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%L 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% "Physics is the hard part$ eEquation of state - includes ionization and molecule formation Radiation - LTE, non-gray, multi-group0Equation of State# Radiative Transfer0  Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. 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% 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/:*!/:*,Q Z  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% "Physics is the hard part$ eEquation of state - includes ionization and molecule formation Radiation - LTE, non-gray, multi-group0Equation of State# Radiative Transfer0  Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling 6Entropy7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  8  Vorticity 9!Turbulent downdrafts1 Granulation20Spectrum of granulation agrees with observations3 Line Profiles4VConvection produces line shifts, changes in line widths. No microturbulence, damping.=   |*( @ |x | c $E8`  & r | S G80P & H | 0޽h ? ̙33"  b(  x  c $J8p~  & $  C Aentropy_snap_all.jpg 0002666EHDPBG4 ABA78158:`~  0J8 rGreen & blue are low entropy downflows, red is high entropy upflows Low entropy plasma rains down from the surface\s., /H  0޽h ? ̙33m  ( ,`T   S bI80e0eAvel_temp_slice.gif 0001F7E3HDPBG4 ABA78158:` &  ~  s *tI8  &   0F8 CVelocity arrows, temperature fluctuation image (red hot, blue cool)`D. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% "Physics is the hard part$ eEquation of state - includes ionization and molecule formation Radiation - LTE, non-gray, multi-group:" Energy Fluxes0Equation of State# Radiative Transfer0  Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling 7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  6Entropy8  Vorticity 9!Turbulent downdrafts1 Granulation20Spectrum of granulation agrees with observations3 Line Profiles4VConvection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% "Physics is the hard part$ eEquation of state - includes ionization and molecule formation Radiation - LTE, non-gray, multi-group:" Energy Fluxes0Equation of State# Radiative Transfer0  Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. 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%} 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/:*!/:*,Q Z  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% "Physics is the hard part$ eEquation of state - includes ionization and molecule formation Radiation - LTE, non-gray, multi-group:" Energy Fluxes0Equation of State# Radiative Transfer0  Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling 7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  6Entropy8  Vorticity 9!Turbulent downdrafts<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 min/= , ((  r  S Ի7  & $  C Agran_size_spectrum1.gif 00016CCCHDPBG4 ABA78158:`` $  C Aintensity_hist.jpg 00016CCCHDPBG4 ABA78158: 0  0t7 Simulated intensity spectrum and distribution agree with observations after smoothing with telescope+seeing point spread function. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% "Physics is the hard part$ eEquation of state - includes ionization and molecule formation Radiation - LTE, non-gray, multi-group:" Energy Fluxes0Equation of State# Radiative Transfer0  Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling 7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V96Entropy8  Vorticity 9!Turbulent downdrafts<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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% 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/:*!/:*,Q Z  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% "Physics is the hard part$ eEquation of state - includes ionization and molecule formation Radiation - LTE, non-gray, multi-group:" Energy Fluxes0Equation of State# Radiative Transfer0  Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling 7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V96Entropy8  Vorticity 9!Turbulent downdrafts<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=X:!Simulation Results: B Field linesE3Magnetic Field StrengthY;Field Distribution(F'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 min/8=W  LD(  x  c $d8`  & $  C A4fieldlines.30G.JPG 0002783CHDPBG4 ABA78158:H  0޽h ? ̙33 h ( > x  c $8  & $  C A5Bz_hist_obs-hagenaar.01.pict 0002783CHDPBG4 ABA78158:t z  ,8A6  47bz_hist.gif 0001F7E3HDPBG4 ABA78158:pp     08P  : simulation    0d8  8observed    0ą8 EBoth simulated and observed distributions are stretched exponentials. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% ""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% ""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surfaceLK =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. 