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Nonsmooth frameworks for an extended Budyko model

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  • 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 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.

    Mathematics Subject Classification: Primary:34A12, 34A36;Secondary:35Q86, 86A40, 86A60.

    Citation:

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  • Figure 1.  Bifurcation diagrams from energy balance models illustrating hysteresis in the climate system. In each figure, solid lines correspond to stable steady states while dashed lines correspond to unstable steady states. The positive horizontal axis can be thought of as increasing atmospheric carbon dioxide, and the vertical axis is the latitude of the ice line. Stability of snowball and ice-free states is inferred; these are physical boundaries and not true equilibria of the equations

    Figure 2.  The physical region of the phase space and possible fixed points of the system given by the $\eta$-nullcline, $h=0$. The location of the equilibrium is determined by the critical effective area of exposed land $0<\eta_c<1$. Solid black portions of the curve represent stable equilibria while the dashed lines denote unstable equilibria. Solid black portions of the boundary are attractive sliding regions and dashed boundaries are crossing regions

    Figure 3.  Attractors of the system when (a) $\eta_c=0.85$ and (b) $\eta_c=0.6$. The $+$ symbol marks the initial condition and the horizontal long-dashed line is the $A$-nullcline. In (a), the orbit reaches the ice-free state and slides until it reaches the intersection of the folded curve with this boundary. It then enters the physical region and approaches the small ice cap equilibrium. In (b), the fixed point is unstable and the orbit oscillates between the ice-free and ice-covered boundaries. Parameters are as in Table 1 and $\delta=0.01$. Simulations were performed using Mathematica 9

    Figure 4.  Periodic orbits of the Jormungand system when (a) $\eta_c=0.8$ and (b) $\eta_c=0.15$. The folded curve is the $\eta$-nullcline $h_J(A, \eta)=0$ and dashing is as in Figures 2 and 3. Parameters are as in Table 3 and $\delta=0.01$

    Table 1.  Parameter values as in [1]

    Parameters Value Units
    $Q$ 321 $\text{W}\text{m}^{-2}$
    $s_1$ 1 dimensionless
    $s_2$ -0.482 dimensionless
    $B$ 1.5 $\text{W}\text{m}^{-2}\text{K}^{-1}$
    $C$ 2.5B $\text{W}\text{m}^{-2}\text{K}^{-1}$
    $\alpha_1$ 0.32 dimensionless
    $\alpha_2$ 0.62 dimensionless
    $T_c$ $-10$
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    Table 2.  Functions as in [26]

    Functions
    $s(y) = 1 - \frac{0.482}{2} (3 y^2 - 1)$
    $h(A, \eta)=\rho\left(112.88+56.91\eta-24.31\eta^2-11.05\eta^3-\frac{A}{1.5}\right)$
    $g(A, \eta)=\delta(\eta-\eta_c)$
    $\alpha(\eta, y)=\begin{cases} &\alpha_1 \text { when } y< \eta \\ & \frac{\alpha_1+\alpha_2}{2} \text{ when } y=\eta\\ &\alpha_2 \text{ when } y>\eta \end{cases}$
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    Table 3.  Parameter values as in Table 1 unless specified above. Additional values taken from [1]

    Parameters Value Units
    $T_c$ 0
    $M$ 25 dimensionless
    $\alpha_w$ $0.35$ dimensionless
    $\alpha_i$ $0.45$ dimensionless
    $\alpha_s$ $0.8$ dimensionless
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