Energy budget and vertical temperature distribution of a 2-layer graybody atmosphere over a blackbody surface including convective heat fluxes

Consider a simple planet with two isothermal layers of graybody atmosphere in radiative equilibrium with a blackbody surface.

Assume the atmosphere is transparent to solar radiation, the lower atmospheric layer has broadband thermal emissivity denoted by \(\epsilon_1\), and the upper atmospheric layer has broadband thermal emissivity denoted by \(\epsilon_2\). Let the planetary albedo be \(\alpha\), and the solar flux at teh top of the atmosphere be designated \(S_0\).

The temperature of the surface, the lower atmospheric layer, and the upper atmospheric layer are denoted \(T_S\), \(T_1\), and \(T_2\) respectively.

Assume that all nonradiative energy fluxes (e.g., turbulence, convection, evaporation, condensation) can be represented as generalized net “convective” heat fluxes \(H_1\) (from the surface to the lower atmosphere) and \(H_2\) (from the lower to the upper atmosphere. (Of course, there can’t be any convective fluxes out the top of the upper atmosphere!)

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Emission:

All emission is isotropic. By the Stefan-Boltzmann Law, the surface emits thermal infrared radiation at the rate \(\sigma T_S^4\). The lower atmospheric layer emits energy both upward and downward at the rate \(\epsilon_1 \sigma T_1^4\), and the upper atmospheric layer emits energy at the rate \(\epsilon_2 \sigma T_2^4\).

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Absorption:

By Kirchoff’s Law the absorptivity of each atmospheric layer in thermal infrared wavelengths is precisely the same as its emissivity. Since the surface is a blackbody, it absorbs all radiation incident on it.

The surface absorbs energy from the Sun at a rate \(\frac{S_0}{4}(1-\alpha)\). It also absorbs energy at a rate \(\epsilon_1 \sigma T_1^4\) from the lower atmospheric layer. Since a fraction \(\epsilon_2\) of the longwave radiation emitted downward by the upper atmospheric layer is absorbed by the lower atmospheric layer, the surface absorbs this radiation at a rate \((1-\epsilon_1)\epsilon_2 \sigma T_2^4\).

The lower atmospheric layer is transparent to solar radiation, but absorbs a fraction \(epsilon_1\) of the longwave radiation incident upon it. From the surface, it absorbs upwelling energy at the rate \(\epsilon_1 \sigma T_S^4\), and from the upper layer it absorbs downwelling energy at the rate \(\epsilon_1 \epsilon_2 \sigma T_2^4\).

The upper atmospheric layer is also transparent to solar radiation, but absorbs a fraction \(\epsilon_2\) of the longwave radiation incident upon it. Since a fraction \(\epsilon_1\) of the longwave radiation emitted upward by the surface is absorbed by the lower atmospheric layer (see previous paragraph), the upper layer absorbs upwelling energy emitted from the surface at the rate \(\epsilon_2 (1-\epsilon_1) \sigma T_S^4\). From the lower atmospheric layer, the upper layer absorbs upwelling energy at the rate \(\epsilon_1 \epsilon_2 \sigma T_1^4\).

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Radiation Budgets:

Using the assumption that the surface and each atmospheric layer is in radiative equilibrium, we can now write a radiation flux budget for the surface, for each atmospheric layer, and for the planet as a whole (defined at the top of the atmosphere). In each of the energy balances that follow, sources of energy for the layer are grouped on the left-hand sides, and losses to the right of the equal signs:

\[ \begin{array}{lrclr} Surface: & \frac{S_0}{4}(1-\alpha) + \epsilon_1 \sigma T_1^4 + (1-\epsilon_1) \epsilon2 \sigma T_2^4 &=& \sigma T_S^4 + H_S & (1)\\ Lower Layer: & \epsilon_1 \sigma T_S^4 + \epsilon_1 \epsilon_2 \sigma T_2^4 + H_S &=& 2 \epsilon_1 \sigma T_1^4 + H_L & (2)\\ Upper Layer:& (1-\epsilon_1) \epsilon_2 \sigma T_S^4 + \epsilon_1 \epsilon_2 \sigma T_1^4 + H_L &=& 2 \epsilon_2 \sigma T_2^4 & (3)\\ \end{array} \]

The energy budgets (1) through (3) constitute a system of three independent equations in three unknowns: \(T_S\), \(T_1\), and \(T_2\).

Adding equations (1) through (3) provides a statement that for the planet as a whole the absorbed solar flux is balanced by the total outgoing longwave radiation (OLR) at the top of the atmosphere (TOA):

\[ \begin{array}{lrclr} Planet (TOA):& \frac{S_0}{4}(1-\alpha) &=& (1-\epsilon_1) (1-\epsilon_2) \sigma T_S^4 + \epsilon_1 (1-\epsilon_2) \sigma T_1^4 + \epsilon_2 \sigma T_2^4. & (4)\\ \end{array} \]

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Solution:

We can rearrange equations (1) through (3) to obtain an equivalent matrix equation:

\[ \begin{bmatrix} -1 & \epsilon_1 & (1-\epsilon_1)\epsilon_2 \\ \epsilon_1 & -2 \epsilon_1 & \epsilon_1 \epsilon_2 \\ (1-\epsilon_1) \epsilon_2 & \epsilon_1 \epsilon_2 & -2 \epsilon_2 \end{bmatrix} \times \left[ \begin{array}{c} T_S^4 \\ T_1^4 \\ T_2^4 \end{array} \right] = \frac{1}{\sigma} \left[ \begin{array}{c} H_s-\frac{S_0}{4}(1-\alpha) \\ H_L-H_s \\ -H_L \end{array} \right]. \]

To solve, we multiply the inverse of the square matrix by the vector on the right-hand side and then take the fourth root of each element of the resulting vector to obtain each temperature.

This is not as complicated as it sounds. To see how easy this is to set up and solve in R, see the “Website Code” tab to the right!