| 1 |
\subsection{KPP: Nonlocal K-Profile Parameterization for |
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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 |
| 15 |
of \cite{lar-eta:94} unifies the treatment of a variety of |
| 16 |
unresolved processes involved in vertical mixing. |
| 17 |
To consider it as one mixing scheme is, in the view of the authors, |
| 18 |
somewhat misleading since it consists of several entities |
| 19 |
to deal with distinct mixing processes in the ocean's surface |
| 20 |
boundary layer, and the interior: |
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% |
| 22 |
\begin{enumerate} |
| 23 |
% |
| 24 |
\item |
| 25 |
mixing in the interior is goverened by |
| 26 |
shear instability (modeled as function of the local gradient |
| 27 |
Richardson number), internal wave activity (assumed constant), |
| 28 |
and double-diffusion (not implemented here). |
| 29 |
% |
| 30 |
\item |
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a boundary layer depth $h$ or \texttt{hbl} is determined |
| 32 |
at each grid point, based on a critical value of turbulent |
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processes parameterized by a bulk Richardson number; |
| 34 |
% |
| 35 |
\item |
| 36 |
mixing is strongly enhanced in the boundary layer under the |
| 37 |
stabilizing or destabilizing influence of surface forcing |
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(buoyancy and momentum) enabling boundary layer properties |
| 39 |
to penetrate well into the thermocline; |
| 40 |
mixing is represented through a polynomial profile whose |
| 41 |
coefficients are determined subject to several contraints; |
| 42 |
% |
| 43 |
\item |
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the boundary-layer profile is made to agree with similarity |
| 45 |
theory of turbulence and is matched, in the asymptotic sense |
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(function and derivative agree at the boundary), |
| 47 |
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; |
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% |
| 51 |
\item |
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a ``non-local'' term $\hat{\gamma}$ or \texttt{ghat} |
| 53 |
which is independent of the vertical property gradient further |
| 54 |
enhances mixing where the water column is unstable |
| 55 |
% |
| 56 |
\end{enumerate} |
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% |
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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 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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and diagnostics output of KPP-derived diffusivities, viscosities |
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and boundary-layer/mixed-layer depths. |
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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 |
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% |
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\begin{itemize} |
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% |
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\item |
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using the \texttt{packages.conf} file by adding \texttt{kpp} to it, |
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% |
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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 |
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% |
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\end{itemize} |
| 89 |
(see Section \ref{sect: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}[h!] |
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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}[h!] |
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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 & |
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~ \\ |
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\hline |
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\multicolumn{3}{|c|}{\textit{Boundary layer parameters (S/R \texttt{bldepth})} } \\ |
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\hline |
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Ricr & 0.3 & |
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critical bulk Richardson number \\ |
