Simple models for the H2O and CO greenhouse...

SimplemodelsfortheH2OandCO2greenhouseeffects

withapplications

NadirJeevanjeeHessFellow

PrincetonUniversity

w/StephanFueglistaler (Princeton),JacobT.Seeley (Berkeley),DavidPaynter(GFDL),andDavidRomps(Berkeley)

OLR

Oceansurface

Space

Atmosphere

Sunlight

BoundaryLayerCo

nvectio

nEvap

Whatis`the’greenhouseeffect?

Qnet =LP(W/m2)

Quantity Value Simple Model Insight

2XCO2forcing 4 W/m2 ?? ??

Atmosphericradiativecooling

2K/day ?? ??

Mean precipitationchange

2-3W/m2/K(2-3%/K)

?? ??

Agenda:aspectsofthegreenhouseeffect

ToolsLine-by-line(LBL)radiationmodel

• Accurateandcomprehensiveinfra-redradiativetransfercalculationusingdetailedmolecularspectroscopy

• Calculatesopticaldepthsτk andupwardanddownwardfluxesFkasfunctionofwavenumberk

Pencil+paperestimates• Someapproximationsrequired…

OLR

Oceansurface

Space

Atmosphere

Sunlight

BoundaryLayer

Evap

Convectio

n

IdealizedtropicalcolumnwithTs=300K,𝚪 =7k/kmRH=0.75CO2=280ppmv

z

UODA:theunitopticaldepthapproximation

Optical depth: ⌧(z) = |{z}abs. coe↵

(m

2/kg)

Z 1

z⇢vdz

0

| {z }(kg/m2

)

=Total e↵ective area

Actual area

z


(m

2/kg)

Z 1

z⇢vdz

0

| {z }(kg/m2

)


Actual area

⌧ < 1 optically thin

⌧ ⇡ 1 just right

⌧ > 1 optically thick




UODA:theunitopticaldepthapproximation

z


(m

2/kg)

Z 1

z⇢vdz

0

| {z }(kg/m2

)


Actual area




UODA:theunitopticaldepthapproximationUODA:Emissiontospacecomesfromunitopticaldepth

200 600 1000 14001e−0

41e

+00

1e+0

4 Absorption spectrum

κ (m

2kg

)

k (cm−1)200 600 1000 140010

0060

020

0

log of optical depth τk

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0

Cooling−to−space (W m2 hPa cm−1)

Pres

sure

(hPa

)

−0.0015

−0.0010

−0.0005

0.0000

0.0005

0.0010

0.0015

k (cm−1)

AspectralviewofUODA

200 600 1000 14001e−0

41e

+00

1e+0

4 Absorption spectrum

κ (m

2kg

)

k (cm−1)200 600 1000 140010

0060

020

0

log of optical depth τk

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0

Cooling−to−space (W m2 hPa cm−1)

Pres

sure

(hPa

)

−0.0015

−0.0010

−0.0005

0.0000

0.0005

0.0010

0.0015

k (cm−1)

AspectralviewofUODA

UseUODA tobuildsimplemodelforCO2forcing

Step1:Parameterize absorption coeffsandopticaldepth

500 600 700 8001e−0

51e−0

11e

+03

Absorption spectrum, LBL

k (cm−1)

κ (m

2kg

)

500 600 700 8001000

200

5010

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)

500 600 700 8001e−0

51e−0

11e

+03

Absorption spectrum, fit

k (cm−1)

κ (m

2kg

)

k0

κ0

500 600 700 8001000

200

5010

ln τk, Theory

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)

(k) = 0 exp

✓� |k � k0|

lk

◆

500 600 700 8001e−0

51e−0

11e

+03


k (cm−1)

κ (m

2kg

)

500 600 700 8001000

200

5010

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)

500 600 700 8001e−0

51e−0

11e

+03


k (cm−1)

κ (m

2kg

)

k0

κ0

500 600 700 8001000

200

5010

ln τk, Theory

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)

Step1:Parameterize absorption coeffsandopticaldepth

(k) = 0 exp

✓� |k � k0|

lk

◆⌧k = (k)

qp2

2gpref

500 600 700 8001e−0

51e−0

11e

+03


k (cm−1)

κ (m

2kg

)

500 600 700 8001000

200

5010

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)

500 600 700 8001e−0

51e−0

11e

+03


k (cm−1)

κ (m

2kg

)

k0

κ0

500 600 700 8001000

200

5010

ln τk, Theory

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)

500 600 700 8001e−0

51e−0

11e

+03


k (cm−1)

κ (m

2kg

)

