1996MNRAS.283.1153W
Mon. Not. R. Astron. Soc. 283, 1153-1164 (1996)
Photospheric convection in strong magnetic fields
N. O. Weiss, D. P. Brownjohn, P. C. Matthews* and M. R. E. Proctor
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW
Accepted 1996 July 17. Received 1996 July 11; in original form 1996 March 4
ABSTRACT
The effect of magnetic fields on convection at the surfaces of cool stars can be explored by
comparing the results of detailed numerical experiments with high-resolution solar observations. We have investigated non-linear three-dimensional magnetoconvection in a fully
compressible perfect gas. In this paper we study the effect of an imposed magnetic field on
the pattern of convection in a deep stratified layer. When the field is strong enough to dominate
the motion we find steady convection with rising plumes on a deformed hexagonal lattice, and
a magnetic network at the upper boundary. This gives way to spatially modulated oscillations
for weaker fields. As the field strength is further reduced the oscillations become more violent and
irregular, and their horizontal scale increases. Magnetic flux moves rapidly along the network that
encloses the ephemeral plumes; when the imposed field is relatively weak, intense fields appear at
junctions in the network, where the magnetic pressure is comparable to the gas pressure and an
order of magnitude greater than the dynamic pressure. This behaviour is related to convection in
sunspots and plages and to the structure of intergranular magnetic fields on the Sun.
Key words: convection - MHD - Sun: granulation - Sun: magnetic fields - sunspots - stars:
magnetic fields.
1 INTRODUCTION
The magnetic field of a late-type star is generated by dynamo action
deep in its convection zone. The detailed structure of the fields that
are observed depends, however, upon their interaction with convection near the surface of the star. Th.e most prominent features are
starspots. They are dark because the magnetic field is so strong that
convective transport is substantially inhibited. At the other extreme,
weak fields are transported passively: magnetic flux is swept to the
boundaries of convection cells and moves along them to accumulate at nodes in an evolving network. Fields of intermediate strength
alter the pattern of convection in the photosphere, an effect that can
be observed by studying small-scale structures on the Sun.
High-resolution solar observations show that, outside sunspots
and pores, nearly all of the magnetic flux is confined to isolated
sheets or tubes, with fields that are locally intense. At the photospheric level these features are almost completely evacuated and
the field strength approaches the value Bp .. 1500 G for which the
magnetic pressure is equal to the ambient gas pressure. The mean
flux density in a plage region or the magnetic network is Bo = f B p '
where f is a local filling factor. As Bo and f increase, the Lorentz
force becomes dynamically more powerful and the pattern of
granular convection changes (ritle et al. 1992). For Bo < 150 G
(f < 0.1) normal granulation is scarcely affected. The bright cores
of granules, where hot gas is rising, have a spacing of around
* Present
address: Department of Theoretical Mechanics, University of
Nottingham, Nottingham NG7 2RD.
1.8 Mm and are enclosed by dark intergranular lanes, with downward motion, where magnetic fields are located. For 150
< Bo < 600 G (0.1 < f < 0.4) Title et al. find that the granulation
is abnormal, with a spacing of only 1.1 Mm, while magnetic fields
form a perforated network along which flux moves like a 'magnetic
fluid'. Strong fields are associated with bright points in line
emission and downward velocities. Only for Bo > 600 G are
magnetic features dark, with a diminished downward flow.
If enough magnetic flux accumulates, a dark pore is formed.
Pores have diameters of 1.5-7.0 Mm and fields of around 2000 G.
Within them are bright features that correspond to umbral dots in
sunspots (Muller 1992; Bonet, Sobotka & Vazquez 1995). If the
total flux 4> exceeds a critical value 4>c'" 7 TWb (or 7 x 1020 mx),
the pore develops a penumbra and becomes a sunspot (Thomas &
Weiss 1992); indeed, spots can form with diameters of only 3.6 Mm
and fluxes of 2 TWb (Rucklidge, Schmidt & Weiss 1995). In the
umbra of a sunspot the field is roughly vertical, with a strength of up to
3000 G. Within the umbra there are small bright points called umbral
dots, visible against a dark background with weak fluctuations (Muller
1992). Umbral dots are present in all sunspots but large spots contain
dark nuclei that are free ofthem (Maltby 1992; Muller 1992; Sobotka,
Bonet & Vazquez 1993). There is, however, a distinction between
peripheral umbral dots, which are related to bright features moving in
from the penumbra, and central umbral dots, which are convective
features (Sobotka et al. 1993, 1995).
To understand these different patterns of magnetoconvection we
need to probe beneath the surface of a star. Since observations only
penetrate to a continuum optical depth TO.5 .. 10 and experiments
© 1996RAS
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
1154
N. O. Weiss et al.
cannot reproduce the relevant parameters, we have to rely on
theory, with a combination of analysis and computation. A local
stability analysis indicates that behaviour depends on the ratio r of
the magnetic to the thermal diffusivity. If r > 1, convection sets in
as steady overturning motion but, if r < 1 and the magnetic field is
sufficiently strong, the initial instability leads to oscillatory convection. In stellar interiors heat is carried by radiation and the
radiative diffusivity is inversely proportional to the opacity. At the
surface of a late-type star r« 1, so we might expect to find
oscillatory convection, but the opacity rises rapidly with depth,
owing to ionization of hydrogen, and r consequently increases. In
the Sun there is a layer from 2 to 20 MIn below the surface where
r> 1 (Meyer et al. 1974; Weiss et al. 1990) and we might expect
convection to be steady. If we consider a stratified layer, in which r
increases with depth, with a strong magnetic field, it is no longer
obvious whether convection is steady or oscillatory at onset. In fact,
linear theory shows that there is a stationary bifurcation that gives
rise to overturning motion, as in the non-dissipative case (Moreno
Insertis & Spruit 1989; Weiss et al. 1990).
