The Astrophysical Journal, 611:568–574, 2004 August 10
# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
LIMITS TO PENETRATION OF MERIDIONAL CIRCULATION BELOW THE SOLAR CONVECTION ZONE
Peter A. Gilman and Mark S. Miesch
High Altitude Observatory, National Center for Atmospheric Research,1 Boulder,
CO 80307-3000; [email protected], [email protected]
Received 2004 February 4; accepted 2004 April 14
ABSTRACT
We show that meridional circulation, such as that observed at the top of and in the solar convection zone (CZ)
by direct doppler and helioseismic techniques, cannot penetrate significantly below the bottom (0.7 R) of the
overshoot layer at the bottom of the CZ. Therefore, solar dynamo models that rely on penetration as deep as to
0.6 R are ruled out. The analysis we carried out to reach this conclusion elucidates two boundary layers, one of
which we have not seen applied to astrophysical problems before. This analysis should be relevant to understanding interfaces between convective and radiative zones in stellar interiors generally.
Subject headingg
s: hydrodynamics — stars: interiors — stars: rotation — Sun: interior — Sun: rotation
1. INTRODUCTION
The existence of meridional circulation in the solar convection zone has recently been demonstrated to be very important for understanding the solar dynamo. The most
successful current solar dynamo models, the so-called flux
transport models (Dikpati & Charbonneau 1999; Dikpati &
Gilman 2001; Dikpati et al. 2002), depend on the presence of
poleward flow in the upper part of the convection zone, coupled with weaker equatorward flow near the bottom, to produce
a number of dynamo features that compare well to properties of
the solar cycle. Perhaps the most significant feature is the solar
cycle period, which has been shown to be produced by equatorward flow near the bottom of the convection zone, and
which is consistent with the observed 20 m s1 poleward
flow in the photosphere and the upper convection zone
(Dikpati & Charbonneau 1999). The period of these dynamos
is much more sensitive to the meridional flow speed than it is to
assumed magnetic diffusivities or even to the details of the
differential rotation profile (Dikpati & Charbonneau 1999).
Previous mean-field dynamo models had to rely much more on
‘‘tuning’’ the diffusivity and /or differential rotation to get the
correct period.
Nandy & Choudhuri (2002) have recently demanded even
more of the meridional circulation, namely that it penetrate
well below the bottom of the convection zone, down to a
depth of perhaps 0.6 R. This requirement is dictated by their
particular solutions to the flux-transport dynamo equations, in
order to produce the best agreement with observations of the
latitudinal distribution of sunspots (the butterfly diagram).
There is no agreement among dynamo modelers at this time
that such a deep meridional circulation is in fact necessary. We
do not pursue that question further here, but instead focus on
the question of whether it is physically realistic that such a
deep penetration should occur. Since the dynamo models in
question are kinematic, the meridional circulation can be
simply assumed to reach down this far. But when the more
realistic problem is solved, with all the relevant forces included in the equations of motion, even without a magnetic
field present, can such penetration be achieved? We show that
the answer is no for all reasonable assumptions.
There are additional problems with a meridional circulation
that penetrates as deep as 0.6 R. Such a flow would transport
light elements such as lithium and berylium from the convection zone to deeper layers, where they would be destroyed
Recently, both surface doppler and helioseismic measurements have demonstrated the existence of a meridional flow
on the Sun (Hathaway 1996; Giles et al. 1997; Braun & Fan
1998; Schou & Bogart 1998; Haber et al. 2002; Basu & Antia
2003). This flow is predominantly toward the poles at all
observed latitudes in both hemispheres in and near the photosphere, although since 1996, when this flow has been detected
by multiple means, there have been years in which an equatorward flow has been seen in high latitudes in the northern
hemisphere (Haber et al. 2002). The magnitude of the poleward flow is 20 m s1, but fluctuates significantly about this
value.
