Turbulence closures
The turbulence closure selected by the user determines the form of stress divergence
Constant isotropic diffusivity
In a constant isotropic diffusivity model, the kinematic stress tensor is defined
where
Similarly, the diffusive tracer flux is
Each tracer may have a unique diffusivity
Constant anisotropic diffusivity
A constant anisotropic diffusivity implies a constant tensor diffusivity
and diffusive flux divergence
in terms of the horizontal viscosities and diffusivities,
Scalar biharmonic diffusivity
A constant biharmonic diffusivity implies a constant tensor diffusivity
With this form the kinematic stress divergence becomes
and diffusive flux divergence
in terms of the horizontal biharmonic viscosities and diffusivities,
Smagorinsky-Lilly turbulence closure
In the turbulence closure proposed by Lilly (1962) and Smagorinsky (1963), the subgrid stress associated with unresolved turbulent motions is modeled diffusively via
where
where
where
where
The effect of subgrid turbulence on tracer mixing is also modeled diffusively via
where the eddy diffusivity
Both
Anisotropic minimum dissipation (AMD) turbulence closure
The anisotropic minimum dissipation (AMD) model proposed by Verstappen (2018) and was described and tested by Vreugdenhil and Taylor (2018). The AMD model uses an eddy diffusivity hypothesis similar the Smagorinsky-Lilly model. In the AMD model, the eddy viscosity and diffusivity for each tracer are defined in terms of eddy viscosity and diffusivity predictors
to ensure that
while the eddy diffusivity predictor for tracer
In the definitions of the eddy viscosity and eddy diffusivity predictor,
A velocity gradient, for example, is therefore
The filter width
The constant
Convective adjustment vertical diffusivity
This closure aims to model the enhanced mixing that occurs due to convection. At every point and for every time instance, the closure diagnoses the gravitational stability of the fluid and applies the vertical diffusivities (i) background_νz
to u, v
and background_κz
to all tracers if the fluid is gravitationally neutral or stable with convective_νz
and convective_κz
if
This closure is a plausible model for convection if convective_κz
background_κz
and convective_νz
background_νz
.