Non-linear behaviour can only be investigated through numerical
models. Here there are two different approaches. The first, and more
ambitious, is to simulate a stellar atmosphere, including ionization
and radiative transfer, as realistically as possible; this has been
attempted by Nordlund & Stein (1990; also Nordlund, Galsgaard &
Stein 1994). The alternative, which we shall follow, is to construct
idealized models and to explore the effects of systematically
varying parameters such as the superadiabatic gradient or the
strength of the magnetic field. In this paper we are concerned
with the spatiotemporal structure of three-dimensional magnetoconvection in a stratified compressible layer. As the mean flux
density is increased, there is a transition from violent time-dependent motion (as in a plage) to highly ordered but less vigorous
convection (corresponding to the dark nuclei of sunspots).
The earliest studies of non-linear magnetoconvection relied on
the Boussinesq approximation (proctor & Weiss 1982; Cattaneo
1995). Some three-dimensional simulations in the anelastic approximation were carried out by Nordlund (1984), but recent work has
dealt with fully compressible behaviour (Weiss 1991; Proctor
1992). Numerical experiments on convection in a shallow,
weakly stratified layer have revealed a variety of waves and
oscillations in two (Hurlburt & Toomre 1988; Hurlburt et al. 1989;
Proctor et al. 1994; Brownjohn et al. 1995) and three (Matthews,
Proctor & Weiss 1995) dimensions. Weiss et al. (1990) investigated
two-dimensional magnetoconvection in a deep layer, with r < 1 at the
top but r > 1 at the bottom. Here we extend that treatment to three
dimensions. Our systematic survey is restricted to a mildly non-linear
regime; more extreme behaviour has been studied in order to model
stellar dynamos (Nordlund et al. 1992; Brandenburg et al. 1996).
In the next section we describe our model configuration and
summarize its stability properties (which are the same in three or
two dimensions). Then, in Section 3, we present results of numerical experiments as the imposed field strength is varied for a fixed
atmosphere and superadiabatic gradient. In Section 4 we investigate
the effects of the imposed symmetry and geometry on our idealized
system, and in the concluding section we discuss the implications of
these model calculations for magnetoconvection in stars like the
Sun. A more detailed treatment of the underlying bifurcation
structure will be presented elsewhere (proctor, Weiss & Matthews
1996; Weiss et al., in preparation). The rich variety of spatiotemporal structures that we have found is best presented as videos,
which can be obtained from the World-Wide Web site http://
www.damtp.cam.ac.uk/user/afdnld/movies/
2 SETTING UP THE MODEL PROBLEM
The system that we shall investigate is a straightforward threedimensional extension of the fully compressible two-dimensional
configuration that was studied earlier (Weiss et al. 1990, which will
be referred to as Paper I). Thus we take a layer of depth d containing
a perfect monatomic gas, with fixed temperatures To and To + !l.T at
its upper and lower boundaries respectively. The gas is electrically
conducting and there is an imposed magnetic field such that the
mean flux density corresponds to a uniform vertical field Bo. We
assume that the z-axis points downwards, in the direction of the
gravitational acceleration g. The origin is chosen so that z = Zo at
the upper boundary, where Zo = Tod/!l.T, and we restrict attention
to the region {O:s:x:s: M; O:s:y:s: M;Zo :s:z :S:Zo + d}; that is to say,
we choose a box with square cross-section and aspect ratio A. This
geometry naturally imposes constraints on the solutions that we can
find.
2.1 The background atmosphere
In the absence of any motion there is a uniformly stratified
equilibrium solution, corresponding to a polytrope of index
m = (gd/UT) - 1, where 1{ is the gas constant. Then the temperature T(z) and the density p(z) are given by
T
= !l.Tz/d,
p
= po(zlzo)m,
(2.1)
where Po = p(zo), and the superadiabatic gradient
1
(V - V.d)
7- 1
(2.2)
= (m+ 1) - - 7 - '
with 7 = 5/3.
We assume that the thermal conductivity K, the electrical
conductivity (/1{)'I/)-1, the shear viscosity p., the magnetic permeability /1{) and the heat capacity cp are all constant. Then the
magnetic diffusivity '1/ is likewise constant but the thermal diffusivity K = K/cpp and the viscous diffusivity ,,= p./p both vary
inversely with p and therefore decrease with increasing depth. (It
might be preferable to keep " rather than p. constant, but that is
computationally less convenient.) While the Prandtl number U" "/K
remains constant, the crucial diffusivity ratio
(2.3)
with r 0 55 t(zo) = cpPo'l//K, is proportional to the density and thus an
increasing function of depth. We shall find it convenient to
characterize the layer by the value 55 r(zo + !d) at its mid-point.
Since we want to model a strongly stratified atmosphere, we set
m = l,zo =
Then the density and the temperature increase by
a factor of 11 across the layer, while the pressure P increases as their
product. As we are interested in having the diffusivity ratio r < 1 at
the top but r > 1 at the bottom, we shall take r 0 = 0.2. Then r lies in
the range 0.2 :s: r(z) :s: 2.2 and f = 1.2.
f
-lad.
2.2 Equations and boundary conditions
As in our earlier two-dimensional investigations, we scale lengths
with the layer depth d, density with Po, temperature with !l.T,
magnetic fields with Bo and time with the reduced acoustic travel
JUT. Note that this scaling for time differs by a factor of
To/!l.T from that adopted by Matthews et al. (1995). Then we
introduce a dimensionless thermal conductivity k and F, the square
of the ratio of the Alfven speed to the reduced sound speed, as a
measure of the field strength, where
j-e d/
-
K
= K/cppody.~
UT,
F
2
= Bo/(p.oPoUT).
(2.4)
© 1996 RAS, MNRAS 283, 1153-1164
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
Photospheric convection
We now have to solve the dimensionless equations
P
ap
at = -V'(pu),
= pT,
aB
- = V x (u xB -
V'B=O,
i(pu)
at
at
= -V'(puu -
(2.5)
-
(2.6)
toKV xB),
FBB) - V(P + lf1B12) + (m
+ l)pz + V'T
(2.7)
and
p
+
("I - l)K
2
(!un+FtoIVxBI)
p
in the region {O s x S A; 0 S Y S A; Zo S Z S Zo
velocity and the viscous stress tensor
Tij
= uK- (aUi
- + -aUj aXj
aXi
2 -aUk) .
-Oij
3 aXk
(2.8)
+ 1}.
