Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... ·...

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Physics of planetary atmospheres Lecture 3: Introduction to Radiative Energy Transport Part II

Transcript of Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... ·...

Page 1: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Physicsofplanetaryatmospheres

Lecture3:Introductionto

RadiativeEnergyTransportPartII

Page 2: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Coordinatesystem

•  Itisconvenienttoexploitsymmetriesinthesetupforhandlingtheangles:

•  Whatworksforincomingradiationbutmaynotworksowellfordescribingphasevariations:

τ

to observer

θ

Page 3: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Two-streamsconcept

•  Whendescribingradiativeenergytransportinplanetaryatmospherewecanintuitivelyidentifytwoflows:“incoming”or“down”and“outgoing”or“up”.

•  Smallthicknessoftheatmosphererelativetotheradiushelpssimplifyingitevenmore.

•  Inthiscaseforanypointintheplanettheexternalsource(star)isatsomezenithangleθwhileoutgoingradiationisdistributedover2πsolidangle.

Page 4: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Momentsofintensity

•  TotalintensityJ:

•  Flux:

•  HighermomentsareoccasionallyusedforRT.

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Page 5: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

WritingRTequationsfortwofluxes•  Let’sintroducetwonewvariables,totalflux:andnetflux:

•  NowwearegoingtoreformulatetheRTequationforthesevariables.

•  Let’sstartwithRTequationforintensitywithisotropicscattering.ThescatteringpartofthesourcefunctionissimplysetbyJ/4π.

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� (1� !0) ·B�<latexit sha1_base64="2Bwlpou39EjjbMgRQjSCq46y6KQ=">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</latexit><latexit sha1_base64="DthEA9g7tk8D80h6AyArOm5zb90=">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</latexit><latexit sha1_base64="DthEA9g7tk8D80h6AyArOm5zb90=">AAACU3icbVHPaxNBFJ5dW21Tq6keRLwMLUJ7SNkVob0IJV7UUwSTFjJheTs7m047s7PMvBXCsv9jKPSgF6/+BV560EnSEG19MPDx/eDNfJOWSjqMom9B+GBt/eGjjc3W1uPtJ0/bO88GzlSWiz43ytizFJxQshB9lKjEWWkF6FSJ0/Ty/Uw//Sqsk6b4gpNSjDSMC5lLDuippH3BdMVyC7zO6MeEKZ/MoKkzhlA19N2Kox3KjBZjSCLGM4OL0KdV5C0rZdPZjztL28HcR7tLS9Leiw6j+dD7IL4FeyfHP6Yvvv982UvaVywzvNKiQK7AuWEclTiqwaLkSjQtVjlRAr+EsRh6WIAWblTPO2noa89kNDfWnwLpnP07UYN2bqJT79SA5+6uNiP/pw0rzI9HtSzKCkXBF4vySlE0dFYwzaQVHNXEA+BW+rtSfg6+LPTf0PIlxHeffB8M3hzGHn/2bXTJYjbIK7JL9klMjsgJ+UB6pE84mZJf5HdAguvgJgzDtYU1DG4zz8k/E27/AVC4t1U=</latexit><latexit sha1_base64="+5os1BqYOH6kWcU4F9UqUJfKYMk=">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</latexit>

Page 6: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

DerivingRTequationsforupadownfluxes

•  Let’sintegratedtheRTequationoverμandφ:

•  Wehavetwoequationsandmanyunknownfunctionseventhoughtheyarenotindependent.

•  Additionalconstraintsareneeded.

dF"d⌧

= J" � !0 ·J

2� 2⇡(1� !0) ·B�

dF#d⌧

= �J# + !0 ·J

2+ 2⇡(1� !0) ·B�

<latexit sha1_base64="C2VXb9CPbJV/rx0kgmIzC+BuysQ=">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</latexit><latexit sha1_base64="C2VXb9CPbJV/rx0kgmIzC+BuysQ=">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</latexit><latexit sha1_base64="C2VXb9CPbJV/rx0kgmIzC+BuysQ=">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</latexit><latexit sha1_base64="C2VXb9CPbJV/rx0kgmIzC+BuysQ=">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</latexit>

Page 7: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Eddingtoncoefficients

•  Definition:

•  Thebigleapistoassumethatthesecoefficientsareconstantinτ.