Since mass is not observed to be piling up near the solar
poles over time, there must be a return flow somewhere inside
the Sun. Helioseismic measures have so far failed to reveal
this return flow, down to a depth of 0.8 R (Braun & Fan
1998). The deeper the flow is, the smaller it needs to be,
because of the substantial rise of density through the solar
convection zone. In fact, the poleward mass flux seen in the
photosphere is tiny compared to that occurring down to the
depth of helioseismic measurements, for the same reason.
There is currently no satisfactory theory for this meridional
flow, although all global convection models for solar differential rotation produce such flows. This is because these
models tend to produce much more structured meridional
flows, with typically several ‘‘cells’’ between the equator and
poles, probably because of the larger influence of rotation on
the flow in these models than is apparently occurring on the
Sun (Miesch et al. 2000; Elliot et al. 2000; Brun & Toomre
2002). Also, since the meridional flow predicted is inevitably
the result of small imbalances among large meridional and
radial forces, namely coriolis, pressure gradient, and buoyancy, as well as turbulent stresses, high accuracy is needed to
get a reliable prediction. In any case, the driving mechanisms
for meridional flows in the solar convection zone are likely to
be far stronger than those in the radiative zone, be they of
thermal or mechanical origin, or both (see x 5).
1
The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research under sponsorship of the National Science Foundation.
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SOLAR MERIDIONAL CIRCULATION PENETRATION
by nuclear burning. Observations of elemental abundances in
the solar photosphere and helioseismic sound speed inversions
rule out such deep mixing (e.g., Michaud & Charbonneau
1991; Chaboyer et al. 1995; Pinsonneault 1997; Brun et al.
2002). We show that fluid dynamical effects should limit the
penetration into the radiative interior to such an extent that
there is no conflict with this compositional constraint. In
qualitative terms, penetration will be limited to the depth of
the so-called overshoot layer immediately below the convection zone, which layer contains at least part of the solar
tachocline. We show that there is effectively no penetration
into the radiative domain below.
If the only effect to consider in estimating the extent of
penetration were how far the kinetic energy of radial motion
(1 m s1 or less) in the meridional circulation would produce
work done against the subadiabatic stratification of the overshoot layer, or below it the radiative layer, the penetration
would be extremely short indeed: at most a kilometer in the
overshoot layer, and a meter in the radiative layer! Of course,
the depth of the overshoot layer itself is reasonably estimated
by converting the kinetic energy of downward plumes into
such work. But for such plumes the velocities are probably at
least 2 orders of magnitude larger, leading to 104 times as
much kinetic energy available to do the work associated with
penetration. Even so, the overshoot layer thickness estimated
by this means is no more than a few percent of the solar radius
(Zahn 1991; Stix 2002; Rempel 2004; the latter shows how
the thickness varies with penetration model, and lists many
earlier references).
That might be the end of the argument about the feasibility
of penetration to depths of 0.6 R but for the fact that the
equatorward meridional flow that must occur at or near the
bottom of the convection zone can in principle drag the stably
stratified material below along with it, through turbulent
transfer of momentum or wave interactions, or perhaps through
other mechanisms. This drag can provide an additional form
of work to be used to achieve further penetration, since in the
confined domain between the equator and the pole some radial motion is inevitably also caused. Therefore, it is necessary to examine a more detailed model for this part of the
Sun, with which we solve for the flow below the convection
zone that arises in response to an imposed equatorward meridional flow at the interface. This is what we do in the
sections that follow.
In carrying out this calculation we encounter two boundary
layers that have been extensively studied in the geophysical
literature, the Ekman layer (e.g., Pedlosky 1987) and a
buoyancy layer (Barcilon & Pedlosky 1967; Veronis 1967),
although not with the particular boundary layer placement
seen here. Our result is therefore of more general fluid dynamical interest than just its application to the Sun. In addition, of course, if the results apply to the Sun they should
apply to the bottoms of most, if not all, stellar convective
envelopes.