Here u is the
(2.9)
We impose the standard free boundary conditions at the top and
bottom of the layer, together with periodic lateral boundary conditions. Thus T = zo, Zo + 1 at the upper and lower boundaries,
respectively, while Uz = aux / az = ally / az = Bx = By = O. All
quantities are assumed to be periodic in x and y with period A, so
that T(O,y,z) = T(A,y,Z), T(x, O,z) = T(x, A,Z), etc. These idealized boundary conditions are appropriate for our astrophysically
motivated problem, as well as being mathematically convenient.
2.3 The model system
to,
For this paper, we retain a fixed reference atmosphere with Zo =
and set the diffusivity ratio = 1.2, with the Prandtl number u = 1.
The state of the system is then defined by choosing the superadiabatic gradient, fixing F, which measures the importance of the
Lorentz force, and adopting some aspect ratio A. The superadiabatic
gradient at any level is conventionally measured by the dimensionless Rayleigh number
t
2
R(z) '" (m
+ 1)
(V - V.d)
J-m-l
- 2_?m'
uK ZO"
(2.10)
We shall refer to the value R = R(zo +!) defined at the middle of
the layer. Then, since R = 241K2 , the Rayleigh number can only be
increased by reducing the thermal conductivity k. As a measure of
the field strength, we shall use the Chandrasekhar number
- 2
5
'
Q",FlutoK ='24FR.
magnetoconvection, the value of a for which R is a minimum
increases with increasing Q (cf. Matthews et al. 1995). Fig. 2 of
Paper I shows the bifurcation value R(e) as a function of the
wavenumber a for Q = 1000. With this field strength the minimum
'(e)
value R min ... 28 000 occurs for a ... 6.6, which corresponds to
A... 1.0; as expected, the favoured aspect ratio has decreased.
For a given aspect ratio A, two-dimensional rolls with solutions
that vary as cos(2TrxlA) or as cos(2Tr)'1A) have the same wavenumber as square cells with solutions that vary as cos(2TrxIA)+
cos(2TrylA) = 2cos[Tr(x + y)lA] cos[Tr(x - y)IA]. Linear theory
cannot tell us whether rolls or squares are preferred. However,
with A = and Q = 1000, the values used in Paper I, square cells
with eigenfunctions proportional to cos(2TrxIA) cos(2TrYf}..), corresponding to a wavenumber a' = 2.j2m/A = 6.66, as well as diagonally oriented rolls, appear before two-dimensional rolls with
a = 4.71. Hence we should expect that three-dimensional behaviour will differ from what has previously been described.
In the weakly non-linear regime, planform selection is affected
by stratification. For Boussinesq magnetoconvection, steady rolls
are preferred to squares (Clune & Knobloch 1994), but the pattern
changes when up-down symmetry is broken. In a weakly stratified
layer, the rolls give way to squares for sufficiently strong fields
(Matthews et al. 1995). For these solutions, which appear in a
subcritical bifurcation, there is a topological difference between
rising fluid (in isolated plumes) and sinking fluid (in sheets around
the edges of a cell). So squares (or even hexagons), with a strong
asymmetry between upward and downward motion, are likely to be
preferred in the deep layer that we have prescribed.
Any detailed investigation of fully non-linear behaviour requires
computation. We use the same mixed finite-difference/pseudospectral code as Matthews et al. (1995); this code is based on that
developed by Cattaneo et al. (1991) for the non-magnetic problem,
where numerical techniques have been employed with great success
(Brummell, Cattaneo & Toomre 1995). The numerical procedure
preserves the conservative formulation of equations (2.5)-(2.7). It
takes advantage of the periodic lateral boundary conditions by using
a spectral representation in the horizontal x- and y-directions,
coupled with fourth-order finite differences in the vertical z-direction. Time is advanced by the explicit second-order AdamsBashforth method, with the time-step limited by the Courant
condition and its diffusive analogue. For most of our numerical
experiments a resolution of 32 points (16 complex Fourier modes)
in each of the x- and y-directions, with 40 mesh intervals (41 points)
in the z-direction, provides sufficient accuracy. Where necessary,
the mesh was refined to 48 x 48 x 61. This resolution is adequate for
the mildly non-linear regime, with R < 6R(e), that will be explored
here. All computations were carried out on Sun, Hewlett-Packard
and Silicon Graphics workstations.
1
aT
"IK 2
- = -u'VT - ("I -l)TV'u +-V T
at
1155
(2.11)
For the runs described in Section 3 the ratio of gas pressure to
magnetic pressure at the middle of the layer, the plasma beta
&= 7.2IF, lies in the range 75 < &< 300: this is the strong-field
regime, although effects of compressibility are limited.
The stability of the static equilibrium solution can be investigated
by linearizing the equations and calculating the growth-rate of the
eigenfunctions. We find that convection sets in at a stationary
(pitchfork) bifurcation, giving rise to weakly non-linear solutions
that are steady, regardless of whether there is a magnetic field or
not. The bifurcation value R(e) is a function of the horizontal
wavenumber a = 2TrIA of the fundamental modes. In the absence
of a magnetic field, R(e) attains its minimum value (R~ = 1189)
for a = 2.42, which corresponds to A = 2.6. As in Boussinesq
3 THREE·DIMENSIONAL RESULTS:
VARYING THE FIELD STRENGTH
For a given aspect ratio, one can obtain a single-parameter family of
solutions either by increasing the Rayleigh number for fixed Q (or
&) or by decreasing the field strength for fixed R. If the aim is to
investigate the bifurcation structure, it is probably more convenient
to vary R, and that approach will be followed elsewhere (Weiss et
al., in preparation). Here we want to model the effect of strong fields
on the pattern of convection, so we shall decrease Q while R is held
constant. The two approaches are qualitatively equivalent in the
mildly non-linear regime that we are able to explore.
In this section we describe numerical experiments with a fixed
© 1996 RAS, MNRAS 283, 1153-1164
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
1156
N. O. Weiss et al.
Table 1. Runs with fixed aspect ratio (R = 100000, A = 2).