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Sp      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLTNOPQRSUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}atial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% ""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . 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N [~9:( ;  >d@Microsoft Organization Chart 2.0 MSOrgChart.20@Microsoft Organization Chart 2.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0~0d>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0~0~0~0~0/ 0DTimes7$\<0~\<0~\G>Ibd4_yB$!(õ.'JֆiyB$ vVϼzEYyGyB$O0rᲙC39yB$t@>2^o!yB$ ĔR< ͒QZ|k'yB$; ^X G*yb$;Sm".X./yB$6EHNyB$D:';zF=NJEKDyB$k@$3Mwu)/v 9Hyb$ e.iw;Ϙ[KSyb$5^sb-2gyB$88- xqBq|_~yB$q UWmƠ ۄyB$ӸmeUra4[ǎyb$qnI=7Z"yb$!SJ[S*bA|yR$&$Z+7{Yyb$l] ^neIyb$߀ QΑkh߲Yyb$M5Vɨp՗d?Oq5yR$'~UIDiQuYyb$@[D@ЁT`AU{yR$Z"{YOi1szb yB$hEhMٕ yR$''C%f:zE yb$7~F]@>/_rx$yb$ 3qV !EXyXyB$n@v )-WyB$L-eD!\c,Q 8 yB$D&7P7Q=DZ?"yB$Eoo!_pZ3mImq|yB$^#5JN~O\ayB$ոaf6WS&y7\LOyB$৿ۏ[ fX s۫yB$upĔz"`Uk8pyB$& [OE-\+ch'gyB$X$Ѧ4qdFyB$vTT×W_cHZyB$_HG#VwZј+dEyB$Y*X}T&p]cp"yB$î}T>9+#^;\Dyb$legUJ |%PhVyb$=mć;ũyB$Z%8MW:DyB$iG$'cfZAyB$q1ǿidK\XbyR$ xH@3n`20yB$Q;\eyb$N,C(5ۃR)apyc $ f@fg4kdkd@w74`7ppp@  <4BdBd0~\< {\<? %t 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/:*!/:*,Q Z  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% ""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% ""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling 7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V96Entropy8  Vorticity 9!Turbulent downdrafts<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=X:!Simulation Results: B Field linesE3Magnetic Field StrengthY;Field Distribution(F'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 min/8=W  d\ (( wY (l ( C 7`    ( 07f : ` ( c $A7 ???@2 ?` ( c $A8 @??0 @  @` ( c $A9 A?? 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% g>""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V96Entropy8  Vorticity 9!Turbulent downdrafts<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=X:!Simulation Results: B Field linesE3Magnetic Field StrengthY;Field Distribution(F'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 min/8=W  H@ ,( @ p ,r , S 7    , 0dR8fP : ` , c $A; B?? \   B` , c $A= C??  C` , c $A< D?? X  D` , c $A> E??@ E  , <R8a  x Original Top Boundary Conditions<!    H , 0޽h ? # i( wY `  c $AA G??@ <O  G  <!R8 ]#Original Bottom Boundary Conditions$#B  s *f޽h ? d    ( wY  l   C $ R8P   H   0޽h ? 4R \(  $  C A: domain.jpg 0002783CHDPBG4 ABA78158:H  0޽h ? ̙33xWKoUwƉy^>oD^3"ЫJ'ۈωNϕ#>. 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% 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/:*!/:*,Q Z  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% g>""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% g>iA"Characteristic Boundary Conditions""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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%  9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/:*!/:*,Q Z  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% g>iA"Characteristic Boundary Conditions""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group? 1:" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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%  9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/:*!/:*>Q "5  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% g>iA"Characteristic Boundary Conditions""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% g>iA"Characteristic Boundary Conditions""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% g>qBiA"Characteristic Boundary Conditions""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% g>qBiA"Characteristic Boundary ConditionsrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Fluid reaching surface comes from small area below the surfaceQPIH  0޽h ? ̙33 KP k c ` ( @   c h2K80e0eA025Stein+Nordlund89-fig4L.gif 0001F7E3HDPBG4 ABA78158:`      03K8P$  :  ]  3 A2?!moovlmvhd/--YX9y>@atrak\tkhd/-/-9@$edtselst9mdia mdhd/-/-_fJhdlrmhlrm1v applS)Apple elementary video MPEG Media Handler[minf gmhdgmin@9hdlrdhlralisappl@:Apple Alias Data HandlerdinfdrefalisHDPBG4XBD\ xz_corks.mpgh_MPEGmMPGGW)f X< E | "0`vn 0 Dۺ0%,$0`0 0R00_0M30xBD0›P0x(03n60l|00 ,timefsttsfstscstszstco udta udta ,   0d3K8Vv <Downflows are fast. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA"Characteristic Boundary ConditionsrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &D Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W=  p4}(  4x 4 c $I8`   $ 4 C AD aw_refl.gif 00004796HDPBG4 ABA78158: < $ 4 C AE gw_refl.gif 00004796HDPBG4 ABA78158:@   4 <J8v fs G Acoustic Wave  4 <u8  6 F Gravity wave H 4 0޽h ? ̙33 hd  ;(  x  c $d7`   $  C A fluxes_z.jpg 0002666EHDPBG4 ABA78158:P  0$7V <  q#ionization energy 3X larger energy2$ dB  <Do p p dB  <Do   07 P  F than thermal H  0޽h ? ̙33r=:0s"=Ks~ H( ; J d0\yR]!