In latitude-dependent energy balance models, ice-free and icecovered 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 nonsmooth frameworks based on physically relevant boundary dynamics. Within these frameworks, 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.
|Original language||English (US)|
|Number of pages||17|
|Journal||Discrete and Continuous Dynamical Systems - Series B|
|State||Published - Aug 2017|
Bibliographical noteFunding Information:
This research was supported in part by the Mathematics and Climate Research Network and NSF grants DMS-0940366, DMS-0940363. AB was also supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation. We thank the members of the MCRN Paleoclimate and Nonsmooth Systems seminar groups for many useful discussions, especially Mary Lou Zeeman and Emma Cutler. We are also grateful to Mike Jeffrey, Rachel Kuske, Andrew Roberts, and the anonymous referee for their insights and suggestions.
- Conceptual climate models
- Nonsmooth dynamical systems
- Snowball Earth