1 |
\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}} |
13 |
|
14 |
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: |
21 |
% |
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 |
31 |
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 |
33 |
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 |
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to penetrate well into the thermocline; |
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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), |
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to the interior thus fixing the polynomial coefficients; |
48 |
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 |
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enhances mixing where the water column is unstable |
55 |
% |
56 |
\end{enumerate} |
57 |
% |
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The scheme has been extensively compared to observations |
59 |
(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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%---------------------------------------------------------------------- |
73 |
|
74 |
\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 |
78 |
% |
79 |
\begin{itemize} |
80 |
% |
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\item |
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using the \texttt{packages.conf} file by adding \texttt{kpp} to it, |
83 |
% |
84 |
\item |
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or using \texttt{genmake2} adding |
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\texttt{-enable=kpp} or \texttt{-disable=kpp} switches |
87 |
% |
88 |
\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} |
98 |
{\footnotesize |
99 |
\begin{tabular}{|l|l|} |
100 |
\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} |
127 |
} |
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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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|
134 |
\subsubsection{Run-time parameters |
135 |
\label{sec:pkg:kpp:runtime}} |
136 |
|
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Run-time parameters are set in files |
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\texttt{data.pkg} and \texttt{data.kpp} |
139 |
which are read in \texttt{kpp\_readparms.F}. |
140 |
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} |
146 |
~ \\ |
147 |
% |
148 |
The KPP package is switched on at runtime by setting |
149 |
\texttt{useKPP = .TRUE.} in \texttt{data.pkg}. |
150 |
|
151 |
\paragraph{Required MITgcm flags} |
152 |
~ \\ |
153 |
% |
154 |
The following flags/parameters of the MITgcm dynamical |
155 |
kernel need to be set in conjunction with KPP: |
156 |
|
157 |
\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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|
162 |
|
163 |
\paragraph{Package flags and parameters} |
164 |
~ \\ |
165 |
% |
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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!] |
171 |
\centering |
172 |
\label{tab:pkg:kpp:runtime_flags} |
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{\footnotesize |
174 |
\begin{tabular}{|l|c|l|} |
175 |
\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 |
180 |
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 \\ |
184 |
kpp\_taveFreq & \texttt{taveFreq} & |
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Averaging and dump frequency of KPP fields \\ |
186 |
KPPmixingMaps & \texttt{.FALSE.} & |
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include KPP diagnostic maps in STDOUT \\ |
188 |
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 & |
208 |
~ \\ |
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\hline |
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\multicolumn{3}{|c|}{\textit{Boundary layer parameters (S/R \texttt{bldepth})} } \\ |
211 |
\hline |
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Ricr & 0.3 & |
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critical bulk Richardson number \\ |
214 |
cekman & 0.7 & |
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coefficient for Ekman depth \\ |
216 |
cmonob & 1.0 & |
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coefficient for Monin-Obukhov depth \\ |
218 |
concv & 1.8 & |
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ratio of interior to entrainment depth buoyancy frequency \\ |
220 |
hbf & 1.0 & |
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fraction of depth to which absorbed solar radiation contributes \\ |
222 |
~ & ~ & |
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to surface buoyancy forcing \\ |
224 |
Vtc & \texttt{~} & |
225 |
non-dim. coeff. for velocity scale of turbulant velocity shear \\ |
226 |
~ & ~ & |
227 |
( = function of concv,concs,epsilon,vonk,Ricr) \\ |
228 |
\hline |
229 |
\multicolumn{3}{|c|}{\textit{Boundary layer mixing parameters (S/R \texttt{blmix})} } \\ |
230 |
\hline |
231 |
cstar & 10. & |
232 |
proportionality coefficient for nonlocal transport \\ |
233 |
cg & ~ & |
234 |
non-dimensional coefficient for counter-gradient term \\ |
235 |
~ & ~ & |
236 |
( = function of cstar,vonk,concs,epsilon) \\ |
237 |
\hline |
238 |
\multicolumn{3}{|c|}{\textit{Interior mixing parameters (S/R \texttt{Ri\_iwmix})} } \\ |
239 |
\hline |
240 |
Riinfty & 0.7 & |
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gradient Richardson number limit for shear instability \\ |
242 |
BVDQcon & -0.2E-4 1/s$^2$ & |
243 |
Brunt-V\"ai\"sal\"a squared \\ |
244 |
difm0 & 0.005 m$^2$/s & |
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viscosity max. due to shear instability \\ |
246 |
difs0 & 0.005 m$^2$/s & |
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tracer diffusivity max. due to shear instability \\ |
248 |
dift0 & 0.005 m$^2$/s & |
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heat diffusivity max. due to shear instability \\ |
250 |
difmcon & 0.1 & |
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viscosity due to convective instability \\ |
252 |
difscon & 0.1 & |
253 |
tracer diffusivity due to convective instability \\ |
254 |
diftcon & 0.1 & |
255 |
heat diffusivity due to convective instability \\ |
256 |
\hline |
257 |
\multicolumn{3}{|c|}{\textit{Double-diffusive mixing parameters (S/R \texttt{ddmix})} } \\ |
258 |
\hline |
259 |
Rrho0 & not used & |
260 |
limit for double diffusive density ratio \\ |
261 |
dsfmax & not used & |
262 |
maximum diffusivity in case of salt fingering \\ |
263 |
\hline |
264 |
\hline |
265 |
\end{tabular} |
266 |
} |
267 |
\caption{~} |
268 |
\end{table} |
269 |
|
270 |
|
271 |
|
272 |
%---------------------------------------------------------------------- |
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}. |
280 |
|
281 |
\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} |
330 |
\gamma_m \, = \, 0 \\ |
331 |
~ \\ |
332 |
\gamma_s \, = \, C_s |
333 |
\frac{\overline{w s_0}}{w_s(\sigma) h} \\ |
334 |
~ \\ |
335 |
\gamma_{\theta} \, = \, C_s |
336 |
\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 |
|
347 |
\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 |
|