SOLAR CONVECTION AND MHD
A. NORDLUND
Theoretical Astrophysics Center, and Niels Bohr Institute for
Astronomy, Physics, and Geophysics
Juliane Maries Vej 30, Dk-2100 Copenhagen , Denmark
AND
R.F. STEIN
Dept. of Physics and Astronomy, Michigan State University,
East Lansing, MI 48823, U.S.A.
abstract
We review the current status of eorts to model, in considerable detail, the
entropy structure of the solar convection zone, in particular the layers near
the surface of the Sun. Given an accurate, tabular equation of state, tables
of continuous and binned spectral line opacities, suitable methods for handling radiative energy transport, and numerical methods that attempt to
minimize viscous eects for given numerical resolution, while still capturing
the shocks that develop, it is possible today to produce numerical models of
the solar surface layers that have no arbitrary, free parameters other than
those that specify the luminosity, surface acceleration of gravity, and the
solar abundance. These models are reasonably converged at numerical resolutions that are aordable today, and they may be used as \test benches"
for studying, e.g., the interaction of the turbulent solar convection with
magnetic elds, and helioseismic oscillations. We also briey review recent
progress in understanding boundary driven magnetic dissipation and its
consequences for the dissipation of magnetic energy in the solar corona.
1.
Introduction
The Sun is an excellent \laboratory" for studying the ability of numerical
models to accurately represent realistic (i.e. turbulent) astrophysical plasmas. Both direct observations of the solar surface (Berger & Title, 1996;
Schrijver et al., 1997) and indirect observations of the solar interior through
helioseismic methods (Christensen-Dalsgaard et al., 1996; Gough et al.,
A. NORDLUND AND R.F. STEIN
294
1996) are providing quantitative (sometimes exceedingly precise) information about the detailed structure of the Sun.
A number of current issues relate to the very surface layers of the Sun.
These are characterized by intense (in places transsonic) convectively driven
turbulent motions that signicantly aect the shapes and strengths of photospheric spectral lines, generate the noise that excites the solar oscillations,
and also interact strongly with the magnetic elds present at the solar surface. Small and large scale photospheric motions are able to exert work on
the coronal plasma by the small and large scale horizontal motions of the
footpoints of the coronal magnetic eld.
Numerical simulations of stratied convection have proved very useful,
both when attempting to understand these mechanisms qualitatively (Graham, 1977; Hossain & Mullan, 1991; Singh & Chan, 1993; Atroshchenko &
Gadun, 1994; Porter & Woodward, 1994; Brummell et al., 1995), and when
attempting to predict the average thermal structure of these layers quantitatively (Nordlund, 1982; Steen et al., 1989; Stein & Nordlund, 1989;
Nordlund & Dravins, 1990; Rast et al., 1993; Kim et al., 1996; Stein &
Nordlund, 1998a).
These simulations have led to a shift in our paradigm for compressible convection, from that of a cascade of energy from large driving eddies to small dissipative eddies, to that of non-local driving by low entropy downdrafts produced on the intermediate scale of granulation by
radiative cooling at the extremely thin surface thermal boundary layer,
which drive both larger scale cellular ows and smaller scale turbulent
motions (Nordlund, 1985; Spruit et al., 1990; Nordlund & Stein, 1997;
Spruit, 1997).
In Section 2 we review current eorts to model solar and stellar surface
layer convection, and in Section 3 we review some recent progress in the
understanding of boundary driven dissipation in magnetically dominated
plasmas. Color versions of the gures included here, and further illustrations
and animations are available on the World Wide Web (Nordlund, 1998).
2.
Convection
Solar convection is characterized by a range of scales, both in space and
time. Fig. 1a illustrates the range of spatial scales involved qualitatively,
while Fig. 2 shows the quantitative background; the mass density and gas
pressure stratication cover about ten and fourteen orders of magnitude,
respectively. The size (\outer scale") of convective motions scale roughly
with the (density or pressure) scale height, that is again roughly proportional to the ratio of pressure to density. From the bottom of the convection
zone to the photosphere, this ratio varies with about two and a half orders
Solar Convection and MHD
295
a) The huge range of convective scales of motion in the solar convection zone is
illustrated in this \artists view" of the Sun. Images of entropy from detailed simulations
of the solar surface layers and from \toy models" of the bulk of the convection zone have
been appropriately scaled, and warped onto the spherical and annular surfaces (see also
Nordlund, 1998). b) A perspective view of the solar surface layers, with surface radiation
intensity shown on the top surface, and images of entropy shown on the sides and on the
\drop-down" of the bottom surface.
Figure 1.
Figure 2.
