A&A 375, 149154 (2001)
DOI: 10.1051/00046361:20010849
Lithium as a passive tracer probing the rotating solar
tachocline turbulence
G. Rüdiger^{1,3}  V. V. Pipin^{1,2}
1  Astrophysikalisches Institut Potsdam, An der
Sternwarte 16, 14482 Potsdam, Germany
2 
Institute for SolarTerrestrial Physics,
PO Box 4026, Irkutsk 664033, Russia
3 
Department of Mathematics, University of Newcastle upon Tyne,
NE1 7RU, UK
Received 25 February 2000 / Accepted 7 June 2001
Abstract
The rotational influence on the eddydiffusivity tensor D_{ij} for
anisotropic turbulence fields is considered in order to explain
the lithium decay law during the spindown process of
solartype stars. Rotation proves to be highly effective in
the transfer of chemicals through the solar tachocline (beneath the
convection zone) which is assumed to contain only turbulence
with horizontal motions. The effect is so strong that the
tachocline turbulence must not exceed a limit of
of the rms velocity in the convection
zone in order to let the lithium survive after
Gigayears. Such long depletion times can also be explained by
a very small rotational influence upon the eddydiffusion
tensor if it is realized with correlation times shorter
than 15 min. It is argued that such slow and/or
shortliving turbulence beneath the convection zone could hardly
drive the solar dynamo.
In our theory the diffusion remains small for rapid rotation due to the
rotational quenching of the turbulence. In young stellar clusters like
Pleiades, there should be a (positive) correlation between rotation rate and
lithium abundance, where the fastest stars should have maximal lithium. First inspections of the data seem to confirm this finding.
Key words: turbulence  stars: abundances  stars: rotation
According to Duncan (1981), the lithium at the
surface of cool mainsequence stars decays exponentially, unlike the stellar
spindown, which follows a power law
t^{1/2} (Skumanich 1972). The characteristic decay time is
about 1 Gyr. The primordial lithium is destroyed by
nuclear reactions at temperatures in excess of
K (see Michaud & Charbonneau 1991; Ahrens et al. 1992), i.e.40000 km below the lower edge of the
convection zone (Fig. 1). There must be a drift process for the
chemicals from the bottom of the convection zone through the
solar "tachocline'' to the burning domain. The effect must be
small, however, in order to allow the existence of
lithium in the solar atmosphere even after 4.6 Gyr. The
lithium decay time is about 10^{7} times the convection zone
diffusion time of 100 yrs. We are thus looking for a
rather small effect which, however, cannot simply be
microscopic diffusion (Schatzman & Maeder 1981; Spruit 1984; Zahn 1989, 1992).
In the following, the
consequences are presented of a quasilinear meanfield
approximation of an anisotropic turbulence field which might be
located in the solar "tachocline'', which is stably stratified
unlike the unstablystratified solar convection zone. However,
to form reduction
factors of
10^{7...8} for any eddy diffusivity is a
nontrivial problem. In other words, the desired eddy
diffusion coefficient only exceeds the molecular one by two
orders of magnitude (Baglin et al. 1985; Lebreton & Maeder 1987)
and this is not easy to explain (Vincent et al. 1996).

Figure 1:
The outer structure of the Sun during its mainsequence life.
R_{i} limits the unstably stratified outer convection zone
while R_{0} displays the lithium burning zone with a temperature
of 2.6 Mio K. Computed by
Kämmerer (2000) with the solar model of Stix & Skaley (1990). 
Open with DEXTER 
The transport of a passive scalar is governed by the diffusion equation,

(1) 
with D as the microscopic diffusion coefficient. In the sense
of the anelastic approximation we shall always apply the
sourcefree condition of the mass flux, i.e.

(2) 
If the field
describes a turbulent flow pattern, the fields are
split into a mean and a fluctuating part,

(3) 
Averaging (1) we get the wellknown diffusion equation in
the presence of turbulence

(4) 
The influence of a meridional circulation considered by
Charbonneau & Michaud (1991), Zahn (1992) and
Chaboyer & Zahn (1992) is neglected here.
The
fluctuations of the chemical concentration in a quasilinear
approximation follow the relation

(5) 
From this we have to compute the turbulent concentration
flux vector
which must be inserted
into (4) in order to find the meanfield diffusion
equation. In particular, we are interested in the effect of the
basic stellar rotation and its
consequences during the stellar spindown process (see Charbonnel et al. 1992).
In a corotating
frame of reference the equation for the fluctuating part of the velocity field
is

(6) 
with
as the random turbulence force and
as
some background eddy viscosity. The desired correlation between
concentration fluctuations and velocity fluctuations,
,
can be found by Fourier
transforming (6) using

(7) 
As usual, the original anisotropic turbulence is assumed to be stationary,
homogeneous and anisotropic, i.e.




