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\subsection{KPP: Nonlocal K-Profile Parameterization for |
2 |
Vertical Mixing} |
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|
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\label{sec:pkg:kpp} |
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\begin{rawhtml} |
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<!-- CMIREDIR:package_kpp: --> |
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\end{rawhtml} |
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|
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Authors: Dimitris Menemenlis and Patrick Heimbach |
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|
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\subsubsection{Introduction |
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\label{sec:pkg:kpp:intro}} |
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|
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The nonlocal K-Profile Parameterization (KPP) scheme |
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of \cite{lar-eta:94} unifies the treatment of a variety of |
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unresolved processes involved in vertical mixing. |
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To consider it as one mixing scheme is, in the view of the authors, |
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somewhat misleading since it consists of several entities |
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to deal with distinct mixing processes in the ocean's surface |
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boundary layer, and the interior: |
21 |
% |
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\begin{enumerate} |
23 |
% |
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\item |
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mixing in the interior is goverened by |
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shear instability (modeled as function of the local gradient |
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Richardson number), internal wave activity (assumed constant), |
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and double-diffusion (not implemented here). |
29 |
% |
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\item |
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a boundary layer depth $h$ or \texttt{hbl} is determined |
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at each grid point, based on a critical value of turbulent |
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processes parameterized by a bulk Richardson number; |
34 |
% |
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\item |
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mixing is strongly enhanced in the boundary layer under the |
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stabilizing or destabilizing influence of surface forcing |
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(buoyancy and momentum) enabling boundary layer properties |
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to penetrate well into the thermocline; |
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mixing is represented through a polynomial profile whose |
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coefficients are determined subject to several contraints; |
42 |
% |
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\item |
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the boundary-layer profile is made to agree with similarity |
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theory of turbulence and is matched, in the asymptotic sense |
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(function and derivative agree at the boundary), |
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to the interior thus fixing the polynomial coefficients; |
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matching allows for some fraction of the boundary layer mixing |
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to affect the interior, and vice versa; |
50 |
% |
51 |
\item |
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a ``non-local'' term $\hat{\gamma}$ or \texttt{ghat} |
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which is independent of the vertical property gradient further |
54 |
enhances mixing where the water column is unstable |
55 |
% |
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\end{enumerate} |
57 |
% |
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The scheme has been extensively compared to observations |
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(see e.g. \cite{lar-eta:97}) and is now coomon in many |
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ocean models. |
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|
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The current code originates in the NCAR NCOM 1-D code |
