Energy Balance Model and Radiosity

An Energy Balance Model (EBM) is a simple tool used to account of the energy flowing in and out of a system. Mathematically, this relationship can be described simply as energy in equals energy out. A proper understanding of the EBM is essential in order to fully understand and appreciate the role the sky plays as a selective surface.

We will accomplish this by creating a Zero Dimensional EBM. This simply represents the energy balance for Earth’s climate if there were no sky. In this model, we treat the entire Earth as a single point with the goal of accounting for a steady state exchange of the energy flux that goes into the Earth and the energy flux that goes out of the earth.

Energy from the Sun to the Earth ( Energy Influx)

$$\text{ Solar Flux }(\mathrm{W})={\mathrm{G}}_{\mathrm{SC}}\times \left(\pi {\mathrm{r}}_{\mathrm{e}}^2\right)$$

Energy from Earth (Energy outflux)

$$\text{ Earth Flux }=\sigma {\mathrm{T}}_{\mathrm{e}}^4\times \left(4\pi {\mathrm{r}}_{\mathrm{e}}^2\right)$$

The EBM dictates that we equate the energy inflow and outflow.

$${\mathrm{G}}_{\mathrm{SC}}\times \left(\pi {\mathrm{r}}_{\mathrm{e}}^2\right)=\sigma {\mathrm{T}}_{\mathrm{e}}^4\times \left(4\pi {\mathrm{r}}_{\mathrm{e}}^2\right)$$

With the above equation, we can easily solve for the steady state temperature (${\mathrm{T}}_{\mathrm{e}}$)

Greybody Accounting

A greybody is one that has a non-zero reflectance. This means that some of the incident irradiance will be reflected. An important concept in developing a model involving a greybody is radiosity (J_e). Radiosity is described as the sum of emitted and reflected radiant energy.

$${\mathrm{J}}_{\mathrm{e}}=\varrho {\mathrm{G}}_{\mathrm{SC}}+ϵ{\mathrm{E}}_{\mathrm{e}}$$

Where $\rho$ is the reflectance of the earth, otherwise known as albedo, and $ϵ$ is the emittance.

In this case, the Earth flux is described by the following equation

$$\text{ Earth Flux }=\varrho {\mathrm{G}}_{\mathrm{SC}}+ϵ{\mathrm{E}}_{\mathrm{e}}\times \left(4\pi {\mathrm{r}}_{\mathrm{e}}^2\right)$$