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  \subsection{KPP: Nonlocal K-Profile Parameterization for
- Diapycnal Mixing}
+ Vertical Mixing}
  \label{sec:pkg:kpp}
  \begin{rawhtml}
 Line 11 
 Authors: Dimitris Menemenlis and Patrick
  \subsubsection{Introduction
  \label{sec:pkg:kpp:intro}}
+ The nonlocal K-Profile Parameterization (KPP) scheme
+ of \cite{lar-eta:94} unifies the treatment of a variety of
+ unresolved processes involved in vertical mixing.
+ To consider it as one mixing scheme is, in the view of the authors,
+ somewhat misleading since it consists of several entities
+ to deal with distinct mixing processes in the ocean's surface
+ boundary layer, and the interior:
+ %
+ \begin{enumerate}
+ %
+ \item
+ mixing in the interior is goverened by
+ shear instability (modeled as function of the local gradient
+ Richardson number), internal wave activity (assumed constant),
+ and double-diffusion (not implemented here).
+ %
+ \item
+ a boundary layer depth $h$ or \texttt{hbl} is determined
+ at each grid point, based on a critical value of turbulent
+ processes parameterized by a bulk Richardson number;
+ %
+ \item
+ mixing is strongly enhanced in the boundary layer under the
+ stabilizing or destabilizing influence of surface forcing
+ (buoyancy and momentum) enabling boundary layer properties
+ to penetrate well into the thermocline;
+ mixing is represented through a polynomial profile whose
+ coefficients are determined subject to several contraints;
+ %
+ \item
+ the boundary-layer profile is made to agree with similarity
+ theory of turbulence and is matched, in the asymptotic sense
+ (function and derivative agree at the boundary),
+ to the interior thus fixing the polynomial coefficients;
+ matching allows for some fraction of the boundary layer mixing
+ to affect the interior, and vice versa;
+ %
+ \item
+ a ``non-local'' term $\hat{\gamma}$ or \texttt{ghat}
+ which is independent of the vertical property gradient further
+ enhances mixing where the water column is unstable
+ %
+ \end{enumerate}
+ %
+ The scheme has been extensively compared to observations
+ (see e.g. \cite{lar-eta:97}) and is now coomon in many
+ ocean models.
+ The following sections will describe the KPP package
+ configuration and compiling (\ref{sec:pkg:kpp:comp}),
+ the settings and choices of runtime parameters
+ (\ref{sec:pkg:kpp:runtime}),
+ more detailed description of equations to which these
+ parameters relate (\ref{sec:pkg:kpp:equations}),
+ and key subroutines where they are used (\ref{sec:pkg:kpp:subroutines}),
+ and diagnostics output of KPP-derived diffusivities, viscosities
+ and boundary-layer/mixed-layer depths.
  %----------------------------------------------------------------------
- \subsubsection{KPP configuration and compiling}
+ \subsubsection{KPP configuration and compiling
+ \label{sec:pkg:kpp:comp}}
  As with all MITgcm packages, KPP can be turned on or off at compile time
  %
-Line 34 
 via CPP preprocessor flags. These option
+Line 93 
 via CPP preprocessor flags. These option
  \texttt{KPP\_OPTIONS.h}. Table \ref{tab:pkg:kpp:cpp} summarizes them.
  \begin{table}[h!]
+ \centering
    \label{tab:pkg:kpp:cpp}
    {\footnotesize
      \begin{tabular}{|l|l|}
-Line 69 
 via CPP preprocessor flags. These option
+Line 129 
 via CPP preprocessor flags. These option
  \end{table}
  %----------------------------------------------------------------------
  \subsubsection{Run-time parameters
-Line 104 
 kernel need to be set in conjunction wit
+Line 163 
 kernel need to be set in conjunction wit
  \paragraph{Package flags and parameters}
  ~ \\
  %
+ Table \ref{tab:pkg:kpp:runtime_flags} summarizes the
+ runtime flags that are set in \texttt{data.pkg}, and
+ their default values.
  \begin{table}[h!]
+ \centering
    \label{tab:pkg:kpp:runtime_flags}
    {\footnotesize
      \begin{tabular}{|l|c|l|}
-Line 210 
 kernel need to be set in conjunction wit
+Line 274 
 kernel need to be set in conjunction wit
  \subsubsection{Equations
  \label{sec:pkg:kpp:equations}}
+ We restrict ourselves to writing out only the essential equations
+ that relate to main processes and parameters mentioned above.