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and was kindly provided by Bill Large and Jan Morzel. |
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It has been adapted first to the MITgcm vector code and |
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subsequently to the current parallel code. |
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Adjustment were mainly in conjunction with WRAPPER requirements |
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(domain decomposition and threading capability), to enable |
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automatic differentiation of tangent linear and adjoint code |
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via TAMC. |
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|
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The following sections will describe the KPP package |
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configuration and compiling (\ref{sec:pkg:kpp:comp}), |
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the settings and choices of runtime parameters |
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(\ref{sec:pkg:kpp:runtime}), |
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more detailed description of equations to which these |
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parameters relate (\ref{sec:pkg:kpp:equations}), |
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and key subroutines where they are used %(\ref{sec:pkg:kpp:subroutines}), |
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(\ref{sec:pkg:kpp:flowchart}), |
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and diagnostics output of KPP-derived diffusivities, viscosities |
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and boundary-layer/mixed-layer depths |
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(\ref{sec:pkg:kpp:diagnostics}). |
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|
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%---------------------------------------------------------------------- |
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|
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\subsubsection{KPP configuration and compiling |
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\label{sec:pkg:kpp:comp}} |
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|
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As with all MITgcm packages, KPP can be turned on or off at compile time |
89 |
% |
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\begin{itemize} |
91 |
% |
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\item |
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using the \texttt{packages.conf} file by adding \texttt{kpp} to it, |
94 |
% |
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\item |
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or using \texttt{genmake2} adding |
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\texttt{-enable=kpp} or \texttt{-disable=kpp} switches |
98 |
% |
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\item |
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\textit{Required packages and CPP options:} \\ |
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No additional packages are required, but the MITgcm kernel flag |
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enabling the penetration of shortwave radiation below |
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the surface layer needs to be set in \texttt{CPP\_OPTIONS.h} |
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as follows: \\ |
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\texttt{\#define SHORTWAVE\_HEATING} |
106 |
% |
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\end{itemize} |
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(see Section \ref{sec:buildingCode}). |
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|
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Parts of the KPP code can be enabled or disabled at compile time |
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via CPP preprocessor flags. These options are set in |
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\texttt{KPP\_OPTIONS.h}. Table \ref{tab:pkg:kpp:cpp} summarizes them. |
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|
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\begin{table}[!ht] |
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\centering |
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\label{tab:pkg:kpp:cpp} |
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{\footnotesize |
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\begin{tabular}{|l|l|} |
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\hline |
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\textbf{CPP option} & \textbf{Description} \\ |
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\hline \hline |
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\texttt{\_KPP\_RL} & |
