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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 current code originates in the NCAR NCOM 1-D code
+ and was kindly provided by Bill Large and Jan Morzel.
+ It has been adapted first to the MITgcm vector code and
+ subsequently to the current parallel code.
+ Adjustment were mainly in conjunction with WRAPPER requirements
+ (domain decomposition and threading capability), to enable
+ automatic differentiation of tangent linear and adjoint code
+ via TAMC.
+ 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}),
+ (\ref{sec:pkg:kpp:flowchart}),
+ and diagnostics output of KPP-derived diffusivities, viscosities
+ and boundary-layer/mixed-layer depths
+ (\ref{sec:pkg:kpp:diagnostics}).
  %----------------------------------------------------------------------
- \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 26 
 using the \texttt{packages.conf} file by
+Line 96 
 using the \texttt{packages.conf} file by
  or using \texttt{genmake2} adding
  \texttt{-enable=kpp} or \texttt{-disable=kpp} switches
  %
+ \item
+ \textit{Required packages and CPP options:} \\
+ No additional packages are required, but the MITgcm kernel flag
+ enabling the penetration of shortwave radiation below
+ the surface layer needs to be set in \texttt{CPP\_OPTIONS.h}
+ as follows: \\
+ \texttt{\#define SHORTWAVE\_HEATING}
+ %
  \end{itemize}
- (see Section \ref{sect:buildingCode}).
+ (see Section \ref{sec:buildingCode}).
  Parts of the KPP code can be enabled or disabled at compile time
  via CPP preprocessor flags. These options are set in
  \texttt{KPP\_OPTIONS.h}. Table \ref{tab:pkg:kpp:cpp} summarizes them.
- \begin{table}[h!]
+ \begin{table}[!ht]
+ \centering
    \label{tab:pkg:kpp:cpp}
    {\footnotesize
      \begin{tabular}{|l|l|}
-Line 69 
 via CPP preprocessor flags. These option
+Line 148 
 via CPP preprocessor flags. These option
  \end{table}
  %----------------------------------------------------------------------
  \subsubsection{Run-time parameters
-Line 104 
 kernel need to be set in conjunction wit
+Line 182 
 kernel need to be set in conjunction wit
  \paragraph{Package flags and parameters}
  ~ \\
  %
- \begin{table}[h!]
+ Table \ref{tab:pkg:kpp:runtime_flags} summarizes the
+ runtime flags that are set in \texttt{data.pkg}, and
+ their default values.
+ \begin{table}[!ht]
+ \centering
    \label{tab:pkg:kpp:runtime_flags}
    {\footnotesize
      \begin{tabular}{|l|c|l|}
-Line 207 
 kernel need to be set in conjunction wit
+Line 290 
 kernel need to be set in conjunction wit
  %----------------------------------------------------------------------
- \subsubsection{Equations
+ \subsubsection{Equations and key routines
  \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}.
- \subsubsection{Key subroutines
+ \paragraph{KPP\_CALC:} Top-level routine. \\
- \label{sec:pkg:kpp:subroutines}}
+ ~
- \paragraph{kpp\_calc:} Top-level routine. \\
+ \paragraph{KPP\_MIX:} Intermediate-level routine \\
  ~
- \paragraph{kpp\_mix:} Intermediate-level routine \\
+ \paragraph{BLMIX: Mixing in the boundary layer} ~ \\
+ %
  ~
- \paragraph{ri\_iwmix:} ~ \\
+ 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}
+ In practice, the routine peforms the following tasks:
+ %
+ \begin{enumerate}
+ %
+ \item
+ compute velocity scales at hbl
+ %
+ \item
+ find the interior viscosities and derivatives at hbl
+ %
+ \item
+ compute turbulent velocity scales on the interfaces
+ %
+ \item
+ compute the dimensionless shape functions at the interfaces
+ %
+ \item
+ compute boundary layer diffusivities at the interfaces
+ %
+ \item
+ compute nonlocal transport term
+ %
+ \item
+ find diffusivities at kbl-1 grid level
+ %
+ \end{enumerate}
+ \paragraph{RI\_IWMIX: Mixing in the interior} ~ \\
  %
  Compute interior viscosity and diffusivity coefficients due to
  %
-Line 234 
 shear instability (dependent on a local
+Line 408 
 shear instability (dependent on a local
  to background internal wave activity, and
  %
  \item
- to static instability (local Richardson number < 0).
+ to static instability (local Richardson number $<$ 0).
  %
  \end{itemize}
+ TO BE CONTINUED.
- \paragraph{bldepth:} ~ \\
+ \paragraph{BLDEPTH: Boundary layer depth calculation:} ~ \\
  %
  The oceanic planetary boundary layer depth, \texttt{hbl}, is determined as
  the shallowest depth where the bulk Richardson number is
-Line 262 
 stable/ustable forcing conditions, and w
+Line 437 
 stable/ustable forcing conditions, and w
  to grid points (caseA), so that conditional branches can be
  avoided in later subroutines.
- \paragraph{blmix:} ~ \\
+ TO BE CONTINUED.
- %
- Compute boundary layer mixing coefficients.
- Mixing coefficients within boundary layer depend on surface
- forcing and the magnitude and gradient of interior mixing below
- the boundary layer ("matching").
- %
- \begin{enumerate}
- %
- \item
- compute velocity scales at hbl
- %
- \item
- find the interior viscosities and derivatives at hbl
- %
- \item
- compute turbulent velocity scales on the interfaces
- %
- \item
- compute the dimensionless shape functions at the interfaces
- %
- \item
- compute boundary layer diffusivities at the interfaces
- %
- \item
- compute nonlocal transport term
- %
- \item
- 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:} ~ \\
+ TO BE CONTINUED.
+ \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.
+ TO BE CONTINUED.
+ \paragraph{Implicit time integration} ~ \\
+ %
+ TO BE CONTINUED.
+ \paragraph{Penetration of shortwave radiation} ~ \\
+ %
+ TO BE CONTINUED.
+ %----------------------------------------------------------------------
+ \subsubsection{Flow chart
+ \label{sec:pkg:kpp:flowchart}}
  {\footnotesize
  \begin{verbatim}
-Line 341 
 c  o
+Line 506 
 c  o
  Diagnostics output is available via the diagnostics package
  (see Section \ref{sec:pkg:diagnostics}).
- Available output fields are summarized in
+ Available output fields are summarized here:
- Table \ref{tab:pkg:kpp:diagnostics}.
- \begin{table}[h!]
- \label{tab:pkg:kpp:diagnostics}
- {\footnotesize
  \begin{verbatim}
  ------------------------------------------------------
   <-Name->|Levs|grid|<--  Units   -->|<- Tile (max=80c)
-Line 359 
 Table \ref{tab:pkg:kpp:diagnostics}.
+Line 520 
 Table \ref{tab:pkg:kpp:diagnostics}.
   KPPmld  |  1 |SM  |m               |Mixed layer depth, dT=.8degC density criterion
   KPPfrac |  1 |SM  |                |Short-wave flux fraction penetrating mixing layer
  \end{verbatim}
- }
- \caption{~}
- \end{table}
  %----------------------------------------------------------------------
-Line 374 
 natl\_box:
+Line 532 
 natl\_box:
  %----------------------------------------------------------------------
  \subsubsection{References}
+ \subsubsection{Experiments and tutorials that use kpp}
+ \label{sec:pkg:kpp:experiments}
+ \begin{itemize}
+ \item{Labrador Sea experiment, in lab\_sea verification directory }
+ \end{itemize}

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