  Mesoscale eddies are of utmost importance in understanding ocean dynamics and
the transport of heat, salt, and nutrients. Accurate representation of these
eddies in ocean models is essential for improving model predictions. However,
accurately representing these mesoscale features in numerical models is
challenging due to their relatively small size. In this study, we propose a
convolutional neural network (CNN) that combines data-driven techniques with
physical principles to develop a robust and interpretable parameterization
scheme for mesoscale eddies in ocean modeling. We first analyze a
high-resolution reanalysis dataset to extract subgrid eddy momentum and use
machine learning algorithms to identify patterns and correlations. To ensure
physical consistency, we have introduced conservation of momentum constraints
in our CNN parameterization scheme through soft and hard constraints. The
interpretability analysis illustrate that the pre-trained CNN parameterization
shows promising results in accurately solving the resolved mean velocity at the
local scale and effectively capturing the representation of unresolved subgrid
turbulence processes at the global scale. Furthermore, to validate the CNN
parameterization scheme offline, we conduct simulations using the MITgcm ocean
model. A series of experiments is conducted to compare the performance of the
model with the CNN parameterization scheme and high-resolution simulations. The
offline validation using MITgcm simulations demonstrates the effectiveness of
the CNN parameterization scheme in improving the representation of mesoscale
eddies in the ocean model. Incorporating the CNN parameterization scheme leads
to better agreement with high-resolution simulations and a more accurate
representation of the kinetic energy spectra.
