\chapter{Vertical Mixing Scheme:~\citet{Pacanowski81}}
\label{app:Pacanowski}

Vertical viscosity and diffusivity $A_{v}$ and $K_{v}$ are given by
the formulae:
\begin{align}
A_{v}&=0.02+min(10,100R^2)+R|W|dz/Re\label{eq:PPAV} \\
K_{v}&=0.02+Rmin(10,100R^2)+R|W|dz/Re\label{eq:PPKV}
R&=\fc{1}{1+5Ri}
\end{align}
$|W|$ is the vertical velocity magnitude, $dz$ is the vertical grid interval,
$Ri$ is the gradient Richardson number, and $Re$ is the specified vertical
cell Reynolds number. This is simply a non-dimensional ratio of advection
to turbulent diffusion for variations having the scale of order one model grid
interval.

The first term in Eqns~(\ref{eq:PPAV}) and~(\ref{eq:PPKV}) represent laminar
diffusivity. The second one is a standard Pacanowski and Philander mixing
parameterization. The third js a generally small mixed physical/numerical
term, assigned a value of 10, which avoids cell Reynolds number overshoots by
vertical advection, such as produced by internal waves. It allows extremely
small vertical mixing necessary to maintain the cold intermediate layer
throughout the model year. The physical processes represented by terms two
and three are the vertical Reynolds stress terms representing the nonlinear
interaction of subgrid-scale flow components with resolved scales. These
diffusion terms are conventional ad hoc representations of the subgrid-scale
effects which have limited basis outside of boundary layers.