500 600 700 8001000

200

5010

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)

500 600 700 8001e−0

51e−0

11e

+03


k (cm−1)

κ (m

2kg

)

k0

κ0

500 600 700 8001000

200

5010

ln τk, Theory

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)

⌧k(p) = (k)qp2

2gps

=) p1(k) =

r2gpsq0| {z }

p0(q)

exp

✓|k � k0|

2lk

◆

Step2.Find𝜏=1emissionlevels,denoted p1(k):

⌧k(p) = (k)qp2

2gps

=) p1(k) =

r2gpsq0| {z }

p0(q)

exp

✓|k � k0|

2lk

◆


600 650 700 7501000

200

5020

CO2 emission levels

k ( cm−1 )

p 1 (hPa)

CO2 (ppmv)280

⌧k(p) = (k)qp2

2gps

=) p1(k) =

r2gpsq0| {z }

p0(q)

exp

✓|k � k0|

2lk

◆


600 650 700 7501000

200

5020

CO2 emission levels

k ( cm−1 )

p 1 (hPa)

CO2 (ppmv)2801120

⌧k(p) = (k)qp2

2gps

=) p1(k) =

r2gpsq0| {z }

p0(q)

exp

✓|k � k0|

2lk

◆


600 650 700 7501000

200

5020

CO2 emission levels

k ( cm−1 )

p 1 (hPa)

∆k = lk ln4

CO2 (ppmv)2801120

Step3.ConstructapictureforCO2 forcing:

Negativestratosphericcontribution

Noforcingcontribution

Positivesurfacecontribution

CO2 forcinglogarithmic,andonlydependsonsurface-stratospheretemperaturecontrast!

F4⇥ = 2 lk ln 4| {z }

�k

2

64 ⇡B(k0

, Ts)| {z }surface

�⇡B(k0

, T (p0

))| {z }stratosphere

3

75

Step4.EstimateCO2 forcing(Wilson2012):600 650 700 75010

00200

5020

CO2 emission levels

k ( cm−1 )

p 1 (hPa)

∆k = lk ln4

CO2 (ppmv)2801120

ValidationofformulaF4⇥ = 2 lk ln 4| {z }

�k

2

64 ⇡B(k0

, Ts)| {z }surface

�⇡B(k0

, T (p0


3

75

OLR

Oceansurface

Space

Atmosphere

Sunlight

BoundaryLayer

Evap

Convectio

n

VaryTsinidealizedtropicalcolumn

0 5 10 15

05

1015

CO2 Forcing

F4xLBL ( W m2 )

F 4xTheory

( W

m2 )

●●

●

●

●

●

●●

●

●

●

●

●

●

Ts (K)250260270280290300310

Caveats:• Clear-skyonly

(noclouds)• RH=0,CO2only

(noH2Ooverlap)• Idealizedatmosphere

(constantRHetc.)• Nostratospheric

adjustment

Back-of-the-envelope

F2⇥ ⇡ 2lk ln 2| {z }15 cm�1

[ ⇡B(k0, 288 K)| {z }0.4 W/m2/cm�1

� ⇡B(k0, 220 K)| {z }0.15 W/m2/cm�1

]

⇡ 3.7 W/m2


2XCO2forcing 4 W/m2 2𝑙%ln2 𝐵% 𝑇+ − 𝐵% 𝑇-./0. Logscaling,Ts-Tstrattemp contrast


2K/day ?? ??


2-3W/m2/K(2-3%/K)

?? ??


2K/dayremarkablyrobust

−50 0 501000

600

200

ECMWF radiative heating (K/day)

Latitude (deg)

Pres

sure

(hPa

)

−3

−2

−1

0

1

2

3

−6 −4 −2 0

1000

600

200

0

Radiative heating profiles

Heating (K/day)

Pres

sure

(hPa

) ECMWF


−50 0 501000

600

200


Latitude (deg)

Pres

sure

(hPa

)

−3

−2

−1

0

1

2

3

−6 −4 −2 0

1000

600

200

0


Heating (K/day)

Pres

sure

(hPa

) ECMWF

OLR

Oceansurface

Space

Atmosphere

Sunlight

BoundaryLayer

Evap

Convectio

n

Canemulatewithsingle-columnLBL,H2Oonly,nocont.


−50 0 501000

600

200


Latitude (deg)

Pres

sure

(hPa

)

−3

−2

−1

0

1

2

3

−6 −4 −2 0

1000

600

200

0


Heating (K/day)

Pres

sure

(hPa

) ECMWFLBL

OLR

Oceansurface

Space

Atmosphere

Sunlight

BoundaryLayer

Evap

Convectio

n

Canemulatewithsingle-columnLBL,H2Oonly,nocont.

