Modelling photospheric magnetoconvection
Mon. Not. R. Astron. Soc. 301, 593±608 (1998)
Modelling photospheric magnetoconvection
S. M. Blanch¯ower, A. M. Rucklidge and N. O. Weiss
Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Cambridge CB3 9EW
Accepted 1998 April 24. Received 1998 March 27; in original form 1997 October 17
A B S T R AC T
Key words: convection ± MHD ± Sun: granulation ± Sun: magnetic ®elds ± sunspots ± stars:
magnetic ®elds.
1
INTRODUCTION
Magnetic ®elds interfere with convective transport in the photospheres of late-type stars. This interaction can be observed in detail
at the surface of the Sun, where features that are only a few hundred
kilometres across can now be resolved, revealing a variety of ®ne
structure that depends on the local strength of the magnetic ®eld. At
the same time, rapid advances in computing power have made it
possible to model non-linear magnetoconvection in regimes where
numerical experiments can be contrasted with solar observations. In
this paper we study the effects of varying the geometry and
boundary conditions in idealized models, and identify different
patterns of behaviour when the ®elds are weak or strong. These
regimes are then related to convective structures on the Sun.
In the solar photosphere, the strongest vertical ®elds are found in
pores and sunspot umbrae, where convective plumes show up as
`umbral dots' (Danielson 1964). These small bright features are
present in all sunspots, though large spots contain isolated regions
(dark nuclei) that are free of them (Muller 1992; Sobotka, Bonet &
VaÂzquez 1993; Sobotka 1997). Until very recently it was thought
that umbral dots had diameters of 180±300 km and a ®lling factor of
3±10 per cent. With improved resolution (Sobotka 1997; Sobotka,
Brandt & Simon 1997a,b) it is now clear that there is no typical
diameter; rather, the number density of umbral dots increases with
decreasing size, down to the limit of resolution at 0.28 arcsec
(200 km). An average specimen has a diameter of 300 km and a
lifetime of 14 min but the lifetimes range from 2 h for the largest
q 1998 RAS
bright dots to a few minutes for the smallest. `Light bridges' are
bright linear features that cut across sunspots and exhibit a ®ne
granular structure (Muller 1992; Sobotka et al. 1993; Sobotka
1997). The ®eld within a light bridge is weaker than in the
surrounding umbra, so the bridge resembles a slot, contained
between magnetic walls that resist distortion, within which more
vigorous convection can occur. Rimmele (1997) has followed the
evolution of convective plumes within such a slot for a full hour and
con®rmed that bright granules are associated with upward motion.
He also found that the velocity and intensity varied in a manner
consistent with oscillatory convection.
Outside sunspots, there is a distinction between plage regions
(with average ®eld strengths greater than 150 G) and quiet Sun.
Plages are characterized by abnormal granulation, corresponding to
convection with a smaller horizontal scale. The ®elds form a
perforated network, including ®ne magnetic structures that give
rise to isolated bright points (Title et al. 1992; Muller 1994;
Sobotka, Bonet & VaÂzquez 1994). These appear most strikingly
in the CH G-band and their dynamic behaviour indicates that
magnetic ¯ux moves rapidly through the intergranular network,
forming ephemeral concentrations rather than isolated ¯ux tubes
(Berger et al. 1995; Berger & Title 1996; Berger et al. 1998).
Within the photospheric network, magnetic structures are smaller
and more nearly isolated, with diameters less than 1000 km and
®elds of 1±2 kG (Muller 1994).
There are two approaches to modelling the non-linear interaction
between convection and magnetic ®elds at the surface of a star like
The increasing power of computers makes it possible to model the non-linear interaction
between magnetic ®elds and convection at the surfaces of solar-type stars in ever greater detail.
We present the results of idealized numerical experiments on two-dimensional magnetoconvection in a fully compressible perfect gas. We ®rst vary the aspect ratio l of the
computational box and show that the system runs through a sequence of convective patterns,
and that it is only for a suf®ciently wide box (l $ 6) that the ¯ow becomes insensitive to
further increases in l. Next, setting l 6, we decrease the ®eld strength from a value strong
enough to halt convection and ®nd transitions to small-scale steady convection, next to
spatially modulated oscillations (®rst periodic, then chaotic) and then to a new regime of ¯ux
separation, with regions of strong ®eld (where convection is almost completely suppressed)
separated by broad convective plumes. We also explore the effects of altering the boundary
conditions and show that this sequence of transitions is robust. Finally, we relate these model
calculations to recent high-resolution observations of solar magnetoconvection, in plage
regions as well as in light bridges and the umbrae of sunspots.
594
S. M. Blanch¯ower, A. M. Rucklidge and N. O. Weiss
boxes. As l is increased there are transitions from steady convection to travelling waves and then to spatially modulated oscillations.
In wide boxes (l $ 6) a totally new phenomenon appears: the
magnetic ®eld separates from the motion, leaving broad, dynamically active convective plumes and strong isolated concentrations of
magnetic ¯ux.
Next, in Section 4, we take a box that is wide enough (l 6) to
allow suf®cient freedom, and vary the ®eld strength to reveal new
patterns of behaviour. For very strong ®elds convection is completely suppressed but as Q is decreased there are again transitions to
steady and oscillatory convection, culminating in ¯ux separation for
Q < 750. The effects of choosing different boundary conditions are
explored in Section 5 and the implications of our results for 3D
models, as well as for photospheric convection in the Sun and other
late-type stars, are discussed in the ®nal section.
2
THE MODEL PROBLEM
Our basic con®guration has already been used for both 2D (Hurlburt
& Toomre 1988; Weiss et al. 1990) and 3D (Weiss et al. 1996)
investigations. Once again, we introduce cartesian co-ordinates
with the z-axis pointing downwards and two-dimensional ®elds
such that the velocity u and the magnetic ®eld B lie in the xz-plane
and are independent of y. Then we cast the equations into dimensionless form and consider a perfect gas occupying the region
f0 # x # l; z0 # z # z0 1g. In the absence of any motion there is
a uniformly strati®ed equilibrium solution corresponding to a
polytrope with index m, for which the temperature T z and the
density r vzm, where v 1=z0 . This strati®cation is superadiabatic if m < 1= g 1, where g is the ratio of the speci®c heat at
constant pressure to that at constant volume.
The non-linear partial differential equations that govern the
evolution of r x; z; t, T x; z; t, B x; z; t and u x; z; t with time
t are as g (...truncated)
