A&A 441, 763-772 (2005)
DOI: 10.1051/0004-6361:20053170
Eun-jin Kim
Department of Applied Mathematics, University of Sheffield, Sheffield, S3 7RH, UK
Received 1 April 2005 / Accepted 24 June 2005
Abstract
We present a self-consistent theory of turbulent transport
in the solar tachocline by taking into account the
effect of the radial differential rotation on
turbulent transport. We show that
the shearing by the radial differential rotation leads to
reduction in turbulent transport of
particles and momentum and the amplitude of turbulent
flow via shear stabilization. The degree of reduction
depends on the direction as well as the quantity that
is transported. Specifically,
particle transport in the vertical (radial) direction,
orthogonal to the shear flow,
is reduced with the scaling
while it is less reduced in the horizonal plane with
the scaling
.
Here,
is shearing rate, representing the radial
differential rotation.
A similar, but weaker, anisotropy also develops in the amplitude
of turbulent flow.
The results suggest that the radial differential rotation in the tachocline
can cause anisotropy in turbulence intensity and particle transport
with weaker turbulence in the radial direction even
in the absence of density stratification and
even when the turbulence is mainly driven radially
by plumes from the convection zone.
We also assess the efficiency of the transport by a meridional
circulation by taking into account the interaction
with the radial differential rotation. Implications for mixing
and angular momentum transport in the solar interior is discussed.
Key words: turbulence - Sun: interior - Sun: rotation - Sun: abundances
One of the outstanding problems in solar physics is to understand the processes by which the transport of the angular momentum and mixing of chemical species take place. To be consistent with observations, these processes should occur on time scales much shorter than those determined by molecular values (i.e., viscous or diffusive time scales) and also in regions where no fast transport is expected according to the standard solar model (e.g. Stix 1989). According to the latter, fast, turbulent transport is possible only in the convection zone ( ), where a vigorous convection takes place due to a unstable background stratification. Below the convection zone, a stable stratification is considered to lead to a quiet radiative interior without turbulent motion. Between these two regions lies a thin boundary layer - the so-called tachocline - of thickness , through which the latitudinal differential rotation in the convection zone smoothly matches onto the uniform rotation in the radiative interior (e.g., Charbonneau et al. 1999). As a shear boundary layer, the tachocline contains both latitudinal and (strong) radial differential rotation. The density stratification in this region is stable in the lower portion while possibly almost adiabatic in the upper part due to plumes penetrating from the convection zone (e.g. Gilman 2000; Rogers & Glazmaier 2005)
In the case of the angular momentum transport, one of the greatest challenges is to explain how the radiative interior maintains the uniform rotation despite constant losses of the angular momentum via solar wind, etc from the surface over the course of solar evolution. This would require an efficient transport outside the convection zone on a time scale much shorter than solar age. On the other hand, the depletion of light element lithium on the solar surface, which is about a factor of 100 smaller than on Earth (Greenstein & Richardson 1951), demands the existence of some kind of mixing outside the convection zone, which carries lithium down to , little below the bottom of the tachocline, where lithium can be destroyed (e.g. Barnes et al. 1999; Pinsonneault 1997). Note that these light elements burn at high temperature, surviving only in the vicinity of the convection zone where the temperature is below the critical value for its destruction.
To understand these problems, it is necessary to identify physical processes, which lead to transport outside the convection zone, especially in the tachocline near the convection zone. In particular, how the transport in the convection zone is linked to that in the interior through the tachocline is crucial to understanding not only the present Sun but also the entire solar evolution. This is true even for the case of solar dynamos since the tachocline is considered to be the very site where toroidal fields are generated from poloidal magnetic fields via shearing ( effect) (Moffatt 1978). Therefore, the dynamics of the tachocline, such as the presence and/or the origin of possible turbulence seems to play an unique role in the overall solar evolution.
