Variational techniques for a one-dimensional energy balance model

Del Sarto, G. ORCID: https://orcid.org/0009-0006-3257-6903, Bröcker, J. ORCID: https://orcid.org/0000-0002-0864-6530, Flandoli, F. ORCID: https://orcid.org/0000-0003-2837-3784 and Kuna, T. (2024) Variational techniques for a one-dimensional energy balance model. Nonlinear Processes in Geophysics, 31 (1). pp. 137-150. ISSN 1607-7946 doi: 10.5194/npg-31-137-2024

Abstract/Summary

A one-dimensional climate energy balance model (1D EBM) is a simplified climate model for the zonally averaged global temperature profile, based on the Earth's energy budget. We examine a class of 1D EBMs which emerges as the parabolic equation corresponding to the Euler–Lagrange equations of an associated variational problem, covering spatially inhomogeneous models such as with latitude-dependent albedo. Sufficient conditions are provided for the existence of at least three steady-state solutions in the form of two local minima and one saddle, that is, of coexisting “cold”, “warm” and unstable “intermediate” climates. We also give an interpretation of minimizers as “typical” or “likely” solutions of time-dependent and stochastic 1D EBMs. We then examine connections between the value function, which represents the minimum value (across all temperature profiles) of the objective functional, regarded as a function of greenhouse gas concentration, and the global mean temperature (also as a function of greenhouse gas concentration, i.e. the bifurcation diagram). Specifically, the global mean temperature varies continuously as long as there is a unique minimizing temperature profile, but coexisting minimizers must have different global mean temperatures. Furthermore, global mean temperature is non-decreasing with respect to greenhouse gas concentration, and its jumps must necessarily be upward. Applicability of our findings to more general spatially heterogeneous reaction–diffusion models is also discussed, as are physical interpretations of our results.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/115721
Item Type	Article
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics Science > School of Mathematical, Physical and Computational Sciences > Department of Meteorology
Publisher	European Geosciences Union
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