8.2 Instabilities

Penetrative convection occupies only the upper portion of the tachocline, if it overlaps at all (see Section 3.2). The lower portion of the tachocline is convectively stable. However, a variety of other instabilities are likely to occur, driven by shear, buoyancy, and magnetism.

Shear instabilities have been well studied for many years in light of their important geophysical and engineering applications. Undular perturbations in the direction of the mean velocity gradient grow by extracting kinetic energy from the shear flow, eventually overturning and spreading into a turbulent mixing layer. If the shear is vertical, such perturbations are suppressed by sub-adiabatic stratification to a degree which may be quantified by the Richardson number, $Ri = (N/ |dU/dz |)2$ . If $Ri > 0.25$ , the vertical shear is hydrodynamically stable¹⁷ . For the lower tachocline, $N ~ 10-3$ s and $S ~ 10 -6 s- 1$ (see Section 3.2), implying very large Richardson numbers $6 Ri ~ 10$ . Vertical shear instabilities should therefore be strongly suppressed. In the overshoot region, $N$ , and therefore $Ri$ , is much smaller, approaching zero at the base of the convection zone. Taking into account the destabilizing influence of thermal diffusion, Schatzman et al. (2000 ) investigated this problem and concluded that the vertical shear may be hydrodynamically unstable near the base of the convection zone at $r = 0.713Ro .$ , but that this region of instability is confined to low latitudes and does not extend deeper than $r ~ 0.695$ . Note that this is a global constraint; stably-stratified flows may still exhibit intermittent turbulence¹⁸ even if $Ri » 1$ due to wave breaking and horizontal layering which can drive the local Richardson number below 0.25 (Anders Pettersson Reif et al., 2002 ; Fritts et al., 2003 ; Petrovay, 2003 ; Hanazaki and Hunt, 2004 ). Note also that magnetism and baroclinicity may act to destabilize the vertical shear. We will return to this issue toward the end of this section. Although the angular velocity gradient in the tachocline is mainly vertical (Section 3.2), stratification does little to suppress horizontal shear instabilities so we might expect that the latitudinal component of the differential rotation is more likely to be unstable. In the absence of magnetic fields, the latitudinal differential rotation will be linearly unstable if the corresponding latitudinal potential vorticity gradient (see Appendix A.6) changes sign somewhere in the domain of interest. This is a variation of Fjortoft’s criterion for a stably-stratified flow, which is in turn related to Rayleigh’s well-known inflexion-point criterion (e.g., Knobloch and Spruit, 1982 ; Vallis, 2005 ). Nonlinear stability is another matter; a shear flow which is linearly stable may still be unstable to finite-amplitude perturbations, particularly at high Reynolds numbers (a familiar example is pipe flow; see Drazin and Reid (1981 ); Tritton (1988 ); Richard and Zahn (1999 )).

In light of the extremely large Reynolds numbers in the solar interior (Section 5.1), Zahn (1992 , 1994 ) has argued that the latitudinal differential rotation should be hydrodynamically unstable to finite-amplitude perturbations. If efficient enough, this nonlinear instability may suppress the latitudinal shear entirely, leading to a state of shellular rotation in which angular velocity is independent of latitude. However, due to the possibly insurmountable difficulties of a complete nonlinear stability analysis, these are mainly empirical arguments based on analogies with laboratory flows. Linear analyses indicate that the latitudinal differential rotation in the tachocline is marginally stable to 2D (latitude/longitude) hydrodynamic perturbations (Charbonneau et al., 1999b ) and perhaps only weakly unstable to 3D perturbations near the base of the convection zone (Dikpati and Gilman, 2001c ; Cally, 2003 ). Furthermore, these linear instabilities saturate readily, mixing potential vorticity only enough to smooth local extrema and thus stabilize the flow (Garaud, 2001 ). It appears then that linear hydrodynamic instabilities, even if they occur, are far too weak to establish uniform rotation on horizontal surfaces. However, the addition of even a weak magnetic field profoundly changes everything.

