HClimRep-Ocean: A Global Ocean Emulator on an Unstructured Mesh

Kacper Nowak; Aleksei Koldunov; Nikolay Koldunov; Savvas Melidonis; Ankit Patnala; Simon Grasse; Julius Polz; Christian Lessig

physics.ao-phSep 23, 2026

HClimRep-Ocean: A Global Ocean Emulator on an Unstructured Mesh

Authors: Kacper Nowak, Aleksei Koldunov, Nikolay Koldunov, Savvas Melidonis, Ankit Patnala, Simon Grasse, Julius Polz, Christian Lessig, +2 more

Organizations: Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany · Forschungszentrum Jülich GmbH, Jülich Supercomputing Center, Jülich, Germany · Karlsruhe Institute of Technology, Karlsruhe, Germany · European Center for Medium-Range Weather Forecasts, Bonn, Germany · University of Cologne, Department of Mathematics and Computer Science, Cologne, Germany · Department of Physics and Electrical Engineering, University of Bremen, Bremen, Germany

Abstract

Machine-learning (ML) emulators for atmospheric processes have advanced rapidly in recent years, transforming weather forecasting. Although early ML ocean forecasting models now exist, they remain less developed than their atmospheric counterparts. Unlike the atmosphere, much of the ocean's kinetic energy resides in mesoscale eddies whose characteristic spatial scales are approximately an order of magnitude smaller than those of comparable atmospheric features. Moreover, complex coastlines, narrow straits, and ice-covered seas make boundary representation a central challenge that atmospheric models do not face. Consequently, numerical ocean simulations commonly use locally refined or even completely unstructured meshes. However, their data-driven counterparts have so far been built around latitude-longitude grids. We present HClimRep-Ocean, an ocean emulator that operates directly on the native unstructured mesh of FESOM2. The emulator is trained on a 209-year AWI-CM3 control integration and is run without atmospheric forcing, receiving the atmospheric state only at initialisation time, which isolates the predictability carried by the ocean state itself. Skill is strongly field-dependent: for currents, HClimRep-Ocean outperforms every reference at 30 day forecast, whereas for temperature and salinity a damped-anomaly persistence forecast remains the more accurate estimator. This behaviour is physically interpretable: current variability is largely geostrophic and internally generated, whereas sea-surface temperature and salinity fluctuations are driven by atmospheric forcing through weather state. Evaluated independently on the OceanBench benchmark, a reanalysis-trained variant of HClimRep-Ocean achieves the lowest RMSE against GLORYS reanalysis among all assessed systems, confirming the competitiveness of the native-mesh approach.

Figures & tables

Region	Nominal resolution
High latitudes (poleward of $60^{\circ}$ )	$20$ km
Tropics ( $30^{\circ}$ S– $30^{\circ}$ N)	$25$ km
Mid - latitudes ( $30^{\circ}$ – $60^{\circ}$ )	$35$ km
Remaining ocean	$50-120$ km

Table 1: Typical horizontal resolution of the CORE2 mesh.

Figure 1: The HClimRep-Ocean model makes use of the WeatherGenerator model prototype which comprises of a series of attention-based transformer blocks. There are three main groups (engines) of such blocks: the local assimilation engine, the global assimilation engine, and the forecasting engine. Each of those engines attends to a different level of the model’s latent representation: the local assimilation engine attends within each cell to combine embedded input tokens from different streams into a fixed number of latent vectors per cell; then the global assimilation engine, by using dense attention over local neighborhoods on the sphere which are defined through the HEALPix subdivision, processes these vectors across cells combining them in a latent Earth system state; the forecasting engine attends to this global state to advance the temporal window by temporal window $\Delta t$ .

