| $L_{min}$ | $L_{max}$ | ||||
| $a$ | $b$ | ||||
| $k_{t}$ | Number of latitude bands = | ||||
| Variable albedo: $T_c$ = | Constant albedo: α = | ||||
| Variable albedo: α$_{ice}$ = |
Changing the parameters may lead to a small time delay before the figures adjust to the new values.
Model description
Energy Balance Models (EBM) are simple models of the Earth's climate (Budyko, 1969; Sellers, 1969). 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. The Earth's surface is divided in latitude bands, each referred to as by its index $i$. The model is governed by the following equation:
| $${\sp 1.2em S_i(1-α(T_i)) \sp 1.2em}↙{⏟}↙\text"shortwave in"={\sp 1.2em A↑(T_i) \sp 1.2em}↙{⏟}↙\text"longwave out"+{\sp 1.2em ∆F_{transp}(T_i) \sp 1.2em}↙{⏟}↙\text"transport"$$ | (1) |
where $T_i$ is the mean surface temperature in the band $i$ (measured in °C). $S_i$ and $α(T_i)$ are respectively the mean annual radiation incident (measured in Wm$^{-2}$ and the albedo for the latitude band $i$, $A↑(T_i)$ is the energy emitted by the Earth and $∆F_{transp}(T_i)$ is the energy exchanged between the latitude band $i$ and the surrounding bands. The value for the total solar irradiance used here equals 1360 Wm$^{-2}$.
The energy emitted $A↑(T_i)$ 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, $A↑(T_i)$ can be evaluated using a linear relation function of the temperature $T_i$. The radiation emitted to space is therefore approximated by:
| $$A↑_i≣A↑(T_i)=a+bT_i$$ | (2) |
where $a$ and $b$ are empirically determined constants designed to account for the greenhouse effect of clouds, water vapour and $CO_2$ (McGuffie K. and A. Henderson-Sellers, 2014). $a$ is expressed in Wm$^{-2}$ while $b$ is expressed in Wm$^{-2}$°C$^{-1}$. The rate of transport of energy $∆F_{transp}(T_i)$ is set proportional to the difference between the zonal temperature $T_i$ and the global mean temperature $\ov T$:
| $$F_i≣F(T_i)=k_t(T_i-\ov T)$$ | (3) |
where $k_t$ is the transport coefficient (measured in Wm$^{-2}$°C$^{-1}$). Incorporating equations (2) and (3) into (1) gives:
| $$T_i={S_i(1-α_i)+k_t\ov T-a}/{b+k_t}$$ | (4) |
with $α_i≣α(T_i)$. In this exercise, two configurations for the albedo $α_i$ are proposed. In the first one, the albedo $α_i$ depends on the temperature $T_i$ as follow:
| $$α_i=\{\,\cl"ma-join-align"{\table{α_{ice}}, \, ,{\text"if " T_i ≤T_c}; {α_{land}}, \, ,{\text"if " T_i >T_c}$$ | (5) |
where $α_{ice}$ is the albedo of ice (its default value is 0.6) and $α_{land}$ is the albedo of land (its default value is 0.3). The goal is to parameterize in a very simple way the snow and ice albedo feedback (also referred to as temperature-albedo feedback).
The second configuration uses a constant albedo, equal for all the latitude bands.
In order to study the behaviour of the global temperature as a response to changes in the total solar irradiance, we introduce a new parameter, the fraction of solar irradiance compared to present-day value, referred to as the luminosity. Equation (4) becomes:
| $$T_i={LS_i(1-α_i)+k_t\ov T-a}/{b+k_t}$$ | (6) |
where $L$
is the luminosity.
Resolution of model equations
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 $\ov T$ 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 for all latitude bands) and with the minimum value of the luminosity $L_{min}$. 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 $L$.
The global temperature is calculated using equation (6) and the methodology described above. Two configurations of the model are proposed:
– The albedo $α_i$ depends 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 $α_{ice}$.
– 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.
Exercises
In order to answer the questions in the following quiz, you can modify several parameters of the model and identify their effect on the state variables. After answering each question, please check it using the box on the left before going to the next question.
Useful references
Budyko M.I. (1969). The effect of solar radiation variations on the climate of the Earth. Tellus, 21: 611–619. doi: 10.1111/j.2153-3490.1969.tb00466.x
McGuffie K. and A. Henderson-Sellers (2014). A climate modeling primer (fourth edition). John
Wiley & Sons, 456pp.
Sellers W. D. (1969). A Global Climatic Model Based on the Energy Balance of the Earth-Atmosphere System. J. Appl. Meteor., 8, 392–400.
doi: 10.1175/1520-0450(1969)008&<0392:AGCMBO&>2.0.CO;2