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\section{Potential vorticity Matlab Toolbox} |
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\label{sec:diag:pv} |
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\begin{rawhtml} |
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%<!-- CMIREDIR:pvdiag_matlab: --> |
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\end{rawhtml} |
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|
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Author: Guillaume Maze |
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|
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\bigskip |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Introduction} |
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|
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\noindent |
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This section of the documentation describes a Matlab package that aims to provide useful |
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routines to compute vorticity fields (relative, potential and planetary) and its related |
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components. This is an offline computation. It was developed to be used in mode water |
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studies, so that it comes with other related routines, in particular ones computing |
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surface vertical potential vorticity fluxes. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Equations} |
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|
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\subsubsection{Potential Vorticity} |
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\noindent |
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The package computes the three components of the relative vorticity defined by: |
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\begin{eqnarray} |
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\omega &= \nabla \times {\bf U} = \left( \begin{array}{c} |
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\omega_x\\ |
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\omega_y\\ |
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\zeta |
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\end{array}\right) |
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\simeq &\left( \begin{array}{c} |
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-\frac{\partial v}{\partial z}\\ |
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-\frac{\partial u}{\partial z}\\ |
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\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} |
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\end{array}\right) \label{sec:diag:pv:eq1} |
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\end{eqnarray} |
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where we omitted (like all across the package) the vertical velocity component. |
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|
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The package then computes the potential vorticity as: |
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\begin{eqnarray} |
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Q &=& -\frac{1}{\rho} \omega\cdot\nabla\sigma_\theta\\ |
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Q &=& -\frac{1}{\rho}\left(\omega_x \frac{\partial \sigma_\theta}{\partial x} + |
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\omega_y \frac{\partial \sigma_\theta}{\partial y} + |
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\left(f+\zeta\right) \frac{\partial \sigma_\theta}{\partial z}\right) \label{sec:diag:pv:eq2} |
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\end{eqnarray} |
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where $\rho$ is the density, $\sigma_\theta$ is the potential density (both eventually |
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computed by the package) and $f$ is the Coriolis parameter. |
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|
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The package is also able to compute the simpler planetary vorticity as: |
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\begin{eqnarray} |
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splQ &=& -\frac{f}{\rho}\frac{\sigma_\theta}{\partial z} \label{sec:diag:pv:eq3} |
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\end{eqnarray} |
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|
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\subsubsection{Surface vertical potential vorticity fluxes} |
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These quantities are useful in mode water studies because of the impermeability theorem |
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which states that for a given potential density layer (embedding a mode water), the |
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integrated PV only changes through surface input/output. |
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|
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Vertical PV fluxes due to frictional and diabatic processes are given by: |
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\begin{eqnarray} |
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J^B_z &=& -\frac{f}{h}\left( \frac{\alpha Q_{net}}{C_w}-\rho_0 \beta S_{net}\right) \label{sec:diag:pv:eq14a}\\ |
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J^F_z &=& \frac{1}{\rho\delta_e} \vec{k}\times\tau\cdot\nabla\sigma_m \label{sec:diag:pv:eq15a} |
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\end{eqnarray} |
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These components can be computed with the package. Details on the variables definition and |
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the way these fluxes are derived can be found in section |