The stratication of mass density and gas pressure in the Sun, from the
photospheric temperature minimum to the center of the Sun. The thin `surface' layer is
crucial for determining the depth at which the stratication changes from nearly adiabatic
convection to radiative diusion.
of magnitude. The time scales span an even larger range, since the velocities
are over an order of magnitude smaller near the bottom of the convection
zone than at the top.
Fortunately, in order to accurately compute the structure of stars similar
to the Sun, one only needs to model a relatively thin surface layer in great
detail; the rest of the structure can be computed with one dimensional
stellar envelope codes that solve the equations for simultaneous hydrostatic
and energy equilibrium (Rosenthal et al., 1998). In convectively unstable
layers, such codes typically employ some variation of the \mixing length
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A. NORDLUND AND R.F. STEIN
Figure 3.
Samples of the run of temperature (a) and entropy (b) with depth, from a
numerical simulation of the solar surface layers (Stein & Nordlund, 1998a). Also shown
are the mean (dashed), mode (dashed-dotted) and extrema (dotted).
recipe" (Bohm-Vitense, 1958; Cox & Giuli, 1968) to describe convective
energy transport. The resulting stratication is very close to adiabatic in
the bulk of the convection zone, and hence the details of the convection
treatment are of little importance there (Gough & Weiss, 1976).
The thin surface layer where convection is of crucial importance is illustrated in Fig. 1b. Even though the depth of this layer is only about 3
Mm (as compared to the total depth of about 200 Mm of the solar convection zone), the layer still covers about ve orders of magnitude in pressure,
because of the low temperatures (and hence small scale heights) of these
surface layers.
The low temperatures, and the presence of the optical surface of the
Sun in this layer imply that, in order to produce quantitative predictions,
it is necessary to use elaborate equations of state, and to include methods
that can accurately represent radiative transfer of energy in the presence
of a large number of spectral lines. Spectral lines block about 12% of the
luminosity of the Sun, as compared to a case with only continuum opacity.
This means that a model that does not account for the spectral line blocking
will have to be about 3% cooler in the optical surface layers, in order to
have the right solar luminosity. That would again result in an interior with
a lower temperature at a given pressure (a lower entropy), and ultimately
to a convection zone that is too deep (Christensen-Dalsgaard, 1997).
As indicated in Fig. 1b, the root mean square temperature uctuations
are quite large near the top of the layer; at the optical surface of the Sun,
temperatures in a horizontal layer vary from about 4,000 K to about 10,000
K. This is further illustrated in Fig. 3, that shows the temperature and entropy as functions of depth, for a sample of horizontal points in a numerical
model (Stein & Nordlund, 1998a).
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297
Figure 4.
Histograms showing the distribution of entropy at the surface (a) and at a
depth of 1 Mm (b), from the numerical simulations of Stein & Nordlund (1998a). The
various curves illustrate the numerical convergence; the labels in the panels indicate the
number of zones in the horizontal directions.
The statistical distribution of the entropy in the very surface layers
is shown in Fig. 4a. It is somewhat bimodal, with a \hot" component at
temperatures around 10,000 K and a \cold" component with temperatures
around 5,000 K. The upper limit of the distribution is exceedingly sharp;
this reects the almost constant entropy of ascending material (Stein &
Nordlund, 1989). At a depth of 1 Mm (less than the outer scale of convection
cells but still several scale heights), there is still a sharp upper limit of
the distribution at the same entropy, but the distribution towards lower
entropy is now exponential, signalling that descending material is subject
to a mixing process. Descending material is indeed mixed with overturning
ascending material at a rate that is almost entirely determined by rst
principles; conservation of mass enforces overturning of ascending material,
and the small scale heights imply that the rate is large, and thus dominates
over turbulent mixing (Nordlund, 1975).
The turbulent and inhomogeneous surface layer is crucial for the solar
p-mode oscillations. On the one hand it produces the main part of the noise
that is responsible for the stochastic excitation of the modes (Goldreich &
Keeley, 1977; Stein & Nordlund, 1991; Goldreich et al., 1994; Nordlund
& Stein, 1998; Stein & Nordlund, 1998b), on the other hand it extends
the acoustical cavity and hence lowers the frequency of the p-modes that
extend into these layers (Rosenthal et al., 1998). In addition, the jump in
average entropy across the turbulent surface convection layers determines
the entropy of the bulk of the convection zone and hence ultimately, as
mentioned above, the depth of the solar convection zone.
