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\subsection{KPP: Nonlocal K-Profile Parameterization for |
\subsection{KPP: Nonlocal K-Profile Parameterization for |
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Diapycnal Mixing} |
Vertical Mixing} |
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\label{sec:pkg:kpp} |
\label{sec:pkg:kpp} |
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\begin{rawhtml} |
\begin{rawhtml} |
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\subsubsection{Introduction |
\subsubsection{Introduction |
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\label{sec:pkg:kpp:intro}} |
\label{sec:pkg:kpp:intro}} |
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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: |
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\begin{enumerate} |
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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). |
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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; |
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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; |
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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; |
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\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 |
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\end{enumerate} |
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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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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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\subsubsection{KPP configuration and compiling} |
\subsubsection{KPP configuration and compiling |
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\label{sec:pkg:kpp:comp}} |
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As with all MITgcm packages, KPP can be turned on or off at compile time |
As with all MITgcm packages, KPP can be turned on or off at compile time |
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\texttt{KPP\_OPTIONS.h}. Table \ref{tab:pkg:kpp:cpp} summarizes them. |
\texttt{KPP\_OPTIONS.h}. Table \ref{tab:pkg:kpp:cpp} summarizes them. |
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\begin{table}[h!] |
\begin{table}[h!] |
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\centering |
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\label{tab:pkg:kpp:cpp} |
\label{tab:pkg:kpp:cpp} |
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{\footnotesize |
{\footnotesize |
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\begin{tabular}{|l|l|} |
\begin{tabular}{|l|l|} |
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\end{table} |
\end{table} |
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%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
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\subsubsection{Run-time parameters |
\subsubsection{Run-time parameters |
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\paragraph{Package flags and parameters} |
\paragraph{Package flags and parameters} |
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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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\begin{table}[h!] |
\begin{table}[h!] |
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\centering |
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\label{tab:pkg:kpp:runtime_flags} |
\label{tab:pkg:kpp:runtime_flags} |
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{\footnotesize |
{\footnotesize |
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\begin{tabular}{|l|c|l|} |
\begin{tabular}{|l|c|l|} |
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\subsubsection{Equations |
\subsubsection{Equations |
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\label{sec:pkg:kpp:equations}} |
\label{sec:pkg:kpp:equations}} |
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We restrict ourselves to writing out only the essential equations |
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that relate to main processes and parameters mentioned above. |
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We closely follow the notation of \cite{lar-eta:94}. |
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\paragraph{Mixing in the boundary layer} ~ \\ |
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~ |
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The vertical fluxes $\overline{wx}$ |
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of momentum and tracer properties $X$ |
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is composed of a gradient-flux term (proportional to |
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the vertical property divergence $\partial_z X$), and |
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a ``nonlocal'' term $\gamma_x$ that enhances the |
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gradient-flux mixing coefficient $K_x$ |
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\begin{equation} |
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\overline{wx}(d) \, = \, -K_x \left( |
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\frac{\partial X}{\partial z} \, - \, \gamma_x \right) |
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\end{equation} |
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\begin{itemize} |
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\item |
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\textit{Boundary layer mixing profile} \\ |
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It is expressed as the product of the boundary layer depth $h$, |
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a depth-dependent turbulent velocity scale $w_x(\sigma)$ and a |
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non-dimensional shape function $G(\sigma)$ |
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\begin{equation} |
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K_x(\sigma) \, = \, h \, w_x(\sigma) \, G(\sigma) |
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\end{equation} |
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with dimensionless vertical coordinate $\sigma = d/h$. |
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For details of $ w_x(\sigma)$ and $G(\sigma)$ we refer to |
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\cite{lar-eta:94}. |
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\item |
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\textit{Nonlocal mixing term} \\ |
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The nonlocal transport term $\gamma$ is nonzero only for |
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tracers in unstable (convective) forcing conditions. |
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Thus, depending on the stability parameter $\zeta = d/L$ |
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(with depth $d$, Monin-Obukhov length scale $L$) |
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it has the following form: |
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\begin{eqnarray} |
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\begin{array}{cl} |
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\gamma_x \, = \, 0 & \zeta \, \ge \, 0 \\ |
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~ & ~ \\ |
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\left. |
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\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} \\ |
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~ \\ |
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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} \\ |
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\end{array} |
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\right\} |
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& |
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\zeta \, < \, 0 \\ |
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\end{array} |
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\end{eqnarray} |
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\end{itemize} |
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\paragraph{Mixing in the interior} ~ \\ |
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~ |
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\paragraph{Implicit time integration} ~ \\ |
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~ |
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%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
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\subsubsection{Key subroutines |
\subsubsection{Key subroutines |
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\end{enumerate} |
\end{enumerate} |
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\paragraph{kpp\_calc\_diff\_t/s, kpp\_calc\_visc:} ~ \\ |
\paragraph{kpp\_calc\_diff\_t/\_s, kpp\_calc\_visc:} ~ \\ |
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Add contribution to net diffusivity/viscosity from |
Add contribution to net diffusivity/viscosity from |
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KPP diffusivity/viscosity. |
KPP diffusivity/viscosity. |
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\paragraph{kpp\_transport\_t/s/ptr:} ~ \\ |
\paragraph{kpp\_transport\_t/\_s/\_ptr:} ~ \\ |
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Add non local KPP transport term (ghat) to diffusive |
Add non local KPP transport term (ghat) to diffusive |
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temperature/salinity/passive tracer flux. |
temperature/salinity/passive tracer flux. |
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The nonlocal transport term is nonzero only for scalars |
The nonlocal transport term is nonzero only for scalars |
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in unstable (convective) forcing conditions. |
in unstable (convective) forcing conditions. |
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\paragraph{Flow chart:} ~ \\ |
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{\footnotesize |
{\footnotesize |
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\begin{verbatim} |
\begin{verbatim} |
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Table \ref{tab:pkg:kpp:diagnostics}. |
Table \ref{tab:pkg:kpp:diagnostics}. |
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\begin{table}[h!] |
\begin{table}[h!] |
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\centering |
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\label{tab:pkg:kpp:diagnostics} |
\label{tab:pkg:kpp:diagnostics} |
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{\footnotesize |
{\footnotesize |
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\begin{verbatim} |
\begin{verbatim} |
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%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
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\subsubsection{References} |
\subsubsection{References} |
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