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No microturbulence, damping.=  Never See Hot Gas#c3D atmosphere is hotter than 1D atmosphere emitting the same flux, 3D atmosphere is more extended9BExtended atmosphere gives better agreement with p-mode frequenciesNStokes Profiles - Micropore Synthetic Observation - Perfect Telescope & Seeing<O& 2 YStokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Moderate SeeingZBY  [Stokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Excellent Seeing\D[  LStokes Image - Quiet Sun Synthetic Observation - Perfect Telescope & SeeingM5L ; !P-Mode Intensity - Velocity Phase8NAt fixed height V & DT have same profile. 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FrN˚$ۇ@|o6sW-d9ȓ.@">?EdK9Enϥl]sНR2ۗḰݼ$KA[ F6t1Z@jj9Z@jj9Z@jjcIENDB`d Equation 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 3.0dR-Equation Equation.30,Microsoft Equation 3.0/ 00DTimess 93S9S9S9:\uC6S9C6S9jp^DArials 93S9S9S9:\uC6S9C6S9jp^ DSymbol 93S9S9S9:\uC6S9C6S9jp^0DBauhaus 93S9S9S9:\uC6S9C6S9jp^   @n?" dd@  @@`` x.$         !"#$b$u_[<~tKpb$) T*Pd{CKpb$s1mˑg=pb$ DIsC3OEҔ8+;pb$=#6ubep<92pb$;Sm".Xqlpb$Um_5+"}pb$7~F]@>/_rx$kpb$X9 ǛJPi%pB$h17xf@QNrLpB$; ^X GpB$D:';zF=NJE[pB$k@$3Mwu)/v IpB$6EHN[pb$9p dk&2;'( pb$&ڮj^[&G.pb$ e.iw;Ϙ[TpB$q UWmƠ S pB$88- xqBq|pb$ DssN pb$qnI=7ZpB$ӸmeUra4[n pb$/lVkzR@5pFa pb$ژV>a|28! g pb$5^sb-& pb$l] ^neI? pb$߀ QΑkh߲ pb$M5Vɨp՗d?Oq5 pb$!SJ[S*bAu pR$&$Z+7{Y pb$1SZh58E U͇| rpb$̒ 6ԜR][QHdpb$99ye\Ϋpc $Pff@>g4KdKdC64$3Y6p@ppp@  <4BdBdS9S9pg4MdMdC64$3Y6p@ pp? %O =:.Radiative Transfer in 3D Numerical Simulations /.( &Robert Stein Department of Physics and Astronomy Michigan State University ke Nordlund Niels Bohr Institute for Astronomy, Physics, and Geophysics University of Copenhagenn $= $TRL1 Radiative Transfer$  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  JFluid Parcels that reach the surface Radiate away their Energy and Entropy( Granulation:0Spectrum of granulation agrees with observations6 Line Profiles7VConvection produces line shifts, changes in line widths. No microturbulence, damping.=  Never See Hot Gas#c3D atmosphere is hotter than 1D atmosphere emitting the same flux, 3D atmosphere is more extended9BExtended atmosphere gives better agreement with p-mode frequenciesNStokes Profiles - Micropore Synthetic Observation - Perfect Telescope & Seeing<O& 2 YStokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Moderate SeeingZBY  [Stokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Excellent Seeing\D[  LStokes Image - Quiet Sun Synthetic Observation - Perfect Telescope & SeeingM5L ; !P-Mode Intensity - Velocity Phase8NAt fixed height V & DT have same profile. Radiation reduces DT - more at low nHO'QHeight of t = 1 varies more at low frequencies where DT larger - Reduces DT moredR *  <!P-Mode Excitation="P-Mode Excitation># Thank You$ /8  @80(  l  C Tj6`  ` $  C A!work_integrand_im.gif 00016CCCHDPBG4 ABA78158:00h@H  0޽h ? ̙33p   @( wY x  c $j6  ` H  0޽h ? ̙33r~= N#}^>@'(  l dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d(&Equation Equation.30,Microsoft Equation 3.0dR-Equation Equation.30,Microsoft Equation 3.0/ 00DTimess 93S9S9S9:\uC6S9C6S9jp^DArials 93S9S9S9:\uC6S9C6S9jp^ DSymbol 93S9S9S9:\uC6S9C6S9jp^0DBauhaus 93S9S9S9:\uC6S9C6S9jp^   @n?" dd@  @@`` x.