Stars over a wide range of parameters have surface layers that are qualitatively similar to the solar surface layers, and hence numerical models that
can successfully predict the properties of the well observed solar oscillations
298
A. NORDLUND AND R.F. STEIN
may be used to predict the envelope structure and oscillation properties of
other stars as well (Ludwig et al., 1997; Trampedach et al., 1997).
In summary, the requirements for accurate, quantitative modeling of
stellar surface layers and predictions of convection zone depths are (Rosenthal et al., 1998):
0 Continuum opacities good to the percent level.
0 The inclusion accurate ionization and dissociation equilibria to an extent sucient to account for all major electron sources and sink.
0 Inclusion of the eects of spectral line opacities to an extent sucient
for predicting the emergent surface luminosity to within about a percent.
0 Numerical resolution sucient to utilize the above; in practice this
means of the order of 2003 mesh points, and a vertical resolution of
about a tenth of the local scale height.
With these requirements fullled, it is possible to accurately predict
the convection zone depth, to predict surface intensity statistics, to predict spectral line widths and shapes, to predict the excitation and damping
of global oscillations, and to predict the inuence of the surface layers on
the frequency spectrum of the global oscillations, using as input parameters only the fundamental stellar parameters (eective temperature, surface
acceleration of gravity, chemical abundances).
3.
Magnetic Dissipation and Coronal Heating
Magnetically dominated astrophysical plasmas, such as the solar corona or
the corona above a magnetized accretion disc, are likely to be continuously
inuenced by braiding motions on its boundaries. The solar photosphere,
for example, is known to be threaded by magnetic ux concentrations that
connect to the solar corona. Because the photosphere and the convection
zone below are much denser than the solar corona, the photospheric ux
concentrations are moved around more or less passively by the convective
motions, which causes the coronal extension of the magnetic eld lines to be
continuously braided. A similar relation is likely to exist between accretion
disk coronae and the much denser and turbulent layers near the equatorial
plane of the disk (Matsumoto & Tajima, 1995; Brandenburg et al., 1995;
Stone et al., 1996).
In such situations, the magnetically dominated (\low beta") part of the
plasma is forced to dissipate the work done by the boundary (\high beta")
plasma, since otherwise the angles of inclination (and hence the work) would
increase indenitely (Parker, 1972, 1983, 1987, 1988).
Analytical work and numerical experiments have claried the mechanisms through which balance between the boundary work and the dissipa-
Solar Convection and MHD
299
Isosurface visualizations of 3-D current sheets, from an experiment with
boundary driven magnetic dissipation (Galsgaard & Nordlund, 1996). The right hand
side panel shows a blow-up of the small box outlined near the bottom in the left hand
side panel. The driving boundaries are to the left and right. The numerical resolution is
1363 . For color renderings, see Nordlund (1998).
Figure 5.
tion in the interior is established (van Ballegooijen, 1986; Mikic et al., 1989;
Strauss, 1991; Gomez & Fontan, 1992; Heyvaerts & Priest, 1992; Biskamp,
1993; Longcope & Sudan, 1994; Galsgaard & Nordlund, 1996, 1997; Ricca
& Berger, 1996).
Figure 5 (from Galsgaard & Nordlund, 1996) shows an example of the
complex, hierarchical current sheet structure that develops in boundary
driven low beta plasmas, even when the driving only consists of large scale,
incompressible shearing motions.
Such a hierarchical set of current sheets has dissipation properties that
dier dramatically from those of the assumed, monolithic current sheets
that have been at the center of attention for decades, in the quest for understanding \fast magnetic reconnection" (Petschek, 1964; Sonnerup, 1988;
Priest & Forbes, 1992). Indeed, as the numerical resolution is increased in
these experiments, the amount of resolved structure increases correspondingly, but the average level of dissipation stays approximately constant
(Galsgaard & Nordlund, 1996).
This is a remarkable result, that may be taken as an indication that
chaotic MHD phenomena obey scaling relations analogous to Kolmogorov
scaling in turbulence (Kolmogorov, 1941; Hunt & Williams, 1991). A scaling relation between the average level of magnetic dissipation and the parameters that characterize the boundary driving is shown in Fig. 6 (from
Galsgaard & Nordlund, 1996). The dissipation in a number of experiments,
with varying driving amplitudes and aspect ratios, may all be tted with a
simple scaling relation. The scaling relation does not depend on the value
A. NORDLUND AND R.F. STEIN
300
800
Frequency
600
400
200
0
-2
-1
0
1
Winding Number
2
Figure 6. The left hand side panel shows the scaling law for magnetic dissipation derived
by Galsgaard & Nordlund (1996). The mean magnetic dissipation is a function of the
magnetic eld strength B0 , the size of the box L, the characteristic velocity Vd and
duration td of shearing events at the boundary, and the Alfven speed VA . The right hand
side panel shows a histogram of the \local winding number" (Nordlund & Galsgaard,
1997), dened as the number of times two neighboring eld lines wind around each
other, as they pass from one boundary to another. The inset shows an image rendering
of the winding number.
of the resistivity, just as the level of viscous dissipation in Kolmogorov
turbulence does not depend on the value of the molecular viscosity.