Thecooling-to-spaceapproximation

Hk =g

Cp@pFk| {z }Flux

div.

⇡ � g

Cp⇡B(k, T )| {z }

Planck

d⌧kdp|{z}

emissivity

gradient

e�⌧k|{z}trans-

missivity

200 600 1000 14001000

600

200

Radiative heating Hk (K day cm−1)

Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)200 600 1000 140010

0060

020

0

Cooling−to−space (K day cm−1)

Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

Thecooling-to-spaceapproximation

200 600 1000 14001000

600

200

Radiative heating Hk (K day cm−1)

Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)200 600 1000 140010

0060

020

0

Cooling−to−space (K day cm−1)

Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

Hk =g

Cp@pFk| {z }Flux

div.

⇡ � g

Cp⇡B(k, T )| {z }

Planck

d⌧kdp|{z}

emissivity

gradient

e�⌧k|{z}trans-

missivity

SimplemodelforH2Ocooling

200 600 1000 14001e−0

41e

+00

1e+0

4 Absorption spectrum, LBL

k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0

Hk, LBL ( K day cm−1 )

Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

200 600 1000 14001e−0

41e

+00

1e+0

4 Absorption spectrum, fit

k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, TheoryPr

essu

re (h

Pa)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0

Hk, Theory ( K day cm−1 )

Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

(k) ⌘ rot

exp

✓�k � k

rot

lrot

◆

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, TheoryPr

essu

re (h

Pa)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)


200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, TheoryPr

essu

re (h

Pa)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

(k) ⌘ rot

exp

✓�k � k

rot

lrot

◆⌧k = (k)

p

prefWVP0 exp

✓� L

RvT

◆

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, TheoryPr

essu

re (h

Pa)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)


200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, TheoryPr

essu

re (h

Pa)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

(k) ⌘ rot

exp

✓�k � k

rot

lrot

◆⌧k = (k)

p

prefWVP0 exp

✓� L

RvT

◆Hk ⇡ � g

Cp⇡B(k, T )

d⌧kdp

e�⌧k

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, TheoryPr

essu

re (h

Pa)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

Simplemodel `emulates’comprehensivemodel (Jeevanjeeetal.2017,JAMES)

Canwedothespectral integral?

H ⇡ g

Cp

Z⇡B(k, T )

d⌧kdp

e�⌧kdk (K/day)

⇡ g

Cp

Z⇡B(k, T )

d ln ⌧kdp| {z }�/p

⌧ke�⌧k

| {z }�(⌧k�1)

dk

d⌧k|{z}lk

d⌧k

⇡ g

Cp⇡B(k1, T )

�

plk

Canwedothespectral integral?

H ⇡ g

Cp

Z⇡B(k, T )

d⌧kdp

e�⌧kdk (K/day)

⇡ g

Cp

Z⇡B(k, T )

d ln ⌧kdp| {z }�/p

⌧ke�⌧k

| {z }�(⌧k�1)

dk

d⌧k|{z}lk

d⌧k

⇡ g

Cp⇡B(k1, T )

�

plk

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, LBL

Pres

sure

(hPa

)

−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

200 600 1000 14001e−0

41e

+00

1e+0


k (cm−1)

κ (m

2kg

)

200 600 1000 14001000

600

200

ln τk, Theory

Pres

sure

(hPa

)−20

−10

0

10

20

k (cm−1)200 600 1000 140010

0060

020

0


Pres

sure

(hPa

)

−0.010

−0.005

0.000

0.005

0.010

k (cm−1)

k1

PlanckEmissivitygradientSpectralwidth

H ⇡ � g

Cp⇡B(k1, T )

�

plkTestformula

−50 0 501000

600

200


Latitude (deg)

Pres

sure

(hPa

)

−3

−2

−1

0

1

2

3

−6 −4 −2 010

0060

020

00


Heating (K/day)

Pres

sure

(hPa

) ECMWFLBL

H ⇡ � g

Cp⇡B(k1, T )

�

plkTestformula

−50 0 501000

600

200


Latitude (deg)

Pres

sure

(hPa

)

−3

−2

−1

0

1

2

3

−6 −4 −2 010

0060

020

00


Heating (K/day)

Pres

sure

(hPa

) ECMWFLBLTheory

Back-of-the-envelope

H ⇡ �✓

g

Cp

◆(⇡B(k1, T ) lk)

✓�H2O

p

◆

⇡ �✓10�4 K/s

W/m2/hPa

◆(20 W/m2)

✓1

100 hPa

◆

= � 2⇥ 10�5 K/s

⇡ � 2 K/day .


