A&A 465, 633-639 (2007)
Eun-jin Kim - N. Leprovost
Department of Applied Mathematics, University of Sheffield, Sheffield S3 7RH, UK
Received 7 September 2006 / Accepted 12 January 2007
Aims. We investigate the confinement and long-term dynamics of the magnetised solar tachocline.
Methods. Starting from first principles, we derive the values of turbulent transport coefficients in the magnetised solar tachocline and then explore the implications for the confinement and long-term dynamics of the tachocline.
Results. For reasonable parameter values, the turbulent eddy viscosity is found to be negative, with turbulence enhancing the radial shear in the tachocline. Both magnetic diffusivity and thermal diffusivity are severely quenched, with values much smaller than the magnitude of the eddy viscosity. The effect of the meridional circulation on momentum transport via the hyperviscosity becomes important when the radial shear becomes large (larger than the presently inferred value) due to negative viscosity. The results imply that the tachocline develops too strong radial shear to be a stationary Hartmann layer. In the limit of strong radiative damping where the turbulence is active on very small scales (<10 ), the eddy viscosity can become positive although its effect is likely to be dominated by the hyperviscosity. In comparison with the momentum transport, the transport of magnetic field, heat, and passive particles is more severely quenched. The results imply that the thickness of the tachocline is of order , independent of the strength of magnetic fields. In addition, the momentum transport is much more efficient than the particle mixing in the tachocline, consistent with the observations.
Key words: turbulence - magnetohydrodynamics (MHD) - Sun: interior - Sun: magnetic fields - Sun: rotation - waves
A consistent theory of transport in the solar interior (in particular, the tachocline) is essential to the understanding of the evolution of solar rotation and magnetic fields and the distribution of chemical species. While in the standard solar model (Stix 1989) turbulence is assumed to be absent in the interior, observations (e.g., Pinsonneault et al. 1989) and numerical simulations (e.g., Rüdiger & Kitchatinov 1996) suggest that transport in this region - although not so fast as turbulent transport (e.g., such as in the convection zone) - should be faster than molecular processes to be consistent with the current rotational profile and surface depletion of light elements. Such a modest transport could be due to waves via dissipative processes (e.g., radiative damping of gravity waves). Another interesting possibility, which has not received much attention, is that turbulence is present in the interior due to a variety of instabilities (e.g., see Spruit 1999, and references therein), but that the overall transport due to this turbulence is considerably reduced as a result of turbulence regulation. Our previous works (Kim 2005; Kim & Leprovost 2007) have shown that stable stratification as well as shearing by the radial differential rotation in the tachocline can precisely do this as the excitation of gravity waves reduces the stochasticity in turbulent flow while shearing enhances the overall dissipation (see also Kim 2004).
The turbulent transport reduction can also be caused by magnetic fields (e.g., Cattaneo & Vainshtein 1991; Gruzinov & Diamond 1994; Kim & Dubrulle 2001; Kim 2006). In the tachocline, a strong toroidal magnetic field of the strength 104-105 G is believed to be present, which can easily be generated when a weak poloidal magnetic field is sheared by differential rotation in the tachocline. A poloidal magnetic field here could be either of primordial origin evolving on a long evolutionary time scale (i.e., slow tachocline), or generated by dynamo process operating on a fast time scale of the solar cycle (i.e. fast tachocline) (see e.g., Gilman 2000; Petrovay 2003). Thus, magnetic fields can potentially play a crucial role in the transport of momentum, chemical species, and magnetic flux on long and/or short time scales. In particular, on an evolutionary time scale, the tachocline may be considered as a boundary layer between the uniformly rotating radiative interior and differentially rotating convection zone with latitudinal variation (Rüdiger & Kitchatinov 1996; Gough & McIntyre 1998; MacGregor & Charbonneau 1999), and the dynamics of this boundary layer crucially depends on the values of the effective magnetic diffusivity, eddy viscosity, etc. The understanding of this boundary layer thus requires the prediction of these turbulent transport coefficients, derived from first principles. Furthermore, momentum transport and chemical mixing across the tachocline play a crucial role in the evolution of solar differential rotation and the distribution of chemical species. In particular, the present solar rotational profile and surface depletion of light elements (lithium) (Schatzman 1993) indicate that the angular momentum transport must have been more efficient than the particle mixing in the solar interior (e.g. Pinsonneault et al. 1989). This should be explained by a consistent theory of momentum transport and chemical mixing in the tachocline, rather than invoking a crude parameterization as has often been done by previous authors.