A popular view is that turbulence in the tachocline, if any, should be strongly anisotropic with much less transport in the radial direction than in the horizontal plane due to the stable density stratification. While this anisotropic turbulence, interestingly, was invoked as a very mechanism for maintaining the identity of the tachocline itself against radiative broadening over the age of the Sun (Spiegel & Zahn 1992), various (hydrodynamic and magnetohydrodynamic) instabilities of the latitudinal differential rotation (e.g. Gilman & Fox 1997; Dikpati & Gilman 1999; Gilman & Dikpati 2000; Cally et al. 2003) have been proposed as a source of anisotropic turbulence.
In the absence of turbulence, transport can still be mediated by meridional circulation. In particular, the latter may carry light elements from the convection zone down to radiative interior on time scale which is sufficiently short. While a recent work by Gilman & Miesch (2004) suggested that the meridional circulation, which is observed near the solar surface (Haber et al. 2000), is unlikely to persist deep below the convection zone in the present Sun, it is important to understand, in general, how the transport by meridional circulation is affected by turbulence and vice versus. The particle transport by meridional circulation can be particularly important for the depletion of light elements in massive stars such as lithium dip in F stars (e.g., Zahn 1992; Schatzman 1993; Pinsonneault 1997) or in the early evolution of the Sun (Pinsonneault et al. 1989) where the transport by a meridional circulation - the so-called Eddington-Sweet circulation (Eddington 1925; Vogt 1925; Sweet 1950) - is likely to be efficient in the interiors due to fast rotation. Interestingly, to obtain agreement with observed surface depletion of lithium, and at the same time to explain the angular momentum transport which seems to take place on a shorter time scale as compared to the mixing of lithium (e.g., Pinsonneault et al. 1989), anisotropic turbulence with a stronger horizontal turbulence was invoked to reduce the mixing by a meridional flow (Chaboyer & Zahn 1992; Zahn 1992).
There is one key missing physics in studying these problems, which has been overlooked by virtually all the previous authors and in the traditional solar modelling. This is the remarkable effect of a stable shear flow on turbulent transport, namely turbulence regulation (or shear stabilization) (e.g. Burrell 1997; Hahm 2002; Kim 2004). When a shear flow acts on a (turbulent) eddy, it advects its different parts at different rates, distorting its shape and generating small scales in the direction orthogonal to the flow. This process continues until the eddy becomes disrupted by dissipation (see Fig. 1). As a result of this shearing process, the transport orthogonal to the shear flow, as well as the intensity of turbulence, is reduced (Kim & Dubrulle 2001; Kim & Diamond 2003; Kim et al. 2004; Kim & Diamond 2004). This shear stabilization has been recognized to be critical to regulating turbulence in many physical systems, including laboratory plasmas (Burrell 1997; Hahm 2002; Kim 2004), geophysial systems (Busse 1983), earth atmosphere (McIntyre 1989), etc. In particular, it is now thought to be the most promising mechanism to achieve an economic future reactor with a good confinement. Similar effect of a shear flow is likely to play a crucial role in the transport in the tachocline, especially by the radial shear which is thought to be stable due to stable stratification. Furthermore, as shearing is most effective in reducing the transport in the direction orthogonal to the shear flow (i.e., in the x direction in Fig. 1), it can cause the anisotropy in turbulent transport as well as turbulence intensity in the tachocline.
Figure 1: Tilting and break-up of (isotropic) turbulent eddy due to the shearing by shear flow . The wavenumber in the x direction increases as , linearly proportional to time t. | |
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In addition to the shearing by the radial differential rotation, Coriolis forces due to the average rotation may also quench the transport and lead to anisotropic turbulence in the tachocline. This effect of Coriolis forces has been studied in the context of the transport of angular momentum (e.g. Rüdiger 1983; Kichatinov 1987; Rüdiger 1989; Kichatinov & Rüdiger 1993) and heat (Rüdiger 1989; Kichatinov et al. 1994; Kichatinov & Rüdiger 1995) in the convection zone to understand the prominent latitudinal differential rotation in that region. In particular, in the limit of strong rotation such that the rotation rate exceeds the background turbulence decorrelation rate, which is the case for the Sun and most single main-sequence stars (Basri 1985), the turbulent viscosity (eddy viscosity) and heat diffusivity are shown to be reduced inversely proportional to the average rotation rate while their values parallel to the rotation axis are a factor of 2 and 4 larger than those in the perpendicular directions, respectively (e.g. Kichatinov et al. 1994). Furthermore, Coriolis forces can give rise to the so-called effect (similar to the effect in dynamos) from the non-diffusive part of Reynolds stress which is proportional to the rotation itself if the background turbulence in the absence of Coriolis forces is anisotropic (e.g. Rüdiger 1983) or inhomogeneous (Kichatinov 1987). In the case of strong rotation, the effect due to the anisotropy in the background turbulence also decreases as (Rüdiger 1983). Note that in these previous works, the effect of rotation shear was treated perturbatively.
