Nonsmooth frameworks for an extended Budyko model

TL;DR

This paper extends an energy balance climate model to include dynamic carbon dioxide levels, developing a nonsmooth mathematical framework that accounts for physical boundaries and proves key properties like existence, uniqueness, and invariance of solutions.

Contribution

It introduces a novel nonsmooth framework for climate models with dynamic carbon dioxide, incorporating physically relevant boundary dynamics and establishing fundamental mathematical properties.

Findings

Proved existence and uniqueness of solutions.

Established invariance of phase space boundaries.

Developed a physically consistent nonsmooth framework.

Abstract

In latitude-dependent energy balance models, ice-free and ice-covered conditions form physical boundaries of the system. With carbon dioxide treated as a bifurcation parameter, the resulting bifurcation diagram is nonsmooth with curves of equilibria and boundaries forming corners at points of intersection. Over long time scales, atmospheric carbon dioxide varies dynamically and the nonsmooth diagram becomes a set of quasi-equilibria. {However, when introducing carbon dynamics, care must be taken with the physical boundaries and appropriate boundary motion specified. In this article, we extend an energy balance model to include slowly varying carbon dioxide and develop a nonsmooth framework based on physically relevant boundary dynamics. Within this framework, we prove existence and uniqueness of solutions, as well as invariance of the region of phase space bounded by ice-free and ice-covered states.