Sellers-Type Energy Balance Model

• Sellers-type Energy Balance Model is a parabolic PDE framework that models Earth’s zonally averaged surface temperature with meridional diffusion and radiative processes.
• It incorporates nonlinear ice-albedo feedbacks using hybrid co-albedo schemes, capturing both smooth transitions and sharp ice-line behavior.
• The model analyzes climate bistability, stochastic transitions, and coupled slow-fast dynamics, informing studies on climate sensitivity and stability.

The Sellers-Type Energy Balance Model (EBM) is a parabolic partial differential equation framework for climate dynamics that rigorously represents zonally averaged surface temperature evolution, including meridional diffusion, radiative fluxes, and nonlinear ice-albedo feedbacks. Sellers-type EBMs are used to model Earth’s climate and its stability properties, and provide mathematically tractable settings for analyzing deterministic and stochastic transitions between climate states.

1. Mathematical Formulation and Fundamental Components

A prototypical Sellers-type diffusive EBM in meridional (cosine-latitude) coordinates $x=\cos\phi\in(-1,1)$ governs the evolution of surface temperature $u(x,t)$ via: $$\frac{\partial u}{\partial t} \;-\;\frac{\partial}{\partial x}\Bigl((1-x^2)\frac{\partial u}{\partial x}\Bigr) \;+\;g(u(x,t)) \;=\;Q\,S(x)\,\beta(u(x,t)),\qquad x\in(-1,1),\,t>0$$ Key model elements:

• Meridional diffusion: $-\partial_x[(1-x^2)\,u_x]$ represents heat transport across latitudes, naturally vanishing at the poles due to geometry.
• Longwave emission: $g(u)$, typically a continuous increasing function (e.g., $g(u)=A+B\,u$), parameterizes outgoing thermal radiation.
• Absorbed shortwave: $Q$ is the global-mean solar constant, $S(x)$ the normalized insolation profile, and $\beta(u)$ is co-albedo (fraction of solar energy absorbed).
• Boundary conditions: Degenerate at $x=\pm1$ (poles), no explicit boundary condition required.

The model admits both smooth Sellers-type and discontinuous Budyko-type representations of $\beta(u)$ to capture feedbacks at the ice line. In recent work, hybrid co-albedo functions have been constructed to merge the mathematical advantages of both (Díaz et al., 27 Sep 2025). Typical parameter choices are $Q\sim340\,\mathrm{W\,m}^{-2}$, $A\sim202\,\mathrm{W\,m}^{-2}$, $B\sim2.17\,\mathrm{W\,m}^{-2}\mathrm{K}^{-1}$, ice co-albedo $\beta_i\sim0.3$, water co-albedo $\beta_w\sim0.6$, and transition widths $\delta\sim0.1\,\mathrm{K}$.

2. Ice-Albedo Feedbacks and Hybrid Co-Albedo Schemes

Ice-albedo feedback is central to climate bistability:

• Sellers smooth feedback: $\beta_{\text{Sellers}}(u)$ is Lipschitz, yields analytic, globally well-posed PDE theory, but lacks precise ice-line detection.
• Budyko step feedback: $\beta_{\text{Budyko}}(u)$ is discontinuous at $u=-10^\circ$C, allowing sharp ice-line transitions but losing PDE regularity.
• Hybrid form: Recent advances (Díaz et al., 27 Sep 2025) construct $\beta_{-10}(u)$, continuous everywhere, but with infinite derivative at the critical ice-transition temperature. This retains PDE well-posedness and delivers exact polar ice cap location via the slope singularity.

$$\beta_{-10}(u)=\begin{cases} \beta_i, & u<-10 \ (\beta_w-\beta_i)\,\theta_\delta(u+10)+\beta_i, & -10\le u\le -10+\delta \ \beta_w, & u>-10+\delta \end{cases}$$

with $\theta_\delta(s)=\frac{s\,\ln s}{\delta\ln\delta}$, $s\ge0$, so that $\beta'_{-10}(-10^+)=+\infty$.

3. Bistability, Edge States, and Bifurcation Structure

Sellers-type EBMs generally admit multiple equilibria due to nonlinear ice-albedo feedback and meridional heat transport (Bodai et al., 2014):

• Warm state ($T_W$): Earth-like, low ice cover, stable.
• Snowball state ($T_{SB}$): high ice cover, stable.
• Edge state ($T_U$ or Melancholia state): unstable intermediate, partitions the basins of attraction.

These branches arise in a fold (saddle-node) bifurcation structure: $$\text{Near fold:}\qquad \mu-\mu_f \approx \frac{1}{2}\left.\frac{d^2\mu}{d\bar{T}^2}\right|_{\bar{T}_f} (\bar{T}-\bar{T}_f)^2$$ Observables such as the large-scale temperature gradient

$\Delta T = T(\phi=0) - T(\phi=\frac{\pi}{2})$

and the material entropy production

$$\dot S_\mathrm{mat} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{D}{T^2(\phi)}[\partial_\phi T(\phi)]^2\cos\phi \,d\phi$$

organize the bifurcation diagram and nonequilibrium thermodynamic properties. Edge-tracking algorithms (Bodai et al., 2014) can compute the unstable manifold by phase-space bisection.