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~ \\ |
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\texttt{FRUGAL\_KPP} & |
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~ \\ |
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\texttt{KPP\_SMOOTH\_SHSQ} & |
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~ \\ |
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\texttt{KPP\_SMOOTH\_DVSQ} & |
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~ \\ |
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\texttt{KPP\_SMOOTH\_DENS} & |
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~ \\ |
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\texttt{KPP\_SMOOTH\_VISC} & |
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~ \\ |
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\texttt{KPP\_SMOOTH\_DIFF} & |
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~ \\ |
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\texttt{KPP\_ESTIMATE\_UREF} & |
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~ \\ |
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\texttt{INCLUDE\_DIAGNOSTICS\_INTERFACE\_CODE} & |
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~ \\ |
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\texttt{KPP\_GHAT} & |
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~ \\ |
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\texttt{EXCLUDE\_KPP\_SHEAR\_MIX} & |
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~ \\ |
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\hline |
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\end{tabular} |
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} |
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\caption{~} |
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\end{table} |
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|
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|
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%---------------------------------------------------------------------- |
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|
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\subsubsection{Run-time parameters |
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\label{sec:pkg:kpp:runtime}} |
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|
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Run-time parameters are set in files |
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\texttt{data.pkg} and \texttt{data.kpp} |
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which are read in \texttt{kpp\_readparms.F}. |
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Run-time parameters may be broken into 3 categories: |
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(i) switching on/off the package at runtime, |
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(ii) required MITgcm flags, |
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(iii) package flags and parameters. |
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|
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\paragraph{Enabling the package} |
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~ \\ |
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% |
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The KPP package is switched on at runtime by setting |
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\texttt{useKPP = .TRUE.} in \texttt{data.pkg}. |
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|
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\paragraph{Required MITgcm flags} |
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~ \\ |
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% |
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The following flags/parameters of the MITgcm dynamical |
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kernel need to be set in conjunction with KPP: |
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|
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\begin{tabular}{ll} |
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\texttt{implicitViscosity = .TRUE.} & enable implicit vertical viscosity \\ |
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\texttt{implicitDiffusion = .TRUE.} & enable implicit vertical diffusion \\ |
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\end{tabular} |
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|
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|
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\paragraph{Package flags and parameters} |
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~ \\ |
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% |
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Table \ref{tab:pkg:kpp:runtime_flags} summarizes the |
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runtime flags that are set in \texttt{data.pkg}, and |
