8.3 Rotating, stratified turbulence

Penetrative convection and instabilities will induce motions in the lower, stably-stratified portion of the tachocline which will, in general, undergo further nonlinear interactions. The resulting dynamics are likely to be turbulent in nature as a result of the low molecular dissipation. Shear instabilities and gravity wave breaking, in particular, can generate vigorous turbulence (e.g., Townsend, 1976 ; Tritton, 1988

; Staquet and Sommeria, 2002

Turbulence in the lower tachocline will be highly anisotropic due to the strong stable stratification and the large rotational influence. Both effects tend to make the dynamics quasi-2D, but in very different ways. The rotational influence will induce vertical coherence, organizing the flow into vortex columns aligned with the rotation vector (e.g., Bartello et al., 1994 ; Cambon et al., 1997 ). This is another manifestation of the Taylor-Proudman theorem which was also discussed in Section 4.3.2. Meanwhile, stable stratification inhibits vertical flows and tends to decouple horizontal layers, favoring pancake-like vortices with large vertical shear (e.g., Métais and Herring, 1989 ; Riley and Lelong, 2000 ; Godoy-Diana et al., 2004 ). The relative influence of these two competing effects can be gauged by the Rossby deformation radius, $LD$ , defined by

where $N$ is the Brunt-Väisälä frequency. The Rossby number and Froude number are defined as $Ro = U/(2_O_0rt)$ and $Fr = U/(N Dt)$ where $U$ is a characteristic velocity scale. For motions on scales $< L D$ , stratification breaks the vertical coherence induced by rotation (Dritschel et al., 1999 ). In the lower tachocline $LD ~ 5R o.$ so stratification dominates but $LD$ approaches zero at the base of the convection zone.

Two-dimensional turbulence has been studied extensively both theoretically and numerically. It is now well known that nonlinear interactions involving triads of wavevectors in 2D turbulence conserve enstrophy (vorticity squared) as well as energy, and that this gives rise to an inverse cascade of energy from small to large scales (e.g., Lesieur, 1997 ; Pope, 2000 )²⁰ . This is in stark contrast to 3D turbulence which exhibits a forward cascade of energy from large to small scales where dissipation occurs. The inverse cascade is manifested as small vortices interact and coalesce into larger vortices. The inverse cascade in 2D turbulence will proceed to the largest scales unless some mechanism suppresses it, such as surface drag in the oceans and atmosphere. Another mechanism for halting the inverse cascade which is more relevant for solar applications occurs in geometries which admit Rossby waves such as rotating spherical shells or $b$ -planes. If the rotation is rapid enough, patches of vorticity can propagate as Rossby wave packets and disperse before they coalesce. Since the phase speed of a Rossby wave increases with the wavelength (see Appendix A.6), this occurs only for wavenumbers below a critical value $kb$ , often referred to as the Rhines wavenumber after Rhines (1975 ). At scales above $k-b1$ , the flow has a Rossby-wave character and at scales below $k-1 b$ , it has the character of 2D turbulence.

The most notable thing about the arrest of the inverse cascade by Rossby wave dispersion is that it is anisotropic (Rhines, 1975 ; Vallis and Maltrud, 1993 ). Low latitudinal wavenumbers are suppressed, but the cascade can proceed to low longitudinal wavenumbers²¹ . This tends to produce banded zonal flows as observed in the jovian planets (Yoden and Yamada, 1993 ; Nozawa and Yoden, 1997 ; Huang and Robinson, 1998 ; Danilov and Gurarie, 2004 ). Similar processes also occur in shallow-water and two-layer systems, in both freely decaying and forced configurations (Panetta, 1993 ; Rhines, 1994 ; Cho and Polvani, 1996a ,b ; Peltier and Stuhne, 2002 ; Kitamura and Matsuda, 2004 ). The number of bands, or jets, is roughly given by $Ro- 1/2$ . Taking $-1 U ~ 10 m s$ yields $Ro ~ 0.004$ in the solar tachocline, which implies as many as 15 jets. Does a quasi-2D inverse cascade occur in real 3D flows? It does in the so-called quasi-geostrophic limit first studied by Charney (1971 ). He showed that in the limit of strong stratification and rapid rotation ( $2 Fr « Ro « 1$ ), nonlinear interactions conserve potential enstrophy (potential vorticity squared; see Appendix A.6) as well as energy, again giving rise to an inverse cascade of energy (see also Salmon, 1978 ; Vallis, 2005 ). This has been demonstrated in 3D simulations by Métais et al. (1996 ). However, the quasi-geostrophic limit does not strictly apply to global-scale motions in spherical shells. It is plausible that similar dynamics occur in spherical systems but this has not yet been rigorously demonstrated.

Thus far in our discussion we have neglected magnetic fields, which can have a profound influence on self-organization processes in turbulent fluids. MHD turbulence does not conserve enstrophy even in the 2D limit, so there is nothing to inhibit a forward cascade of kinetic energy (e.g., Biskamp, 1993 ; Kim and Dubrulle, 2002 ). An inverse cascade does occur, but it involves a different ideal invariant, namely magnetic helicity (or in 2D, the magnetic potential). Thus, the physical mechanisms described above which can create banded zonal flows probably do not operate globally in the tachocline, although related dynamics likely occur in relatively field-free regions. Self-organization in MHD turbulence generally proceeds by creating large-scale magnetic structures which can then feed back on mean flows through the Maxwell stress.

