At 50 km grid spacing there is no land in the Lesser Antilles. Every island in the chain is ocean to the model. This note sets out the conservation law I think is missing from physics-constrained downscaling, the decomposition geometry that makes it enforceable, and the experiment that would show I am wrong.
Between 12 and 20 degrees north and 64 and 56 degrees west, a climate model at CORDEX resolution contains no land grid point. Antigua, Dominica, Saint Lucia, Barbados, Grenada, Saint Vincent: all sea. The nearest land the model knows about is roughly 500 km away, in Puerto Rico or on the Venezuelan coast. Cantet, Déqué, Palany and Maridet stated this plainly in Tellus A in 2014, and it is still the cleanest one-sentence account of the problem I work on.
Jamaica does better. The island is big enough to occupy four to six cells at that spacing, depending on where the lattice happens to fall. It keeps its existence and loses its shape. Blue Mountain Peak reaches 2,256 m; a 50 km cell holds the average of everything inside it, which along the line of the trade winds comes to about 851 m. The mountain survives as a bump.
That matters because the shape is the climate. The north-eastern slopes of the Blue Mountains take 3,000 to 5,000 mm of rain a year. The southern coastal plains of St Catherine and Clarendon take under 1,500 mm. Twenty kilometres apart, a factor of three to four in annual rainfall, both inside the same model cell and both replaced by the same number.
The dynamical route does not close this gap
The standard answer is to nest a regional model inside the global one and run it at 4 km or better. It works. A 4 km convection-permitting simulation over Puerto Rico added value over its ERA5 forcing in up to 84.0 per cent of grid cells, which is a real result, and the authors were straightforward that the remaining errors were still large.
The obstacle is arithmetic. Halving the grid spacing quarters the cell area, so the cell count goes as the inverse square. A dynamical model also has to shorten its timestep to keep the Courant condition, Δt ≤ Δx/c, or the numerics come apart. So the step count goes as the inverse first power on top of that, and with vertical refinement the exponent reaches four.
Going from 50 km to 1 km is therefore a factor of 125,000 in two dimensions, or 6.25 million in three. That is the whole reason AI emulation exists as a field. It is also why a Caribbean meteorological service cannot simply buy its way to kilometre-scale projections: the ensembles needed to sample uncertainty multiply the figure again.
What the mechanism actually is
Before arguing about model architecture it is worth being concrete about the physics a coarse cell is throwing away, because the physics is simple and the loss is large.
Take a parcel of air leaving the sea on Jamaica's north-eastern coast at 28 °C with a dewpoint of 24 °C. The trade wind pushes it up the slope. It cools at the dry adiabatic rate, 9.8 K per kilometre, until temperature and dewpoint meet. Using the usual rule of thumb, 125 m for every kelvin of dewpoint depression, that happens at 500 m. From there it is saturated and cools more slowly, around 5 K per kilometre in air this warm, reaching about 14.3 °C at the 2,256 m summit.
The mixing ratio falls from 19.0 to 13.3 grams per kilogram over that ascent. The difference, 5.6 grams per kilogram of air, is what falls out as rain on the windward slope. The parcel that comes down the other side has been wrung out, so it warms at the dry rate all the way down and arrives on the leeward plain near 36 °C. That is the rain shadow, and it is why Kingston and the St Elizabeth plains are dry while Portland is the wettest parish in the country.
A 50 km cell cannot hold any of this. It has one elevation, so it has one lapse-rate path. The ascent, the condensation, the descent and the drying all happen inside a single number.
What emulation got right, and the assumption underneath it
Generative downscaling models have become good. The most interesting recent development is that people started enforcing physics on them instead of hoping the network would learn it. The constraint that has been used, and it works, is that the coarsened prediction should equal the coarse input:
where 𝒞 is the area-weighted coarsening operator. Enforce that as a soft penalty, handle the gradient conflict properly, and both physical consistency and out-of-region generalisation improve. Reported gains include conservation error down by about 40 per cent in-domain and a halving of the precipitation wet bias when the model is moved to a region it was not trained on.
There is an assumption underneath it, and the authors who introduced the constraint said so themselves. The relation holds by construction in a controlled experiment, because the coarse input was made by coarsening the fine data. In operational downscaling the coarse field comes from an independent global model with its own dynamics and its own biases. Enforcing the constraint then drags the fine field towards a biased mean.
This is not hypothetical for the Caribbean. CMIP-class models carry well-documented precipitation biases over this basin, tied to sea-surface temperature errors in the tropical North Atlantic. A constraint that binds my kilometre-scale output to that coarse mean makes the model more confidently wrong, and it does so in a way that looks like better physics on every diagnostic I would normally run.