2. MODEL FORMULATION
The solar tachocline may be no more than 3% of the
solar radius, so it may be accurately treated as a thin shell
(Charbonneau et al. 1999). The overshoot layer of the convection zone, partially congruent with the tachocline, is similarly thin. Therefore, to a very good approximation the radius
of the shell we consider can be taken to be constant, with the
radial coordinate replaced by a local vertical coordinate z. In
addition, the shell is thin compared to scale heights of all the
569
Fig. 1.—Schematic showing the geometry of our domain as a rotating,
stably stratified Cartesian slab that extends from the equator to the north pole
in latitude. A meridional circulation (v; w) is imposed at the upper surface,
which enters the domain at high latitudes, moves equatorward, and exits at
low latitudes. The latitudinal boundary conditions prohibit the flow from
crossing the equator or pole.
thermodynamic variables, so it can be treated as a gas in the
so-called Boussinesq approximation (Spiegel & Veronis 1960;
Lantz & Fan 1999; Cally 2003). The meridional circulation
flowing equatorward is almost certainly of very large latitudinal scale compared to the thickness of the shell, so this class
of motion can be taken to be in hydrostatic balance in the radial
direction.
While in the Sun this shell contains both molecular and
turbulent viscous and thermal and radiative diffusion, we limit
consideration to a fluid of constant viscous and thermal diffusivities and respectively, which is sufficient to do
plausible boundary layer physics. Since the vertical scale of
the motion and thermodynamic variables is very small compared to the horizontal scale, only diffusion in the vertical
direction need be included.
The real Sun is also spherical, but we first consider instead a
shallow channel of fluid in Cartesian geometry, periodic or
infinitely long in the x direction (corresponding to solar longitude) but bounded in y (corresponding to solar latitude) by
lateral, stress-free thermally insulating walls, meant to correspond to the pole and the equator of the shell. The bottom of
the channel is rigid and stress free (but may be taken to great
depth compared to the boundary layers we find), while the top
boundary is allowed to exchange momentum and thermal
energy as well as mass with the fluid below. The Cartesian
geometry allows us to highlight the relevant boundary layer
thicknesses, which also apply to the spherical case, as we
show in x 4. With the forcing of equatorward ( y) flow at the
top, which is assumed to be zero at both lateral boundaries,
latitudinal flow will in general be induced within the channel
and accompanied by vertical flow for mass conservation, including usually flow into the channel from above at ‘‘high
latitudes’’ and flow out again at ‘‘low latitudes.’’
The channel is allowed to rotate about the z-axis at a rate .
Magnetic fields are omitted in this first study. The channel
fluid is always subadiabatically stratified, with the difference
between the actual temperature gradient and the adiabatic
gradient denoted by . A schematic diagram of the domain we
have defined is shown in Figure 1.
We look then for steady, x-independent solutions for the
flow in the interior of this channel forced by latitudinal flow
at the top. The variables include u; v; and w, the velocities in
the x; y; and z directions, respectively; p, the perturbation
pressure divided by the mean density; and T, the perturbation
temperature. Perturbations are defined as departures from a
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GILMAN & MIESCH
horizontally uniform background state that is in hydrostatic
balance. Given all of the assumptions above, the linearized
dimensional equations that govern this problem are given by
mass continuity
@v @w
þ
¼ 0;
@y @z
ð1Þ
the horizontal equations of motion
2
v þ @2u
¼ 0;
@z2
2
u @p
@2v
þ 2 ¼ 0;
@y
@z
ð2Þ
vertical hydrostatic balance
@p
¼ g T
@z
1
¼
T
ð3Þ
(where T is the mean temperature), and the thermodynamic
equation
w @2T
¼ 0:
@z2
ð4Þ
In the above, we have eliminated the perturbation density from
the hydrostatic relation by expanding the equation of state and
linearizing and subtracting out the equation of state for the
background, horizontally uniform thermodynamic variables.