Q
2000
1500
1400
1000
500
RIR(e)
(u2)~
N
r
1.62
2.14
2.28
3.14
5.90
0.043
0.059
0.064
0.095
0.13
6
6
6
2.5
1.5
0.46
0.46
0.46
0.71
0.92
steady
aperiodic oscillations
aperiodic oscillations
chaotic
chaotic
las
Rayleigh number R =
and aspect ratio A = 2. The effects of
varying the geometry will be considered in Section 4. As explained
above, the initial bifurcation leads to steady convection. We start
with a field that is strong enough to enforce steady motion, and
illustrate the different time-dependent structures that appear as Q is
reduced. The results of these calculations are summarized in Table 1.
3.1 Steady convection for Q = 2000
When Q is sufficiently large, convection is inhibited by the Lorentz
force. For this case we obtain a feebly convecting solution: the root
mean square velocity U = (u2)~ (where the average is over space
and time) is small compared with the reference Alfven speed of
unity (based on the mean field at the top), and the magnetic
Reynolds number Rm ... 323U ... 14. Motion is steady and the pattern
is three-dimensional. The solution is illustrated in Fig. l(a) (opposite), which shows the computational box with a reflected image of
its lower surface. Only fields at the edges of the box are displayed.
Here, and in Fig. 2 (opposite p. 1157), we represent three-dimensional scalar and vector fields by a combination of colour coding
and arrows. In subsequent figures, colours are replaced by a greyscale. The arrows represent the tangential components of the
velocity U, in magnitude and direction; that is to say, the length
of each arrow is proportional to the value of lui at its foot, while its
direction is everywhere that of the local velocity u. The arrows
show the actual velocity on the upper and lower surfaces (where
Uz = 0) but only the projected velocity (after the normal component
has been eliminated) on the sidewalls. The colours (grey-scale) are
used for different fields on the different types of box surface. On the
top and bottom, they denote the value of IBI2 = B;; the spectral
scale ranges from violet (dark) for very weak fields to red (light) for
the strongest fields. On the sidewalls, they correspond to deviations
of the temperature T from the reference atmosphere (so the
polytropic stratification is eliminated); here cold gas is violet
(dark) while hot gas is red (light), so rising plumes appear more
prominent.
The pattern in Fig. l(a) is dominated by rising plumes, which
expand as they impinge upon the upper boundary. As a result,
magnetic flux is swept outwards and confined to a broad network
which encloses the plumes at the top of the layer. Within this
network the field is strong enough to halt all components of the
horizontal motion. At the base of the layer, on the other hand, the
magnetic field is weak except in flux tubes centred on the rising
plumes, where the horizontal inflow produces fields that are locally
intense. This magnetic structure is similar to that found for kinematic magnetoconvection in an idealized hexagonal cell (Galloway
& Proctor 1983; Proctor 1992), except that the field strength is
limited dynamically, through effects of the Lorentz force, rather
than by resistivity.
Fig. 3(a) shows how the magnetic structure varies with height
across the layer. Here the grey-scaling is proportional to the value of
IBI2 and the panels show the field strength at the top, middle and
bottom of the box. From these horizontal slices we see that the
strong flux concentrations at the base of the layer swell and expand
into interconnected ring-like structures that form a network at the
top. The geometry of the corresponding vector field (represented by
individual lines of force) is extremely intricate and complicated
(Galloway & Proctor 1983).
In Fig. 3(b) the grey scaling shows values of the density p at
depths corresponding to those in Fig. 3(a). Within the layer, the
rising plumes are buoyant because p within them is reduced below
its horizontally averaged value. At the top and bottom, however,
other effects come into play. As the rising plume impinges on the
upper boundary, the pressure rises and buoyancy braking produces
a corresponding increase in p (Massaguer & Zahn 1980). In
addition, magnetohydrostatic equilibrium leads to a reduction in
pressure and density in the network where the field is strong. At the
lower boundary, on the other hand, the gas pressure and density are
reduced at the centre of the rising plume, where the field is intense
and the magnetic pressure is high. Owing to stratification, the
effective value of {3 at the bottom of the layer is greater by a
factor of 121 than at the top, so much more flux can accumulate
before the resulting magnetic pressure has a significant influence.
These effects can be observed in the panels of Fig. 3.
As a result of the strong up-down asymmetry resulting from
stratification, the convection cells form an almost hexagonal pattern
that is periodic, with period A, in the x- and y-directions. Each cell
has a rising plume at its centre and sinking fluid around its
periphery, and there are six cells fitted into the square box. Given
the positions of the upward plumes (which lie on an almost rhombic
lattice, actually composed of parallelograms with sides in the ratio
0.943:1), it is possible to construct a Voronoi tesselation of the
plane (cf. Simon, Title & Weiss 1991), and so to generate a
deformed hexagonal grid with weak sinking plumes at the comers.
For up-hexagons there are therefore twice as many downward
plumes as upward plumes.
To calculate the mean spacing, A = 2r, between the plumes we
first obtain (or estimate if necessary) the mean, time-averaged
number N of plumes in a periodic box and then form the rms
radius r = Nv;N. From Fig. l(a) we have N = 6 and so r ... 0.5,
giving a value of A that is consistent with the actual spacing; it is
also possible to find a steady solution withN = 7. These relatively
ordered patterns, with a narrow spacing, are a consequence of the
strong field, which dominates the motion. If convection is driven
more vigorously, withR = 2 x 105 , we find that the motion is timedependent and aperiodic, with r .. 0.6. The effect of doubling R is
similar to that produced by halving Q.
3.2 Spatially modulated oscillations
The steady solution found for R = 105 and Q = 2000 undergoes an
oscillatory bifurcation as Q is reduced. For Q = 1500 we find
aperiodic oscillations; the plumes still lie on an almost hexagonal
grid but they are spatially modulated so that alternate plumes wax
and wane in strength. Although the amplitude of the variations in
(u 2 ) is small, the modulation is clearly visible in a video. The
amplitude is more apparent for lower field strengths. The panels in
Fig. l(c) show solutions for Q = 1400 at opposite phases of the
cycle, and demonstrate how one row of plumes expands while the
adjacent rows contract. These oscillations are a three-dimensional
analogue of the two-dimensional spatial modulation that was
described in Paper I. As Q is further reduced, the oscillations
grow wilder until the plumes merge to create a different pattern
with a larger horizontal scale.
