Simulating Solar Convection

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Bob Stein - MSUDavid Benson - MSUAake Nordlund - Copenhagen Univ.Mats Carlsson - Oslo Univ.

Simulated Emergent Intensity

METHOD• Solve conservation equations for:

mass, momentum, internal energy & induction equation

• LTE non-gray radiation transfer

• Realistic tabular EOS and opacities

No free parameters (except for resolution & diffusion model).

Conservation Equations

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∂ρ∂t

= −∇ • ρu

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∂ρui

∂t= −

∂

∂x j

ρuiu j + Pδij + ρυ∂ui

∂x j

+∂u j

∂x i

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥+ ρgi + J × B( )i

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∂ρe∂t

= −∇ • ρeu− P∇ • u+ ρν∂ui

∂x j

+∂u j

∂x i

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+ηJ 2 +Qrad

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∂B∂t

= −∇ × E, E = −u× B +ηJ, J = ∇ × B /μ0

Mass

Momentum

Energy

Magnetic Flux

Simulation Domain

48 Mm

48 M

m

20 M

m

500 x 500 x 500 -> 2000 x 2000 x 500

Variables

Spatial Derivatives

Spatial differencing– 6th-order finite difference, non-uniform mesh

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∂U∂x

⎛

⎝ ⎜

⎞

⎠ ⎟j−1/ 2

=

a U j( ) −U j −1( )[ ]

+b U j +1( ) −U j − 2( )[ ]

+c U j + 2( ) −U j − 3( )[ ]

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟J j( )

c = (-1.+(3.**5-3.)/(3.**3-3.))/(5.**5-5.-5.*(3.**5-3))b = (-1.-120.*c)/24., a = (1.-3.*b-5.*c)

Time Advance

Time advancement– 3rd order Runga-Kutta

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∂U∂t

=α i

∂U

∂t,

∂U

∂t=∂U

∂t+ f U( ),

U =U + β i

∂U

∂tdt

α = 0,−0.64,−1.3[ ], β = 0.46,0.92,0.39[ ]

For i=1,3 do

Radiation Heating/Cooling

• LTE• Non-gray, 4 bin multi-group• Formal Solution

Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface.

• Produces low entropy plasma whose buoyancy work drives convection

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Qrad = 4π κ λλ

∫ (Jλ − Sλ )dλ

Solve Feautrier equations along rays through each grid point at

the surfaced2Pdτλ

2 =Pλ −Bλ

Pλ =12

[I λ(Ω)+Iλ(−Ω)]

Actually solve for q = P - B

qλ =Pλ −Bλ

d2qλdτλ

2 =qλ −d2Bλdτλ

2

Simplifications• Only 5 rays• 4 Multi-group opacity bins• Assume L C

5 Rays Through Each Surface Grid Point

μ=cosθ=1,1/3, wμ =1/4, 3/4, ϕ rotates15oeachtimestep

Interpolate source function to rays at each height

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φ€

Θ

Opacity is binned, according to its magnitude, into 4 bins.

Line opacities are assumed proportional to the continuum opacity

Weight = number of wavelengths in bin

κ i =10i κ0, i=0(continuum),2,3,4(strongestlines)

wi = wλ jj(i )∑ , j(i) =wavelengthsλ j inbini

Bi = Bλ jj(i )∑ wλ j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ wi

Solve Transfer Equation for each bin i

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qi = Pi − Bi

d2qi

dτ i2

= qi −d2Bi

dτ i2

Finite Difference Equationqj−1

1τj −τ j−1

2τ j+1 −τj−1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−qj 1+1

τj −τj−1

+1

τ j+1 −τj

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2τj+1 −τ j−1

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

+qj+1

1τj+1 −τ j

2τj+1 −τ j−1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ =Sj−1

1τj −τj−1

2τj+1 −τ j−1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−Sj1

τj −τj−1

+1

τ j+1 −τj

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2τj+1 −τ j−1

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ +Sj+1

1τ j+1 −τj

2τ j+1 −τj−1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Ajqj−1 +Bjqj +C jqj+1 =Dj

• Problem: at small optical depth the 1 is lost re 1/2 in B

• Solution: store the value -1, (the sum of the elements in a

row) and calculate B = - (1+A+B)

