The Nature of the Dependence of Equilibrium Temperature on the Concentration of Greenhouse Gases

San José State University Department of Economics

applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

The Nature of the Dependence of Equilibrium
Temperature on the Concentration of Greenhouse Gases

This is an investigation of how temperature changes when the concentration of greenhouse gases increases. A linear relationship would say that if the concentration of greenhouse gases increases by an amount x and the temperature increases by y then doubling the increase in greenhouses gases to 2x would produce an increase of 2y. A logarithmic relationship would say that if a doubling of the concentration produced an increase in temperature of z degrees, it would take an additional doubling of the concentrations to produce an increase of an additional z degrees. In effect what is sought is the functional form of the relationship between the concentration of greenhouse gases and surface temperature. The concentration of various greenhouse gases must be reduced to a single figure that represents the amounts of the various gases in terms of the equivalent amount of one single greenhouse gas, say water vapor.
The Dynamics of Temperature
Let T be the absolute temperature, t time and H(t) the heat energy input to a body. A body with an absolute surface temperature T radiates an amount of energy proportional to the fourth power of T, T⁴. The equation for the dynamics of temperature is then
C(dT/dt) = H(t) − Aσ T⁴.

where C is the heat capacity of the body, A is the area of the body and σ is the Stefan-Boltzmann constant. This is the dynamics of temperature in the absence of a greenhouse effect.
With a greenhouse effect some proportion b of the outgoing thermal radiation is absorbed and some proportion c is radiated back to the surface. The value of c could be ½ but at this point it will be left arbitrary. This radiation could go up, down or sideways.
With a greenhouse effect the dynamic equation is then
C(dT/dt) = H(t) − AσT⁴ + bcAT_a⁴.

wherer T_a is the temperature of the atmosphere.
The dynamic equation for the temperature of the atmosphere is
C_a(dT_a/dt) = bAσT⁴ − AσT_a⁴.

From this point on the body in question will be the surface of a sphere, such as the Earth.
The amount of energy absorbed from sunlight, H(t), is proportional to the cross section area of the sphere, which is one fourth of the total area A of the sphere. If a is the albedo, the proportion of incoming solar radiation which is reflected, and ψ is the solar intensity then H(t)=¼A(1-a)ψ. The dynamic equation for temperature is then
C(dT/dt) = ¼A(1-a)ψ − AσT⁴ + cbAσT_a⁴
or, equivalently
(C/A)(dT/dt) = ¼(1-a)ψ − σT⁴ + cbσT_a⁴

Clearly a relevant parameter for the dynamics of temperature is the heat capacity per unit area, (C/A).
The dynamic equation of the temperature of the atmosphere is
(C_a/A)(dT_a/dt) = bσT⁴ −T_a⁴.

The Relationship Between the Amount
of Radiation Absorbed and the
Concentration of Greenhouses Gases

The Beer-Lambert Law implies that the amount of radiation absorbed is
(1 − exp(−∫₀^Dkg(z)dz))

where k is a parameter characteristic of the medium the thermal radiation is passing through and D is the depth of the medium. The concentration g(z) is the molecular density of the greenhouse gases weighted for their radiative efficiencies. The above formula can be rearranged to
(1 − exp(−kG))

where G=∫₀^Dg(z)dz is the weighted amount of greenhouse gases in the atmosphere above a unit area of surface. The amount of energy not absorbed by the medium and that escapes to space is then exp(−kG) . The amount absorbed and radiated back to the surface is then c(1 − exp(−kG)). The dynamic equations for the temperatures are then
(C/A)(dT/dt) = ¼(1-a)ψ −σT⁴ + c(1 − exp(−kG))σT_a⁴

(C_a/A)(dT_a/dt) = (1 − exp(−kG))σT⁴ − σT_a⁴.

Equilibrium Temperatures

If (dT_a/dt)=0 then
(1 − exp(−kG))σT⁴ = σT_a⁴.

The expression on the left may be substituted for the expression on the right in the first dynamic equation. Thus if (dT/dt)=0 then
¼(1-a)ψ = σT⁴ − c(1 − exp(−kG))²σT⁴
or, equivalently
T⁴ = (¼(1-a)ψ/σ)/[1 − c(1 − exp(−kG))²]

The equilibrium temperature is then given by
T = (¼(1-a)ψ/σ)^1/4 /[1 − c(1 − exp(−kG))²]^1/4

The shape of this function is shown below.

This is then the functional relationship between the concentration of greenhouse gases and the absolute temperature of the surface of a sphere. Note that as the concentration of greenhouse gas increases without limit the surface temperature approaches a limit.
Since by a previous formula
T_a = T[1-exp(-kG)]^1/4

the absolute temperature of the atmosphere is
T_a =
(¼(1-a)ψ/σ)^1/4[1-exp(-kG)]^1/4 /[1 − c(1 − exp(−kG))²]^1/4

The plot of this function is as follows.

