The resolution of climate models is limited by computational cost.
Therefore, we must rely on parameterizations to represent processes
occurring below the scale resolved by the models. Here, we focus on
parameterizations of ocean mesoscale eddies and employ machine learning
(ML), namely relevance vector machines (RVM) and convolutional neural
networks (CNN), to derive computationally efficient parameterizations
from data, which are interpretable and/or encapsulate physics. In
particular, we demonstrate the usefulness of the RVM algorithm to reveal
closed-form equations for eddy parameterizations with embedded
conservation laws. When implemented in an idealized ocean model, all
parameterizations improve the statistics of the coarse-resolution
simulation. The CNN is more stable than the RVM such that its skill in
reproducing the high-resolution simulation is higher than the other
schemes; however, the RVM scheme is interpretable. This work shows the
potential for new physics-constrained interpretable ML turbulence
parameterizations for use in ocean climate models.