%t9gbo6smGxc·qFkk:euz]_;__ :]xt95t py6s=8by5S<$}d }E:fm0/EBBY`d0HhNJl1U O4 τ9Shw-fL:1Jb><)K Jxȅu(¤Zg"p/_LBZ~ȣM1UMQv`hDIX/h.=ОG{v0^Yx\E,_h2ltϞ,2 Bet{U;Odf{11-"6HfqiYŽ)hwHYv2rIG3gG҄|[Ÿ;JrS\x2{ZjiY&Yb8l̽ ^wcg#=.|DŽqWj"RfUOW^wo?__t`cqQ)-p{~YOK}4y4s_܏_lޫ~ Bдoy8~lrpKi;^?a$#;]#'_ )CHkgZ:BwPJ8`{4]$ ^klzur/ ^!`$8@p7AJR'pc{n Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0~0d>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0~0~0~0~0d?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0/ 0DTimes7D\<0~\<0~\~L+k}0,MhyB$t@>2^o!yB$ ĔR< ͒QZ|k'yB$; ^X G*yb$;Sm".X./yB$6EHNyB$D:';zF=NJEKDyB$k@$3Mwu)/v 9Hyb$ e.iw;Ϙ[KSyb$5^sb-2gyB$88- xqBq|_~yB$q UWmƠ ۄyB$ӸmeUra4[ǎyb$qnI=7Z"yb$!SJ[S*bA|yR$&$Z+7{Yyb$l] ^neIyb$߀ QΑkh߲Yyb$M5Vɨp՗d?Oq5yR$'~UIDiQuYyb$@[D@ЁT`AU{yR$Z"{YOi1szb yB$hEhMٕ yR$''C%f:zE yb$7~F]@>/_rx$yb$ 3qV !EXyXyB$n@v )-WyB$L-eD!\c,Q 8 yB$D&7P7Q=DZ?"yB$Eoo!_pZ3mImq|yB$^#5JN~O\ayB$ոaf6WS&y7\LOyB$৿ۏ[ fX s۫yB$upĔz"`Uk8pyB$& [OE-\+ch'gyB$X$Ѧ4qdFyB$vTT×W_cHZyB$_HG#VwZј+dEyB$Y*X}T&p]cp"yB$î}T>9+#^;\Dyb$legUJ |%PhVyb$=mć;ũyB$Z%8MW:DyB$iG$'cfZAyB$q1ǿidK\XbyR$ xH@3n`20yB$Q;\eyb$N,C(5ۃR)apyB$j=5BXsɦOyB$hbD0uQ9M#yB$?9#$>&G(yR$k~xǸ2!1xD,yB$8(L;O3LzyB$("c˕;HQGtyB$+Hȯ2pΥ vyB$jV PJv#1yb$̒ 6ԜR][QHyB$W6aSm!Z1>ʘٗyB$[mqGZϳ0yB$ҏ .ɞh7S8yB$xZi\N yb$v̋e2a yb$6\ޯ;v% y$yB$#  Ոyc $ f@fg4kdkd@w77{\<z\<ppp@  <4BdBd0~\< {\<? %! 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/:*!/:*>Q "5  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA"Characteristic Boundary ConditionsrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations H  rCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$      - !"#$%&'()*+,s./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrytuvwz{|}~Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W&  Df(  Dl D C TD8`   R D s *f` D c $AK  ??P  H D 0޽h ?   >6L( wY Ll L C tD8    R L s *f` L c $AJ ??8 `  ` L c $AL ??3 H L 0޽h ?   NF T(  Tl T C TD8`    T 0D8f : ` T c $AM ??0P ` T c $AI ??   T <ĮC8FFmf w=Characteristic equation for incoming waves at bottom & top is>=  T <vE8 P&  i/So boundary condition for no reflected waves is0/H T 0޽h ? 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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%& 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Staggering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=Wm  @( ,`T   S b 70e0eAvel_temp_slice.gif 0001F7E3HDPBG4 ABA78158:`   ~  s *7     0$7 CVelocity arrows, temperature fluctuation image (red hot, blue cool)`D.f H  0޽h ? ̙33xX]lW3ıi㶴]Pfp}H ޝMgD x'7c Z ! 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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 'd@Microsoft Organization Chart 2.0 MSOrgChart.20@Microsoft Organization Chart 2.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0~0d>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0~0~0~0~0d?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0d zEquation Equation.30,Microsoft Equation 3.0d}Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d#Equation Equation.30,Microsoft Equation 3.0d$Equation Equation.30,Microsoft Equation 3.0/ 0DTimes8dq\<0~\<0~\9 "/0123456689:&%'().=#";"  HIJKM N O PQoB$Ȣ&FvrIaϚyB${571PhwYyB$o P=Xv&h3# yB$qPQ%8\Ku)yB$3@jyB$ vVϼzEYyGyB$>~L+k}0,MhyB$t@>2^o!yB$ ĔR< ͒QZ|k'yB$; 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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%& 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,=Opacity is binned, according to its magnitude, into 4 bins.  --fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. 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%& 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$(Solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)3 28*:Interpolate q=P-B from slanted grid back to Cartesian grid+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin6g@$% . Advantage  [Wavelengths with same t(z) are grouped together, so integral over t and sum over l commute\/Radiative Heating/Cooling `=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W   <:( > <x < c $T7    <0 6A ?}pt  jB < s *f޽h ? ̙33  rjT(  Tr T S 7   $ T C Aopac_lambda.gif 00016CCCHDPBG4 ABA78158: $ T C A opac_bins.gif 00016CCCHDPBG4 ABA78158: H T 0޽h ? ̙33xWMYv x?803^Q"rCBogX1S$lr㆐l q,&C8ps{6-PD~}^^|m׬ĮmͲLJI݌)mkk+o? h6])5SݵqwƏ3؉;ֆ|y $ "Gz˿~(Jv{L&q?b5qr*Hsi|ge/SgcO:[O>0<_Kf:~7нLyF5`>ЃDn 0砽_~ zA=z$)P T-5@@i*~52Rٝ!%=$T\7ھy0n̏2h^s"ǏP[em'Gˇ[bɾy<s4ps!I楋<\ Y ϒs)~Zawɜ>@BH.br ]?