4. Stochastic Forcing and Noise-Induced Transitions

Stochastic Sellers-type EBMs model radiative forcing variability (solar, atmospheric) as multiplicative or additive stochastic processes (Díaz et al., 27 Sep 2025, Díaz et al., 2021, Lucarini et al., 2021):

• Additive or multiplicative Wiener (Gaussian) noise: Forcing term includes $\varepsilon Q S(x)\beta(u)\,dW_t$ (cylindrical Wiener process). Existence, uniqueness, and comparison theorems are established for mild solutions in $L^2$ spaces (Díaz et al., 27 Sep 2025, Díaz et al., 2021).
• α-stable Lévy noise: Solar constant is perturbed by $\dot L^\alpha(t)$ (symmetrical α-stable Lévy process, $0<\alpha<2$). This produces fundamentally different transition statistics and paths between attractors (Lucarini et al., 2021).

Transition time laws:

• Gaussian noise ($\alpha=2$): Mean residence time scales exponentially (Kramers law):

  $$\mathbb{E}[\tau]\approx \exp\left(\frac{2\Delta\Phi}{\varepsilon^2}\right)$$

  where $\Delta\Phi$ is the quasi-potential barrier at the edge state.

• Lévy noise ($0<\alpha<2$): Residence time scales algebraically:

  $\mathbb{E}[\tau] \propto \varepsilon^{-\alpha}$

  The escape mechanism is governed by rare, large jumps (compound Poisson process). Most probable paths under Lévy noise do not pass through the edge state but traverse the "closest boundary region" to the outgoing attractor—unlike the Gaussian case, which always transitions via the unique saddle $M$ (Lucarini et al., 2021).

5. Moving Ice-Line Dynamics and Low-Dimensional Reductions

Extensions with moving ice lines couple a slow ODE for the ice-line latitude to the high-dimensional temperature field (Pavlyukevich et al., 2023, Widiasih, 2011):

• Coupled slow-fast dynamics:

  $$\frac{d\eta}{dt} = \varepsilon \left[T(t,\eta(t)) - T_{\text{crit}}\right], \quad T_{\text{crit}} \sim -10^\circ \mathrm{C}$$

  Averaging theory rigorously reduces the system in the $\varepsilon \to 0$ limit to an effective SDE for the ice line, revealing metastability and noise-induced transitions (Pavlyukevich et al., 2023).

• Invariant manifold reduction: The temperature field collapses rapidly onto a one-dimensional graph in function space. The resulting ODE for $\eta$ admits explicit calculation of bifurcations and stability regimes, recovering canonical equilibria like snowball, large/small ice cap, and ice-free climate (Widiasih, 2011).

6. Memory Effects and Time-Dependent Forcing

Recent Sellers-type models incorporate memory and nonautonomous (time-dependent, chaotic) forcing (Cannarsa et al., 2018, Longo et al., 3 Mar 2025):

• Memory operator: Surface temperature and radiative response depend on past histories via convolution terms:

  $H(t,x,u) = \int_{-T}^0 k(s,x) u(t+s,x)\,ds$

  This influences solution regularity and enables inverse problem formulations to reconstruct climate forcings from observational data.

• Nonautonomous equilibria and climate sensitivity: In models with rapidly varying solar or cloud coefficients, the pullback attractor framework identifies three bounded (upper, middle, lower) nonautonomous solutions. Averaging approximations estimate the error between solutions to the full nonautonomous system and their time-averaged counterpart, justifying the reduction to effective centennial-scale models (Longo et al., 3 Mar 2025).

7. Applications and Extensions

Sellers-type EBMs have been extended beyond Earth's climate:

• Exoplanet climates: Latitudinal and tidally locked energy-balance models (e.g., HEXTOR) adopt Sellers-type diffusion but incorporate radiative-convective lookup tables for albedo and infrared flux, validated against general circulation models (GCMs) for both rapidly and synchronously rotating terrestrial planets (Haqq-Misra et al., 2022).
• Coupling with ocean/atmosphere fluid dynamics: Models with dynamic boundaries and full primitive equations admit strong time-periodic solutions under arbitrarily large periodic forcing, ensuring physical robustness (Sarto et al., 4 Dec 2025).
• Pleistocene glacial cycles: Flip-flop ice-line models derived from Sellers-type EBMs capture key transitions like the Mid-Pleistocene Transition, revealing the role of ice-albedo and temperature-precipitation feedbacks in regulating glacial–interglacial periodicity as bifurcation parameters are slowly varied (Widiasih et al., 2018).

---

In conclusion, the Sellers-type Energy Balance Model provides a rigorous and flexible framework for climate dynamics, admitting analytic characterization of bistability, ice-line feedbacks, stochastic transitions, and extensions to diverse climate systems. The model underpins both conceptual and applied studies of climate sensitivity and resilience, with modern mathematical developments supporting existence, uniqueness, and quantitative analysis under a range of deterministic, stochastic, and time-dependent forcings (Díaz et al., 27 Sep 2025, Bodai et al., 2014, Pavlyukevich et al., 2023, Longo et al., 3 Mar 2025, Lucarini et al., 2021, Haqq-Misra et al., 2022, Cannarsa et al., 2018, Sarto et al., 4 Dec 2025, Widiasih et al., 2018, Díaz et al., 2021, Widiasih, 2011).