This result is interesting also from a practical point of view, since it
was obtained with relatively modest numerical resolutions (up to 1363 ). We
may surmise that, unless the geometry is such that a range of \outer" scales
are involved, and/or extreme aspect ratios are required, one can expect to
obtain reasonably realistic levels of dissipation in numerical MHD models
that are aordable today, even on ordinary workstations.
4.
Conclusions
Numerical models of the solar surface layers and the adjacent layers of
the solar convection zone are now in many respects accurate enough for
quantitative predictions, for example of the convection zone depth, and of
the amount of excitation and damping of global oscillations. The models
are \parameter-free," in the sense that the main physical parameters (effective temperature, surface acceleration of gravity, and chemical composition) are known from observations, and the main physical characteristics of
the plasma (equation of state, continuous and spectral line opacities) are
obtainable from standard packages. The remaining, numerical parameters
(numerical resolution and viscosities, etc.) have been pushed into a regime
where they only have a marginal eect, even quantitatively.
It is indeed fortunate that the asymptotic behavior of turbulent systems,
both hydrodynamical and magnetohydrodynamical, is so benign that currently aordable numerical models are near, or already inside, the asymp-
Solar Convection and MHD
301
totic regime where details of the numerical dissipation become unimportant.
Certainly, such models can be used with great benet to increase our understanding of the hydrodynamic and magnetohydrodynamic mechanisms
at work, in the surface layers of stars as well as in other astrophysical contexts. Apart from its intrinsic value, such an understanding is indeed also
necessary when attempting to obtain similarly quantitative predictions in
other circumstances; only if we understand the mechanisms at work can
we assess if the models are providing fair numerical representations of the
systems under study.
Acknowledgements
The work of A.N was supported in part by the Danish Research Foundation, through its establishment of the Theoretical Astrophysics Center. RFS
was supported by NASA grant NAG 5-4031 and NSF grant AST 9521785.
Calculations were performed at the National Center for Supercomputer
Applications, which is supported by the National Science Foundation, at
Michigan State University and at UNIC, Denmark.
References
Atroshchenko, I. N., Gadun, A. S. 1994, A&A, 291, 635
Berger, T. E., Title, A. M. 1996, ApJ, 463, 365
Biskamp, D. 1993, Nonlinear Magnetohydrodynamics, Cambridge Monograms on Plasma
Physics
Bohm-Vitense, E. 1958, Zs. f. Ap., 46, 108
Brandenburg, A., Nordlund, A., Stein, R. F., Torkelsson, U. 1995, ApJ, 446, 741
Brummell, N. H., Cattaneo, F., Toomre, J. 1995, Sci, 269, 1313
Christensen-Dalsgaard, J. 1997, in F. P. Pijpers, J.Christensen-Dalsgaard, C. S. Rosenthal (eds.), Solar Convection, Oscillations and their Relationship; SCORe'96, Kluwer
Academic Press, Dordrecht, p. 3
Christensen-Dalsgaard, J., Dappen, W., et al. 1996, Science, 272, 1286
Cox, J., Giuli, R. 1968, Priciples of stellar structure I, Gordon and Breach, New York
Galsgaard, K., Nordlund, A. 1996, Journal of Geophysical Research, 101(A6), 13445
Galsgaard, K., Nordlund, A. 1997, Journal of Geophysical Research, 102, 231
Goldreich, P., Keeley, D. A. 1977, ApJ, 212, 243
Goldreich, P., Murray, N., Kumar, P. 1994, ApJ, 424, 466
