# A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport

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## Abstract

**:**

## 1. Introduction

## 2. Results

## 3. Model Equations

#### 3.1. The Zonal Average Surface Temperature Equation

#### 3.2. The Spectral Approach

#### 3.3. A Dynamic Ice Line

#### 3.4. Incorporating Accumulation and Ablation Zones

## 4. The Flip-Flop and Associated Filippov Flow

## 5. A Unique Nonsmooth Limit Cycle

#### 5.1. Stable Virtual Equilibria

**Definition**

**1.**

#### 5.2. Tangencies on the Switching Manifold

#### 5.3. The Filippov Return Map and a Unique Limit Cycle

**Theorem**

**1.**

**Proof.**

## 6. Feedbacks and Meridional Flux at the Albedo Line

## 7. Bifurcation Scenarios

## 8. Forcing with Milankovitch Cycles

## 9. Discussion and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**${\delta}^{18}$O is a proxy for global ice volume, with larger values corresponding to more ice and colder periods. Data from https://lorraine-lisiecki.com/stack.html.

**Figure 2.**Plots of ${h}_{G}$ (blue) and ${h}_{I}$ (red), where the diffusion coefficients satisfy ${D}_{G}<{D}_{I}$. (

**a**,

**b**) $N=1,{D}_{G}=0.3,{D}_{I}=0.394$; (

**c**) $N=3,{D}_{G}=0.3$ (blue), ${D}_{I}=0.394$ (dashed), ${D}_{I}=0.43$ (red, solid).

**Figure 3.**The ice line position at equilibrium as a function of the diffusion coefficient D when $N=1$. Solid: Stable equilibria. Dashed: Unstable equilibria.

**Figure 4.**The snow line $\eta $ separates the ablation zone from the accumulation zone. The variable $\xi $ denotes the ice edge.

**Figure 6.**The Filippov flow set-up. Above $\Sigma $ and to the right of ${L}_{1}$, ${\varphi}_{G}$-trajectories move left and downward (blue arrows). Below $\Sigma $ and to the left of ${L}_{2}$, ${\varphi}_{I}$-trajectories move right and upward (red arrows). The blue and red dashed lines are the $\xi $-nullcines for systems (20) and (21), respectively. Vector fields ${V}_{G}$ (blue) and ${V}_{I}$ (red) are schematically indicated on $\Sigma $. We refer the reader to the text for further details.

**Figure 7.**The limit cycle $\Gamma $ for system (23) ($\rho =0.1R,\u03f5=0.03R$). (

**a**) $N=1,{D}_{G}=0.3,{D}_{I}=0.394$. (

**b**) $N=3,{D}_{G}=0.3,{D}_{I}=0.43$. The magenta curve is a plot of the critical mass balance expression $a(1-\eta )-b(\eta -\xi )$ evolving with time.

**Figure 8.**Red: The meridional transport flux (watts) at the albedo line $\eta $ during the limit cycle pictured in Figure 6a. The flux function has been scaled and translated for this plot. Brown: $\eta $. Green: $\xi $.

**Figure 9.**(

**a**) the limiting maximum and minimum values of the norm of trajectories as a function of ${b}_{G}$ for the discontinuous model; (

**b**) as in (

**a**), but for the smooth approximation of the model with $M=10$ in (28).

**Figure 10.**The maximum ice sheet size increases as ${D}_{G}$ decreases. Blue: ${D}_{G}=0.385$. Red: ${D}_{G}=0.36$. Green: ${D}_{G}=0.3$. Brown: ${D}_{G}=0.23$.

**Figure 11.**A typical trajectory with eccentricity and obliquity forcing, computed from 2 million years ago to the present. The model continues to exhibit glacial cycle behavior. ${D}_{G}=0.3,{D}_{I}=0.38.$

**Figure 12.**

**Top**: $\eta $ (brown) and $\xi $ (green) over the past 1 million years, forced by both obliquity and eccentricity.

**Bottom**: Obliquity (blue) and the poleward flux $\ell (\eta )$ (27) (red, scaled) at the albedo line $\eta $. ${D}_{G}=0.3,{D}_{I}=0.38.$

Parameter | Value | Units | Parameter | Value | Units |
---|---|---|---|---|---|

Q | 343 | Wm${}^{-2}$ | ${T}_{c}$ | $-10$ | ${}^{\circ}$C |

A | 202 | Wm${}^{-2}$ | ${b}_{G}$ | 1.5 | dimensionless |

B | 1.9 | Wm${}^{-2}{{(}^{\circ}\mathrm{C})}^{-1}$ | b | 1.75 | dimensionless |

${D}_{G}$ | 0.3 | Wm${}^{-2}{{(}^{\circ}\mathrm{C})}^{-1}$ | ${b}_{I}$ | 4 | dimensionless |

${D}_{I}$ | 0.394 | Wm${}^{-2}{{(}^{\circ}\mathrm{C})}^{-1}$ | a | 1.05 | dimensionless |

${\alpha}_{1}$ | 0.32 | dimensionless | R | $0.5\times {10}^{9}$ | J/(m${}^{2}{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C) |

${\alpha}_{2}$ | 0.62 | dimensionless | $\beta $ | 23.4${}^{\circ}$ |

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Walsh, J.; Widiasih, E. A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport. *Mathematics* **2020**, *8*, 316.
https://doi.org/10.3390/math8030316

**AMA Style**

Walsh J, Widiasih E. A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport. *Mathematics*. 2020; 8(3):316.
https://doi.org/10.3390/math8030316

**Chicago/Turabian Style**

Walsh, James, and Esther Widiasih. 2020. "A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport" *Mathematics* 8, no. 3: 316.
https://doi.org/10.3390/math8030316