(8) 
(cf. Rüdiger 1989).
is
the isotropic part of the turbulence spectrum,
and
is the vertical unit vector. The radial
turbulence intensity may be denoted by
while for the azimuthal turbulence intensity
.
An anisotropy parameter s is defined by
v^{2} = s w^{2}, so that a large s denotes horizontaltype turbulences. It
is

(9) 
The vertical vector
represents the basic anisotropy, which
is described by the spectrum
of additive horizontal motions

(10) 
The result for the turbulent concentrationflux vector may be
written as an anisotropic diffusion in terms of the mean concentration
gradient, i.e.

(11) 
(cf. Dolginov & Silantev 1992). We shall compute in the following the diffusion tensor
without rotation and with rotation for the isotropic and anisotropic
parts of the turbulence fields.
The result is very simple without rotation. It follows

(12) 
or just

(13) 
with the eddy diffusivity
formed here only with the vertical turbulence intensity.
More structure results if the turbulence is subject to a basic
rotation. The expressions are given here only in the
socalled approximation which is very close to the mixinglength
approximation (Kitchatinov 1986). We find

(14) 
with
as the vector
parallel to the rotation axis. The components of the tensor are
(see Hathaway 1984). The
ensures a latitudinal transport of the
chemical composition.
Without rotation
and
.
The rotational quenching
functions result as

(18) 
and

(19) 
with the Coriolis number
.
As usual we consider the temperature profile below the convection zone as
stable so that any vertical fluctuations there are strongly suppressed. Any
turbulence which possibly exists in the tachocline must be anisotropic. As a model of such
an anisotropic turbulence, in this section a strictly horizontal turbulence is
considered, i.e., there are, for zero rotation, no vertical
fluctuations.
For horizontal turbulence without rotation the diffusion
tensor becomes

(20) 
without any vertical components. But with rotation and within the
approximation we find



(21) 
which yields in spherical coordinates the tensor components
The amplitude s is introduced as the intensity of the horizontal turbulence
in units of the intensity of the isotropic turbulence in the bulk of the
convection zone.
The rotational quenching functions result as
Without rotation
and
,
only the latitudinal diffusion
coefficient
exists. With rotation, the appearance of the vertical diffusion D_{rr} and the offdiagonal components
can be observed (Fig. 2), which are playing  as we shall see
 key roles in the
meanfield equation for the largescale concentration distributions in
rotating turbulence fields. One can demonstrate that this equation for all
remains of the elliptical type.
Note that the influence of rotation is so strong that for
the vertical diffusion  which is only rotationally created  equals the latitudinal diffusion. Note also that the new diffusion coefficients are strongly dependent, resulting in dependent profiles of the mean concentration (see below).

Figure 2:
The diffusion tensor components (22) for midlatitudes (
)
in their dependence on the Coriolis number .
Note that for fast rotation (
)
all components are rotationally quenched. 
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The meanfield diffusion equation without
largescale circulation reads



(24) 
The boundary conditions are

(25) 
at the lower boundary (x_{0}=0.6), where the lithium may be destroyed, and

(26) 
at the solar surface (x=1) where no radial flux may be allowed.
For the correlation time in the definition of
a
radial profile is used very similar to the radial profile of
the turnover time in the mixinglength theory of the solar
convection zone, i.e.

(27) 
where
is the value at
.
Within the
surface layer, x>0.95, the Coriolis number
was put to zero.
Equation (24) is used in dimensionless form, distances measured in units
of radius, R, and times measured in units of the
diffusion time,
.

