Although the distinction is sometimes blurred, the tachocline and the overshoot region are empirically two very different things. Whereas the tachocline is defined helioseismically from rotational inversions (Section 3.2), the overshoot region is usually defined in terms of the mean stratification and must be probed instead with structural inversions (Section 3.6). The tachocline encompasses the overshoot region but appears to be wider; whereas the upper tachocline may extend substantially into the convective envelope at high latitudes, the lower tachocline lies below the overshoot region at all latitudes (Section 3.2). What the two have in common is that they are both thin — according to current estimates, the tachocline extends roughly a few percent of the solar radius while the overshoot region occupies less than one percent. Thus, local-area and thin-shell models are particularly useful here (Section 5).
Convective penetration
Due to its wide applicability in astronomy and geophysics, there is a large body of literature on convective penetration. Much of this work, particularly in a solar context, is concerned with the structure of the overshoot region and how the penetration depth varies with the vigor of the convection and the stiffness of the transition from subadiabatic to superadiabatic stratification.
Figure 22 illustrates the structure of the overshoot region at the base of the solar envelope as suggested by Zahn (1991). In the convection zone, the radial entropy gradient, \(d\overline{S}/dr\), is negative but nearly adiabatic due to the efficient mixing of entropy by turbulent convection. The convective enthalpy flux is positive (outward) and the radiative heat flux normalized by the total flux, L⊙/4πr2 is less than unity. To a good approximation, the normalized convective enthalpy flux and radiative heat flux sum to unity, with smaller contributions from the other terms in Equation (3).
In many theoretical studies, the base of the convection zone is defined as the point where \(d\overline{S}/dr\) changes sign and becomes positive (subadiabatic). The inertia of convective downflows takes them beyond this point into the stably-stratified interior. Here the enthalpy flux becomes negative (inward) and the outward radiative flux must increase to compensate. Downward motions will be quickly decelerated by buoyancy but the turbulent mixing may still be efficient enough to establish a nearly adiabatic penetration region where \(d\overline{S}/dr\geq 0\). Eventually, downflows will be decelerated enough such that their effective Péclet number, Pe = UL/κ, becomes small and turbulent mixing becomes inefficient relative to thermal diffusion. This occurs in a thin thermal adjustment layer where the enthalpy flux falls to zero and the stratification becomes strongly subadiabatic. Deeper in the interior, the radiative heat flux carries the entire solar luminosity.
Sometimes a distinction is made between convective overshoot and convective penetration. The former is used to describe any convective motions which are carried into a region of stable stratification by their own inertia. By contrast, the latter term often has a more specific meaning, implying that the convection is efficient enough to establish a nearly adiabatic penetration region as indicated in Figure 22. From the perspective of solar structure modeling and helioseismic probing, it is often more convenient to define the base of the convection zone as the bottom of the well-mixed, nearly adiabatic penetration region rather than where the entropy gradient changes sign.
The presence of a nearly adiabatic penetration region in the Sun is currently a matter of some debate. Although many early models and relatively low-resolution 2D and 3D simulations produced a true penetration region where \(d\overline{S}/dr\geq 0\) (reviewed by Brummell et al., 2002b; Rempel, 2004), recent high-resolution simulations of turbulent penetrative convection by Brummell et al. (2002b) exhibited only strongly subadiabatic overshoot. They attributed the absence of a nearly adiabatic penetration region to the small filling factor of downflow plumes, which dominate the flow field in turbulent parameter regimes (see Section 5.2). However, reduced models based on the dynamics of intermittent plumes suggest that such numerical simulations may exhibit more adiabatic penetration if they could achieve more solar-like parameter regimes (Zahn, 1991; Rempel, 2004). In particular, higher Péclet numbers and a lower imposed heat flux may modify the balance between advective and diffusive heat transport enough to produce a nearly adiabatic stratification.