The purpose of this paper is to provide a consistent theory of turbulent transport in the magnetised tachocline and then investigate its implications for a long-term dynamics of the (slow) tachocline. Special attention is paid to the elucidation of different effects on transport of shearing, stable stratification, and magnetic fields, identifying what is most likely to be the main mechanism for turbulence regulation in the tachocline. The remainder of the paper is organized as follows. We elucidate the effects of gravity-Alfven waves on turbulent transport in Sect. 2. In Sect. 3, we incorporate the effect of shear flow given by the radial differential rotation and provide the theoretical predictions for turbulent coefficients in the stratified magnetised tachocline. We elaborate on implications of the results for a long-term dynamics of the tachocline in Sect. 4. Section 5 is devoted to the discussion of the limit of strong radiative damping. Section 6 contains the conclusions and discussions.
To elucidate the role of magnetic fields and stable stratification
it is illuminating to examine their effect on the diffusion
of magnetic flux. To this end, we ignore the large-scale
shear flow U0 and forcing f and
recast Eqs. (1)-(3) as follows:
Since in the tachocline the radiative damping is much larger than ohmic diffusivity and viscosity, it is instructive to consider the limit of strong radiative damping (with ) to elucidate the effect of radiative damping (recall that the result (9) is valid in the limit of small dissipation ( )).
In this case, Eq. (6) can be approximated
analysis then gives us the magnetic diffusivity
the coupled Eqs. (1)-(3) can
easily be combined to form the following equation for
The solution to Eq. (13) in the limit of strong magnetic
be found as
For clarity, we now examine the behavior of as a function of . For large (strong stratification/weak magnetic field), , showing that the effect of magnetic fields tends to make eddy viscosity positive. In the opposite limit of small (weak stratification/strong magnetic field), , whose absolute magnitude is small compared to the 2D HD case where . This again reflects the tendency of magnetic fields making the eddy viscosity less negative (i.e. more positive). Note that for parameter values typical of the tachocline, the cross-over scale LNfrom to is roughly LN = 107 - 108 cm.
To recapitulate, the result (16) shows a tendency of a negative eddy viscosity in a stratified medium despite the presence of a rather strong magnetic field. Thus, the turbulent transport in the tachocline would amplify the shear provided by the radial differential rotation for reasonable values of parameters. We emphasize that the negative eddy viscosity represents the amplification of a large-scale shear flow at the expense of small-scale turbulence. Note that if the scale of the mean flow continues to decrease and becomes comparable to the characteristic scale of small-scale (due to the negative viscosity), the concept of eddy viscosity becomes invalid.
The value of magnetic diffusivity, defined by
depends on whether
is larger or smaller
than unity. First, in the case
which is valid
cm (1011 cm)
G (105 G)
or on scales
we can obtain
Equation (17) shows that the turbulent diffusion of magnetic field ( ) can severely be quenched by a strong mean magnetic field and stratification, proportional to 1/B02 for and to 1/B0 N for , respectively. We note that was observed in numerical simulation (e.g. Cattaneo & Vainshtein 1991) of 2D unstratified MHD turbulence (). As the stratification becomes stronger for a fixed B0 with a further increase in , the diffusion is now reduced as 1/N2 [see Eq. (18)]. It is important to note that in all cases, magnetic diffusivity has a much smaller magnitude than the eddy viscosity [in Eq. (16)], with a small value of . A similar tendency was also found in the stably stratified shear turbulence without magnetic fields (Kim & Leprovost 2007). Specifically, in the strong magnetic field and weak stratification region with and , . Furthermore, the heat diffusivity in Eq. (19), although larger than by a factor of , is yet much smaller than the magnitude of . For instance, for and , this ratio becomes for typical parameter values. These results have very interesting implications for a long-term dynamics of the tachocline, as discussed in Sect. 4.
