Variational Structure of Energy-Balance Climate Models with Heterogeneous Insolation
Abstract
We analyse stationary Sellers-type energy-balance climate models with heterogeneous insolation Qε = Q*+ε Q1, both on the interval (−1,1) with degenerate diffusion κ(x)=1−x2 and on 𝕊2 with uniformly elliptic diffusion. Assuming that the homogeneous model corresponding to ε = 0 admits three constant equilibria θS <θM < θW 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$, uMε, and uWε. The cold and warm branches, uSε and uWε respectively, are strict local minimisers of the associated energy functional, while the intermediate branch uMε is non-degenerate and has Morse index one. We further establish the Palais-Smale condition and obtain a mountain-pass critical point uεMP 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 uMε. 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.
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BibTeX
@misc{sarto2026variational,
title = {{Variational Structure of Energy-Balance Climate Models with Heterogeneous Insolation}},
author = {Gianmarco Del Sarto and Giuseppe Floridia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33797},
eprint = {2610.33797},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33797},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Variational Structure of Energy-Balance Climate Models with Heterogeneous Insolation
AU - Gianmarco Del Sarto
AU - Giuseppe Floridia
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33797
UR - https://arxiv.org/abs/2610.33797
AB - We analyse stationary Sellers-type energy-balance climate models with heterogeneous insolation Q_ε = Q_*+ε Q_1, both on the interval (−1,1) with degenerate diffusion κ(x)=1−x^2 and on 𝕊^2 with uniformly elliptic diffusion. Assuming that the homogeneous model corresponding to ε = 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^ε, and u_W^ε. The cold and warm branches, u_S^ε and u_W^ε respectively, are strict local minimisers of the associated energy functional, while the intermediate branch u_M^ε is non-degenerate and has Morse index one. We further establish the Palais-Smale condition and obtain a mountain-pass critical point u^ε_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^ε. 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.
ER -