^ lymKeq>/]nJ짤~%\H .R#D2#]sd-J}lfy *lfᮒSίg!fJ*$mFj8ޜD'>82(|R49d (jS JdL<:sqdRڲ*JH*!UjxI>@%ށ+A1:)U ɚZkzw~ٞ5V`v'رcc1%MC5k8qռ9LBu}0>{o(蘄W߉`_>?>)f-$S~9f'XYq\f tҞ+Y?F)|/ۿ;tG@f' ' `3rGXӏB>yD!B>$SOCg |y _T *4d%*J|ŽJat^ K7/NE>\`)%М2o1 ?#L12@sϲ~FT3uJ ry#2Nz&\&H/KuS KG$t+ |% fK8wPB$Ѝxmbq2U*Ͳ[GD[2154/ti(g? .Ot/WZ\y[yV`[7;܈|V_/;t췙1> v(MN@@Q/պiv.ex=ϐIn{zEUaeyLr+Gq$Yo?M*U[(eدvjKJ&4 &bSԇ=1V\m:~]*Yc]®0} @,1ZP,(lگd1dkPl(VйhvF2\J$h~j޵XdKa,jmnш==O6A-E$kՎt`rEEm} *^5\ ߟo>_͆u{%Pj`w A BZ#)٠Y(H#-=*ڈ(0bw1h جtJ\JjBcH#b0(,xZr%F8uay:$i ]m1=GpiX1,]jaXG VT8Û~: 'r<릓-P8 ` $n) Mqi)`ڨ wjY T TahZS#` i  ^JAsܔ Ӝ$v殓),!FD9/}=?7 /TXKEl;|ۗ&,=CJU9cED M[9| <˲ ~ta禅H%8C;iIFm]tҼ/6Ly}AtGtk(z@*FfJ woT?+^7iݜ.K9ZŢ3kȖ@CjnyCtYRLK,N5\o\q%Q-wb|-cGOu(%m7P"Rdǰq7nڑ0bۑ]B'ZR7e"1ܥqQX4:jӭ :u|?q雌׿HMq x8Vx9+;w;<%pLyby-Vv؆MI9xXd!i$ lD4Jyki*CDp-)o]Hjy9C ?@I˖SY2^ԔE%e](Tae˓2yGT㱛M^T?_ճ6dU&x~ T)C/CnPA#dwHxMoTp7ߩ&))m)}/[UR i­rvbkomo6TBJ8 'nDПPoH¼ TT]ϼy~o:w]p 3 C]<"BJod읗B#iw|8m pm81->~8dbg~?#Ʋwʇ@ = G hB XOw ˩_L>+B?ëib=;v=[dtxJ~AN=m;uh+hAt0;󠢤XwJGNɣ`yu5ibPI7h$k ,+tITF~D7]t N7Viqm̥:uQdV()J2UtZ5٪R)k\)szK(>S^IoQU#4MbhS)\ JMF`dQyiXT:l(:pˁGU< 5B*TjZU=5Vk_+z3XHbbyՐT"HLk \G9)  *g=ARU55+zx:oRIͶ{ħ׏JRU;Ait]a4-iue;9V+5ɆUU#rmy Jd)}oЎp[c-}G/!',ܴ x *'7 aXe"|=I''V\8$B" w瞦-ܝ\`BaEשNr'e{-y>. h 1?;Z_*u'P#' /%Vr,f ` w&, fc~rQ( ;  quation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,MicrosoAft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0pd>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0ppppd?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.3osoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0d zEquation Equation.30,Microsoft Equation 3.0d}Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d#Equation Equation.30,Microsoft Equation 3.0d$Equation Equation.30,Microsoft Equation 3.0/ 0DTimes7t\2^o!B$ ĔR< ͒QZ|k'$b$;Sm".X./B$6EHN$$b$ e.iw;Ϙ[KSb$5^sb-2g$$$b$qnI=7Z"b$!SJ[S*bA|R$&$Z+7{Yb$l] ^neIb$߀ QΑkh߲Yb$M5Vɨp՗d?Oq5R$'~UIDiQuYb$@[D@ЁT`AU{R$Z"{YOi1szb B$hEhMٕ R$''C%f:zE b$7~F]@>/_rx$b$ 3qV !EXyXB$n@v )-WB$L-eD!\c,Q 8 B$D&7P7Q=DZ?"B$Eoo!_pZ3mImq|B$^#5JN~O\aB$ոaf6WS&y7\LOB$৿ۏ[ fX s۫B$upĔz"`Uk8pB$& [OE-\+ch'gB$X$Ѧ4qdFB$vTT×W_cHZB$_HG#VwZј+dEB$Y*X}T&p]cp"B$î}T>9+#^;\Db$legUJ |%PhVb$=mć;ũB$Z%8MW:DB$iG$'cfZAB$q1ǿidK\XbR$ xH@3n`20B$Q;\eb$N,C(5ۃR)apB$j=5BXsɦOB$hbD0uQ9M#B$?9#$>&G(R$k~xǸ2!1xD,B$8(L;O3LzB$("c˕;HQGtB$+Hȯ2pΥ vB$jV PJv#1b$̒ 6ԜR][QHB$W6aSm!Z1>ʘٗB$;^ίz%4IRY}B$ҏ .ɞh7S8B$@I`8D'`clb$v̋e2a b$6\ޯ;v% B$pܧ%"dҲ#DB$#  ՈB$58/O)bSw bB$ #=yE0)B$tHfw)GI+Lo B$%뗤~OxAU &drB$eTڬc0LLb}6B$ћK-lDoo?B$X9`R!HB$ fLNT~Y?-$LB$ ,Ծ6OB$s^;/Tt1 |$TB$.8g 27XB$qެ{%_Y{tt[B$Y.>ކ+9$^B$Ƽ<&D @f5 #bB$8)| bW"]} Cec $ f@fg4kdkd"x7( \< \<ppp@  <4BdBdp\<`\<0 ? %! 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E`=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=Wr5-rQ( ;  d@Microsoft Organization Chart 2.0 MSOrgChart.20@Microsoft Organization Chart 2.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0pd>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0ppppd?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0d zEquation Equation.30,Microsoft Equation 3.0d}Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d#Equation Equation.30,Microsoft Equation 3.0d$Equation Equation.30,Microsoft Equation 3.0/ 0DTimes7t\2^o!B$ ĔR< ͒QZ|k'$b$;Sm".X./B$6EHN$$b$ e.iw;Ϙ[KSb$5^sb-2g$$$b$qnI=7Z"b$!SJ[S*bA|R$&$Z+7{Yb$l] ^neIb$߀ QΑkh߲Yb$M5Vɨp՗d?Oq5R$'~UIDiQuYb$@[D@ЁT`AU{R$Z"{YOi1szb B$hEhMٕ R$''C%f:zE b$7~F]@>/_rx$b$ 3qV !EXyXB$n@v )-WB$L-eD!\c,Q 8 B$D&7P7Q=DZ?"B$Eoo!_pZ3mImq|B$^#5JN~O\aB$ոaf6WS&y7\LOB$৿ۏ[ fX s۫B$upĔz"`Uk8pB$& [OE-\+ch'gB$X$Ѧ4qdFB$vTT×W_cHZB$_HG#VwZј+dEB$Y*X}T&p]cp"B$î}T>9+#^;\Db$legUJ |%PhVb$=mć;ũB$Z%8MW:DB$iG$'cfZAB$q1ǿidK\XbR$ xH@3n`20B$Q;\eb$N,C(5ۃR)apB$j=5BXsɦOB$hbD0uQ9M#B$?9#$>&G(R$k~xǸ2!1xD,B$8(L;O3LzB$("c˕;HQGtB$+Hȯ2pΥ vB$jV PJv#1b$̒ 6ԜR][QHB$W6aSm!Z1>ʘٗB$;^ίz%4IRY}B$ҏ .ɞh7S8B$@I`8D'`clb$v̋e2a b$6\ޯ;v% B$pܧ%"dҲ#DB$#  ՈB$58/O)bSw bB$ #=yE0)B$tHfw)GI+Lo B$%뗤~OxAU &drB$eTڬc0LLb}6B$ћK-lDoo?B$X9`R!HB$ fLNT~Y?-$LB$ ,Ծ6OB$s^;/Tt1 |$TB$.8g 27XB$qެ{%_Y{tt[B$Y.>ކ+9$^B$Ƽ<&D @f5 #bB$8)| bW"]} Cec $ f@fg4kdkd"x7( \< \<ppp@  <4BdBdp\<`\<0 ? %! 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E`=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=Wri-E-~R( ; * d@Microsoft Organization Chart 2.0 MSOrgChart.2n<} ʯg :p[x3hUE:n;L8'8_9^Sa#V>as&S6M(֙XlV4%JL?qqM>JN,°kBh$9 d߷fAky_Ȕ< }|Ye|x?f%-8A${aXf<φ˒yAr'1J̌";NR|4eA|tC>N2!V"S=/摆Ed AWhΊq%/"k4,q/;ˠUz^op/A4xʖu[ⓥV -M$"WZE,pZ[nBV"Tj99glЬv Mkb 'qÄwAUh3k[:ޱeLLlu[i4N^ׂ}$1ZE Qӻ|t5{FJTt,*]ku8OKkUXoW .N@yQJlBxaZHia+y{v [`mdK]f.o 5*>p\.0`5vZL|voL~sj[u%0kO|cL&'|o?>5W[UKL]ؒn6+LNh߾ڗI2-07:-J,%h{U9PKzHUh_UlY$]T iEqѱy`)++t TS]n=QwxRʐ9Il4 !xUIoVGq,vl M:Pt\؄(R nNvm9;y$iBA77D2!`/_.'{~bm(W2~bJK"-Kd,˂QRwvxoAR`$/ K ,2 =^:uX,1;xk&LQef(\d^ D <_mXLC[vK#[\-JUX2XfbtOUy2sF.#zNir6.??Џ(k<1KcLLaZ6iQms\ze<20@Microsoft Organization Chart 2.