•  Thereisnoreasonwhyitshouldbetruebutoneshouldexpectcorrelationbetweenfluxandmeanintensity.

✏" ⌘ F"J"

, ✏ ⌘ F+

J<latexit sha1_base64="jncYft1rmGsgZzQ0AGqKJdjh3BA=">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</latexit><latexit sha1_base64="jncYft1rmGsgZzQ0AGqKJdjh3BA=">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</latexit><latexit sha1_base64="jncYft1rmGsgZzQ0AGqKJdjh3BA=">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</latexit><latexit sha1_base64="jncYft1rmGsgZzQ0AGqKJdjh3BA=">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</latexit>

Page 8: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

RTwithconstantEddingtoncoefficients

•  Thetransformationfortheup/downfluxes:

•  Nowwehavetwoequationsandtwounknownfunctions.

dF"d⌧

= J" � !0 ·J

2� 2⇡(1� !0) ·B�

dF#d⌧

= �J# + !0 ·J

2+ 2⇡(1� !0) ·B�

dF"d⌧

= F" ·✓

1

✏"� !0

2✏

◆� !0F#

2✏� 2⇡(1� !0) ·B�

dF#d⌧

= �F# ·✓

1

✏#� !0

2✏

◆+

!0F"2✏

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Page 9: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Furtherassumptions•  Iftheratiosoffluxtomeanintensityforupanddowndirectionsareconstant,thereisnoreasonwhytheyshouldbedifferent(samerays,samelocalconditions).

•  Thenwecandenoteandsimplifyourequations:Note,thatequationsarecoupledbyscattering!

