.945G .243. The Astrophysical Journal, 243:945-953, 1981 February 1 . © 1981. The American Astronomical Society. All rights reserved. Printed in U.S.A. 198lApJ. CONVECTION AND MAGNETIC FIELDS IN STARS D. J. Galloway High Altitude Observatory, National Center for Atmospheric Research1 AND N. O. Weiss Sacramento Peak Observatory2 Received 1980 March 10; accepted 1980 August 18 ABSTRACT Recent observations have demonstrated the unity of the study of stellar and solar magnetic fields. Results from numerical experiments on magnetoconvection are presented and used to discuss the concentration of magnetic flux into isolated ropes in the turbulent convective zones of the Sun or other late-type stars. Arguments are given for siting the solar dynamo at the base of the convective zone. Magnetic buoyancy leads to the emergence of magnetic flux in active regions, but weaker flux ropes are shredded and dispersed throughout the convective zone. The observed maximum field strengths in late-type stars should be comparable with the field (87rp)1/2 that balances the photospheric pressure. Subject headings: convection — hydrodynamics — stars: magnetic — Sun : magnetic fields I. INTRODUCTION In § II we summarize the results obtained for simplified The Sun is unique in having a magnetic field that can be models of hydromagnetic convection. The structure of mapped in detail, but magnetic activity seems to be a turbulent magnetic fields is then described in § III. standard feature of stars with deep convective envelopes. Theory and observation are brought together in § IV, Fields of 2000 gauss or more have been found in two where we discuss the formation of isolated flux tubes in late-type main sequence stars (Robinson, Worden, and the interior of the Sun. Photospheric fields are covered in Harvey 1980). Wilson (1978) has detected chromospheric § V. Finally, we consider the origin of magnetic fields Ca ii emissions from many G, K, and M stars, some of in the Sun and other stars with vigorous convective zones. which show cyclic variations of activity. Some, perhaps Our conclusions emphasize the need for observational all, dMe stars flare, and the unexpectedly high X-ray programs, covering stars of differing mass, that can luminosities of K and M stars (e.g., Vaiana 1979) suggest establish the correlation between magnetic activity and that they have strong magnetic fields. Although the Sun’s rotation or convective velocity; in addition, we need field is comparatively feeble, all attempts to explain better estimates of the field strengths in starspots and the magnetic activity in stars are based on solar observations. areas they occupy. Over the last decade the structure of large-scale solar magnetic fields has been clarified by measurements from II. CONCENTRATION OF MAGNETIC FLUX space, while high resolution ground-based observations have revealed the interaction between small-scale mag- In this section we treat the interaction between laminar netic features and cellular convection in the photosphere. convection and an externally imposed magnetic field. The Theory also has made considerable progress (Parker calculations are restricted to Boussinesq fluids with 1979a). Since magnetic activity is associated with turbu- idealized geometries. In a conducting medium the mag- lent convection, it is important to study the interaction netic field B is governed by the induction equation between magnetic fields and convection (for a review, see c)B 2 Weiss 1977). Nonlinear magnetoconvection has recently — = curl (« x Z?) + r}V B , (1) been investigated in several series of numerical experi- ments (Peckover and Weiss 1978; Galloway and Moore where r¡ is the magnetic diffusivity. If the velocity u is 1979; Weiss 1981a, b). The aim of this paper is to relate prescribed (the kinematic problem) the behavior of the these idealized model calculations to the structure of field depends on the magnetic Reynolds number, magnetic fields at the surface and in the interior of the Sun Rm = UL/r¡, where U, L are a characteristic speed and and other late-type stars. length scale for the motion (Moffatt 1978 ; Parker 1979a). Figure 1 shows the distortion of an initially uniform field B0 by the two-dimensional velocity 1 The National Center for Atmospheric Research is sponsored by the National Science Foundation. u= C/(—sin (nx/L) cos (nz/L), 0, 2 Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation. cos (nx/L) sin (nz/L)), (2) 945 © American Astronomical Society • Provided by the NASA Astrophysics Data System .945G .243. 198lApJ. Fig. 1.—Expulsion of magnetic flux : the evolution of the magnetic field in a kinematic calculation with Rm = 250. Lines offeree are shown at times T/t! = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 20, where t, = 5t0/8. © American Astronomical Society • Provided by the NASA Astrophysics Data System .945G .243. CONVECTION AND MAGNETIC FIELDS . 