Another challenge to numerical simulations of penetrative convection is achieving a high stiffness parameter St, which is a measure of the subadiabatic stratification in the stable zone relative to the superadiabatic stratification in the convection zone. In the Sun this ratio is roughly 105 whereas simulations consider values of at most 10–100. Thus, the depth of penetration, Δp, in simulations is artificially high and much work has focused on establishing scaling relations between Δp and S in order to extrapolate the results to solar conditions. Analytic estimates by Hurlburt et al. (1994) suggest that the extent of the nearly adiabatic penetration region, if present, scales as \(S_{\rm t}^{-1}\) whereas the depth of the thermal adjustment layer scales as \(S_{\rm t}^{-1/4}\). Numerical simulations are generally consistent with these scaling estimates (Hurlburt et al., 1994; Singh et al., 1995; Brummell et al., 2002b). However, Rogers and Glatzmaier (2005a) have recently achieved stiffness values of over 500 in high-resolution simulations of 2D penetrative convection and they find a much shallower scaling law, \(\Delta_{\rm p}\sim S_{\rm t}^{-0.04}\) for \(S_{\rm t}\geq 10\). When extrapolated to solar conditions, most simulations and models imply penetration depths ranging from about 0.01–1 pressure scale heights HP, implying a Δp of a few percent of the solar radius or less (see, e.g., Rempel, 2004; Stix, 2002). By comparison, upper limits from helioseismology suggest that the overshoot region is no more than about 0.05HP, which is less than 0.01R⊙ (Section 3.6). Helioseismic inversions can also set limits on how abruptly the entropy gradient changes at the base of the convection zone, ruling out a very thin thermal adjustment layer (Monteiro et al., 1994; Basu et al., 1994; Roxburgh and Vorontsov, 1994).
Brummell et al. (2002b) also considered the variation of the penetration depth with rotation and latitude, under the f-plane approximation. They found that rotation generally has a stabilizing effect because plumes are tilted away from the vertical by turbulent alignment and weakened by vortex interactions. Similar results were also reported by Julien et al. (1996a, 1999); see Section 5.2. The penetration depth was greatest at the equator and poles, and least at mid-latitudes. The smaller penetration at mid-latitudes relative to high latitudes was attributed to turbulent alignment because tilted plumes have less downward momentum. The enhanced penetration at low latitudes was attributed to the formation of horizontal convective rolls which are analogous to the north-south aligned downflow lanes typically seen in global convection simulations (Section 6.2). Global simulations of penetrative convection by Miesch et al. (2000) do indeed exhibit deeper penetration at the equator, but there is less evidence for enhanced penetration at the poles in turbulent parameter regimes. However, the simulations by Miesch et al. (2000) used a realistic value for the solar luminosity so it was impractical to cover a full thermal equilibration timescale (∼ 105 yr; see Section 5.1). Thus, any conclusions made about the detailed structure of the overshoot region must be regarded as tentative.
Investigating convective penetration with global models remains an important challenge for the near future. Although global models can say little about the thermal structure of the overshoot region at present, they have already produced provocative and robust results regarding its dynamics. In particular, they have indicated that penetrative convection in the Sun is likely to induce equatorward meridional circulation and poleward angular momentum transport in the overshoot region (see Sections 6.3 and 6.4).
Another aspect of penetrative convection which has important implications for solar interior dynamics is the generation of gravity waves. Figure 23 illustrates wave excitation in simulations of penetrative convection by Rogers and Glatzmaier (2005b). The geometry is a 2D circular annulus with the inner boundary placed very near the origin to minimize spurious wave reflection. Gravity waves appear as rings of vorticity in the stable zone propagating outward. This outward phase velocity implies an inward group velocity, and is therefore consistent with wave generation at the base of the convection zone (see Appendix A.7).