The aim of this paper is to develop a self-consistent theory of turbulent transport incorporating the effect of shearing on transport, for the first time in the solar context. We focus on the shearing due to the radial different rotation in the tachocline and study how it affects the turbulent transport. In contrast to most of previous works which often crudely parameterized turbulent coefficients and then adjusted them to obtain agreement between prediction and observations, we shall self-consistently compute them under the physically plausible assumption that turbulence arises either from plumes penetrating from convection zone or from the instabilities of the latitudinal differential rotation. We will demonstrate that in both cases the transport can significantly be reduced by shear stabilization, with the different reduction in the horizontal plane and vertical direction. Specifically, we show that the shearing by the radial differential rotation leads to much stronger reduction in the particle transport in the radial direction, orthogonal to the shear flow (with the scaling with the shearing rate ) than in the horizonal plane (with the scaling ). A similar, but weaker, anisotropy also develops in the amplitude of turbulent flow. In the case where the turbulence is driven by the instability of the latitudinal differential rotation, the overall amplitude of vertical particle transport as well as vertical turbulent flow becomes negligible. Special attention will be paid to the distinction between turbulence intensity and turbulent transport which have different physical origins and thus different scalings with , as shall be shown. These results suggest that even without density stratification, the radial differential rotation alone can give rise to anisotropic turbulent transport in the tachocline. We will also provide a self-consistent theory of the transport by a meridional flow by incorporating the effect of shear flow on turbulence, and assess its efficiency by comparing it with turbulent transport. To understand the effect of shear flows on the transport in the simplest way, the analysis in the present paper shall be limited to the case of non-magnetized tachocline with incompressible fluid, with no background density stratification, and with only radial differential shear. To keep the analysis tractable, the effect of Coriolis forces will also be ignored. While the average rotation rate is comparable to radial shear in the tachocline (i.e., ), the incorporation of Coriolis forces is unlikely to fundamentally change the results obtained in the paper for the following reason. Since the turbulent heat conductivity tensor considered by the previous authors (e.g. see Eq. (7) in Kichatinov et al. 1994) takes the same form as turbulent particle diffusivity tensor (in the absence of shearing), a similar quenching and anisotropy found in heat diffusivity is expected for turbulent diffusivity of particles due to Coriolis forces. Thus, the quenching in turbulent (eddy) viscosity and particle diffusivity due to Coriolis forces would become proportional to , which is much weaker than the reduction caused by radial shear, found in this paper. Furthermore, while Coriolis forces alone favor the transport in the direction parallel to the rotation axis, the resulting anisotropy in eddy viscosity and turbulent diffusivity will be very weak, being only a factor of 2 and 4. In comparison, the anisotropy due to the radial shear depends on a small parameter inversely proportional to , and can thus become very large for strong shear . The effects of the magnetic fields, density stratification, and latitudinal differential rotation as well as Coriolis forces will be considered in the subsequent papers.
The remainder of the paper is organized as follows. We shall first consider the case where transport takes place due to turbulence with the negligible effect from meridional circulation and formulate the problem in Sect. 2. In Sect. 3, the effect of radial shear on turbulence intensity and momentum and particle transport will be discussed. We then include the meridional flow and investigate its effect on particle transport in Sect. 4. Section 5 contains discussions and conclusions.
We consider incompressible fluid
with no average rotation
and local cartesian coordinates x, y, and z for
radial, azimuthal, and latitudinal directions, respectively.
