Thermal Structure of the Troposphere
© Bob Field 2007
The preliminary Excel model of the thermal structure of the troposphere is based on a global energy budget.
The model calculates the flow of energy between the troposphere and the surface of the Earth based on incident
sunlight and atmospheric and surface optical and thermal properties specified by the analyst. The model also
estimates the thermal structure of the troposphere which is defined as the temperature as a function of altitude
including the surface of the planet and ranging to 10 km. The textbook standard temperature profile is described
by the normal lapse rate of 6.5K per km. The global average surface temperature is reported to be 288K and
temperature decreases 6.5K/km typically in the troposphere before rising and falling and rising again in the thin
upper atmosphere where extreme ultraviolet is absorbed by ozone and other gases.
The model fills the niche between simple global energy budgets that treat the atmosphere as a single layer and
advanced research models that are extremely complex. Most of the mass of the atmosphere is in the troposphere
and the troposphere model has many simplifying assumptions especially concerning the upper atmosphere. The
model is preliminary because there are many effects that can be incorporated if time allows and because many
assumptions and approximations need to be investigated further. Specifically, the model ignores the relationship
between composition and optical and thermal properties, distribution of water vapor with altitude, the upper
atmosphere, wavelength-dependent effects other than the broad bands of short vs. long wavelength, latitude
dependent effects, and variations due to clouds and due to surface features like oceans vs. land.
The model was used to investigate the Earth’s atmosphere with and without the benefit of latent heat transport,
the effect of variations in greenhouse gas long wavelength absorption, the effect of a faint young Sun (four
billion years ago), the effect of a highly reflecting surface (Snowball Earth nearly one billion years ago), and the
effects of higher solar flux and extreme greenhouses as found on Venus.
2000
Spectrum of Sunlight observed on Earth
visible
window
scattering by N2, O2 and aerosols
Intensity
1500
sun is directly overhead
no clouds
direct beam only
1000
500
absorption by ozone,
water, and CO2
0
0.3
0.5
UV Visible
1
1.5
Infrared
2
2.5
Wavelength
3
Figure 1 The spectrum of sunlight includes ultraviolet, visible, and near infrared wavelengths much of which is
absorbed or scattered by the atmosphere even without clouds present
Figure 1 shows the spectrum of sunlight incident on the Earth’s atmosphere when the Sun is directly overhead.
On a cloudless day, the direct beam that reaches the Earth’s surface is reduced by scattering losses shown in
blue and by absorption shown in black. The atmosphere of the planet is continuously intercepting a solar flux of
1372 watts per square meter if the slight ellipticity of the Earth’s orbit is ignored. Locally this flux varies from
zero on the side of the rotating planet that is facing away from the Sun to 1372 w/m2 where the Sun is directly
overhead, such as the Equator at local noon on an Equinox.
The nearly spherical planet intercepts a disk of sunlight equal to πR2, where R is the average radius of planet,
which bulges at the Equator due to the rapid rotational velocity. Although this solar power is non-uniformly
distributed over a sphere of surface area 4πR2, the average global flux is 343 w/m2 at all times, 24 hours a day
all year long. The local daily average at the Equator on an Equinox is 1372w/m 2/2 or 686 w/m2. In the winter,
the daily average at a pole is zero. The global average flux is therefore one fourth of the peak or 343 w/m2.
Some of the flux incident on the top of the atmosphere is absorbed or scattered before reaching the surface of
the planet. Absorption and scattering varies with path length through the atmosphere which varies with altitude,
latitude, seasons, and time of day. Absorption and scattering by clouds and by clear skies varies with
wavelength and with composition, especially moisture content. Absorption by the Earth’s surface varies from
deep blue oceans to bare land to vegetation.
Blackbody Radiation
300
5800K
solar energy
absorbed
by Earth
Intensity
250
200
373K
water
boils
150
100
273K
water
freezes
255K
atmosphere
50
0
0
5
10
Wavelength
15
20
25
30

Figure 2 All of the solar energy absorbed by the Earth’s surface and atmosphere is radiated as long wavelength
infrared “blackbody” radiation
Figure 2 shows that the energy radiated in the long wavelength infrared has the same area as the energy
absorbed by the Earth even though their spectral distributions are very different. All of the flux intercepted by
the Earth is returned to space in order to maintain thermodynamic equilibrium. Since the Earth’s albedo is about
30%, the remaining 70% of the incident flux is absorbed by the atmosphere or surface which radiates it away as
blackbody radiation into space. The atmosphere and surface of the Earth transfer energy by radiation,
convection, conduction, and evaporation/condensation of moisture, but only radiation can transport energy into
space. Figure 3 below shows the absorption spectrum of several major triatomic greenhouse gases that trap
outgoing long wavelength radiation and help elevate the average surface temperature of the Earth to 288K well
above the effective atmospheric temperature of 255K.
Greenhouse Gases Absorb Blackbody Radiation
30
288K Earth's surface
Intensity
25
20
CO2
O3
15
H2O
10
255K
atmosphere
5
0
0
5
15
10
20
25
30

Wavelength (microns)
Figure 3 The Earth’s average surface temperature is 288K because of long wavelength absorption by
greenhouse gases
Average Global Energy Budget (W/m2)
343
incident
shortwave flux
ISSW 343
planet
(90 + 16) + (22 + 215) = 343
reflected
outgoing
shortwave flux longwave flux
90 16
22
215
surfaceatmosphere
heat transfer
OLW tropo
OSW space
absorbed and
reflected by
clouds, H2O, O3,
aerosols
atmosphere
emitted by
surface
368
368
absorbed and
emitted by
clouds, H2O,
CO2, aerosols
106
106
68
68
ASW tropo
latent and
sensible heat
flux
reflected
by surface
169
AISW
surface
390
surface
(68) + (368 + 106) = 542
after Salby45, etc.
ASWLW surface
OLW
surface
atmosphere
327
106
ILW
surface
(215) + 327 = 542
169 + 327 = 496 surface 390 + 106 = 496
ILW + OLW tropo
Figure 4 Simplified Average Global Energy Flow Budget annotated with model parameters
Figure 4 is an average global energy budget diagram (simplified from Salby page 45) that quantifies the flow of
energy on Earth. Unlike Salby, this diagram combines the properties of the clear sky and the clouds to provide
an unambiguous interpretation of the budget factors. It is possible to estimate the surface temperature of the
Earth from the energy flux radiated by the surface. Figure 5 shows the thermal structure of the atmosphere
including the normal lapse rate of the troposphere which is the focus of attention in the preliminary model.
Temperature vs. Altitude

Sun
60 miles
Thermosphere
Temperature
of the atmosphere
-130F
50
Mesosphere
40
30 miles
-70F
32F
ozone
layer
20
Stratosphere
10
75% of air
Troposphere
60F
sea level
ocean
After Tarbuck
Figure 5 Thermal structure of the atmosphere showing the 6.5K per km normal lapse rate in the troposphere
Density vs. Altitude in the Troposphere
Troposphere altitude (km)
10
9
8
7
6
5
4
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Density (kg/m3)
Figure 6 Density vs. altitude in the troposphere
The preliminary model of the thermal structure of the troposphere includes inputs for the incident solar flux and
the absorption and scattering properties of the atmosphere and the surface of the Earth. It also has a factor to
account for ocean evaporation which transports latent heat to the atmosphere and a preliminary formula to
represent non-radiative transport within the atmosphere due to convection. The troposphere includes about the
80% of the mass of the atmosphere that is below 10 km altitude and it is a fully convective region unlike the
upper atmosphere.The density chart in Figure 6 shows that most of the atmosphere is in the troposphere and is
the justification for the current emphasis on this region. The twenty points shown in the density chart are used to
weight absorption and scattering fractions in the Excel model.
