tion in the umbra, and our higher Rayleigh number results with convection

in the penumbra. These regimes correspond in turn to our vertical and hor-

izontal convection branches, i.e., the vertical convection mode is the umbral

mode while the horizontal mode corresponds to the penumbral mode. If this

interpretation of our results is correct it suggests a natural explanation for

the abrupt change in the properties of the umbra and penumbra even when

Nonlinear magnetoconvection in the presence of a strong oblique ¬eld 355

the e¬ective Rayleigh number varies gradually across the spot. Furthermore,

putting the question of pattern selection aside, observations of parallel (ra-

dial) rolls in the penumbra are consistent with our ¬nding that these are the

structures that transport heat most e¬ciently and are therefore the most

luminous.

23.5 Conclusion

We have summarized the derivation of a new class of reduced PDEs (23.5)

valid for magnetoconvection in the presence of a strong oblique ¬eld. These

¬lter out the fast, small scale Alfv´n waves and relax the need to resolve the

e

horizontal mechanical and magnetic boundary layers. This reformulation

promises to have signi¬cant applications in astrophysics and in particular

for studies of convection in sunspots.

It is shown that the reduced set of equations admits exact fully nonlinear,

single-mode solutions which are determined from the nonlinear eigenvalue

problem (23.6). This is possible when the ¬eld strength is large and the

distortion of the ¬eld by the ¬‚ow remains small. When the imposed mag-

netic ¬eld is vertical the solutions of the nonlinear eigenvalue problem with

ω = 0 can be used to construct a variety of three-dimensional patterns, all

of which have the same Nusselt number in the strong ¬eld regime. A sim-

ilar degeneracy characterizes all oscillatory patterns in this regime. Weak

selection among these patterns is due to subdominant terms not computed

here. An inclined ¬eld breaks this degeneracy and the theory then describes

two-dimensional structures only. Whenever these distort the magnetic ¬eld

(i.e., k0 — (g — ˆ) = 0) the behaviour of the system falls into two possible

r

regimes. For small tilt angles the magnetic ¬eld plays a relatively minor

role in inhibiting convection, and the Nusselt number is an increasing func-

tion of the Rayleigh number. If the tilt angle is increased past a threshold

value (which depends on the value of ζ) a hysteretic transition may take

place with increasing Rayleigh number from this “vertical ¬eld” regime to a

“horizontal ¬eld” regime in which the ¬eld plays a major role in inhibiting

the heat transport. The possible connection with observation of sunspots is

discussed.

The work presented here can be extended to incorporate the e¬ects of

depth-dependent pro¬les of the di¬usivities thereby breaking the Boussinesq

midplane symmetry (Julien et al. 1999, 2000). In addition it is possible

to explore the transition from steady to overstable motion with increasing

depth. The procedure outlined above extends readily to the more realistic

Julien, Knobloch & Tobias

356

anelastic formulation of the basic equations of motion, as detailed in Julien

et al. (2002).

Acknowledgments. This work was supported by NASA under SEC grant

MAS-99026 (KJ), the Department of Energy under Grant No. DE-FG03-

95ER-25251 (EK) and NASA under SPTP grant NAG5-2256 (SMT).