The Effect of an Infinitesimal Increase
in Greenhouse Gas Concentration

The effect of an infinitesimal increase in G can most conveniently be obtained by differentiating
T⁴ = (¼(1-a)ψ/σ)/[1 − c(1 − exp(−kG))²]

Thus
4T³(dT/dG) = (¼(1-a)ψ/σ)[−2(−k*exp(-kG))]/[1 − c(1 − exp(−kG))³]
or, equivalently
(dT/dG) = (¼(1-a)ψ/σ)[k*exp(-kG))]/{2T³[1 − c(1 − exp(−kG))³]}

This relationship is as shown below.

Thus T increases with increases in G but at an ever decreasing rate. The relationship expressed as the change in T for a proportional increases in G, dG/G, is
G(dT/dG) = ((1/8)(1-a)ψ/σ)[kG*exp(-kG))]/{T³[1 − c(1 − exp(−kG))³]}

The term kG*exp(-kG) rises to a peak and then falls asymptotically to zero. The term T⁴ in the denominator further decreases the proportional effect on T of a proportional increase in G. This is shown below.
(To be continued.)

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins

The Dynamics of Temperature

C(dT/dt) = H(t) − Aσ T4.

C(dT/dt) = H(t) − AσT4 + bcATa4.

Ca(dTa/dt) = bAσT4 − AσTa4.

C(dT/dt) = ¼A(1-a)ψ − AσT4 + cbAσTa4 or, equivalently (C/A)(dT/dt) = ¼(1-a)ψ − σT4 + cbσTa4

(Ca/A)(dTa/dt) = bσT4 −Ta4.

The Relationship Between the Amount of Radiation Absorbed and the Concentration of Greenhouses Gases

(1 − exp(−∫0Dkg(z)dz))

(1 − exp(−kG))

(C/A)(dT/dt) = ¼(1-a)ψ −σT4 + c(1 − exp(−kG))σTa4 (Ca/A)(dTa/dt) = (1 − exp(−kG))σT4 − σTa4.

Equilibrium Temperatures

(1 − exp(−kG))σT4 = σTa4.

¼(1-a)ψ = σT4 − c(1 − exp(−kG))²σT4 or, equivalently T4 = (¼(1-a)ψ/σ)/[1 − c(1 − exp(−kG))²]

T = (¼(1-a)ψ/σ)1/4 /[1 − c(1 − exp(−kG))²]1/4

Ta = T[1-exp(-kG)]1/4

Ta = (¼(1-a)ψ/σ)1/4[1-exp(-kG)]1/4 /[1 − c(1 − exp(−kG))²]1/4

The Effect of an Infinitesimal Increase in Greenhouse Gas Concentration

T4 = (¼(1-a)ψ/σ)/[1 − c(1 − exp(−kG))²]

4T3(dT/dG) = (¼(1-a)ψ/σ)[−2(−k*exp(-kG))]/[1 − c(1 − exp(−kG))³] or, equivalently (dT/dG) = (¼(1-a)ψ/σ)[k*exp(-kG))]/{2T3[1 − c(1 − exp(−kG))³]}

G(dT/dG) = ((1/8)(1-a)ψ/σ)[kG*exp(-kG))]/{T3[1 − c(1 − exp(−kG))³]}

C(dT/dt) = H(t) − Aσ T⁴.

C(dT/dt) = H(t) − AσT⁴ + bcAT_a⁴.

C_a(dT_a/dt) = bAσT⁴ − AσT_a⁴.

C(dT/dt) = ¼A(1-a)ψ − AσT⁴ + cbAσT_a⁴
or, equivalently
(C/A)(dT/dt) = ¼(1-a)ψ − σT⁴ + cbσT_a⁴

(C_a/A)(dT_a/dt) = bσT⁴ −T_a⁴.

The Relationship Between the Amount
of Radiation Absorbed and the
Concentration of Greenhouses Gases

(1 − exp(−∫₀^Dkg(z)dz))

(C/A)(dT/dt) = ¼(1-a)ψ −σT⁴ + c(1 − exp(−kG))σT_a⁴

(C_a/A)(dT_a/dt) = (1 − exp(−kG))σT⁴ − σT_a⁴.

(1 − exp(−kG))σT⁴ = σT_a⁴.

¼(1-a)ψ = σT⁴ − c(1 − exp(−kG))²σT⁴
or, equivalently
T⁴ = (¼(1-a)ψ/σ)/[1 − c(1 − exp(−kG))²]

T = (¼(1-a)ψ/σ)^1/4 /[1 − c(1 − exp(−kG))²]^1/4

T_a = T[1-exp(-kG)]^1/4

T_a =
(¼(1-a)ψ/σ)^1/4[1-exp(-kG)]^1/4 /[1 − c(1 − exp(−kG))²]^1/4

The Effect of an Infinitesimal Increase
in Greenhouse Gas Concentration

T⁴ = (¼(1-a)ψ/σ)/[1 − c(1 − exp(−kG))²]

4T³(dT/dG) = (¼(1-a)ψ/σ)[−2(−kexp(-kG))]/[1 − c(1 − exp(−kG))³]
or, equivalently
(dT/dG) = (¼(1-a)ψ/σ)[kexp(-kG))]/{2T³[1 − c(1 − exp(−kG))³]}

G(dT/dG) = ((1/8)(1-a)ψ/σ)[kG*exp(-kG))]/{T³[1 − c(1 − exp(−kG))³]}