If we eliminate all dependent variables in favor of the
meridional flow v that we are forcing at the upper boundary,
we get the single equation
@ 6 v 4
2 @ 2 v g @ 2 v
þ 2
þ
¼ 0:
@z6
@y2
@z2
ð5Þ
For this equation to be dimensionally correct, the coefficients of the second derivative terms in equation (5) must
both have dimensions of length to the negative fourth power.
These lengths define the two boundary layer thicknesses of
the problem, namely dE ¼ (=2
)1=2, the classical Ekman
depth, and dBD ¼ (=g )1=4 , a thickness for what we call a
buoyancy-diffusion layer, present in the nonrotating case and
discussed in a geophysical context in Barcilon & Pedlosky
(1967) and Veronis (1967). With these definitions, equation (5)
becomes
@6v
1 @2v
1 @2v
þ 4 2þ 4
¼ 0:
6
@z
dE @z
dBD @y2
ð6Þ
3. SOLUTIONS
Having all constant coefficients, equation (6) clearly separates in y and z. The simplest forcing we can impose at the
top that is consistent with our constraints is of the form
v 0 sinðy=LÞ, in which L is the actual width of the channel.
From equation (6), this forcing leads to the same functional
form in y for all depths. So if we assume a separated solution
of the form
v ¼ v 0 e kz sin
y
;
L
ð7Þ
Vol. 611
then substitution of equation (7) into equation (6) yields the
following equation for k:
k 6 þ dE4 k 2 2
L
4
dBD
¼ 0;
ð8Þ
a bicubic equation already in standard form to be solved
according to formulas in Burington (1962). Solving the
bicubic equation for k 2 yields one real root and two roots
that are complex conjugates of one another. The six roots of
equation (8) are then obtained by taking the positive and
negative square roots of these three solutions for k 2, yielding
two real and four complex roots. Since there are six roots, a
total of six boundary conditions at top and bottom can be
satisfied, and so far we have applied only one. Three of the six
solutions have a positive real part, implying a boundary layer
structure with profiles exponentially declining away from the
upper boundary in distances determined by combinations of
the two lengths dE and dBD . A linear superposition of these
three decaying roots (one real, two complex) yields the general solution for a semi-infinite domain. If a bottom boundary
is included, all six solutions are required and a similar
boundary layer will form near the bottom surface. Since four
of the six roots are complex, the general solution will exhibit
periodic structure (multiple cells) as well as exponential
envelopes.
Two limiting cases are particularly useful. When ! 0,
dE ! 1, and equation (8) gives solutions of the form
1=3
2=3
dBD ; n ¼ 0; 1; 2; 3; 4; 5:
ð9Þ
k ¼ e in=3
L
There is some real part to k for all , so all solutions have
some exponential envelope. Except for numerical factors of
order unity, the decay length k 1 of a typical solution is
1=6
1
1=3 2=3
1=3
:
ð10Þ
k L dBD L
g It is instructive to evaluate equation (10) in the neighborhood of the solar tachocline. At tachocline depths, L 7:7 ; 1010 cm for the distance between the equator and a pole.
The 1/6 power makes the boundary layer depth rather insensitive to all the other parameters. For the overshoot layer, the
largest ; we can possibly assume is ~1012 cm2 s1, which is
not much less than in the convection zone, but ; could be as
low as 109 cm2 s1 each. Here g is related to the effective
gravity of the layer, which is determined by the fraction that
the actual subadiabatic temperature gradient is of the adiabatic
gradient. For the overshoot layer, this fraction f is probably
no larger than 104 and might be as small as 105 or even
106. Here g at tachocline depths is 5 ; 104 cm s2. In more
detail, g g f ð 1= Þ=Hp , in which Hp is the pressure
scale height at tachocline depths and is the ratio of specific
heats. With these numbers, the range of estimates for k 1 is
1600 k 1 3:4 ; 104 km (overshoot layer).