© 1996 RAS, MNRAS 283,1153-1164
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
Opposite p. 1156, MNRAS, 283
(b)
(a)
(c)
Figure 1. Patterns of non-linear magnetoconvection (R = 105 , t.. = 2). (a) Steady convection for Q = 2000, in the magnetically dominated regime. (b) Snapshot
of turbulent convection for Q = 500. (c) Spatially modulated oscillations for Q = 1400: the two frames show the structure at opposite phases of an oscillation.
The colour coding, ranging from violet (low) to red (high), denotes the value of IBI2 at the upper and lower boundaries and the temperature fluctuation on the
sidewalls of the box. The arrows represent the component of velocity paraliel to the wall of the box.
© 1996 RAS, MNRAS 283,1153-1164
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
(a)
(b)
(c)
(d)
Figure 2. Aperiodic convection: as for Fig. 1 but with Q = 1000. The sequence of four panels shows the chaotic evolution of broad rising plumes enclosed by a
magnetic network at times separated by an interval ill = 4.7.
© 1996 RAS, MNRAS 283,1153-1164
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
Photospheric convection
( a)
1157
(b)
Figure 3. Variation of field strength and density with height for steady convection (Q = 20(0). The grey-scales indicate the magnitude of (a) IBI2, with
strong(weak) fields in light(dark) regions, and (b) p (light regions light, dark regions heavy) in horizontal planes at Z = Zo (top), Zo +! (middle) and Zo + 1
(bottom). Note how the intense fields centred on rising plumes at the base of the layer develop into a magnetic network at the top. At the bottom the density is
reduced in buoyant rising plumes; at the top, buoyancy braking leads to enhanced density above the rising plumes, while the magnetic network is partially
evacuated.
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
1158
N. O. Weiss et al.
( a)
(b)
I
Figure 4. Evolution of magnetic structures in turbulent magnetoconvection (Q = 500): the horizontal variation of IBI2 at (a) the top and (b) the bottom of the
layer, at times separated by t::.t = 3.8, running downwards. The last pair of panels correspond to Fig. 1(b). Note the presence of an intense field at the junction in
the network, and the rapid transfer of magnetic flux as the pattern evolves.
© 1996 RAS, MNRAS 283, 1153-1164
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
1159
Photospheric convection
2.5
(a)
2.0
~
;::l
1.5
'"'"~
0.. 1.0
0.5
/
0.0
0
20
10
30
40
30
40
y
2.5
(b)
2.0
~
;::l
1.5
'"'"~
0.. 1.0
0.5
0.0
0
20
10
y
(c)
2.5
2.0
~
;::l
1.5
'"'"~
0.. 1.0
0.5
.....
0.0
0
20
10
30
40
x
Figure 5. Profiles of magnetic pressurePm (full lines), gas pressureP (dotted lines) and dynamic pressurePd (broken lines) across the top of the box for Q = 500,
at the time corresponding to Fig. 1(b) and the bottom panel in Fig. 4(a), where the positions of the profiles are indicated. (a) Variation withy for x = ~ A. (b)
Variation with y for x = ~ A. (c) Variation with x for y = A. All pressures are normalized with respect to Po.
H
3.3 Aperiodic convection for Q = 1000
With a weaker field, convection is both more vigorous and more
obviously chaotic, while the hexagonal arrangement is replaced by
widely spaced irregular plumes. Such aperiodic spatiotemporal
behaviour is best studied by following the changing structures in
a video. The sequence in Fig. 2 shows an evolving pattern of broad
plumes, which is altered as old plumes amalgamate or new ones
form. At the top of the layer, magnetic flux is confined to a network
that continuously changes. Within this network there are flows that
sweep the field into new configurations. Indeed, strong horizontal
velocities are produced when fluid is squirted out of the gap
between two expanding plumes, as in Fig. 2(c); this effect is not
present in steady solutions or simple kinematic models, nor is it
possible in a restricted two-dimensional or axisymmetric geometry.
At the lower boundary the flux is still concentrated into compact
regions with fields that are intense. As they reach a greater height in
the box, these flux tubes change their form into sheets along which
the field can move to form new local concentrations. At any instant
there are typically two or three plumes in the box, so r .. 0.7.
Weiss et al. (in preparation) describe a series of runs as R is
increased from 40 000 to 100000. At first convection is steady, with
a deformed hexagonal pattern (N = 4, r = 0.56); then there are
periodic, spatially modulated oscillations which rapidly become
chaotic. The pattern in Fig. 2 seems to be characteristic of situations
where the magnetic field remains strong but is no longer dominant.
© 1996 RAS, MNRAS 283,1153-1164
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N. O. Weiss et al.
3.4 Turbulent convection for Q
= 500
When Q is halved again, convection becomes more violently timedependent (although the fluid velocity remains subsonic). The
plumes expand to such an extent that there is frequently only a
single broad upwelling in the box. At the top, magnetic flux is
confined to narrow channels between expanding plumes, but there
is now a much greater tendency for intense fields to be produced at
junctions in the network, as in the frame displayed in Fig. 1(b). Fig.
4(a) shows the changing structure of the magnetic field at the top of
the layer. As the pattern evolves, magnetic flux moves rapidly along
the sheets at cell boundaries and then accumulates at new stagnation
points, where strong fields are built up. Comparing the steady
pattern in Fig. l(a) with the instantaneous field in Fig. 1(b), we
see that the broad, uniform magnetic network has been replaced by
local flux tubes, linked by narrow sheets. The sheets are located in
regions of converging flow, where the downward velocity is large;
indeed, one can identify jet-like flows at greater depths, presumably
guided and kept coherent by the concentrated field. They are also
sites of particularly rapid horizontal motion, which sweeps magnetic flux from one comer to another. As a result, the mixed
magnetic structure changes very quickly.
The magnetic pattern at the bottom of the layer is also different.
There are still compact regions where the field is intense, but these
regions move rapidly about as magnetic flux migrates from one flux
concentration to another along a linear feature, as illustrated in Fig.
