\[ \begin{align}\begin{aligned}\newcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\r}[1]{\mathrm{#1}} \newcommand{\bz}{\b{z}} \newcommand{\bu}{\b{u}} \newcommand{\bcdot}{\b{\cdot}} \newcommand{\d}{\partial}\\\newcommand{\p}{\, .} \newcommand{\c}{\, ,}\\\newcommand{\bnabla}{\b{\nabla}} \newcommand{\bcdot}{\b{\cdot}}\end{aligned}\end{align} \]

Anisotropic minimum dissipation

The anisotropic minimum dissipation (AMD) model, like Constant Smagorinsky, models unresolved turbulent subgrid stress \(F^\bu_{ij}\) as the product of the resolved rate of strain \(S_{ij} = \tfrac{1}{2} \left ( \d_i u_j + \d_j u_i \right )\) and an eddy viscosity \(\nu_e\). The subgrid tracer flux, \(F^b\), is modeled similarly in terms of the resolved tracer gradient \(\d_i \theta\) and an eddy viscosity \(\kappa_e\). Unlike Constant Smagorinsky, however, which introduces a turbulent Prandtl number (typically \(O(1)\)) to relate eddy viscosity to eddy diffusivity, the AMD eddy diffusivity \(\kappa_e\) is determined by an analog of the method used to determine eddy visosity.

Basic form

The AMD model is an eddy viscosity model in that the relationship between subgrid stress and rate of strain is

\[F^\bu_{ij} = 2 \nu_e S_{ij} \c\]

while the relationship between subgrid tracer gradients and subgrid tracer flux is

\[\b{F}^\theta = -\kappa_e \bnabla \theta \c\]

where \(\nu_e\) and \(\kappa_e\) are the eddy viscosity and eddy diffusivity. The eddy viscosity and diffusivity are defined in terms of eddy viscosity and diffusivity ‘predictors’ \(\nu_e^\dagger\) and \(\kappa_e^\dagger\), such that

\[\nu_e = \r{max} \left [ 0, \nu_e^\dagger \right ] \c\]

and

\[\kappa_e = \r{max} \left [ 0, \kappa_e^\dagger \right ] \c\]

These formulas ensure that \(\nu_e\) and \(\kappa_e\) are always greater than \(0\).

The eddy viscosity predictor

The buoyancy-modified eddy viscosity predictor \(\nu_e^\dagger\) is determined via the direction-dependent formula (Abkar et al 2017)

\[\nu_e^\dagger = - C \frac{ \left ( \hat{\d}_k u_i \right ) \left ( \hat{\d}_k u_j \right ) S_{ij} - \left ( \hat{\d}_k w \right ) \hat{\d}_k b} {\left ( \d_{\ell} u_m\right )^2} \c\]

where \(\hat{\d}_k\) is the anisotropic scaled gradient operator,

\[\hat{\d}_i = \Delta_i \d_i\]

for ‘filter width’ \(\Delta_i\), and \(C\) is the Poincaré constant. A key feature of the AMD scheme is the direction-dependent, anisotropic filter width \(\Delta_i\). The filter width is typically defined as a multiple of the grid spacing in the \(i^{\r{th}}\) direction. \(\nu_e^\dagger\) then becomes

\[\nu_e^\dagger = - C \frac{ \Delta_k^2 u_{i,k} u_{j,k} S_{ij} - \Delta_k^2 w_{,k} b_{,k}}{\r{tr}(\bnabla \bu)} \c\]

where the tracer, or first invariant of the tensor \(\bnabla \bu\) is

\[\r{tr}(\bnabla \bu) = u_x^2 + v_x^2 + w_x^2 + u_y^2 + v_y^2 + w_y^2 + u_z^2 + v_z^2 + w_z^2\]

Explicitly, \(\Delta_k u_{i,k} u_{j,k} S_{ij}\) is

\[\begin{split}\begin{split} \Delta_k u_{i,k} u_{j,k} S_{ij} &= \, \Delta_1^2 \left (u_x^2 S_{11} + v_x^2 S_{22} + w_x^2 S_{33} + 2 u_x v_x S_{12} + 2 u_x w_x S_{13} + 2 v_x w_x S_{23} \right ) \\ \, & + \Delta_2^2 \left (u_y^2 S_{11} + v_y^2 S_{22} + w_y^2 S_{33} + 2 u_y v_y S_{12} + 2 u_y w_y S_{13} + 2 v_y w_y S_{23} \right ) \\ \, & + \Delta_3^2 \left (u_z^2 S_{11} + v_z^2 S_{22} + w_z^2 S_{33} + 2 u_z v_z S_{12} + 2 u_z w_z S_{13} + 2 v_z w_z S_{23} \right ) \\ \end{split}\end{split}\]

while \(\Delta_k^2 w_{,k} b_{,k}\) is

\[\Delta_k^2 w_{,k} b_{,k} = \Delta_1^2 w_x b_x + \Delta_2^2 w_y b_y + \Delta_3^2 w_z b_z \p\]

The eddy diffusivity predictor

The eddy diffusivity predictor \(\kappa_e\) for a quantity \(\theta\) is

\[\kappa_e^\dagger = - C \frac{ \Delta_k^2 u_{i,k} \theta_{,k} \theta_{,i}}{ \theta_{,\ell}^2 }\]

Note that \(\Delta_k^2 u_{i,k} \theta_{,k} \theta_{,i}\) expands to

\[\Delta_k^2 u_{i,k} \theta_{,k} \theta_{,i} = \bnabla \theta \bcdot \big ( \Delta_1^2 \theta_x \bu_x + \Delta_2^2 \theta_y \bu_y + \Delta_3^2 \theta_z \bu_z \big ) \p\]

Abkar et al 2016 recommend \(C=1/12\) for a spectral method.