In our model, the troposphere is divided into 20 layers represented by 20 rows in the Excel spreadsheet. The
model could have been divided into 10 layers or 40 layers or any other number. Energy is transported by
convection as well as electromagnetic radiation throughout the troposphere, so the concept of layers is a
numerical modeling term, not a description of physical features. The model calculates the energy flow in each
layer and estimates the temperature of the Earth’s surface and each layer of the troposphere by using a modified
version of the Stefan Boltzmann blackbody radiation law that accounts for the partial transparency of the
atmosphere. The outputs of the model include all of the fluxes in the energy budget and a temperature profile in
the troposphere which can be compared to the standard lapse rate of 6.5K per km.
The model inputs can be varied to analyze the effect of variations of in the properties of the atmosphere, the
surface of the planet, and the output of the Sun which is known to change over billions of years. The model
relates properties of the planet to energy flow but it does not derive the properties from fundamental
characteristics like the density and composition of the atmosphere, oceans, and land. A more advanced model
could include these relationships but would probably need to include wavelength dependent effects. A more
advanced model could include local variations from the global averages and could include horizontal energy
flow on the Earth’s surface.
Altitude dependent properties can be modeled in different rows of the current model, but the cases analyzed in
this study had properties that only varied with relative atmospheric density. The main goal of the current effort
was to develop a model and to exercise it to investigate the potential for using Excel to model global energy
flow.
energy flow in
the atmosphere
incoming
SW
outgoing
LW
outgoing
SW
convection
conduction
absorb and scatter SW
absorb and emit LW
evaporate water
latent heat
absorb and scatter SW
absorb and emit LW
condense water
convection
absorb and scatter SW
absorb and emit LW
convect moist air
convection
absorb and scatter SW
absorb and emit LW
convect moist air
convection
incoming
LW
conduct heat
troposphere diagram SWLWLH
Figure 7 Schematic diagram of incoming and outgoing SW and LW energy flow, latent heat, and convection
Figure 7 shows a diagram of energy flow summarizing the processes that were modeled. The diagram shows
how quickly a multi-layer model becomes complex. The particles in the atmosphere scatter short wavelength
(ultraviolet, visible, and near infrared) sunlight in all directions, some of it generally upward and some of it
generally downward. The model assumes that each layer scatters SW equal amounts up and down and that this
energy is simply included in the general flow of incoming or outgoing SW radiation and subject to additional
scattering and absorption.
The incoming SW flux is the flux exiting the bottom of a layer and equals the incoming flux from the adjacent
layer above it reduced by the total absorption and half of the scattered incoming of the layer itself. The outgoing
SW flux is the flux exiting the top of a layer and equals the outgoing flux from the adjacent layer below it
reduced by the total absorption and half of the scattered outgoing SW flux of the layer itself. The incoming SW
flux for the top layer is the incident SW flux from the Sun. The outgoing SW flux for the lowest layer is the SW
scattered by the surface.
Layers of the atmosphere absorb SW flux, long wavelength (LW) flux, sensible heat from conduction, and
latent heat (LH) from evaporation/condensation. Regardless of the source, energy absorbed by a layer of the
atmosphere is re-radiated in all directions as LW infrared, some of it generally upwelling and some of it
generally downwelling. In the model, radiant LW flux either goes up or down and it is simply included in the
general flow of LW radiation that is either incoming or outgoing. In the model, LW is either transmitted or
absorbed and re-radiated; it is not scattered by aerosols or particulates or by the surface of the Earth.
The incoming LW flux is the flux exiting the bottom of a layer and equals the incoming flux transmitted and
radiated downward by the adjacent layer above it reduced by the total absorption of the layer itself. The
outgoing LW flux is the flux exiting the top of a layer and equals the outgoing flux transmitted and radiated by
the adjacent layer below it reduced by the total absorption of the layer itself. The incoming LW flux for the top
layer is zero because no LW flux is incident from space. The outgoing LW flux for the lowest layer is the LW
radiated by the surface.
The model includes an input for radiative transport of LW flux from the surface. In the absence of sensible and
latent heat, this factor would be unity. For the baseline global energy budget model, this factor is approximately
0.79 and the remaining 0.21 is transported by latent heat (0.18) and sensible heat (0.03). For a planet with a
solid surface, latent heat is zero and the factor is 0.97. As in all of the preliminary analyses, these factors are
taken as fixed properties of a system and are not related to the composition or other fundamental characteristics
of the system.
Since the model only has a moderate number of layers and they are only partially transmitting, there is a
temperature gradient across the layer which influences the fraction of absorbed energy radiated up vs. down.
There is no simple way to determine the fraction that is upwelling, so reasonable estimates are based on the
