Energy balance model

Definition

Let $S_0 > 0$ be the solar constant, the flux of solar radiation at the mean distance of the Earth from the Sun. A zero-dimensional energy balance model is the ordinary differential equation $$C \frac{dT}{dt} = \frac{S_0}{4} \big( 1 - \alpha(T) \big) - E(T).$$ Here $T(t)$ is the global mean surface temperature and $C > 0$ is a heat capacity per unit area. The planetary albedo is $\alpha(T) \in [0, 1]$, and the outgoing infrared flux $E(T)$ is an increasing function of $T$. Common choices are $E(T) = \varepsilon \sigma T^4$, with $\sigma$ the Stefan–Boltzmann constant and an effective emissivity $0 < \varepsilon \le 1$, or the linear fit $E(T) = A + B T$ with $B > 0$.

In a one-dimensional energy balance model, $T(y, t)$ depends on $y = \sin(\text{latitude}) \in [-1, 1]$, and $$C \frac{\partial T}{\partial t} = \frac{S_0}{4} s(y) \big( 1 - \alpha(y, T) \big) - E(T) + \frac{\partial}{\partial y} \Big( D (1 - y^2) \frac{\partial T}{\partial y} \Big).$$ Here $s(y) \ge 0$ is the distribution of annual mean insolation, with mean $1$ over $[-1, 1]$. The constant $D > 0$ is a diffusion coefficient for the transport of heat toward the poles.