$         !"#$b$u_[<~tKpb$) T*Pd{CKpb$s1mˑg=pb$ DIsC3OEҔ8+;pb$=#6ubep<92pb$;Sm".Xqlpb$Um_5+"}pb$7~F]@>/_rx$kpb$X9 ǛJPi%pB$h17xf@QNrLpB$; ^X GpB$D:';zF=NJE[pB$k@$3Mwu)/v IpB$6EHN[pb$9p dk&2;'( pb$&ڮj^[&G.pb$ e.iw;Ϙ[TpB$q UWmƠ S pB$88- xqBq|pb$ DssN pb$qnI=7ZpB$ӸmeUra4[n pb$/lVkzR@5pFa pb$ژV>a|28! g pb$5^sb-& pb$l] ^neI? pb$߀ QΑkh߲ pb$M5Vɨp՗d?Oq5 pb$!SJ[S*bAu pR$&$Z+7{Y pb$1SZh58E U͇| rpb$̒ 6ԜR][QHdpb$99ye\Ϋpc $Pff@>g4KdKdC64$3Y6p@ppp@  <4BdBdS9S9g4MdMdC64$3Y6p@ pp? %O =:.Radiative Transfer in 3D Numerical Simulations /.( &Robert Stein Department of Physics and Astronomy Michigan State University ke Nordlund Niels Bohr Institute for Astronomy, Physics, and Geophysics University of Copenhagenn $= $TRL1 Radiative Transfer$  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  JFluid Parcels that reach the surface Radiate away their Energy and Entropy( Granulation:0Spectrum of granulation agrees with observations6 Line Profiles7VConvection produces line shifts, changes in line widths. No microturbulence, damping.=  Never See Hot Gas#c3D atmosphere is hotter than 1D atmosphere emitting the same flux, 3D atmosphere is more extended9BExtended atmosphere gives better agreement with p-mode frequenciesNStokes Profiles - Micropore Synthetic Observation - Perfect Telescope & Seeing<O& 2 YStokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Moderate SeeingZBY  [Stokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Excellent Seeing\D[  LStokes Image - Quiet Sun Synthetic Observation - Perfect Telescope & SeeingM5L ; !P-Mode Intensity - Velocity Phase8NAt fixed height V & DT have same profile. Radiation reduces DT - more at low nHO'QHeight of t = 1 varies more at low frequencies where DT larger - Reduces DT moredR *  <!P-Mode Excitation="P-Mode Excitation># Thank You$ /8rz>>'(  l dRdO) SNC @PictureszNPowerPoint Document(l"SummaryInformation(                         ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; A E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d  D        !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcd{fghijklmnopqrstuvwxy|}~eezDocumentSummaryInformation8Current User#_"Bob.Radiative Transfer in 3D Numerical Simulations /.( &Robert Stein Department of Physics and Astronomy Michigan State University ke Nordlund Niels Bohr Institute for Astronomy, Physics, and Geophysics University of Copenhagenn $= $TRL1 Radiative Transfer$  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  JFluid Parcels that reach the surface Radiate away their Energy and Entropy( Granulation:0Spectrum of granulation agrees with observations6 Line Profiles7VConvection produces line shifts, changes in line widths. No microturbulence, damping.