assumption that the effective temperature of downwelling radiation is lower than the effective temperature of
upwelling radiation of the layer directly below it.
A
B
C
D
D
W
Density? (kg/m3)
Relative Density
FSSW
SW
Scattering
Fraction
FASW
SW
Absorption
Fraction
E
0.141
F
G
H
I
J
K
L
M
N
H
ISW
SISW
AISW
OSW
SOSW
AOSW
Incoming SW
ISW Scattering
ISW Absorbed
Outgoing SW
OSW Scattering
OSW Absorbed
0
FT
FU
FD
LW
LW Upwelling
LW Transmission Fraction
Downwelling
Fraction
Fraction
altitude
(km)
space
space
0.416
0.441
0.466
0.4975
0.529
0.5625
0.596
0.6335
0.671
0.708
0.745
0.779
0.813
0.8575
0.902
0.9535
1.005
1.0555
1.106
1.15
surface
=AVERAGE(A6:A25)
=A6/A$27
=A7/A$27
=A8/A$27
=A9/A$27
=A10/A$27
=A11/A$27
=A12/A$27
=A13/A$27
=A14/A$27
=A15/A$27
=A16/A$27
=A17/A$27
=A18/A$27
=A19/A$27
=A20/A$27
=A21/A$27
=A22/A$27
=A23/A$27
=A24/A$27
=A25/A$27
1
=AVERAGE(B6:B25)
O
0.045
=C6
=C7
=C8
=C9
=C10
=C11
=C12
=C13
=C14
=C6
=C16
=C17
=C18
=C19
=C20
=C21
=C22
=C23
=C24
0.135
0.0103
=D6
=D7
=D8
=D9
=D10
=D11
=D12
=D13
=D14
=D6
=D16
=D17
=D18
=D19
=D20
=D21
=D22
=D23
=D24
=1-C26
=1-B6*E$2
=1-B7*E$2
=1-B8*E$2
=1-B9*E$2
=1-B10*E$2
=1-B11*E$2
=1-B12*E$2
=1-B13*E$2
=1-B14*E$2
=1-B15*E$2
=1-B16*E$2
=1-B17*E$2
=1-B18*E$2
=1-B19*E$2
=1-B20*E$2
=1-B21*E$2
=1-B22*E$2
=1-B23*E$2
=1-B24*E$2
=1-B25*E$2
0
=E28*E29
=PRODUCT(E6:E15)
=PRODUCT(E16:E25)
0.495
=F6+F$2
=F7+F$2
=F8+F$2
=F9+F$2
=F10+F$2
=F11+F$2
=F12+F$2
=F13+F$2
=F14+F$2
=F15+F$2
=F16+F$2
=F17+F$2
=F18+F$2
=F19+F$2
=F20+F$2
=F21+F$2
=F22+F$2
=F23+F$2
=F24+F$2
0.8
=1-F6
=1-F7
=1-F8
=1-F9
=1-F10
=1-F11
=1-F12
=1-F13
=1-F14
=1-F15
=1-F16
=1-F17
=1-F18
=1-F19
=1-F20
=1-F21
=1-F22
=1-F23
=1-F24
=1-F25
0
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
surface
tropo
Total
balance
343
=I5-J6/2-K6+M6/2
=I6-J7/2-K7+M7/2
=I7-J8/2-K8+M8/2
=I8-J9/2-K9+M9/2
=I9-J10/2-K10+M10/2
=I10-J11/2-K11+M11/2
=I11-J12/2-K12+M12/2
=I12-J13/2-K13+M13/2
=I13-J14/2-K14+M14/2
=I14-J15/2-K15+M15/2
=I15-J16/2-K16+M16/2
=I16-J17/2-K17+M17/2
=I17-J18/2-K18+M18/2
=I18-J19/2-K19+M19/2
=I19-J20/2-K20+M20/2
=I20-J21/2-K21+M21/2
=I21-J22/2-K22+M22/2
=I22-J23/2-K23+M23/2
=I23-J24/2-K24+M24/2
=I24-J25/2-K25+M25/2
=I25
=I5-L5-Q5
=B6*C6*I5
=B7*C7*I6
=B8*C8*I7
=B9*C9*I8
=B10*C10*I9
=B11*C11*I10
=B12*C12*I11
=B13*C13*I12
=B14*C14*I13
=B15*C15*I14
=B16*C16*I15
=B17*C17*I16
=B18*C18*I17
=B19*C19*I18
=B20*C20*I19
=B21*C21*I20
=B22*C22*I21
=B23*C23*I22
=B24*C24*I23
=B25*C25*I24
=B26*C26*I26
=SUM(J6:J25)
=L6
=L7-M6/2-N6+J6/2
=L8-M7/2-N7+J7/2
=L9-M8/2-N8+J8/2
=L10-M9/2-N9+J9/2
=L11-M10/2-N10+J10/2
=L12-M11/2-N11+J11/2
=L13-M12/2-N12+J12/2
=L14-M13/2-N13+J13/2
=L15-M14/2-N14+J14/2
=L16-M15/2-N15+J15/2
=L17-M16/2-N16+J16/2
=L18-M17/2-N17+J17/2
=L19-M18/2-N18+J18/2
=L20-M19/2-N19+J19/2
=L21-M20/2-N20+J20/2
=L22-M21/2-N21+J21/2
=L23-M22/2-N22+J22/2
=L24-M23/2-N23+J23/2
=L25-M24/2-N24+J24/2
=L26-M25/2-N25+J25/2
=J26
=B6*D6*I5
=B7*D7*I6
=B8*D8*I7
=B9*D9*I8
=B10*D10*I9
=B11*D11*I10
=B12*D12*I11
=B13*D13*I12
=B14*D14*I13
=B15*D15*I14
=B16*D16*I15
=B17*D17*I16
=B18*D18*I17
=B19*D19*I18
=B20*D20*I19
=B21*D21*I20
=B22*D22*I21
=B23*D23*I22
=B24*D24*I23
=B25*D25*I24
=B26*D26*I26
=SUM(K6:K25)
=B6*C6*L7
=B7*C7*L8
=B8*C8*L9
=B9*C9*L10
=B10*C10*L11
=B11*C11*L12
=B12*C12*L13
=B13*C13*L14
=B14*C14*L15
=B15*C15*L16
=B16*C16*L17
=B17*C17*L18
=B18*C18*L19
=B19*C19*L20
=B20*C20*L21
=B21*C21*L22
=B22*C22*L23
=B23*C23*L24
=B24*C24*L25
=B25*C25*L26
0
=N29-Q5
P
Q
R
S
T
U
ASWLW
AT
OLW
ILW
TU
TD
TD-TU
SW+LW absorbed flux
SW+LW+NR absorbed flux
upwelling temperature (K)
downwelling temperature (K)
temperature
difference (K)
0.04
=K6+N6+(1-E6)*(Q7+R5)
=K7+N7+(1-E7)*(Q8+R6)
=K8+N8+(1-E8)*(Q9+R7)
=K9+N9+(1-E9)*(Q10+R8)
=K10+N10+(1-E10)*(Q11+R9)
=K11+N11+(1-E11)*(Q12+R10)
=K12+N12+(1-E12)*(Q13+R11)
=K13+N13+(1-E13)*(Q14+R12)
=K14+N14+(1-E14)*(Q15+R13)
=K15+N15+(1-E15)*(Q16+R14)
=K16+N16+(1-E16)*(Q17+R15)
=K17+N17+(1-E17)*(Q18+R16)
=K18+N18+(1-E18)*(Q19+R17)
=K19+N19+(1-E19)*(Q20+R18)
=K20+N20+(1-E20)*(Q21+R19)
=K21+N21+(1-E21)*(Q22+R20)
=K22+N22+(1-E22)*(Q23+R21)
=K23+N23+(1-E23)*(Q24+R22)
=K24+N24+(1-E24)*(Q25+R23)
=K25+N25+(1-E25)*(Q26+R24)
=K26+(1-E26)*R25
=SUM(O6:O25)
=B6*D6*L7
=B7*D7*L8
=B8*D8*L9
=B9*D9*L10
=B10*D10*L11
=B11*D11*L12
=B12*D12*L13
=B13*D13*L14
=B14*D14*L15
=B15*D15*L16
=B16*D16*L17
=B17*D17*L18
=B18*D18*L19
=B19*D19*L20
=B20*D20*L21
=B21*D21*L22
=B22*D22*L23
=B23*D23*L24
=B24*D24*L25
=B25*D25*L26
0
=SUM(N6:N25)
=K27+N27
=K26+N28
0.0000000567
=(1-P$2)*O6+B6*(10-H6)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O7+B7*(10-H7)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O8+B8*(10-H8)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O9+B9*(10-H9)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O10+B10*(10-H10)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O11+B11*(10-H11)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O12+B12*(10-H12)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O13+B13*(10-H13)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O14+B14*(10-H14)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O15+B15*(10-H15)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O16+B16*(10-H16)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O17+B17*(10-H17)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O18+B18*(10-H18)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O19+B19*(10-H19)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O20+B20*(10-H20)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O21+B21*(10-H21)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O22+B22*(10-H22)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O23+B23*(10-H23)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O24+B24*(10-H24)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=(1-P$2)*O25+B25*(10-H25)*(P$2*SUM(O$6:O$25)+(1-F$26)*P$26)/112.29865
