# Nonsmooth Invariant Manifolds in a Conceptual Climate Model

## TL;DR

This paper investigates a conceptual climate model to understand the stability of tropical ice sheets and the conditions under which open water can persist at the equator during ice ages, using advanced mathematical techniques.

## Contribution

It introduces a novel analysis of a coupled temperature-ice line model, proving the existence of a stable tropical ice line state through nonsmooth invariant manifolds.

## Key findings

- Existence of a stable tropical ice line state with open water at the equator.
- Identification of bifurcations related to CO2 levels and heat transport efficiency.
- Application of spectral and Hadamard graph transform methods to analyze nonsmooth dynamics.

## Abstract

There is widespread agreement that ice sheets flowed into the ocean in tropical latitudes at sea level during the Earth's past. Whether these extreme ice ages were snowball Earth events, with the entire surface covered in ice, or whether ocean water remained ice free in regions about the equator, continues to be controversial. For the latter situation to occur, the effect of positive ice albedo feedback would have to be damped to stabilize an advancing ice sheet shy of the equator. In this paper we analyze a conceptual model comprised of a zonally averaged surface temperature equation coupled to a dynamic ice line equation. This difference equation model is aligned with the cold world of these great glacial episodes through an appropriately chosen albedo function. Using the spectral method, the analysis leads to a nonsmooth singular perturbation problem. The Hadamard graph transform method is applied to prove the persistence of an invariant manifold, thereby providing insight into model behavior. A stable climate state with the ice line resting in tropical latitudes, but with open water about the equator, is shown to exist. Also presented are local smooth and nonsmooth bifurcations as parameters related to atmospheric CO$_2$ concentrations and the efficiency of meridional heat transport, respectively, are varied.