4(b). Thus the magnetic structures at the upper and lower boundaries have developed a greater similarity. The regions of intense
magnetic field do not, however, coincide: at the bottom, flux is
swept into a rising plume, which evolves within one turnover time,
so that the anticorrelation between the magnetic structures is
weaker than before.
Although these results, with Rm ... 40, are near the limit of our
numerical resolution, we believe that they are still reliable; in a
strongly chaotic flow the evolution of the system is, however,
sensitive to discretization. Convection is now so vigorous that the
magnetic field is dynamically important only where it is locally
intense. Thus there is no longer any coherent pattern at the base of
the layer, and the stratification of r has ceased to be significant.
Nevertheless, the Lorentz force has a global effect on the flow:
compared with a field-free solution for the same R, the kinetic
energy is less, the flow is more ordered and the plumes are more
closely spaced. For the run illustrated in Fig. 1(b) there are typically
one or two dominant plumes in the box, giving r .. 1. Any further
decrease in Q would lead to an expanded horizontal scale of motion
that could only be accommodated with a larger aspect ratio.
We can quantify the dynamical balances in the ephemeral
magnetic structure at any instant of time by comparing the magnetic
pressure Pm = !FIBI2 and the dynamic pressure P d = !plul 2 with
the gas pressure P (or, equivalently, comparing the magnetic and
kinetic energy densities with the thermal energy of the gas). We
shall restrict our attention to the situation illustrated in Fig. 1(b) and
the last pair of panels in Fig. 4. At the top of the layer (z = Zo = fo)
there is a single large rising plume, centred at (x, y) .. (0.9>', O. Th)
and enclosed by lanes of sinking fluid. Magnetic flux is confined to
these lanes, and the strongest fields are near their intersection at
(x, y) .. (0.3>', 0.4>.). In Fig. 5 we display selected profiles of
Pm = !FB;,Pd = !p(u; + U;) and P = PZo at the upper boundary,
normalized with respect to the value of the pressure (Po = zo) in the
reference atmosphere.
We first show, in Fig. 5(a), these three quantities as functions ofy
along the line x =
which passes through the centre of the rising
a}"
plume and crosses the lane that surrounds it at y .. 0.2. Note that the
overall stratification is altered by convection so as to increase the
density at the upper boundary; as a result the average gas pressure is
doubled and P rises to a peak value of 2.4P0 along this line. The gas
pressure varies smoothly over the rising plume, decreasing from a
maximum at its centre (a stagnation point), and then drops sharply
to a minimum in the lane at y = >.. The kinetic energy density is
small near the centre of the plume and highest on a ring near its
periphery. The profile shows two asymmetrical maxima, with a
peak value P.:'ax ... 0.6Po. P d then drops rapidly to a minimum in the
lane (approximately parallel to the x-axis), where Uy" 0 and Ux is
locally very small. By contrast, the magnetic energy density rises to
a sharp peak in the lane, with a maximum value ?::ax .. 2.2Po.
Elsewhere, the magnetic pressure is negligible. Along this cut,
therefore, we find that P.:'ax ~?::ax ... P.
Next, in Fig. 5(b), we display the three pressures as functions of y
along the line x = >., which runs along the lane enclosing the
plume and through the principal flux concentration. Here the
dynamic pressure is always small (P.:'ax ... 0.03Po). The gas pressure
is systematically lower than in Fig. 5(a), with a base value
P ... 1.2Po. The magnetic pressure, on the other hand, is higher
and Pm hovers in the range 1.4 s PmlPO s 2.0. The profiles in
Fig. 5(c) are taken along the line y = ~ >., which passes along the
lane and through the flux concentration in a perpendicular direction,
before entering the plume at x ... 0.6>.; as was apparent from Fig.
1(b), the magnetic feature is elongated in the x-direction. These
profiles again show similar behaviour. The dynamic pressure is very
small outside the plume and rises to a modest peak atx ... 0.8}., while
P andPmare anticorrelated, with?,::ax = 2.5Po. The overall picture
bears some resemblance to the results of Hurlburt & Toomre
(1988), who looked at steady-state two-dimensional flux concentrations; P and Pm are certainly anticorrelated, and the variation in
total pressure is substantially less than that in either P or Pm taken
separately.
From these profiles we can draw several conclusions. First of all,
the dynamic pressure is relatively small. The peak value of Pm is six
times greater than the peak value of Pd: in other words, the field
strength is two to three times greater than its equipartition value.
Secondly, the total pressure remains fairly uniform, although there
is no reason to expect magnetohydrostatic balance in a timedependent calculation. However, the gas pressure and magnetic
pressure individually show substantial variations. Where the field is
strong the pressure and density are halved - that is to say, the flux
concentration is partially evacuated. Moreover, the peak value of
Pm actually exceeds that ofP, while the profiles in Figs 5(a) and (b)
suggest that the dynamic balance depends on magnetic tension as
well as on magnetic pressure.
f4
H
4 GEOMETRICAL EFFECTS: VARYING THE
ASPECT RA TIO
In any of our numerical experiments the pattern selected depends
upon the lateral boundary conditions. Imposing a square lattice
automatically favours square cells or rolls - but since asymmetric
hexagons were found for>. = 2 and Q = 2000 we might expect to
find more nearly regular hexagons if the boundary conditions were
relaxed (e.g. in a wider box). Indeed, we could have obtained
perfect hexagons by taking a rectangular domain with sides in the
ratio V3:1 (cf. Matthews et al. 1995). The pattern found is also
affected by the choice of aspect ratio: ideally we should require that
>. ~ A = 2r, although computational constraints tend to favour
experiments with>. - A instead. So it is essential for us to establish
© 1996 RAS, MNRAS 283,1153-1164
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1996MNRAS.283.1153W
Photospheric convection
Table 2. Runs with varying aspect ratio (R =
100000 , Q = 1000).
A
4
3
2
8
3
RIR(e)
(u2 )1
3.55
3.14
2.53
0.079
0.095
0.10
N
2.5
4.5
r
0.71
0.71
stable to three-dimensional perturbations. Thus the pattern of
convection changes dramatically if A< 2r, as might be expected.
