= Turbulence closure =

== Deardorff subgrid-scale model ==

One of the main challenges in LES modeling is the turbulence closure. The filtering process yields four SGS covariance terms (see the first five equations Sect. [wiki:doc/tec/gov governing equations]) that cannot be explicitly calculated. In PALM, these SGS terms are parametrized using a 1.5-order closure after [#deardorff Deardorff (1980)]. PALM uses the modified version of [#moeng Moeng and Wyngaard (1988)] and [#saiki Saiki et al. (2000)]. The closure is based on the assumption that the energy transport by SGS eddies is proportional to the local gradients of the mean quantities and reads
{{{
#!Latex
\begin{align*}
&   \overline{u_i^{\prime\prime} u_j^{\prime\prime}} - \frac{2}{3} e \delta_{ij} = -K_\mathrm{m} \left(\frac{\partial u_{i}} {\partial x_{j}} + \frac{\partial u_j} {\partial x_{i}}\right),\,\\
&  \overline{u^{\prime\prime}_{i}\theta^{\prime\prime}} = -K_\mathrm{h} \:\frac{\partial \theta} {\partial x_{i}},\,\\
&  \overline{u^{\prime\prime}_{i}q^{\prime\prime}_\mathrm{v}} = -K_\mathrm{h} \:\frac{\partial q_\mathrm{v}} {\partial x_{i}},\,\\
&  \overline{u^{\prime\prime}_{i}s^{\prime\prime}} = -K_\mathrm{h} \:\frac{\partial s}{\partial x_{i}},\,
\end{align*}
}}}
where ''K'',,m,, and ''K'',,h,, are the local SGS eddy diffusivities of momentum and heat, respectively. They are related to the SGS-TKE as follows
{{{
#!Latex
\begin{align*}
&K_\mathrm{m} = c_\mathrm{m}\;l\sqrt{e},\\
&  K_\mathrm{h} = \left(1+\frac{2l}{\boldsymbol{\varDelta}}\right)  K_\mathrm{m}.
\end{align*}
}}}
Here, ''c'',,m,, = 0.1 is a model constant and '''''Δ''' = (Δx Δy Δz)^1/3^'' with ''Δx'', ''Δy'', ''Δz'' being the grid spacings in ''x'', ''y'' and ''z'' direction, respectively. The SGS mixing length ''l'' depends on height ''z'' (distance from the wall when topography is used), '''''Δ''''', and stratification and is calculated as
{{{
#!Latex
\begin{align*}
  & l =
\begin{cases}
\min\left(1.8z,\boldsymbol{\varDelta},
  0.76\sqrt{e}\left(\frac{g}{\theta_{\mathrm{v},0}}
 \frac{\partial{\theta_{\mathrm{v}}}}{\partial z}\right)^{-\frac{1}{2}} \right) &\text{for~}
    \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} > 0, \\
 \min\left(1.8z, \boldsymbol{\varDelta}\right) &\text{for~}    \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} \leq 0.
\end{cases}
\end{align*}
}}}
Moreover, the closure includes a prognostic equation for the SGS-TKE:
{{{
#!Latex
\begin{align*}
  & \frac{\partial{e}}{\partial t} = - u_j\frac{\partial
    e}{\partial x_j} - \left(\overline{u_i^{\prime\prime}
      u_j^{\prime\prime}}\right)\frac{\partial u_i} {\partial x_j} +
  \frac{g}{\theta_{\mathrm{v},0}}\overline{u_3^{\prime\prime}
    {\theta_{\mathrm{v}}}^{\prime\prime}}-\frac{\partial}{\partial x_j} \left[\overline{u_j^{\prime\prime}
      \left(e + \frac{p^{\prime\prime}}{\rho_0}\right)}\right] -
  \epsilon.
\end{align*}
}}}
The pressure term in the equation above is parametrized as
{{{
#!Latex
\begin{align*}
  &\left[\overline{u_j^{\prime\prime} \left(e +
        \frac{p^{\prime\prime}}{\rho_0}\right)}\right] = -2
  K_\mathrm{m} \frac{\partial e}{\partial x_j}
\end{align*}
}}}
and ''ε'' is the SGS dissipation rate within a grid volume, given by
{{{
#!Latex
\begin{align*}
  &  \epsilon=\left(0.19 +
    0.74\frac{l}{\boldsymbol{\varDelta}}\right)\frac{e^{\frac{3}{2}}}{l}.
\end{align*}
}}}
Since ''θ'',,v,, depends on ''θ'', ''q'',,v,,, and ''q'',,l,, (see last equation Sect. [wiki:doc/tec/gov governing equations]), the vertical SGS buoyancy flux depends on the respective SGS fluxes ([#stull Stull, 1988, Chap. 4.4.5]):
{{{
#!Latex
\begin{align*}
  & \overline{w^{\prime\prime}
    {\theta_{\mathrm{v}}}^{\prime\prime}}=K_1\,\cdot\,\overline{w^{\prime\prime}
    {\theta}^{\prime\prime}} + K_2\,\cdot\,\overline{w^{\prime\prime}
    {q_\mathrm{v}}^{\prime\prime}}-
  \theta\,\cdot\,\overline{w^{\prime\prime}
    {q_\mathrm{l}}^{\prime\prime}},
\end{align*}
}}}
with
{{{
#!Latex
\begin{align*}
  & K_1 =1+\left(\frac{R_\mathrm{v}}{R_\mathrm{d}}-1\right) q_\mathrm{v}  - q_\mathrm{l},\\
  & K_2 =\left(\frac{R_\mathrm{v}}{R_\mathrm{d}}-1\right)\,\theta,
\end{align*}
}}}
and the vertical SGS flux of liquid water, calculated as
{{{
#!Latex
\begin{align*}
  & \overline{w^{\prime\prime} {q_\mathrm{l}}^{\prime\prime}} = -K_\mathrm{h} \:\frac{\partial q_\mathrm{l}} {\partial z}.
\end{align*}
}}}
Note that this parametrization of the SGS buoyancy flux differs from that used with bulk cloud microphysics (see Sect. [wiki:doc/tec/microphysics#Turbulenceclosure turbulence closure in cloud microphysics]).


