Turbulence closures

The turbulence closure selected by the user determines the form of stress divergence   and diffusive flux divergence   in the momentum and tracer conservation equations.

Constant isotropic diffusivity

In a constant isotropic diffusivity model, the kinematic stress tensor is defined

where is a constant viscosity and   is the strain-rate tensor. The divergence of is then

Similarly, the diffusive tracer flux is    for tracer diffusivity , and the diffusive tracer flux divergence is

Each tracer may have a unique diffusivity .

Constant anisotropic diffusivity

A constant anisotropic diffusivity implies a constant tensor diffusivity and stress   with non-zero components    and  . With this form the kinematic stress divergence becomes

and diffusive flux divergence

in terms of the horizontal viscosities and diffusivities, and , and the vertical viscosity and diffusivities, and . Each tracer may have a unique diffusivity components and .

Scalar biharmonic diffusivity

A constant biharmonic diffusivity implies a constant tensor diffusivity and stress  with non-zero components    and  .

With this form the kinematic stress divergence becomes

and diffusive flux divergence

in terms of the horizontal biharmonic viscosities and diffusivities, and , and the vertical biharmonic viscosity and diffusivities, and . Each tracer may have a unique diffusivity components and .

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   is the resolved strain rate. The eddy viscosity is given by

where is the "filter width" associated with the finite volume grid spacing and is a user-specified model constant,  . The factor reduces in regions of strong stratification via

where   is the squared buoyancy frequency for stable stratification with   and is a user-specified constant. Lilly (1962) proposed  , where is a turbulent Prandtl number. The filter width for the Smagorinsky-Lilly closure is

where , , and are the grid spacing in the , , and directions at location  .

The effect of subgrid turbulence on tracer mixing is also modeled diffusively via

where the eddy diffusivity is

Both and may be set independently for each tracer.

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 and , such that

to ensure that   and  , where and are the constant isotropic background viscosity and diffusivities for each tracer. The eddy viscosity predictor is

while the eddy diffusivity predictor for tracer is

In the definitions of the eddy viscosity and eddy diffusivity predictor, and are user-specified model constants, is a "filter width" associated with the finite volume grid spacing, and the hat decorators on partial derivatives, velocities, and the Kronecker delta are defined such that

A velocity gradient, for example, is therefore  , while the normalized strain tensor is

The filter width in that appears in the viscosity and diffusivity predictors is taken as the square root of the harmonic mean of the squares of the filter widths in each direction:

The constant permits the "buoyancy modification" term it multiplies to be omitted from a calculation. By default we use the model constants   and  .

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  , or (ii) convective_νz and convective_κz if  .

This closure is a plausible model for convection if convective_κz background_κz and convective_νz background_νz.