A02_sgs_scheme_details.tex

% Deardorff 1980 implementation in PALM
%	Prognostic equation for SGS TKE
%	\begin{equation} \label{eq:eprog}
%	\frac{\partial e}{\partial t} = 
%	-u_j \frac{\partial e}{\partial x_j} 
%	- (\overline{u''_i u''_j})\frac{\partial u_i}{\partial x_j}
%	+ \frac{g_i}{\rho_0}\overline{u''_i \rho''}
%	- \frac{\partial}{\partial x_j}[\overline{u_j''(e+\frac{p''}{\rho_0})}]
%	- \varepsilon
%	\end{equation}

%	modified perturbation pressure:
%	\begin{equation} \label{eq:pi}
%	\pi^* = p^* + \frac{2}{3}\rho_0 e
%	\end{equation}
%	where $p^*$ is the perturbation pressure and the turbulent kinetic energy $e$ is

%	\begin{equation} \label{eq:sadiff}
%	\overline{u''_i Sa''} = -K_h \frac{\partial Sa}{\partial x_i}
%	\end{equation}
	
%	Eddy diffusivity of momentum
%	\begin{equation} \label{eq:Km}
%	K_m = c_m l \sqrt{e}
%	\end{equation}

%	Eddy diffusivity of heat and salt
%	\begin{equation} \label{eq:Kh}
%	K_h = (1+\frac{2l}{\Delta})K_m
%	\end{equation}
%    where
%	\begin{equation} \label{eq:gridl}
%	\Delta = \sqrt[3]{\Delta x \Delta y \Delta z}
%	\end{equation}

%Turbulence closure by the gradient mean approximation
%\begin{equation} \label{eq:momdiff}
%\overline{u''_i u_j''} - \frac{2}{3}e \delta_{ij} = -K_m(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})
%\end{equation}

%\begin{equation} \label{eq:ptdiff}
%\overline{u''_i \theta''} = -K_h \frac{\partial \theta}{\partial x_i}
%\end{equation}

%\begin{equation} \label{eq:e}
%e = \frac{1}{2}\overline{u'_i u'_i}
%\end{equation}