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their default values. |
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|
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\begin{table}[!ht] |
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\centering |
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\label{tab:pkg:kpp:runtime_flags} |
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{\footnotesize |
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\begin{tabular}{|l|c|l|} |
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\hline |
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\textbf{Flag/parameter} & \textbf{default} & \textbf{Description} \\ |
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\hline \hline |
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\multicolumn{3}{|c|}{\textit{I/O related parameters} } \\ |
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\hline |
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kpp\_freq & \texttt{deltaTClock} & |
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Recomputation frequency for KPP fields \\ |
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kpp\_dumpFreq & \texttt{dumpFreq} & |
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Dump frequency of KPP field snapshots \\ |
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kpp\_taveFreq & \texttt{taveFreq} & |
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Averaging and dump frequency of KPP fields \\ |
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KPPmixingMaps & \texttt{.FALSE.} & |
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include KPP diagnostic maps in STDOUT \\ |
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KPPwriteState & \texttt{.FALSE.} & |
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write KPP state to file \\ |
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KPP\_ghatUseTotalDiffus & \texttt{.FALSE.} & |
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if \texttt{.T.} compute non-local term using total vertical diffusivity \\ |
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~ & ~ & |
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if \texttt{.F.} use KPP vertical diffusivity \\ |
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\hline |
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\multicolumn{3}{|c|}{\textit{Genral KPP parameters} } \\ |
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\hline |
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minKPPhbl & \texttt{delRc(1)} & |
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Minimum boundary layer depth \\ |
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epsilon & 0.1 & |
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nondimensional extent of the surface layer \\ |
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vonk & 0.4 & |
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von Karman constant \\ |
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dB\_dz & 5.2E-5 1/s$^2$ & |
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maximum dB/dz in mixed layer hMix \\ |
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concs & 98.96 & |
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~ \\ |
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concv & 1.8 & |
227 |
~ \\ |
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\hline |
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\multicolumn{3}{|c|}{\textit{Boundary layer parameters (S/R \texttt{bldepth})} } \\ |
230 |
\hline |
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Ricr & 0.3 & |
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critical bulk Richardson number \\ |
233 |
cekman & 0.7 & |
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coefficient for Ekman depth \\ |
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cmonob & 1.0 & |
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coefficient for Monin-Obukhov depth \\ |
237 |
concv & 1.8 & |
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ratio of interior to entrainment depth buoyancy frequency \\ |
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hbf & 1.0 & |
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fraction of depth to which absorbed solar radiation contributes \\ |
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~ & ~ & |
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to surface buoyancy forcing \\ |
243 |
Vtc & \texttt{~} & |
244 |
non-dim. coeff. for velocity scale of turbulant velocity shear \\ |
245 |
~ & ~ & |
246 |
( = function of concv,concs,epsilon,vonk,Ricr) \\ |
247 |
\hline |
248 |
\multicolumn{3}{|c|}{\textit{Boundary layer mixing parameters (S/R \texttt{blmix})} } \\ |
249 |
\hline |
250 |
cstar & 10. & |
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proportionality coefficient for nonlocal transport \\ |
252 |
cg & ~ & |
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non-dimensional coefficient for counter-gradient term \\ |