In a series of papers, Gilman and collaborators have shown that the combination of latitudinal differential rotation and a toroidal field in the tachocline is linearly unstable to 2D perturbations for a wide range of field amplitudes and configurations, from broad distributions which occupy an entire hemisphere to localized bands of flux which span only a few degrees of latitude (Gilman and Fox, 1997 , 1999a ,b ; Dikpati and Gilman, 1999 ; Gilman and Dikpati, 2000 ). Possible modes of instability for a toroidal band are illustrated in Figure 24 . Similar modes of instability also occur for broad fields.

Figure 24:

Schematic illustration of (a) $m = 0$ , (b) $m = 1$ , and (c) $m = 2$ instabilities for a toroidal band of flux on a 2D spherical surface in the presence of a latitudinal differential rotation (from Dikpati et al., 2004

A band of toroidal flux will experience a magnetic tension force which will tend to make it contract and move toward the poles (Figure 24

, panel a). This is the poleward slip instability first studied using the thin flux tube approximation (Spruit and van Ballegooijen, 1982 ). In perfectly conducting, 2D, incompressible flow this axisymmetric mode (longitudinal wavenumber $m = 0$ ) is excluded because of mass conservation; the ring cannot push fluid uniformly poleward. However, the ring can tip as shown in panel b of Figure 24

. This is the $m = 1$ tipping instability and it generally has the largest growth rate for solar parameter regimes, with timescales of order a few months¹⁹ . Higher-wavenumber instabilities may also occur for weak fields ( $< 104 G$ ) which deform the ring as shown in panel b of Figure 24

. Unstable modes grow by extracting energy from the differential rotation or from the magnetic energy of the initial toroidal field, the latter of which becomes significant only for strong fields. The nonlinear saturation and evolution of these 2D instabilities was investigated by Cally (2001 ) and Cally et al. (2003

). It was found that for broad fields, the tipping instability could lead to several different behaviors depending on the relative phases of the northern and southern hemispheres. If they tip out of phase, this leads to a clam-shell instability in which field lines spread out one side of the shell and reconnect on the other, eventually achieving a poloidal configuration. If the tipping occurs in phase, oscillatory solutions are possible in which field lines remain parallel and no reconnection occurs. The clam-shell instability does not occur for banded field profiles, but bands do tip, eventually equilibrating at a tilt angle which increases with the latitude of the initial band (high-latitude bands tip more).

There is little evidence for clam-shell patterns and highly tilted toroidal field bands in the Sun so it is interesting to explore possible mechanisms which may suppress or alter these instabilities. One possibility is that the instabilities may not be as efficient for the more complex toroidal field profiles which are likely to exist in the Sun. Cally et al. (2003 ) found one mixed profile in particular with low-latitude toroidal bands superposed on a broad field which did not exhibit a clam-shell instability. Another suppression mechanism may arise from the coupling of adjacent horizontal layers by turbulent mixing. This was recently incorporated into the 2D calculations of Dikpati et al. (2004 ) as an effective kinetic and magnetic drag. Results indicated that the clam-shell instability was indeed suppressed for large magnetic drag in particular, but that the tipping instabilities for toroidal bands still equilibrated at tilt angles comparable to the nondiffusive cases.

An efficient mechanism for suppressing the poleward slip instability as well as the tipping instability of a toroidal band arises if the band possesses a coincident prograde zonal jet which provides a gyroscopic inertia (Rempel et al., 2000 ; Dikpati et al., 2003 ). Such a jet could be established by conservation of angular momentum in a band which begins to slip poleward and is then stabilized. The resulting centrifugal force can fully or partially balance the latitudinal component of the magnetic tension force in an equilibrium state, with the remaining contribution coming from pressure gradients. Jet formation is indeed observed in nonlinear simulations and contributes to a net flattening of the differential rotation profile (Cally et al., 2003 ). This flattening is achieved mainly by the Maxwell stress, which transport angular momentum poleward as a result of shear-induced correlations; $< ' '> B hB f$ .