Then, main governing equations for the total velocity
and
density n of chemical elements are as follows:
To simplify the analysis, we shall limit ourselves
to the case of unit Prandtl number ()
and to
the quasi-linear evolution (see, e.g. Moffatt 1978)
of mean fields and fluctuations.
By taking average of Eq. (3), we obtain the following
equations for the mean fields n_{0} and U_{0}:
For the evolution of fluctuations, we approximate
the radial differential rotation
by a linear shear flow with
(here,
is shearing rate which is assumed to be positive without
loss of generality) and obtain the
following equations for
and n' from
Eqs. (1)-(3):
With the help of Eq. (9),
the solutions to Eqs. (6)-(8)
can be obtained after a long, but straightforward algebra.
Referring the readers to Appendix A for intermediate steps,
in this section, we simply provide the solutions, which are
In this section, we shall demonstrate that the radial shear reduces turbulence amplitude in the vertical direction and in the horizontal plane, and vertical transport of momentum and particles by shear stabilization. The degree of reduction by shearing can be different in each case; in particular, turbulence amplitude depends on the direction, suggesting anisotropic turbulence.
From Eqs. (9)-(12) and (15)-(18),
we obtain the following:
Therefore, the reduction in turbulence amplitude by the radial shear is more severe in the radial direction than in the horizontal plane, with the anisotropy in turbulence amplitude. This is because a shear flow in the y direction, varying in x, shears the radial turbulent flow v_{x} directly while its shearing only indirectly influences the horizontal flow v_{z} through the incompressible condition ( ) and enhanced dissipation for finite D. Thus, shearing regularizes the horizontal turbulence amplitude , with its value remaining finite even in the absence of dissipation (D=0). In contrast, it is a finite dissipation () that keeps finite. The anisotropy in turbulence amplitude should, however, be distinguished from that in turbulent transport, as will be discussed in Sect. 4. Equations (23)-(24) also indicate that the precise value of depends on the characteristics of the forcing, such as power spectrum and typical wavenumber . We briefly discuss this dependence in the following.
If the turbulence is mainly driven from plumes
penetrating from the convection zone (see Fig. 2),
the power spectrum is likely to be anisotropic with
.
Thus,
and
with
.
For example, for the parameter values of cm^{2} s^{-1},
s^{-1},
and
cm
,
.
In contrast, if turbulence is due to the instability of
latitudinal differential rotation, the forcing is likely act
only on the horizontal plane with
.
Thus,
,
rendering
and
in Eq. (22).
Thus, in this case, there is no turbulence in the radial
direction with
while the horizontal turbulent flow is reduced for large
as
.
This result of the reduction in the horizontal turbulence (parallel to the
shear flow) in the absence of vertical flows, which may seem surprising
to some readers, is
basically due to the enhanced dissipation by shearing for ,
as noted previously. That is, even if the forcing drives
only horizontal turbulent flows, the shear flow creates fine scales in the
vertical direction in these flows, enhancing their dissipation
rate.
Therefore, in both cases, the turbulence amplitude
in the radial direction is always much weaker than
that in the horizontal plane (see Fig. 2).
It is also interesting to consider
the special case of an isotropic forcing where
,
with
Figure 2: Tilting and break-up of turbulent eddy driven by plumes due to the shearing by radial differential rotation . The wavenumber in the x direction again increases as , linearly proportional to time t. The turbulent eddy is almost uniform in x with initially at t=0, but gradually develops radial dependences as the shearing continues. | |
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In order to compute momentum and particle fluxes (thereby turbulent
viscosity
and diffusivity
), we
go through a similar procedure by using
Eqs. (9)-(13), (15)-(17), and (21)
(see Appendix B for details). The results are
Let us examine, in some detail, other important implications of Eqs. (26) and (27), excluding the case where f_{x} = 0. First, Eq. (26) shows that turbulent viscosity can be either positive or negative depending on characteristic scales of the forcing. In the 2D limit where (i.e. in the x-y plane), becomes negative, recovering the well-known result of inverse cascade in 2D hydrodynamic turbulence due to the conservation of enstropy (see, e.g., Eq. (33) in Kim & Dubrulle 2001). Negative viscosity means that turbulence acts as a source of large-scale shear flow. Equation (26) clearly shows how this negative viscosity becomes positive in 3D hydrodynamic turbulence for , with turbulence now damping the shear flow. Interestingly, the result leads to a non-linear equation for in Eq. (5), with the possibility of causing time-dependence in the evolution of the mean flow . Note that time-dependence of the radial differential rotation can also be caused by the interaction with gravity waves (Kim & MacGregor 2001), with time variability being a generic feature of a nonlinear system. Furthermore, since turbulence (with ) becomes inefficient in smoothing out the profile of as it develops larger gradient, the (angular) momentum transport by turbulence within this hydrodynamical turbulence does not seem to be easily reconciled with the apparent uniform rotation in the solar radiative interior.