== Dynamic subgrid-scale model ==

The dynamic SGS model, based on an idea of [#germano Germano et al. (1991)], can be used as an alternative to the Moeng-Wyngaard version of the Deardorff model. In this case, ''K'',,m,, is calculated as
{{{
#!Latex
\begin{align*}
   K_\mathrm{m} = c_*\;\Delta_\mathrm{max}\;\sqrt{e},
\end{align*}
}}}
where ''Δ'',,max,, being the maximum of ''Δx'', ''Δy'', ''Δz''.
The calculation of ''c'',,*,, is based on an idea of [#germano Germano et al. (1991)] to use a test filter, which is
{{{
#!Latex
\begin{align*}
   \Delta_T = 2\Delta_\mathrm{max}
\end{align*}
}}}
in our case. The subgrid stress on the test filter scale is then
{{{
#!Latex
\begin{align*}
   T_{ij} = \widehat{\overline{u_iu_j}} - \widehat{\overline{u}}_i\widehat{\overline{u}}_j
\end{align*}
}}}
(the hat denotes a filter operation on the test filter scale) which is also an unknown. The difference between subgrid stress on the test filter level and test filtered subgrid stress is described by the Germano identity
{{{
#!Latex
\begin{align*}
   L_{ij} = T_{ij} - \widehat{\tau}_{ij} = \widehat{\overline{u}_i\overline{u}_j} - \widehat{\overline{u}}_i\widehat{\overline{u}}_j
\end{align*}
}}}
and can be calculated directly by application of the test filter on resolved quantities. ''c'',,*,, is then calculated via
{{{
#!Latex
\begin{equation*}
   c_*=-\frac{L_{ij}^d\widehat{\overline{S}}^d_{ij}}{2\nu_t^T\widehat{\overline{S}}_{lk}^d\widehat{\overline{S}}_{kl}^d},
\end{equation*}
}}}
where
{{{
#!Latex
\begin{align*}
   \overline{S}_{ij} = \frac{1}{2}\left(\frac{\partial\overline{u_i}}{\partial x_j} + \frac{\partial\overline{u_j}}{\partial x_i} \right)
\end{align*}
}}}
the strain tensor and ''ν,,t,,'' the SGS viscosity. Unlike other dynamic models this formulation of ''c'',,*,, is not derived using model assumptions for the subgrid stress and the stress on the test filter level, but is based on proven turbulence properties ([#heinz2008 Heinz, 2008]; [#heinz2012 Heinz and Gopalan, 2012]). Furthermore, the stability of the simulation is ensured by using dynamic bounds that keep the values of ''c'',,*,, in the range
{{{
#!Latex
\begin{equation*}
   |c_*| \leq \frac{23}{24\sqrt{3}}\frac{e^{1/2}}{\Delta|\overline{S}|},
\end{equation*}
}}}
as was derived by [#mokhtarpoor2017 Mokhtarpoor and Heinz (2017)]. This model does not need artificial clipping for stable runs and allows the occurence of backscatter (negative values of ''ν,,t,,'').


== References ==
* [=#deardorff]'''Deardorff JW.''' 1980. Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Lay. Meteorol. 18: 495–527.

* [=#germano]'''Germano M., Piomelli U., Moin P., Cabot WH.''' 1991. A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A 3: 1760–1765.

* [=#heinz2008]'''Heinz S.''' 2008. Realizability of dynamic subgrid-scale stress models via stochastic analysis. Monte Carlo Methods Appl. 14: 311-329.

* [=#heinz2012]'''Heinz S., Gopalan H.''' 2012. Realizable versus non-realizable dynamic subgrid-scale stress models. Physics of Fluids 24: 115105.

* [=#moeng]'''Moeng CH, Wyngaard JC.''' 1988. Spectral analysis of large-eddy simulations of the convective boundary layer. J. Atmos. Sci. 45: 3573–3587.

* [=#mokhtarpoor2017]'''Mokhtarpoor R., Heinz S.''' 2017. Dynamic large eddy simulation: Stability via realizability. Physics of Fluids 29: 105104.

* [=#saiki]'''Saiki EM, Moeng CH, Sullivan PP.''' 2000. Large-eddy simulation of the stably stratified planetary boundary layer. Bound.-Lay. Meteorol. 95: 1–30.

* [=#stull]'''Stull RB.''' 1988. An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers. Dordrecht. 666 pp.