2XCO2forcing 4 W/m2 2𝑙1ln2[𝐵% 𝑇+ − 𝐵% 𝑇-./0. ] Logscaling,Ts-Tstrattemp contrast


2K/day −𝑔𝐶6𝜋𝐵 𝑘9 𝑇 , 𝑇

𝛽𝑝𝑙/=.

(Planck) x(emissivity)x

(spectralwidth)


2-3W/m2/K(2-3%/K)

?? ??


OLR

Oceansurface

Space

Atmosphere

Sunlight

BoundaryLayerEvap

Convectio

n

Whatis`the’greenhouseeffect?

Qnet =LP(W/m2)

Modelsrobustlypredict>?@1ABC>DE

≈ 2%KI9 .Why?

Toanswerthis,notethatopticaldepth𝜏% = 𝜅 𝑘 6

6LBMWVPQexp(−

VWXD

)isdominatedbyT-dependence!

Butheatingrate𝐻 = [\]𝜕6𝐹 isaflux-divergence inp-coordinates.

WhatifwechangetoT-coordsanduseCTSapprox?

𝜕D𝐹 ≈ −𝜋𝐵 𝑘9(𝑇),𝑇𝑑ln𝜏%a𝑑𝑇 𝑙%

ThisdependsalmostentirelyonTalone!

!"# isTs-invariant

−5 −4 −3 −2 −1 0

300

260

220

LBL

∂TF (W m2 K)

Tem

pera

ture

(K)

−5 −4 −3 −2 −1 030

026

022

0

Theory

∂TF (W m2 K)

Tem

pera

ture

(K)

OLR

Oceansurface

Space

AtmosphereSunligh

t

BoundaryLayer

Evap

Convection

single-columnLBL,H2Oonly,nocont.

!"# isTs-invariant

−5 −4 −3 −2 −1 0

300

260

220

LBL

∂TF (W m2 K)

Tem

pera

ture

(K)

Ts (K)

270

−5 −4 −3 −2 −1 030

026

022

0

Theory

∂TF (W m2 K)

Tem

pera

ture

(K)

Ts (K)

270

OLR

Oceansurface

Space

AtmosphereSunligh

t

BoundaryLayer

Evap

Convection


!"# isTs-invariant

−5 −4 −3 −2 −1 0

300

260

220

LBL

∂TF (W m2 K)

Tem

pera

ture

(K)

Ts (K)

270280

−5 −4 −3 −2 −1 030

026

022

0

Theory

∂TF (W m2 K)

Tem

pera

ture

(K)

Ts (K)

270280

OLR

Oceansurface

Space

AtmosphereSunligh

t

BoundaryLayer

Evap

Convection


!"# isTs-invariant

−5 −4 −3 −2 −1 0

300

260

220

LBL

∂TF (W m2 K)

Tem

pera

ture

(K)

Ts (K)

270280290

−5 −4 −3 −2 −1 030

026

022

0

Theory

∂TF (W m2 K)

Tem

pera

ture

(K)

Ts (K)

270280290

OLR

Oceansurface

Space

AtmosphereSunligh

t

BoundaryLayer

Evap

Convection


!"# isTs-invariant

−5 −4 −3 −2 −1 0

300

260

220

LBL

∂TF (W m2 K)

Tem

pera

ture

(K)

Ts (K)

270280290300

−5 −4 −3 −2 −1 030

026

022

0

Theory

∂TF (W m2 K)

Tem

pera

ture

(K)

Ts (K)

270280290300

OLR

Oceansurface

Space

AtmosphereSunligh

t

BoundaryLayer

Evap

Convection


ASimplePicturefordQ/dTs

Q

(�@TF )(Ts)

Q

�Q�@TFTs

T

dQ

dTs⇡ (�@TF )(Ts) (W/m2/K)

�Ts

T

�@TF

�Ts

OLR

Oceansurface

Space

Atmosphere

Sunlight

BoundaryLayer

Evap

Convectio

nRadiative-convectiveequilibrium(RCE)

Qnet =LP(W/m2)

Model:DAM(Romps2008)MovieCredit:JacobT.SeeleySurfacecolors=low-levelair temp

Test formula,butinwhatvenue?

Limitedareacloud-resolvingmodel

Howdoesourformulado?