Secondly, Eq. (27) shows that turbulent diffusivity is proportional to , but with the proportionality , depending on . That is, particle transport is not exactly proportional to turbulence amplitude, as has often been assumed. If turbulent flow has the characteristic velocity v_{x} with characteristic length scale l_{x} and correlation time in the x direction, one would expect that . The comparison with our result then implies that and . That is, the characteristic length scale and correlation time of the turbulent flow are set by shearing time , both of them becoming shorter as increases. This is a clear manifestation that the decorrelation rate of two near-by points in the fluid is enhanced by a shear flow (the so-called enhanced decorrelation) with a shorter correlation length (Biglari et al. 1990), as a result of tilting and distortion of eddies. This point will again be emphasized in Sect. IV in considering particle transport in the horizontal plane ( ).
Finally, we estimate the minimum velocity of forcing (e.g.,
the velocity of plumes) in order
for the turbulent transport
to be at least larger
than molecular value
cm^{2} s^{-1}. To this end,
we approximate Eq. (27) as
So far we have focused on the effect of shearing by the radial differential rotation on turbulent transport and studied how much reduction it causes in turbulence amplitude and vertical transport of particle and momentum. In this section, we will examine how the particle transport by meridional circulation is affected by the radial differential rotation, assess its efficiency compared to vertical turbulent transport of particles, and discuss the anisotropic transport of particles by computing horizontal turbulent transport of particles. As noted previously, the meridional circulation can be crucial to explaining the depletion of light elements in the Sun or massive stars.
We assume the meridional circulation to take the form
and include it
in the total velocity
.
The density perturbation n_{m}(x,z) due to
meridional flow also contributes to the total density
n=n_{0}
+n_{m} (x,z) + n'.
We denote the average over fluctuations of total density and velocity
by single angular brackets
as before. That is
Thus, shearing results in the anisotropy in particle transport, as well as in turbulence intensity, with less transport in the radial direction. Physically, this result makes sense since it is only through diffusion that the radial shear has any effect on the particle transport in the horizontal plane. In fact, in the limit , as density fluctuation accumulates on small-scales in the absence of diffusion while , suffering directly from shearing, remains finite. For finite D, the transport in the horizontal plane is affected by shearing only through the enhanced dissipation, and thus less reduced.
As previously noted, the anisotropy in the particle transport is, however, different from that in turbulence amplitude . Furthermore, in comparison with Eq. (24), Eq. (37) indicates . Thus, the proportionality between and is not constant, but depends on . We recall that we have found a similar result for the transport of particle in the vertical direction with . To see how the shearing affects the effective correlation time and length scale of turbulent flow l_{z} in the horizontal plane, we let the characteristic velocity of turbulent flow v_{z} in that plane. Then, by comparing the conventional expectation with our result , we obtain and . These results indicate that horizontal turbulence is also subjected to the enhanced decorrelation by shear flow, but only through dissipation (). This is also true in the case of a purely horizontal forcing with f_{x} = 0, similarly to (see the discussion before Eq. (25) in Sect. 3.1). Therefore, while in the presence of non-vanishing dissipation (), the shear flow shortens the effective length scale l_{z} and decorrelation time in the horizontal plane, the decorrelation time for horizontal turbulence is still larger than for vertical turbulence for which the radial shear operates more efficiently.
When turbulence is driven by plumes,
.