✏0 ⌘ ✏" = ✏#<latexit sha1_base64="6D+Av1QPsoh9iv0938dB6VwcC1M=">AAACHXicbVDLSgMxFM3UV62vqks3g0V0VWakoBuh6MZlBfuAzjBk0ts2NJOMSaallP6IG3/FjQtFXLgR/8a0HVFbDwROzrmH5J4wZlRpx/m0MkvLK6tr2fXcxubW9k5+d6+mRCIJVIlgQjZCrIBRDlVNNYNGLAFHIYN62Lua+PU+SEUFv9XDGPwIdzhtU4K1kYJ8yYNYUSb4sQd3Ce1/XwMvibGUYnDxo7TEgE+1IF9wis4U9iJxU1JAKSpB/t2ESRIB14RhpZquE2t/hKWmhME45yUKYkx6uANNQzmOQPmj6XZj+8goLbstpDlc21P1d2KEI6WGUWgmI6y7at6biP95zUS3z/0R5XGigZPZQ+2E2VrYk6rsFpVANBsagomk5q826WKJiTaF5kwJ7vzKi6R2WnQNvykVypdpHVl0gA7RCXLRGSqja1RBVUTQPXpEz+jFerCerFfrbTaasdLMPvoD6+MLgLKkCA==</latexit><latexit sha1_base64="6D+Av1QPsoh9iv0938dB6VwcC1M=">AAACHXicbVDLSgMxFM3UV62vqks3g0V0VWakoBuh6MZlBfuAzjBk0ts2NJOMSaallP6IG3/FjQtFXLgR/8a0HVFbDwROzrmH5J4wZlRpx/m0MkvLK6tr2fXcxubW9k5+d6+mRCIJVIlgQjZCrIBRDlVNNYNGLAFHIYN62Lua+PU+SEUFv9XDGPwIdzhtU4K1kYJ8yYNYUSb4sQd3Ce1/XwMvibGUYnDxo7TEgE+1IF9wis4U9iJxU1JAKSpB/t2ESRIB14RhpZquE2t/hKWmhME45yUKYkx6uANNQzmOQPmj6XZj+8goLbstpDlc21P1d2KEI6WGUWgmI6y7at6biP95zUS3z/0R5XGigZPZQ+2E2VrYk6rsFpVANBsagomk5q826WKJiTaF5kwJ7vzKi6R2WnQNvykVypdpHVl0gA7RCXLRGSqja1RBVUTQPXpEz+jFerCerFfrbTaasdLMPvoD6+MLgLKkCA==</latexit><latexit sha1_base64="6D+Av1QPsoh9iv0938dB6VwcC1M=">AAACHXicbVDLSgMxFM3UV62vqks3g0V0VWakoBuh6MZlBfuAzjBk0ts2NJOMSaallP6IG3/FjQtFXLgR/8a0HVFbDwROzrmH5J4wZlRpx/m0MkvLK6tr2fXcxubW9k5+d6+mRCIJVIlgQjZCrIBRDlVNNYNGLAFHIYN62Lua+PU+SEUFv9XDGPwIdzhtU4K1kYJ8yYNYUSb4sQd3Ce1/XwMvibGUYnDxo7TEgE+1IF9wis4U9iJxU1JAKSpB/t2ESRIB14RhpZquE2t/hKWmhME45yUKYkx6uANNQzmOQPmj6XZj+8goLbstpDlc21P1d2KEI6WGUWgmI6y7at6biP95zUS3z/0R5XGigZPZQ+2E2VrYk6rsFpVANBsagomk5q826WKJiTaF5kwJ7vzKi6R2WnQNvykVypdpHVl0gA7RCXLRGSqja1RBVUTQPXpEz+jFerCerFfrbTaasdLMPvoD6+MLgLKkCA==</latexit><latexit sha1_base64="6D+Av1QPsoh9iv0938dB6VwcC1M=">AAACHXicbVDLSgMxFM3UV62vqks3g0V0VWakoBuh6MZlBfuAzjBk0ts2NJOMSaallP6IG3/FjQtFXLgR/8a0HVFbDwROzrmH5J4wZlRpx/m0MkvLK6tr2fXcxubW9k5+d6+mRCIJVIlgQjZCrIBRDlVNNYNGLAFHIYN62Lua+PU+SEUFv9XDGPwIdzhtU4K1kYJ8yYNYUSb4sQd3Ce1/XwMvibGUYnDxo7TEgE+1IF9wis4U9iJxU1JAKSpB/t2ESRIB14RhpZquE2t/hKWmhME45yUKYkx6uANNQzmOQPmj6XZj+8goLbstpDlc21P1d2KEI6WGUWgmI6y7at6biP95zUS3z/0R5XGigZPZQ+2E2VrYk6rsFpVANBsagomk5q826WKJiTaF5kwJ7vzKi6R2WnQNvykVypdpHVl0gA7RCXLRGSqja1RBVUTQPXpEz+jFerCerFfrbTaasdLMPvoD6+MLgLKkCA==</latexit>

dF"d⌧

= F" · �a � F# · �s �B� · �BdF#d⌧

= �F# · �a + F" · �s +B� · �B<latexit sha1_base64="y4x5I/qLtlcX9/z4hSYJ4uS2pG8=">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</latexit><latexit sha1_base64="y4x5I/qLtlcX9/z4hSYJ4uS2pG8=">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</latexit><latexit sha1_base64="y4x5I/qLtlcX9/z4hSYJ4uS2pG8=">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</latexit><latexit sha1_base64="y4x5I/qLtlcX9/z4hSYJ4uS2pG8=">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</latexit>

Page 10: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Stellaranalogy

•  RTequations:

•  Feautriersolution:

µdI"d⌧

= I" � !0 ·J

4⇡� (1� !0) ·B�

µdI#d⌧

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4⇡+ (1� !0) ·B�

+ :µdI+d⌧

= I�

� :µdI�d⌧

= I+ � !0 ·J

2⇡� 2(1� !0) ·B�

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µ2 d2I+d⌧2

= I+ � !0 ·J

2⇡� 2(1� !0) ·B�

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Page 11: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Feautriermethod

•  Itisstableandfast•  Itrequirestwoboundaryconditionsandwedohavethem:deepinsideastarI+isgivenbyPlanckfunction.AtthesurfaceI+ isequaltoI .