The second axis
The cross-scale constraint is vertical. It ties the fine field to the coarse field, cell by cell, across scales. It says nothing whatsoever about lateral exchange.
A predicted field can satisfy it exactly at every coarse cell while carrying arbitrary flux discontinuities across interior faces, because a cell mean is preserved regardless of how mass moves inside or between cells. The second law, the one I want to enforce, is that the flux of a conserved quantity leaving subdomain b across face f equals the flux entering the neighbouring subdomain b′ across the same face, at every timestep:
The two properties are independent. Neither implies the other. The figure below constructs the case: two adjacent subdomains whose cell means are exactly right, and whose face fluxes disagree by 2.0 kg m−2 s−1. Mass is created at the face, the cross-scale diagnostic reads clean, and nothing in the current toolkit notices.
The reason I care about this beyond tidiness is the biased-driver problem above. Lateral conservation constrains the fine field against itself. It is a statement about internal consistency, not about agreement with an external driver. A field can be laterally conservative and still correct a biased large-scale mean. That is the operating regime a Caribbean forecast has to survive.
Where the faces go
Decomposing a domain into subdomains is old. Extended physics-informed neural networks did it with explicit interface conditions; finite basis PINNs replaced those conditions with overlapping subdomains whose summed solutions are continuous by construction. Tiled super-resolution has been used for downscaling, and the result there is worth knowing: a tiled transformer and an overlapping-patch transformer over Iberia had identical root-mean-square error to two decimal places. Overlap removed the seam artefacts and recovered none of the skill.
What nobody has varied is the geometry. Every decomposed method in this literature partitions on a regular lattice, for the same reason every image model does, which is that tensors are rectangular. The subdomain boundaries have never been allowed to follow the physical topology.
Here is the argument for why that should matter, and it has to be an argument, because the claim on its own is worth nothing. The interface constraint acts on faces. Face placement therefore decides where the strongest physical constraint in the model does its work. A face placed along a ridge line separates two air masses with genuinely different dynamics, and the constraint is describing a real discontinuity. A face placed arbitrarily across a windward slope cuts a single coherent orographic process in half and forces the model to reconstruct it through a penalty term. Same constraint, different physics, decided entirely by where the line was drawn.
What the output is actually for
A kilometre-scale wind and rainfall field is an input, not an answer. The reason for wanting it is that at that resolution you can put it into a three-dimensional scene and ask what happens to a specific place.
Two relations do most of the work. Wind load is dynamic pressure, q = ½ρV², multiplied by a pressure coefficient that depends on where you are on the building. Because it goes as the square of speed, a 20 per cent error in the predicted wind is a 44 per cent error in the predicted load. And the wind over terrain is not the wind on the plain: the standard topographic speed-up for a ridge of moderate slope raises the crest wind by around 40 per cent, which doubles the load on anything standing there.
Water is routed with equally old arithmetic. Peak discharge from the rational method, channel velocity from Manning. For a 4 km² catchment under 80 mm of rain an hour, that is 62 m³ s−1 moving at 5.3 m s−1, which gives a settlement 2.8 km downstream about nine minutes.
Nine minutes is a number a district officer can act on. A basin-wide rainfall anomaly is not. The whole case for sub-kilometre emulation rests on that difference, and on being able to run it often enough and cheaply enough that the answer is available before the event.
The experiment, and the result that would kill it
The claim is testable, which is the only thing that makes it worth making. The design is a bias sweep: inject a bias of known magnitude and sign into the driving field, then measure how conservation error and probabilistic skill degrade under each constraint separately and under both together.
If the cross-scale arm degrades with driver bias and the lateral arm stays flat, the distinction has operational value. If both degrade together, the lateral constraint inherits driver bias after all and the central claim is wrong.
The ablation ladder that goes with it holds one backbone, one parameter budget and one training budget fixed, and varies only the decomposition: full domain, regular tiles, regular tiles with overlap, partition-of-unity summation, the interface penalty alone, both together, and finally terrain-aligned faces. Two of those comparisons are designed to be able to come back flat.
- Does the flux penalty beat partition-of-unity summation? Overlapping subdomains that sum to a global solution are already continuous at the interface. Continuity and conservation are different properties: a smoothly varying field can still create mass. If the gap between them is zero, I will report zero.
- Do terrain-aligned faces beat regular faces at matched subdomain count? This is the contribution most likely to fail, and I would rather say so before a reviewer does.
