CO 5 BOLD WORKSHOP 2006, 12 14 June First MHD simulations of the chromosphere Oskar Steiner Kiepenheuer-Institut für Sonnenphysik, Freiburg i.br., Germany
Collaborators The work presented in this talk is done in cooperation with: Sven Wedemeyer-Böhm, KIS Freiburg, Werner Schaffenberger, IGAM Graz, and Bernd Freytag, LANL Los Alamos.
1. MHD-simulation from the convection zone to the chromosphere At the Kiepenheuer-Institut, we carried out a three-dimensional MHD-simulation that encompasses the integral layers from the top of the convection zone to the mid-chromosphere. We use CO 5 BOLD, a finite volume code for solving the hydrodynamic equations in two or three spatial dimensions. It is based on Riemann solvers and higher order reconstruction schemes. For MHD we use a constrained transport scheme for the magnetic field and a 2nd-order accurate HLL Riemann solver scheme.
1.1. MHD equations The MHD-equations can be written in conservative form as: U t + F = S, where the vector of conserved variables U, the source term S due to gravity and radiation, and the flux tensor F are U = (ρ, ρv, B, E), S = (0, ρg, 0,ρg v + q rad ), F = 0 B @ ρv ρvv + p + B B I 8π BB 4π E + p + B B 8π vb Bv v 1 4π (v B) B 1 C A. The MHD-equations are (although not strictly) hyperbolic.
MHD equations (cont.) The total energy E is given by E = ρǫ + ρ v v 2 + B B 8π, where ǫ is the thermal energy per unit mass. The additional solenoidality constraint, B = 0, must also be fulfilled. The MHD equations must be closed by an equation of state which gives the gas pressure as a function of the density and the thermal energy per unit mass The radiative source term is given by p = p(ρ, ǫ). Z q rad = 4πρ κ ν (J ν B ν )dν, J(r) = 1 4π I I(r, n)dν.
1.2. Initial and boundary conditions The three-dimensional computational domain extends from 1400 km below the surface of optical depth unity to 1400 km above it and it has a horizontal dimension of 4800 x 4800 km. The domain is subdivided into 120 3 computational cells. B 0 2.8 Mm 4.8 Mm 1.4 Mm 1.4 Mm τ = 1 4.8 Mm The simulation starts with a homogeneous vertical magnetic field of a flux density of 10 G superposed on a previously computed, relaxed model of thermal convection. This flux density ought to mimic magneonvection in a network-cell interior.
Initial and boundary conditions (cont.) Boundary conditions: v x,y z = 0 ; v z = 0 ; ǫ z = 0 ; B x,y = 0 ; B z z = 0 z y x periodic periodic v x,y z = 0 ; ρv z dσ = 0 ; outflow: s z = 0 B inflow: s = const ; B x,y = 0 ; z z = 0
1.3. Results z = -1210 km z = 60 km z = 1300 km 4000 4000 y [km] 3000 2000 y [km] 3000 2000 1000 1000 1000 2000 3000 4000 x [km] 1000 2000 3000 4000 x [km] 1000 2000 3000 4000 x [km] 1000 2000 3000 4000 x [km] 0 1 2 log B (G) 0 1 2 log B (G) 0.2 0.4 0.6 0.8 1.0 1.2 log B (G) Horizontal sections through 3-D computational domain. Color coding displays log B with individual scaling for each panel. Left: Bottom layer at a depth of 1210 km. Middle: Layer 60 km above optical depth τ c = 1. Right: Top, chromospheric layer in a height of 1300 km. White arrows indicate the horizontal velocity on a common scaling. Longest arrows in the panels from left to right correspond to 4.5, 8.8, and 25.2 km/s, respectively. Rightmost: Emergent intensity.
Results (cont.) 3.0 z [km] 1000 500 0-500 -1000 2.5 2.0 1.5 1.0 0.5 log B (G) 1000 2000 3000 4000 x [km] 0.0 Snapshot of a vertical section showing log B (color coded) and velocity vectors projected on the vertical plane (white arrows). The b/w dashed curve shows optical depth unity and the dot-dashed and solid black contours β = 1 and 100, respectively. movie with β = 1 surface Schaffenberger, Wedemeyer-Böhm, Steiner & Freytag, 2005, in Chromospheric and Coronal Magnetic Fields, Innes, Lagg, Solanki, & Danesy (eds.), ESA Publication SP-596
Results (cont.) z [km] 1200 1000 800 600 400 200 400 600 800 1000 1200 1400 x [km] 40 30 20 10 B [G] z [km] 1300 1200 1100 1000 900 800 700 3400 3600 3800 4000 4200 4400 x [km] 35 30 25 20 15 10 5 B [G] Two instances of shock induced magnetic field compression. Absolute magnetic flux density (colors) with velocity field (arrows), Mach = 1-contour (dashed) and β = 1-contour (white solid).
