neto-Hydrodynamic Equations conservation / t = −∇ · ( u) ntum conservation u)/ t =−∇·( uu)−∇ − g+J×B−2Ω×u−∇· visc gy conservation t =−∇·( u)− (∇· u)+ rad + visc +J 2 ction equation t = −∇ × E, E=−u×B+J+ (1/en e ) (J×B−∇ e ), Numerical Method Spatial differencing 6th-order centered finite- difference staggered Time advancement 3rd order Runga-Kutta Equation of state Tabular including ionization H, He + abundant elements Radiative transfer 3D, LTE 4 bin multi-group Simulation Setup The computational domain is 2016x500x2016. It extends 48 Mm wide by20 Mm deep, which is 10% of the geometric depth but 2/3 of the scale heights of the convection zone. Vertical boundary conditions: Extrapolate lnρ; Velocity -> constant @ top, zero derivative @ bottom; energy/mass -> average value @ top, extrapolate @ bottom; B tends to potential field @ top, Inflows at bottom (20 Mm) advect in Weak (1 kG), minimally structured (horizontal, uniform, untwisted) magnetic field . Initial state – non-magnetic convection. Granules in the Quiet and Magnetic Sun Robert F. Stein, Michigan State University Valentyna Abramenko, Big Bear Solar Observatory Åke Nordlund, Niels Bohr Institute, University of Copenhagen Quiet Sun: (left) Vertical velocity image (light is down, gray and dark up) is turbulent at granule edges. (right) Fluid streamlines with volume rendering of magnetic field strength. Horizontal vortex tubes are common, vertical vortex tubes occur at some granule lane vertices. Plasma reaching the surface originates from the centers of underlying larger cells a depth. Rising plasma diverges and turn sover like a fountain and heads back down. Magnetic Sun: vertical vortex tubes along intergranular lanes. Plasma turning over into intergranular lanes where there are strong magnetic concentrations wraps around the the magnetic field creating a vertical vortex tube. Vertical velocity image at continuum optical depth 0.1 with magnetic field contours at 300 & 1000 G. Granule boundaries are corrugated in quiet Sun, but smoother with swirls at boundaries of magentic regions. TiO band intensity image from New Solar Telescope (Big Bear Observatory) Continuum intensity image from 12x12x6 Mm simulation, convolved with an 1.5 m airy psf. The scale is not exactly the same as in the observed snapshot. Granules in field free regions have scalloped edges, whereas in magnetic locations granule boundaries are smoother with swirly strings of points. These are bright (as pointed out by Henk Spruit) because where the field is strong, the density is lower and radiation escapes from deeper where the plasma gets heated by the deeper hotter walls of the ascending granules. Both the observations with NST and the degraded simulation i show a very similar behavior in both the quiet and magnetic locations. esolution simulations and observations reveal ranule properties are very different in quiet Sun age regions. In the quiet Sun granules have ped edges with turbulent vertical velocity at their In plage granules have swirlingvertical vortex at their edges. Diverging upflows approach the ow intergranular lanes, are deflected by the magnetic field and flow around the field creating al vortex tubes. The best solar observations tly clearly show the scalloped edges of granules quiet Sun intensity images. Small vortex tubes intergranular lanes at the edges of strong fields rderline visible currently. Simulation Results

Magneto-Hydrodynamic Equations Mass conservation /t = − ∇ · (u) Momentum conservation (u)/t =− ∇ ·(uu)− ∇ −g+J×B−2Ω×u− ∇ · visc Energy conservation /t

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Page 1: Magneto-Hydrodynamic Equations Mass conservation /t = − ∇ · (u) Momentum conservation (u)/t =− ∇ ·(uu)− ∇ −g+J×B−2Ω×u− ∇ · visc Energy conservation /t

Magneto-Hydrodynamic EquationsMass conservation𝜕𝜌/ t = − · (𝜕 ∇ 𝜌u)Momentum conservation

(𝜕 𝜌u)/ t =− ·(𝜕 ∇ 𝜌uu)− −∇𝑃 𝜌g+J×B−2𝜌Ω×u− ·∇ 𝜏visc

Energy conservation/ t =− ·(𝜕𝑒 𝜕 ∇ 𝑒u)− ( ·𝑃 ∇ u)+𝑄rad +𝑄visc +𝜂J2

Induction equation𝜕B/ t = − × 𝜕 ∇ E, E=−u×B+𝜂J+ (1/ene) (J×B−∇𝑃e),

Numerical MethodSpatial differencing6th-order centered finite-difference staggeredTime advancement3rd order Runga-KuttaEquation of stateTabular including ionization H, He + abundant elementsRadiative transfer3D, LTE 4 bin multi-group

Simulation SetupThe computational domain is 2016x500x2016.It extends 48 Mm wide by20 Mm deep, which is 10% of the geometric depth but 2/3 of the scale heights of the convection zone.Vertical boundary conditions: Extrapolate lnρ; Velocity -> constant @ top, zero derivative @ bottom; energy/mass -> average value @ top, extrapolate @ bottom;B tends to potential field @ top,Inflows at bottom (20 Mm) advect in

Weak (1 kG), minimally structured (horizontal, uniform, untwisted) magnetic field .

Initial state – non-magnetic convection.

Granules in the Quiet and Magnetic SunRobert F. Stein, Michigan State University

Valentyna Abramenko, Big Bear Solar ObservatoryÅke Nordlund, Niels Bohr Institute, University of Copenhagen

Quiet Sun: (left) Vertical velocity image (light is down, gray and dark up) is turbulent at granule edges. (right) Fluid streamlines with volume rendering of magnetic field strength. Horizontal vortex tubes are common, vertical vortex tubes occur at some granule lane vertices. Plasma reaching the surface originates from the centers of underlying larger cells a depth. Rising plasma diverges and turn sover like a fountain and heads back down.

Magnetic Sun: vertical vortex tubes along intergranular lanes. Plasma turning over into intergranular lanes where there are strong magnetic field concentrationswraps around the the magnetic field creating a vertical vortex tube.

Vertical velocity image at continuum optical depth 0.1with magnetic field contours at 300 & 1000 G.Granule boundaries are corrugated in quiet Sun, butsmoother with swirls at boundaries of magentic regions.

TiO band intensity image from New Solar Telescope (Big Bear Observatory) Continuum intensity image from 12x12x6 Mm simulation,convolved with an 1.5 m airy psf. The scale is not exactlythe same as in the observed snapshot.

Granules in field free regions have scalloped edges, whereas in magnetic locations granule boundaries are smoother with swirly strings of bright points.These are bright (as pointed out by Henk Spruit) because where the field is strong, the density is lower and radiation escapes from deeper layers wherethe plasma gets heated by the deeper hotter walls of the ascending granules. Both the observations with NST and the degraded simulation intensityshow a very similar behavior in both the quiet and magnetic locations.

High resolution simulations and observations reveal that granule properties are very different in quiet Sunand plage regions. In the quiet Sun granules have scalloped edges with turbulent vertical velocity at theiredges. In plage granules have swirlingvertical vortex tubes at their edges. Diverging upflows approach the downflow intergranular lanes, are deflected by the strong magnetic field and flow around the field creating vertical vortex tubes. The best solar observations currently clearly show the scalloped edges of granules in the quiet Sun intensity images. Small vortex tubes in the intergranular lanes at the edges of strong fields are borderline visible currently.

Simulation Results