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Numerical Simulation of Magnetoconvection in a Stellar Envelope

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Numerical Simulation of Magnetoconvection in a Stellar Envelope Center for Turbulence Research 281 Annual Research Briefs 2002 Numerical simulation of magnetoconvection in a stellar envelope By S. D. Ustyugov AND A. N. Andrianov y 1. Introduction One of the main problems in the physics of the Sun is the interaction between con- vection and magnetic ¯elds. Convective motions in stellar envelopes span a substantial radial distance from the convectively unstable region into the adjacent stable zones and can distort the magnetic ¯eld into concentrated flux sheets and tubes in which the mag- netic pressure is comparable to the gas pressure (see Hurlburt & Toomre (1988)). On the other hand, magnetic ¯elds can be su±ciently strong to suppress convection on granu- lar and supergranular scales. The dynamics of such couplings is strongly nonlinear, and the flow spans many scale heights in the vertical direction and weakens the stable ther- mal strati¯cation. Studies of convection in the presence of magnetic ¯elds have already shown e®ects arising from compressibility, revealing distinctive asymmetries between up- ward and downward flows. Convection penetrates into the underlying stable layers as downward-directed plumes (see Hurlburt et.al. (1986)). Here we consider compressible convection in three spatial dimensions in the presence of an externally imposed magnetic ¯eld. We study penetrative convection within relatively simple con¯gurations consisting of two layers with well-posed boundary conditions. We include all di®usive processes and do not model the unresolved scales. 2. Initial model and equations We shall consider penetrative convection in a compressible stellar envelope in the pres- ence of an imposed magnetic ¯eld. We assume that this envelope experiences uniform gravitational acceleration directed downward, and possesses constant thermal conductiv- ity, a constant magnetic di®usivity, and a constant shear viscosity. The strati¯cation consists of an unstable layer bounded below by a stable polytrope. This con¯guration mimics the lower part of the solar convection zone and the upper radiative zone. The initial distribution of temperature, density, and pressure are expressed as m® m®+1 z ¡ z® ½ T P T T = T® + ; = ; = (2.1) K® ½® µT® ¶ P® µT® ¶ for the unstable (® = 1) and stable (® = 2) layers. Here the polytropic indices m® are given by m® + 1 = gK®=RFT , where R is the gas constant, K® is the coe±cient of thermal conductivity, and FT is the total energy flux. Convective instability occurs when m® < mc, where mc, the critical polytropic index, depends on the ratio of speci¯c heats y Keldysh Institute of Applied Mathematics, Moscow, Russia 282 S. D. Ustyugov & A. N. Andrianov γ. For perfect a monoatomic gas (γ = 5=3), mc = 3=2. We take m1 = 1 in the unstable layer and m2 = 3 in the stable layer. We have solved the equations of three-dimensional, fully compressible nonlinear mag- netoconvection. We use cartesian coordinates x = x1; y = x2; z = x3, where the positive z-axis points downwards. The equations of motion are as follows: conservation of mass, @½ @ + (½ui) = 0; (2.2) @t @xi conservation of momentum, @½ui @ 1 + ±ij P + BkBk + ½uiuj ¡ ¿ij ¡ BiBj ¡ ½g±i3 = 0; (2.3) @t @xj · µ 2 ¶ ¸ conservation of energy, @½E @ 1 P @T + ½uj uiui + ² + ¡ gx3 ¡ ui¿ji ¡ K @t @xj · µ2 ½ ¶ @xj ¸ (2.4) @ @Bi @Bj ¡ ´Bi ¡ + Bi (Bjui ¡ Biuj) = 0; @xj · µ @xj @xi ¶ ¸ and the induction equation, @Bi @ @Bi @Bj + ujBi ¡ uiBj ¡ ´ ¡ = 0: (2.5) @t @xj · µ @xj @xi ¶¸ These are augmented by the equation of state for a perfect gas, P = R½T; (2.6) and expressions for the total energy per unit mass, 1 1 BiBi E = ² + u u ¡ gx3 + ; (2.7) 2 i i 2 ½ and the viscous stress tensor, @ui @uj 2 @ul ¿ij = ¹ + ¡ ±ij : (2.8) µ@xj @xi 3 @xl ¶ Here ½, T , P , and ² are density, temperature, pressure, and speci¯c internal energy, and Bi denotes components of the magnetic ¯eld. We write the equations in dimensionless form and take our unit of length to be the depth of the unstable layer d. The time unit d= (Rd¯1) is related to the sound travel time across the unstable layer, where ¯1 is the initial temperature gradient within the unstable layer. The magnetic ¯eld is scaled by B^, the value of the initially uniform vertical magnetic ¯eld. The evolution of the system depends on some physical and geometrical