\varepsilon\varepsilon= Turbulence Paramereization =

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Since r????, 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.

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

The //TKE-l// model uses the following prognostic equation to calculate //e//:
{{{
#!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//,,m,,, //K//,,h,,, and //K//,,e,, are the eddy diffusivities of momentum and heat, and the diffusivity coefficient of the TKE, respectively, which are calculated via
{{{
#!Latex
\begin{align*}
   K_\mathrm{m} &= c_0 \ l \ \sqrt{e}, \\
   K_\mathrm{h} &= \frac{K_\mathrm{m}}{\mathrm{Pr}}, \\
   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/inipar#rans_const_sigma rans_const_sigma].
Here, //Pr// and //l// denote the Prandtl number and mixing length, respectively.
The model constant //c//,,0,, is set to 0.55 by default, but can be altered via the namelist parameter [/wiki/doc/app/inipar#rans_const_c rans_const_c].
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 Monin-Obukhov length, and the height, respectively.

The dissipation rate of the TKE, ε, is calculated via
{{{
#!Latex
\begin{equation*}
   \varepsilon = c_0^3 \ e \ \frac{\sqrt{e}}{l}.
\end{equation*}
}}}

== [=#tkee_model TKE-ε model] ==

The TKE-ε model uses an additional prognostic equation to calculate the dissipation rate ε of the TKE:
{{{
#!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*}
}}}
The diffusivity coefficient //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/inipar#rans_const_sigma rans_const_sigma].