By these estimates, then, the penetration ignoring rotation
effects will be between 0.1% of the solar radius and the
depth of the overshoot layer, which is very unlikely to be as
large as 3:4 ; 104 km (5% of the solar radius). Various
estimates (Rempel 2004, and references therein) put it closer
to 1% of the solar radius. In the radiative layer below the
overshooting, we expect ; to be much smaller and the effective gravity to be much larger, so the penetration into this
No. 1, 2004
SOLAR MERIDIONAL CIRCULATION PENETRATION
571
Fig. 2.—General solution of eq. (8) for a semi-infinite layer in solar parameter regimes. Of the three roots that decay with depth (positive real part), one is real (k1 )
and two are complex conjugates (k2 , k3 ). The left panel shows the amplitude of k1 as a function of the two parameters dE and dBD (the influence of rotation increases
downward and the influence of stratification increases toward the left). The center and right panels show similarly the amplitude of the real and imaginary parts of k2
(and, in turn, k3 ). The values that correspond to the colors and contour levels are indicated by the color bar at right.
layer is much smaller still. For f 101 and ; 106 cm2
s1, we get k 1 50 km! Clearly, from this boundary layer
process no effective penetration is possible below the overshoot layer. From helioseismic inferences, the bottom of the
nearly adiabatic layer associated with the convection zone is
no further in than 0.71 R (Christensen-Dalsgaard et al.
1991; Basu & Antia 1997; Charbonneau et al. 1999). Thus it is
not nearly deep enough to allow the meridional circulation to
reach down to 0.6 R, as Nandy & Choudhuri (2002) require.
The boundary layer characterized by equation (10) has been
discussed in, e.g., Barcilon & Pedlosky (1967) and Veronis
(1967), but there the boundary layer occurs on the side wall of
the fluid container, rather than its top. Steady boundary-layer
type solutions are possible because, even without rotation,
vertical diffusion of temperature is balanced by vertical advection along the subadiabatic temperature gradient, while the
viscous forces are balanced by pressure work, which links
back to the stable stratification also. In Lagrangian terms,
steady flow occurs because the residence time of every fluid
particle that enters the domain at high latitudes and exits at
low latitudes is finite. The boundary layer is thicker for larger
and/or k because more downward diffusion can take place
during this residence time. The boundary layer is thinner when
the stratification is more subadiabatic because this combats
downward diffusion of temperature and it requires more work
to alter the pressure contours to drive the flow.
How does the presence of rotation change the amount of
penetration? When rotation dominates, the term involving dBD
in equation (8) can be discarded, and we reduce the problem to
four solutions of the form
k ¼ e i=2 dE1 ;
ð11Þ
which is the classical Ekman boundary layer problem
(Pedlosky 1987), for which the boundary layer depth k 1 dE . For the solar tachocline, 2:6 ; 106 s1 and 109 1012 cm2 s1, as we assumed earlier, the range for this
depth is 140 k 1 4400 km.
Thus, the Ekman depth is generally smaller than in the
buoyancy-diffusion depth in the overshoot layer, so penetration is only constrained further.
The nature of the boundary layer is governed by the product
PrBu, where Pr is the Prandtl number, Pr ¼ =, and Bu is
the Burger number, Bu ¼ (2
L=ND)2 , which measures the
relative influence of rotation and stratification (D is the
thickness of the layer and N is the Brunt-Vaisala frequency;
N 2 ¼ g ). If PrBu 3 1 the solution will be dominated by
the buoyancy-diffusion layer, but if PrBuT1 then the Ekman
layer will prevail. In the solar interior below the convective
envelope, PrBu is estimated to be between 104 and 101, so
the Ekman layer is expected to play the biggest role, except
near the equator where the vertical component of the rotation
vector vanishes.
As with the buoyancy-diffusion layer associated with dBD ,
the Ekman depth in the radiative interior is much less than in
the overshoot layer. If we use ¼ 106 cm2 s1 as earlier, we
get k 1 5 km there. So again, there is no effective penetration into the radiative layer.
