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 †

1. Introduction One of the main problems in the physics of the Sun is the interaction between convection and magnetic fields. 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 field into concentrated flux sheets and tubes in which the magnetic pressure is comparable to the gas pressure (see Hurlburt & Toomre (1988)). On the other hand, magnetic fields can be sufficiently 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 thermal stratification. Studies of convection in the presence of magnetic fields have already shown effects arising from compressibility, revealing distinctive asymmetries between upward 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 field. We study penetrative convection within relatively simple configurations consisting of two layers with well-posed boundary conditions. We include all diffusive 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 presence of an imposed magnetic field. We assume that this envelope experiences uniform gravitational acceleration directed downward, and possesses constant thermal conductivity, a constant magnetic diffusivity, and a constant shear viscosity. The stratification consists of an unstable layer bounded below by a stable polytrope. This configuration 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

mα mα+1 z − zα ρ T P T T = Tα + , = , = (2.1) Kα ρα µTα ¶ Pα µTα ¶ for the unstable (α = 1) and stable (α = 2) layers. Here the polytropic indices mα are given by mα + 1 = gKα/RFT , where R is the gas constant, Kα is the coefficient of thermal conductivity, and FT is the total energy flux. Convective instability occurs when mα < mc, where mc, the critical polytropic index, depends on the ratio of specific heats † 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 magnetoconvection. 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, ∂ρ ∂ + (ρui) = 0, (2.2) ∂t ∂xi conservation of momentum,

∂ρui ∂ 1 + δij P + BkBk + ρuiuj − τij − BiBj − ρgδi3 = 0, (2.3) ∂t ∂xj · µ 2 ¶ ¸ conservation of energy,

∂ρE ∂ 1 P ∂T + ρuj uiui + ² + − gx3 − uiτji − K ∂t ∂xj · µ2 ρ ¶ ∂xj ¸ (2.4) ∂ ∂Bi ∂Bj − ηBi − + Bi (Bjui − Biuj) = 0, ∂xj · µ ∂xj ∂xi ¶ ¸ and the induction equation,

∂Bi ∂ ∂Bi ∂Bj + ujBi − uiBj − η − = 0. (2.5) ∂t ∂xj · µ ∂xj ∂xi ¶¸ These are augmented by the equation of state for a perfect gas,

P = RρT, (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 ρ and the viscous stress tensor,

∂ui ∂uj 2 ∂ul τij = µ + − δij . (2.8) µ∂xj ∂xi 3 ∂xl ¶ Here ρ, T , P , and ² are density, temperature, pressure, and specific internal energy, and Bi denotes components of the magnetic field. 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/ (Rdβ1) is related to the sound travel time across the unstable layer, where β1 is the initial temperature gradient within the unstable layer. The magnetic field is scaled by Bˆ, the value of the initially uniform vertical magnetic field. The evolution of the system depends on some physical and geometrical parameters. We define 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 χ2 is the ratio of the density at the bottom of the unstable layer (z = z2) to that at the top:

ρ (z2) χ2 = (2.9) ρ (z1)

The Prandtl number, which gives the ratio of viscous to thermal diﬀusivity, is deﬁned as µc 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 λ z , (2.11) where λ is the ratio of the sound speed to the thermal diﬀusion 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 Rˆ = R z1 + . (2.12) a a µ 2¶ as a nominal Rayleigh number.

The strength of the imposed magnetic ﬁeld and its eﬀect on convective stability can be measured by the Chandrasekhar number d2 Q = B2 (2.13) µη

The magnetic Prandtl number (the ratio of magnetic to thermal diﬀusivity) is deﬁned as

ζ = ηρcp/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 diﬀusion 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 deﬁned as 1 F +1 2 = F +1 + F + P +1 2Φ +1 2 . (3.4) i / 2 i i i / i / ¡ ¢ The last term Pi+1/2Φi+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 φi+1/2 of vector Φi+1/2 can be written as 1 φl = ψ(al )(gl + gl) − ψ(al + γl )αl , (3.5) i+1/2 2 i+1/2 i+1 i i+1/2 i+1/2 i+1/2 1 γl = ψ(al )(gl − gl)/αl if αl 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 αi+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 ψ is an entropy correction to | ai+1/2 |. l −1 The αi+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 ﬁxed 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. The main parameters are ﬁxed at the commonly used values of

γ = 5/3, σ = 1, ζ = 0.25, K = 0.05, Q = 144 (4.4)

5. Results

We study the effects of compressibility and magnetic fields on the penetration of convection into the region of stable stratification that lies below a stellar convection zone. Our results show that penetrative convection exhibits a distinct asymmetry between upflow and downflow: the downward flow is concentrated into strong localized plumes which deeply penetrate into the lower stable zones and involve large positive density fluctuations (Figure. 1). Compressibility leads to significant pressure fluctuations and results in enhanced buoyancy-driving in regions of downflow. The upward flows span significantly wider regions and are distinctly weaker than downflows but not negligible. Convection tends to sweep the initially uniform vertical magnetic field into concentrated magnetic flux tubes and sheets with significant magnetic pressure (see also Weiss et.al. (1996)). The motion in downdrafts in the stable region are considerably less vigor- ous in the regions of magnetic sheets (Figure 2). Buoyancy and the Lorentz force retard downward flow, and magnetic pressure produces partial evacuation of matter which en- hances buoyancy-braking in flux sheets (Figure 3). In the upper unstable region, variations of velocity are much weaker. Velocity magni- tudes indicate slowly modulated convection. The cellular flows display prominent narrow regions with downward flows surrounded by broader regions of upflow. The magnetic fields concentrate into flux tubes and sheets also, but the motion in downdrafts in this region is slower than in the stable layers. There are large variations of magnetic field in- tensity between regions with flux sheets and those with upward flow. In the penetrative region these variations are weaker and the magnetic field is concentrated mainly in flux tubes. (Figure 4).

6. Conclusions

In this report we have studied formation and development of penetrative convection in stellar envelopes in the presence of imposed magnetic fields. We have revealed a complex, time-varying flow field with compressible convection interacting with the magnetic field 286 S. D. Ustyugov & A. N. Andrianov

Figure 1. Velocity ﬁelds and levels of the vertical component of velocity (denoted by color).

Figure 2. Distribution levels of the vertical component of velocity. Magnetoconvection in a stellar envelope 287

Figure 3. Distribution levels of the vertical component of magnetic field. in an unstable layer bounded below by stably a stratified polytrope. On one side, convection sweeps the magnetic field into flux tubes and sheets, and on the other side magnetic pressure regulates the velocity of flow in downdrafts. In the future we will explore magnetoconvection for other initial parameters: aspect ratio, strength of magnetic field, Prandtl number, etc. We also plan to include a real-gas equation of state and opacity relations in our computer program.

7. Acknowledgments

This work has been supported by CTR. The authors thank Alexander G. Kosovichev from HEPL, Stanford University, Nagi N. Mansour and Alan Wray from NASA Ames Research Center, for help and very useful discussions.

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X Y

Figure 4. Iso-surface of the vertical component of magnetic ﬁeld.

Shu, S.- W. 1989 Eﬃcient implementation of essentially non-oscillatory shock-capturing schemes.II JCP 83, 32. Zhukov, V. T., Zabrodin, A. V., & Feodoritova, O. B. 1993 Comp. Maths. Math. Phys. 33, 1099. Andrianov, A. N., Efimkin, K. N., Levashov, V. Y. & Shishkova, I. N. 2001 Lecture Notes in Computer Science 2073, July. Ustyugov, S. D. & Andrianov, A. N. 2002 HYP 2002 (in press).