deeper into the solar interior. Eventually, these waves too will dissipate, resulting in an exchange of angular momentum between the convective envelope and the radiative interior. In a steady state the net transport must vanish but over evolutionary timescales the Sun is not steady. Rather, the solar envelope is continually losing angular momentum via the solar wind. In this situation, Talon et al. (2002) argue that gravity waves will systematically extract angular momentum from the radiative interior over the lifetime of the Sun. The resulting coupling between the convection zone and radiative interior may help to explain why the mean rotation rate of these two regions is comparable (Section 3.1).
The dynamical influence of a toroidal magnetic field on gravity wave propagation is similar in some ways to that of a zonal flow. Here a magnetic critical layer exists where the horizontal group velocity of the wave approaches the Alfvén speed relative to the mean flow (Barnes et al., 1998; McKenzie and Axford, 2000; MacGregor, 2003). This is analogous to a hydrodynamic critical layer in that the vertical wavenumber increases without bound but the dynamics in the vicinity of the critical layer can be notably different. The presence of a toroidal field significantly limits the range of wavenumbers which can propagate without becoming evanescent. The Dopplershifted frequency no longer vanishes in the critical layer; rather, it approaches where is the Alfvén speed. If the field is strong, waves are Alfvénic in nature and propagate along the field lines. Gravity waves may therefore be absorbed by the critical layer (dissipated) or they may be converted to Alfvén modes which propagate horizontally. Such filtering by strong toroidal fields in the tachocline may greatly enhance the shear filtering described above (Kim and MacGregor, 2003).
Shear and magnetic fields not only filter waves by selective dissipation, but they can also reflect waves. In some cases, overreflection can occur wherein there is a net transfer of energy from the field or shear to the waves. This can increase the amplitude of a wave to the point of nonlinear breaking. Since gravity waves are evanescent in the convection zone, wave reflection by angular velocity shear and toroidal fields in the lower tachocline may essentially create a waveguide, channeling gravity and Alfvén waves into a narrow horizontal layer, where they eventually dissipate by wave breaking or radiative diffusion (MacGregor, 2003).
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