Variational Structure of Energy-Balance Climate Models with Heterogeneous Insolation

Abstract

We analyse stationary Sellers-type energy-balance climate models with heterogeneous insolation $Q_\varepsilon = Q_*+\varepsilon Q_1$, both on the interval $(-1,1)$ with degenerate diffusion $κ(x)=1-x^2$ and on $\mathbb{S}^2$ with uniformly elliptic diffusion. Assuming that the homogeneous model corresponding to $\varepsilon = 0$ admits three constant equilibria $θ_S <θ_M < θ_W$ satisfying suitable non-degeneracy and spectral conditions, we prove that, for sufficiently small $\vert \varepsilon \vert $, these equilibria continue to three stationary branches $ u_S^\varepsilon$, $u_M^\varepsilon$, and $u_W^\varepsilon$. The cold and warm branches, $u_S^\varepsilon$ and $u_W^\varepsilon$ respectively, are strict local minimisers of the associated energy functional, while the intermediate branch $u_M^\varepsilon$ is non-degenerate and has Morse index one. We further establish the Palais-Smale condition and obtain a mountain-pass critical point $u^\varepsilon_{MP}$ at the minimax level between the two local minima. Under an additional isolation assumption for the homogeneous problem at the minimax level, this critical point coincides with $u_M^\varepsilon$. Lastly, a Legendre-Galerkin discretisation of the one-dimensional model numerically illustrates the three stationary branches, their discrete Morse indices, and the corresponding energy landscape.