Although gravity waves are present in all simulations of penetrative convection, little is known about the details of wave excitation in the Sun. Unless steps are taken to avoid it, numerical simulations generally suffer from wave reflection at the lower boundary and imposed horizontal periodicities which can substantially alter the spectra, energetics, and transport properties of the waves. Furthermore, obtaining a reliable estimate of gravity wave amplitudes and spectra in a high-resolution simulation of penetrative convection is not a trivial undertaking (e.g., Dintrans and Brandenburg, 2004). The most straightforward method is based on spectral transforms of the velocity or density field in space and time, but this can be unwieldy in a 3D simulation because it requires storing a substantial volume of data at a high temporal cadence and a long enough duration to achieve stable statistics. To date, most investigations of gravity wave excitation in simulations of penetrative convection have been restricted to 2D flows (Hurlburt et al., 1986; Andersen, 1996; Dintrans et al., 2003; Kiraga et al., 2003; Rogers and Glatzmaier, 2005a, b). Theoretical estimates of wave excitation are sensitive to assumptions made about the structure of the convection which are difficult to justify (Goldreich and Kumar, 1990; Fritts et al., 1998; Kumar et al., 1999).
Despite this uncertainty, some general comments can be made. We expect that the gravity wave spectra will peak at spatial and temporal frequencies which correspond to the characteristic scales of the convection which drives them. These are currently uncertain but may be estimated from numerical simulations (Section 6.2). Modes with very small wavelengths (≤ 1 Mm) will be efficiently dissipated by thermal diffusion while modes with horizontal phase velocities comparable to the local differential rotation will be filtered out by critical level absorption and radiative diffusion (Fritts et al., 1998; Kumar et al., 1999; Talon et al., 2002). If the motions are indeed gravity waves, their frequencies will be bounded from above by the Brunt-Väisälä frequency, N, which corresponds to a period of a few hours in the solar interior. However, since the Sun is rotating and magnetized, we might expect a wide variety of waves to be generated by penetrative convection, including inertial gravity waves, Rossby waves, and Alfvén waves. Characteristic velocity amplitudes will vary substantially with radius but may be ∼ 1–10 m s−1 near the overshoot region based on estimates for the vertical velocity in downward plumes, which may reach 100 m s−1, and a moderate conversion efficiency.
No discussion of penetrative convection would be complete without some mention of transport processes. It is well established that turbulent penetrative convection can efficiently pump magnetic fields out of the convection zone into to the overshoot region, and possibly deeper (Brandenburg et al., 1996; Tobias et al., 1998, 2001; Dorch and Nordlund, 2001; Ziegler and Rüdiger, 2003). This is thought to play an integral role in the solar dynamo by continually supplying the tachocline with disordered field which can then be organized and amplified by rotational shear (Section 4.5). Transport of chemical tracers by penetrative convection and the waves it generates can has important implications for solar structure models and spectroscopic measurements of stellar compositional abundances (Montalbán, 1994; Schatzman, 1996; Hurlburt et al., 1994; Pinsonneault, 1997; Fritts et al., 1998; Brummell et al., 2002b; Ziegler and Rüdiger, 2003). Furthermore, angular momentum transport by gravity waves has important implications for understanding the structure and evolution of the solar internal rotation profile as we will discuss further in Sections 8.4 and 8.5.
We emphasize that convective penetration in the Sun is a very intermittent process, dominated by extreme, impulsive events; particularly strong plumes or ensembles of plumes which penetrate deeper than average and then quickly lose coherence. A jackhammer is a better analogy than a drill. Thus, the transport of magnetic fields, chemical tracers, and momentum, is generally deeper than might be expected from average measures such as the mean stratification or the mean kinetic energy density (e.g., Brummell et al., 2002b).
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 stableFootnote 17.
For the lower tachocline, N ∼ 10−3 s and S ∼ 10−6 s−1 (see Section 3.2), implying very large Richardson numbers Ri ∼ 106. 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 = 0.713R⊙, 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 turbulenceFootnote 18 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.
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 monthsFootnote 19. 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 clamshell 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′θB′ϕ〉.
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 (≤ 104 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.
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 (∂Ω/∂λ > 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 ∂Ω/∂λ > 0 or in the equatorial plane where buoyancy resists motions perpendicular to the rotation axis. The MRI criterion ∂Ω/∂λ yt; 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: \(\partial{\cal L}/\partial\lambda < 0\) (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).