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0pd>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0ppppd?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0d zEquation Equation.30,Microsoft Equation 3.0d}Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d#Equation Equation.30,Microsoft Equation 3.0d$Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0/ 0DTimes7t\2^o!B$ ĔR< ͒QZ|k'$b$;Sm".X./B$6EHNB$6H>&ՇŻAv$b$ e.iw;Ϙ[KSb$5^sb-2g$$$b$qnI=7Z"b$!SJ[S*bA|R$&$Z+7{Yb$l] ^neIb$߀ QΑkh߲Yb$M5Vɨp՗d?Oq5R$'~UIDiQuYb$@[D@ЁT`AU{R$Z"{YOi1szb B$hEhMٕ R$''C%f:zE b$7~F]@>/_rx$b$ 3qV !EXyXB$n@v )-WB$L-eD!\c,Q 8 B$D&7P7Q=DZ?"B$Eoo!_pZ3mImq|B$^#5JN~O\aB$ոaf6WS&y7\LOB$৿ۏ[ fX s۫B$upĔz"`Uk8pB$& [OE-\+ch'gB$X$Ѧ4qdFB$vTT×W_cHZB$_HG#VwZј+dEB$Y*X}T&p]cp"B$î}T>9+#^;\Db$legUJ |%PhVb$=mć;ũB$Z%8MW:DB$iG$'cfZAB$q1ǿidK\XbR$ xH@3n`20B$Q;\eb$N,C(5ۃR)apB$j=5BXsɦOB$hbD0uQ9M#B$?9#$>&G(R$k~xǸ2!1xD,B$8(L;O3LzB$("c˕;HQGtB$+Hȯ2pΥ vB$jV PJv#1b$̒ 6ԜR][QHB$W6aSm!Z1>ʘٗB$;^ίz%4IRY}B$ҏ .ɞh7S8B$@I`8D'`clb$v̋e2a b$6\ޯ;v% B$pܧ%"dҲ#DB$#  ՈB$58/O)bSw bB$ #=yE0)B$tHfw)GI+Lo B$%뗤~OxAU &drB$eTڬc0LLb}6B$ћK-lDoo?B$X9`R!HB$ fLNT~Y?-$LB$ ,Ծ6OB$s^;/Tt1 |$TB$.8g 27XB$qެ{%_Y{tt[B$Y.>ކ+9$^B$Ƽ<&D @f5 #bB$8)| bW"]} Cec $ f@fg4kdkd"x7( \< \<ppp@  <4BdBdp\<`\<0 ? %! 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E`=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W/  4u(  4r 4 S 7  g  40 6A ?` Y   h 4 <7O- qRadiative Heating/Cooling"   4 <7   FJ is average over angles of integrals along rays through entire domainGF` 4 c $A ??E  B 4 s *f޽h ? ̙33MxWoD]ҒRXZ vCr"A?D#&˝\ISJ(" H7_ SgHYIU*:yfgg~3e{Nv$G6HO 4H$ͭT0,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Eq^( O0~?="Ww+&?%绌?F־9?=S.&q9~H+]!?CB2OZc|Eg h]_!mFDQ~~SI خ`Ad` y$5= *׀| 7['O #u G|&g& ݐv}\z;B/fkx ^.t`|h̅+y+CmX}#\u|?D9rMu}|A:&aUx2ĭb.yp"9r(/d`T*҉9"}9fP`w `aF/@'` &h#Oz߫.RWsT1JGQ r'F N(8Be W}1sndӦiszS)*ïSE=HUfSjSQ'x2 jh^:-< tFy6@$ zrL=F﫣zaTT[r4\YkM3~:f1fE,pg@rՀKjT:l,WGZkAԬ.s,۲\g l5[,r7Mƺ`\0K|!3L|q.HQ\٪TNryԨTlEM7󃆻Fm`!Ma-[Pk1ZSN`i3ywl~8U^F cWtH)+ȵZ~ Z'UtuPA 8(OJEl2CeK*`Ye0Yѱg0Pu,ѫ|-t*3IDG^DžkV2\* YpvfvƪjJ6,IahZ L<1nt|+SSIxɑ-ޞJkMx9x:^ x> ۍjW#imQ R.эvS#˨ Q;*)vs?$F¨_Z-u_=v&2_)#|ţsd,|]Үt{b71UT]^?oz ,Ըr$1h ~R( ; * d@Microsoft Organization Chart 2.0 MSOrgChart.20@Microsoft Organization Chart 2.0dEquation Equation.30,Microsoft Equation 3.0d8,27cPH)iPE M08JȧPͩPJN6$4yPILH E(x- 8HNH1Jಁ n$奓-䬠ٹUM03}&QxE]U&$H0Ibf wݷ֙~3BYC]H/tVʲ h+bcI`K18xVzaMHrfpNU?A{Rqaˊ"w&1eeO#k ā=hVu2rҲgn:Z5 pM:v2?d[tC[~˝M ù8fo_ʄW4ƐJ542A*:N=0lA+MKkI.+ei J-:m嘚A*ߌ ozCĎ8>=l'J5(CV*;aT]-]+Vvckح3^w˧g2ٶ5t3,blam; r(z5=CPu8@wf`I1l@È$} EzV K5Tz.ys8Knk8#+6nGsQA0~_ Tp5{lϝ+{4 YxUnVK/ٌcIP ."+"/"^hօ P|؄(R&lj!(v/t K;RM+Cf̙s\P,oW_F9&{~$Ю!6>$JWx1OrW(yi9TnǢe% Z KFXKB'Kd_~,S6V.V["Q &F8Cj&bZH=2)9 .#V81I]q8dOiwWZ Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0pd>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0ppppd?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0d zEquation Equation.30,Microsoft Equation 3.0d}Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d#Equation Equation.30,Microsoft Equation 3.0d$Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0/ 0DTimes7t\2^o!B$ ĔR< ͒QZ|k'B$]n=QwxRʐ9[Ub$;Sm".X./B$6EHNB$6H>&ՇŻAv$b$ e.iw;Ϙ[KSb$5^sb-2g$$$b$qnI=7Z"b$!SJ[S*bA|R$&$Z+7{Yb$l] ^neIb$߀ QΑkh߲Yb$M5Vɨp՗d?Oq5R$'~UIDiQuYb$@[D@ЁT`AU{R$Z"{YOi1szb B$hEhMٕ R$''C%f:zE b$7~F]@>/_rx$b$ 3qV !EXyXB$n@v )-WB$L-eD!\c,Q 8 B$D&7P7Q=DZ?"B$Eoo!_pZ3mImq|B$^#5JN~O\aB$ոaf6WS&y7\LOB$৿ۏ[ fX s۫B$upĔz"`Uk8pB$& [OE-\+ch'gB$X$Ѧ4qdFB$vTT×W_cHZB$_HG#VwZј+dEB$Y*X}T&p]cp"B$î}T>9+#^;\Db$legUJ |%PhVb$=mć;ũB$Z%8MW:DB$iG$'cfZAB$q1ǿidK\XbR$ xH@3n`20B$Q;\eb$N,C(5ۃR)apB$j=5BXsɦOB$hbD0uQ9M#B$?9#$>&G(R$k~xǸ2!1xD,B$8(L;O3LzB$("c˕;HQGtB$+Hȯ2pΥ vB$jV PJv#1b$̒ 6ԜR][QHB$W6aSm!Z1>ʘٗB$;^ίz%4IRY}B$ҏ .ɞh7S8B$@I`8D'`clb$v̋e2a b$6\ޯ;v% B$pܧ%"dҲ#DB$#  ՈB$58/O)bSw bB$ #=yE0)B$tHfw)GI+Lo B$%뗤~OxAU &drB$eTڬc0LLb}6B$ћK-lDoo?B$X9`R!HB$ fLNT~Y?-$LB$ ,Ծ6OB$s^;/Tt1 |$TB$.8g 27XB$qެ{%_Y{tt[B$Y.>ކ+9$^B$Ƽ<&D @f5 #bB$8)| bW"]} Cec $ f@fg4kdkd"x7 g( \< \<ppp@  <4BdBdp\<`\<0 ? %! 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.,^Opacity is rapidly varying function of wavelength. 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No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W   <:( > <x < c $О7  g  <0 6A ?}pt  B < s *f޽h ? ̙33xWoUwv0@[+~J݇ -I>L;tfJbB%&<ß/$! ƿ߽Z%ٜ9s~~rmɑ!ţ 8B){s^*80> p(o1ËreH?/_{suYLzn"ў|v GI y??yNӲKIITgj~y獤o;.ު@xxW&9< <| x:  <| vǻ}XL}xq@sE$|s[>ioGlڲ9ǴhK3|҅q9sCs.IDԳ) `HixN1wɷRc|_ &A`yQwHK~9rKEl+El!%;O/qrt6E 'h$Ӓ0vl)bȴR?2'Ip3ؙ@Q2N'Ra'\<[c!:= eBUbL 96_ԡ98O뢲}sb1ղlWXiFlZp3),]̳bȚ^5лMӵzMF酮Y[\fUxP`kag]wZ`%2~S+0Ef%Ŋ|hNUAQiFlMf][ejqY[Bi5΍DZֳr`t8m'.^t f7b#mϿ1O (Κ+!+~Rٌno?d:~GlgAq}YiXSD%roVnh[$o M-+l͎UFh'^֚]t' 1W<,YTdNي\lZcCR.^,adĢVYݨJj*Y,WT%!>Ve^˴Rp_2=_TYkhh,۪<)/UV.썥ye*kZJ* ŒҚh Wg^d ^_/FV4J}ŠP67c6A@3%V|i˶&*u@W* [ZvwbuPJ)*uu+(M.