254 |
~ & ~ & |
255 |
( = function of cstar,vonk,concs,epsilon) \\ |
256 |
\hline |
257 |
\multicolumn{3}{|c|}{\textit{Interior mixing parameters (S/R \texttt{Ri\_iwmix})} } \\ |
258 |
\hline |
259 |
Riinfty & 0.7 & |
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gradient Richardson number limit for shear instability \\ |
261 |
BVDQcon & -0.2E-4 1/s$^2$ & |
262 |
Brunt-V\"ai\"sal\"a squared \\ |
263 |
difm0 & 0.005 m$^2$/s & |
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viscosity max. due to shear instability \\ |
265 |
difs0 & 0.005 m$^2$/s & |
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tracer diffusivity max. due to shear instability \\ |
267 |
dift0 & 0.005 m$^2$/s & |
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heat diffusivity max. due to shear instability \\ |
269 |
difmcon & 0.1 & |
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viscosity due to convective instability \\ |
271 |
difscon & 0.1 & |
272 |
tracer diffusivity due to convective instability \\ |
273 |
diftcon & 0.1 & |
274 |
heat diffusivity due to convective instability \\ |
275 |
\hline |
276 |
\multicolumn{3}{|c|}{\textit{Double-diffusive mixing parameters (S/R \texttt{ddmix})} } \\ |
277 |
\hline |
278 |
Rrho0 & not used & |
279 |
limit for double diffusive density ratio \\ |
280 |
dsfmax & not used & |
281 |
maximum diffusivity in case of salt fingering \\ |
282 |
\hline |
283 |
\hline |
284 |
\end{tabular} |
285 |
} |
286 |
\caption{~} |
287 |
\end{table} |
288 |
|
289 |
|
290 |
|
291 |
%---------------------------------------------------------------------- |
292 |
|
293 |
\subsubsection{Equations and key routines |
294 |
\label{sec:pkg:kpp:equations}} |
295 |
|
296 |
We restrict ourselves to writing out only the essential equations |
297 |
that relate to main processes and parameters mentioned above. |
298 |
We closely follow the notation of \cite{lar-eta:94}. |
299 |
|
300 |
\paragraph{KPP\_CALC:} Top-level routine. \\ |
301 |
~ |
302 |
|
303 |
\paragraph{KPP\_MIX:} Intermediate-level routine \\ |
304 |
~ |
305 |
|
306 |
\paragraph{BLMIX: Mixing in the boundary layer} ~ \\ |
307 |
% |
308 |
~ |
309 |
|
310 |
The vertical fluxes $\overline{wx}$ |
311 |
of momentum and tracer properties $X$ |
312 |
is composed of a gradient-flux term (proportional to |
313 |
the vertical property divergence $\partial_z X$), and |
314 |
a ``nonlocal'' term $\gamma_x$ that enhances the |
315 |
gradient-flux mixing coefficient $K_x$ |
316 |
% |
317 |
\begin{equation} |
318 |
\overline{wx}(d) \, = \, -K_x \left( |
319 |
\frac{\partial X}{\partial z} \, - \, \gamma_x \right) |
320 |
\end{equation} |
321 |
|
322 |
\begin{itemize} |
323 |
% |
324 |
\item |
325 |
\textit{Boundary layer mixing profile} \\ |
326 |
% |
327 |
It is expressed as the product of the boundary layer depth $h$, |
328 |
a depth-dependent turbulent velocity scale $w_x(\sigma)$ and a |
329 |
non-dimensional shape function $G(\sigma)$ |
330 |
% |
331 |
\begin{equation} |
332 |
K_x(\sigma) \, = \, h \, w_x(\sigma) \, G(\sigma) |
333 |
\end{equation} |
334 |
% |
335 |
with dimensionless vertical coordinate $\sigma = d/h$. |
336 |
For details of $ w_x(\sigma)$ and $G(\sigma)$ we refer to |
337 |
\cite{lar-eta:94}. |
338 |
|
339 |
% |
340 |
\item |
341 |
\textit{Nonlocal mixing term} \\ |
342 |
% |
343 |
The nonlocal transport term $\gamma$ is nonzero only for |
344 |
tracers in unstable (convective) forcing conditions. |
345 |
Thus, depending on the stability parameter $\zeta = d/L$ |
346 |
(with depth $d$, Monin-Obukhov length scale $L$) |
347 |
it has the following form: |
348 |
% |
349 |
\begin{eqnarray} |
350 |
\begin{array}{cl} |
351 |
\gamma_x \, = \, 0 & \zeta \, \ge \, 0 \\ |
352 |
~ & ~ \\ |
353 |
\left. |
354 |
\begin{array}{c} |
355 |
\gamma_m \, = \, 0 \\ |
356 |
~ \\ |
357 |
\gamma_s \, = \, C_s |
358 |
\frac{\overline{w s_0}}{w_s(\sigma) h} \\ |
359 |
~ \\ |
360 |
\gamma_{\theta} \, = \, C_s |
361 |
\frac{\overline{w \theta_0}+\overline{w \theta_R}}{w_s(\sigma) h} \\ |
362 |
\end{array} |
363 |
\right\} |
364 |
& |
365 |
\zeta \, < \, 0 \\ |
366 |
\end{array} |
367 |
\end{eqnarray} |
368 |
|
369 |
\end{itemize} |
370 |
|
371 |
In practice, the routine peforms the following tasks: |
372 |
% |
373 |
\begin{enumerate} |
374 |
% |
375 |
\item |
376 |
compute velocity scales at hbl |
377 |
% |
378 |
\item |
379 |
find the interior viscosities and derivatives at hbl |
380 |
% |
381 |
\item |
382 |
compute turbulent velocity scales on the interfaces |
383 |
% |
384 |
\item |
385 |
compute the dimensionless shape functions at the interfaces |
386 |
% |
387 |
\item |
388 |
compute boundary layer diffusivities at the interfaces |
389 |
% |
390 |
\item |
391 |
compute nonlocal transport term |
392 |
% |
393 |
\item |
394 |
find diffusivities at kbl-1 grid level |
395 |
% |
396 |
\end{enumerate} |
397 |
|
398 |
\paragraph{RI\_IWMIX: Mixing in the interior} ~ \\ |
399 |
% |
400 |
Compute interior viscosity and diffusivity coefficients due to |