280 290 300 310

4060

8010

012

014

0

Net cooling and precip vs. Ts

Ts (K)

Qne

t, P

(

Wm

2 )

dQ

dTs⇡ (�@TF )(Ts)

280 290 300 310

4060

8010

012

014

0


Ts (K)

Qne

t, P

(

Wm

2 )

●

●

●

●● Qnet


dQ


280 290 300 310

4060

8010

012

014

0


Ts (K)

Qne

t, P

(

Wm

2 )

●

●

●

●● Qnet

P


dQ


280 290 300 310

4060

8010

012

014

0


Ts (K)

Qne

t, P

(

Wm

2 )

●

●

●

●● Qnet

PTheory


dQ


1

P

dP

dTs(300 K) ⇡ 1

Q

dQ

dTs(300 K)

⇡ (�@TF )(300 K)

Q(300 K)

=3 W/m2/K

100 W/m2

= 3% K�1

Intuitionfor2-3%perK?

0 1 2 3 4 5

300

260

220

180

Flux Divergence

−∂TF ( W m2 K )

Tem

pera

ture

(K)

Q

�@TF ⇠ (T � Ttp)� , � ⇡ 1

=) Q(Ts) ⇠ (Ts � Ttp)�+1

=) d lnQ

dTs=

� + 1

Ts � Ttp⇡ 2% K�1 !

Ttp


0 1 2 3 4 5

300

260

220

180

Flux Divergence

−∂TF ( W m2 K )

Tem

pera

ture

(K)

Q

�@TF ⇠ (T � Ttp)� , � ⇡ 1

=) Q(Ts) ⇠ (Ts � Ttp)�+1

=) d lnQ

dTs=

� + 1

Ts � Ttp⇡ 2% K�1 !

Ttp

�@TF ⇠ (T � Ttp)� , � ⇡ 1

=) Q(Ts) ⇠ (Ts � Ttp)�+1

=) d lnQ

dTs=

� + 1

Ts � Ttp⇡ 2% K�1 !


0 1 2 3 4 5

300

260

220

180

Flux Divergence

−∂TF ( W m2 K )

Tem

pera

ture

(K)

Q

�@TF ⇠ (T � Ttp)� , � ⇡ 1

=) Q(Ts) ⇠ (Ts � Ttp)�+1

=) d lnQ

dTs=

� + 1

Ts � Ttp⇡ 2% K�1 !

Ttp

�@TF ⇠ (T � Ttp)� , � ⇡ 1

=) Q(Ts) ⇠ (Ts � Ttp)�+1

=) d lnQ

dTs=

� + 1

Ts � Ttp⇡ 2% K�1 !

Keyfact:1Kincrease inTs is1%increase inatmosphericdepthTs-Ttp





𝛽𝑝𝑙/=.


(spectralwidth)


2-3W/m2/K(2-3%/K)

?? ??






𝛽𝑝𝑙/=.


(spectralwidth)


2-3W/m2/K(2-3%/K)

(−𝜕D𝐹@b.) cDEJeevanjee andRompsPNAS2018

Troposphericdeepening


Thankyou!

ValidationforCO2only

50 150 250 350

−50

050

F4X , CO2 only, LBL

Longitude (deg)

Latit

ude

(deg

)

0

5

10

15Global mean = 7.7 W m2

50 150 250 350

−50

050

F4X , CO2 only, Theory

Longitude (deg)

Latit

ude

(deg

)

0

5

10


0 5 10 15

−50

050

F4X , CO2 only

F4X (W m2)La

titud

e (d

eg)

LBLTheory

F4⇥ = 2 lk ln 4| {z }

�k

2

64 ⇡B(k0

, Ts)| {z }surface

�⇡B(k0

, T (p0


3

75

F4x variationsdeterminedbyTs variations

F4⇥ = 2 lk ln 4| {z }

�k

2

64 ⇡B(k0

, Ts)| {z }surface

�⇡B(k0

, T (p0


3

75

50 150 250 350

−50

050

F4X , CO2 only, LBL

Longitude (deg)

Latit

ude

(deg

)

0

5

10

15W m2

50 150 250 350

−50

050

Surface Temperature

Longitude (deg)

Latit

ude

(deg

)

220240260280300320340

K

Futurework– H2Oeffects

50 150 250 350

−50

050

F4X , CO2 only

Longitude (deg)

Latit

ude

(deg

)

0

5

10


50 150 250 350

−50

050

F4X , all gases

Longitude (deg)

Latit

ude

(deg

)

0

5

10

15Global mean = 5 W m2

0 5 10 15

−50

050

F4X

F4X (W m2)

Latit

ude

(deg

)

CO2 onlyall gases

Simple models for the H2O and CO greenhouse...

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