FILLET: Ice Line Latitudinal EBM Tenacity

• FILLET is a standardized framework that quantifies ice-line tenacity in latitudinal EBMs, benchmarking resilience via hysteresis metrics under variable forcings.
• It integrates classical Budyko-Sellers feedbacks and diffusive dynamics to delineate stable climate states and identify bifurcation thresholds.
• The protocol supports ensemble-based diagnostics and model intercomparisons to assess planetary habitability under diverse climatic and stellar conditions.

The Functionality of Ice Line Latitudinal EBM Tenacity (FILLET) framework is a rigorously defined protocol for benchmarking, comparing, and analyzing the resilience and hysteresis of ice-line behavior in latitudinal energy balance models (EBMs) under varying planetary and stellar forcing. FILLET focuses on the quantification of ice-line “tenacity”—the persistence or resistance of the ice margin to transition under external forcings such as insolation, obliquity, or greenhouse gas concentration. Originally developed within the CUISINES exoplanet model intercomparison project (exoMIP), FILLET enables the identification of robust versus model-dependent features, establishment of standardized experimental protocols, and ensemble-based diagnostics of future planetary climate and habitability predictions (Deitrick et al., 2023, Barnes et al., 15 Nov 2025).

1. Governing Principles and Mathematical Formulation

FILLET is anchored in the latitudinally averaged EBM formalism, representing planetary climate as a system governed by the balance between radiative forcing, outgoing longwave loss, and meridional diffusive heat transport. The foundational EBM equation on a sphere for surface temperature $T(\phi, t)$ at latitude $\phi$ is: $$C(\phi)\,\frac{\partial T(\phi,t)}{\partial t} = S(\phi)\,[1 - \alpha(T(\phi,t), \phi)] - I(T(\phi,t)) + D\,\nabla^2 T(\phi, t)$$ where:

• $C(\phi)$: effective specific heat capacity,
• $S(\phi)$: seasonally and diurnally averaged insolation,
• $\alpha$: latitude- and temperature-dependent albedo,
• $I(T) = A + B T$: outgoing longwave radiation, linearized,
• $D$: diffusivity describing meridional heat transport.

The ice line $\phi_i$ is defined as the latitude at which $T(\phi_i) = T_{\mathrm{frz}}$, separating ice-covered from ice-free regions. Tenacity quantifies the width or robustness of the hysteresis loop—range of control parameters (instellation, $\mathrm{CO}_2$, etc.) for which multiple ice-line equilibria exist and the ice edge is resistant to displacement (Deitrick et al., 2023, Barnes et al., 15 Nov 2025).

2. Ice-Albedo Feedbacks, Diffusive Stabilization, and the Tenacity Mechanism

The classical Budyko-Sellers EBMs couple strong positive ice-albedo feedback with meridional heat transport. When the ice line advances equatorward (or retreats poleward), the jump in albedo at the ice edge modifies the absorbed shortwave flux: $$\alpha(y;\eta) = \begin{cases} \alpha_1, & 0 \leq y < \eta \ \alpha_2, & \eta < y \leq 1, \end{cases}$$ with $\alpha_1 < \alpha_2$. This drives further cooling (advance) or warming (retreat), potentially generating multiple equilibria such as snowball, small ice cap, and ice-free states (Walsh, 2016). Diffusive heat transport, represented by $D\,\partial_y[(1-y^2)\partial_y T]$, acts to suppress extreme latitudinal gradients, thereby limiting the amplitude of the positive feedback, stabilizing the ice line against small perturbations, and setting the critical thresholds for bifurcation (Walsh, 2016).

FILLET formalizes “tenacity” as the resistance of the system to abrupt shifts in ice extent. Quantitatively, this is measured by the slope of the reduced one-dimensional dynamical equation for the ice line at each equilibrium: $$\dot{\eta} = H(\eta) = \sum_{n=0}^N f_{2n}(\eta) p_{2n}(\eta) - T_c$$ where $H'(\eta^*)$ at an equilibrium $\eta^*$ determines both its stability and tenacity; more negative $H'(\eta^*)$ values correspond to higher tenacity (stronger restoring force) (Walsh, 2016).

3. Fast-Slow Dynamics, Invariant Manifolds, and Reduced Scalar Maps

The coupled EBM and moving ice-line system exhibits a fast-slow structure: temperature variables relax rapidly toward a slow manifold $\mathcal{M}_I$, while the ice-line $\eta$ evolves on a much slower timescale set by small parameter $\epsilon$ (Widiasih, 2011, Walsh, 2016). The generalized temperature field can be expanded as $T(y,t) = \sum_{n} T_{2n}(t) p_{2n}(y)$ (spectral representation), and for $\epsilon \to 0$,

$$T_{2n} \to f_{2n}(\eta) = \frac{Q (s_{2n} - \bar{\alpha}_{2n}(\eta))}{B + 2n(2n+1) D}$$

The slow evolution of the ice line then reduces to a scalar ODE. Linear stability analysis about fixed points ($H(\eta^*) = 0$) dictates tenacity via the sign and magnitude of $H'(\eta^*)$. Fast dynamics ensure that perturbations transverse to $\mathcal{M}_I$ decay rapidly, while the evolution of $\eta$ is governed by the local restoring force in $H(\eta)$ (Widiasih, 2011).

