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Magneto-Hydrodynamic Equations Mass conservation 𝜕𝜌/𝜕t = −∇ · (𝜌 u )PowerPoint Presentation

Magneto-Hydrodynamic Equations Mass conservation 𝜕𝜌/𝜕t = −∇ · (𝜌 u )

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Granules in the Quiet and Magnetic Sun

Robert F. Stein, Michigan State University

ValentynaAbramenko, Big Bear Solar Observatory

ÅkeNordlund, Niels Bohr Institute, University of Copenhagen

Simulation Results

Magneto-Hydrodynamic Equations

Mass conservation

𝜕𝜌/𝜕t = −∇ · (𝜌u)

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

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

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

High resolution simulations and observations reveal

that granule properties are very different in quiet Sun

and plage regions. In the quiet Sun granules have

scalloped edges with turbulent vertical velocity at their

edges. In plage granules have swirlingvertical vortex

tubes at their edges. Diverging upflowsapproach the

downflowintergranular 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.

Numerical Method

Spatial differencing

6th-order centered finite-differencestaggered

Time advancement

3rd order Runga-Kutta

Equation of state

Tabular including ionization H, He + abundant elements

Radiative transfer

3D, LTE 4 bin multi-group

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.

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.

- 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.

Magnetic Sun: vertical vortex tubes along intergranular lanes. Plasma turning over into intergranular lanes where there are strong magnetic field concentrations

wraps around the the magnetic field creating a vertical vortex tube.

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.

TiOband intensity image from New Solar Telescope (Big Bear Observatory)

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 HenkSpruit) because where the field is strong, the density is lower and radiation escapes from deeper layers where

the plasma gets heated by the deeper hotter walls of the ascending granules. Both the observations with NST and the degraded simulation intensity

show a very similar behavior in both the quiet and magnetic locations.

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