	RMSE		Skill vs pers.
Field	+15 d	+30 d	+15 d	+30 d
SST	$0.61{\pm}0.07$ ∘ C	$0.69{\pm}0.08$ ∘ C	$0.30{\pm}0.12$	$0.50{\pm}0.11$
SSS	$0.25{\pm}0.01$ psu	$0.31{\pm}0.02$ psu	$0.08{\pm}0.11$	$0.20{\pm}0.11$
SSH	$0.045{\pm}0.005$ m	$0.050{\pm}0.005$ m	$0.20{\pm}0.07$	$0.18{\pm}0.07$
$T$ @ 25 m	$0.55{\pm}0.03$ ∘ C	$0.65{\pm}0.04$ ∘ C	$0.23{\pm}0.11$	$0.41{\pm}0.12$
$T$ @ 95 m	$0.37{\pm}0.01$ ∘ C	$0.46{\pm}0.02$ ∘ C	$0.31{\pm}0.04$	$0.37{\pm}0.04$

Table 2: Forecast accuracy of the FESOM2 emulator at the 15- and 30-day horizons. Root-mean-square errors and standard deviations are area-weighted global means over wet mesh points, verified against the FESOM2 reference simulation. Skill is measured against a persistence forecast that carries the initial state forward, and is formed for each initialisation before averaging rather than as a ratio of the averaged errors. Uncertainties are the standard deviation across the 329 initialisations.

Figure 2: Surface forecast skill of the emulator against three references, versus lead time (mean over 329 daily initializations of the validation year; shading shows

\pm 1\sigma

across those initializations). Columns: sea-surface temperature (SST), salinity (SSS), height (SSH) and current speed

|U|

at 2 m; top row root-mean-square error (RMSE, weighted by the FESOM2 node and element areas), bottom row anomaly correlation coefficient (ACC) relative to the 30-year day-of-year climatology. References are persistence, damped persistence and climatology. The emulator beats raw persistence and climatology at all but the shortest leads, but only the currents clearly beat the damped-persistence baseline; for SST and SSS the emulator does not, and for SSH only marginally near two weeks.

Figure 3: Where the forecast beats the seasonal cycle. Each panel maps the skill against the day-of-year climatology, $1-\mathrm{RMSE}/\mathrm{RMSE}_{\mathrm{clim}}$ , evaluated at every mesh point over the $N=329$ held-out initialisations at the 15-day horizon; red denotes a forecast more accurate than climatology, blue less accurate and white equally accurate, and the scale is clipped at $-1$ . Velocity skill is computed from the vector error, $\mathrm{RMSE}_{|U|}^{2}=\mathrm{RMSE}_{u}^{2}+\mathrm{RMSE}_{v}^{2}$ , applied to both the forecast and the climatology. The layout matches Fig. 7 so that the two may be compared directly: that figure shows where the forecast carries the right amount of variability, this one where it is actually more accurate than the seasonal cycle. Skill is broadly positive in the tropics and subtropics for all three fields and negative over the Arctic and parts of the subpolar gyres for surface temperature.

Figure 4: Forecast skill as a joint function of lead time and depth, for temperature, salinity and the two horizontal velocity components (columns). The upper row shows the skill against the day-of-year climatology, $1-\mathrm{RMSE}/\mathrm{RMSE}_{\mathrm{clim}}$ , so that positive values indicate a forecast more accurate than climatology, zero indicates equal accuracy and negative values indicate that climatology would have been the better estimate; the scale is clipped at $-1$ . The lower row shows the anomaly correlation on a fixed $0$ – $1$ scale.

Figure 5: Vertical structure of the forecast, for temperature, salinity and the two horizontal velocity components (columns). Top row: area-weighted mean profiles of the reference simulation and of the emulator at +15 and +30 d. Second and third rows: root-mean-square error against depth at +15 and +30 d, for the emulator and the three baselines. Bottom row: mean error (bias, emulator

-

reference) at the same two leads. Shading shows

\pm 1\sigma

across 60 initialisations spanning the held-out year; statistics use the FESOM2 node and element areas and exclude cells below the local bathymetry. The mean stratification is preserved to +30 d, and the emulator improves on every baseline for the velocity components at all depths, but below roughly 100 m at +15 d — and below 35 m at +30 d — its salinity error exceeds that of climatology.