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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 \\ |
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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 \\ |
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Vtc & \texttt{~} & |
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non-dim. coeff. for velocity scale of turbulant velocity shear \\ |
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~ & ~ & |
| 227 |
( = function of concv,concs,epsilon,vonk,Ricr) \\ |
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\hline |
| 229 |
\multicolumn{3}{|c|}{\textit{Boundary layer mixing parameters (S/R \texttt{blmix})} } \\ |
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\hline |
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cstar & 10. & |
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proportionality coefficient for nonlocal transport \\ |
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cg & ~ & |
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non-dimensional coefficient for counter-gradient term \\ |
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~ & ~ & |
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( = function of cstar,vonk,concs,epsilon) \\ |
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\hline |
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\multicolumn{3}{|c|}{\textit{Interior mixing parameters (S/R \texttt{Ri\_iwmix})} } \\ |
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\hline |
| 240 |
Riinfty & 0.7 & |
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gradient Richardson number limit for shear instability \\ |
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BVDQcon & -0.2E-4 1/s$^2$ & |
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Brunt-V\"ai\"sal\"a squared \\ |
| 244 |
difm0 & 0.005 m$^2$/s & |
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viscosity max. due to shear instability \\ |
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difs0 & 0.005 m$^2$/s & |
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tracer diffusivity max. due to shear instability \\ |
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dift0 & 0.005 m$^2$/s & |
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heat diffusivity max. due to shear instability \\ |
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difmcon & 0.1 & |
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viscosity due to convective instability \\ |
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difscon & 0.1 & |
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tracer diffusivity due to convective instability \\ |
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diftcon & 0.1 & |
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heat diffusivity due to convective instability \\ |
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\hline |
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\multicolumn{3}{|c|}{\textit{Double-diffusive mixing parameters (S/R \texttt{ddmix})} } \\ |
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\hline |
| 259 |
Rrho0 & not used & |
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limit for double diffusive density ratio \\ |
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dsfmax & not used & |
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maximum diffusivity in case of salt fingering \\ |
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\hline |
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\hline |
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\end{tabular} |
| 266 |
} |
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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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%---------------------------------------------------------------------- |
| 273 |
|
| 274 |
\subsubsection{Equations |
| 275 |
\label{sec:pkg:kpp:equations}} |
| 276 |
|
| 277 |
We restrict ourselves to writing out only the essential equations |
| 278 |
that relate to main processes and parameters mentioned above. |
| 279 |
We closely follow the notation of \cite{lar-eta:94}. |
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|
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\paragraph{Mixing in the boundary layer} ~ \\ |
| 282 |
% |
| 283 |
~ |
| 284 |
|