•  Noadditionalclosureassumptionsareneeded.•  JisI+ integratedoverμ.Solvingthisequationformanydirectionsisnearlyaseasyasforthesingledirection.

•  Discretisationleadstoa3-diagonalsystemoflinearequations.

Page 12: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Purescattering

•  Equilibriummeansthenetenergyfluxthroughanylayerisconstant:

•  Inpurescatteringregimeweget:Noabsorption/emission,noheatingorcoolingofgas!

!0 = 1, �B = 0 ) �a ⌘✓1

✏0� !0

2✏

◆=

!0

2✏⌘ �s

) ✏ = ✏0<latexit sha1_base64="8Zdx2VpKZUn0/DhBvO84Lo+YmwM=">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</latexit><latexit sha1_base64="m9j1S03epV4fAdE9pjrXqq3B9ok=">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</latexit><latexit sha1_base64="m9j1S03epV4fAdE9pjrXqq3B9ok=">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</latexit><latexit sha1_base64="Wppna0E9WKSlABn0sojoa25/uuw=">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</latexit>

F� = F" � F#, F+ = F" + F#ZdF�d⌧

d� = 0 =

Z(�a � �s)F+d�� 2

ZB� · �Bd�

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Page 13: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Pureabsorption

!0 = 0 ) �a =1

✏0, �s = 0 and �B = 2⇡

dF"d⌧

= F" · �a �B� · �BdF#d⌧

= �F# · �a +B� · �B<latexit sha1_base64="2PhppYx5Ta+FPADTeopQZ14o0r8=">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</latexit><latexit sha1_base64="2PhppYx5Ta+FPADTeopQZ14o0r8=">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</latexit><latexit sha1_base64="2PhppYx5Ta+FPADTeopQZ14o0r8=">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</latexit><latexit sha1_base64="2PhppYx5Ta+FPADTeopQZ14o0r8=">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</latexit>

Let’sseeifwecanestimatethevalueofε’inaverysimplifiedcase.Let’stakeisothermalmedium.ThenBisconstantinτ.

F"1 = F"2e��a�⌧ +

�B�a

B�1� e��a�⌧

F#2 = F#1e��a�⌧ +

�B�a

B�1� e��a�⌧

�<latexit sha1_base64="/igruA6/fKvhTH3f5BmwvTbmhtM=">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</latexit><latexit sha1_base64="/igruA6/fKvhTH3f5BmwvTbmhtM=">AAAC4XictVJNaxRBEO0Zv+L6kVWPXhoXISIJM0HQSyCsIh4juElgezPU9NbsNunuGbprDMswNy8eFPHqv/LmP/Fo72YC+dCjBQ2v36viVVd1XmnlKUl+RfG16zdu3lq73btz99799f6Dh/u+rJ3EkSx16Q5z8KiVxREp0nhYOQSTazzIj18v9YOP6Lwq7QdaVDgxMLOqUBIoUFn/99usEXUFzpUnPG13zl+3W45HzaaYgTGQARdvUBMIgrp9LgoHsumkYXuGoB0KjQVtpJv/KhVOzeb0jAvRW5pNyxN7ZrdzkUj/p3/WHyRbySr4VZB2YMC62Mv6P0NrsjZoSWrwfpwmFU0acKSkxrYnao8VyGOY4ThACwb9pFltqOVPAzPlRenCscRX7PmKBoz3C5OHTAM095e1Jfk3bVxT8WrSKFvVhFaeGhW15lTy5br5VDmUpBcBgHQq9MrlHMLsKHyKXhhCevnJV8H+9lYa8PsXg91hN4419pg9YRssZS/ZLnvH9tiIyego+hR9ib7GMv4cf4u/n6bGUVfziF2I+McfYK3ppA==</latexit><latexit sha1_base64="/igruA6/fKvhTH3f5BmwvTbmhtM=">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</latexit><latexit sha1_base64="/igruA6/fKvhTH3f5BmwvTbmhtM=">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</latexit>

Page 14: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Pureabsorptioninopticallythickmedium

•  Furthersimplificationisachievedassumingthepathbetweenpoints1and2tobeopticallythick:

•  Fromthemomentequationsweknowthatinisothermal,absorbing,opticallythickcasefluxisequalto½oftotalintensity.Thismeansthatε=ε‘=½.

