A Basic One-Dimensional Energy Balance Model

Model Description

As presented in section 3.2.1, Energy Balance Models (EBM) are simple models of the Earth's climate. Their equations are based on the planetary radiation budget. EBM can be zero-dimensional, i.e. they consider the quantities averaged over the whole Earth, or they can include spatial dimensions. The EBM used in the framework of this exercise is one-dimensional. The surface temperature of the Earth is latitudinally resolved.

As shown in fig. 3.3, the Earth's surface is divided in latitude zones in which the temperature is averaged. The index $i$ refers to these areas. The model is governed by the following equation:

$$\underset{\text{ shortwave in}}{\underbrace{S_i\left(1-\alpha \left(T_i\right)\right)}}=\underset{\text{ longwave out}}{\underbrace{R\uparrow \left(T_i\right)}}+\underset{\text{ transport }}{\underbrace{F\left(T_i\right)}}$$

(1)

where $T_i$ is the surface temperature of zone $i$ (measured in $\mathrm{°C}$), $S_i$ and $\alpha \left(T_i\right)$ are respectively the mean annual radiation incident (measured in $W^{}{m}^{-2}$) and the albedo for the latitude zone $i$, $R\uparrow \left(T_i\right)$ is the energy emitted by the Earth and $F\left(T_i\right)$ is the energy transferred between the latitude zone $i$ and its surrounding zones. The value for the solar constant used here equals 1370 $W^{}{m}^{-2}$.

The energy emitted $R\uparrow \left(T_i\right)$ can be evaluated using the Stefan-Boltzmann law for black body radiation. In that case, the radiation emitted to space is proportional to $T^4$. However, since the temperature range of interest (between 250 and 300 K approximately) is relatively small, $R\uparrow \left(T_i\right)$ can be evaluated through a linear relation with the temperature $T_i$. Radiation emitted to space is therefore approximated by:

$$R_i\equiv R\uparrow \left(T_i\right)=A+B{T}_i$$

(2)

where $A$ and $B$ are empirically determined constants designed to account for the greenhouse effect of clouds, water vapour and $C{0}_2$ (McGuffie K. and A. Henderson-Sellers, 2005). $A$ is expressed in $W^{}{m}^{-2}$ while $B$ is expressed in $W^{}{m}^{-2}{\mathrm{°C}}^{-1}$. The rate of transport of energy $F\left(T_i\right)$ is set proportional to the difference between the zonal temperature $T_i$ and the average temperature $\bar{T}$:

$$F_i\equiv F\left(T_i\right)=k_t\left(T_i-\bar{T}\right)$$

(3)

where $k_t$ is the transport coefficient (measured in $W^{}{m}^{-2}{\mathrm{°C}}^{-1}$). Incorporating equations (2) and (3) into (1) gives:

$$T_i=\frac{S_i\left(1-{\alpha }_i\right)+k_t\bar{T}-A}{B+k_t}$$

(4)

with ${\alpha }_i\equiv \alpha \left(T_i\right)$. In this exercise, two configurations for the albedo ${\alpha }_i$ are proposed. In the first one, the albedo ${\alpha }_i$ depends on the temperature $T_i$ as follow:

$${\alpha }_i=\left\{\begin{array}{c}{\alpha }_{\mathrm{ice}}\text{ if }{T}_i\le T_c\hfill \\ {\alpha }_{\mathrm{land}}\text{ if }{T}_i>T_c\hfill \end{array}\right.$$

(5)

where ${\alpha }_{\mathrm{ice}}$ is the albedo of ice (its default value is 0.6) and ${\alpha }_{\mathrm{land}}$ is the albedo of land (its default value is 0.3). The goal is to parameterize in a very simple way the temperature-albedo feedback (see section 4.2.3).

The second configuration uses a constant albedo, equal for all the latitude zone.

In order to study the behaviour of the global temperature as a response to changes in the solar constant, we introduce a new parameter, the fraction of solar constant compared to present-day value, referred to as the luminosity. Equation (4) becomes:

$$T_i=\frac{L{S}_i\left(1-{\alpha }_i\right)+k_t\bar{T}-A}{B+k_t}$$

(6)

where $L$ is the luminosity.

For given values of $A$, $B$ and $k_t$ and for a suitable distribution of the mean annual incoming solar radiation $S_i$ (taking into account the tilt of the Earth's axis), the global temperature $\bar{T}$ is calculated through successive applications of equation (6). This equation is first applied to a first-guess temperature distribution (here we took $T_i=-10$°C $\forall i$) and with the minimum value of the luminosity $L_{\text{ min}}$. After several iterations, a steady state solution is reached. The value of the luminosity $L$ is then increased and the procedure is repeated, starting from the steady state reached previously. Once the maximum value of $L$ has been reached, the procedure is repeated for decreasing values of $L$, starting from the steady state obtained for the maximum value of $L$.

Exercises

After clicking on the button “Launch EBM” below, an applet showing graphs referring to the global temperature should appear. The global temperature is calculated using equation (6) and the methodology described above. Two configurations of the model are proposed:

– Albedo ${\alpha }_i$ depending on the zonal temperature $T_i$, as in equation (5). For that configuration, you can change the values of the temperature $T_c$ and of the albedo of ice ${\alpha }_{\mathrm{ice}}$.
– Constant albedo, equal for all the latitude zones. For that configuration, you can modify the value of the constant albedo.

For each configuration, the global temperature is calculated for increasing as well as for decreasing values of luminosity.

The parameters $A$, $B$, $k_t$, the number of latitude bands and the minimum and maximum values of the luminosity can be modified. These parameters affect the variables of both configurations.

The applet proposed here shoud help you to answer the questions in the following quiz. After answering each question, please check it using the box on the left before going to the next question.

Note: If you are using Internet Explorer or Firefox, you may need to install Java in your browser to run the applet. Here are installation instructions for Internet Explorer and for Firefox.

Useful References

$\left[1\right]$ Budyko M.I. (1969). The effect of solar radiation variations on the climate of the Earth. Tellus 21: 611-619.
$\left[2\right]$ McGuffie K. and A. Henderson-Sellers (2005). A climate modeling primer (third edition). John Wiley & Sons, 280pp.

Acknowledgements

Components of the applet are based on the Java Components for Mathematics developed at Hobart and William Smith Colleges.