| 285 |
The vertical fluxes $\overline{wx}$ |
| 286 |
of momentum and tracer properties $X$ |
| 287 |
is composed of a gradient-flux term (proportional to |
| 288 |
the vertical property divergence $\partial_z X$), and |
| 289 |
a ``nonlocal'' term $\gamma_x$ that enhances the |
| 290 |
gradient-flux mixing coefficient $K_x$ |
| 291 |
% |
| 292 |
\begin{equation} |
| 293 |
\overline{wx}(d) \, = \, -K_x \left( |
| 294 |
\frac{\partial X}{\partial z} \, - \, \gamma_x \right) |
| 295 |
\end{equation} |
| 296 |
|
| 297 |
\begin{itemize} |
| 298 |
% |
| 299 |
\item |
| 300 |
\textit{Boundary layer mixing profile} \\ |
| 301 |
% |
| 302 |
It is expressed as the product of the boundary layer depth $h$, |
| 303 |
a depth-dependent turbulent velocity scale $w_x(\sigma)$ and a |
| 304 |
non-dimensional shape function $G(\sigma)$ |
| 305 |
% |
| 306 |
\begin{equation} |
| 307 |
K_x(\sigma) \, = \, h \, w_x(\sigma) \, G(\sigma) |
| 308 |
\end{equation} |
| 309 |
% |
| 310 |
with dimensionless vertical coordinate $\sigma = d/h$. |
| 311 |
For details of $ w_x(\sigma)$ and $G(\sigma)$ we refer to |
| 312 |
\cite{lar-eta:94}. |
| 313 |
|
| 314 |
% |
| 315 |
\item |
| 316 |
\textit{Nonlocal mixing term} \\ |
| 317 |
% |
| 318 |
The nonlocal transport term $\gamma$ is nonzero only for |
| 319 |
tracers in unstable (convective) forcing conditions. |
| 320 |
Thus, depending on the stability parameter $\zeta = d/L$ |
| 321 |
(with depth $d$, Monin-Obukhov length scale $L$) |
| 322 |
it has the following form: |
| 323 |
% |
| 324 |
\begin{eqnarray} |
| 325 |
\begin{array}{cl} |
| 326 |
\gamma_x \, = \, 0 & \zeta \, \ge \, 0 \\ |
| 327 |
~ & ~ \\ |
| 328 |
\left. |
| 329 |
\begin{array}{c} |
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\gamma_m \, = \, 0 \\ |
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~ \\ |
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\gamma_s \, = \, C_s |
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\frac{\overline{w s_0}}{w_s(\sigma) h} \\ |
| 334 |
~ \\ |
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\gamma_{\theta} \, = \, C_s |
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\frac{\overline{w \theta_0}+\overline{w \theta_R}}{w_s(\sigma) h} \\ |
| 337 |
\end{array} |
| 338 |
\right\} |
| 339 |
& |
| 340 |
\zeta \, < \, 0 \\ |
| 341 |
\end{array} |
| 342 |
\end{eqnarray} |
| 343 |
|
| 344 |
\end{itemize} |
| 345 |
|
| 346 |
|
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\paragraph{Mixing in the interior} ~ \\ |
| 348 |
% |
| 349 |
~ |
| 350 |
|
| 351 |
\paragraph{Implicit time integration} ~ \\ |
| 352 |
% |
| 353 |
~ |
| 354 |
|
| 355 |
%---------------------------------------------------------------------- |
| 356 |
|
| 357 |
\subsubsection{Key subroutines |
| 358 |
\label{sec:pkg:kpp:subroutines}} |
| 359 |
|
| 360 |
\paragraph{kpp\_calc:} Top-level routine. \\ |
| 361 |
~ |
| 362 |
|
| 363 |
\paragraph{kpp\_mix:} Intermediate-level routine \\ |
| 364 |
~ |
| 365 |
|
| 366 |
\paragraph{ri\_iwmix:} ~ \\ |
| 367 |
% |
| 368 |
Compute interior viscosity and diffusivity coefficients due to |
| 369 |
% |
| 370 |
\begin{itemize} |
| 371 |
% |
| 372 |
\item |
| 373 |
shear instability (dependent on a local gradient Richardson number), |
| 374 |
% |
| 375 |
\item |
| 376 |
to background internal wave activity, and |
| 377 |
% |
| 378 |
\item |
| 379 |
to static instability (local Richardson number < 0). |
| 380 |
% |
| 381 |
\end{itemize} |
| 382 |
|
| 383 |
|
| 384 |
\paragraph{bldepth:} ~ \\ |
| 385 |
% |
| 386 |
The oceanic planetary boundary layer depth, \texttt{hbl}, is determined as |
| 387 |
the shallowest depth where the bulk Richardson number is |
| 388 |
equal to the critical value, \texttt{Ricr}. |
| 389 |
|
| 390 |
Bulk Richardson numbers are evaluated by computing velocity and |
| 391 |
buoyancy differences between values at zgrid(kl) < 0 and surface |
| 392 |
reference values. |
| 393 |
In this configuration, the reference values are equal to the |
| 394 |
values in the surface layer. |
| 395 |
When using a very fine vertical grid, these values should be |
| 396 |
computed as the vertical average of velocity and buoyancy from |
| 397 |
the surface down to epsilon*zgrid(kl). |
| 398 |
|
| 399 |
When the bulk Richardson number at k exceeds Ricr, hbl is |
| 400 |
linearly interpolated between grid levels zgrid(k) and zgrid(k-1). |
| 401 |
|
| 402 |