Results for the general case when both boundary layers are
included are shown in Figure 2. These show the form and size
of the vertical attenuation ‘‘rate’’ k from equation (8) for the
three solutions k1 ; k2 ; k3 that decay as z ! 1 as functions of
the two boundary layer lengths. The log scale range for k is
0.01–1000 Mm1. The units on the axes are Mm.
We can see that the domains in which the two boundary
layers dominate are clearly separated. Where the k contours
are horizontal, the Ekman layer dominates, root k1 vanishes,
and k2 and k3 are complex conjugates of each other. Where
the contours are vertical, the buoyancy-diffusion layer dominates, and the decay lengths represented by the real parts of
k1 , k2 , k3 differ only by trigonometric factors of order unity
(see eq. [9]). The purely real root k1 disappears (black) when
the subadiabaticity ! zero, because the thermodynamics becomes decoupled from the equations of motion and the effect
of temperature diffusion is lost. In that limit, the original
equations yield a fourth-order rather than a sixth-order equation for k.
We can see from Figure 2 that the combined boundary
layer thicknesses, as defined by R(k)1, are not that different
from the asymptotic limits. Therefore, the ranges of k 1 estimated above apply in the general case quite well. Therefore,
our conclusion that there is no significant penetration of meridional circulation below the overshoot layer applies equally
572
GILMAN & MIESCH
Vol. 611
well when both boundary layer depths must be taken into
account.
4. IMPROVEMENTS FOR GREATER REALISM
One obvious improvement we can make for solar applications is to go to spherical geometry. When we do that, we can
reduce the problem to a single equation for v, completely
analogous to equation (6), of the form
@6v
cos2 @ 2 v
1 @
1 @
(v
sin
)
¼ 0;
þ
þ
4 @ sin @
@z6
dE4 @z2 rt2 dBD
ð12Þ
in which rt is the radius of the spherical shell at the tachocline
and is the colatitude. Thus the same two boundary layer
lengths appear, but the coefficients of the equation are no
longer constant; rather, they depend on the colatitude .
However, the coefficients are still independent of z, so we still
expect solutions e kz as in the Cartesian problem.
Equation (12) can be solved analytically in the nonrotating
case, where the problem reduces exactly to the Cartesian
problem in z if we assume a top forcing for v sin cos .
This leads to a form for k 1 similar to equation (9) for Cartesian geometry, with the factor (=L)1=3 replaced by (6=rt )1=3.
The nonrotating case is interesting from a mathematical
standpoint because it admits analytic solutions that are directly analogous to the Cartesian case. However, it is not very
realistic because the Sun and other stars do in fact rotate and
this profoundly effects the structure of the meridional circulation. Thus, in the remainder of this section we consider the
rotating, spherical system expressed by equation (12). For
this we need a series of Legendre polynomials in order to
describe the structure of the resulting circulation.
From equation (12), we should expect that the boundary
layer depth dBD dominates near the equator, and the boundary
layer thickness should vary with latitude. We show in Figure 3
a typical example of the flow found in a spherical shell as a
solution to equation (12), obtained using a code that solves the
axisymmetric, nonmagnetic version of the thin-shell equations
derived by Miesch & Gilman (2004), evolving the flow until it
reaches a steady state. Here we have imposed a latitudinal
flow at the top of the form sin cos . The thickness of the
layer corresponds to ~15 Mm in the overshoot layer of the
Sun and ~ 0.2 Mm in the radiative zone. The dashed white
curve gives the Ekman depth dE assumed and the dotted
straight line shows the buoyancy-diffusion layer depth dBD
taken. The top frame depicts the latitudinal flow, v, and the
bottom frame the radial motion, w. The boundary layer, as
expected, is thicker near the equator than near the poles, but is
at all latitudes close to dE and dBD . Penetration is only a
fraction of the depth shown. There is also an induced differential rotation (not shown) inside the shell, even though none
is imposed at the top. This arises from the action of coriolis
forces on the meridional circulation, leading to the so-called
thermal wind. In the linear approximation, however, this has
no effect on the penetration depth of the meridional circulation. Thus the spherical system, while not as simple as the
Cartesian one analyzed above, gives very similar results.
