Abstract
A regional climate emulator must represent processes inside each local atmospheric volume while transmitting information across the boundaries that connect them. This article examines a modular approach in which adjacent three-dimensional volumes exchange prognostic state and moisture flux. It develops the numerical distinction between conserving a coarse-area rainfall total and enforcing an atmospheric budget across a shared face. The first is directly testable from gridded precipitation; the second requires wind, moisture, layer geometry and consistent time integration. A preliminary Jamaican experiment used monthly CHIRPS v2 fields for 2000–2025, with 2022–2025 reserved for retrospective assessment. Reported results for a matched neural-model comparison were 15.36 mm/month RMSE without an exact aggregate constraint and 14.23 mm/month with it, a relative reduction of 7.4%. A seasonal spatial-allocation baseline nevertheless achieved 9.73 mm/month, outperforming both networks. These results provide evidence about the tested constraint, not about a complete three-dimensional world model or forecast skill. The proposed next stage uses ERA5 atmospheric fields, GPM precipitation and topography to examine coupled cells, constrained historical-state reconstruction and forward evolution as separate experiments. The research question is whether physical consistency adds skill and interpretability beyond simpler methods without importing bias from coarse observations.
1. Introduction
Jamaican rainfall is spatially uneven. A relatively small island contains steep relief, narrow coastal plains and slopes with different orientations to prevailing moisture transport. Coarse atmospheric grids smooth the variations that matter to local flood management and regional climate analysis. Reducing grid spacing after a model run does not, by itself, recover the physics removed by that averaging operation. A method must establish what information is available, what information is reconstructed and what information cannot be inferred from the data.
The proposed research treats the atmosphere as a set of coupled computational volumes. A local model estimates how its state evolves, and neighbouring models communicate through shared boundaries. An observation-informed inverse step attempts to estimate earlier states within a known historical window. A prospective forecast, by contrast, may use only inputs available at the declared forecast issue time. Calling both operations a “rewind” obscures that important temporal distinction.
There are two immediate tests of this approach. First, can a local model reproduce the aggregate precipitation imposed by a coarse observation without discarding useful fine-scale structure? Second, can adjoining models exchange a moisture flux that is physically consistent on both sides of an interface? The present research programme has evidence for the first question from a preliminary component experiment, while the second remains a proposed coupled-atmosphere test.
2. Relevant literature and scientific context
The CHIRPS product combines infrared precipitation estimates, climatological information and station observations, providing a high-resolution historical rainfall record suitable for examining monthly spatial patterns [1]. It is an observationally informed gridded estimate rather than independent rain-gauge truth at every location. ERA5 is a reanalysis that combines atmospheric models and assimilated observations into globally coherent historical state fields [2]. Its wind, temperature, humidity and vertical-motion variables can constrain a retrospective atmospheric reconstruction, but a reanalysis is not the same as an untouched future forecast.
Physics-informed neural networks augment data-fitting objectives with differential-equation residuals and boundary or initial-condition terms [3]. Their appeal does not establish that every physical penalty improves prediction. Constraints can amplify errors when the target forcing is biased, when a field is not a conserved quantity, or when the equations are only approximately valid at the chosen resolution.
Recent extreme-precipitation work has demonstrated that embedding aspects of physical evolution in a generative model can improve short-range storm detection relative to some numerical forecasts. A study in the Tennessee Valley found substantial differences between NowcastNet and HRRR for extreme grid-cell precipitation, while documenting overestimated area-wide rainfall and increasing spatial-pattern errors at longer lead times [4]. Those results cannot be transferred directly to Jamaica: data availability, terrain, storm climatology, verification scale and forecast horizon all differ. They instead motivate stringent comparison against local baselines and reporting both grid-level and spatial-object errors.
3. Mathematical formulation
3.1 Atmospheric state and transport
Let an atmospheric cell Ωi contain a state vector of resolved wind components, temperature, pressure and specific humidity, qi(x,y,z,t). In continuous form, a simplified conservation statement for water vapour follows from the divergence theorem. Flux entering the volume, flux leaving it, and local sources determine its rate of change:
Integration over one cell gives a finite-volume representation. The outward-normal flux is counted positively, so water transported out of one cell must enter its neighbour with the opposite sign, apart from explicitly modelled interface sources or numerical error.
For adjacent cells sharing face Γij, ideal consistent coupling imposes Fij + Fji = 0. Achieving this numerically requires compatible units, face areas, coordinate transformations and boundary quadrature. Overlapping learned volumes need particular care because simply averaging two predicted fields can conceal an unbalanced flux.