+ We closely follow the notation of \cite{lar-eta:94}.
+ \paragraph{Mixing in the boundary layer} ~ \\
+ %
+ ~
+ The vertical fluxes $\overline{wx}$
+ of momentum and tracer properties $X$
+ is composed of a gradient-flux term (proportional to
+ the vertical property divergence $\partial_z X$), and
+ a ``nonlocal'' term $\gamma_x$ that enhances the
+ gradient-flux mixing coefficient $K_x$
+ %
+ \begin{equation}
+ \overline{wx}(d) \, = \, -K_x \left(
+ \frac{\partial X}{\partial z} \, - \, \gamma_x \right)
+ \end{equation}
+ \begin{itemize}
+ %
+ \item
+ \textit{Boundary layer mixing profile} \\
+ %
+ It is expressed as the product of the boundary layer depth $h$,
+ a depth-dependent turbulent velocity scale $w_x(\sigma)$ and a
+ non-dimensional shape function $G(\sigma)$
+ %
+ \begin{equation}
+ K_x(\sigma) \, = \, h \, w_x(\sigma) \, G(\sigma)
+ \end{equation}
+ %
+ with dimensionless vertical coordinate $\sigma = d/h$.
+ For details of $ w_x(\sigma)$ and $G(\sigma)$ we refer to
+ \cite{lar-eta:94}.
+ %
+ \item
+ \textit{Nonlocal mixing term} \\
+ %
+ The nonlocal transport term $\gamma$ is nonzero only for
+ tracers in unstable (convective) forcing conditions.
+ Thus, depending on the  stability parameter $\zeta = d/L$
+ (with depth $d$, Monin-Obukhov length scale $L$)
+ it has the following form:
+ %
+ \begin{eqnarray}
+ \begin{array}{cl}
+ \gamma_x \, = \, 0 & \zeta \, \ge \, 0 \\
+ ~ & ~ \\
+ \left.
+ \begin{array}{c}
+ \gamma_m \, = \, 0 \\
+  ~ \\
+ \gamma_s \, = \, C_s
+ \frac{\overline{w s_0}}{w_s(\sigma) h} \\
+  ~ \\
+ \gamma_{\theta} \, = \, C_s
+ \frac{\overline{w \theta_0}+\overline{w \theta_R}}{w_s(\sigma) h} \\
+ \end{array}
+ \right\}
+ &
+ \zeta \, < \, 0 \\
+ \end{array}
+ \end{eqnarray}
+ \end{itemize}
+ \paragraph{Mixing in the interior} ~ \\
+ %
+ ~
+ \paragraph{Implicit time integration} ~ \\
+ %
+ ~
  %----------------------------------------------------------------------
  \subsubsection{Key subroutines
-Line 294 
 find diffusivities at kbl-1 grid level
+Line 436 
 find diffusivities at kbl-1 grid level
  %
  \end{enumerate}
- \paragraph{kpp\_calc\_diff\_t/s, kpp\_calc\_visc:} ~  \\
+ \paragraph{kpp\_calc\_diff\_t/\_s, kpp\_calc\_visc:} ~  \\
  %
  Add contribution to net diffusivity/viscosity from
  KPP diffusivity/viscosity.
- \paragraph{kpp\_transport\_t/s/ptr:} ~ \\
+ \paragraph{kpp\_transport\_t/\_s/\_ptr:} ~ \\
  %
  Add non local KPP transport term (ghat) to diffusive
  temperature/salinity/passive tracer flux.
  The nonlocal transport term is nonzero only for scalars
  in unstable (convective) forcing conditions.
+ \paragraph{Flow chart:} ~ \\
+ %
  {\footnotesize
  \begin{verbatim}
-Line 345 
 Available output fields are summarized i
+Line 489 
 Available output fields are summarized i
  Table \ref{tab:pkg:kpp:diagnostics}.
  \begin{table}[h!]
+ \centering
  \label{tab:pkg:kpp:diagnostics}
  {\footnotesize
  \begin{verbatim}
-Line 374 
 natl\_box:
+Line 519 
 natl\_box:
  %----------------------------------------------------------------------
  \subsubsection{References}

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