Where this is weakest
Three places, in order of how much they worry me.
Lateral conservation is necessary and not sufficient. A field can be perfectly flux-consistent across every face and still be wrong about the weather. The constraint removes a failure mode; it does not supply skill. Anyone selling it as a source of accuracy is overselling it, and that includes me if I am careless.
The face-flux estimator is approximate. I am estimating a derivative quantity from a model output, so the estimator's own error enters the penalty. That needs its convergence properties written down explicitly, and it is the part of the method I am least sure of.
Validation is thin. Jamaica's station network is sparse and clustered, gridded precipitation products over the Caribbean are dominated by station density more than by grid resolution, and the effect is worst under steep altitude gradients, and that is the terrain I am claiming to resolve. My answer is to lean on tests that need no stations at all, conservation residuals, cross-variable admissibility, spectra, and on process-based checks against the published Jamaican climatology, the north-east wettest and the southern plains driest, the bimodal May to June and October rainfall seasons. That is a weaker form of validation than a dense observing network would give.
Questions I get asked
How is this different from finite basis PINNs with terrain added?
FBPINNs decompose a domain in order to solve a known partial differential equation whose solution is unknown. This decomposes in order to downscale, where the coarse field is given and the fine field is underdetermined, so physics constrains the relationship instead of determining the answer. FBPINN subdomains are geometric and uniform; these follow terrain, so the decomposition itself encodes geophysical structure. And the FBPINN contribution is optimisation-theoretic, addressing spectral bias and scaling. This one is physical.
Partition of unity already gives continuity. Why add a penalty?
Continuity and conservation are different properties. A partition-of-unity sum guarantees the field varies smoothly across the interface. It does not guarantee that net flux across the face balances, and a smoothly varying field can create mass. The comparison between those two is one of the experiments, and it is designed so that a null result is publishable.
Physics-constrained generative downscaling already exists. What is new?
The quantity being conserved. Existing constraints are cross-scale, binding the fine field to the coarse input. This one is lateral, binding adjacent fine regions to each other at the same scale. The cross-scale constraint rests on the coarse field being an unbiased aggregate of the fine field, which holds by construction in a controlled experiment and fails with an independent driver. The lateral constraint does not have that dependency. The bias sweep tests whether that difference survives contact with data.
Why small islands and not a larger domain?
Partly because it is where I live and where the decisions are, and partly because it is the hardest available test. Existing geographic generalisation experiments have moved models between mid-latitude continental regions. A tropical maritime island domain is a much larger distributional shift, and the scale gap there is categorical: at 50 km the islands are not poorly resolved, they are absent.
What does this cost to run?
That is the number I most want to report and cannot yet. Compute cost per downscaled simulated year, in currency, at each subdomain size, is almost never published in this literature, and it is the figure a meteorological service actually needs, because the question they ask is whether a method fits in a line item. It will be in the results.
References
- Cantet, P., Déqué, M., Palany, P. and Maridet, J.-L. (2014). The importance of using a high-resolution model to study the climate change on small islands: the Lesser Antilles case. Tellus A, 66, 24065. tellusjournal.org
- Climate Studies Group, Mona (2012, 2017). State of the Jamaican Climate. Produced for the Planning Institute of Jamaica. Policymakers' summary (PDF)
- Evaluation of precipitation across the contiguous United States, Alaska, and Puerto Rico in multi-decadal convection-permitting simulations (2024). Scientific Reports, 14, 1420. nature.com
- Moseley, B., Markham, A. and Nissen-Meyer, T. (2023). Finite basis physics-informed neural networks. Advances in Computational Mathematics, 49, 62.
- Jagtap, A. D. and Karniadakis, G. E. (2020). Extended physics-informed neural networks. Communications in Computational Physics, 28(5), 2002–2041.
- Beucler, T. et al. (2021). Enforcing analytic constraints in neural networks emulating physical systems. Physical Review Letters, 126, 098302.
- Pérez, A., Santa Cruz, M., San Martín, D. and Gutiérrez, J. M. (2024). Transformer-based super-resolution downscaling for regional reanalysis: full domain vs tiling approaches. arXiv:2410.12728.
- Bolton, D. (1980). The computation of equivalent potential temperature. Monthly Weather Review, 108, 1046–1053. Used for the saturation vapour pressure in the orographic cross-section.
Adrian Dunkley, Climate Studies Group Mona
Department of Physics, The University of the West Indies, Mona. Correspondence: adrian.dunkley@mymona.uwi.edu
UWI Mona profile