8.5 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, $D t$ , is given by

where $rt ~ 0.7Ro .$ is the tachocline location and $nH$ is the horizontal turbulent viscosity. In the solar tachocline, $_O_ ~ 2.7× 10-6s-1$ , $N ~ 10-3s-1$ , and $kr ~ 107 cm s-2$ . This implies that a turbulent viscosity as low as $nH ~ 3 × 106$ 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 $n » n H V$ . 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 )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, $@_O_/@c < 0$ . 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

where $B0$ is the poloidal field strength at $rt$ . The final equality in Equation (31

) is derived using $r ~ 0.2 g cm -3$ and molecular values for the diffusivities, $n ~ 5$ $cm2 s-1$ and $j ~ 103$ $cm2 s-1$ . A field strength of $B > 10 -6G 0$ 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 $- 4 B0 ~ 2 × 10 G$ 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 )

$MC -- RS \~/ .F = r <vM >. \~/ L = - \~/ .F , (32)$

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, $2 L = c _O_$ . In the Sun, $\~/ L$ 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 $L$ .

Figure 29:

Schematic diagram from Gough and McIntyre (1998

) (

http://www.nature.com), illustrating the proposed tachocline structure. A meridional circulation (black lines) is driven by gyroscopic pumping in the convective envelope (orange) and penetrates into the tachocline (green) along lines of constant specific angular momentum $L$ . A poloidal magnetic field (red lines) in the radiative interior (blue) halts the downward spread of this circulation in a thin boundary layer called the tachopause. In upwelling regions, the field structure is uncertain (dotted lines). The width of the tachocline is exaggerated in this perspective.

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 $30o$ , where the radial shear across the tachocline vanishes. The tachopause occupies only a few percent of the total tachocline width which is given by

This suggests field strengths of $~ 0.1 -1 G$ but a range of values is consistent with helioseismic inversions because $D t$ is relatively insensitive to $B 0$ .

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 ( $d_O_/dc < 0$ ) 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, $1/2 (2jt/wc)$ , where $jt$ 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 Forgács-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.