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
$${L_{\rm{D}}} = {{N{\Delta _{\rm{t}}}} \over {2{\Omega _0}}} = {{{\rm{Ro}}} \over {{\rm{Fr}}}}{r_{\rm{t}}},$$
(28)
where N is the Brunt-Väisälä frequency. The Rossby number and Froude number are defined as Ro = U/(2Ω0rt) and Fr = U/(NΔt) where U is a characteristic velocity scale. For motions on scales ≤ LD, stratification breaks the vertical coherence induced by rotation (Dritschel et al., 1999). In the lower tachocline LD ∼ 5R⊙ 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)Footnote 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 β-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 kβ, often referred to as the Rhines wavenumber after Rhines (1975). At scales above \(k_{\beta}^{-1}\), the flow has a Rossby-wave character and at scales below \(k_{\beta}^{-1}\), 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 wavenumbersFootnote 21. 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−½. Taking U ∼ 10 m s−1 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 (Fr2 ≪ 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 β-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 ≤ kβ) 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 Ri = (N/S)2 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 → ∞ and may be absent altogether in statistically steady flows.
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.
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 threeFootnote 22. 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 Corioliss 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 \(\hat{x}\) and \(\hat{y}\) are the local longitude and latitude coordinates and \(\hat{z}\) 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, \(U_{0}(z)\hat{x}\) is
$$\sigma - {k_x}{U_0} = N\cos \psi,$$
(29)
where σ is the frequency, kx is the component of the wave vector in the direction of the shear and ψ is the angle it makes with the horizontal (see Appendix A.7). The direction of phase propagation is given by the angle ψ but in a stationary medium (U0 = 0), the group velocity is perpendicular to the phase velocity (Appendix A.7). The highest-frequency waves have σ = N and have a horizontal phase velocity (ψ = 0° or 180°).
The intrinsic frequency of the wave, σ 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 (σkx > 0), the wave may encounter a critical layer where the Doppler-shifted frequency σ − 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.
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, \(U_{0}(y)\hat{x}\). 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 yt 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.
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 ±kxυA where υA 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).
Tachocline confinement
One of the most compelling questions about the tachocline is: Why is it so thin? The transition from a ∼ 30% latitudinal variation of angular velocity in the convection zone to nearly uniform rotation in the radiative interior occurs over roughly 5% or less of the solar radius (Section 3.2).
The issue is best illustrated by considering one of the pioneering papers on the subject: The very paper which coined the term tachocline. Soon after the first helioseismic indications that a rotational shear layer exists near the base of the convection zone, Spiegel and Zahn (1992) considered the problem from the perspective of an axisymmetric spin-down scenario. They considered a spherical volume of fluid in hydrostatic and geostrophic balance subject to an imposed latitudinal differential rotation on the upper boundary. This was intended to represent the radiative solar interior under the influence of wind stress from the convective envelope. A meridional circulation was quickly established in which the advective heat flux was balanced by radiative diffusion. If momentum transport by unresolved turbulent motions was neglected, they found that this circulation steadily spread into the radiative interior, redistributing angular momentum away from uniform rotation on a timescale of several billion years. If such a radiative spreading had occurred over the lifetime of the Sun, the differential rotation of the envelope would have spread deep into the solar interior, in marked contrast to the nearly uniform rotation inferred from helioseismic inversions (see Section 3.1). Further numerical calculations were later performed by Elliott (1997), confirming these results.
Thus, the question of why the tachocline is so thin is equivalent to asking what can stop this radiative spreading. Or from a somewhat different perspective, one may instead ask: What process or processes can maintain uniform rotation in the radiative interior, in spite of stresses exerted by the convection zone?