The spherical problem is also straightforward to solve
when a differential rotation, even a large one, is imposed on
the top boundary, such as the solar convection zone imposes
on the solar tachocline. To solve this problem requires
retaining more terms in the equations of motion, and we
save this for a later paper. But adding this differential rotation,
Fig. 3.—Steady-state meridional circulation (satisfying eq. [12]) for a thin
shell subject to a radial and latitudinal velocity imposed on the upper surface.
The upper and lower frames display the latitudinal and vertical velocity, respectively, with contour lines at v ¼ 0 and w ¼ 0 in order to illustrate the
multicellular structure. Yellow/orange tones denote southward and outward
flow while blue/black tones
pffiffiffidenote northward and2=3inward flow. The solution
corresponds to dE ¼ (0:1= 2) D at the poles and d BD L1=3 ¼ 0:17D, where D is
the thickness of the layer. The local values of dE and dBD obtained from the
boundary layer analysis are shown on the upper frame as dashed and dotted
curves respectively.
or thermal wind as it is often called in geophysical fluid dynamics, is unlikely to change our results qualitatively. If the
amplitude of the imposed differential rotation is small, it can be
treated linearly and then it has no effect on the meridional
circulation. A larger differential rotation can alter the circulation by modifying the centrifugal forces, but this effect has to
compete with the negative buoyancy force associated with the
subadiabatic stratification. The effect of rotation compared to
gravity on the Sun is measured by its oblateness, and so is
P104. The part of that from differential rotation would
therefore be 105. This effect would therefore be totally
negligible in the radiative zone, where the effective gravity
associated with the subadiabatic stratification there is 101 g.
In the overshoot layer it might make a quantitative difference,
but would certainly not dominate.
We have also not included effects of Lorentz forces associated with the likely very large amplitude toroidal field in the
tachocline. But this force is certain to oppose penetration, not
enhance it.
5. DISCUSSION AND CONCLUSIONS
We have shown here that there are two boundary layer
thicknesses that govern the amount a meridional circulation
originating in the convection zone can penetrate through the
tachocline to the solar interior. One of these, a buoyancydiffusion layer, which had been previously reported in the
geophysical fluid dynamics literature, is independent of rotation and sets an upper limit to the amount of penetration
possible; hence its value in examining the nonrotating limit.
The second boundary layer, the classical Ekman layer, shrinks
with increasing rotation and, except near the solar equator
where it becomes infinite, further constrains the amount of
No. 1, 2004
SOLAR MERIDIONAL CIRCULATION PENETRATION
penetration. The overall result is that for plausible values of
the subadiabatic stratification in the solar tachocline, the meridional circulation cannot penetrate significantly below the
bottom of the overshoot part of the tachocline, near 0.7 R .
It is important to realize that even though the Cartesian
analysis reduces the problem to algebra, the system is truly
two-dimensional, not one-dimensional, in that a finite length
L in latitude ( y) is required to generate any penetration at all.
From equation (8), if L ! 1 then the problem reduces to the
classical Ekman layer adjacent to an infinite plane boundary,
vertical motions vanish, and the buoyancy-diffusion layer
plays no role. For finite L, the y-bounded Cartesian system is
made to mimic the confinement of a spherical shell such as
the solar tachocline and its neighborhood.
These fundamental boundary layer thicknesses are the
same in Cartesian and spherical geometries, and therefore the
results and conclusions are essentially the same, even though
the rotating spherical case does not lend itself readily to analytical solution. Given that the combination of these two
boundary layers have not to our knowledge been discussed
previously in the astrophysical literature, and since they
should be relevant to the interface between convective and
radiative domains in the interior of stars generally, we use
the Cartesian solutions to elucidate their properties in combination in the clearest possible way. The thin-shell model
used here breaks down for small-scale circulations that
are not necessarily in hydrostatic balance as reflected by
equation (3).