Advantage• Wavelengths with same (z) are grouped together, so

integral over and sum over commute

κλ (J λ −Bλ )dλ∫ λ = κλ j

j (i)∑

i∑ (J λ j −Bλ j )wλ j

= κλ jj (i)∑

i∑ Lτλ j

(Bλ j )wλ j

(J λ −Bλ) =Lτλ (Bλ ) =dμμ0

1

∫ eτλ / μ dte−t /μ

0

∞

∫ Bλ (t)−Bλ

κλ jj (i)∑

i∑ Lτλ j (Bλ j )wλ j ≅ κ i

i∑ Lτi ( Bλ j

j (i)∑ wλ j )

≡ κ ii∑ Lτi (Bi )wi ≡ κ i

i∑ (J i −Bi)wi

Interpolate q=P-B from slanted grid back to Cartesian grid

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φ€

Θ

Radiative Heating/Cooling

Qrad =4πρ κ ii∑

Ω∑ qiwiwΩ

Energy Fluxes

ionization energy 3X larger energy than thermal

Equation of State

• Tabular EOS includes ionization, excitationH, He, H2, other abundant elements

Diffusion stabilizes scheme

• Spreads shocks

• Damps small scale wiggles

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ν =amax −∇ • u,0( )

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ν =b csound + cAlfven( ) + c urms

Boundary Conditions• Current: ghost zones loaded by extrapolation

– Density, top hydrostatic, bottom logarithmic

– Velocity, symmetric

– Energy (per unit mass), top = slowly evolving average

– Magnetic (Electric field), top -> potential, bottom -> fixed value in inflows, damped in outflows

• Future: ghost zones loaded from characteristics normal to boundary(Poinsot & Lele, JCP, 101, 104-129, 1992)modified for real gases

Observables

Gra

nula

tion

Emergent Intensity Distribution

Line Profiles

Line profile without velocities. Line profile with velocities.

simulation

observed

Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence.

Average profile is combination of lines of different shifts & widths.

average profile

Velocity spectrum, (kP(k))1/2

*

* ***

*

MDI doppler (Hathaway) TRACE

correlation tracking (Shine)

MDI correlation tracking (Shine)

3-D simulations (Stein & Nordlund)

Oscillation modes

Simulation MDI Observations

Local Helioseismologyuses wave travel times through the atmosphere

(by former grad. Student Dali Georgobiani)

Dark line is theoretical wave travel time.

P-Modes Excitedby PdV work

Triangles = simulation, Squares = observations (l=0-3)

Excitation decreases at lowfrequencies because oscillationmode inertia increases andcompressibility (dV) decreases.

Excitation decreases at highfrequencies because convectivepressure fluctuations have longperiods.

(by former grad. studentsDali Georgobiani & Regner Trampedach)

P-Mode Excitation

Solar Magneto-Convection

Mean AtmosphereTemperature, Density and Pressure

(105 dynes/cm2)

(10-7 gm/cm2)

(K)

Mean AtmosphereIonization of He, He I and He II

Inhomogeneous T (see only cool gas), & Pturb

Raise atmosphere One scale height

3D atmosphere not same as 1D atmosphere

Never See Hot Gas

Granule ~ Fountain

Granules:diverging warm

upflow at center,

converging cool, turbulent downflows at

edges

Red=diverging flowBlue =converging flowGreen=vorticity

Fluid Parcels

reaching the

surface Radiate away their

Energy and

Entropy

Z

SE

Qρ

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Magnetic Boundary Conditions

Magnetic structure depends on boundary conditions

• Bottom either:1) Inflows advect in horizontal field

or2) Magnetic field vertical

• Top: B tends toward potential

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B Swept to Cell Boundaries

Magnetic Field Lines - fed horizontally

Flux Emergence & Disappearance1 2

3 4

Emerging flux

Disappearing flux

Magnetic Flux Emergence

Magnetic field lines rise up through theatmosphere and open out to space

G-band image & magnetic

field contours

(-.3,1,2 kG)

G-band &

Magnetic Field

Contours: .5, 1, 1.5 kG (gray)20 G (red/green)

Magnetic Field & Velocity (@ surface)

Up Down

G-band Bright Points = large B, but some large B dark

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G-bandimages from simulation

at disk center & towards limb

(by Norwegian collaboratorMats Carlsson)

Notice:Hilly appearance of granulesBright points, where magnetic field is strongStriated bright walls of granules, when looking through magnetic fieldDark micropore, where especially large magnetic flux

Comparison with observationsSimulation, mu=0.6 Observation, mu=0.63

Height where tau=1

Magnetic concentrations:

cool, low ρlow opacity.

Towards limb,radiation

emerges from hot granule

walls behind.

On optical depth scale,

magneticconcentrations

are hot, contrast

increases with opacity

Magnetic Field &Velocity

High velocity sheets at

edges of flux concentration

The End