In this
case,
Eqs. (27), (29), (33), and (37)
suggest that
the transport by
dominates over the transport
by turbulence (
)
when
To assess the importance of the transport by a meridional circulation in the Sun, we take s^{-1}, cm^{-1}, cm^{2} s^{-1}, , and cm; thus and and . Here, H_{0} cm is pressure scale height. We note that in order to be consistent with the depletion of lithium, the total (vertical) diffusivity of chemical elements near the tachocline must be cm^{2} s^{-1}(Barnes et al. 1999). Thus, by requiring that the maximum of and be set by this total diffusivity 20 D, we can consider the following two cases.
The results have very interesting implications for the mixing and diffusion in the tachocline. Since the radial differential rotation is largest near the poles and equator, the quenching of the turbulent transport (such as diffusion of chemical elements) will be most prominent near the poles and equator with the latitudinal dependent diffusion and mixing. They may also have an important implication for the long-term dynamics of the solar tachocline. According to our results, a strong radial shear can cause anisotropic turbulent transport of particles in that regime. If similar anisotropy also develops in momentum transport (i.e. eddy viscosity), it can in turn prevent the radiative spreading of the region (Spiegel & Zahn 1992). That is, the radial shear may have a positive feedback on keeping a strong radial shear in the thin regime. In order to address this problem, the present work should be extended to include the latitudinal differential rotation and to study horizontal turbulent transport of momentum. Of course, there are other alternative mechanisms for maintaining thin tachocline. For instance, it may be through Lorentz force in the case of magnetized tachocline (Rüdiger & Kitchatinov 1996; Gough & McIntyre 1998; MacGregor & Charbonneau 1999). Nevertheless, the aforementioned positive feedback due to shearing can still play an important role in the overall dynamics of the tachocline and thus should be investigated. In the case where the tachocline evolves on a short time scale of order of 1 yr (i.e. fast tachocline), for instance, due to the instability of latitudinal differential rotation, (e.g. Gilman & Fox 1997; Dikpati & Gilman 1999; Gilman & Dikpati 2000; Cally et al. 2003), the overall angular momentum transport in the tachocline should be closely linked to that in the convection zone (Gilman 2000), and other physical effects not included in the paper will play an important role in determining the thickness of the tachocline.
Finally, we note that the analysis in the present paper was limited to purely hydrodynamical turbulence without density stratification and without Coriolis forces in the tachocline in order to clearly identify the effect of radial differential shear. While turbulence quenching and anisotropy in the tachocline can be caused by shearing effect alone, it is important to include the effects of Coriolis forces, density stratification and strong toroidal magnetic fields. Even if the reduction in the transport and anisotropy caused by Coriolis forces alone is likely to be much weaker as compared to those by the radial shear discussed in this paper, it is important to study how Coriolis forces together with radial shear affect the overall dynamics of turbulence and transport in the tachocline. In particular, Coriolis forces may lead to the effect, with an interesting consequence in the angular momentum transport in the tachocline. On the other hand, both density stratification and magnetic fields would enhance the anisotropy in turbulence due to radial shear found in the paper, by further reducing transport in the radial direction. In fact, it is the density stratification which is commonly thought to be responsible for anisotropic turbulence in the tachocline. In addition to causing anisotropy, magnetic fields may also have different effects on the transport of particles and momentum, which can have a significant implication for explaining apparent, more efficient angular momentum transport than the mixing of lithium over the evolution of the Sun. Note that within the hydrodynamical model considered in the paper, the transport of particle and momentum appears to be comparable. Note also that there are alternative mechanisms to explain these problems including gravity waves (e.g., García Lopez & Spruit 1991). Furthermore, in the magnetized tachocline, the problem of transport of magnetic flux itself would become of considerable interest for solar dynamos. The work addressing these issues is in progress and will be published in the subsequent papers.
Acknowledgements
The author thanks M. J. Thompson and anonymous referee for useful comments.
In this Appendix, we show how to derive Eqs. (11)-(14)
from Eqs. (6)-(8). By using Eqs. (9) and (10) in (6)-(8), we obtain the following
set of equations:
To obtain Eq. (27),
we substitute Eq. (11) in (14) and change the
order of time integrals to obtain