=O26
=SUM(P6:P25)
20
outgoing LW
incoming LW
=Q6
=E6*Q7+F6*P6
=E7*Q8+F7*P7
=E8*Q9+F8*P8
=E9*Q10+F9*P9
=E10*Q11+F10*P10
=E11*Q12+F11*P11
=E12*Q13+F12*P12
=E13*Q14+F13*P13
=E14*Q15+F14*P14
=E15*Q16+F15*P15
=E16*Q17+F16*P16
=E17*Q18+F17*P17
=E18*Q19+F18*P18
=E19*Q20+F19*P19
=E20*Q21+F20*P20
=E21*Q22+F21*P21
=E22*Q23+F22*P22
=E23*Q24+F23*P23
=E24*Q25+F24*P24
=E25*Q26+F25*P25
=F26*P26
0
=E6*R5+G6*P6
=E7*R6+G7*P7
=E8*R7+G8*P8
=E9*R8+G9*P9
=E10*R9+G10*P10
=E11*R10+G11*P11
=E12*R11+G12*P12
=E13*R12+G13*P13
=E14*R13+G14*P14
=E15*R14+G15*P15
=E16*R15+G16*P16
=E17*R16+G17*P17
=E18*R17+G18*P18
=E19*R18+G19*P19
=E20*R19+G20*P20
=E21*R20+G21*P21
=E22*R21+G22*P22
=E23*R22+G23*P23
=E24*R23+G24*P24
=E25*R24+G25*P25
=R25+G26*P26
=E27*Q26
=Q5-Q28
=Q29+R26
=(F6*P6/(1-E6)/S$2)^0.25
=(F7*P7/(1-E7)/S$2)^0.25
=(F8*P8/(1-E8)/S$2)^0.25
=(F9*P9/(1-E9)/S$2)^0.25
=(F10*P10/(1-E10)/S$2)^0.25
=(F11*P11/(1-E11)/S$2)^0.25
=(F12*P12/(1-E12)/S$2)^0.25
=(F13*P13/(1-E13)/S$2)^0.25
=(F14*P14/(1-E14)/S$2)^0.25
=(F15*P15/(1-E15)/S$2)^0.25
=(F16*P16/(1-E16)/S$2)^0.25
=(F17*P17/(1-E17)/S$2)^0.25
=(F18*P18/(1-E18)/S$2)^0.25
=(F19*P19/(1-E19)/S$2)^0.25
=(F20*P20/(1-E20)/S$2)^0.25
=(F21*P21/(1-E21)/S$2)^0.25
=(F22*P22/(1-E22)/S$2)^0.25
=(F23*P23/(1-E23)/S$2)^0.25
=(F24*P24/(1-E24)/S$2)^0.25
=(F25*P25/(1-E25)/S$2)^0.25
=(F26*P26/(1-E26)/S$2)^0.25
=AVERAGE(S6:S25)
=(G6*P6/(1-E6)/S$2)^0.25
=(G7*P7/(1-E7)/S$2)^0.25
=(G8*P8/(1-E8)/S$2)^0.25
=(G9*P9/(1-E9)/S$2)^0.25
=(G10*P10/(1-E10)/S$2)^0.25
=(G11*P11/(1-E11)/S$2)^0.25
=(G12*P12/(1-E12)/S$2)^0.25
=(G13*P13/(1-E13)/S$2)^0.25
=(G14*P14/(1-E14)/S$2)^0.25
=(G15*P15/(1-E15)/S$2)^0.25
=(G16*P16/(1-E16)/S$2)^0.25
=(G17*P17/(1-E17)/S$2)^0.25
=(G18*P18/(1-E18)/S$2)^0.25
=(G19*P19/(1-E19)/S$2)^0.25
=(G20*P20/(1-E20)/S$2)^0.25
=(G21*P21/(1-E21)/S$2)^0.25
=(G22*P22/(1-E22)/S$2)^0.25
=(G23*P23/(1-E23)/S$2)^0.25
=(G24*P24/(1-E24)/S$2)^0.25
=(G25*P25/(1-E25)/S$2)^0.25
=T6-S6
=T7-S7
=T8-S8
=T9-S9
=T10-S10
=T11-S11
=T12-S12
=T13-S13
=T14-S14
=T15-S15
=T16-S16
=T17-S17
=T18-S18
=T19-S19
=T20-S20
=T21-S21
=T22-S22
=T23-S23
=T24-S24
=T25-S25
=AVERAGE(T6:T25)
Figure 8 Formulas in preliminary Excel model spreadsheet
Figure 8 shows the preliminary model formulas in their Excel spreadsheet. Each layer of the atmosphere is
represented by a different row, starting at the top of the troposphere and the planetary surface is in the row
below the atmospheric layers. Some of the columns of the spreadsheet have color coded cells to highlight
locations of SW and LW flux formulas.
The inputs are shown in dark red text in pink shaded cells and the key outputs are shown in red text. In addition
to the flux calculations, the last three columns show temperature calculations based on the equation for
blackbody radiation in a partially absorbing medium like the atmosphere. The inputs and outputs are scattered
all over the spreadsheet. It is possible to add altitude dependent effects by altering the rows of some of the
columns instead of equating the rows. In this model, the only altitude dependent effect used is density.
The most important feature of the model is that the cells have many circular references that account for the
feedback loops as LW flux is radiated back and forth between layers of the atmosphere and the surface of the
Earth. Excel has a special option that allows the user to automatically override the circular reference error
message and to iterate the spreadsheet calculation until it meets a convergence criteria or performs a specific
number of iterations.
T1
inputs
FSSW
FSSW surface
FASW
FALW
FULW increment
FULW
FULW surface
ISSW
AT SW+LW+NR
outputs
ISW surface
SISW surface
AISW surface
AISW tropo
OSW space
AOSW tropo
ASW tropo
ASW tropo + surface
ASWLW surface
OLW space
OLW surface
ILW surface
OLW tropo
ILW+OLW tropo
temperature upper tropo
temperature lower tropo
surface temperature
average tropo temperature
SWLWLH
0.045
0.135
0.0103
0.146
0
0.495
0.79
343
0
SWLWLH
195.4
26.4
169.0
55.3
105.8
13.0
68.2
237.2
494.7
237.2
390.8
325.7
223.3
549.0
232.0
283.5
288.1
257.1
inputs
FSSW
FSSW surface
FASW
FALW
FULW increment
FULW
0.79 FULW surface
ISSW
AT SW+LW+NR
SWLWLH
tropo goals
169
106
68
237
496
237
390
327
215
542
220
286
288
255
outputs
ISW surface
SISW surface
AISW surface
AISW tropo
OSW space
AOSW tropo
ASW tropo
ASW tropo + surface
ASWLW surface
OLW space
OLW surface
ILW surface
OLW tropo
ILW+OLW tropo
temperature upper tropo
temperature lower tropo
surface temperature
average tropo temperature
T1
T2
SWLWLH
SWLW
0.045
0.135
0.0103
0.146
0
0.495
0.79
343
0
SWLWLH
195.4
26.4
169.0
55.3
105.8
13.0
68.2
237.2
494.7
237.2
390.8
325.7
223.3
549.0
232.0
283.5
288.1
257.1
0.045
0.135
0.0103
0.146
0
0.495
0.97
343
0
SWLW
195.4
26.4
169.0
55.3
105.8
13.0
68.2
237.2
531.8
237.2
515.8
362.8
218.9
581.7
224.8
295.0
308.8
256.9
G1
G2
2x
0.5x
greenhouse greenhouse
0.045
0.135
0.0103
0.292
0
0.495
0.79
343
0
0.045
0.135
0.0103
0.073
0
0.495
0.79
343
0
2x
0.5x
greenhouse greenhouse
195.4
26.4
169.0
55.3
105.8
13.0
68.2
237.2
717.2
237.2
566.6
548.2
236.9
785.2
227.1
314.5
316.2
275.2
195.4
26.4
169.0
55.3
105.8
13.0
68.2
237.2
367.2
237.2
290.1
198.2
177.6
375.8
242.1
262.6
267.4
250.2
Figure 9 Model inputs and outputs for baseline case T1 and goals and for cases T2, G1, and G2
In order to make it easy to run the model and to interpret the results, the spreadsheet inputs are linked to the
inputs in the first column of a second spreadsheet shown in Figure 9. The cells below the inputs are linked to the
outputs of the spreadsheet. The second column lists ten goals for the output fluxes of the baseline case: these
values correspond to the global energy budget values in Figure 4.
