An eddy-diffusivity mass-flux (EDMF) scheme

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The EDMF family of schemes parameterizes turbulent convection by introducing a conditional average that partitions the subgrid boundary layer flow into a turbulent 'environment' with area $a_0$, and non-turbulent updrafts and downdrafts with areas $a_i$ for $i>0$.

We consider two types of EDMF schemes: those with prognostic equations that model the time-evolution and spatial distribution of turbulent kinetic energy (TKE), and those that parameterize the effect of turbulent environmental mixing with a 'K-profile'.

Turbulent eddy diffusivty and mass flux

In all schemes, the turbulent velocity fluxes are parameterized with an eddy diffusivity. For the $x$-velocity $U$, for example,

\[\beq \overline{w u} = \d_z \left ( K \d_z U \right ) \c \eeq\]

where $K$ is the eddy diffusivity. We consider various models for eddy diffusivity ranging from a model similar to the K-profile parameterizaton (KPP), and a formulation in terms of a prognostic, time- and $z$-dependent turbulent kinetic energy variable, $e$. The turbulent flux of scalars $\phi$ such as temperature and salinity is parameterized by both a turbulent flux and mass transport,

\[\beq \overline{u \phi} = \d_z \left ( K \d_z \Phi \right ) - \d_z \sum_i a_i \tilde W_i \tilde \Phi_i \c \eeq\]

where $\tilde \Phi_i$ is the difference between the average of $\phi$ within domain $i$ and the total horizontal average $\Phi$:

\[\begin{align} \tilde \Phi_i & \defn \left ( \frac{1}{A_i} \int_{A_i} \r{d} A - \frac{1}{A} \int_A \r{d} A \right ) \phi \c \\ &= \Phi_i - \Phi \c \end{align}\]

where we have introduced the notation $\Phi_i$ to denote the average of $\phi$ within the environment or updraft area $A_i$. The terms $a_i \tilde W_i \tilde \Phi_i$ account for the vertical transport of $\phi$ by environment and updraft vertical velocities $W_i$.

Zero-plume, 1.5-order EDMF scheme

A relatively simple EDMF scheme emerges in the limit of vanishing updrafts and downdrafts, in which case $W = W_0 = a_0 = 0$. In the 1.5-order version of this closure, turbulent diffusivity is modeled via the prognostic turbulent kinetic energy (TKE) equation

\[\beq \d_t e = K \left [ \left ( \d_z U \right )^2 + \left ( \d_z V \right )^2 \right ] + \d_z \left ( K \d_z e \right ) - K \d_z B - \C{\ep}{} \frac{e^{3/2}}{\ell} \c \label{TKE} \eeq\]

where $\C{\ep}{} = 2.0$ is a model parameter, $\d_z B = g \left( \alpha \d_z T - \beta \d_z S \right )$ is the buoyancy gradient in terms of gravitational acceleration $g$ and thermal expansion and haline contraction coefficients $\alpha$ and $\beta$, and $K$ is the eddy diffusivity defined in terms of turbulent 'velocity' $\sqrt{e}$ and a mixing length $\ell$:

\[\beq K = \C{K}{} \underbrace{ \C{\kappa}{} z \left ( 1 - \F{a}{} \tfrac{Q_b}{\uwind^3} z \right )^{\F{n}{}}} _{\defn \ell} \, \sqrt{e} \p \label{eddydiffusivity} \eeq\]

In \eqref{eddydiffusivity}, $Q_b = g \left ( \alpha Q_\theta - \beta Q_s \right )$ is the buoyancy flux define in terms of temperature and salinity fluxes $Q_\theta$ and $Q_s$, and $\uwind \defn | \b{Q}_u |^{1/2}$ is the friction velocity defined in terms of velocity flux $\b{Q}_u = \b{\tau} / \rho_0$ or wind-stress $\b{\tau}$ and reference density $\rho_0$. $\C{\kappa}{} = 0.41$ and $\C{K}{} = 0.1$ in \eqref{eddydiffusivity} are the 'Von Karman' and eddy diffusivity model parameters, respectively. $\F{a}{}$ and $\F{n}{}$ in \eqref{eddydiffusivity} are piecewise constant model functions that model the effect of boundary layer stability on the mixing length and are

\[\beq \F{a}{} = \left \{ \begin{matrix} -100 & \text{for unstable boundary layers with } Q_b > 0 \\ 2.7 & \text{for stable boundary layers with } Q_b \le 0 \end{matrix} \right . \c \eeq\]

and

\[\beq \F{n}{} = \left \{ \begin{matrix} 0.2 & \text{for unstable boundary layers with } Q_b > 0 \\ -1 & \text{for stable boundary layers with } Q_b \le 0 \end{matrix} \right . \p \eeq\]

Note that parameterized buoyancy flux $\overline{w b} \defn -K \d_z B$ appears in the TKE equation \eqref{TKE}.