^-vhŝZYHۓRnXU*eVSvc|~խ'èVsznlrPmXrlV}ow">hC#<42y`J_`ܺ;L*RrxS")JwJKPUijc#fEH!3إSz {\I] 6&/CғY&&AjeWOnw {oYCo xNxW_oD]ҒRXZ vCrwE?4yT]b_N*TBi/<$>x+|#7 3s4PE4>ofvo~?zVQ ^ E0xo_ҿGΝ7`Ť<~d|g_Ֆ׾9OϔˤIg_ gtϐ̓&9~96>ڧeDWHotTC|z/}{G0kX>|G"XB^!IM |c#5ac~M` o."pQDŽ%=cB7${EЋٹZ35³>>*7!>::sJ$ca`qW'? {Ajl]%_Pι .~Xp#AFr}vA,%?HbpK2t"GȤwt_6Y4(꧆ P G=$/)uF!/t*:aLmQTB&EIASt tujz_̜ٴi lp  TQRE L <7Dx1NKjn;]Q@ Ъ= \6Fo(|X8Ֆ\/ c~=ftlLgd4 l+d\5`[w>Qt=Z5k9,Y>4Ą[aܸ *a跸0++-6.q _m R9W*Ө\u5*k1[1x f.b@[?FvwuSXV6Zn{֔%·A|o8lLፀa P՗"|z*"VؒF ;3PQV{i?"/QETʘ ̹ۇgxNչF䧗Ca5rVsBVI]cRFQ.da҆ *b|vf:L@tY2L-TKj8_ 8]ʌrR~#E0qᚕ (vJ@bYzZeg 3KRVqOi GT{8r^jr@'r)ZitަivZ/HZ[@f}ei}dKtԈn-2*HwJ{р0VK](bp݆ɫ̗|}c%|Y Fbwpdr]MLx>l״ϛ^]r 0&.5 ><R( ; * d@Microsoft Organization Chart 2.0 MSOrgChart.20@Microsoft Organization Chart 2.0dE /f^(1G&Y!?Pjpt/O#Tp`Xd BDf5=Ŕb\<g+֕Qz.yFUU}>ݸ{(Y/LvWb&, G#O@R[!3WrxguxM$vk;0€tb@圬4 %`3M6e2P QD׽*2ś$0%0$ ;Ұ :g~З"?~z~"5&%vGZ6`W8 H0ὀaL:Ƭ髕W&w0&D7T #2cBϸ MͲkDKGvkysWmӳ;?nzpƅ5(Ķ+ԌԳ'%>?[#ÿTzO *YEBec. !5帣*'JNc~+ <JAI6>"P g Y[oS6(yTAH}*_Fy4̛ YXHX_L-Qaq1]P<      !"#$%&'()*+,-./0123456789:;<=>?@CBDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|~ƼۘӚOɷ T=kn0=1k9 PNG  IHDRL0ASJsBITO PLTE:w`bKGDH IDATx흋* E?ϴHBC{-E+ĈF-iL fZ0-5ӂiգޥբ)9ՠ7ASl,FԄZ`95 }\0: s3ZÔ_Yu_g4&nу_j0E+j#RHm2dzg:fa@~o_cL#阗N&$@M\L0)td'6&Ô_er~S"V>#J&^I˿W4qӦRO86g[i[Fk`Z<}C2mMS4&oTzmZ0-5ӂiiSK5L3'-{jw|mZ]_ x~݉|gv4L`´OS{AS4|jL oai M t&qRF䖽&Tт %`2Aٺ!{Zc!JB]?x5.L~}zSXЕaa0kJ$Q3RB_3X5"zQ &p􀔾$dxubteI`q. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.`=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W/  4u(  4r 4 S 7  g  40 6A ?` J   ؔ 4 <7O- qRadiative Heating/Cooling"   4 <7   FJ is average over angles of integrals along rays through entire domainGF` 4 c $A ??E P B 4 s *f޽h ? ̙33t   0P( > Px P c $7`  g ~ P s *7f00  g ^ P 6Ԕ00 H P 0޽h ? ̙33r<$+ے f<WR( ; * d(pJTi] }'s4` vA=$A7GE"##j4'qjҎA,Ihq~L3bzpk^"_$\,~Z$2ŚJ<չ(1 G$N8S @p(tjfaQvzy+EgK:V8m'hUcD^n3E&`f-LE:C)l=e글yO CG Sf0޴e^Gx=~zv̴qu:kM4,3\^>QE}Dt!?||9g !D@oy`}H3;I`H-I3]iR^jU4)Mi ڑ@z8y ڹn$&4,d9q~o  T-\-pVdb7ut( 0)xVnVK9q4h{yyIJ,ZA :K̇M"ev^6,E76Y -GKCR2$rfΜ93sMz} Ǔ):{|Ǡ,Q't$BQ` %Ma[˽+-,: ?a6[ҺWd; Bu@Y }fp ,&%6Xǀ^* {gy]c =XL0FYoX$\cJQYqdUMڷzkݮ3&2uuU5Q,$եmnJoIWWmlmqd~]-[S|tr&b;n.n O$O] $acOstɨ+kl_>7IE8@W) S\1piI>Ύ"r!הrʋi3A7 ~S$LwZ8 atW,6gZMhrX(K(*xE6}ZٓY-fNPYG97ZUQ|'tON7zEoDxߏ}QT9dG(9!QZ3[$*Sȣ ğs<ծI0xP瑥ܲ|O7\ۜ#Q[]nxpj;r <.\Av?Jy$5 YKTt%<]ȍ 6l hK1/)J(MJsUf8t*ªd[Ftg#*@ۮyHRgS]oZ5EVK?\gMӴ0yk?u*m-3nlc^(K* mW+2EZolOC٠]xDvƷt0Q.LZܡ*R% <^U˨R64M8HZBa +{`6q̷ /r(3ZNB4}U@n}`gYaLH۷]m0ERGt<."B>3)$(k];g~,/7J][U|U.P[{nhϻ9[7ٺ*PV*u% :X^Q TUHIoܽX1՗j1 I(|?`bx}VMoU}o$V]>ky $@R@ObX3U6bJ*5 /. e·g,%=s?ޛuIogO.&sJ J#4jm#֛F I0w)M0UK\jP PGHjM.NO2٢ZJ紵EOwNitf7JR2eF;wʐ=(Te"kfs\5H}^>`xqa}lfvJLC& ϗ]:}UyթNR~cW.+&yٕ@Microsoft Organization Chart 2.0 MSOrgChart.20@Microsoft Organization Chart 2.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0pd>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0ppppd?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0d zEquation Equation.30,Microsoft Equation 3.0d}Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d#Equation Equation.30,Microsoft Equation 3.0d$Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0/ 0DTimes7$\F     oB$Ȣ&FvrIaϚB${571PhwYB$o P=Xv&h3# B$qPQ%8\Ku)B$3@jB$p5{lϝ+B$H'XԷO!9"wB$t@>2^o!B$ ĔR< ͒QZ|k'B$\-pVdb7ut(5Cb$;Sm".X./B$6EHN$$b$ e.iw;Ϙ[KSb$5^sb-2g$$$b$qnI=7Z"b$!SJ[S*bA|R$&$Z+7{Yb$l] ^neIb$߀ QΑkh߲Yb$M5Vɨp՗d?Oq5R$'~UIDiQuYb$@[D@ЁT`AU{R$Z"{YOi1szb B$hEhMٕ R$''C%f:zE b$7~F]@>/_rx$b$ 3qV !EXyXB$n@v )-WB$L-eD!\c,Q 8 B$D&7P7Q=DZ?"B$Eoo!_pZ3mImq|B$^#5JN~O\aB$ոaf6WS&y7\LOB$৿ۏ[ fX s۫B$upĔz"`Uk8pB$& [OE-\+ch'gB$X$Ѧ4qdFB$vTT×W_cHZB$_HG#VwZј+dEB$Y*X}T&p]cp"B$î}T>9+#^;\Db$legUJ |%PhVb$=mć;ũB$Z%8MW:DB$iG$'cfZAB$q1ǿidK\XbR$ xH@3n`20B$Q;\eb$N,C(5ۃR)apB$j=5BXsɦOB$hbD0uQ9M#B$?9#$>&G(R$k~xǸ2!1xD,B$8(L;O3LzB$("c˕;HQGtB$+Hȯ2pΥ vB$jV PJv#1b$̒ 6ԜR][QHB$W6aSm!Z1>ʘٗB$;^ίz%4IRY}B$ҏ .ɞh7S8B$@I`8D'`clb$v̋e2a b$6\ޯ;v% B$pܧ%"dҲ#DB$#  ՈB$58/O)bSw bB$ #=yE0)B$tHfw)GI+Lo B$%뗤~OxAU &drB$eTڬc0LLb}6B$ћK-lDoo?B$X9`R!HB$ fLNT~Y?-$LB$ ,Ծ6OB$s^;/Tt1 |$TB$.8g 27XB$qެ{%_Y{tt[B$Y.>ކ+9$^B$Ƽ<&D @f5 #bB$8)| bW"]} Cec $ f@fg4kdkd"x7< / \< \<ppp@  <4BdBdp\<`\<0 ? %! 9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft E Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.`=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W/  4u(  4r 4 S 7  g  40 6A ?` J   ؔ 4 <7O- qRadiative Heating/Cooling"   4 <7   FJ is average over angles of integrals along rays through entire domainGF` 4 c $A ??E  B 4 s *f޽h ? ̙33   <:( > <x < c $r7  g  <0 6A ?V}p   w4B < s *f޽h ? ̙33xWoGY()}y>RA** n6&HUIQW\zo/Q{h/=+I3(uUޯG6x 'DJͭ /<$3wϼ87wryHX1/~H?I?o>~|(=S> .9hW==?C|DZc|c@^S~v) U|z;sc ~LUG8 #D(= d==<<,r#(e1+W u7o%O&.<kb2{f=C߉J-j3&U1γ?