		SST RMSE ( ∘ C)		SST skill vs pers.		SSH RMSE (m)
Local season (N / S)	$N$	NH	SH	NH	SH	NH	SH
Winter (DJF / JJA)	59 / 92	$0.52$	$0.37$	$0.16$	$0.23$	$0.059$	$0.048$
Spring (MAM / SON)	92 / 86	$0.65$	$0.47$	$0.27$	$0.23$	$0.042$	$0.042$
Summer (JJA / DJF)	92 / 59	$0.99$	$0.69$	$0.25$	$0.21$	$0.037$	$0.039$
Autumn (SON / MAM)	86 / 92	$0.65$	$0.47$	$0.44$	$0.42$	$0.053$	$0.044$

Table 3: Seasonal dependence of forecast error at the 15-day horizon, ordered by local season so that each row compares the two hemispheres in the same physical season; the calendar seasons that this corresponds to are given in parentheses (northern / southern) and

N

is the number of initialisations contributing to each entry. Scores are area-weighted means within each hemisphere, with skill measured against persistence as in Table 2 . Surface temperature error peaks for local summer initializations and skill against persistence peaks for local autumn initializations in both hemispheres, whereas sea-surface height error peaks in local winter: the seasonal cycles are of the same shape in the two hemispheres and opposite in sign between the thermal and the dynamic surface field.

Figure 6: Spatial variability of the forecast. Wavelength spectra computed directly on the native triangular mesh by implicit filtering [ 34 ] , for sea-surface height, sea-surface temperature and surface kinetic energy, comparing the reference simulation with the emulator at the 15- and 30-day horizons; curves are averaged over the initialisations sampled for this diagnostic. Wavelength decreases to the right and the axis stops at 200 km, below which the CORE2 mesh no longer resolves the flow reliably. Forecast and reference coincide at basin scales and separate towards smaller ones for the two tracer fields, whereas kinetic energy is deficient by a roughly constant factor at every resolved scale. The 15- and 30-day curves nearly overlap, indicating that the spectral deficit is established early in the rollout.

Figure 7: Geographical distribution of the variance deficit. Each panel shows $\log_{2}$ of the ratio between the forecast and reference variance across the all initialisations at the 15-day horizon, so that zero denotes a forecast with the correct amount of variability, negative values (blue) too little and positive values (red) too much; eddy kinetic energy is formed from the two horizontal velocity components. Surface temperature variance is close to correct and locally excessive in the western boundary currents and the Southern Ocean, whereas eddy kinetic energy is deficient almost everywhere outside the equatorial band, most severely in the Gulf Stream extension, the Kuroshio, the Antarctic Circumpolar Current and the subtropical gyre interiors. The global mean of this field is therefore dominated by the tropics and understates the deficit in the regions where the mesoscale is most energetic.

Figure 8: Temporal power spectra of the 30-day forecasts (top; truth solid, emulator dashed) and the emulator/truth power ratio (bottom) for SST, SSS, SSH and current speed at 2 m and 107 m. Spectra are computed per node from linearly detrended 30-day lead series, area-weighted over the mesh and averaged over 24 initialisations (two per calendar month), with shading showing

\pm 1\sigma

across those initialisations; resolved periods are 2–30 d. The hatched band (periods

\geq

15 d, i.e. at most two cycles in the record and affected by detrending) is qualitative only; open markers mark the Nyquist period (2 d). The percentage in each lower panel is the sub-weekly (

<

7 d) power retained by the emulator. For currents and SSH the emulator damps the short periods, most strongly at the surface (18 % of sub-weekly power retained at 2 m, 44 % at 107 m, 33 % for SSH). For SST and SSS the loss within the 30-day window is broad-band and largest at periods of 10–30 d (ratio

\approx

0.3), reflecting the decay of anomaly amplitude rather than a high-frequency cut-off; their sub-weekly power is retained at

\approx

50 %.

Figure 9: Behaviour of the emulator over a single 180-day rollout, decomposed through Eq. ( 2 ). (a) Error relative to the day-of-year climatology; the dashed line marks parity with climatology and the dotted line the value

\sqrt{2}

reached by a forecast that has lost all correlation with the reference while keeping its amplitude. (b) Anomaly correlation with the reference: the phase information. (c) Ratio of anomaly standard deviations: the amplitude, on a logarithmic axis so that equal factors above and below unity are equally far from it. Panels (b) and (c) together determine (a). All statistics are area-weighted with the FESOM2 node and element areas.