4.2 A =
2D periodic
chaotic
chaotic
whether the chaotic solution found for Q = 1000 is sensitive to
changes in the aspect ratio. In this section we therefore compare the
run described in Section 3.3 with behaviour in narrower and wider
boxes. The results obtained as A is varied with R = 105 and
Q = 1000 are summarized in Table 2.
1161
J: large aspect ratio behaviour
Doubling the aspect ratio allows three-dimensional structures to
persist. Convection is aperiodic and the pattern of motion is similar
to that found for A = 2; from Table 2, we see that the rms velocity
and plume radius are virtually unchanged. Fig. 7 shows the evolution of the magnetic structure at the upper and lower boundaries.
The overall pattern is very similar to that in Fig. 2, so we can be
confident that this behaviour is indeed robust. Apparently the
convective structure is insensitive to variations in the aspect ratio
provided that A > 3r.
4.1 A = ~: two-dimensional rolls
We start by considering a smaller aspect ratio (with the value,
A = that was used for the two-dimensional experiments in Paper
I). AsR is increased aboveR<el , convection first appears with steady
square cells (rather than hexagons) which eventually give way to
periodic, spatially modulated oscillations, as adumbrated in Paper I
(cf. Weiss et aI., in preparation). These oscillations persist, with
increasing amplitude, for 6.5 x 104 sR s 8.5 x 104 . At R = 105
there is, however, a surprise: the three-dimensional pattern collapses into two-dimensional oscillations that are identical to those
found for the same parameter values in Paper I, when two-dimensional symmetry was explicitly imposed. Fig. 6 shows the structure
of these spatially modulated solutions at two opposite phases of the
cycle. Since this two-dimensional behaviour has already been
described in Paper I (see fig. 14 and also fig. 8 of Proctor 1992),
we shall not discuss it further here. For this aspect ratio the rolls are
1,
4.3 Consequences of the imposed lattice structure
The linearized problem has a discrete set of modes: for a given
aspect ratio the preferred wavenumber is that of the mode that first
'<el
,
becomes unstable, at R = R min . In the non-linear regime the
situation is less straightforward. First of all, there are transcritical
and subcritical bifurcations (Matthews et a1. 1995). Secondly, and
more importantly, the wavenumber of an irregular time-dependent
structure is no longer easily defined. For three-dimensional flows
we can introduce an effective wavenumber keff that is inversely
proportional to r. For any supercritical R there is a preferred value
of keff and the non-linear system struggles to achieve a pattern with
this value that is compatible with the imposed periodic lattice. If
A - r a steady pattern might involve square cells with
N = 1,2,4 or 6 plumes in each box, or deformed hexagons with
N = 4,6, 7 or 8, but the choice depends on A. At R = 4.5 X 104 , for
Figure 6. Two-dimensional rolls for A = ~: spatially modulated oscillation with Q = 1000 at two opposite phases of the cycle. The grey-scale corresponds to the
colour scale in Figs 1 and 2.
© 1996 RAS~~~~\\\Wrolf~mical
Society • Provided by the NASA Astrophysics Data System
1996MNRAS.283.1153W
1162
N. O. Weiss et al.
(a)
(b)
Figure 7. Variation of 1812 at (a) the top and (b) the bottom boundaries for Q = 1000 with A =~, at times separated by an interval
structures have the same horizontal scale as in Fig. 2.
t::.t = 3.9. The magnetic
© 1996 RAS, MNRAS 283,1153-1164
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1996MNRAS.283.1153W
Photospheric convection
1
example, we find square cells (N = 2, r = 0.53) for A = but
deformed hexagons (N = 4, r = 0.56) for A = 2 and again (but
differently arranged, with N = 8, r = 0.53) for A = ~. For R = 105
the situation is more extreme: as we have seen, periodic motion in
two-dimensional rolls is preferred when A = although deformed
hexagons return at larger aspect ratios.
In order to be sure that the pattern found in a numerical
experiment is independent of the aspect ratio, we once again need
to have A :» 2r. This is feasible so long as the value of r is restricted
by the magnetic field. It is possible, however, that for Q = 0 there
may be no preferred length-scale in this problem when R is very
large, whatever the choice of A; Hurlburt (private communication)
discovered, for two-dimensional convection in boxes with R = lOS
and A:S; 12, that plumes gradually merge until, after a very long
time, there are only two rolls in the box. We intend to carry out
further numerical experiments for very small values of Q to see if
this conjecture remains valid in three dimensions, and to investigate
the effects on the preferred aspect ratio of changing the boundary
conditions at the top and/or bottom boundaries.
1
5 STELLAR MAGNETOCONVECTION
The changing patterns in our numerical experiments can be summarized extremely briefly: as the imposed field strength is reduced,
the spatiotemporal structure becomes more complex, while the
horizontal scale increases. More precisely, if Q is sufficiently large
then convection is completely suppressed. As Q is decreased, there
is a stationary bifurcation leading to steady three-dimensional
convection in deformed up-hexagons, with a marked asymmetry
between upward and downward motion. A further reduction in Q
produces an oscillatory (Hop±) bifurcation, followed by spatially
modulated oscillations in which alternate plumes wax and wane
periodically. As Q decreases these oscillations grow more vigorous
and become aperiodic. Eventually the time-dependence is so
violent that adjacent plumes merge to produce weakly turbulent
behaviour with a larger characteristic spacing. For high Q, when the
Lorentz force is dominant, magnetic flux at the top of the layer is
confined to a network enclosing the rising plumes; at the bottom,
flux is swept into the rising plumes to produce fields that are locally
intense. At low Q, strong fields form at junctions in the network but
the magnetic structure changes in response to the evolving pattern
of convection, and the 'magnetic fluid' flows rapidly along cell
boundaries to form new field concentrations, at both the top and
bottom of the layer. The tests in Section 4 confirm that these results
may be regarded as robust.
When the imposed magnetic field is relatively weak, the ephemeral flux concentrations contain fields that are locally intense. We
find that the peak value of the magnetic energy density at the top of
the layer is substantially greater than that of the kinetic energy
density. Thus the conventional estimate of the field strength, based
on assuming equipartition between magnetic and kinetic energies,
is certainly not valid at this level. In fact, the maximum magnetic
pressure is close to the maximum gas pressure (which is found at the
centre of the rising plume) and more than twice the background
value of P in the lanes that separate the plumes. Where the field is
strong the gas pressure drops, and the 'flux tube' is partially
evacuated. An accurate description of time-dependent behaviour
then requires a fully compressible calculation, and cannot be
attained within the anelastic approximation.