The results of the present and previous paper (Kim & Leprovost 2007; Kim & MacGregor 2001) suggest that the uniform rotation in the radiative interior is very unlikely to be explained by hydrodynamical means as the momentum transport in stratified medium accelerates the mean flow, sharpening the gradient of radial differential rotation that has been created during the solar spin-down (see, however, Charbonnel & Talon 2005). Previous authors (Rüdiger & Kitchatinov 1996; Gough & McIntyre 1998; MacGregor & Charbonneau 1999), have however shown that a rather weak poloidal magnetic field in the radiative interior can eliminate the differential rotation, thereby leading to a uniform rotation therein. In this case, the tachocline can be envisioned as a boundary layer where the generation of the toroidal magnetic field by the shearing of the poloidal magnetic field (due to differential rotation) is balanced by the diffusion of the toroidal magnetic field while the dissipation of the radial differential rotation is balanced by the azimuthal Lorentz force associated with the large-scale toroidal and poloidal magnetic fields, i.e., Hartmann layer (Rüdiger & Kitchatinov 1996; Gough & McIntyre 1998; MacGregor & Charbonneau 1999). By adopting the molecular values for viscosity, magnetic diffusivity, and heat diffusivity in the tachocline, these previous authors obtained estimates of the tachocline thickness and the strength of the interior poloidal magnetic field. However, in the case of the tachocline with residual turbulence, for instance, driven externally (e.g. plume penetration) or internally (e.g. via instability), the values of turbulent transport coefficients, instead of molecular values, should be used in the analysis of the Hartmann layer.
For the clarity of the discussion, it is worth recalling that
the Hartmann layer is based on the configuration where
a poloidal magnetic field
is fully contained in the interior, without penetrating
into the convection zone above so that the latitudinal differential
rotation in the convection zone does not leave its footprint
into the radiative interior.
By representing the latitudinal coordinate by z, and
by denoting the poloidal and the toroidal magnetic
fields by Bz and By=B0, respectively, the major
force balance for the toroidal magnetic field in the
tachocline By and the mean shear flow Uy due to
the differential rotation can roughly be expressed as
While the values of turbulent transport ( , , and ) for the coupled system (20)-(21) have conventionally been assumed to be positive, the results in Sect. 3 show that the turbulent momentum transport in the sheared stratified turbulent tachocline is anti-diffusive with negative viscosity (i.e., ), accelerating the mean shear flow Uy (i.e., radial differential rotation). The negative viscosity will amplifies the shear in the differential rotation and consequently toroidal magnetic field since they are coupled via the Lorentz force (via Bz). The crucial question is then how the Hartmann layer is maintained with negative viscosity which tends to become unstable due to . Obviously, if the magnetic diffusivity is large enough to overcome an unstable situation caused by the negative viscosity, the coupled system can find a stable stationary configuration. However, this is very unlikely since the magnetic diffusion is pathetically small compared to the magnitude of the eddy viscosity, as discussed in Sect. 3. Could the hyperviscosity due to the meridional circulation then stabilize the system? To answer this question, we note that the contribution from the hyperviscosity to Eq. (20) is of order while the contribution from the eddy viscosity is of order , where h is of order of the tachocline thickness. Thus, the ratio of the two is roughly (e.g. for and ), which can be shown to be small for the parameter values typical of the present solar tachocline. However, it is important to realize that this ratio ( for and , for instance) becomes large as the shear increases. Therefore, it is plausible that as the shear is amplified via the negative eddy viscosity, the effect of the hyperviscosity becomes important and eventually dominates the negative viscosity, possibly stabilizing the system. In order for this to be the case, the shear in the radial differential rotation however has to be larger than what is observed today. This can be checked by requiring . For instance, in the case and , this demands , where is the characteristic scale of the forcing. Even if we take G and a thin tachocline , for cm-1, and for cm-1, which seem to be rather too large to be reasonable (recall s-1 and s-1are the presently inferred values in the tachocline).
To make this argument more concrete, it is instructive to
examine the behavior of Eqs. (20)-(21)
in more detail.
To this end, we average
them over the space ()
with the approximation
and use the values of
in Sect. 3.
Here, h is again the tachocline thickness.
The resulting envelop equations for
and By, in a properly non-dimensonalized form,
are as follows:
We note that the values of , and are all proportional to the intensity of the forcing , which is a free parameter in our problem. In our non-dimensionlization, this value is fixed by utilizing the observational evidence that the particle diffusivity of light elements (lithium) is about cm2 s-1 to be consistent with the present solar surface lithium depletion (e.g., see Barnes et al. 1999). Since where cm2 s-1 and cm2 s-1, we impose the condition that cm2 s-1 for the parameter values typical of the present sun (i.e. G, s-1, etc). (Note that in Eqs. (17)-(18) depends on both and By, which change in time according to Eq. (22).)
