The greenhouse effect occurs when the heated surface of the Earth radiates infrared (long wavelength) radiation which greenhouse gases such as water vapor and carbon dioxide absorb and heat the air which in turn radiates infrared radiation upward and downward where it is reabsorbed and reradiated. The analysis is carried out first in terms of the simplest version of the model. That simplest version involves a uniform atmosphere in horizon planes. A spherical model is considered elsewhere.

Radiation of intensity I traveling through a medium experiences diminishment according to the Beer-Lambert Law

dI/dz = −κρI

where z is distance through the medium, ρ is a density and κ is an absorption coefficient. The density can be expressed as linear, areal or volume density and the absorption coefficient has to be expressed in the corresponding form. The product κρ is unaffected by the form chosen. For the purposes of later analysis it is convenient to have ρ and κ in terms of volume density.

The level of density depends upon height and the value of κ depends upon the composition of the air; i.e., the proportional amounts of greenhouse gases such as water vapor, carbon dioxide and ozone in the air.

The level of The dimension of the product κρ is inverse length. The values κ and ρ may vary with position z.

The Beer-Lambert Law implies that

I(z) = I(0)exp(−∫₀^zκρds)

The quanty exp(−∫₀^zκρds) is called the transmission coefficient and the quantinty ∫₀^zκρds is called the optical path length. The analysis is improved, conceptually and analyticlly, if the optical path length is used as the length variable. Let x be the optical length (or optical height or optical depth) where

x = ∫₀^zκρds
and
dx = κρdz

The crucial thing is that there is a monotonic function relating x and z.

The atmosphere does not end at some height but trails off with lower and lower density. Thus z may not have have an upper limit but x may nevertheless have a finite level.

Since κρ has dimension of inverse length optical path length x is a dimensionless variable.

In terms of optical path length the Beer-Lambert Law takes the form

dI/dx = −I

The Case of an Absorbing-Emitting Medium

In a medium which absorbs radiant energy the temperature increases and it radiates energy according to the Stefan-Boltzmann formula σT⁴. The Beer-Lambert Law is replaced by the Schwarzchild Equation

dI/dz = −κρI + ½κρR(z)

where R(z)=σT⁴(z). This formula presumes Kirchhoff's Law that the emissivity of a substance is equal to its absorptivity. It also presumes that half of the thermal radiation is in the forward direction.

In terms of the optical path length the Schwarzchild equation takes the form

dI/dx = −I + ½R

Since there is forward flowing radiation and backward flowing radiation the directionality must be taken into account. Let I⁺ be the radiation flux intensity in the direction of increasing optical path length x and I^- the flux intensity in the opposite direction.

The two equations for the flux intensities are

dI⁺/dx = −I⁺ + ½R
and
dI^-/dx = I^- − ½R

For further analysis it is convenient to define two new variables

φ = ½(I⁺ − I^-)
and
μ = ½(I⁺ + I^-)

The reason for these definitions is the equations for the rates of change of these two variables along the optical path take particularly interesting form;

dμ/dx = −φ
dφ/dx = −μ + ½R

Temperature Balance in an Infinitesimal Element

Consider an infinitesimal element of thickness dz and cross-sectional area dA. The net inflow of radiative energy to the element is

dA[(I⁺(z+½dz)−I⁺(z-½dz))−(I^-(z+½dz)−I⁺(z-½dz))]
which reduces to
[(dI⁺/dz)−(dI^-/dz)]dAdz
or further to
(d(I⁺-I^-)/dz)

That net inflow of energy to the infinitesimal element must increase its temperature according to

ρc_p(dT/dt)dAdz

where ρ has to be mass density, c_p is specific heat at constant pressure and t is time.

At equilibrium dT/dt=0 so it must be that

(d(I⁺-I^-)/dz) = 0

This means that (I⁺-I^-) and φ=½(I⁺-I^-) have to be constant over the optical path. The fact that φ is constant is very important because, from above,

dμ/dx = −φ

The solution to this equation for μ is

μ = −φx + constant

The value of the constant can be determined from the knowledge that I^-=0 at the top of the atmosphere, say x=x_max. Since φ=½(I⁺-I^-) this means I⁺=2φ+I^- and hence at x=x_max, I⁺=2φ and hence μ=½(2φ)=φ. Therefore

φ = −φx_max + constant
so
constant = φ[x_max +1]

Therefore the solution for μ is

μ = φ[(x_max−x) +1]

At the Earth's surface x=0 and the value of I⁺ is R₀=σT₀⁴. The value of I^- at x=0 is I⁺−φ so μ=R₀-φ/2 at x=0. The comparison of this value with the solution found above for μ gives

R₀-φ/2 = φ[x_max +1]
which implies that
φ = R₀/[x_max + 3/2]

However the true determining condition is that the outflow of long wavelength energy at the top of the atmosphere must be equal to the net inflow of short wavelength radiation at the top of the atmosphere. At the top of the atmosphere I⁺=2φ and μ=φ. Thus if the net energy input of short wavelength radiation is S then

2φ = S
φ = ½S
which means that
R₀ = ½S[x_max + 3/2]

This latter condition establishes the surface temperature of the Earth.

The fact that φ is constant means that dφ/dx=0 so μ-½R=0. Thus R=2μ. This gives the complete solution to the problem of temperature as a function of optical height; i.e.,

R(x) = S[(x_max−x) +1]
and hence
T(x) = [(S/σ)(x_max−x) +1)]^1/4

where, as stated previously, S is the net energy per unit area received from the Sun.

One interesting implication of the solution is that as x→0, R→ R₀−φ, hence there is a temperature discontinuity at the surface. In other words, the surface is at a higher temperature than the air just above it. This is actually a familiar phenomenon as at the beach where the sand may be too hot to walk upon barefooted while the air temperature is not uncomfortable.

The analysis indicates that the air temperature decreases with altitude always whereas the real atmosphere has temperature decreasing up to a particular level called the tropopause and above that level the temperature increases with altitude (in the stratosphere). The inability to explain the increase in temperature with height is a definite short coming of the model. The model presented above did not take into account the absorption of short wavelength radiation from the sun. This occurs primarily as a result of ozone in the atmosphere. The ozone is more heavily concentrated in the stratosphere. Considering the incoming short wavelength radiation the optical path length would be the optical depth of the atmosphere and the model would predict temperature declining with optical depth which is the same thing as temperature increasing with heighth above the surface. Thus an extension of the model may be able to explain the temperature profile of the stratosphere. The sphericity of the atmosphere may also affect the anomalous behavior of the stratosphere. The Sun's rays which do not intersect the Earth's surface spend more of their time in the stratosphere.

(To be continued.)