947 when Rm = 250, with boundary conditions such that Bx = 0 onx,z = 0, L, so that lines of force can move freely 198lApJ. at the upper and lower boundaries. (For a similar calcula- tion with different choices of velocity and boundary conditions, see Weiss 1966). As the eddy turns over, magnetic flux is quickly concentrated into sheets at the lateral boundaries, while the central field grows progres- sively more distorted until reconnection occurs and flux is expelled from the convective eddy. In the final state the field at the edges of the cell is amplified by a factor of order 1/2 Rm (e.g., Proctor and Weiss 1978). This amplification proceeds rapidly: After one turnover time t0 = L/U, the field can be increased by a factor of up to 200 (Weiss 1966). On the other hand, flux expulsion takes much longer: The process is completed by t ^ 5.5t0 when 1/3 Rm = 250, but this time increases as Rm (Weiss 1966). In a three-dimensional flow, the flux is concentrated into an isolated tube and the field is amplified by a factor of order Rm (Galloway, Proctor, and Weiss 1978 ; Galloway and Moore 1979). These strong fields exert forces which modify the pattern of convection, so it is necessary to consider the dynamical problem and solve the equations of motion. Linear theory has been extensively discussed (Chan- drasekhar 1961; Danielson 1961). In the astrophysically interesting case, when the thermal diffusivity k rj and B0 is sufficiently large, convection sets in as overstable oscillations. Heat transport by oscillatory convection is Fig. 2.—The dynamical effect of the magnetic field on convection : relatively inefficient so it is important to ascertain when steady state with motion excluded from the stagnant flux sheets. Streamlines and lines of force for a model with R = 6500, Q = 2000, steady convection can occur. The configuration is <7 = 1, £ = 0.05. described by dimensionless parameters gocßd4 Bo2d2 and Q = (3) KV 4nprjv the field R0, the nonlinear results indicate that steady convection occurs more readily in wider cells, which (the Rayleigh and Chandrasekhar numbers), and by the allow more space for convective motion between the ratios concentrated fields. The numerical experiments suggest 2 (e) g = v/k and Ç = vj/k (4) that Rmin ~ 7r CQ R , as conjectured by Danielson (1961). For a given Rayleigh number, steady convection of the diffusivities. Here v is the viscous diffusivity, d the occurs for Q<Qmax~R/£; hence only oscillatory layer depth, ß the superadiabatic temperature gradient, motion is possible when the field exceeds a critical value and a the coefficient of expansion. For Ç 1 but (Ô > 1, 2 112 overstability sets in when R = R(0) ~ 7i2<jÇQ/(g + 1) and ßmax ~ (gpocßd ) . (5) steady solutions bifurcate from the static conducting The solutions displayed in Figure 2 are symmetrical (e) 2 solution at R = R ~7r <2. about the center of the cell. As Q approaches ßmax, Nonlinear calculations confirm the existence of subcri- asymmetrical solutions are preferred. Figure 3 shows the (e) tical steady convection in the range Rmin < R < R . As R final steady state for convection in a box whose width is 4 is increased, steady convection first appears with finite times its depth. The run was started from noisy initial amplitude at R = Rmin. Figure 2 shows streamlines and conditions but settled down to give two cells (with the lines of force for steady two-dimensional convection with same sense of rotation) on either side of a stagnant central periodic boundary conditions (cf. Weiss 1981a). Once slab. In this solution the central region contains over 60% again, the flux is confined to narrow sheets, allowing of the total flux, but with more extreme parameters 90% convection to take place in the field-free central region; of the flux can be concentrated on one side of a convection within these sheets, the field is strong enough to exclude cell (Weiss 1981/?). Heat transport is apparently made the motion. Similar results have been obtained for an more efficient by assembling as much magnetic flux as axisymmetric configuration, where the flux is confined to possible into a single isolated sheet or tube, so liberating a stagnant axial tube within which the field is nearly the rest of the region for convection to take place. In a uniform (Galloway and Moore 1979). wider layer this provides a mechanism for amalgamating Although linear theory predicts that convection should small flux tubes into a single concentration when the first appear in narrow cells, elongated in the direction of imposed field is relatively strong. © American Astronomical Society • Provided by the NASA Astrophysics Data System .945G .243. 948 GALLOWAY AND WEISS Vol. 243. flow with a velocity I/o at the flux sheet. The magnetic field generates a counterflow with velocity 2 198lApJ. U1 ~ (B* /47ipv)e (Galloway, Proctor, and Weiss 1978), which is proportional to e.