nu.tex

\section{Viscosity and Diffusion}
\label{Nu}

\subsection{Horizontal viscosity}
The horizontal viscosity and diffusion coefficients are scalars which
are read in from \code{ocean.in}.  Several
factors to consider when choosing these values are:
\begin{klist}
   \kitem{spindown time}  The spindown time on wavenumber $k$ is ${1
   \over k^2 \nu_2}$ for the Laplacian operator and ${1 \over k^4
   \nu_4}$ for the biharmonic operator.  The smallest wavenumber
   corresponds to the length $2 \Delta x$ and is $k = {\pi \over
   \Delta x}$, leading to
\begin{equation}
    \Delta t < t_{damp} = {\Delta x^2 \over \pi^2 \nu_2}
    \mbox{\quad or \quad} {\Delta x^4 \over \pi^4 \nu_4}
\end{equation}
This time should be short enough to damp out the numerical noise
which is being generated but long enough on the larger scales to
retain the features you are interested in.  This time should also be
resolved by the model timestep.
   \kitem{boundary layer thickness} The western boundary layer has a
   thickness proportional to 
\begin{equation}
    \Delta x < L_{BL} = \left( {\nu_2 \over \beta} \right)^{1 \over 3}
    \mbox{\quad and \quad}
    \left( {\nu_4 \over \beta} \right)^{1 \over 5}
\end{equation}
for the Laplacian and biharmonic viscosity, respectively.  We have
found that the model typically requires the boundary layer to be
resolved with at least one grid cell.  This leads to coarse grids
requiring large values of $\nu$.
\end{klist}

\subsection{Horizontal Diffusion}
We have chosen anything from zero to the value of the horizontal
viscosity for the horizontal diffusion coefficient.  One common choice
is an order of magnitude smaller than the viscosity.

\subsection{Vertical Viscosity and Diffusion}
ROMS stores the vertical mixing coefficients in arrays with three
spatial dimensions called
\code{Akv} and \code{Akt}.  \code{Akt} also has a fourth dimension
specifying which tracer, so
that temperature and salt can have differing values.  Both \code{Akt}
and \code{Akv} are stored at $w$-points in the model;
horizontal averaging is done to obtain \code{Akv} at the horizontal $u$
and $v$-points.  The values for these coefficients can be set in a
number of ways, as described in \S\ref{Vmix}.