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\ref{sec:diag:pv:notes-flux-form}. |
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|
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\vspace{.5cm} |
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\noindent |
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We now give some simple explanations about these fluxes and how they can reduce the PV |
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level of an oceanic potential density layer. |
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|
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\paragraph{Diabatic process} |
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Let's take the PV flux due to surface buoyancy forcing from Eq.\ref{sec:diag:pv:eq14a} and |
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simplify it as: |
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\begin{eqnarray} |
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J^B_z &\simeq& -\frac{\alpha f}{hC_w} Q_{net} |
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\end{eqnarray} |
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When the net surface heat flux $Q_{net}$ is upward i.e. negative and cooling the ocean |
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(buoyancy loss), surface density will increase, triggering mixing which reduces the |
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stratification and then the PV. |
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\begin{eqnarray} |
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Q_{net} &<& 0 \,\,\,\hbox{(upward, cooling)} \nonumber \\ |
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J^B_z &>& 0 \,\,\,\hbox{(upward)} \nonumber \\ |
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-\rho^{-1}\nabla\cdot J^B_z &<& 0 \,\,\, \hbox{(PV flux divergence)} \nonumber \\ |
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PV &\searrow& \hbox{where $Q_{net}<0$}\nonumber |
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\end{eqnarray} |
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|
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|
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\paragraph{Frictional process: "Down-front" wind-stress} |
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Now let's take the PV flux due to the "wind-driven buoyancy flux" from |
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Eq.\ref{sec:diag:pv:eq15a} and simplify it as: |
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\begin{eqnarray} |
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J^F_z &=& \frac{1}{\rho\delta_e} \left( \tau_x\frac{\partial \sigma}{\partial y} - \tau_y\frac{\partial \sigma}{\partial x} \right) \\ |
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J^F_z &\simeq& \frac{1}{\rho\delta_e} \tau_x\frac{\partial \sigma}{\partial y} \nonumber |
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\end{eqnarray} |
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When the wind is blowing from the east above the Gulf Stream (a region of high meridional |
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density gradient), it induces an advection of dense water from the northern side of the GS |
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to the southern side through Ekman currents. Then, it induces a "wind-driven" buoyancy |
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lost and mixing which reduces the stratification and the PV. |
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\begin{eqnarray} |
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\vec{k}\times\tau\cdot\nabla\sigma &>& 0 \,\,\, \hbox{("Down-front" wind)} \nonumber \\ |
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J^F_z &>& 0 \,\,\, \hbox{(upward)} \nonumber \\ |
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-\rho^{-1}\nabla\cdot J^F_z &<& 0 \,\,\, \hbox{(PV flux divergence)} \nonumber \\ |
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PV &\searrow& \hbox{where $\vec{k}\times\tau\cdot\nabla\sigma>0$}\nonumber |
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\end{eqnarray} |
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|
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|
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\paragraph{Diabatic versus frictional processes} |
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A recent debate in the community arose about the relative role of these processes. Taking |
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the ratio of Eq.\ref{sec:diag:pv:eq14a} and Eq.\ref{sec:diag:pv:eq15a} leads to: |
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\begin{eqnarray} |
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\frac{J^F_z}{J^B_Z} &=& \frac{ \frac{1}{\rho\delta_e} \vec{k}\times\tau\cdot\nabla\sigma } |
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{-\frac{f}{h}\left( \frac{\alpha Q_{net}}{C_w}-\rho_0 \beta S_{net}\right)} \\ |
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&\simeq& \frac{Q_{Ek}/\delta_e}{Q_{net}/h} \nonumber |
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\end{eqnarray} |
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where appears the lateral heat flux induced by Ekman currents: |
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\begin{eqnarray} |
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Q_{Ek} &=& -\frac{C_w}{\alpha\rho f}\vec{k}\times\tau\cdot\nabla\sigma |
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\nonumber \\ |
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&=& \frac{C_w}{\alpha}\delta_e\vec{u_{Ek}}\cdot\nabla\sigma |
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\end{eqnarray} |
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which can be computed with the package. In the aim of comparing both processes, it will be |