parameters. We de¯ne the computational box to extend in a horizontal plane from x = y = 0 to x = y = A, where A is the aspect ratio of the unstable layer, and vertically from z = z1 Magnetoconvection in a stellar envelope 283 to z = z3. The depth of the unstable layer is unity in our units. The density contrast Â2 is the ratio of the density at the bottom of the unstable layer (z = z2) to that at the top: ½ (z2) Â2 = (2.9) ½ (z1) The Prandtl number, which gives the ratio of viscous to thermal di®usivity, is de¯ned as ¹c σ = p (2.10) K The degree of instability in the unstable layer is given by the Rayleigh number ¡1 ¡1 2 2m1¡1 Ra = (m1 + 1) (mc ¡ m1) (mc + 1) σ1 ¸ z ; (2.11) where ¸ is the ratio of the sound speed to the thermal di®usion time and σ1 is the Prandtl number in the unstable layer. Since Ra depends on vertical position in the unstable layer, we evaluate it at the center, taking 1 R^ = R z1 + : (2.12) a a µ 2¶ as a nominal Rayleigh number. The strength of the imposed magnetic ¯eld and its e®ect on convective stability can be measured by the Chandrasekhar number d2 Q = B2 (2.13) ¹´ The magnetic Prandtl number (the ratio of magnetic to thermal di®usivity) is de¯ned as ³ = ´½cp=K (2.14) 3. Numerical method For the numerical simulation we have used an explicit TVD method, second order ac- curate in space with the time advance using a three-step Runge-Kutta scheme (see Yee et.al. (1990), Shu (1989)). Our algorithm uses equations in conservative form for advec- tion, wave propagation, and di®usion processes. The 3-D compressible time-dependent MHD equation with source term can be written as @U @F @G @H + + + = S; (3.1) @t @x @y @z The dependent variable U is vector of conserved variables, and F; G; H are vectors of flux in the three space directions. Let Ut = L(U)i;j;k; (3.2) 284 S. D. Ustyugov & A. N. Andrianov be the semi-discrete form of (3.1) at point (i; j; k), where L is the spatial discretization operator 1 1 L = ¡ F +1 2 ¡ F ¡1 2 ¡ G +1 2 ¡ G ¡1 2 i;j;k ¢x i = i = ¢y j = j = ¡ ¢ ¡ ¢ (3.3) 1 ¡ H +1 2 ¡ H ¡1 2 + S ; ¢z k = k = i;j;k ¡ ¢ The numerical flux, for example in the x direction, is de¯ned as 1 F +1 2 = F +1 + F + P +1 2© +1 2 : (3.4) i = 2 i i i = i = ¡ ¢ The last term Pi+1=2©i+1=2 is a nonlinear dissipation, and the quantity Pi+1=2 is the right eigenvector matrix @F=@U using, for example, Roe's approximate average state. l The components Ái+1=2 of vector ©i+1=2 can be written as 1 Ál = Ã(al )(gl + gl) ¡ Ã(al + γl )®l ; (3.5) i+1=2 2 i+1=2 i+1 i i+1=2 i+1=2 i+1=2 1 γl = Ã(al )(gl ¡ gl)=®l if ®l 6= 0; (3.6) i+1=2 2 i+1=2 i+1 i i+1=2 i+1=2 l l γi+1=2 = 0 if ®i+1=2 = 0: (3.7) l The ai+1=2; l = 1; :::; 7, are the characteristic speeds of @F=@U evaluated at some l symmetric average of Ui and Ui+1. The function à is an entropy correction to j ai+1=2 j. l ¡1 The ®i+1=2 are elements of Ri+1=2(Ui+1 ¡ Ui). l The limiter function gi can be expressed as l l l 1 l l g = minmod(2a ¡ ; 2a ; (a ¡ + a )): (3.8) i i 1=2 i+1=2 2 i 1=2 i+1=2 For the thermal conductivity we have applied an explicit multistage numerical scheme. This scheme is absolutely stable and is described in detail by Zhukov et.al. (1993). The numerical scheme has been tested widely and used successfully in earlier work (see Ustyu- gov & Andrianov (2002)). We have carried out computations in a cube with 128 mesh points in each direction on 256 processors of the parallel machine with cluster architec- ture MBC 1000M. The computer program was parallelized using the NORMA system (see Andrianov et.al. (2001)). The Norma program was then compiled using Fortran and the MPI library. A typical simulation requires on this cluster 50 hours to advance the solution well past its start-up transient. 4. Boundary and Initial conditions We impose periodic boundary conditions with period A for all variables in x and y. On the upper and lower surfaces we use stress-free boundary conditions: Bx = By = @Bz=@z = uz = @ux=@z = @uy=@z = 0; (4.1) Magnetoconvection in a stellar envelope 285 i.e., vertical velocities, together with the horizontal components of both the viscous and magnetic stresses, vanish at the upper and lower surfaces. The temperature is held ¯xed on the upper surface, and its vertical derivative is imposed on the lower. These conditions require T = T1 at z = z1; (4.2) @T=@z = 1=K2 at z = z3; (4.3) We introduce a small-amplitude velocity perturbation over several wavenumbers within the unstable layer.
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