Subsequent work has shown that similar instabilities also occur in quasi-2D systems under the shallow-water (SW) and thin-shell approximations discussed in Section 5.4 (Dikpati and Gilman, 2001c ; Gilman and Dikpati, 2002 ; Dikpati et al., 2003 ; Cally, 2003 ; Gilman et al., 2004 ). Results again indicate that the tachocline differential rotation is in general unstable and that the $m = 1$ tipping instability is typically the dominant mode for hydromagnetic perturbations. An additional hydrodynamic mode is also present which may be unstable throughout much of the tachocline even in the absence of magnetic fields (Dikpati and Gilman, 2001c ). Although formally allowed, the $m = 0$ poleward slip instability of a toroidal flux band is suppressed by a restoring pressure force which arise as mass is pushed toward the poles, tending to deform the upper boundary into a prolate shape (Dikpati and Gilman, 2001b ).

Growth rates for the $m = 1$ and $m = 2$ SW modes of a toroidal band are shown in Figure 25 for parameter values characteristic of the overshoot region and lower tachocline ( $G$ is the reduced gravity and $s$ is the fractional angular velocity contrast between the equator and pole). In the overshoot region, weak bands ( $4 < 10 G$ ) are unstable at all latitudes. For stronger fields, mid-latitude bands are stabilized by a zonal jet but bands at low and high latitudes remain unstable. In the radiative zone, bands at nearly all field strengths considered are stable at low latitudes but unstable at higher latitudes. Strong bands at all latitudes are stabilized by a zonal jet but weak mid-latitude bands remain unstable.

Figure 25:

Growth rates for magnetic shear instabilities are plotted as a function of the initial latitude (vertical axes) and field strength (horizontal axes) of a toroidal band. Shaded areas indicate instability (growth rates for one or more modes $> 0.0025$ ). The left and right columns correspond to parameter regimes characteristic of the overshoot region and lower tachocline, respectively. The lower plots represent cases in which a zonal jet contributes to the initial force balance as discussed in the text. Cases represented in the upper plots have no such jet. Contour lines represent $m = 1$ and $m = 2$ symmetric (S) and antisymmetric (A) modes as indicated. The nondimensional model is normalized such that a growth rate of 0.01 corresponds to an e-folding growth time of 1 year. The parameter $s$ is the fractional angular velocity contrast between equator and pole and, in our notation, the reduced gravity $( -- ) 2 2 G = d S/dr (g(rt)d )/(2CP _O_ 0)$ (from Dikpati et al., 2003 ).

Using a thin-shell model, Cally (2003 ) has identified a polar twist instability in which high-latitude toroidal loops lift and twist out of the horizontal plane. This is a different type of $m = 1$ instability which does not occur in 2D systems and which can exhibit large growth rates (e-folding timescales of months). However, the polar twist instability only operates at high field strengths ( $> 105 G$ ) and large vertical wavenumbers where it may be suppressed by turbulent diffusion. Furthermore, a poloidal field component may stabilize toroidal flux structures near the poles by essentially forming a twisted tube aligned with the rotation axis.

The magneto-shear instabilities studied by Gilman, Fox, Dikpati, and Cally are concerned with the joint instability of latitudinal differential rotation and strong toroidal fields which are thought to exist in the solar tachocline. They are likely related to the toroidal field instabilities described by Tayler (1973 ), Acheson (1978 ), and Spruit (1999 ) but a precise link has not yet been established. Other classes of hydrodynamic and magnetohydrodynamic (MHD) shear instabilities are also likely to operate in the tachocline and radiative interior. Notable among these is the magneto-rotational instability (MRI) described by Velikhov (1959 ), Chandrasekhar (1961 ) and Balbus and Hawley (1991 ) and applied to stellar interiors by Balbus and Hawley (1994 ). This instability is thought to generate vigorous turbulence in accretion disks which plays an essential role in the global angular momentum balance (Balbus and Hawley, 1998 ).