=  Never See Hot Gas#c3D atmosphere is hotter than 1D atmosphere emitting the same flux, 3D atmosphere is more extended9BExtended atmosphere gives better agreement with p-mode frequenciesNStokes Profiles - Micropore Synthetic Observation - Perfect Telescope & Seeing<O& 2 YStokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Moderate SeeingZBY  [Stokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Excellent Seeing\D[  LStokes Image - Quiet Sun Synthetic Observation - Perfect Telescope & SeeingM5L ; !P-Mode Intensity - Velocity Phase8NAt fixed height V & DT have same profile. Radiation reduces DT - more at low nHO'QHeight of t = 1 varies more at low frequencies where DT larger - Reduces DT moredR *  <!P-Mode Excitation="P-Mode Excitation>#The End$/8r#`>F'(  l ߀߀߀dEquation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0d Equation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0dEquation Equation.30,Microsoft Equation 3.0d!Equation Equation.30,Microsoft Equation 3.0d(&Equation Equation.30,Microsoft Equation 3.0dR-Equation Equation.30,Microsoft Equation 3.0/ 00DTimess 936660i<,1 6@1 6H6DArials 936660i<,1 6@1 6H6 DSymbol 936660i<,1 6@1 6H60DBauhaus 936660i<,1 6@1 6H6   @n?" dd@  @@`` /$          b$u_[<~tKb$) T*Pd{CKb$s1mˑg=b$ DIsC3OEҔ8+;b$=#6ubep<92b$;Sm".Xqlb$Um_5+"}b$7~F]@>/_rx$kb$X9 ǛJPi%B$h17xf@QNrLB$; ^X GB$D:';zF=NJE[B$k@$3Mwu)/v IB$6EHN[b$9p dk&2;'( b$&ڮj^[&G.b$ e.iw;Ϙ[TB$q UWmƠ S B$88- xqBq|b$ DssN b$qnI=7ZB$ӸmeUra4[n b$/lVkzR@5pFa b$ژV>a|28! g b$5^sb-& b$l] ^neI? b$߀ QΑkh߲ b$M5Vɨp՗d?Oq5 b$!SJ[S*bAu R$&$Z+7{Y b$1SZh58E U͇| rb$̒ 6ԜR][QHdb$99ye\Ϋc $Pff@>g4KdKd 14i1@ppp@  <4BdBd6p6g4MdMd 14i1p@ pp? %O =8.Radiative Transfer in 3D Numerical Simulations /.( &Robert Stein Department of Physics and Astronomy Michigan State University ke Nordlund Niels Bohr Institute for Astronomy, Physics, and Geophysics University of Copenhagenn $= $TRL1 Radiative Transfer$  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  JFluid Parcels that reach the surface Radiate away their Energy and Entropy( Granulation:0Spectrum of granulation agrees with observations6 Line Profiles7VConvection produces line shifts, changes in line widths. No microturbulence, damping.=  Never See Hot Gas#c3D atmosphere is hotter than 1D atmosphere emitting the same flux, 3D atmosphere is more extended9BExtended atmosphere gives better agreement with p-mode frequenciesNStokes Profiles - Micropore Synthetic Observation - Perfect Telescope & Seeing<O& 2 YStokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Moderate SeeingZBY  [Stokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Excellent Seeing\D[  LStokes Image - Quiet Sun Synthetic Observation - Perfect Telescope & SeeingM5L ; !P-Mode Intensity - Velocity Phase8NAt fixed height V & DT have same profile. Radiation reduces DT - more at low nHO'QHeight of t = 1 varies more at low frequencies where DT larger - Reduces DT moredR *  <!P-Mode Excitation="P-Mode Excitation>#The End$/8  LD( @ p $  C A rad_rays.gif 00016CCCHDPBG4 ABA78158:P `x  c $2P  a0 H  0޽h ? ̙33r+p">߀RdO)5M> i @PictureszNPowerPoint Document(l"SummaryInformation(                         ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d g DocumentSummaryInformation8Current User#_"Bob