The inputs to the first case T1, named SWLWLH, were manually adjusted to produce outputs that match the
goals displayed in the second column. As implied by the name, T1 is the first troposphere case and includes
short wavelength, long wavelength, and latent heat transport mechanisms. An additional column not shown
includes the fractional differences between the outputs and the goals to help the analyst perform the
optimization. It may be possible to automate this optimization, but the spreadsheet is already extremely
complex due to the iteration of circular references.
The third column of the table in Figure 9 displays the name of the parameter displayed in the row. The
preliminary model includes nine potential inputs, two of which were not used in the cases analyzed. The first
two inputs - FSSW and FSSW surface - are the fraction of short wavelength flux scattered by an atmospheric
layer and by the surface. The atmospheric fraction is subsequently weighted by a relative density factor. The
next two inputs - FASW and FALW – are the fraction of SW and LW flux absorbed by an atmospheric layer.
These are also weighted by a relative density factor in the model.
The FULW increment is an unused parameter intended to increment the fraction of upwelling LW flux between
tropospheric layers. The next two inputs – FULW and FULW surface – specify the fraction of LW flux that
upwells. The downwelling fraction is unity minus the upwelling fraction. ISSW is the incident SW flux from
space, the only input that is not dimensionless but is in watts per square meter and is 343 for the current solar
flux. Finally AT SW+LW+NR is an unused parameter that would enable the analyst to vary the total absorption
incrementally among the atmospheric layers in order to model thermal convection more or less realistically.
The input values for this case are not exactly physically meaningful in that they are averages that do not
correlate with the vertical variation in LW absorption due to water vapor distribution. In addition, they are
averaged over clear skies and clouds, over all latitudes and surface features, and over all short or long
wavelengths. Despite these limitations, the outputs match the ten goals to better than one percent typically.
There may be other combinations of inputs that would produce a similar match especially choices that account
for the variables ignored in this analysis. But this match provides a basis for analyzing other cases in order to
investigate the variation of energy flow with variations in properties of a planet.
The model table has 18 outputs, ten of which correspond to values in the global energy budget. The first two
outputs – ISW surface and SISW surface - are SW fluxes incident on and scattered by the surface, neither of
which appears in budget diagram. The difference between ISW surface and SISW surface is AISW surface, the
incoming SW flux that is absorbed by the surface which matches the 169 w/m2 goal when the SW inputs are
optimally selected. AISW tropo is the incoming SW flux that is absorbed by all of the troposphere layers and is
not explicitly in the budget diagram.
OSW tropo is the outgoing SW flux that reaches space. It nearly matches the 106 w/m 2 goal which corresponds
to an albedo of about 30%. AOSW tropo is the outgoing SW flux absorbed by the troposphere and is not
explicitly in the budget diagram. ASW tropo is the total SW flux absorbed by the troposphere; it equals the sum
of AISW tropo and AOSW tropo and it matches the 68 w/m2 goal. ASW tropo + surface is the sum of AISW
surface and ASW tropo and nearly matches the goal of 237 w/m2.
ASWLW surface is the all important total SW and LW flux absorbed by the surface. It nearly matches the 496
w/m2 goal and it far exceeds the AISW surface value of 169 w/m2 because greenhouse gases trap outgoing
energy and radiate portions of it back to the surface of the Earth producing the global warming effect that is
essential to liquid oceans and to life. Otherwise the average surface temperature would be about 68K colder and
well below freezing. In other words, most of the energy incident on the Earth’s surface is LW from the
atmosphere not sunlight from space. The elevated surface temperature radiates this away and elevates the
temperature of the lower atmosphere, a key to the feedback that makes the LW flux so high.
OLW space and OLW surface are, respectively, the outgoing LW fluxes transmitted to space and radiated from
the surface and match the budget values of 237 and 390 w/m2. The OLW space match is necessary to balance
the global energy flow since OLW space (237) and OSW space (106) must equal ISSW (343) for thermal
equilibrium. Otherwise the planet is warming or cooling in real time not over long time periods.
ILW surface is the incoming LW flux radiated to the surface by the troposphere. OLW tropo is the LW flux
radiated to space by the troposphere and matches the 215 w/m2 goal. This is 22 less than the total radiated to
space because it does not include the portion of OLW space that was originally radiated by the surface and
transmitted by the atmosphere. ILW + OLW tropo is the sum of the incoming LW flux radiated to the surface
by the troposphere and radiated to space by the troposphere and nearly matches the 542 w/m2 goal.
The last four rows of outputs are temperatures rather than fluxes and provide information about the thermal
structure of the troposphere and the planet’s surface. They have corresponding goals but generally matching the
LW flux radiated by the surface determined the surface temperature, so these goals are really known average
values rather than actual goals used to optimize inputs.
The temperature of the upper troposphere and the temperature of the lower troposphere have global average
values of 220K and 286K, which is consistent with the 6.5K normal lapse rate over 10 km. The preliminary
model matches the lower troposphere value well and is somewhat high for the upper troposphere. The surface
temperature in the model matches the global average of 288K as expected. The average troposphere temperature
is close to the known 255K even though the model average is not weighted for relative density.
The other columns in the table summarize all of the inputs and outputs of all of the cases analyzed starting with
the baseline case. Each case can be identified by an alphanumeric number (T1, T2, G1, G2) and a descriptive
name as displayed in the first two rows. A case can be run by pasting its inputs into the first column. Additional
columns can be added and inputs can be modified to represent other cases. The inputs can be pasted into the
first column and the outputs in column one can then be pasted as values into the appropriate column to preserve
the record of the case analysis.
In addition to fluxes, the table also shows the temperatures of the upper and lower troposphere and the planet
surface as calculated by the model using a Stefan Boltzmann type equation (radiated flux is proportional to
temperature to the fourth power). The Excel document has a chart of the thermal structure - temperature
(abscissa) vs. altitude (ordinate) - that can be pasted into a PowerPoint or Word document. Subtables have been
pasted onto other sheets of the document and can also be pasted into other documents. Pasting the charts and
tables as enhanced metafile pictures seems to work well.
The two troposphere cases correspond to the Earth’s average global energy budget and thermal structure as it is
today (T1) and the as it could be if evaporation and condensation did not transfer latent heat from the ocean
surface to the atmosphere (T2). The two cases have identical inputs except for the entry shown in blue which is
in the FULW surface row. The fraction of the long wavelength upwelling from the surface increases from 0.79
to 0.97 if latent heat does not contribute to energy flow from the surface to the atmosphere.
The output table shows that the SW outputs are the same for both cases as expected, but that some of the LW
outputs are higher in the absence of latent heat transport. Most notably is that the outgoing LW surface flux is
estimated to be 516 rather than 391 because the surface is not cooled by evaporation. According to Stefan
Boltzmann’s law, the surface temperature must be more than 7% hotter in order to radiate nearly 32% more heat
away. This explains the nearly 21K difference between the two surface temperatures, 288K vs. 309K, displayed
in the output table. It also explains the elevated temperature of the lower troposphere.