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useful to plot surface net and lateral Ekman-induced heat fluxes together with PV fluxes. |
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|
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|
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Key routines} |
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|
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\begin{itemize} |
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\item {\bf {\ttfamily A\_compute\_potential\_density.m}}: Compute the potential density |
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field. Requires the potential temperature and salinity (either total or anomalous) and |
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produces one output file with the potential density field (file prefix is {\ttfamily |
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SIGMATHETA}). The routine uses {\ttfamily densjmd95.m} a Matlab counterpart of the |
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MITgcm built-in function to compute the density. |
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\item {\bf {\ttfamily B\_compute\_relative\_vorticity.m}}: Compute the three components of |
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the relative vorticity defined in Eq.~(\ref{sec:diag:pv:eq1}). Requires the two |
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horizontal velocity components and produces three output files with the three components |
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(files prefix are {\ttfamily OMEGAX}, {\ttfamily OMEGAY} and {\ttfamily ZETA}). |
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\item {\bf {\ttfamily C\_compute\_potential\_vorticity.m}}: Compute the potential |
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vorticity without the negative ratio by the density. Two options are possible in order |
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to compute either the full component (term into parenthesis in |
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Eq.~\ref{sec:diag:pv:eq2}) or the planetary component ($f\partial_z\sigma_\theta$ in |
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Eq.~\ref{sec:diag:pv:eq3}). Requires the relative vorticity components and the potential |
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density, and produces one output file with the potential vorticity (file prefix is |
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{\ttfamily PV} for the full term and {\ttfamily splPV} for the planetary component). |
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\item {\bf {\ttfamily D\_compute\_potential\_vorticity.m}}: Load the field computed with |
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\mbox{{\ttfamily C\_comp...}} and divide it by $-\rho$ to obtain the |
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correct potential vorticity. Require the density field and after loading, overwrite the |
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file with prefix {\ttfamily PV} or {\ttfamily splPV}. |
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\item {\bf {\ttfamily compute\_density.m}}: Compute the density $\rho$ from the potential |
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temperature and the salinity fields. |
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\item {\bf {\ttfamily compute\_JFz.m}}: Compute the surface vertical PV flux due to |
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frictional processes. Requires the wind stress components, density, potential density |
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and Ekman layer depth (all of them, except the wind stress, may be computed with the |
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package), and produces one output file with the PV flux $J^F_z$ (see |
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Eq.~\ref{sec:diag:pv:eq15a}) and with {\ttfamily JFz} as a prefix. |
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\item {\bf {\ttfamily compute\_JBz.m}}: Compute the surface vertical PV flux due to |
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diabatic processes as: |
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\begin{eqnarray} |
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J^B_z &=& -\frac{f}{h}\frac{\alpha Q_{net}}{C_w} \nonumber |
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\end{eqnarray} |
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which is a simplified version of the full expression given in |
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Eq.~(\ref{sec:diag:pv:eq14a}). Requires the net surface heat flux and the mixed layer depth |
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(of which an estimation can be computed with the package), and produces one output file |
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with the PV flux $J^B_z$ and with {\ttfamily JBz} as a prefix. |
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\item {\bf {\ttfamily compute\_QEk.m}}: Compute the horizontal heat flux due to Ekman |
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currents from the PV flux induced by frictional forces as: |
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\begin{eqnarray} |
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Q_{Ek} &=& - \frac{C_w \delta_e}{\alpha f}J^F_z\nonumber |
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\end{eqnarray} |
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Requires the PV flux due to frictional forces and the Ekman layer depth, and produces one |
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output with the heat flux and with {\ttfamily QEk} as a prefix. |
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\item {\bf {\ttfamily eg\_main\_getPV}}: A complete example of how to set up a master |
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routine able to compute everything from the package. |