	SST		SSS		Merid. current		Zonal current		SSH
Model	d1	d10	d1	d10	d1	d10	d1	d10	d1	d10
HClimRep	0.57	0.64	0.41	0.39	0.11	0.13	0.11	0.13	0.06	0.06
GLO12	0.55	0.66	0.63	0.64	0.13	0.16	0.13	0.17	0.07	0.08
GLONET	0.67	0.91	0.56	0.56	0.13	0.15	0.13	0.15	0.08	0.10
WenHai	0.55	0.61	0.59	0.58	0.13	0.14	0.13	0.15	0.07	0.08
XiHe	0.63	0.79	0.52	0.50	0.13	0.13	0.14	0.14	0.08	0.09

Table 4: Surface RMSD against the GLORYS12 reanalysis for all OceanBench challengers at lead days 1 and 10. Bold entries mark the lowest (best) value in each column. Units: SST in °C, SSS in PSU, currents in m s -1 , SSH in m. LangYa does not report day-10 scores.

System	Grid	Forcing	Depth	Training data	Lead time
HClimRep	Unstructured	No	20	AWI-CM3 / GLORYS12	30 d
XiHe [ 9 ]	Regular $1/12^{\circ}$	Yes	23	GLORYS12	10–60 d
WenHai [ 7 ]	Regular $1/12^{\circ}$	Yes	23	GLORYS12	10 d
GLONET [ 11 ]	Regular $1/4^{\circ}$	No	21	GLORYS12	10 d
LangYa [ 35 ]	Regular $1/12^{\circ}$	Yes	32	GLORYS12	7 d
Njord [ 21 ]	$1/4^{\circ}$ + $k$ -means	Yes	6 + sea ice	GLORYS12 (+GLO12)	10 d

Table 5: Qualitative comparison of global ocean ML systems. “Forcing” indicates whether the model receives atmospheric fields during the forecast rollout. “Grid” indicates the spatial discretisation. “Depth” indicates the number of vertical levels. “Training data” indicates the primary training target.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Region	Longitude	Latitude
Gulf Stream	$-80$ to $-40$	$30$ to $50$
Kuroshio	$130$ to $170$	$25$ to $45$
Agulhas	$10$ to $40$	$-45$ to $-30$
Brazil–Malvinas	$-60$ to $-40$	$-50$ to $-30$
Tropical Pacific	$-180$ to $-80$	$-10$ to $10$
Southern Ocean	all	$-65$ to $-45$

Appendix

Table 6: Region definitions. Longitudes are given in the range

-180^{\circ}

180^{\circ}

	Skill vs persistence				Retained EKE
Region	SST	SSS	SSH	$\|U\|$	2 m	107 m
Gulf Stream	$+0.17{\pm}0.32$	$-0.15{\pm}0.22$	$-0.15{\pm}0.43$	$+0.17{\pm}0.20$	$0.17$	$0.62$
Kuroshio	$+0.19{\pm}0.31$	$-0.62{\pm}0.44$	$+0.06{\pm}0.24$	$+0.17{\pm}0.19$	$0.23$	$0.79$
Agulhas	$-0.02{\pm}0.30$	$-1.36{\pm}0.40$	$+0.06{\pm}0.19$	$+0.21{\pm}0.21$	$0.25$	$0.68$
Brazil–Malvinas	$+0.28{\pm}0.27$	$-0.40{\pm}0.44$	$-0.03{\pm}0.37$	$+0.19{\pm}0.23$	$0.22$	$0.65$
Tropical Pacific	$+0.46{\pm}0.09$	$+0.20{\pm}0.13$	$+0.49{\pm}0.10$	$+0.51{\pm}0.08$	$0.64$	$0.85$

Appendix

Table 7: Regional skill at the 15-day horizon, measured against persistence as in Table 2 . Skill is formed for each of the 329 held-out initialisations and the table gives its mean

\pm

the standard deviation across them; the standard error of the mean is

0.02

or smaller throughout, so the differences between regions are well resolved. The last two columns give the median fraction of the reference eddy kinetic energy retained at the same lead, at 2 and 107 m, computed about the day-of-year climatology over the same initialisations. Regions where the emulator retains more of the mesoscale are also the regions in which it forecasts best.