A
B
C
D
D
W
Density
(kg/m3)
Relative
Density
FSSW
SW
Scattering
Fraction
FASW
SW
Absorption
Fraction
space
0.416
0.441
0.466
0.498
0.529
0.563
0.596
0.634
0.671
0.708
0.745
0.779
0.813
0.858
0.902
0.954
1.005
1.056
1.106
1.150
surface
0.74
0.559
0.592
0.626
0.668
0.711
0.756
0.801
0.851
0.901
0.951
1.001
1.047
1.092
1.152
1.212
1.281
1.350
1.418
1.486
1.545
1
1.00
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.045
0.135
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.0103
0.865
E
0.146
F
G
0.000
FT
FU
FD
LW
LW
LW
Transmission Upwelling Downwelling
Fraction
Fraction
Fraction
0.939
0.933
0.928
0.920
0.913
0.904
0.895
0.885
0.875
0.865
0.854
0.844
0.833
0.819
0.805
0.788
0.771
0.753
0.736
0.720
0
0.036
0.370
0.096
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.790
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0
H
I
J
K
L
M
N
H
ISW
SISW
AISW
OSW
SOSW
AOSW
altitude
(km)
Incoming
SW
ISW
Scattering
ISW
Absorbed
Outgoing
SW
OSW
Scattering
OSW
Absorbed
space
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
surface
tropo
Total
balance
343
338.0
332.8
327.3
321.6
315.6
309.3
302.7
295.8
288.7
281.3
273.6
265.8
257.8
249.6
241.1
232.4
223.4
214.2
204.8
195.4
195.4
8.6
9.0
9.4
9.8
10.3
10.7
11.1
11.6
12.0
12.4
12.7
12.9
13.1
13.4
13.6
13.9
14.1
14.3
14.3
14.2
26.4
241.4
2.0
2.1
2.1
2.3
2.4
2.5
2.6
2.7
2.7
2.8
2.9
2.9
3.0
3.1
3.1
3.2
3.2
3.3
3.3
3.3
169.0
55.3
105.8
105.8
103.4
100.8
98.1
95.3
92.3
89.2
85.9
82.4
78.7
74.8
70.8
66.7
62.4
57.9
53.2
48.3
43.1
37.8
32.2
26.4
2.6
2.7
2.8
2.9
3.0
3.0
3.1
3.2
3.2
3.2
3.2
3.1
3.1
3.0
2.9
2.8
2.6
2.4
2.2
1.8
0.0
0.6
0.6
0.6
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.4
0.0
13.0
68.2
237.2
0.000
0.000
O
S
T
U
5.67E-08
TU
TD
TD-TU
SW+LW SW+LW+NR
upwelling
downwelling temperature
outgoing incoming
absorbed
absorbed
temperature temperature difference
LW
LW
flux
flux
(K)
(K)
(K)
237.2
0
17.3
20.2
237.2
10.2
232.0
233.1
1.2
19.8
22.9
242.0
21.1
234.1
235.2
1.2
22.6
25.8
247.1
32.6
236.3
237.5
1.2
26.1
29.6
252.6
45.0
238.6
239.8
1.2
30.1
33.8
258.5
58.1
241.1
242.3
1.2
34.8
38.7
265.0
72.1
243.7
244.9
1.2
40.0
44.2
271.9
86.9
246.4
247.6
1.2
46.2
50.7
279.2
102.5
249.2
250.5
1.2
53.2
57.9
287.0
118.9
252.2
253.4
1.3
60.8
65.8
295.3
136.0
255.2
256.4
1.3
69.2
74.4
303.9
153.7
258.2
259.5
1.3
78.0
83.4
312.8
171.8
261.3
262.6
1.3
87.5
93.1
321.8
190.2
264.3
265.6
1.3
99.5
105.4
330.8
209.1
267.2
268.6
1.3
112.5
118.8
340.0
228.4
270.1
271.5
1.4
128.0
134.7
349.2
248.1
273.0
274.4
1.4
144.8
151.8
358.4
267.9
275.8
277.2
1.4
162.6
169.9
367.4
287.7
278.5
279.9
1.4
181.4
189.1
376.0
307.1
281.1
282.5
1.4
199.4
207.4
383.9
325.7
283.5
284.9
1.4
494.7
494.7
390.8
325.7
288.1
1613.8
1717.7
257.1
258.4
20
13.9
223.3
549.0
ASWLW
P
0.000
AT
Q
R
OLW
ILW
Figure 10 Numerical values of Excel spreadsheet model for baseline case T1
Figure 10 shows the numerical values of the Excel spreadsheet model for the baseline case. The most interesting
outputs of this table are the upwelling and downwelling temperatures for the 20 model layers of the troposphere.
Figure 11 is an Excel chart for case T1 showing these temperatures plotted on the abscissa with the altitude on
the ordinate. For each layer the upwelling temperature (blue curve) is slightly less than the downwelling
temperature (red curve) because each layer is partially absorbing and has a temperature gradient across it. The
green dotted line show the normal lapse rate which is also graphed on page 237 of Salby. The match is not bad
considering the approximations and simplifications in the preliminary model and its inputs and both curves are
fairly linear over most of the troposphere.
Thermal Structure of Troposphere
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
Salby237
4
3
2
1
0
210
220
230
240
250
260
270
280
290
300
310
320
Temperature (K)
Figure 11 Excel chart for baseline case T1
To be physically reasonable, the red points must be lower than the blue points on the layer below them. The
analyst’s selection of the LW upwelling fraction influences these curves and many other model outputs. For
now these fractions were arbitrarily selected and the curves were checked to insure that they are physically
reasonable. In order to be more realistic, the model will need significant improvements in this area.
Thermal Structure of Troposphere
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
Salby237
4
3
2
1
0
210
220
230
240
250
260
270
280
290
Temperature (K)
Figure 12 Excel chart for case T2 no latent heat
300
310
320
Figure 12 shows the temperature chart for case T2 which does not include any latent heat or convection. The
substantial rise in surface temperature is readily apparent and the lapse rate does seem to be less linear as
indicated by other curves on page 237 of Salby that are not shown on this chart.
Figure 9 also included two cases labeled G1 and G2 in which variations in greenhouse gases are analyzed. The
fractional absorption of LW flux per layer (FALW) for G1 is two times that of T1 whereas G2 is half of T1. No
other input values are different and latent heat is included. Once again since none of the SW inputs have
changed, none of the SW outputs have changed from T1.
The outgoing LW flux to space is still 237 w/m2 because thermal equilibrium with the incident solar flux is still
maintained. What has changed is the LW flux flowing within the troposphere and to and from the surface of the
planet. The incoming LW flux that the surface absorbs goes from the baseline 327 to 548 and 198 respectively
for the two cases. This translates to a surface temperature change from 288K to 316K or 267K, neither of which
would be very favorable to stable liquid oceans and to life on the surface of the planet.
Thermal Structure of 2X Greenhouse
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
Salby237
4
3
2
1
0
210
220
230
240
250
260
270
280
290
300
Temperature (K)
Figure 13 Excel chart for case G1 increased LW absorption
310
320
Thermal Structure of 0.5X Greenhouse
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
Salby237
4
3
2
1
0
210
220
230
240
250
260
270
280
290
300
310
320
Temperature (K)
Figure 14 Excel chart for case G2 decreased LW absorption
Figures 13 and 14 show the dramatic changes in thermal structure resulting from changes in LW absorption.