References

Brownjohn, D. P., Hurlburt, N. E., Proctor, M. R. E., & Weiss, N. O., 1995, JFM,

300, 287

Chandrasekhar, S., 1961, Hydrodynamic and Hydromagnetic Stability, Oxford

University Press

Clune, T., Knobloch, E., 1993, Physica D, 74, 151

Embid, P. F., Majda, A. J., 1998, GAFD, 87, 1

Hughes, D. W., Proctor, M. R. E., 1988, Ann. Rev. Fluid Mech., 20, 187

Julien, K., Knobloch, E. & Tobias, S. M., 1999, Physica D, 128, 105

Julien, K., Knobloch, E. & Tobias, S. M., 2000, JFM, 410, 285

Julien, K., Knobloch, E. & Tobias, S. M., 2002, in Ferriz-Mas, A. &

N´nez-Jim´nez, M., eds, Advances in Nonlinear Dynamos, Taylor and

u˜ e

Francis, in press

Matthews, P. C., Hurlburt, N. E., Proctor, M. R. E. & Brownjohn, D. P., 1992,

JFM, 240, 559

Taylor, G. I., 1923, Proc. Roy. Soc. A, 104, 213

Thomas, J. H. and Weiss, N. O., 1992, in Thomas, J. H. and Weiss, N. O., eds,

Sunspots: Theory and Observations, Kluwer, p. 3

Weiss, N. O., Brownjohn, D. P., Hurlburt, N. E. & Proctor, M. R. E., 1990,

MNRAS, 245, 434

Weiss, N. O., Brownjohn, D. P., Matthews, P. C. & Proctor, M. R. E., 1996,

MNRAS, 283, 1153

24

Simulations of astrophysical ¬‚uids

¨

MARCUS BRUGGEN

Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK, and

International University Bremen, Campus Ring 1, 28759 Bremen, Germany

In this contribution I discuss how recent advances in numerical techniques

and computational power can be applied to problems in astrophysical ¬‚uid

mechanics. As a case in point some results of simulations of radio relics

are presented which have provided strong support for a model that explains

the origin of these peculiar objects. Radio relics are extended radio sources

which do not appear to be associated with any radio galaxy. Here a model

is presented which explains the origin of these relics in terms of old plasma

that has been compressed by a shock wave. Having taken into account syn-

chrotron, inverse Compton and adiabatic energy losses and gains, the rel-

ativistic electron population was evolved in time and synthetic radio maps

were made which reproduce the observations remarkably well. Finally, some

other examples are discussed where hydrodynamical simulations have proven

very useful for astrophysical problems.

24.1 Introduction

With the advent of powerful computers and more accurate algorithms, simu-

lations of astrophysical ¬‚uids have become increasingly useful. Most ¬elds of

astrophysics, such as solar physics, star formation, stellar evolution and cos-

mology have bene¬tted greatly from hydrodynamical simulations and hopes

for further advances are high.

Essentially, there are two main approaches to the numerical solution of the

equations of hydrodynamics: Finite-grid simulations and Smoothed Particle

Hydrodynamics (SPH). In the former approach the equations are discretised

on a computational mesh before they are solved. The latter method avoids

the notion of a mesh and employs particles to track the ¬‚uid. Both methods

have di¬erent strengths and weaknesses but here I will only be concerned

with grid-based codes. These have proven especially useful for discontinuous

357

Br¨ggen

u

358

¬‚ows where shock-capturing advection schemes yield much greater accuracy

than SPH codes. Moreover, it is generally easier to add more physics such

as magnetic ¬elds or radiation to the grid-based codes than to SPH codes.

One of the main challenges in the simulation of astrophysical ¬‚uids is to

bridge the gap between the simulation of the macroscopic ¬‚ow and di¬erent

microphysical processes, such as nuclear reactions, X-ray and radio emission,

just to name a few. The scales of the macroscopic ¬‚ow and the microphysical

processes are separated by many orders of magnitude, yet both ™worlds™ are

tightly interlinked. In this contribution I will show some results from simu-

lations where we have attempted to bridge the gap between the large-scale

¬‚ow and the di¬erent physical processes of radio emission from a relativistic

plasma. These simulations have proven useful for the understanding of radio

relics and radio galaxies. Despite the very general title, I can only give very

few examples of the use of hydrodynamical simulations in astrophysics. In

the course of this meeting we have heard of several applications in the ¬eld

of stellar physics so that here I am going to give some examples from the

realm of extragalactic astronomy.

Many problems in numerical hydrodynamics require a high resolution in

order to describe the evolution of the system accurately enough. In turn, the

use of large grids implies high demands in terms of both, computer mem-

ory and CPU time. One numerical technique which has been developed to

increase the e¬ciency of these ¬nite-di¬erence simulations is the method of