Pacanowski-Philander
\[\newcommand{\c} {\, ,}
\newcommand{\p} {\, .}
\newcommand{\d} {\partial}
\newcommand{\r}[1] {\mathrm{#1}}
\newcommand{\ee} {\mathrm{e}}
\newcommand{\beq} {\begin{equation}}
\newcommand{\eeq} {\end{equation}}
\newcommand{\beqs} {\begin{gather}}
\newcommand{\eeqs} {\end{gather}}
\newcommand{\Ri} {\mathrm{Ri}}\]
In the model proposed by Pacanowski and Philander (1981), turbulent fluxes are diffusive, so that
\[\overline{w \phi} = K_\Phi \d_z \Phi \c\]
where the diffusivity for velocity fields, $K_U$, is
\[\beq \label{momentumdiffusivity}
K_U = \nu_0 + \frac{\nu_1}{\left ( 1 + c \Ri \right )^n} \c
\eeq\]
while the diffusivity for tracer fields is
\[\beq \label{tracerdiffusivity}
K_T = \kappa_0 + \frac{\kappa_1}{\left ( 1 + c \Ri \right )^{n+1}} \p
\eeq\]
In \eqref{momentumdiffusivity} and \eqref{tracerdiffusivity}, the local Richardson number $\Ri$ is defined
\[\beq
Ri = \frac{\d_z B}{\left ( \d_z U \right )^2 + \left ( \d_z V \right )^2} \c
\eeq\]
in terms of the buoyancy $B = - g \rho' / \rho_0$, where $g$ is gravitational acceleration, $\rho_0$ is a reference density, and $\rho'$ is the density deviation therefrom. With the linear equation of state
\[\beq
\rho = \rho_0 \left [ 1 - \alpha \left ( T - T_0 \right ) + \beta \left ( S - S_0 \right ) \right ]
\eeq\]
near some reference temperature $T_0$ and reference salinity $S_0$, buoyancy $B$ is given by
\[\beq
B = g \left [ \alpha \left ( T - T_0 \right ) - \beta \left ( S - S_0 \right ) \right ] \c
\eeq\]
and its vertical derivative is
\[\beq
\d_z B = g \left ( \alpha \d_z T - \beta \d_z S \right ) \p
\eeq\]
Parameters
Typical values for the model parameters in PP (see the text following equation 19 in chapter 3 of CV12) are
| Parameter | Value | Units |
|---|---|---|
| $\nu_0$ | $10^{-4}$ | $\r{m^2 \, s^{-1}}$ |
| $\nu_1$ | $10^{-2}$ | $\r{m^2 \, s^{-1}}$ |
| $\kappa_0$ | $10^{-5}$ | $\r{m^2 \, s^{-1}}$ |
| $\kappa_1$ | $10^{-2}$ | $\r{m^2 \, s^{-1}}$ |
| $c$ | $5$ | none |
| $n$ | $2$ | none |