*L;;sRJ=| y`d}% V@uu}|EP|(ᓦX`r{wQ\_ ۋZ~Ӌ;q8s@^*`H^/`C R=/ߩ=}BT_`qct4/)y䱹FP0\-("[| PFk? _Tj'H*ljE#et44ܶtq:Y&?M\s^>'oS0@t҄CG]to83Vm&L5 Mk8` |e<p ٨lWfYfV*a.n-71۴\6LIqJv6=ZV@7f1TfV4ڑϴU' (^غݺ#5F6ı]Vo]Oe3V"g5Ws*:NEg]zdyd:j |$@I"ע[7Xa-%Au¸XwNNl)Yw9}iXK>zغpߙI;N^\b\s'G5Yg& Mhʅ TUF*Ub*Y%X4'JL3Ք,1t=KಿTD|>KX%]؀jUVV5N]T47@gDYpҼ2G .26UTW)kYjJeEWQ^RIi, ²*Ёё¬ںiYW6 ua!@H2!תmFǔM> %be9-d):hsT1I+xBhZmÐ W +ci(z5eRR}T1SftǺQT=l2{XGdYMLD|j5ӰKqx_μ;^DH )9\SFWսTBMv%{:Ýg H m%6܈p ݚ͇tkbl"vXáH]\C~ʭߩ}db,R#i;ńZ_Cd&m(;E"]x7幾xWۏU?gZX(;/GM]w;o!Ĕv*ݙngv$ } <hL&^5?@e}g.mݍƠ|7r93ӏ;~پ*x항l(a1%odp_iKgzLLfEOćuO?|pxv۰SH>k],* ,!u/?Z~Vv%1]{Ovڷ{:HuZbv0c;YR{dl=|q@|xi3g5` .]EEeJ-=*T"6˸1BGǹa+6Je3!l~{jv&m!4iKUkؒ7E#YY~䷲(.ʖB+ >9̧ddu>9`_5dTdj멍LWv|<5cY1&p\pk8/.q qU *hܣx'|Jf S&J>JagA*7Ye q#ա|ӛ;5L&F"g*r18U4a}QwDW}SF&I b4di"♋îMU".\Szm p$ UC0mǠb4$=`]ijzthXRˇ:,f)ÊnӮQe%!}RjpuyjѦ}uCw1N ]ׅa KNiȵ2Se\)é#Q\ kג%eꮱeYT)?dQ3IzS`Rf,,k(ԡ+m2,51#vE6#mH[P7%hFL˴rz.z˶aU=qy©my^e/h"_୆?q哯3wN?+򕣅8ƕ.%e`\jVz 򕶲UD&Te+eVrX)KsBitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0ppppd?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0d zEquation Equation.30,Microsoft Equation 3.0d}Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d#Equation Equation.30,Microsoft Equation 3.0d$Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0/ 0DTimes7\ C23456769:;<="%#)*+,H1&%>F     oB$Ȣ&FvrIaϚB${571PhwYB$o P=Xv&h3# B$qPQ%8\Ku)B$3@jB$p5{lϝ+B$H'XԷO!9"wB$t@>2^o!B$ ĔR< ͒QZ|k'B$\-pVdb7ut(5Cb$;Sm".X./B$6EHN$$b$ e.iw;Ϙ[KSb$5^sb-2g$$$b$qnI=7Z"b$!SJ[S*bA|R$&$Z+7{Yb$l] ^neIb$߀ QΑkh߲Yb$M5Vɨp՗d?Oq5R$'~UIDiQuYb$@[D@ЁT`AU{R$Z"{YOi1szb B$hEhMٕ R$''C%f:zE b$7~F]@>/_rx$b$ 3qV !EXyXB$n@v )-WB$L-eD!\c,Q 8 B$D&7P7Q=DZ?"B$Eoo!_pZ3mImq|B$^#5JN~O\aB$ոaf6WS&y7\LOB$৿ۏ[ fX s۫B$upĔz"`Uk8pB$& [OE-\+ch'gB$X$Ѧ4qdFB$vTT×W_cHZB$_HG#VwZј+dEB$Y*X}T&p]cp"B$î}T>9+#^;\Db$legUJ |%PhVb$=mć;ũB$Z%8MW:DB$iG$'cfZAB$q1ǿidK\XbR$ xH@3n`20B$Q;\eb$N,C(5ۃR)apB$j=5BXsɦOB$hbD0uQ9M#B$?9#$>&G(R$k~xǸ2!1xD,B$8(L;O3LzB$("c˕;HQGtB$+Hȯ2pΥ vB$jV PJv#1b$̒ 6ԜR][QHB$W6aSm!Z1>ʘٗB$;^ίz%4IRY}B$ҏ .ɞh7S8B$@I`8D'`clb$v̋e2a b$6\ޯ;v% B$pܧ%"dҲ#DB$#  ՈB$58/O)bSw bB$ #=yE0)B$tHfw)GI+Lo B$%뗤~OxAU &drB$eTڬc0LLb}6B$ћK-lDoo?B$X9`R!HB$ fLNT~Y?-$LB$ ,Ծ6OB$s^;/Tt1 |$TB$.8g 27XB$qެ{%_Y{tt[B$Y.>ކ+9$^B$Ƽ<&D @f5 #bB$8)| bW"]} Cec $ f@fg4kdkd"x7}\< \<ppp@  <4BdBdp\<`\<? %!1Numerical Simulations of Solar Magneto-Convection9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.`=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W 0(  l  C t7p g l  C t7 `   g H  0޽h ? ̙33r1MO9R( ; * d@Microsoft Organizatim 8y t}d  on Chart 2.0 MSOrgChart.20@Microsoft Organization Chart 2.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0pd>Bitmap Image Paint.Picture0Bitmap Imaged?Ligning Equation.30,Microsoft Equation 3.0d@Ligning Equation.30,Microsoft Equation 3.0ppppd?\Equation Equation.30,Microsoft Equation 3.0d@]Equation Equation.30,Microsoft Equation 3.0dA^Equation Equation.30,Microsoft Equation 3.0dB_Equation Equation.30,Microsoft Equation 3.0dCbEquation Equation.30,Microsoft Equation 3.0dDcEquation Equation.30,Microsoft Equation 3.0dEdEquation Equation.30,Microsoft Equation 3.0dskEquation Equation.30,Microsoft Equation 3.0dumEquation Equation.30,Microsoft Equation 3.0dwoEquation Equation.30,Microsoft Equation 3.0dxpEquation Equation.30,Microsoft Equation 3.0duEquation Equation.30,Microsoft Equation 3.0d zEquation Equation.30,Microsoft Equation 3.0d}Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d#Equation Equation.30,Microsoft Equation 3.0d$Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0/ 0DTimes7\ C23456769:;<="%#)*+,H1&%>F     oB$Ȣ&FvrIaϚB${571PhwYB$o P=Xv&h3# B$qPQ%8\Ku)B$3@jB$p5{lϝ+B$H'XԷO!9"wB$t@>2^o!B$ ĔR< ͒QZ|k'B$\-pVdb7ut(5Cb$;Sm".X./B$6EHN$$b$ e.iw;Ϙ[KSb$5^sb-2g$$$b$qnI=7Z"b$!SJ[S*bA|R$&$Z+7{Yb$l] ^neIb$߀ QΑkh߲Yb$M5Vɨp՗d?Oq5R$'~UIDiQuYb$@[D@ЁT`AU{R$Z"{YOi1szb B$hEhMٕ R$''C%f:zE b$7~F]@>/_rx$b$ 3qV !EXyXB$n@v )-WB$L-eD!\c,Q 8 B$D&7P7Q=DZ?"B$Eoo!_pZ3mImq|B$^#5JN~O\aB$ոaf6WS&y7\LOB$৿ۏ[ fX s۫B$upĔz"`Uk8pB$& [OE-\+ch'gB$X$Ѧ4qdFB$vTT×W_cHZB$_HG#VwZј+dEB$Y*X}T&p]cp"B$î}T>9+#^;\Db$legUJ |%PhVb$=mć;ũB$Z%8MW:DB$iG$'cfZAB$q1ǿidK\XbR$ xH@3n`20B$Q;\eb$N,C(5ۃR)apB$j=5BXsɦOB$hbD0uQ9M#B$?9#$>&G(R$k~xǸ2!1xD,B$8(L;O3LzB$("c˕;HQGtB$+Hȯ2pΥ vB$jV PJv#1b$̒ 6ԜR][QHB$W6aSm!Z1>ʘٗB$;^ίz%4IRY}B$ҏ .ɞh7S8B$@I`8D'`clb$v̋e2a b$6\ޯ;v% B$pܧ%"dҲ#DB$#  ՈB$58/O)bSw bB$ #=yE0)B$tHfw)GI+Lo B$%뗤~OxAU &drB$eTڬc0LLb}6B$ћK-lDoo?B$X9`R!HB$ fLNT~Y?-$LB$ ,Ծ6OB$s^;/Tt1 |$TB$.8g 27XB$qެ{%_Y{tt[B$Y.>ކ+9$^B$Ƽ<&D @f5 #bB$8)| bW"]} Cec $ f@fg4kdkd"x7}\< \<ppp@  <4BdBdp\<`\<? %!1Numerical Simulations of Solar Magneto-Convection9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.`=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=WrMO)O9R( ; *ivuF-iL ~kj]-ii \ fz&2-/fM\Zhk%=97'LKZ0i9BӴÜՂP JŚe,0=e~&iWNڬ9`z^uAfgqtL+я7T6wNӴ`r?'LSd}ZNuyG=dp퓐`yauɤ+S r펽. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$,^Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes ._D((E+Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC b7.  $.`=-Example: 3D, Compressible Magneto-Convectiona<7CStratified convective flow: diverging upflows, turbulent downflows &DB$(6  R5S6T7U8V9<#HFluid Parcels reaching the surface Radiate away their Energy and Entropy IH(6Entropy8  Vorticity 9!Turbulent downdraftsfX:!Simulation Results: B Field linesF'Magnetic Field Lines, t=0.5 minG(Magnetic Field Lines, t=1.0 minH)Magnetic Field Lines, t=1.5 minI*Magnetic Field Lines, t=2.0 minJ+Magnetic Field Lines, t=2.5 minK,Magnetic Field Lines, t=3.0 minL-Magnetic Field Lines, t=3.5 minM.Magnetic Field Lines, t=4.0 minN/Magnetic Field Lines, t=4.5 minO0Magnetic Field Lines, t=5.0 minP1Magnetic Field Lines, t=5.5 minQ2Magnetic Field Lines: t=6 minh?*Tests: Comparison with Solar Observations!qGranulation Intensity Distribution Horizontal Velocity Spectrum Line Profiles Magnetic Field Distribution P-Modes1 GranulationD4Spectrum of granulationA$Solar velocity spectrum B% Line ProfilesC&^Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence._^,=E3Magnetic Field StrengthY;Field Distribution(j@P-Mode Excitation/8=W   `P(    r  S x7`  g f  s *A> ??`}oP `  c $A ??p`1 `  c $A ??p0  `  c $A ??  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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$,^Opacity is rapidly varying function of wavelength. 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%"1Numerical Simulations of Solar Magneto-Convection9Bob Stein Michigan State University East Lansing, MI, USAOutline[Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples5 ComputationSolve Conservation equations mass, momentum & internal energy Induction equation Radiative transfer equation 3D, Compressible Realistic EOS includes ionization Open boundaries Fix entropy of inflowing plasma at bottom ~!/D*!/D*>Q ,5  Code Flow Equations   Centering Time Advance Time Advance 3rd order, leapfrog:!    Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$,^Opacity is rapidly varying function of wavelength. 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Spatial Derivatives 5th order$ DiffusionQ1D Shock Tube TestRMHD Shock Tube Test 9Boundary Conditions horizontally periodic vertically open$:% sEWave Reflectiong>qBiA6Characteristic Boundary Conditions Physical Conditions&7#jNo Pressure drift Zero net mass flux Minimal reflected waves Entropy of inflowing material at bottom givenH {GCharacteristic z-Derivatives|FBoundary EquationsI,Local, 1D, inviscid Characteristic Equations HOutgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary ConditionsD2''qJ,Incoming Characteristic - No Reflected WavesK+Incoming Characteristic - No Pressure DriftL,Incoming characteristic - Zero net mass fluxM+Incoming characteristic - Specified EntropyN0Incoming characteristics - Horizontal velocitiesrCACharacteristic Magnetic Boundary Condition -- a work in progress""Physics is the time consuming part$# yEquation of state - includes ionization and molecule formation Radiative heating and cooling - LTE, non-gray, multi-group,? :" Energy Fluxes0Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass forL$$ $$G 4Log Pressure Temperature Log Opacity Source Function# Radiative Cooling & Heating0 Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $b [& KSolve Feautrier equations along rays through each grid point at the surfaceLK =')Rays: 5 Through Each Surface Grid Point $*"$,^Opacity is rapidly varying function of wavelength. 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Spatial Derivatives 5th order$ Diffusion 9Boundary Conditions horizontally periodic vertically open$:% "Physics is the hard part$ eEquation of state - includes ionization and molecule formation Radiation - LTE, non-gray, multi-group:" Energy Fluxes0Equation of State# Radiative Transfer0  Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry$ Energy Conservation% Radiation TransferLTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. $d Y& KSolve Feautrier equations along rays through each grid point at the surface =''5 Rays Through Each Surface Grid Point (Actually solve for q = P - B)Finite Difference EquationProblem: at small optical depth the 1 is lost re 1/Dt2 in B Solution: store the value -1, (the sum of the elements in a row) and calculate B = - (1+A+B)32$8$*;Interpolate q=P-B from slanted grid back to Cartesian grid +Simplifications6Only 5 rays 4 Multi-group opacity bins Assume kL a kC J7.$-,=Opacity is binned, according to its magnitude, into 4 bins. $ ,-fLine opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bing@'. 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