Machine Learning based vertical diffusivity in EPBL

These changes enhance the existing vertical diffusivity used in EPBL with machine learned diffusivity that is constrained on a second moment closure. Using symbolic regression and least-squares fitting, a shape function ( g(\sigma) ) responds to changes in the surface forcing (Latitude, wind stress, surface buoyancy flux, boundary layer depth). g(\sigma) goes from zero to 1 and its skewness changes as per surface forcing conditions.

There are two options for the turbulent velocity scale: v0 and v0h.

The velocity scale, v0, is an approximation that depends on three inputs: Coriolis parameter f, ustar u_*, and surface buoyancy flux BFlux. It agrees with diagnosed velocity scale from GLS scheme given by v0 = < { \kappa } max / h >, where h is boundary layer depth and angled brackets denote averaged data for a given forcing: B, u_*, f. 
When v0 is multiplied with g(\sigma) and multiplied by the energetics based boundary layer depth h, i.e \nu = . g(\sigma) \cdot v_0 \cdot h, we get a diffusivity which is constrained on a second moment closure.

The second velocity scale, v0h, depends on  ustar u_*, surface buoyancy flux BFlux, and boundary layer depth. It agrees with diagnosed velocity scale from GLS scheme given by v0^h =  { \kappa } max / h , where h is the boundary layer depth. v0^h agrees with convective velocity scaling under pure convection ( v0^h ~ (Bh)^(1/3) ).

When v0 or v0^h is multiplied with g(\sigma) and multiplied by the energetics based boundary layer depth h, i.e \nu = . g(\sigma) \cdot v_0 \cdot h, we get a diffusivity which is constrained on a second moment closure.

The Machine learned (ML) diffusivity is activated by using the logical flags:
1. EPBL_EQD_DIFFUSIVITY_SHAPE
2. EPBL_EQD_DIFFUSIVITY_VELOCITY
3. EPBL_EQD_DIFFUSIVITY_VELOCITY_H

Logical flag 1 activates the new equation for shape function.
Logical flag 2 activates velocity scale v0
Logical flag 3 activates velocity scale v0^h

To use ML diffusivity, Logical flag 1 has to be set to ‘True’ and either logical flag 2 or 3 should be set to ‘True’. 2 and 3 both cannot be true or false.
The subroutines replace the default EPBL mixing coefficients of Reichl and Hallberg (2018).
The publications for OSBL machine learned diffusivity are: Sane et al. 2023 ( https://doi.org/10.1029/2023MS003890 ) and Sanę et al. 2025.

Author: aakashsane