Advection-Diffusion Equation for local isotropy in the oceanic well-mixed layer

[Darwin123]

My question is whether differential equations using a time
dependent diffusivity can be used to model the diffusivity of a
passive tracer in the ocean.
I have found models where the diffusivity in the ocean is modeled
by a 4/3 power law. The four thirds power law is associated with local
isotropy. However, none of the researchers go and substitute the
diffusivity into the advection-diffusion equations. In fact, I have
found references that claim the advection-diffusion equations are
inappropriate for time varying diffusivity. However, if this is the
case I see no value to a formula for diffusivity. I would like to use
that 4/3 power law to model the evolution of a tracer plume in the
ocean, including patchiness.
I am trying to solve a problem concerning turbulent diffusion of a
passive tracer in the middle ocean in the well mixed layer. I am
interested in a plume which is much smaller than the Lagrangian
distance but larger than the integral scale length. I am assuming that
the wind driven surface waves are driving the turbulence. Therefore,
the integral scale length is on the order of the wave height while the
integral scale velocity is on the order of the maximum velocity of the
surface. I know from my reading that turbulence in the well mixed
layer has a local isotropy, but not a homogeneous isotropy.
Most of the treatments concerning diffusion in the ocean involve
the Naver Stokes equation. Is there any reason that the standard
advection-diffusion equation can't be used as a simplification?