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\end{itemize} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Technical details} |
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|
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\subsubsection{File name} |
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\noindent |
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A file name is formed by three parameters which need to be set up as global variables in Matlab |
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before running any routines. They are: |
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\begin{itemize} |
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\item the prefix, ie the variable name ({\ttfamily netcdf\_UVEL} for example). This |
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parameter is specified in the help section of all diagnostic routines. |
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\item {\ttfamily netcdf\_domain}: the geographical domain. |
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\item {\ttfamily netcdf\_suff}: the netcdf extension (nc or cdf for example). |
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\end{itemize} |
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Then, for example, if the calling Matlab routine had set up: |
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\begin{verbatim} |
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global netcdf_THETA netcdf_SALTanom netcdf_domain netcdf_suff |
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netcdf_THETA = 'THETA'; |
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netcdf_SALTanom = 'SALT'; |
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netcdf_domain = 'north_atlantic'; |
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netcdf_suff = 'nc'; |
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\end{verbatim} |
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the routine {\ttfamily A\_compute\_potential\_density.m} to compute the potential density |
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field, will look for the files: |
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\begin{verbatim} |
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THETA.north_atlantic.nc |
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SALT.north_atlantic.nc |
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\end{verbatim} |
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and the output file will automatically be: {\ttfamily SIGMATHETA.north\_atlantic.nc}. |
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|
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Otherwise indicated, output file prefix cannot be changed. |
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|
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\subsubsection{Path to file} |
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\noindent |
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All diagnostic routines look for input files in a subdirectory (relative to the Matlab |
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routine directory) called {\ttfamily ./netcdf-files} which in turn, is supposed to contain |
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subdirectories for each set of fields. For example, computing the potential density for |
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the timestep 12H00 02/03/2005 will require a |
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subdirectory with the potential temperature and salinity files like: |
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\begin{verbatim} |
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./netcdf-files/200501031200/THETA.north_atlantic.nc |
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./netcdf-files/200501031200/SALT.north_atlantic.nc |
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\end{verbatim} |
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\noindent |
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The output file {\ttfamily SIGMATHETA.north\_atlantic.nc} will be created in {\ttfamily |
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./netcdf-files/200501031200/}. |
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\noindent |
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All diagnostic routines take as argument the name of the timestep subdirectory into |
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{\ttfamily ./netcdf-files}. |
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|
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\subsubsection{Grids} |
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\noindent |
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With MITgcm numerical outputs, velocity and tracer fields may not be defined on the same |
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grid. Usually, UVEL and VVEL are defined on a C-grid but when interpolated from a |
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cube-sphere simulation they are defined on a A-grid. When it is needed, routines allow |
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to set up a global variable which define the grid to use. |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Notes on the flux form of the PV equation and vertical PV fluxes} |
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\label{sec:diag:pv:notes-flux-form} |
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|
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\subsubsection{Flux form of the PV equation} |
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The conservative flux form of the potential vorticity equation is: |
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\begin{eqnarray} |
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\frac{\partial \rho Q}{\partial t} + \nabla \cdot \vec{J} &=& 0 \label{sec:diag:pv:eq4} |