401 |
% |
402 |
\begin{itemize} |
403 |
% |
404 |
\item |
405 |
shear instability (dependent on a local gradient Richardson number), |
406 |
% |
407 |
\item |
408 |
to background internal wave activity, and |
409 |
% |
410 |
\item |
411 |
to static instability (local Richardson number $<$ 0). |
412 |
% |
413 |
\end{itemize} |
414 |
|
415 |
TO BE CONTINUED. |
416 |
|
417 |
\paragraph{BLDEPTH: Boundary layer depth calculation:} ~ \\ |
418 |
% |
419 |
The oceanic planetary boundary layer depth, \texttt{hbl}, is determined as |
420 |
the shallowest depth where the bulk Richardson number is |
421 |
equal to the critical value, \texttt{Ricr}. |
422 |
|
423 |
Bulk Richardson numbers are evaluated by computing velocity and |
424 |
buoyancy differences between values at zgrid(kl) < 0 and surface |
425 |
reference values. |
426 |
In this configuration, the reference values are equal to the |
427 |
values in the surface layer. |
428 |
When using a very fine vertical grid, these values should be |
429 |
computed as the vertical average of velocity and buoyancy from |
430 |
the surface down to epsilon*zgrid(kl). |
431 |
|
432 |
When the bulk Richardson number at k exceeds Ricr, hbl is |
433 |
linearly interpolated between grid levels zgrid(k) and zgrid(k-1). |
434 |
|
435 |
The water column and the surface forcing are diagnosed for |
436 |
stable/ustable forcing conditions, and where hbl is relative |
437 |
to grid points (caseA), so that conditional branches can be |
438 |
avoided in later subroutines. |
439 |
|
440 |
TO BE CONTINUED. |
441 |
|
442 |
\paragraph{KPP\_CALC\_DIFF\_T/\_S, KPP\_CALC\_VISC:} ~ \\ |
443 |
% |
444 |
Add contribution to net diffusivity/viscosity from |
445 |
KPP diffusivity/viscosity. |
446 |
|
447 |
TO BE CONTINUED. |
448 |
|
449 |
\paragraph{KPP\_TRANSPORT\_T/\_S/\_PTR:} ~ \\ |
450 |
% |
451 |
Add non local KPP transport term (ghat) to diffusive |
452 |
temperature/salinity/passive tracer flux. |
453 |
The nonlocal transport term is nonzero only for scalars |
454 |
in unstable (convective) forcing conditions. |
455 |
|
456 |
TO BE CONTINUED. |
457 |
|
458 |
\paragraph{Implicit time integration} ~ \\ |
459 |
% |
460 |
TO BE CONTINUED. |
461 |
|
462 |
|
463 |
\paragraph{Penetration of shortwave radiation} ~ \\ |
464 |
% |
465 |
TO BE CONTINUED. |
466 |
|
467 |
|
468 |
%---------------------------------------------------------------------- |
469 |
|
470 |
\subsubsection{Flow chart |
471 |
\label{sec:pkg:kpp:flowchart}} |
472 |
|
473 |
|
474 |
{\footnotesize |
475 |
\begin{verbatim} |
476 |
|
477 |
C !CALLING SEQUENCE: |
478 |
c ... |
479 |
c kpp_calc (TOP LEVEL ROUTINE) |
480 |
c | |
481 |
c |-- statekpp: o compute all EOS/density-related arrays |
482 |
c | o uses S/R FIND_ALPHA, FIND_BETA, FIND_RHO |
483 |
c | |
484 |
c |-- kppmix |
485 |
c | |--- ri_iwmix (compute interior mixing coefficients due to constant |
486 |
c | | internal wave activity, static instability, |
487 |
c | | and local shear instability). |
488 |
c | | |
489 |
c | |--- bldepth (diagnose boundary layer depth) |
490 |
c | | |
491 |
c | |--- blmix (compute boundary layer diffusivities) |
492 |
c | | |
493 |
c | |--- enhance (enhance diffusivity at interface kbl - 1) |
494 |
c | o |
495 |
c | |
496 |
c |-- swfrac |
497 |
c o |
498 |
|
499 |
\end{verbatim} |
500 |
} |
501 |
|
502 |
%---------------------------------------------------------------------- |
503 |
|
504 |
\subsubsection{KPP diagnostics |
505 |
\label{sec:pkg:kpp:diagnostics}} |
506 |
|
507 |
Diagnostics output is available via the diagnostics package |
508 |
(see Section \ref{sec:pkg:diagnostics}). |
509 |
Available output fields are summarized here: |
510 |
|
511 |
\begin{verbatim} |
512 |
------------------------------------------------------ |
513 |
<-Name->|Levs|grid|<-- Units -->|<- Tile (max=80c) |
514 |
------------------------------------------------------ |
515 |
KPPviscA| 23 |SM |m^2/s |KPP vertical eddy viscosity coefficient |
516 |
KPPdiffS| 23 |SM |m^2/s |Vertical diffusion coefficient for salt & tracers |
517 |
KPPdiffT| 23 |SM |m^2/s |Vertical diffusion coefficient for heat |
518 |
KPPghat | 23 |SM |s/m^2 |Nonlocal transport coefficient |
519 |
KPPhbl | 1 |SM |m |KPP boundary layer depth, bulk Ri criterion |
520 |
KPPmld | 1 |SM |m |Mixed layer depth, dT=.8degC density criterion |
521 |
KPPfrac | 1 |SM | |Short-wave flux fraction penetrating mixing layer |
522 |
\end{verbatim} |
523 |
|
524 |
%---------------------------------------------------------------------- |
525 |
|
526 |
\subsubsection{Reference experiments} |
527 |
|
528 |
lab\_sea: |
529 |
|
530 |
natl\_box: |
531 |
|
532 |
%---------------------------------------------------------------------- |
533 |
|
534 |
\subsubsection{References} |
535 |
|
536 |
\subsubsection{Experiments and tutorials that use kpp} |
537 |
\label{sec:pkg:kpp:experiments} |
538 |
|
539 |
\begin{itemize} |
540 |
\item{Labrador Sea experiment, in lab\_sea verification directory } |
541 |
\end{itemize} |