F"1 =�B�a

B = 2⇡✏0B

F#2 =�B�a

B = 2⇡✏0B<latexit sha1_base64="boEjBavk/BNkZtczStNyT2oQWfM=">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</latexit><latexit sha1_base64="boEjBavk/BNkZtczStNyT2oQWfM=">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</latexit><latexit sha1_base64="boEjBavk/BNkZtczStNyT2oQWfM=">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</latexit><latexit sha1_base64="boEjBavk/BNkZtczStNyT2oQWfM=">AAACYXiclZFLS8QwFIXT+h5fVZdugoPoStpB0I0gI4hLBUeF6VBuM+kYzIskVYfSP+nOjRv/iJmH4GvjhcDHufdwk5Ncc2ZdHL8G4czs3PzC4lJjeWV1bT3a2LyxqjSEdojiytzlYClnknYcc5zeaUNB5Jze5g9no/7tIzWWKXnthpr2BAwkKxgB56Usej7PqrTUYIx6wkl9khYGSJUOQAjI2vUnQd0+aaWapVRbxpXcw22cpo2Rua+e5MTe+o89i5rxQTwu/BuSKTTRtC6z6MWvIqWg0hEO1naTWLteBcYxwmndSEtLNZAHGNCuRwmC2l41TqjGu17p40IZf6TDY/WrowJh7VDkflKAu7c/eyPxr163dMVxr2JSl45KMllUlBw7hUdx4z4zlDg+9ADEMH9XTO7Bh+T8pzR8CMnPJ/+Gm9ZB4vnqsHnansaxiLbRDtpHCTpCp+gCXaIOIugtmA1Wg7XgPVwKo3BzMhoGU88W+lbh9gcKLLbe</latexit>

Page 15: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Transmissionfunction

•  Substitutingexpressionεtoγ‘swefinallyget:•  Transmissionfunctionwasdefinedinlecture2assumingisothermalmedium:

F"1 = F"2T + ⇡B(1� T )

F#2 = F#1T + ⇡B(1� T )<latexit sha1_base64="Kj+Bm7yNdbdO6kSjDJLHwhlnl6I=">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</latexit><latexit sha1_base64="Kj+Bm7yNdbdO6kSjDJLHwhlnl6I=">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</latexit><latexit sha1_base64="Kj+Bm7yNdbdO6kSjDJLHwhlnl6I=">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</latexit><latexit sha1_base64="Kj+Bm7yNdbdO6kSjDJLHwhlnl6I=">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</latexit>

�a = 2� !0, �s = !0 and �B = 2⇡(1� !0)<latexit sha1_base64="AIBq0io4IEgfTR+/V8wXmAee8RQ=">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</latexit><latexit sha1_base64="5jzYQt06QGaZ7FB1LCN+3duQ2FI=">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</latexit><latexit sha1_base64="5jzYQt06QGaZ7FB1LCN+3duQ2FI=">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</latexit><latexit sha1_base64="hwd6fov+GHowl/tHuW9FgbJxSxU=">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</latexit>

Page 16: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Transmissionfunction

•  γaandthereforeεappearinthecoefficientinfrontofopticaldepthintheformalsolution:

•  ComparingthetwosetsofequationswegetmorerealisticexpressionfortransmissionT:

