Milestone 3 - Constant-coefficient EBM

  1. The 0D0D energy balance model
  2. Analytical solution of the 0D0D-EBM

The 0D0D energy balance model

We recall that the EBM we have derived so far is given by

C(x)∂T∂t+A(CO2)+BT=Ssol(x,t). C(x) \frac{\partial { T}}{\partial { t} } + A(CO_2) + B T = S_{sol}(x,t).

This equation is somewhat difficult to solve analytically, because of the complexity in the solar forcing term. To get a feeling for the analytical behaviour, as a first step, we introduce a simplification and consider a constant coefficient approximation by getting rid of the spatial and temporal dependence of the heat capacity and solar forcing coefficients: C(x)C(x) and Ssol(x,t)=(1−α(x))S(x,t)S_{sol}(x,t) = (1 - \alpha(x)) S(x,t). We do this by computing area averages in space and averages in time to obtain

C‾dTAdt+A(CO2)+B TA=Ssol‾^, \overline{C} \frac{d { T_A}}{d { t} } + A(CO_2) + B\, T_A = \widehat{\overline{S_{sol}}},

where C‾\overline{C} is the spatial average of the heat capacity coefficient, Ssol‾^\widehat{\overline{S_{sol}}} is a spatial and temporal average of the solar forcing term, and TAT_A is an approximation to the average of Earth's temperature.

Note that spatial averaging needs to account for the spherical shape of Earth, i.e., the spherical coordinate system. The computation of the spatial and temporal averages is detailed in Milestone 3 - Averages.

Remark: Often in numerical analysis of (partial) differential equations, the important dimensions of the equations are the spatial dimensions (as these make the numerical approximation computationally expensive). Thus, one only counts the spatial dimensions to characterize the problem. We follow in the same vein and note that the simplified model only depends on time and not on any spatial coordinate after averaging. Hence, we also coin this type of model a 0D0D-EBM.

Analytical solution of the 0D0D-EBM

We can define the steady-state solution (also known as constant equilibrium solution) TeqT_{eq} by assuming dTA/dt=0d T_A / d t =0, to obtain

Teq=Ssol‾^−AB. T_{eq} = \frac{\widehat{\overline{S_{sol}}}- A}{B}.
Remark: This is the same temperature formula that we obtained in our simple radiation energy balance model.

The ordinary differential equation (2) can be recast into

C‾dTAdt=B(Teq−TA(t)), \overline{C} \frac{d { T_A}}{d { t} } = B \left( {T_{eq}} - T_A(t) \right),

and solved analytically as

TA(t)=Teq+(TA(t=0)−Teq)e−t/τ, T_A(t) = T_{eq} + (T_A(t=0) - T_{eq}) e^{-t/\tau},

where τ=C‾/B\tau = \overline{C}/B and TA(t=0)T_A(t=0) is the initial temperature of the system.

For large times, t→∞t \rightarrow \infty, the term e−t/τ→0e^{-t/\tau} \rightarrow 0, which shows that the solution will get to an equilibrium for large times. Depending on the choice of TA(t=0)T_A(t=0), we will converge to TeqT_{eq} from below or from above:

Created by Gregor Gassner and Andrés Rueda-Ramírez with contributions by Simone Chiocchetti, Daniel Bach, Sophia Horak, Philipp Baasch, Benjamin Bolm, Erik Faulhaber, and Luca Sommer. Last modified: April 02, 2026. Website built with Franklin.jl and the Julia programming language.