Spiegel and Zahn (1992) were the first to suggest a mechanism. They argued that turbulence arising from nonlinear shear instabilities would mix angular momentum in such a way that horizontal transport would be much more efficient than vertical transport, and would therefore drive the radiative interior toward shellular rotation (see Section 8.2). They modeled this turbulent transport as an anisotropic viscosity in which the horizontal component greatly exceeded the vertical component. Their calculations and subsequent calculations by Elliott (1997) demonstrated that this anisotropic transport could effectively halt the radiative spreading, producing an equilibrium profile in which the width of the tachocline, Δt, is given by
$${{{\Delta _{\rm{t}}}} \over {{r_{\rm{t}}}}}\sim{\left({{\Omega \over N}} \right)^{1/2}}{\left({{{{\kappa _r}} \over {{\nu _{\rm{H}}}}}} \right)^{1/4}},$$
(30)
where rt ∼ 0.7R⊙ is the tachocline location and νH is the horizontal turbulent viscosity. In the solar tachocline, Ω ∼ 2.7 × 10−6s−1, N ∼ 10−3s−1, and κr ∼ 107 cm s−2. This implies that a turbulent viscosity as low as νH ∼ 3 × 106 would be sufficient to confine the tachocline to about 5% of the solar radius, consistent with helioseismic inversions (Section 3.2). The figure cited by Elliott (1997) is about an order of magnitude less.
Although Spiegel and Zahn (1992) identified nonlinear hydrodynamic shear instabilities in particular, other mechanisms may produce a similar confinement, provided they induce down-gradient horizontal angular momentum transport. Or, in other words, provided they act as a positive anisotropic turbulent viscosity with νH ≫ νV. One such alternative mechanism may be provided by the 2D and shallow-water magneto-shear instabilities studied by Gilman, Fox, Dikpati, and Cally which generally transport angular momentum poleward via the Maxwell stress (see Section 8.2). Another possible mechanism might be stratified turbulence induced by penetrative convection (Miesch, 2001, 2003).
These mechanisms may help to explain why the latitudinal differential rotation of the convective envelope does not spread inward, but they do little to explain why the radiative interior as a whole is rotating uniformly. Stratified, rotating turbulence near the base of the convection zone may produce down-gradient angular momentum transport in latitude but this is by no means certain and in any case, the vertical transport is likely to be counter-gradient (see Section 8.3). Deeper in the interior, angular momentum transport by gravity waves would also tend to enhance shear rather than suppress it due to the selective dissipation of prograde and retrograde modes (see Section 8.4). These points have been made repeatedly by McIntyre and others (McIntyre, 1994, 1998, 2003; Gough and McIntyre, 1998; Ringot, 1998). Gravity waves may still play a role in tachocline confinement, but only if there is some additional mechanism such as shear turbulence to provide an effective viscous diffusion (Talon et al., 2002)Footnote 23. Hydrodynamic instabilities alone appear to be too inefficient to maintain uniform rotation (Spruit, 1999; Garaud, 2001; Mathis and Zahn, 2004).
The difficulties in producing diffusive angular momentum transport in rotating, stably-stratified flows by purely hydrodynamical means has led some to suggest that magnetic fields are necessary in order to maintain uniform rotation in the radiative interior (Rüdiger and Kitchatinov, 1997; Gough and McIntyre, 1998). Such fields may arise as a remnant, or fossil, left over from the gravitational collapse of the protostellar cloud from which the Sun formed. An axisymmetric poloidal field will resist differential rotation via magnetic tension. The resulting torques will tend to establish uniform rotation along magnetic field lines on an Alfvénic timescale, a result which is known as Ferraro’s theorem (Cowling, 1957; Mestel and Weiss, 1987; MacGregor and Charbonneau, 1999). Turbulence induced by instabilities may then couple adjacent field lines. For example, as the solar wind spins down the convective envelope, angular velocity profiles may be established which decrease outward with cylindrical radius, ∂Ω/∂λ < 0. These would then be subject to magneto-rotational instabilities (Section 8.2) which, together with the torques implied by Ferraro’s theorem, could establish uniform rotation throughout the radiative interior. Diffusive instabilities may also play a role (Menou et al., 2004).
According to Ferraro’s theorem, the fossil field must be confined entirely to the radiative interior in order to maintain uniform rotation. If poloidal field lines were to extend into the convective envelope, then at least some fraction of the differential rotation there would be transmitted into the interior, which would be inconsistent with helioseismic inversions. This expectation is borne out by numerical calculations (MacGregor and Charbonneau, 1999).
