Learning Depth-Aware Neural Corrections for Baroclinic Instability in a Mesoscale-Resolving Ocean Model

Three-dimensional ocean dynamics are expensive to resolve and expensive to learn: fully three-dimensional convolutional networks scale poorly with depth, while two-dimensional networks cannot represent the vertical structure of the flow. We address this trade-off by adapting a super-resolution U-Net -previously validated on two-dimensional shallow-water systems- with explicit vertical-coordinate conditioning, so that a two-dimensional network produces depth-resolved three-dimensional corrections. The network is coupled to the coarse-resolution primitive-equation ocean model ICON-O at 6-hour checkpoints, predicting additive corrections to the three-dimensional temperature and horizontal-velocity fields while taking the diagnostic vertical velocity as an auxiliary input. We evaluate the approach in a wind-driven baroclinic channel-flow experiment, using a {0.5}{} simulation as reference and comparing a {5}{}+CNN hybrid against free-running {5}{} and {2}{} configurations. On the evaluated trajectory, the hybrid reduces the L₂ error growth rate, the spatial-correlation decay, and the error-spectrum amplitude to levels close to those of the {2}{} simulation, and substantially enhances the eddy-kinetic-energy and vertical-buoyancy-flux distributions relative to the {5}{} free run. Spectral analysis shows the correction is most effective on the velocity field at all scales; temperature corrections are bounded near the grid scale by unresolved vertical advection. Mass and potential-energy conservation are improved relative to the unassimilated coarse simulation; the implicit energy input from the continuous nudging toward CNN-generated targets remained bounded over the integration window, leaving a net kinetic-energy drift comparable to that of the coarse free run. Ultimately, this study demonstrates that machine-learning corrections can recover a substantial portion of the high-resolution dynamical skill: eddy-kinetic-energy statistics reach the {2}{} level, and the vertical buoyancy flux that drives the baroclinic energy conversion is partially restored, though it remains below the {2}{} value.