3.2 Coarse rainfall conservation is a different requirement
A tested monthly downscaling component constrains the area-weighted mean rainfall within each coarse region. If fine cells belong to coarse region Gj, the correction can be written as:
An exact correction may lower the error relative to another network trained on the same target. That is a numerical comparison, not a statement that the coarse input is perfectly observed. If the coarse rainfall is biased, exact agreement can transmit that bias into every reconstructed fine cell. A scientifically useful model must be compared against the same coarse information available to a simple, well-tuned baseline.
3.3 Inverse estimation and forward simulation
Historical-state inference may be expressed as an optimisation over earlier states xt−k:t. The objective combines mismatch with observed variables, physical residuals and a prior on credible atmospheric states. The formulation is constrained smoothing, not the unique reversal of a dissipative chaotic atmosphere.
A forward forecast starts with a fixed issue-time state and may not be updated with withheld later observations. The retrospective reconstruction and the prospective forecast therefore require separate scores, data manifests and leakage tests.
4. Interactive representation
5. Preliminary experiment and planned validation
The recorded component study used monthly CHIRPS version 2 over Jamaican land cells from 2000 through 2025. A fine grid comprising 360 land cells was assigned to 29 coarse regions. The reported study compared matched neural networks with no conservation restriction, a soft penalty and an exact coarse-aggregate constraint, then evaluated a retrospective period of 2022–2025. This is an evaluation of conditional spatial rainfall reconstruction, not a real-time nowcast and not evidence that the full architecture has been trained.
| Method | Reported RMSE | Interpretation |
|---|---|---|
| Unconstrained neural model | 15.36 mm/month | Matched neural reference |
| Exact aggregate-conservation model | 14.23 mm/month | 7.4% lower than unconstrained model |
| Seasonal spatial-allocation baseline | 9.73 mm/month | Best of these three reported values |
Table 1. Source-manuscript results from the existing Jamaican component benchmark. The original experiment and complete data provenance have not been independently reproduced for this web article. Lower RMSE is better.
The exact-conservation fit achieved small numerical aggregate residuals in the original report, but numerical closure is not proof of climatological skill. Subsequent source-archive follow-up checks reproduced saved predictions and score calculations while identifying model-optimisation limits. Independent station evidence remains sparse. Publication of comparative skill claims requires a full raw-data rerun, checks of observation vintage and split integrity, and independent rainfall validation.
The next stage requires vertical ERA5 pressure-level wind, temperature, moisture, pressure-related variables and consistent coordinates. Surface rainfall from GPM IMERG and a documented elevation model must be aligned to the same event windows and spatial reference. A July 2024 Hurricane Beryl case study is planned, accompanied by a second genuinely held-out Caribbean weather event. The evaluation will compare coarse interpolation, climatological and persistence references, a modest neural operator, the same architecture without interface losses, and the full constrained variant. The outcomes will include RMSE, bias, rare-event detection, fractions skill score, spatial displacement, probabilistic calibration where ensembles are available, interface residual and compute cost.
6. Discussion and limitations
The main finding from the recorded component analysis is deliberately narrower than the long-term research ambition: enforcing a coarse rainfall total changed the error of a matched neural downscaler, yet the simpler seasonal allocation baseline was appreciably more accurate. This result limits any claim that physics-informed architecture automatically provides greater predictive skill. It suggests that seasonal structure is a strong comparator and that constrained learning requires a justified observation model.
Several scientific obstacles remain. Precipitation at the surface is not the same state variable as water vapour in an atmospheric volume. ERA5 reanalysis includes assimilated information that may be unavailable to an operational forecaster. Historical-state estimation may be non-unique, especially for rapid convective change. Grid-scale conservation depends on variables, coordinate surfaces and parameterised sources; forcing exact numerical agreement with an inaccurate field can worsen local forecasts. Finally, an attractive three-dimensional visualisation is not evidence that physical equations have been solved.
The proposed experiment will succeed scientifically even if the constrained emulator does not outperform simpler baselines, provided it identifies where conservation improves consistency, where it imports bias, and what information is needed to close the interface budget. The contribution will be determined by reproducible comparisons rather than by the complexity of the model.
7. Conclusion
Coupled atmospheric volumes offer a testable representation of Caribbean storm dynamics, but the separate tasks of fine-scale rainfall reconstruction, historical-state inference and future forecasting must not be conflated. Preliminary Jamaican results support a limited claim about a rainfall aggregation constraint and, more importantly, demonstrate the need to benchmark against strong seasonal methods. Physical interface consistency, 3D reconstruction and forecast skill remain questions for the proposed ERA5- and satellite-informed experiments.
References
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- Hersbach, H. et al. (2020). The ERA5 global reanalysis. Quarterly Journal of the Royal Meteorological Society, 146, 1999–2049. https://doi.org/10.1002/qj.3803.
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