The temperature of the 2X case rises rapidly (from left to right) with decreasing altitude near the Earth’s surface
while the temperature of the 0.5X case rises very slowly with decreasing altitude.
single layer
T
A
0.99
0.01
0.98
0.02
0.96
0.04
0.92
0.08
0.84
0.16
0.68
0.32
0.36
0.64
10 layers
T^10
Atotal10
0.90
0.10
0.82
0.18
0.66
0.34
0.43
0.57
0.17
0.83
0.02
0.98
3.7E-05 0.99996
20 layers
T^20
Atotal20
0.82
0.18
0.67
0.33
0.44
0.56
0.19
0.81
0.03
0.97
0.0004
0.9996
1.3E-09 1.000000
Figure 15 Doubling the absorption per layer does not double the total absorption
Figure 15 is a table that is not directly related to the Excel model but is intended to clarify the meaning of a 2X
case. The case does not involve doubling of greenhouse gases necessarily because the preliminary model is not
based on actual optical properties of atmospheric constituents. The 2X is the absorption per layer and the effect
on thermal structure and energy flow depends on the number of layers chosen for the model and on the
magnitude of the absorption.
For low absorption media such as the atmosphere in the visible portion of the solar spectrum, the total
atmospheric absorption varies nearly linearly with layer absorption. For highly absorbing media such as the
atmosphere across most of the long wavelength infrared spectrum, very little flux is transmitted through
multiple layers and doubling the absorption per layer greatly reduces the transmission but does not double the
already very high absorption. Because these effects are highly non-linear it is difficult to use intuition to
interpret the meaning of these input changes without a lot of thoughtful consideration.
inputs
FSSW
FSSW surface
FASW
FALW
FULW increment
FULW
FULW surface
ISSW
AT SW+LW+NR
outputs
ISW surface
SISW surface
AISW surface
AISW tropo
OSW space
AOSW tropo
ASW tropo
ASW tropo + surface
ASWLW surface
OLW space
OLW surface
ILW surface
OLW tropo
ILW+OLW tropo
temperature upper tropo
temperature lower tropo
surface temperature
average tropo temperature
T1
E1
E2
S1
SWLWLH
4BYA
4BYA + 2x
greenhouse
1BYA
snowball
0.045
0.135
0.0103
0.146
0
0.495
0.79
343
0
SWLWLH
195.4
26.4
169.0
55.3
105.8
13.0
68.2
237.2
494.7
237.2
390.8
325.7
223.3
549.0
232.0
283.5
288.1
257.1
0.045
0.135
0.0103
0.146
0
0.495
0.79
250
0
4BYA
142.4
19.2
123.2
40.3
77.1
9.4
49.7
172.9
360.6
172.9
284.9
237.4
162.8
400.2
214.3
261.9
266.2
237.5
0.045
0.135
0.0103
0.292
0
0.495
0.79
250
0
4BYA + 2x
greenhouse
142.4
19.2
123.2
40.3
77.1
9.4
49.7
172.9
522.8
172.9
413.0
399.6
172.7
572.3
209.9
290.6
292.1
254.3
0.045
0.6
0.0103
0.146
0
0.495
0.97
320
0
1BYA
snowball
209.2
125.5
83.7
54.1
154.1
28.1
82.2
165.9
324.1
165.9
314.3
240.4
154.7
395.1
209.0
264.4
272.9
235.1
S2
1BYA
snowball +
1.4x
greenhouse
0.045
0.6
0.0103
0.204
0
0.495
0.97
320
0
1BYA
snowball +
1.4x
greenhouse
209.2
125.5
83.7
54.1
154.1
28.1
82.2
165.9
411.8
165.9
399.4
328.1
162.9
491.0
206.4
281.6
289.7
243.2
V
Venus
0.310
0.1
0.006
0.6
0
0.470
0.97
656
0
Venus
117.6
11.8
105.9
45.8
477.6
26.7
72.5
178.4
2378.6
178.4
2307.3
2272.8
178.4
2451.2
207.9
431.7
449.1
310.4
Figure 16 Model inputs and outputs for baseline case T1 and cases E1, E2, S1, S2, and V
Figure 16 is a table that includes the baseline case T1 and the five other cases analyzed. E1 and E2 represent
conditions that could have been prevalent on early Earth 4 billion years ago (4BYA) at a time when the Sun was
about 25% less intense and the composition of the atmosphere was far different than today. Both cases show
250 w/m2 of incident solar flux instead of 343. The second case E2 also shows the same doubling of the LW
flux absorption fraction per layer that was analyzed in G1 to represent a hypothetical atmosphere that had more
carbon dioxide because bacterial photosynthesis had not yet removed most of the carbon dioxide from the
atmosphere.
All of the SW fluxes have changed as a result of the lower solar flux, but the 30% albedo is unchanged. For case
E1 the surface only absorbs 360 w/m2 of SW and LW flux because both are lower. In fact E1 resembles the
0.5X greenhouse case previously analyzed and the surface temperature of 266K is very close to the G2 case of
267K. This time the E2 case with higher LW absorption has a surface absorption of 523 which is similar to the
T1 baseline and its surface temperature is a more favorable 292K. These points are also illustrated by the
thermal structure charts of Figures 17 and 18 for E1 and E2 respectively.
Thermal Structure of 4BYA
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
Salby237
4
3
2
1
0
210
220
230
240
250
260
270
280
290
300
310
320
310
320
Temperature (K)
Figure 17 Excel chart for case E1 Early Earth
Thermal Structure of 4BYA + 2X Greenhouse
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
Salby237
4
3
2
1
0
210
220
230
240
250
260
270
280
290
300
Temperature (K)
Figure 18 Excel chart for case E2 Early Earth with enhanced LW absorption
The next two cases in Figure 16 are S1 and S2 and represent a period in the Earth’s history possibly a billion
years ago when the oceans were apparently completely frozen. The Snowball Earth scenarios require three or
four changes in inputs. First and foremost is the high reflectance of ice. In the model the typical 0.135
reflectance (FSSW surface) that has been used as an average of the deep blue sea and the somewhat lighter land
is replaced by 0.6. Second the latent heat mechanism is unavailable if water is frozen and cannot evaporate so
FULW surface is 0.97 instead of 0.79. Third the Sun was slightly less intense a billion years ago, so ISSW is
only 320 w/m2.
With the freezing of oceans, photosynthesis must have decreased sharply and carbon dioxide emitted by
volcanic activity would have increased LW absorption in the atmosphere over time, bringing about a lifesaving
global warming recovery from Snowball Earth. Case S2 represents this later condition with an arbitrarily
selected 1.4X increase in LW flux absorption which just happens to raise the surface temperature from 272.9K
which is freezing to 289.7K which resembles current values and would gradually melt ice in the tropics and in
the temperate zone.
The SW flux incident on the surface is slightly higher than baseline despite the dimmer Sun because S1 and S2
reflect so much light from the surface that more of it is scattered back by the atmosphere. The SW flux scattered
into space is much higher as expected but the albedo only increases from 30% to 48% despite the surface
reflection of 60% because of all of the losses in the atmosphere for the incoming and outgoing paths of sunlight.
In fact the high reflectance of sunlight increases the absorption of sunlight in the atmosphere despite the lower
incident solar flux. The lower surface absorption of SW flux reduces the LW fluxes for S1. The surface only
absorbs 324 w/m2 of SW and LW flux instead of the baseline 496 and the lack of evaporative cooling only
partially offsets this shortfall. For case S2, the increased LW absorption in the atmosphere provides sufficient
surface heating to raise the total absorbed flux to 411.8, but absent latent heat losses, this is sufficient to elevate
the surface temperature. At some point, the melting of surface waters will have two effects: it will permit latent
heat to cool the surface, but it will have pools of water that absorb more sunlight than ice does. It does not take
much water depth for the warming effect of absorption to exceed the cooling effect of evaporation.