If the fossil field is confined to the radiative interior and meridional circulation is neglected, the tachocline which develops is essentially a classical Hartmann layer in which magnetic tension balances viscous diffusion (Rüdiger and Kitchatinov, 1997; MacGregor and Charbonneau, 1999). In this case the tachocline width is given by
$${{{\Delta _{\rm{t}}}} \over {{r_{\rm{t}}}}}\sim{\left({{{4\pi \rho} \over {r_{\rm{t}}^2}}{{\nu \eta} \over {B_0^2}}} \right)^{1/4}}\sim 5 \times {10^{- 5}}B_0^{- 1/2},$$
(31)
where B0 is the poloidal field strength at rt. The final equality in Equation (31) is derived using ρ ∼ 0.2 g cm−3 and molecular values for the diffusivities, ν ∼ 5 cm2 s−1 and η ∼ 103 cm2 s−1. A field strength of B0 ≥ 10−6G would confine the tachocline to less than 4% of the solar radius, well within helioseismic limits. If there is enough vertical mixing to act as a turbulent viscosity and diffusivity, a larger magnetic field would be needed.
In all likelihood, there will be a significant meridional circulation in the tachocline. In the Spiegel and Zahn (1992) scenario discussed above, for example, the differential rotation spreads not by viscous diffusion but by advection due to a radiatively-driven circulation. In this case, MacGregor and Charbonneau (1999) estimate that a field of B0 ∼ 2 × 10−4 G would be required for confinement, about two orders of magnitude larger than the viscous estimate implied by Equation (31).
Meridional circulation also plays an essential role in the tachocline model proposed by Gough and McIntyre (1998). Here the circulation is driven by the Reynolds stress in the convection zone through what may be called gyroscopic pumping (McIntyre, 1998). Consider an axisymmetric ring of fluid. If the ring is subject to a prograde longitudinal force it will tend to drift away from the rotation axis due to the Coriolis force. If the force is retrograde, the ring will drift inward. In the solar convection zone, the Reynolds stress act to accelerate the equator relative to the poles, which would tend to establish a global circulation.
Further insight into how this operates can be obtained by considering the angular momentum balance expressed by Equation (8)
$${\bf{\nabla}} \cdot {{\bf{F}}^{{\rm{MC}}}} = \bar \rho \langle {{{\bf{v}}_{\rm{M}}}} \rangle \cdot {\bf{\nabla}} {\cal L} = - {\bf{\nabla}} \cdot {{\bf{F}}^{{\rm{RS}}}},$$
(32)
where we have also used Equation (6). The Reynolds stress produces a flux convergence and divergence at low and high latitudes respectively. By Equation (32), this induces a meridional circulation across lines of constant specific angular momentum, \({\cal L}=\lambda^{2}\Omega\). In the Sun, \({\bf{\nabla }}{\cal L}\) is approximately perpendicular to the rotation axis and directed outward (Figure 6, panel b), so Equation (32) implies a flux divergence at mid-latitudes in the convection zone. Below the convection zone the Reynolds stress is neglected and the circulation follows surfaces of constant \({\cal L}\).
In the Gough & McIntyre model, this gyroscopic circulation is prevented from burrowing deep into the radiative interior by a fossil poloidal field as illustrated in Figure 29. The tachocline itself is non-magnetic but there exists a thin boundary layer at its base, called the tachopause, where the circulation is diverted horizontally by the interior field. This gives rise to a horizontal convergence and an associated upwelling at latitudes of about 30°, where the radial shear across the tachocline vanishes. The tachopause occupies only a few percent of the total tachocline width which is given by
$${{{\Delta _{\rm{t}}}} \over {{r_{\rm{t}}}}}\sim 3 \times {10^{- 2}}B_0^{- 1/9}.$$
(33)
This suggests field strengths of ∼ 0.1–1 G but a range of values is consistent with helioseismic inversions because Δt is relatively insensitive to B0.