F"1 = F"2e��a�⌧ +

�B�a

B�1� e��a�⌧

F#2 = F#1e��a�⌧ +

�B�a

B�1� e��a�⌧

�<latexit sha1_base64="/igruA6/fKvhTH3f5BmwvTbmhtM=">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</latexit><latexit sha1_base64="/igruA6/fKvhTH3f5BmwvTbmhtM=">AAAC4XictVJNaxRBEO0Zv+L6kVWPXhoXISIJM0HQSyCsIh4juElgezPU9NbsNunuGbprDMswNy8eFPHqv/LmP/Fo72YC+dCjBQ2v36viVVd1XmnlKUl+RfG16zdu3lq73btz99799f6Dh/u+rJ3EkSx16Q5z8KiVxREp0nhYOQSTazzIj18v9YOP6Lwq7QdaVDgxMLOqUBIoUFn/99usEXUFzpUnPG13zl+3W45HzaaYgTGQARdvUBMIgrp9LgoHsumkYXuGoB0KjQVtpJv/KhVOzeb0jAvRW5pNyxN7ZrdzkUj/p3/WHyRbySr4VZB2YMC62Mv6P0NrsjZoSWrwfpwmFU0acKSkxrYnao8VyGOY4ThACwb9pFltqOVPAzPlRenCscRX7PmKBoz3C5OHTAM095e1Jfk3bVxT8WrSKFvVhFaeGhW15lTy5br5VDmUpBcBgHQq9MrlHMLsKHyKXhhCevnJV8H+9lYa8PsXg91hN4419pg9YRssZS/ZLnvH9tiIyego+hR9ib7GMv4cf4u/n6bGUVfziF2I+McfYK3ppA==</latexit><latexit sha1_base64="/igruA6/fKvhTH3f5BmwvTbmhtM=">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</latexit><latexit sha1_base64="/igruA6/fKvhTH3f5BmwvTbmhtM=">AAAC4XictVJNaxRBEO0Zv+L6kVWPXhoXISIJM0HQSyCsIh4juElgezPU9NbsNunuGbprDMswNy8eFPHqv/LmP/Fo72YC+dCjBQ2v36viVVd1XmnlKUl+RfG16zdu3lq73btz99799f6Dh/u+rJ3EkSx16Q5z8KiVxREp0nhYOQSTazzIj18v9YOP6Lwq7QdaVDgxMLOqUBIoUFn/99usEXUFzpUnPG13zl+3W45HzaaYgTGQARdvUBMIgrp9LgoHsumkYXuGoB0KjQVtpJv/KhVOzeb0jAvRW5pNyxN7ZrdzkUj/p3/WHyRbySr4VZB2YMC62Mv6P0NrsjZoSWrwfpwmFU0acKSkxrYnao8VyGOY4ThACwb9pFltqOVPAzPlRenCscRX7PmKBoz3C5OHTAM095e1Jfk3bVxT8WrSKFvVhFaeGhW15lTy5br5VDmUpBcBgHQq9MrlHMLsKHyKXhhCevnJV8H+9lYa8PsXg91hN4419pg9YRssZS/ZLnvH9tiIyego+hR9ib7GMv4cf4u/n6bGUVfziF2I+McfYK3ppA==</latexit>