Another important factor in a tachocline context is the presence of rotational shear imposed by large-scale stresses from the overlying convective envelope. If the turbulence is itself driven by instabilities of this rotational shear, one may expect it to have a diffusive influence, extracting energy from the shear flow by reducing its amplitude. One may also expect a diffusive behavior if the turbulence is small-scale, isotropic, and homogeneous across horizontal surfaces. In other words, if there is a scale separation with local turbulent mixing. Alternatively, if the flow is dominated by waves, one might expect non-local transport which is in general non-diffusive (e.g., McIntyre, 1998 , 2003 ).

The influence of an imposed differential rotation on 2D turbulence in a $b$ -plane was studied by Shepherd (1987 ). He found that the shearing of vortices by the differential rotation substantially altered the nonlinear transfer rates among spectral modes. In forced-dissipative simulations, small-scale turbulent motions tended to extract energy from the mean shear but the shear-induced Reynolds stress from the larger-scale wave field ( $k < k b$ ) tended to amplify the mean flow. The net transfer between the mean flow and the fluctuations about it depended sensitively on the parameters of the problem. Shepherd concluded that this complex interaction could not be modeled with a simple linear parameterization, diffusive or otherwise. More recent simulations by Williams (2003 ) in 2D spherical shells have also shown that the interaction between Rossby wave turbulence and horizontal shear flows can act either to suppress or enhance the shear, depending on the particular details of the problem.

Research into the interaction between a shear flow and 3D, stably-stratified turbulence has focused mainly on the case of non-rotating Cartesian domains with vertical shear. Here an important parameter is the Richardson number $2 Ri = (N/S)$ where $S$ is the mean shear (cf. Section 8.2). At small $Ri$ (shear-dominated), the turbulent transport of momentum and buoyancy tends to be down-gradient (diffusive) but at large $Ri$ (buoyancy-dominated), turbulent transport is generally oscillatory and can be persistently counter-gradient (Holt et al., 1992 ; Galmiche et al., 2002 ; Jacobitz, 2002 ). These studies are based on numerical simulations of freely-evolving (decaying) turbulence with homogeneous and isotropic initial conditions and an imposed shear. An effective time-dependent viscosity and diffusivity can be defined based on the instantaneous turbulent fluxes and the mean gradients as shown in Figure 26 . Counter-gradient transport is manifested as a negative turbulent viscosity after about 2.5 shear timescales in the strongly-stratified simulation (Figure 26 , panel b). Although oscillatory, counter-gradient transport is a robust result of these numerical experiments, it may be a consequence of how they are set up; turbulent fluctuations are sheared by a mean flow which is switched on at some arbitrary time. An analysis in terms of rapid distortion theory by Hanazaki and Hunt (2004 ) suggests that the counter-gradient fluxes become very weak as $Ri --> oo$ and may be absent altogether in statistically steady flows.

Figure 26:

Results are shown from simulations of freely-evolving stably-stratified turbulence with imposed shear. The effective turbulent viscosity $nut$ (black lines) and turbulent thermal diffusivity $kappat$ (red lines) are plotted as a function of time for simulations with vertical shear (solid lines) and horizontal shear (dashed lines). The time is normalized with respect to the shear rate, $- 1 | \~/ U |$ , and $nut$ and $kt$ are normalized with respect to the molecular values. Frames (a) and (b) correspond respectively to moderately stratified ( $Ri = 0.2$ ) and strongly stratified ( $Ri = 2$ ) cases (from Jacobitz, 2002

Jacobitz (2002 ) also considered the case of horizontal shear in a vertically stratified domain. In this case the turbulent transport was generally down-gradient (diffusive) even for strong stratification (Figure 26

). Similar conclusions were also reached by Miesch (2003

) who found down-gradient horizontal angular momentum transport and counter-gradient vertical transport in simulations of rotating, stably-stratified turbulence in thin spherical shells with random external forcing.

Counter-gradient transport in stably-stratified flows is often associated with the presence of waves (although this is not the only mechanism, (e.g., Holt et al., 1992 ; Galmiche and Hunt, 2002 )). Waves carry pseudo-momentum which is conserved until they dissipate, giving rise to long-range transport as described in Section 8.4.

Magnetic fields generally tend to induce down-gradient momentum transport in turbulent shear flows by suppressing upscale kinetic energy transfer (cf. inverse cascades) and by imposing rigidity via magnetic tension. However, the transport efficiency can be reduced due to the partial offset of Reynolds and Maxwell stresses, which often have opposite senses (e.g., Kim et al., 2001 ). Magnetic fields can also suppress turbulent magnetic diffusion (Cattaneo and Vainshtein, 1991 ; Yousef et al., 2003 ). Still, magnetism can also have non-diffusive effects. For example, the balance between the Lorentz and Coriolis forces in toroidal field bands can induce zonal jets (see Section 8.2).

In summary, turbulent transport and self-organization in the tachocline is complex and not well understood. A variety of processes can act to establish or to suppress mean flows. Which of these prevail will depend on the subtleties of how the tachocline couples to the convection zone and radiative interior, a topic which will likely occupy researchers for many years to come.