J. Fluid Mech. zyxwvutsrq (1979), wol. 90, part 2, pp. 275-287 Printed i n zyxwvutsr Great Britain 27 3 The dynamic effect of flux ropes on Rayleigh-Benard convection By M. R. E. PROCTOR Department of Applied Mathematics and Theoretical Physics, University of Cambridge AND D. J. GALLOWAY Astronomy Centre, ‘LTniversity of Sussex, Brighton? (Received 24 March 1978) zyxwv The interaction between magnetic fields and convection in a fluid heated from below is investigated in an axisymmetric cylindrical geometry. When R,, the magnetic Reynolds number, is large the field is concentrated into a thin rope on the axis of the cylinder. FOP weak magnetic fields a larger Rayleigh number is necessary to produce a flux rope than that needed for infinitesimal convection. For larger total fluxes, how- ever, the opposite is true and the system is subcritically unst’able to steady motions. The results are contrasted with those found by Busse (1975) for the corresponding two-dimensional roll problem. 1. Introduction The theory of the interaction between magnetic fields and thermal convection is of great importance in the study of the solar convective zone. Recent observations (Stenflo 1976; Harvey 1977) have shown that the photospheric granulation and super- granulation are threaded by intermittent but intense flux concentrations that are formed by, and react upon, the convective motions (for a discussion see Galloway, Proctor zyxwvuts & Weiss zyxwvu 1977); the precise nature of the dynamic balance that is attained is the main goal of inquiry. There have been numerous studies of idealized model problems. Thompson (1951), Chandrasekhar (1961) and Danielson (1961) have studied the linear instability of a Boussinesq fluid layer heated uniformly from below (the Rayleigh- BBnard or RB problem) with an imposed vertical magnetic field. They found that if zyx p, = K/T < 1, where K is the thermal conductivity and 7 the magnetic diffusivity, motion occurs first as steady convection as the Rayleigh number (a dimensionless measure of the temperature difference across the layer) is increased. If p3 > 1, however, and there is sufficient magnetic flux present, then convection occurs first as overstable oscil- lations. All these linear results are independent of the convection planform. More recent investigations of the nonlinear regime have shown that the linear results are rather unrepresentative, When p, b 1 the effects of advection of the magnetic field quickly come to dominate those of diffusion (i.e. the magnetic Reynolds number R, E PL/q 9 1, where L and P are typical length and velocity scales); then j- Present address: High Altitude Observatory, Boulder, Colorado. 10 FLM 90