Figure 3:
The overall turbulence model. Below the convection
zone with isotropic turbulence there is a stable "tachocline''
layer with a horizontal turbulence field. At its bottom
lithium is burned. 
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We start with a model close to that of Vincent et al. (1996).
The diffusivity
may be constant in the whole
integration region,
.
Between the burning
zone at x_{0} and the bottom of the convection zone,
,
there is a turbulence field with motions
only in the horizontal directions (Fig. 3). Its intensity is
given by the parameter s. Following Spiegel & Zahn (1992) we
shall refer to the region
as the solar
"tachocline''. The probing of its turbulence with respect to the lithium problem is the
main scope of the present study.
The general influence of rotation on the spatial distribution of
chemicals inside the star is given in Fig. 4 for both
and
.
The rotational influence
produces a distinct dependence of the concentration on latitude.

Figure 4:
Isolines of the chemicals without (Left)
and with (Right) rotation feedback on the turbulence,
.
In both cases s= 100. Note that the rotation produces
latitudinal profiles in the composition. 
Open with DEXTER 

Figure 5:
The decay of chemicals at the surface of the convection zone
for isotropic turbulence in the whole computational region, x_{0}<x<1.
No rotation (solid), with rotation (
,
dashed), extra horizontal
turbulence in the tachocline beneath the convection zone (
,
s=100, dotted). 
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Figure 5 shows the decay of chemicals at the
surface. The solid line corresponds to nonrotating
turbulence for any value of s. The same result is obtained even
for s=0, i.e.horizontal turbulence.
The result of Michaud & Charbonneau
(1991) and Charbonneau (1992) is confirmed in that "horizontal turbulence cannot reduce the effect of a given vertical turbulent diffusion coefficient''.
The dashed curve describes the depletion of the chemicals under
the action of isotropic but rotating turbulence. Note the
rotational slowdown of the chemical depletion.
Faster depletion, however, is provided by the extra
influence of rotating anisotropic turbulence beneath the
convection zone. By this result the influence of the rotation
on the turbulence field is reflected. The basic phenomenon is
the formation of a latitudinal profile of
the concentration as shown in Fig. 4.
We have to take from Fig. 5 that the additional anisotropy due to the
basic rotation acts in an unexpected way: rotation suppresses
the mixing of isotropic turbulence but, on the other hand, it
enables a horizontal turbulence (which without rotation would not be
active) to accelerate the vertical mixing. The rotation in the mixing process
which is demonstrated in Fig. 5 has a very complex character: it both
suppresses and deforms the turbulence. The interplay between both procedures
yields the resulting effect. We shall see that for slow rotation enhanced diffusion
dominates while for fast rotation the diffusion reduces the suppression of
turbulence. As a result for a given rotation rate a maximal diffusion rate exists
between slow and fast rotation.
Now the turbulence in the convection zone is considered as isotropic
turbulence under the influence of
rotation described in Sect. 2.1. Below the convection zone
the turbulence may be so anisotropic
that vertical motions do not exist. The basic rotation, however,
produces offdiagonal components in the diffusivity tensor as described in Sect. 2.2.
Thus we have 2 free parameters for the
tachocline turbulence: intensity s of the
horizontal motion and correlation time
of the
eddies. For
the turbulence
completely disappears and for
the
rotational influence disappears. In both limits the decay time of
the chemicals at the solar surface must become infinite.
More information is given in Fig. 6, where the
radial dependencies of the concentration profiles inside
the star are given for three different latitudes and two
different horizontal intensities.
The diffusion is now much slower than with isotropic turbulence in the whole
computational space, as in Fig. 5.
The
differential concentrations are produced in the tachocline and
are directly imprinted at the surface and are therefore not screened by the convective pattern within the convection zone. This is the
same situation as for differential rotation but different from turbulent heat transport (cf. Stix 1981). The
timedependence of the surface differences of the concentration
is so weak that our snapshots are always characteristic
the situation.

Figure 6:
Radial profiles of the chemical concentration along the equator (solid), midlatitudes (dashed) and the poles (dotted) for s=1 (Left)
and s=0.01 (Right). Only for small horizontal intensities does the
poleequator difference become rather small. The
timedependence is now very weak,
at the base of the
convection zone and also within the tachocline. 
Open with DEXTER 

Figure 7:
Depletion time in units of the diffusion time. For weak
horizontal turbulence intensities it approximately varies with 1/s.
at the base of the convection zone and also
within the tachocline. 
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In Fig. 7 the decay time of an initially uniform
concentration is plotted for various horizontal turbulence
intensities in the tachocline. For s=0, of course, there is no depletion,
but the depletion time is strikingly short for nearly homogeneous turbulence fields with (say)
s=1. The correlation time profile and the basic rotation
are taken from a solar model by Stix & Skaley with
at the base of the
convection zone. We find that rotation is highly effective in
transporting the chemicals to the burning zone at x_{0}. A scaling such as