The water column and the surface forcing are diagnosed for |
| 403 |
stable/ustable forcing conditions, and where hbl is relative |
| 404 |
to grid points (caseA), so that conditional branches can be |
| 405 |
avoided in later subroutines. |
| 406 |
|
| 407 |
\paragraph{blmix:} ~ \\ |
| 408 |
% |
| 409 |
Compute boundary layer mixing coefficients. |
| 410 |
Mixing coefficients within boundary layer depend on surface |
| 411 |
forcing and the magnitude and gradient of interior mixing below |
| 412 |
the boundary layer ("matching"). |
| 413 |
% |
| 414 |
\begin{enumerate} |
| 415 |
% |
| 416 |
\item |
| 417 |
compute velocity scales at hbl |
| 418 |
% |
| 419 |
\item |
| 420 |
find the interior viscosities and derivatives at hbl |
| 421 |
% |
| 422 |
\item |
| 423 |
compute turbulent velocity scales on the interfaces |
| 424 |
% |
| 425 |
\item |
| 426 |
compute the dimensionless shape functions at the interfaces |
| 427 |
% |
| 428 |
\item |
| 429 |
compute boundary layer diffusivities at the interfaces |
| 430 |
% |
| 431 |
\item |
| 432 |
compute nonlocal transport term |
| 433 |
% |
| 434 |
\item |
| 435 |
find diffusivities at kbl-1 grid level |
| 436 |
% |
| 437 |
\end{enumerate} |
| 438 |
|
| 439 |
\paragraph{kpp\_calc\_diff\_t/\_s, kpp\_calc\_visc:} ~ \\ |
| 440 |
% |
| 441 |
Add contribution to net diffusivity/viscosity from |
| 442 |
KPP diffusivity/viscosity. |
| 443 |
|
| 444 |
\paragraph{kpp\_transport\_t/\_s/\_ptr:} ~ \\ |
| 445 |
% |
| 446 |
Add non local KPP transport term (ghat) to diffusive |
| 447 |
temperature/salinity/passive tracer flux. |
| 448 |
The nonlocal transport term is nonzero only for scalars |
| 449 |
in unstable (convective) forcing conditions. |
| 450 |
|
| 451 |
\paragraph{Flow chart:} ~ \\ |
| 452 |
% |
| 453 |
{\footnotesize |
| 454 |
\begin{verbatim} |
| 455 |
|
| 456 |
C !CALLING SEQUENCE: |
| 457 |
c ... |
| 458 |
c kpp_calc (TOP LEVEL ROUTINE) |
| 459 |
c | |
| 460 |
c |-- statekpp: o compute all EOS/density-related arrays |
| 461 |
c | o uses S/R FIND_ALPHA, FIND_BETA, FIND_RHO |
| 462 |
c | |
| 463 |
c |-- kppmix |
| 464 |
c | |--- ri_iwmix (compute interior mixing coefficients due to constant |
| 465 |
c | | internal wave activity, static instability, |
| 466 |
c | | and local shear instability). |
| 467 |
c | | |
| 468 |
c | |--- bldepth (diagnose boundary layer depth) |
| 469 |
c | | |
| 470 |
c | |--- blmix (compute boundary layer diffusivities) |
| 471 |
c | | |
| 472 |
c | |--- enhance (enhance diffusivity at interface kbl - 1) |
| 473 |
c | o |
| 474 |
c | |
| 475 |
c |-- swfrac |
| 476 |
c o |
| 477 |
|
| 478 |
\end{verbatim} |
| 479 |
} |
| 480 |
|
| 481 |
%---------------------------------------------------------------------- |
| 482 |
|
| 483 |
\subsubsection{KPP diagnostics |
| 484 |
\label{sec:pkg:kpp:diagnostics}} |
| 485 |
|
| 486 |
Diagnostics output is available via the diagnostics package |
| 487 |
(see Section \ref{sec:pkg:diagnostics}). |
| 488 |
Available output fields are summarized in |
| 489 |
Table \ref{tab:pkg:kpp:diagnostics}. |
| 490 |
|
| 491 |
\begin{table}[h!] |
| 492 |
\centering |
| 493 |
\label{tab:pkg:kpp:diagnostics} |
| 494 |
{\footnotesize |
| 495 |
\begin{verbatim} |
| 496 |
------------------------------------------------------ |
| 497 |
<-Name->|Levs|grid|<-- Units -->|<- Tile (max=80c) |
| 498 |
------------------------------------------------------ |
| 499 |
KPPviscA| 23 |SM |m^2/s |KPP vertical eddy viscosity coefficient |
| 500 |
KPPdiffS| 23 |SM |m^2/s |Vertical diffusion coefficient for salt & tracers |
| 501 |
KPPdiffT| 23 |SM |m^2/s |Vertical diffusion coefficient for heat |
| 502 |
KPPghat | 23 |SM |s/m^2 |Nonlocal transport coefficient |
| 503 |
KPPhbl | 1 |SM |m |KPP boundary layer depth, bulk Ri criterion |
| 504 |
KPPmld | 1 |SM |m |Mixed layer depth, dT=.8degC density criterion |
| 505 |
KPPfrac | 1 |SM | |Short-wave flux fraction penetrating mixing layer |
| 506 |
\end{verbatim} |
| 507 |
} |
| 508 |
\caption{~} |
| 509 |
\end{table} |
| 510 |
|
| 511 |
%---------------------------------------------------------------------- |
| 512 |
|
| 513 |
\subsubsection{Reference experiments} |
| 514 |
|
| 515 |
lab\_sea: |
| 516 |
|
| 517 |
natl\_box: |
| 518 |
|
| 519 |
%---------------------------------------------------------------------- |
| 520 |
|
| 521 |
\subsubsection{References} |
| 522 |
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