The dynamical balance in the tachopause not only keeps the circulation from spreading inward, but it also keeps the fossil field confined to the radiative interior. This can only occur in downwelling regions; upwelling regions are likely to be more complex and may alter this simple picture. In the most recent incarnation of the Gough & McIntyre model (McIntyre, private communication), some of the magnetic field lines in upwelling regions follow the circulation streamlines into the convection zone. Regions in which the angular velocity decreases outward (dΩ/dλ < 0) would then be subject to magneto-rotational instabilities (MRI; see Section 8.2) which would alter the local tachocline structure, still maintaining thermal wind balance.
Although magnetic confinement models are compelling, there are many aspects which need further verification and clarification. Among these is the configuration of the fossil field. Axisymmetric poloidal fields are likely to be unstable over evolutionary timescales so any fossil field which may exist in the solar interior today is probably of mixed poloidal and toroidal topology (Mestel and Weiss, 1987; Spruit, 1999). This has been incorporated into the Gough & McIntyre model, but still only in a schematic way. Another open question is whether a circulation which is driven in the convection zone can overcome the stiff subadiabatic stratification in the lower tachocline and penetrate all the way to the tachopause Gilman and Miesch (2004).
Some aspects of the Gough & McIntyre model have been investigated numerically by Garaud (2002) who solved the axisymmetric MHD equations under the Boussinesq approximation. The circulation in Garaud’s model was driven by Ekman pumping and bore little resemblance to the baroclinic circulations considered by either Gough and McIntyre (1998) or Spiegel and Zahn (1992). Nevertheless, the results did demonstrate that a circulation is capable of confining a poloidal field largely to the radiative interior. Furthermore, the field was able to establish nearly uniform rotation in the interior over an intermediate range of field strengths.
A common feature in nearly all magnetic confinement models is the presence of a polar pit. This is a region near the magnetic poles where the poloidal field is primarily radial and therefore cannot confine the tachocline. Here the meridional circulation and consequently the differential rotation spreads much deeper into the radiative interior. This could in principle be probed with helioseismology, although the low sensitivity of frequency splittings to angular velocity variations near the rotation axis would make it difficult to detect. Currently there is little helioseismic evidence either supporting or refuting the presence of a polar pit.
An alternative to tachocline confinement by a weak fossil field in the radiative interior is tachocline confinement by a strong dynamo field originating in the convection zone. This possibility has been explored by Forgács-Dajka and Petrovay (2002) and Forgács-Dajka (2004) who consider a thin, axisymmetric shell of fluid under the anelastic approximation. A latitudinal differential rotation is imposed on the upper boundary along with an oscillatory poloidal field intended to represent dynamo processes in the convective envelope. The characteristic penetration depth of the field is the electromagnetic skin depth for a conductor, (2ηt/ωc, where ηt is a turbulent diffusivity and wc is the frequency of the oscillation. If the turbulent diffusivity is large enough (∼ 1010 cm s−2) and if the imposed field is strong enough (∼ 103 G), then the field can penetrate deep enough to suppress the spread of differential rotation into the interior.
It is an open question how the relatively weak Lorentz force and circulations associated with magnetic confinement by a fossil field may coexist with and couple to the much stronger forces and motions which exist in the convection zone. In this context, a distinction is often made between fast tachocline dynamics which occur on timescales of weeks to decades and slow tachocline dynamics which occur on much longer timescales (e.g., Gilman, 2000a). Nearly all of the processes discussed in Sections 8.1 and 8.4 fall under the category of fast dynamics. Although they involve relatively weak circulations, the tachocline models of Forgács-Dajka and Petrovay (2002) and Forgás-Dajka (2004) may also be classified as fast because they require efficient turbulent mixing to operate and because they are concerned with dynamo-generated fields with an oscillation period of 22 years. The remaining magnetic confinement models discussed in this section represent slow dynamics. For example, the overturning time scale for the tachocline circulation in the Gough & McIntyre model is of order a million years. Fast dynamics are likely to dominate in the upper tachocline which probably overlaps with the convection zone and overshoot region. However, slow dynamics may be ultimately responsible for the nearly uniform rotation of the radiative interior and may therefore determine the lower boundary of the tachocline.