|Figure 1: The plot of the amplitude of as a function of ( hyperviscosity) for . The cross symbols represent the results of the numerical simulation of the toy model Eq. (22). The dashed line denotes the asymptotic behavior, given by .|
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The results from the numerical simulation of Eq. (22) for are plotted by the cross symbols in Fig. 1, which shows for various values of . The results nicely show that the amplitude of S decreases as (hyperviscosity) becomes large, as expected. However, for small (hyperviscosity), an amplitude of is too large to be reasonable. For this large value of , which is comparable to the Brunt-Väisälä frequency N, the shear flow is likely to be unstable against a shear instability (see, e.g., Drazin 1981). As increases, the amplitude of decreases and approaches the asymptotic value given by the dashed line , which is obtained by using |S|=|M|. The asymptotic value indicates that for while for . By using the definition of , one can easily show that a reasonable value of shear is possible only for an extremely thin tachocline with . Since |S|=|M| along the dashed line in Fig. 1, the toroidal magnetic field also seems too strong (>105 G) to be stable (e.g. against the magnetic buoyancy instability). To summarize, the results from our toy model (22) clearly demonstrate that for reasonable parameter values, a stable stationary Hartmann layer is very unlikely in the tachocline. Thus, a large-scale shear flow is likely to become time-dependent (similarly to the behaviour found in Kim & MacGregor 2001), or to develop a secondary instability.
Eqs. (1)-(2) can easily be combined to yield
The comparison of Eqs. (25) and (26) reveals that the magnetic diffusivity is smaller than eddy viscosity roughly by a factor of (recall is a small parameter characterizing the strong shear limit, where is the characteristic scale of the forcing). The ratio is however larger than the value in the case of weak damping in Sect. 3. This is an interesting result since this ratio is crucial in the estimate of the tachocline thickness, which has been taken to be large by using the molecular values of and in previous works. That is, our predicted value () is much smaller than based on molecular values. Furthermore, the ratio of the effect of the hyperviscosity to that of eddy viscosity is now roughly given by for reasonable parameter values. That is, the effect of hyperviscosity due to meridional circulation is crucial in maintaining the momentum balance in the tachocline.
Based on these observations, we now seek to obtain the estimate on the
thickness of the tachocline
and the strength By and Bz
and by ignoring
in Eq. (20).
A simple analysis of
with the help of Eqs. (26) and (27)
In the limit of strong radiative damping where the temperature (density) fluctuation is almost stationary, the turbulent momentum transport is found to be diffusive, down the gradient. This requires the turbulence to be on very small scales such that . Here, is the characteristic length scale of the forcing, and s-1is the radial shear. Thus, the forcing scale has to be smaller than 106 - 107 cm, which is about . In this case, the tachocline may be viewed as a Hartmann layer with possibly stable configurations of the radial differential rotation and toroidal magnetic field. A simple analysis by using the predicted values of , , and suggested that the effect of the hyperviscosity is likely to dominate over the eddy viscosity. However, since the magnetic diffusion is severely quenched compared with heat diffusivity, the momentum transport is effectively much more efficient than the magnetic field diffusion. Consequently, the major force balance in the tachocline leads to the estimate of the tachocline thickness ( ) for s-1, which tends to be larger than the previously estimated value by Gough & McIntyre (1998). As and depend on the strength of the toroidal magnetic fields and radial shear, the tachocline thickness is found to be independent of the strength of (toroidal and/or poloidal) magnetic field (cf. Gough & McIntyre 1998). The strength of poloidal magnetic field in the radiative interior is estimated to be of order 10-4 - 10-2 G for a value of the toroidal magnetic field 104 - 105 G, comparable to the previously estimated value. Importantly, the results provide a natural explanation for a more efficient momentum transport than particle mixing in the tachocline. However, a severe reduction in the magnetic field diffusion could be problematic for the solar dynamo (e.g., the interface dynamo, Parker 1993).
While the discussion of our results is focused on the applications to the present sun, they might also have interesting implications for other stars. In particular, our predicted values of turbulent transport coefficients have different dependences on thestrength of magnetic field, stratification, and radial shear as well as on the molecular values of viscosity, ohmic diffusivity, and radiative diffusivity. As the values of these parameters vary from one star to another, it would be of interest to explore the implications of these results for other stars. For example, it might be possible to utilise the results to infer the value of the radial shear or the strength of magnetic field in other stars, and also to gain some insight into the presence of a tachocline-like shear layer in those. Of course, in the case of more massive stars, the rotation rate is much faster than that of the sun, demanding the prediction of turbulent transport coefficients by taking into account the effect of average rotation (Leprovost & Kim 2007). Ultimately, it will be interesting to investigate a consistent model incorporating the spin-down of the sun (and other stars) due to the angular momentum loss. Finally, we note that our analysis, limited to 2D, should be extended to 3D, in particular, to study solar dynamos (e.g. the effect). These issues are currently under investigation and will be addressed in future publications.
This work was supported by the PPARC Grant PP/B501512/1.