8.4 Internal waves

8.4 Internal waves

Waves are ubiquitous in rotating, stratified flows. In the tachocline, they may be driven by penetrative convection (Section 8.1), shear, or instabilities (Section 8.2). Restoring forces may be provided by buoyancy (gravity waves), the Coriolis force (Rossby and other inertial waves; see Appendix A.6), magnetic tension (Alfvén waves), or some combination of the three²² . We will refer to these modes collectively as internal waves. Linear, non-dissipative waves cannot redistribute momentum in a time-averaged sense. However, waves can redistribute momentum if they dissipate by wave breaking or by thermal or viscous diffusion. Thus, waves induce a momentum transport from regions of excitation to regions of dissipation which is, in general, long-range (non-local) and can be counter-gradient (non-diffusive). There are multiple examples of wave-driven flows in the Earth’s atmosphere where such non-local momentum transport is reasonably well-established (McIntyre, 1998

; Shepherd, 2000 ; Baldwin et al., 2001

).

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 ( $N - 1$ ) of a few hours then rotation may be neglected. For illustration, we consider a Cartesian domain defined such that $^x$ and $^y$ are the local longitude and latitude coordinates and $z^$ is the height (antiparallel to $g$ ). Of particular interest in a tachocline context is the interaction of gravity waves with a vertical shear. The dispersion relation for small-wavelength internal gravity waves in a vertically-sheared zonal flow, $U0(z)^x$ is

where $s$ is the frequency, $kx$ is the component of the wave vector in the direction of the shear and $y$ is the angle it makes with the horizontal (see Appendix A.7). The direction of phase propagation is given by the angle $y$ but in a stationary medium ( $U = 0 0$ ), the group velocity is perpendicular to the phase velocity (Appendix A.7). The highest-frequency waves have $s = N$ and have a horizontal phase velocity ( $y = 0o$ or $180o$ ).

The intrinsic frequency of the wave, $s$ 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, $U0$ . For illustration, we will assume $U0 > 0$ . If the zonal phase speed of the wave is parallel to the mean flow ( $skx > 0$ ), the wave may encounter a critical layer where the Doppler-shifted frequency $s - kxU0$ approaches zero. The resulting dynamics are illustrated in panel a of Figure 27 . In a solar context, the vertical coordinate $z$ may be regarded as increasing downward, with $z = 0$ at the base of the convection zone.

Figure 27:

Resulting dynamics when an internal gravity wave encounters (a) a critical layer $zc$ and (b) a trapping plane $yt$ , indicated by dashed lines (see text). Curved lines represent ray paths while thin and bold arrows indicate the wavevector $k$ and the group velocity including advection by the background flow $c' = c + U ^x g g 0$ where $c = @s/@k g$ . Ray paths are everywhere parallel to $c' g$ . In (a) the zonal velocity gradient is vertical, $U0(z)$ and the perspective shows a longitude-depth ( $x$ , $z$ ) plane. Two ray paths are shown. As each wave asymptotically approaches $zc$ the vertical wavenumber increases and the group velocity becomes parallel to $x^$ . In (b) the zonal velocity gradient is latitudinal $U0(y)$ and the perspective shows a horizontal ( $x$ , $y$ ) plane. The $k$ and $c' g$ vectors are shown at several points along a single ray path. As $y t$ is approached, $c' g$ again becomes parallel to $^x$ (from Staquet and Sommeria (2002

); see also Staquet and Huerre (2002

)).

As the wave approaches the critical layer $zc$ , its vertical wavenumber increases and its group velocity slows, making it more susceptible to viscous and thermal diffusion (see Appendix A.7). If it is not dissipated first by diffusion, the wave will increase in amplitude and eventually break before encountering the critical layer. Thus, there is generally a transfer of momentum from waves to the mean flow near a critical layer, a phenomenon which is often referred to as critical layer absorption.

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 $y$ , the local latitudinal coordinate, $U0(y)^x$ . If a wave propagates horizontally against the mean flow, it may encounter a trapping plane at $yt$ where the Doppler-shifted frequency approaches the Brunt-Väisälä frequency, $N$ . The horizontal group velocity again approaches the mean flow speed, and the latitudinal wavenumber, $ky$ increases without limit according to WKB theory. The wave will again break or dissipate by thermal or viscous diffusion before $y t$ 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 $z$ ) while lower layers (higher $z$ ) decelerate again as a result of the reduced wave flux. In this way, oscillating zonal flows can be established which are analogous to the Quasi-Biennial Oscillation (QBO) in the Earth’s stratosphere (Baldwin et al., 2001 ).

Wave-driven 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 $U0(z)$ , in which momentum transport by radiatively-damped gravity waves is offset by viscous diffusion. Two waves were included in the model, prograde and retrograde, with horizontal velocities parallel and anti-parallel 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 quasi-periodic, 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 counter-gradient angular momentum transport, accelerating the mean flow.

View Image

Figure 28:

An oscillating zonal flow driven by gravity waves is shown based on the two-wave model described by Kim and MacGregor (2001 ) and MacGregor (2003

). The left column illustrates the zonal velocity $u$ as a function of height $z$ at several instants in the evolution, with time increasing downward as indicated. The right column illustrates the corresponding rate of change of $u$ induced by prograde waves (solid lines), retrograde waves (dashed lines), and viscous dissipation (dotted lines). All quantities are normalized with respect to a characteristic velocity and vertical length scales $u0$ and $H0$ . As time proceeds, waves propagating with the same sense as $u$ accelerate the flow in such a way that velocity extrema shift upward (toward the right) while new extrema appear deeper down. Vertical dotted lines in the left column are included as a reference point to illustrate the phase of the oscillation (courtesy K. MacGregor).

The selective dissipation of waves with horizontal phase speeds parallel to the mean zonal flow acts as a filtering mechanism, removing these modes from the wave field. This filtering is latitude-dependent, since the radial angular velocity gradient in the tachocline varies from positive values at the equator to negative values at the poles (Section 3.1). Fritts et al. (1998

) argue that the momentum redistribution resulting from this inhomogeneous wave filtering will establish a residual meridional circulation which may have implications for chemical transport and the low abundance of Lithium in the solar envelope relative to cosmic abundances. Chemical transport by gravity waves has also been studied by other authors from the perspective of light-element depletion in stars, and is often parameterized in terms of an effective diffusion (Montalbán, 1994 ; Schatzman, 1996 ; Pinsonneault, 1997 ).

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 Doppler-shifted frequency no longer vanishes in the critical layer; rather, it approaches $± kxvA$ where $vA$ 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, over-reflection 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 ).