Turbulence closures

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

Constant isotropic diffusivity

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

τij=νΣij,

where ν is a constant viscosity and Σij12(vi,j+vj,i) is the strain-rate tensor. The divergence of τ is then

τ=ν2v.

Similarly, the diffusive tracer flux is qc=κc for tracer diffusivity κ, and the diffusive tracer flux divergence is

qc=κ2c.

Each tracer may have a unique diffusivity κ.

Constant anisotropic diffusivity

A constant anisotropic diffusivity implies a constant tensor diffusivity νjk and stress τij=νjkui,k with non-zero components ν11=ν22=νh and ν33=νz. With this form the kinematic stress divergence becomes

τ=[νh(x2+y2)+νvz2]v,

and diffusive flux divergence

qc=[κh(x2+y2)+κvz2]c,

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

Scalar biharmonic diffusivity

A constant biharmonic diffusivity implies a constant tensor diffusivity νjk and stressτij=νjkk3ui with non-zero components ν11=ν22=νh and ν33=νz.

With this form the kinematic stress divergence becomes

τ=[νh(x2+y2)2+νvz4]v,

and diffusive flux divergence

qc=[κh(x2+y2)2+κvz4]c,

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

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

τij=2νeΣij,

where Σij=12(vi,j+vj,i) is the resolved strain rate. The eddy viscosity is given by

(1)νe=(CΔf)2Σ2ς(N2/Σ2),

where Δf is the "filter width" associated with the finite volume grid spacing and C is a user-specified model constant, Σ2ΣijΣij. The factor ς(N2/Σ2) reduces νe in regions of strong stratification via

ς(N2/Σ2)=1min(1,CbN2/Σ2),

where N2=max(0,zb) is the squared buoyancy frequency for stable stratification with zb>0 and Cb is a user-specified constant. Lilly (1962) proposed Cb=1/Pr, where Pr is a turbulent Prandtl number. The filter width for the Smagorinsky-Lilly closure is

Δf(x)=(ΔxΔyΔz)1/3,

where Δx, Δy, and Δz are the grid spacing in the x^, y^, and z^ directions at location x=(x,y,z).

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

qc=κec,

where the eddy diffusivity κe is

κe=νePr+κ.

Both Pr and κ 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 νe and κe, such that

νe=max(0,νe)+νandκe=max(0,κe)+κ,

to ensure that νe0 and κe0, where ν and κ are the constant isotropic background viscosity and diffusivities for each tracer. The eddy viscosity predictor is

(2)νe=CΔf2(^kv^i)(^kv^j)Σ^ij+Cbδ^i3(^kvi^)(^kb)(^lv^m)(^lv^m),

while the eddy diffusivity predictor for tracer c is

(3)κe=CΔf2(^kv^i)(^kc)(^ic)(^lc)(^lc).

In the definitions of the eddy viscosity and eddy diffusivity predictor, C and Cb are user-specified model constants, Δf is a "filter width" associated with the finite volume grid spacing, and the hat decorators on partial derivatives, velocities, and the Kronecker delta δ^i3 are defined such that

^iΔii,v^i(x,t)vi(x,t)Δi,andδ^i3δi3Δ3.

A velocity gradient, for example, is therefore ^iv^j(x,t)=ΔiΔjivj(x,t), while the normalized strain tensor is

Σ^ij=12[^iv^j(x,t)+^jv^i(x,t)].

The filter width Δf 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:

1Δf2=13(1Δx2+1Δy2+1Δz2).

The constant Cb permits the "buoyancy modification" term it multiplies to be omitted from a calculation. By default we use the model constants C=1/12 and Cb=0.

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 b/z0, or (ii) convective_νz and convective_κz if b/z<0.

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