(28) 
is found compatible with the idea that the diffusion of horizontal
turbulence under the influenc of rotation with
can
be considered approximately as diffusion of isotropic turbulence. In reality, both the characteristic times differ by 7 orders of
magnitude. Therefore, the
horizontal rms velocity of the tachocline turbulence must not
exceed
of the convection zone turbulence. That there
is still lithium at the solar surface is only compatible
here with the given concept of rotating turbulence with very small
s of the tachocline turbulence. Consequently, the horizontal
turbulence intensities must be very low, i.e.
of the order of 1 cm/s. There is certainly no
chance of maintaining magnetic fields of the order of kGauss by such a
slow flow.
But there is also the possibility that the tachocline
turbulence has a rather short correlation time, e.g. similar to
that of
granulation at the top of the convection zone. Then the
Coriolis number, ,
is smaller than unity and the
rotational influence can only be very small. For a given intensity
(s=1) the resulting depletion times are shown in
Fig. 8. We find a relation
with
in seconds. Similarly, for s=0.01 we find

(29) 
from our simulation.
In the latter case with
s the
lithium decay time will approach the order of 10^{9} yrs. The corresponding Coriolis number is always smaller than 0.05.
Again, with such a small value there is no hope of forming an effective
effect for an appropriate dynamo process. In this approach, we find a
close relation between the lithium problem and the theory of the solar dynamo.
Within this framework, the observations do not favour the existence of an overshoot dynamo below the bottom of the convection zone.

Figure 8:
Depletion time
(in units of the diffusion time) for horizontal turbulence
intensities with s=1 strongly depends on the correlation
time of the tachocline turbulence, taken here
for a rotation period of 25 days. 
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For a given stellar model the basic rotation also forms a
free parameter which can be varied.
One can vary the basic rotation rate such as it
varies in a young stellar cluster for all stars with the same
age. The question is how the lithium varies.
The same turbulence model can be
used within the sample. We take s=0.01 and
= 12 days. Figure 9
shows the results. Due to the rotational quenching of the eddy diffusivity
(see Boubnov & Golitsin 1995), the faster the rotation the more lithium
remains at the surface. The rotational quenching of the eddy
diffusivity dominates the effect of the large latitudinal
gradients of the concentration. The slower rotators are thus more
effective in mixing the chemicals downwards. The plot,
however, does not display that for still slower rotation, in the
limit for
,
the mixing again becomes slower and slower and therefore the decay
of the primordial lithium, too. The observation of the
lithiumrotation correlation in young stellar clusters seems to
confirm our prediction.

Figure 9:
The general situation for young stellar clusters: the same turbulence model with s=0.01 but the angular
velocity varied. The surface concentration at the equator
is shown after 100 diffusion times. Due to the rotational quenching of the
eddy diffusivity, the fast rotators possess the
highest surface concentration of chemicals. 
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Rotation easily produces offdiagonal elements and other new terms in the diffusion tensor.
We have shown with our computations that under the rotational influence, even a
horizontal turbulence field will be very effective in transporting chemical
components in the vertical direction. Two main consequences of this finding have
been formulated, i.e.
 even a strictly horizontal turbulence in the solar tachocline must be very weak (or
very shortlived) to ensure the observed slow diffusion of lithium downward to
the 2.6 Mio K layer where it burns;
 for Gstars in young stellar clusters a correlation should exist
between the lithium abundance and the actual rotation rate. The faster the
rotation, the stronger the eddy diffusivity is quenched and the higher the
lithium concentration in the convection zone must be.
Our concept also provides statements for the longterm evolution
of the lithium abundance under the influence of the general spindown of
solartype stars. To this end a timedependent code (which is in preparation) must simultaneously include
both the time and depthdependent effects of lithium burning and the decay law
of the rotation rate with time.
Acknowledgements
V. V. Pipin acknowledges the kind support by the Deutsche
Forschungsgemeinschaft (436 RUS 113/255). The referee, J. P. Zahn, is acknowledged for many valuable suggestions concerning the concept of the paper.

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Copyright ESO 2001