= Turbulence Parameterization =

Since r2696, PALM can be operated as a RANS (Reynolds-averaged Navier-Stokes) model.
When running PALM as a RANS model, a different turbulence closure is used compared to the LES model where the turbulence kinetic energy (TKE) //e// is completely parameterized.

Two different turbulence models are available:
* [#tkel_model TKE-l model],
* [#tkee_model TKE-ε model],
which are described below.

PALM is automatically switched into RANS mode if one of the above listed turbulence parameterizations is selected.
The turbulence parameterization can be selected via the namelist parameter //[/wiki/doc/app/initialization_parameters#turbulence_closure turbulence_closure]//.

Prior to r3545, the namelist parameter //rans_mode// needs to be set to //.TRUE.// in addition of selecting one of the turbulence parameterizations if the RANS mode shall be used.


== [=#tkel_model TKE-l model] ==

The //TKE-l// model calculates the eddy diffusivities via the turbulence kinetic energy //e// and the mixing length //l//:
{{{
#!Latex
\begin{align*}
   K_\mathrm{m} &= c_0 \ l \ \sqrt{e}, \\
   K_\mathrm{h} &= \frac{K_\mathrm{m}}{\mathrm{Pr}},
\end{align*}
}}}
where //Pr// denotes the Prandtl number.
The model constant //c//,,0,, is set to 0.55 by default, but can be altered via the namelist parameter [/wiki/doc/app/initialization_parameters#rans_const_c rans_const_c].

The TKE is calculated using the following prognostic equation:
{{{
#!Latex
\begin{equation*}
   \frac{\partial{e}}{\partial t} =
      - u_j\frac{\partial e}{\partial x_j}
      + K_\mathrm{m} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \frac{\partial u_i}{\partial x_j}
      - \frac{g}{\theta_{\mathrm{v},0}} K_\mathrm{h} \frac{\partial \theta_{\mathrm{v},0}}{\partial z}
      + K_\mathrm{e} \frac{\partial^2 e}{\partial x_j^2}
      - \varepsilon,
\end{equation*}
}}}
where //K//,,e,, is defined as
{{{
#!Latex
\begin{align*}
   K_\mathrm{e} &= \frac{K_\mathrm{m}}{\sigma_e},
\end{align*}
}}}
with //σ//,,e,, = 1.
This can be altered via the namelist parameter [/wiki/doc/app/initialization_parameters#rans_const_sigma rans_const_sigma].
The dissipation rate //ε// of the TKE is calculated via
{{{
#!Latex
\begin{equation*}
   \varepsilon = c_0^3 \ e \ \frac{\sqrt{e}}{l}.
\end{equation*}
}}}

The mixing length is defined using the mixing length //l//,,B,, according to Blackadar (1962) and the Dyer-Businger function //Φ//,,m,,
{{{
#!Latex
\begin{align*}
   l &= \min\left( \frac{l_\mathrm{B}}{\Phi_\mathrm{m}}, l_\mathrm{wall} \right), \\
   l_\mathrm{B} &= \dfrac{\kappa z}{1+\frac{\kappa z}{\lambda}}, \\
   \lambda &= 2.7 \cdot 10^{-4} |U_g| f, \\
   \Phi_\mathrm{m} &= 1+5\frac{z}{L},
\end{align*}
}}}
where //κ//, //f//, //U//,,g,,, //L//, and //z// denote the von-Karman constant, the Coriolis parameter, the geostrophic wind, the Obukhov length, and the height, respectively.
//l//,,wall,, defines an upper limit of the mixing length as the distance to the nearest solid surface (wall).


== [=#tkee_model TKE-ε model] ==

The //TKE-ε// model calculates the eddy diffusivities via the turbulence kinetic energy //e// and the dissipation rate //ε// of the TKE:
{{{
#!Latex
\begin{align*}
   K_\mathrm{m} &= c_0^4 \ \frac{e^2}{\varepsilon}, \\
   K_\mathrm{h} &= \frac{K_\mathrm{m}}{\mathrm{Pr}},
\end{align*}
}}}
where //Pr// denotes the Prandtl number.
The model constant //c//,,0,, is set to 0.55 by default, but can be altered via the namelist parameter [/wiki/doc/app/initialization_parameters#rans_const_c rans_const_c].

The TKE is calculated using the same prognostic equation as the //TKE-l// model.
The dissipation rate //ε// is calculated via the following prognostic equation:
{{{
#!Latex
\begin{equation*}
   \frac{\partial{\varepsilon}}{\partial t} =
      - u_j\frac{\partial \varepsilon}{\partial x_j}
      + c_1 \frac{\varepsilon}{e} K_\mathrm{m} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \frac{\partial u_i}{\partial x_j}
      - c_3 \frac{\varepsilon}{e} \frac{g}{\theta_{\mathrm{v},0}} K_\mathrm{h} \frac{\partial \theta_{\mathrm{v},0}}{\partial z}
      + K_\varepsilon \frac{\partial^2 \varepsilon}{\partial x_j^2}
      - c_2 \frac{\varepsilon^2}{e},
\end{equation*}
}}}
where //K//,,ε,, is defined as
{{{
#!Latex
\begin{equation*}
   K_\varepsilon = \frac{K_\mathrm{m}}{\sigma_\varepsilon}
\end{equation*}
}}}
with //σ//,,ε,, = 1.3.
This can be altered via the namelist parameter [/wiki/doc/app/initialization_parameters#rans_const_sigma rans_const_sigma].
The model constants //c//,,1,,, //c//,,2,,, and //c//,,3,, are set to 1.44, 1.92, and 1.44, respectively.
These values can be altered via the namelist parameter [/wiki/doc/app/initialization_parameters#rans_const_c rans_const_c].

Boundary conditions for //e//, //ε//, and //K//,,m,, are
{{{
#!Latex
\begin{align*}
   e &= \left( \frac{u_*}{c_0} \right)^2, \\
   \varepsilon &= \frac{u_*^3}{\kappa z_\mathrm{MO}}, \\
   K_\mathrm{m} &= \kappa \ u_* \ z_\mathrm{MO} \ \Phi_\mathrm{m}^{-1}(\frac{z_\mathrm{MO}}{L}),
\end{align*}
}}}
where //u//,,*,, denote the friction velocity. Note that //Φ//,,m,, is only used for horizontal boundaries.