Unlike the quasi-2D instabilities studied by Gilman, Fox & Dikpati, the MRI operates mainly on relatively weak poloidal fields which tether axisymmetric rings of fluid to a particular point in the meridional plane. When these rings are perturbed, magnetic tension tends to resist shearing by the differential rotation. If the angular velocity decreases outward from the rotation axis ( $@_O_/@c < 0$ ), the resulting torques act to amplify the perturbations, leading to instability. When applied to the radiative interior of the Sun, (Balbus and Hawley, 1994 ) found that the instability was mainly confined to horizontal surfaces by the subadiabatic stratification, producing equatorward angular momentum transport which tends to drive the system toward shellular rotation. Toroidal fields are also subject to MRI as long as the perturbations allow for a poloidal component. However MRI cannot occur in strictly 2D spherical shells so it is distinct from the Gilman-Fox-Dikpati instabilities even in the toroidal field case. Furthermore, the MRI does not operate in regions where $@_O_/@c > 0$ or in the equatorial plane where buoyancy resists motions perpendicular to the rotation axis. The MRI criterion $@_O_/@c < 0$ is more limiting than its hydrodynamic analogue, the Rayleigh instability criterion, which states that a differential rotation profile is unstable if the specific angular momentum decreases outward: $@L/@c < 0$ (e.g., Knobloch and Spruit, 1982 ).

As we have discussed, buoyancy in the subadiabatic radiative interior generally has a stabilizing influence on vertical shear but they can also have a destabilizing effect in the presence of rotation and magnetic fields. Rotation can induce baroclinicity, which refers to a state in which isosurfaces of constant density and pressure do not coincide. Fluid particles can tap the gravitational potential energy in such a configuration if they are allowed to move horizontally as well as vertically, in effect circumventing the Schwarzschild criterion for convective stability which applies only to vertical gradients. If a vertical shear is in thermal wind balance as is likely in the lower tachocline (Section 4.3.2), it may be subject to baroclinic instabilities. Such instabilities represent the main driver of weather systems on the Earth despite the large atmospheric Richardson numbers which suggest that the vertical shear would be stable in the absence of baroclinic effects (e.g., Vallis, 2005 ). Baroclinic instability in a stellar context was studied by Spruit and Knobloch (1984 ) who concluded that it is probably only significant very near the base of the convection zone where the stratification is relatively weak and where more standard shear instabilities may also occur. However, this work predated the discovery of the tachocline and should perhaps be revisited.

Cally (2000 ) has argued that a strong uniform toroidal field can further stabilize the vertical shear in a stably-stratified medium. However, if the field strength decreases with height then the fluid is top-heavy and is susceptible to magnetic buoyancy instabilities. Such instabilities likely play an essential role in tachocline dynamics but they have been comprehensively reviewed elsewhere in these volumes by Fan (2004 ), so we will not address them again here. We merely note that although shear can inhibit magnetic buoyancy instabilities (Tobias and Hughes, 2004 ), it can also induce them by forming concentrated magnetic structures (Brummell et al., 2002a ; Cline et al., 2003a ,b ).

The presence of a small but finite thermal, magnetic, and viscous diffusion can also induce secular instabilities such as the Goldreich-Schubert-Fricke (GSF) instability (Knobloch and Spruit, 1982 ; Menou et al., 2004 ). These generally operate either on small spatial scales or on long temporal scales so they have little bearing on global-scale dynamics which occur over the course of a solar activity cycle. However, they may play a role in tachocline confinement (Section 8.5). Secular instabilities and rotational shear instabilities may also be important for chemical mixing in the radiative interior and light-element depletion in the solar envelope (Zahn, 1994 ; Pinsonneault, 1997 ; Barnes et al., 1999 ; Mathis and Zahn, 2004 ).