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

Neural networks offer novel ways to parameterize unresolved ocean mixing, but they are challenging to interpret. We present a two‑step approach to parameterize vertical diffusivity for the ocean surface boundary layer. First, we train neural networks that predict diffusivity constrained on second moment closure. Second, symbolic regression and least-squares give equations fitted on network inferred data. The resulting expressions match neural‑network performance in global forced simulations while providing interpretation. 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. Stabilizing forcing shifts the mixing peak towards the surface while convective forcing moves it to mid‑boundary layer depth. The amplitude of diffusivity set by the velocity scale arises from the balance between surface - shear and buoyancy flux. Equations yield a transparent, efficient, and physically grounded vertical diffusivity applicable for ocean models.