Figures 19 and 20 illustrate the thermal structure of S1 and S2. The most interesting feature is that the surface
warms up while the upper atmosphere becomes slightly cooler in case S2 compared to case S1.
Thermal Structure of Snowball
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
Salby237
4
3
2
1
0
210
220
230
240
250
260
270
280
290
300
Temperature (K)
Figure 19 Excel chart for case S1 Snowball Earth
310
320
Thermal Structure of Snowball + 1.4X Greenhouse
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
Salby237
4
3
2
1
0
210
220
230
240
250
260
270
280
290
300
310
320
Temperature (K)
Figure 20 Excel chart for case S2 Snowball Earth with enhanced LW absorption
The last case in Figure 16 is labeled V for Venus. In order to exercise the preliminary model under extremely
non-Earth-like conditions, the approximate state of Venus was analyzed. The results were not terribly accurate
but the trend is reasonable and the inputs were not necessarily realistic anyway. For Venus all seven non-zero
inputs were altered from the baseline Earth.
The obvious change is that Venus is much closer to the Sun so the incident solar flux was increased to 656 w/m2
since flux varies as the square of distance. The second effect is that Venus is so hot that the surface is dry so
latent heat was not included. The third effect is that Venus has a much more massive atmosphere so all of the
absorption factors were increased. The SW scattering was adjusted to match a high albedo of Venus of 73%,
which is typical of recently reported values. Surface scatter and absorption were reduced but scattering by dense
clouds is high. Because the atmosphere is approximately 90% carbon dioxide, LW absorption is very high. The
large resulting temperature gradient across highly absorbing layers required that the layer upwelling flux
fraction be reduced somewhat to avoid non-physical output fluxes.
It proved to be very difficult to develop a consistent and reasonable set of inputs. The results are quite
interesting. The surface of Venus is known to be visibly dim despite the proximity to the Sun and the 117.6
w/m2 flux incident on the surface is slightly more than half that of Earth. Despite that, the greenhouse effect
deposits 2378.6 w/m2 to the surface in case V which is nearly five times the Earth’s 496 and explains the greatly
elevated surface temperature despite the low level of sunlight.
Case V predicted a surface temperature of 449K, far higher than Earth but far lower than the known value for
Venus of about 750K. This outcome is apparent from Figure 21 which required a major scale change at the high
temperature end and surprisingly a slight decrease at the low end to accommodate the cool upper atmosphere
predicted by case V.
Thermal Structure of Venus
downwelling temperature (K)
upwelling temperature (K)
10
9
altitude (km)
8
7
6
5
4
3
2
Salby237
1
0
200 220 240 260 280 300 320 340 360 380 400 420 440 460 480
Temperature (K)
Figure 21 Excel chart for case V Venus
For the record the Excel spreadsheet is also included in Figure 22 for this case. An accurate model of Venus
will require much more consideration.
0.600
D
W
Density
(kg/m3)
Relative
Density
space
0.416
0.441
0.466
0.498
0.529
0.563
0.596
0.634
0.671
0.708
0.745
0.779
0.813
0.858
0.902
0.954
1.005
1.056
1.106
1.150
surface
0.74
0.559
0.592
0.626
0.668
0.711
0.756
0.801
0.851
0.901
0.951
1.001
1.047
1.092
1.152
1.212
1.281
1.350
1.418
1.486
1.545
1
1.00
FSSW
SW
Scattering
Fraction
FASW
SW
Absorption
Fraction
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.310
0.100
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.0060
0.900
0.000
FT
FU
FD
LW
LW
LW
Transmission Upwelling Downwelling
Fraction
Fraction
Fraction
0.749
0.726
0.703
0.672
0.641
0.606
0.570
0.529
0.486
0.443
0.399
0.358
0.315
0.258
0.200
0.130
0.059
-0.013
-0.087
-0.152
0
0.000
0.006
0.000
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.470
0.970
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0.530
0
H
ISW
SISW
AISW
OSW
SOSW
AOSW
altitude
(km)
Incoming
SW
ISW
Scattering
ISW
Absorbed
Outgoing
SW
OSW
Scattering
OSW
Absorbed
space
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
surface
tropo
Total
balance
656
637.0
617.3
596.8
575.3
552.9
529.5
505.3
480.2
454.1
427.3
399.8
371.8
343.3
314.0
283.8
252.6
220.4
187.2
152.9
117.6
117.6
113.7
117.0
119.8
123.6
126.7
129.5
131.4
133.3
134.2
133.9
132.6
129.7
125.9
122.6
117.9
112.7
105.7
96.9
86.2
73.2
11.8
2366.7
2.2
2.3
2.3
2.4
2.5
2.5
2.5
2.6
2.6
2.6
2.6
2.5
2.4
2.4
2.3
2.2
2.0
1.9
1.7
1.4
105.9
45.8
477.6
477.6
462.3
446.4
429.9
412.4
394.1
375.0
355.1
334.2
312.5
289.9
266.6
242.6
217.9
192.3
165.7
137.7
108.4
77.7
45.4
11.8
80.1
82.0
83.4
85.4
86.8
87.9
88.1
88.2
87.3
85.5
82.7
78.7
73.8
68.7
62.2
54.7
45.4
34.1
20.9
5.6
0.0
1.6
1.6
1.6
1.7
1.7
1.7
1.7
1.7
1.7
1.7
1.6
1.5
1.4
1.3
1.2
1.1
0.9
0.7
0.4
0.1
0.0
26.7
72.5
178.4
0.000
0.000
ASWLW
0.000
AT
OLW
SW+LW SW+LW+NR
outgoing
absorbed
absorbed
LW
flux
flux
178.4
54.6
56.5
178.4
74.8
76.9
202.7
99.3
101.5
229.2
131.4
133.8
258.3
170.7
173.2
290.7
219.5
222.2
326.8
278.6
281.4
367.0
353.0
356.0
411.7
443.0
446.3
462.0
550.8
554.2
518.5
679.7
683.3
582.0
828.5
832.3
653.4
1004.5
1008.4
733.3
1237.7
1241.8
823.2
1523.5
1527.8
927.9
1899.4
1904.0
1051.1
2377.0
2381.8
1200.8
2988.3
2993.3
1385.6
3790.9
3796.2
1617.9
4821.4
4827.0
1917.5
2378.6
2378.6
2307.3
23526.5
23597.9
20
0.0
178.4
5.67E-08
TU
TD
TD-TU
upwelling
downwelling temperature
incoming
temperature temperature difference
LW
(K)
(K)
(K)
0
30.0
207.9
214.3
6.3
62.5
219.7
226.4
6.7
97.7
230.7
237.7
7.0
136.6
241.2
248.5
7.4
179.3
251.4
259.1
7.7
226.4
261.5
269.4
8.0
278.2
271.4
279.7
8.3
335.9
281.3
289.9
8.6
399.9
291.3
300.2
8.9
471.0
301.4
310.6
9.2
550.2
311.6
321.1
9.5
637.9
321.9
331.7
9.8
735.4
332.4
342.5
10.1
848.0
343.2
353.7
10.5
979.0
354.7
365.5
10.8
1136.5
367.0
378.2
11.2
1329.1
380.6
392.2
11.6
1569.0
395.6
407.7
12.1
1875.9
412.5
425.1
12.6
2272.8
431.7
444.8
13.2
2272.8
449.1
310.4
319.9
ILW
2451.2
Figure 22 Numerical values of Excel spreadsheet model for case V Venus