T = e��a�⌧ = e��⌧/✏<latexit sha1_base64="agzD6BnO5mlE8RB5thmng1BCQDU=">AAACG3icbZDLSgMxFIYz3q23qks3g0Vwo84UQTeCqAuXFVoVOrWcSU9rMMkMyRmhDH0PN76KGxeKuBJc+DamF8TbD4Ev/zmH5PxxKoWlIPjwxsYnJqemZ2YLc/MLi0vF5ZVzm2SGY40nMjGXMViUQmONBEm8TA2CiiVexDfH/frFLRorEl2lbooNBR0t2oIDOatZLFcP8CrfijqgFDQhOkFJEBFkvaH/dd+JMLVCJrrXLJaC7WAg/y+EIyixkSrN4lvUSnimUBOXYG09DFJq5GBIcIm9QpRZTIHfQAfrDjUotI18sFvP33BOy28nxh1N/sD9PpGDsrarYtepgK7t71rf/K9Wz6i938iFTjNCzYcPtTPpU+L3g/JbwiAn2XUA3Aj3V59fgwFOLs6CCyH8vfJfOC9vh47PdkuHR6M4ZtgaW2ebLGR77JCdsgqrMc7u2AN7Ys/evffovXivw9YxbzSzyn7Ie/8EgjChuQ==</latexit><latexit sha1_base64="agzD6BnO5mlE8RB5thmng1BCQDU=">AAACG3icbZDLSgMxFIYz3q23qks3g0Vwo84UQTeCqAuXFVoVOrWcSU9rMMkMyRmhDH0PN76KGxeKuBJc+DamF8TbD4Ev/zmH5PxxKoWlIPjwxsYnJqemZ2YLc/MLi0vF5ZVzm2SGY40nMjGXMViUQmONBEm8TA2CiiVexDfH/frFLRorEl2lbooNBR0t2oIDOatZLFcP8CrfijqgFDQhOkFJEBFkvaH/dd+JMLVCJrrXLJaC7WAg/y+EIyixkSrN4lvUSnimUBOXYG09DFJq5GBIcIm9QpRZTIHfQAfrDjUotI18sFvP33BOy28nxh1N/sD9PpGDsrarYtepgK7t71rf/K9Wz6i938iFTjNCzYcPtTPpU+L3g/JbwiAn2XUA3Aj3V59fgwFOLs6CCyH8vfJfOC9vh47PdkuHR6M4ZtgaW2ebLGR77JCdsgqrMc7u2AN7Ys/evffovXivw9YxbzSzyn7Ie/8EgjChuQ==</latexit><latexit sha1_base64="agzD6BnO5mlE8RB5thmng1BCQDU=">AAACG3icbZDLSgMxFIYz3q23qks3g0Vwo84UQTeCqAuXFVoVOrWcSU9rMMkMyRmhDH0PN76KGxeKuBJc+DamF8TbD4Ev/zmH5PxxKoWlIPjwxsYnJqemZ2YLc/MLi0vF5ZVzm2SGY40nMjGXMViUQmONBEm8TA2CiiVexDfH/frFLRorEl2lbooNBR0t2oIDOatZLFcP8CrfijqgFDQhOkFJEBFkvaH/dd+JMLVCJrrXLJaC7WAg/y+EIyixkSrN4lvUSnimUBOXYG09DFJq5GBIcIm9QpRZTIHfQAfrDjUotI18sFvP33BOy28nxh1N/sD9PpGDsrarYtepgK7t71rf/K9Wz6i938iFTjNCzYcPtTPpU+L3g/JbwiAn2XUA3Aj3V59fgwFOLs6CCyH8vfJfOC9vh47PdkuHR6M4ZtgaW2ebLGR77JCdsgqrMc7u2AN7Ys/evffovXivw9YxbzSzyn7Ie/8EgjChuQ==</latexit><latexit sha1_base64="agzD6BnO5mlE8RB5thmng1BCQDU=">AAACG3icbZDLSgMxFIYz3q23qks3g0Vwo84UQTeCqAuXFVoVOrWcSU9rMMkMyRmhDH0PN76KGxeKuBJc+DamF8TbD4Ev/zmH5PxxKoWlIPjwxsYnJqemZ2YLc/MLi0vF5ZVzm2SGY40nMjGXMViUQmONBEm8TA2CiiVexDfH/frFLRorEl2lbooNBR0t2oIDOatZLFcP8CrfijqgFDQhOkFJEBFkvaH/dd+JMLVCJrrXLJaC7WAg/y+EIyixkSrN4lvUSnimUBOXYG09DFJq5GBIcIm9QpRZTIHfQAfrDjUotI18sFvP33BOy28nxh1N/sD9PpGDsrarYtepgK7t71rf/K9Wz6i938iFTjNCzYcPtTPpU+L3g/JbwiAn2XUA3Aj3V59fgwFOLs6CCyH8vfJfOC9vh47PdkuHR6M4ZtgaW2ebLGR77JCdsgqrMc7u2AN7Ys/evffovXivw9YxbzSzyn7Ie/8EgjChuQ==</latexit>

Page 17: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Two-streamapproximationsolution(isotropicscattering)

•  Westartwithexpressionforupanddownfluxes:

•  Addingandsubtractingtheseequationsgives:

dF"d⌧

= F" · �a � F# · �s �B� · �BdF#d⌧

= �F# · �a + F" · �s +B� · �B<latexit sha1_base64="y4x5I/qLtlcX9/z4hSYJ4uS2pG8=">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</latexit><latexit sha1_base64="y4x5I/qLtlcX9/z4hSYJ4uS2pG8=">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</latexit><latexit sha1_base64="y4x5I/qLtlcX9/z4hSYJ4uS2pG8=">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</latexit><latexit sha1_base64="y4x5I/qLtlcX9/z4hSYJ4uS2pG8=">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</latexit>

dF+

d⌧= (�a + �s) · F�

dF�d⌧

= (�a � �s) · F+ � 2�B ·B<latexit sha1_base64="ViuYT2WM8A30sK3SSXtP6Dn+ytU=">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</latexit><latexit sha1_base64="ViuYT2WM8A30sK3SSXtP6Dn+ytU=">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</latexit><latexit sha1_base64="ViuYT2WM8A30sK3SSXtP6Dn+ytU=">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</latexit><latexit sha1_base64="ViuYT2WM8A30sK3SSXtP6Dn+ytU=">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</latexit>

Page 18: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Two-streamapproximationsolution

•  Substitutingtheexpressionforthenetfluxtothederivativein2ndequationweget:

•  Thebigassumptionhereisthatγ’sarenotfunctionsofτ.

•  Thusanatmospheremustbedividedinlayers,thinenoughforthistobetrue.

d2F+

d⌧2= D · F+ � 2�B(�a + �s) ·B

Di↵usivity factor D ⌘p(�a + �s)(�a � �s)

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Page 19: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Stepstorealism

•  Firststep:dividetheatmosphereinmanyisothermallayers.Inthiscasewecanusethesolutionsforthetwostreamsdirectlyassumingthevalueoftemperature.

•  Secondstep:UselinearapproximationforBineachlayerB(τ)=B1+(τ-τ1)(B2-B1)/(τ2-τ1).ThataddssmallcomplicationbuttheintegralofBcanstillbecalculatedanalytically.

•  Thirdstep:non-isotropicscattering.

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Non-isotropicscattering

•  Ofcourse,scatteringinopticallythinpartofplanetaryatmospheresisnon-isotropic.

•  Thescatteringpatternisexpectedtobesymmetricrelativetotheirradiationdirection:

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Singlescatteringgeometry

•  Let’sdefinetwodirections:onetothestarθandonetotheobserverθ‘.

•  Thecos(θ-θ’)=μ‘’isknownfromthesphericalgeometrycoursethatnobodytakesanylonger:

•  IfthephasefunctionPdependsonlyonθ-θ’thenitmustbeinvariantfortranspositionoftheimage.

µ00 = µµ0 +p

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Page 22: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Constructingscatteringphasefunction

•  AnotherimportantpropertyofPisthatitshouldbeinvariantofcoordinatesystemtransformation.

•  InsteadofderivingananalyticalformofPwecanconstructitoutofafamilyofsphericalfunctionswiththerightpropertiesorbyselectingananalyticalexpressionwithfittingparameters(Henyey-Greenstein).

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Deepinside

•  Intheopticallydeeppartoftheatmospherescatteringbecomesequivalenttoabsorptionbecausephotonstravelveryshortdistancestayinginequilibriumwiththegas.

•  ThusJ=BandS=B.•  FluxisthendefinedbythegradientofB–diffusionapproximation.

Page 24: Lecture 3: Introduction to Radiative Energy Transport Part IIpiskunov/TEACHING/PLANETARY... · planetary atmosphere we can intuitively identify two flows: “incoming” or “down”

Realthing

•  Usingfluxconceptkillsourabilityoftreating3Datmospheres,

•  Wetotallyignoredconvectionandhorizontalenergytransportbywinds,surfacegradientoftemperatureandalbedo.

•  InrealclimatemodelRTissolvedforintensityalongmanyraysbutthetwo-streamorFeautriermethodarefrequentlyused.