FILLET Protocol v1.0: EBM Benchmarking

• FILLET Protocol v1.0 is a systematic framework for evaluating one- and two-dimensional energy-balance models by prescribing controlled experiments and unified output formats.
• The protocol standardizes numerical experiments and boundary conditions, enabling consistent parameter sweeps in instellation, obliquity, and CO₂ levels to diagnose climate state classifications.
• By benchmarking EBMs with reproducible diagnostics and ensemble metrics, FILLET v1.0 enhances model intercomparisons and aids in transparent assessments of planetary climate predictions.

The Functionality of Ice Line Latitudinal EBM Tenacity (FILLET) Protocol Version 1.0 defines a systematic approach for evaluating and intercomparing one- and two-dimensional energy-balance models (EBMs) of Earth-like planets. Developed under the auspices of the CUISINES exoplanet model intercomparison project (exo-MIP), FILLET v1.0 establishes standardized experimental designs and reporting protocols, enabling rigorous diagnostics of model-dependent behaviors in planetary climate predictions under varying climate regimes and forcing parameters (Deitrick et al., 2023).

1. Scope and Objectives

FILLET Protocol Version 1.0 is designed to address intrinsic differences among EBMs by prescribing controlled numerical experiments and unified output formats. Its fundamental aims are:

1. To define a suite of benchmark climates (notably reproductions of pre-industrial Earth) and structured parameter sweeps (in instellation, obliquity, and atmospheric CO₂), thereby sampling the principal ice-albedo feedback regimes—namely snowball, ice caps, ice belts, and ice-free planets.
2. To enforce consistent boundary conditions across participating models, facilitating forensic diagnosis of differences in parameterizations such as albedo transitions, heat transport, and radiative transfer.
3. To construct an ensemble of EBM outputs—including mean and standard deviation metrics—for adoption by the exoplanet science community, analogous to the role of CMIP in GCM intercomparisons (Barnes et al., 15 Nov 2025).

2. Governing Equations and Parameterizations

At its core, FILLET v1.0 mandates that all EBMs solve the annual-mean, one-dimensional (latitude, φ) energy balance equation:

$$C(\phi)\,\frac{\partial T(\phi,t)}{\partial t} = S(a)\;Q(\phi,\varepsilon)\,\bigl[1-\alpha\bigl(\phi,T\bigr)\bigr] - I\bigl(T(\phi)\bigr) + \frac{1}{\cos\phi} \,\frac{\partial}{\partial \phi}\! \Bigl[D(\phi)\,\cos\phi\,\frac{\partial T(\phi)}{\partial \phi}\Bigr]$$

where:

• $C(\phi)$: Effective heat capacity (J m⁻² K⁻¹), dependent on surface type (land, ocean, sea ice).
• $S(a)$: Top-of-atmosphere instellation, $S_\oplus/a^2$, with $S_\oplus = 1361\,\text{W\,m}^{-2}$.
• $Q(\phi,\varepsilon)$: Normalized diurnal-mean insolation for latitude $\phi$ and obliquity $\varepsilon$.
• $\alpha(\phi,T)$: Broadband planetary albedo, typically with a temperature-dependent parameterization to capture the ice–albedo feedback. The default "smooth transition" is

$$\alpha(T)=\alpha_i + (\alpha_w-\alpha_i)\,\frac{1}{2}\,[1+\tanh((T-T_{crit})/\Delta T)]$$

with $\alpha_w \approx 0.30$ (warm surface), $\alpha_i \approx 0.60$ (ice), $T_{crit} \approx 273\,\text{K}$, and $\Delta T \approx 2\,\text{K}$.

• $I(T)$: Outgoing longwave radiation, linearized as $I(T)=A+B\,T$, with defaults $A \approx 203\,\text{W\,m}^{-2}$, $B \approx 2.09\,\text{W\,m}^{-2}\,\text{K}^{-1}$.
• $D(\phi)$: Meridional heat diffusion coefficient, typically constant ($D=0.6\,\text{W\,m}^{-2}\,\text{K}^{-1}$) or latitude-dependent.

Reference parameter values are stipulated, but participants may employ alternative, documented values. The framework explicitly allows for different functional forms provided full disclosure and consistent reporting are maintained (Deitrick et al., 2023).

3. Experimental Matrix and Benchmarks

FILLET v1.0 specifies two climatological benchmarks and four primary experiments:

Label	Instellation ($S/S_\oplus$)	$X_{\rm CO_2}$ (ppm)	$a$ (AU)	Obliquity ($\varepsilon$) (°)
Benchmark 2	1.00	280	1.0	23.5
Benchmark 3	1.00	280	1.0	60
Exp. 1	0.80–1.25 [0.025]	280	1.0	0–90 [10]
Exp. 1a	$S(a)=S_\oplus/a^2$	280	0.875–1.10 [0.0125]	0–90 [10]
Exp. 2	1.05–1.50 [0.025]	280	1.0	0–90 [10]
Exp. 2a	$S(a)=S_\oplus/a^2$	280	0.80–0.975 [0.0125]	0–90 [10]
Exp. 3	0.80–1.50 [0.0125]	280	1.0	23.5
Exp. 4	1.00	50–5050 [25 log-spaced]	1.0	23.5

Brackets [ ] indicate the increment in parameter sweeps. All experiments assume an Earth-mass, 1 bar N₂-dominated background and annual steady-state solutions (Barnes et al., 15 Nov 2025).