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\end{eqnarray} |
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where the potential vorticity $Q$ is given by the Eq.\ref{sec:diag:pv:eq2}. |
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|
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The generalized flux vector of potential vorticity is: |
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\begin{eqnarray} |
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\vec{J} &=& \rho Q \vec{u} + \vec{N_Q} |
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\end{eqnarray} |
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which allows to rewrite Eq.\ref{sec:diag:pv:eq4} as: |
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\begin{eqnarray} |
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\frac{DQ}{dt} &=& -\frac{1}{\rho}\nabla\cdot\vec{N_Q} \label{sec:diag:pv:eq5} |
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\end{eqnarray} |
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where the nonadvective PV flux $\vec{N_Q}$ is given by: |
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\begin{eqnarray} |
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\vec{N_Q} &=& -\frac{\rho_0}{g}B\vec{\omega_a} + \vec{F}\times\nabla\sigma_\theta |
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\label{sec:diag:pv:eq6} |
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\end{eqnarray} |
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|
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Its first component is linked to the buoyancy forcing\footnote{ |
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Note that introducing B into Eq.\ref{sec:diag:pv:eq6} yields to: |
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\begin{eqnarray} |
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\vec{N_Q} &=& \omega_a \frac{D \sigma_\theta}{dt} + \vec{F}\times\nabla\sigma_\theta |
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\end{eqnarray}}: |
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\begin{eqnarray} |
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B &=& -\frac{g}{\rho_o}\frac{D \sigma_\theta}{dt} |
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\end{eqnarray} |
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and the second one to the nonconservative body forces per unit mass: |
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\begin{eqnarray} |
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\vec{F} &=& \frac{D \vec{u}}{dt} + 2\Omega\times\vec{u} + \nabla p |
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\end{eqnarray} |
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|
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\subsubsection{Determining the PV flux at the ocean's surface} |
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In the context of mode water study, we're particularly interested in how the PV may be |
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reduced by surface PV fluxes because a mode water is characterised by a low PV level. |
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Considering the volume limited by two $iso-\sigma_\theta$, PV |
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flux is limited to surface processes and then vertical component of $\vec{N_Q}$. It is |
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supposed that $B$ and $\vec{F}$ will only be nonzero in the mixed layer (of depth $h$ and |
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variable density $\sigma_m$) exposed to mechanical forcing by the wind and buoyancy fluxes |
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through the ocean's surface. |
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|
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Given the assumption of a mechanical forcing confined to a thin surface Ekman layer (of |
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depth $\delta_e$, eventually computed by the package) and of hydrostatic and geostrophic |
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balances, we can write: |
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\begin{eqnarray} |
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\vec{u_g} &=& \frac{1}{\rho f} \vec{k}\times\nabla p \\ |
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\frac{\partial p_m}{\partial z} &=& -\sigma_m g \\ |
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\frac{\partial \sigma_m}{\partial t} + \vec{u}_m\cdot\nabla\sigma_m &=& -\frac{\rho_0}{g}B \label{sec:diag:pv:eq7} |
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\end{eqnarray} |
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where: |
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\begin{eqnarray} |
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\vec{u}_m &=& \vec{u}_g + \vec{u}_{Ek} + o(R_o) \label{sec:diag:pv:eq8} |
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\end{eqnarray} |
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is the full velocity field composed by the geostrophic current $\vec{u}_g$ and the Ekman |
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drift: |
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\begin{eqnarray} |
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\vec{u}_{Ek} &=& -\frac{1}{\rho f}\vec{k}\times\frac{\partial \tau}{\partial z} \label{sec:diag:pv:eq9} |
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\end{eqnarray} |
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(where $\tau$ is the wind stress) and last by other ageostrophic components of $o(R_o)$ |
| 305 |
which are neglected. |
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|
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Partitioning the buoyancy forcing as: |
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\begin{eqnarray} |
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B &=& B_g + B_{Ek} \label{sec:diag:pv:eq10} |