Results (cont.) s v x log ρ p 2 1 v 3.2 0.8 3.8 10.4 8.6 10.7 2.5 v 2 v 1 ρ 2 B2 v 1 ρ B 1 1 B z 12.6 5.2 cs 6.0 3.9 v 2 2 1 > 1 + 2.4 7.0 c s c A 1 c A 4.0 2.4 β 1.31 2.14 v 2 2
Results (cont.) 3.0 z [km] 1000 500 0-500 2.5 2.0 1.5 1.0 log B (G) -1000 1000 2000 3000 4000 x [km] 0.5 0.0 Snapshot of a vertical section showing log B (color coded) and B projected on the vertical plane (white arrows). The b/w dashed curve shows optical depth unity and the dot-dashed and solid black contours β = 1 and 100, respectively. Schaffenberger, Wedemeyer-Böhm, Steiner & Freytag, 2005, in Chromospheric and Coronal Magnetic Fields, ESA Publication SP-596
Results (cont.) The formation of the small-scale canopy field proceeds by the action of the expanding flow above granule centers. This flow transports shells of horizontal magnetic field to the upper photosphere and lower chromosphere, where layers of different field directions may come close together, leading to a complicated meshwork of current sheets in a height range from approximately 400 to 900 km.
Results (cont.) Logarithmic current density, log j, in a vertical cross section (top panel) and in four horizontal cross sections in a depth of 1180 km below, and at heights of 90 km, 610 km, and 1310 km above the average height of optical depth unity from left to right, respectively. The arrows in the top panel indicate the magnetic field strength and direction.
2. Discussion and looking ahead 1st critical comment: Closed upper boundary The closed upper boundary may lead to forced horizontal flows and, together with the boundary condition for the magnetic field to the generation of perpendicular shocks. 3.0 z [km] 1000 500 0-500 -1000 2.5 2.0 1.5 1.0 0.5 log B (G) 1000 2000 3000 4000 x [km] 0.0
Discussion and looking ahead (cont.) 2nd critical comment: Homogeneous initial B z The condition of an initial homogeneous vertical magnetic field tends to force magnetic canopies τ = 1
Discussion and looking ahead (cont.) 2nd critical comment: Homogeneous initial B z The condition of an initial homogeneous vertical magnetic field tends to force magnetic canopies τ = 1 τ = 1
Discussion and looking ahead (cont.) 3rd critical comment: Vanishing horizontal magnetic field This condition does badly apply to the bottom boundary. It inhibits the transport of magnetic field and any Poynting flux across the boundaries. It impedes magnetic pumping taking place. Stein & Nordlund use theore th following boundary condition: In outflows the boundary conditions for the vector potential is 2 A x z 2 = 2 A y z 2 = A z z = 0 B x z = B y z = B2 z z 2 = 0 consistent with B = 0. In inflows they set A x = A z = 0; A y z = B x0.
Discussion and looking ahead (cont.) Combined volume renderings of enstrophy (purple-white) and of magnetic energy (blue-green-yellow), in which high values appear as opaque and bright. (a) Initial configuration with a layer of magnetic field inserted in the unstable convection zone. (b) A later time, showing concentrations of the magnetic field in the stable region. (c e) Volume renderings for a subvolume of the full domain centered around a coherent downflow. The strong plumes pump magnetic fields downward and amplify them by local dynamo action. From Tobias et al. (1998), ApJ, L177
Discussion and looking ahead (cont.) Horizontal average of magnetic energy (top) and B y (bottom) as functions of z and time. The initial and final states are shown as thicker lines. Both, magnetic energy and flux are transported downward with time into the stable region. From Tobias et al. (1998), ApJ, L177
Table of content 1. MHD-simulation from the convection zone to the chromosphere 1.1. MHD equations 1.2. Initial and boundary conditions 1.3. Results 2. Discussion and looking ahead 3. References
3. References Schaffenberger, W., Wedemeyer-Böhm, S., Steiner, O., and Freytag, B.: 2005, Magnetohydrodynamic simulation from the convection zone to the chromosphere, in Chromospheric and Coronal Magnetic Fields, ESA Publication, SP-596 Schaffenberger, W., Wedemeyer-Böhm, S., Steiner, O., and Freytag, B.: 2006, Holistic MHD-simulation from the convection zone to the chromosphere, in Solar MHD: Theory and Observations a High Spatial Resolution Perspective, J. Leibacher, H. Uitenbroek, & R.F. Stein (eds.), ASP Conference Series