We have not specified a driving mechanism for the meridional flow in the convection zone, because none is required for
the theory. It may well be that such meridional flow arises
because the Sun is rotating, but rotation is not a requirement
for the existence of meridional circulation. In any case, only
meridional flows within the bulk of the convection zone and
the tachocline are relevant to flux-transport dynamo theories
for the Sun. Futhermore, they must have sufficient amplitude
to play a role on a timescale less than a solar cycle. This
amplitude is set empirically with reference to surface doppler
and helioseismic measurements.
In radiative regions of rotating stars, the distortion of the
gravitational potential by centrifugal forces results in a thermal imbalance that tends to drive global motions often referred to as Eddington-Sweet circulations (e.g., Tassoul 1978).
A variety of other mechanisms can also drive or alter meridional circulations near the base of the convection zone, including compositional gradients (e.g., Maeder & Zahn 1998),
rotational and baroclinic instabilities (e.g., Zahn 1992; Maeder
& Meynet 2000), gravity waves (e.g., Fritts et al. 1998), and
magnetic fields (e.g., Gough & McIntyre 1998). However,
these circulations are generally very weak relative to those in
the convection zone, with turnover timescales of thousands or
millions of years, and are therefore of less relevance to solar
dynamo theory.
573
Observations of elemental abundances in the Sun reveal a
depletion of lithium relative to cosmic abundances, which is
attributed at least in part to nuclear burning over the Sun’s
main-sequence lifetime (e.g., Michaud & Charbonneau 1991;
Pinsonneault 1997). Lithium is destroyed at temperatures
above 2:5 ; 106 K, which in the Sun corresponds to a radius
of ~ 0.68 R , well below the convective envelope. Thus, the
observed depletion has been attributed to enhanced mixing
below the convection zone, which may be due to meridional
circulation or to a number of alternative mechanisms such as
rotational instabilities or gravity waves (e.g., Michaud &
Charbonneau 1991; Pinsonneault 1997). However, the observed depletion of other light elements that are burned at
higher temperatures is much less, suggesting that the mixing
does not reach much deeper into the radiative interior. The
most important of these is beryllium, which burns at 3:5 ;
106 K, suggesting that efficient mixing by meridional circulation or other processes cannot extend deeper than 0.55 R.
More sophisticated estimates obtained from applying elemental abundance observations to one-dimensional stellar
structure models favor no mixing down as far as r 0:6 R
(e.g., Michaud & Charbonneau 1991; Chaboyer et al. 1995;
Pinsonneault 1997; Brun et al. 2002). However, this constraint on meridional circulation is not as strict as the one we
have found, which is r 0:7 R .
Our fundamental conclusion from this analysis is that the
physics of the solar tachocline and neighboring regions
does not allow penetration of meridional circulation originating in the solar convection zone below the overshoot layer.
However, some clarification is necessary here because there
are several alternative ways in which the extent of the overshoot layer in the Sun may be defined. For example, the
penetration depth may be determined based on the kinetic
energy or enthalpy flux of the convection or on the efficiency
of tracer particle transport (e.g., Zahn 1991; Brummell et al.
2002; Rempel 2004). For our purposes we define the base of
the overshoot region rb as the radius above which the stratification is nearly adiabatic and below which the stratification
is substantially subadiabatic. Although little is known about
the detailed structure of the overshoot region at the base of
the solar convection zone, rb is reasonably well established
from helioseismic inversions, with a value of 0.713 R
(Christensen-Dalsgaard et al. 1991; Basu 1997). Thus, our
results indicate that penetration inward to 0.6 R , needed to
make certain flux-transport dynamo models apply to the Sun,
is ruled out.
We thank Matthias Rempel and the anonymous reviewer
for helpful suggestions, which have improved the manuscript.
This work was partly supported by NASA through the SEC
theory program, work orders W-10,177 and W-10,175.
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