The results from our idealized model can be compared with the
actual behaviour of magnetic fields in stellar photospheres, as
revealed by high-resolution solar observations (although our para-
1163
meter values are far removed from reality). When the mean flux
density is extremely low, the field is transported passively: flux is
swept into the network, where it accumulates at junctions, as has
been demonstrated by kinematic modelling (Simon, Title & Weiss
1995). Outside plage regions on the Sun, magnetic fields are
confined to the intergranular network, where local concentrations
can be detected in continuum observations through the creation of
filigree and bright points or, much more effectively, in the CH G
band at 4304 A(Berger et al. 1995; Berger & Title 1996). Where the
field is intense, the magnetic pressure is comparable to the external
gas pressure, and an order of magnitude greater than the dynamic
pressure. All this is similar to the behaviour found in our computations for Q = 500, where magnetic flux was confined to the !1etwork
and moved rapidly along it to form intense but ephemeral fields at
the comers where several cells meet. The fields in plage regions,
although globally still weak (150 < Bo < 600 G), are locally strong
enough to be dynamically significant: observations show that the
magnetic field forms a thick network, with holes that correspond to
the convection cells, and that this network is continually changing
(Title et al. 1992; see especially the accompanying video). This
corresponds to the pattern that was found for Q = 1000.
Strong vertical fields occur in pores and sunspot umbrae, where
convection is substantially impeded. Umbral dots appear sporadically: some (the peripheral umbral dots) are probably associated
with flux tubes moving in from the penumbra, but central umbral
dots seem to be local convective features, with a radius of only 90150 km (Sobotka et al. 1993). We suggest that they correspond to
more vigorous versions of the spatially modulated plumes that were
found for Q = 1400 in Section 3.2. In the dark cores or nuclei of
large sunspots there are no umbral dots to be seen. It seems likely
that these are regions where the field is particularly strong and
dominates convection. The pattern should then be compared with
the steady solution for Q = 2000. In a real star, where there are no
rigid boundaries, convection is unlikely to be steady but will
gradually evolve; so we might expect a less ordered version of
the deformed hexagons in Fig. l(a), without the vigorous timedependent plumes that develop for smaller values of Q. Nevertheless, there will still be an important distinction between a
magnetic network at the photosphere, where plumes expand as
they impinge upon a stably stratified atmosphere, and intense but
isolated fields where the plumes are formed by radial inflow at
greater depths.
The numerical experiments described in Sections 3 and 4 raise an
apparent contradiction. The reduced energy flux from the umbra of
a spot cannot be supplied by radiative transport alone (Jalm 1992),
so there must be vigorous convection below the photosphere. As our
results demonstrate, if convective transport is effective it inevitably
leads to distorted fields and significant variations in temperature.
Yet there is no sign of any significant fluctuations in field strength
within the umbra of a sunspot, while umbral dots (the intensity of
which is about three times that of the dark background) only occur
sporadically and occupy less than 10 per cent of the central umbra
(Sobotka et al. 1993). This paradox can only be resolved if the
variations in B and T at the top of the convecting region are
obscured by the blanketing effect of a radiative layer that is
optically thick and lies immediately below the photosphere.
Semi-empirical models of sunspot umbrae, based on infrared
observations, do indeed imply that convective transport only
becomes important at an optical depth 70.5 -10, which is about
100 km below the level of the photosphere, where 70.5 = 1 (Maltby
et al. 1986; Maltby 1992). Is this layer deep enough for variations to
be imperceptible at the photosphere? There are two physical effects
© 1996 RAS, MNRAS 283, 1153-1164
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1996MNRAS.283.1153W
1164
N. O. Weiss et al.
that might be involved. First there is the skin effect: owing to
diffusion, the amplitude of a time-dependent temperature variation
falls off with increasing height. However, this only yields a 25 per
cent reduction for a variation with a period of 1 h, and is therefore
relatively unimportant. The only alternative is diffusive smoothing:
for a pattern with a horizontal wavenumber k the steady solution, if
we assume a uniform diffusivity, will vary as exp(ik· r + kz), and
this can be used to set an upper limit to the horizontal scale of the
convective pattern. If we consider square cells of side 2r,
k = V2:rr/r, and if the amplitude is reduced by an order of
magnitude in a layer of thickness h we have kh > 2.5 or r < 1.8h;
taking h = 100 kIn, this gives r < 180 kIn. This is consistent with
the observed radii of the plumes that just succeed in penetrating the
radiative blanket in order to produce umbral dots. So it seems that
the radiative layer is responsible for the comparative lack of umbral
structure. In the dark cores of sunspot umbrae, where the photosphere is cooler and no dots are found, the field is presumably
stronger and any inhomogeneities caused by the less vigorous
convection are obscured by a deeper radiative layer.
In this paper we have only considered the effects of vertical fields
on convection at the surface of a star. Sunspots pose many other
problems. Below their umbrae, convection extends to deeper levels
than have been modelled here, although not necessarily to the base
of the convection zone. Instead, it is probable that the energy
radiated from the umbral photosphere is drawn from the internal
energy of the underlying flux tube at depths of order 50 Mm
(Thomas & Weiss 1992), and that the properties of such flux
tubes vary systematically during the solar cycle, as indicated by
the observed variations in umbral intensity (Maltby 1992). A more
immediate problem is raised by recent observations of fine structure
in sunspot penumbrae, where the field is strongly inclined and the
inclinations of bright and dark filaments are significantly different.
There is an urgent need to model three-dimensional convection in a
flux tube with a mean field whose inclination to the vertical
increases from zero at the central axis to 70° at the surface that
separates it from the surrounding field-free plasma (Rucklidge et al.
1995). That requires a much more ambitious calculation than we
have attempted here.
ACKNOWLEDGMENTS
We thank Alastair Rucklidge and Alan Title for helpful comments
and suggestions, and we are grateful to SERC and PPARC for
grants that have supported this research.
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n5
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