4. Model Intercomparison, Hysteresis Protocols, and Diagnostic Outputs

FILLET, as operationalized in the CUISINES exoplanet intercomparison project, prescribes a suite of standardized numerical experiments across variations in instellation ($S$), obliquity ($\varepsilon$), and $\mathrm{CO}_2$ mixing ratio. Benchmarks include perturbations from both warm and cold initial states to resolve the extent of climate hysteresis. Diagnostic outputs are strictly defined:

• Annual mean surface temperature $T_{\text{surf}}(\phi)$
• Surface and top-of-atmosphere albedo ($A_{\text{surf}},A_{\text{TOA}}$)
• Outgoing longwave flux $OLR(\phi)$
• Ice-line latitude extrema, $\lambda_{\max}$ and $\lambda_{\min}$, per hemisphere and surface type (land/sea), four per hemisphere (Barnes et al., 15 Nov 2025)

Tenacity is quantified as $T = |\lambda_{\max} - \lambda_{\min}|$ in latitude, or equivalently, as the width between forward and reverse transition points in control parameter space (e.g., $\tau_S = S_\uparrow - S_\downarrow$) (Barnes et al., 15 Nov 2025, Deitrick et al., 2023).

Diagnostic	Definition	Role in FILLET
$\lambda_{\max}$, $\lambda_{\min}$	Max/min ice-edge latitude per hemisphere & Tenacity metric; hysteresis width
$D$	Diffusion coefficient (W m⁻² K⁻¹)	Meridional heat transport
$A,B$	OLR linearization parameters	Radiative damping

5. Bifurcation Structure, Critical Thresholds, and Stability Regimes

FILLET elucidates the bifurcation structure of the coupled temperature–ice-line system, identifying parameter regions with multiple coexisting attractors. For instance, increasing diffusivity $D$ beyond a critical $D_c$ annihilates the small ice cap equilibria via saddle-node bifurcation, yielding only snowball or ice-free states. In the "Jormungand" configuration (intermediate bare-ice albedo), additional tropical cap states are permitted, depending on $D$ and albedo assignment (Walsh, 2016).

Under stochastic forcing, the dynamics of the ice line are described by a slow-fast SDE: $$d\eta_t = \widehat{f}(\eta_t)\,dt + \widehat{\sigma}(\eta_t)\,dW_t$$ Metastable climate states correspond to zeros of $\widehat{f}(\eta)$; their tenacity under noise is quantified by the mean escape time from the potential well: $$T_i \sim \exp\{ \Delta U_i \} / \sigma^2, \qquad \Delta U_i = U(\eta_{\text{saddle}}) - U(\eta_i^*)$$ where $U(\eta)$ is the effective potential constructed from the drift term (Pavlyukevich et al., 2023).

6. Physical and Scientific Implications of Ice-Line Tenacity

FILLET tenacity metrics are central to understanding planetary climate resilience and habitability boundaries. Wide hysteresis regions (large tenacity) indicate that the climate system is resistant to parameter changes, requiring large perturbations to induce ice-line migration. This has direct implications for the stability and recoverability of habitable climates under stellar evolutionary forcing or internal geochemical cycling (Deitrick et al., 2023, Barnes et al., 15 Nov 2025).

FILLET output standards and protocol facilitate ensemble-based climate prediction, diagnosing which outcomes (e.g., presence of ice belts, latitude of threshold transitions) are robust across models and which reflect parameterization choices. Differences in critical thresholds (e.g., the bifurcation value of $S$ or $D$) can be traced to model-specific treatments of radiative transfer, albedo parametrization, or diffusion (Barnes et al., 15 Nov 2025).

7. Extensions, Limitations, and Prospects

FILLET as currently formulated is restricted to one- and two-dimensional aquaplanet EBMs with a single or double ice line and relies on linearized OLR and annual-mean conditions. The protocol does not encompass explicit couplings to carbon-cycle feedbacks or spatially resolved continentality, though extension to such systems is recommended for future work (Widiasih, 2011, Deitrick et al., 2023). The standardized output and intercomparison structure provided by FILLET, combined with the tenacity metric, lays the groundwork for reproducible, transparent advances in climate modeling and robust characterization of planetary habitability (Barnes et al., 15 Nov 2025, Deitrick et al., 2023).