Gomez, D., Fontan, C. 1992, ApJ, 394, 662
Gough, D. O., Kosovichev, A. G., Toomre, J., et al. 1996, Science, 272, 1296
Gough, D. O., Weiss, N. O. 1976, MNRAS, 176, 589
Graham, E. 1977, in Problems of Stellar Convection, Vol. 71 of Lecture Notes in Physics ,
Springer-Verlag, Berlin, p. 151
Heyvaerts, J., Priest, E. R. 1992, ApJ, 390, 297
Hossain, M., Mullan, D. J. 1991, ApJ, 380, 631
Hunt, J. C. R. Phillips, O. M., Williams, D. (eds.) 1991, Turbulence and stochatic
processes: Kolmogorov's idaes 50 years on, Vol. 434, The Royal Society, University
Press, Cambridge
Kim, Y.-C., Fox, P. A., Demarque, P., Soa, S. 1996, ApJ, 461, 499
Kolmogorov, A. N. 1941, Dokl. Akad. Nauk. SSSR, 30, 301, Reprinted in Friedlander
302
A. NORDLUND AND R.F. STEIN
S.K., Topper L., eds., 1961, Turbulence: Classic Papers on Statistical Theory, New
York, Interscience
Longcope, D. W., Sudan, R. N. 1994, ApJ, 437, 491
Ludwig, H., Freytag, B., Steen, M. 1997, in Solar Convection, Oscillations and their
Relationship; SCORe'96, p. 59
Matsumoto, R., Tajima, T. 1995, ApJ, 445, 767
Mikic, Z., Schnack, D. D., van Hoven, G. 1989, ApJ, 338, 1148
Nordlund, A. 1975, A&A, 50, 23
Nordlund, A. 1982, A&A, 107, 1
Nordlund, A. 1985, Solar Physics, 100, 209
Nordlund, A. 1998, http://www.astro.ku.dk/~aake/talks/NAP98
Nordlund, A., Dravins, D. 1990, A&A, 228, 155
Nordlund, A., Galsgaard, K. 1997, in G. Simnett, C. E. Alissandrakis, L. Vlahos (eds.),
Proc. 8th European Solar Physics Meeting, Vol. 489 of Lecture Notes in Physics ,
Springer, Heidelberg, p. 179
Nordlund, A., Stein, R. F. 1997, in F. P. Pijpers, J.Christensen-Dalsgaard, C. S. Rosenthal (eds.), Solar Convection, Oscillations and their Relationship; SCORe'96, Kluwer
Academic Press, Dordrecht, p. 79
Nordlund, A., Stein, R. F. 1998, in F. L. Deubner, J. Christensen-Dalsgaard (eds.), Proc.
IAU Symp. 185, Vol. 185 of IAU Symp., Reidel, Dordrecht, (in press)
Parker, E. N. 1972, ApJ, 174, 499
Parker, E. N. 1983, ApJ, 264, 642
Parker, E. N. 1987, ApJ, 318, 876
Parker, E. N. 1988, ApJ, 330, 474
Petschek, H. E. 1964, in Physics of Solar Flares, NASA Spec. Publ. SP-50, 425
Porter, D. H., Woodward, P. R. 1994, ApJS, 93, 309
Priest, E. R., Forbes, T. G. 1992, J. Geophys. Res., 97, 16757
Rast, M. P., Nordlund, A., Stein, R. F., Toomre, J. 1993, ApJ, 408, L53
Ricca, R. L., Berger, M. A. 1996, Physics Today, 49(12), 28
Rosenthal, C. S., Christensen-Dalsgaard, J., Nordlund, A., Stein, R. F., Trampedach, R.
1998, A&A, (in press)
Schrijver, C. J., Hagenaar, H. J., Title, A. M. 1997, ApJ, 475, 328
Singh, H. P., Chan, K. L. 1993, A&A, 279, 107
Sonnerup, B. U. 'O. 1988, Comp. Phys. Commun., 49, 142
Spruit, H. C. 1997, Mem. Soc. Astron. It.
Spruit, H. C., Nordlund, A., Title, A. 1990, ARA&A, 28, 263
Steen, M., Ludwig, H.-G., Kruss, A. 1989, A&A, 213, 371
Stein, R. F., Nordlund, A. 1989, ApJ, 342, L95
Stein, R. F., Nordlund, A. 1991, in D. Gough, J. Toomre (eds.), Challenges to Theories of
the Structure of Moderate Mass Stars, Vol. 388 of Lecture Notes in Physics , Springer,
Heidelberg, p. 195
Stein, R. F., Nordlund, A. 1998a, ApJ, 499, 914
Stein, R. F., Nordlund, A. 1998b, ApJ, (in preparation)
Stone, J. M., Hawley, J. F., Gammie, C. F., Balbus, S. A. 1996, ApJ, 463, 656
Strauss, H. R. 1991, ApJ, 381, 508
Trampedach, R., Christensen-Dalsgaard, J., Nordlund, A., Stein, R. F. 1997, in F. P. Pijpers, J.Christensen-Dalsgaard, C. S. Rosenthal (eds.), Solar Convection, Oscillations
and their Relationship; SCORe'96, Kluwer Academic Press, Dordrecht, p. 73
van Ballegooijen, A. A. 1986, ApJ, 311, 1001