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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
€
∂ρ∂t
= −∇ • ρu
€
∂ρui
∂t= −
∂
∂x j
ρuiu j + Pδij + ρυ∂ui
∂x j
+∂u j
∂x i
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥+ ρgi + J × B( )i
€
∂ρe∂t
= −∇ • ρeu− P∇ • u+ ρν∂ui
∂x j
+∂u j
∂x i
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
+ηJ 2 +Qrad
€
∂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
€
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
€
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
€
ν =amax −∇ • u,0( )
€
€
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
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