Figure 10: Distribution of the skill against persistence across 329 held-out initializations, by region and field, at 15-day horizon (box: interquartile range, whiskers: 5th to 95th percentile, line: median). Dashed lines mark parity with persistence. The distributions are strongly asymmetric — skill cannot exceed one but is unbounded below — and their width varies as much between regions as their mean does: in tropical Pacific every field is positive in

94

–

100\,\%

of initializations, whereas in Gulf Stream surface temperature is positive in

71\,\%

and sea-surface height in only

44\,\%

. Salinity in the Agulhas is negative in every single initialisation.

Figure 11: Fraction of the reference eddy kinetic energy retained by the emulator, by region. Eddy kinetic energy is formed about the day-of-year climatology and averaged within each region with the FESOM2 element areas; each box shows the distribution over the 329 held-out initialisations (box: interquartile range, whiskers: 5th to 95th percentile, line: median). Colour denotes depth and fill denotes lead time. The deficit is severe at the surface in the eddy-active regions — a median of

0.17

in the Gulf Stream against

0.64

in the tropical Pacific — and three to four times smaller at 107 m. It is also established early: the medians change little between +15 and +30 days, except in the Agulhas and the Southern Ocean, where the deficit continues to deepen. The spread between initialisations is itself informative: in the western boundary currents the retained fraction ranges from below

0.1

to above

0.4

depending on the start date, whereas the tropical Pacific and the Southern Ocean behave consistently. The Kuroshio at 107 m is the only case in which the emulator exceeds the reference, carrying more deep eddy energy than it should in

24\,\%

of initialisations at +30 days.

Stage	Horizon	$\eta_{\max}$	Trainable weights	Steps
Pre-training	4	$5\times 10^{-5}$	all ( $1.138\times 10^{9}$ )	$\sim$ 32 900
Forecast fine-tuning	15	$10^{-5}$	forecasting engine	$\sim$ 6 700
Final refinement	15	$10^{-5}$	global aggregation engine	1 408

Appendix

Table 8: Training stages. Each stage is initialised from the preceding checkpoint; “steps” are optimiser steps at a global batch of eight rollouts, the horizon is given in daily forecast steps and

\eta_{\max}

is the peak learning rate.

Model Component	Configuration	Value
Embedding	embed_unembed_mode	block
	HEALPix level	5
	Dropout rate	0.1
Local Assimilation Engine	Attention blocks	4
	Attention heads	16
	Embedding dimension	2048

Appendix

Table 9: Model configuration of HClimRep-Ocean prototype model.

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Skillful Global Ocean Emulation and the Role of Correlation-Aware Loss

Niraj Agarwal, Timothy A. Smith, Sergey Frolov, Laura C. Slivinski

CIRES, University of Colorado, Boulder, CO, USA · NOAA Physical Sciences Laboratory, Boulder, CO, USA

Data Assimilation Ocean Role

Open paper AskPDF arXiv

Machine learning emulators have shown extraordinary skill in forecasting atmospheric states, and their application to global ocean dynamics offers similar promise. Here, we adapt the GraphCast architecture into a dedicated ocean-only emulator, driven by prescribed atmospheric conditions, for medium-range predictions. The emulator is trained on NOAA's UFS-Replay dataset. Using a 24 hour time step, single initial condition, and without using autoregressive training, we produce an emulator that provides skillful forecasts for 10-15 day lead times. We further demonstrate the use of Mahalanobis distance as loss that improves the forecast skill compared to the Mean Squared Error loss by explicitly accounting for the correlations between tendencies of the target variables. Using spatial correlation analysis of the forecasted fields, we also show that the proposed correlation-aware loss acts as a statistical-dynamical regularizer for the slow, correlated dynamics of the global oceans, offering a better background forecast for downstream tasks like data assimilation.

Abstract

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