4. Output Specification and Reporting

FILLET v1.0 mandates comprehensive reporting of both latitudinal profiles and global scalar diagnostics for every submitted simulation:

Latitudinal outputs (as a function of φ):

• Latitude, φ (°)
• Annual mean surface temperature, $T_{surf}(\phi)$ (K)
• Annual mean surface albedo, $A_{surf}(\phi)$
• Annual mean TOA albedo, $A_{\text{TOA}}(\phi)$
• Annual mean outgoing longwave radiation, $OLR(\phi)$ (W m⁻²)

Global outputs (single line per experiment-parameter case):

• Instellation, $S/S_\oplus$
• Obliquity, $\varepsilon$ (°)
• $X_{\rm CO_2}$ (ppm)
• Global mean surface temperature, $T_{glob}$ (K)
• Two ice-edge latitudes per hemisphere: maximum and minimum $\phi$ defining northern (IceLineNMax/IceLineNMin) and southern (IceLineSMax/IceLineSMin) ice extents
• Meridional heat diffusion coefficient, $D$ (W m⁻² K⁻¹)
• Global mean outgoing longwave radiation, $OLR_{glob}$ (W m⁻²)

Accurate identification of ice-edge extremes in both hemispheres allows unambiguous differentiation between all four canonical climate states (snowball, ice caps, ice belts, and ice-free) during post-processing.

5. Numerical Implementation and Best Practices

Spatial discretization employs either equal-area or equal-spacing in latitude, typically with 18–150 grid cells. Temporal stepping uses $\Delta t$ values sufficient to resolve annual cycles, commonly $10^3$–$10^5$ s, with convergence defined by $<0.01$ K yr$^{-1}$ change in annual mean temperature per latitude.

Initialization protocols are specified:

• Warm start (Benchmark 2–4, Exp 1/1a): $T(\varphi,0)=T_0-7.35+20[1-2\sin^2\varphi]\,^\circ$C, $T_0\approx15$°C.
• Cold start (Exp 2/2a): Subtract 40 K from the warm-start profile.

Boundary conditions are Neumann (zero meridional heat flux) at the poles.

A reference workflow is provided:

Read protocol inputs: S, XCO2, a, ε, C, D, α_tunings Initialize latitude grid {φ_i}, time step Δt, total steps Nstep Initialize T_i according to warm or cold start For n=1 to Nstep: compute daily/seasonal insolation S_i(t) compute α_i = α(T_i) using parameterization compute OLR_i = A(XCO2)+B(XCO2)*T_i compute diffusion flux F_i = D*∂T/∂φ (discrete divergence) update T_i ← T_i + Δt/C * [S_i(1−α_i) − OLR_i + divergence(F_i)] end loop Annually-average T_i, α_i, A_TOA_i Compute ice-line latitude(s): φ at which T≤T_frz Output lat-profiles and global diagnostics

Sanity checks include reproducing pre-industrial Earth conditions with tuned parameters, comparing outputs for low/high obliquity cases, and verifying numerical stability and resolution adequacy. Convergence criteria and standard pitfalls (large $\Delta t$, low spatial resolution, neglect of seasonality at high obliquity) are noted (Deitrick et al., 2023).

6. Diagnostic Analysis and Climate State Classification

The FILLET v1.0 protocol emphasizes construction of hysteresis curves (ice-line vs. $S$ or $X_{\rm CO_2}$) from sequence runs under warm and cold initializations. By systematically varying instellation and CO₂ abundance, the protocol facilitates the identification of bifurcation thresholds, quantification of climate state stability, and detection of multiple equilibria.

Community analysis scripts ingest the reported outputs, classify each climate state, and compute ensemble averages and standard deviations across models for quantities such as $T_{surf}(\phi)$, ice lines, and $OLR(\phi)$. A plausible implication is that this enables transparent benchmarking of EBM physical assumptions and enhances reproducibility for exoplanet surface climate prediction.

7. Significance and Applications

FILLET Protocol v1.0 marks a substantial advance in EBM rigor by providing the exoplanet modeling community with:

• A unified set of protocols facilitating like-for-like comparison of EBM predictions.
• Quantitative ensemble metrics (mean, standard deviation) for key climate diagnostics.
• A transparent process analogous to global GCM intercomparisons (e.g., CMIP) but optimized for EBMs and exoplanetary parameter space (Barnes et al., 15 Nov 2025).

Adherence to FILLET v1.0 enables robust attribution of model-dependent effects, underpins systematic exploration of planetary habitability, and accelerates the identification of parameter regimes where model physics, rather than physical principles, drive uncertainties in climate predictions.