| 310 |
\end{eqnarray} |
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and using Eq.\ref{sec:diag:pv:eq8} and Eq.\ref{sec:diag:pv:eq9}, the Eq.\ref{sec:diag:pv:eq7} becomes: |
| 312 |
\begin{eqnarray} |
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\frac{\partial \sigma_m}{\partial t} + \vec{u}_g\cdot\nabla\sigma_m &=& -\frac{\rho_0}{g} B_g |
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\end{eqnarray} |
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revealing the "wind-driven buoyancy forcing": |
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\begin{eqnarray} |
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B_{Ek} &=& \frac{g}{\rho_0}\frac{1}{\rho f}\left(\vec{k}\times\frac{\partial \tau}{\partial z}\right)\cdot\nabla\sigma_m |
| 318 |
\end{eqnarray} |
| 319 |
Note that since: |
| 320 |
\begin{eqnarray} |
| 321 |
\frac{\partial B_g}{\partial z} &=& \frac{\partial}{\partial z}\left(-\frac{g}{\rho_0}\vec{u_g}\cdot\nabla\sigma_m\right) |
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= -\frac{g}{\rho_0}\frac{\partial \vec{u_g}}{\partial z}\cdot\nabla\sigma_m |
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= 0 |
| 324 |
\end{eqnarray} |
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$B_g$ must be uniform throughout the depth of the mixed layer and then being related to |
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the surface buoyancy flux by integrating Eq.\ref{sec:diag:pv:eq10} through the mixed layer: |
| 327 |
\begin{eqnarray} |
| 328 |
\int_{-h}^0B\,dz &=\, hB_g + \int_{-h}^0B_{Ek}\,dz \,=& \mathcal{B}_{in} \label{sec:diag:pv:eq11} |
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\end{eqnarray} |
| 330 |
where $\mathcal{B}_{in}$ is the vertically integrated surface buoyancy (in)flux: |
| 331 |
\begin{eqnarray} |
| 332 |
\mathcal{B}_{in} &=& \frac{g}{\rho_o}\left( \frac{\alpha Q_{net}}{C_w} - \rho_0\beta S_{net}\right) |
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%\label{sec:diag:pv:eq12} |
| 334 |
\end{eqnarray} |
| 335 |
with $\alpha\simeq 2.5\times10^{-4}\, K^{-1}$ the thermal expansion coefficient (computed |
| 336 |
by the package otherwise), $C_w=4187J.kg^{-1}.K^{-1}$ the specific heat of seawater, |
| 337 |
$Q_{net}[W.m^{-2}]$ the net heat surface flux (positive downward, warming the ocean), |
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$\beta[PSU^{-1}]$ the saline contraction coefficient, and |
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$S_{net}=S*(E-P)[PSU.m.s^{-1}]$ the net freshwater surface flux with $S[PSU]$ |
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the surface salinity and $(E-P)[m.s^{-1}]$ the fresh water flux. |
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|
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Introducing the body force in the Ekman layer: |
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\begin{eqnarray} |
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F_z &=& \frac{1}{\rho}\frac{\partial \tau}{\partial z} |
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\end{eqnarray} |
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the vertical component of Eq.\ref{sec:diag:pv:eq6} is: |
| 347 |
\begin{eqnarray} |
| 348 |
\vec{N_Q}_z &=& -\frac{\rho_0}{g}(B_g+B_{Ek})\omega_z |
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+ \frac{1}{\rho} |
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\left( \frac{\partial \tau}{\partial z}\times\nabla\sigma_\theta \right)\cdot\vec{k} |
| 351 |
\nonumber \\ |
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&=& -\frac{\rho_0}{g}B_g\omega_z |
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-\frac{\rho_0}{g} |
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\left(\frac{g}{\rho_0}\frac{1}{\rho f}\vec{k}\times\frac{\partial \tau}{\partial z} |
| 355 |
\cdot\nabla\sigma_m\right)\omega_z |
| 356 |
+ \frac{1}{\rho} |
| 357 |
\left( \frac{\partial \tau}{\partial z}\times\nabla\sigma_\theta \right)\cdot\vec{k} |
| 358 |
\nonumber \\ |
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&=& -\frac{\rho_0}{g}B_g\omega_z |
| 360 |
+ \left(1-\frac{\omega_z}{f}\right)\left(\frac{1}{\rho}\frac{\partial \tau}{\partial z} |
| 361 |
\times\nabla\sigma_\theta \right)\cdot\vec{k} |
| 362 |
\end{eqnarray} |
| 363 |
and given the assumption that $\omega_z\simeq f$, the second term vanishes and we obtain: |
| 364 |
\begin{eqnarray} |
| 365 |
\vec{N_Q}_z &=& -\frac{\rho_0}{g}f B_g %\label{sec:diag:pv:eq12} |
| 366 |
\end{eqnarray} |
| 367 |
Note that the wind-stress forcing does not appear explicitly here but is implicit in $B_g$ |
| 368 |
through Eq.\ref{sec:diag:pv:eq11}: the buoyancy forcing $B_g$ is determined by the |
| 369 |
difference between the integrated surface buoyancy flux $\mathcal{B}_{in}$ and the |
| 370 |
integrated "wind-driven buoyancy forcing": |
| 371 |
\begin{eqnarray} |
| 372 |
B_g &=& \frac{1}{h}\left( \mathcal{B}_{in} - \int_{-h}^0B_{Ek}dz \right) \nonumber \\ |
| 373 |
&=& \frac{1}{h}\frac{g}{\rho_0}\left( \frac{\alpha Q_{net}}{C_w} - \rho_0 \beta S_{net}\right) |
| 374 |
- \frac{1}{h}\int_{-h}^0 |
| 375 |
\frac{g}{\rho_0}\frac{1}{\rho f}\vec{k}\times \frac{\partial \tau}{\partial z} \cdot\nabla\sigma_m dz |
| 376 |
\nonumber \\ |
| 377 |
&=& \frac{1}{h}\frac{g}{\rho_0}\left( \frac{\alpha Q_{net}}{C_w} - \rho_0 \beta S_{net}\right) |
| 378 |
- \frac{g}{\rho_0}\frac{1}{\rho f \delta_e}\vec{k}\times\tau\cdot\nabla\sigma_m |
| 379 |
\end{eqnarray} |
| 380 |
Finally, from Eq.\ref{sec:diag:pv:eq6}, the vertical surface flux of PV may be written as: |
| 381 |
\begin{eqnarray} |
| 382 |
\vec{N_Q}_z &=& J^B_z + J^F_z \label{sec:diag:pv:eq13} \\ |
| 383 |
J^B_z &=& -\frac{f}{h}\left( \frac{\alpha Q_{net}}{C_w}-\rho_0 \beta S_{net}\right) \label{sec:diag:pv:eq14}\\ |
| 384 |
J^F_z &=& \frac{1}{\rho\delta_e} \vec{k}\times\tau\cdot\nabla\sigma_m \label{sec:diag:pv:eq15} |
| 385 |
\end{eqnarray} |
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