Machine Learned Equations for Vertical Mixing Coefficients in the Ocean Surface Boundary Layer

Neural networks offer novel ways to parameterize unresolved ocean mixing but are challenging to interpret. Here, we derive compact equations that reproduce the behavior of neural networks trained on a second-moment closure data set. The resulting interpretable expressions employed in a physics-based first order closure scheme match neural-network performance in global forced simulations. They expose a structural error in the baseline physics-based scheme and describe how surface friction velocity, buoyancy flux, rotation, and boundary layer depth regulate diffusivity. The equations reveal a shift in the mixing peak toward the surface under stabilizing conditions and toward the mid-boundary layer depth under convective conditions. The diffusivity amplitude (set by the velocity scale) is controlled by surface shear and buoyancy flux. Equations yield a transparent, efficient, and physically grounded vertical diffusivity applicable for ocean models.