simulating solar convection bob stein - msu david benson - msu aake nordlund - copenhagen univ. mats...
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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
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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).
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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
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Simulation Domain
48 Mm
48 M
m
20 M
m
500 x 500 x 500 -> 2000 x 2000 x 500
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Variables
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Spatial Derivatives
Spatial differencing– 6th-order finite difference, non-uniform mesh
€
∂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)
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Time Advance
Time advancement– 3rd order Runga-Kutta
€
∂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
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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
€
Qrad = 4π κ λλ
∫ (Jλ − Sλ )dλ
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Solve Feautrier equations along rays through each grid point at
the surfaced2Pdτλ
2 =Pλ −Bλ
Pλ =12
[I λ(Ω)+Iλ(−Ω)]
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Actually solve for q = P - B
qλ =Pλ −Bλ
d2qλdτλ
2 =qλ −d2Bλdτλ
2
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Simplifications• Only 5 rays• 4 Multi-group opacity bins• Assume L C
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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
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Opacity is binned, according to its magnitude, into 4 bins.
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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
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Solve Transfer Equation for each bin i
€
qi = Pi − Bi
d2qi
dτ i2
= qi −d2Bi
dτ i2
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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)
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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
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Interpolate q=P-B from slanted grid back to Cartesian grid
€
φ€
Θ
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Radiative Heating/Cooling
Qrad =4πρ κ ii∑
Ω∑ qiwiwΩ
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Energy Fluxes
ionization energy 3X larger energy than thermal
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Equation of State
• Tabular EOS includes ionization, excitationH, He, H2, other abundant elements
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Diffusion stabilizes scheme
• Spreads shocks
• Damps small scale wiggles
€
ν =amax −∇ • u,0( )
€
€
ν =b csound + cAlfven( ) + c urms
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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
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Observables
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Gra
nula
tion
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Emergent Intensity Distribution
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Line Profiles
Line profile without velocities. Line profile with velocities.
simulation
observed
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Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence.
Average profile is combination of lines of different shifts & widths.
average profile
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Velocity spectrum, (kP(k))1/2
*
* ***
*
MDI doppler (Hathaway) TRACE
correlation tracking (Shine)
MDI correlation tracking (Shine)
3-D simulations (Stein & Nordlund)
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Oscillation modes
Simulation MDI Observations
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Local Helioseismologyuses wave travel times through the atmosphere
(by former grad. Student Dali Georgobiani)
Dark line is theoretical wave travel time.
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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)
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P-Mode Excitation
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Solar Magneto-Convection
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Mean AtmosphereTemperature, Density and Pressure
(105 dynes/cm2)
(10-7 gm/cm2)
(K)
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Mean AtmosphereIonization of He, He I and He II
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Inhomogeneous T (see only cool gas), & Pturb
Raise atmosphere One scale height
3D atmosphere not same as 1D atmosphere
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Never See Hot Gas
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Granule ~ Fountain
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Granules:diverging warm
upflow at center,
converging cool, turbulent downflows at
edges
Red=diverging flowBlue =converging flowGreen=vorticity
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Fluid Parcels
reaching the
surface Radiate away their
Energy and
Entropy
Z
SE
Qρ
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QuickTime™ and aPhoto - JPEG decompressor
are needed to see this picture.
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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
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Magnetic Field Lines - fed horizontally
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Flux Emergence & Disappearance1 2
3 4
Emerging flux
Disappearing flux
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Magnetic Flux Emergence
Magnetic field lines rise up through theatmosphere and open out to space
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G-band image & magnetic
field contours
(-.3,1,2 kG)
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G-band &
Magnetic Field
Contours: .5, 1, 1.5 kG (gray)20 G (red/green)
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Magnetic Field & Velocity (@ surface)
Up Down
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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
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Comparison with observationsSimulation, mu=0.6 Observation, mu=0.63
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Height where tau=1
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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
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Magnetic Field &Velocity
High velocity sheets at
edges of flux concentration
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The End