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

ParameterValueUnits
$\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