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% $Header: /u/gcmpack/mitgcmdoc/part3/case_studies/carbon_outgassing_sensitivity/co2sens.tex,v 1.3 2001/11/13 18:19:18 adcroft Exp $ |
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% $Name: $ |
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|
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\section{Example: Centennial Time Scale Sensitivities} |
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|
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\bodytext{bgcolor="#FFFFFFFF"} |
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|
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%\begin{center} |
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%{\Large \bf Using MITgcm to Look at Centennial Time Scale |
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%Sensitivities} |
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% |
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%\vspace*{4mm} |
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% |
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%\vspace*{3mm} |
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%{\large May 2001} |
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%\end{center} |
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|
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\subsection{Introduction} |
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|
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This document describes the fourth example MITgcm experiment. |
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This example illustrates the use of |
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the MITgcm to perform sensitivity analysis in a |
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large scale ocean circulation simulation. |
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|
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\subsection{Overview} |
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|
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This example experiment demonstrates using the MITgcm to simulate |
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the planetary ocean circulation. The simulation is configured |
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with realistic geography and bathymetry on a |
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$4^{\circ} \times 4^{\circ}$ spherical polar grid. |
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Twenty vertical layers are used in the vertical, ranging in thickness |
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from $50\,{\rm m}$ at the surface to $815\,{\rm m}$ at depth, |
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giving a maximum model depth of $6\,{\rm km}$. |
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At this resolution, the configuration |
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can be integrated forward for thousands of years on a single |
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processor desktop computer. |
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\\ |
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|
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The model is forced with climatological wind stress data and surface |
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flux data from Da Silva \cite{DaSilva94}. Climatological data |
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from Levitus \cite{Levitus94} is used to initialize the model hydrography. |
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Levitus data is also used throughout the calculation |
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to derive air-sea fluxes of heat at the ocean surface. |
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These fluxes are combined with climatological estimates of |
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surface heat flux and fresh water, resulting in a mixed boundary |
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condition of the style described in Haney \cite{Haney}. |
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Altogether, this yields the following forcing applied |
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in the model surface layer. |
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|
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\begin{eqnarray} |
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\label{EQ:global_forcing} |
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\label{EQ:global_forcing_fu} |
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{\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}} |
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\\ |
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\label{EQ:global_forcing_fv} |
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{\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}} |
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\\ |
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\label{EQ:global_forcing_ft} |
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{\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} ) |
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- \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q} |
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\\ |
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\label{EQ:global_forcing_fs} |
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{\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} ) |
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+ \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R}) |
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\end{eqnarray} |
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|
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\noindent where ${\cal F}_{u}$, ${\cal F}_{v}$, ${\cal F}_{\theta}$, |
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${\cal F}_{s}$ are the forcing terms in the zonal and meridional |
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momentum and in the potential temperature and salinity |
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equations respectively. |
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The term $\Delta z_{s}$ represents the top ocean layer thickness. |
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It is used in conjunction with the reference density, $\rho_{0}$ |
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(here set to $999.8\,{\rm kg\,m^{-3}}$), the |
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reference salinity, $S_{0}$ (here set to 35ppt), |
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and a specific heat capacity $C_{p}$ to convert |
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wind-stress fluxes given in ${\rm N}\,m^{-2}$, |
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\\ |
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|
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|
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The configuration is illustrated in figure \ref{simulation_config}. |
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|
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|
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\subsection{Discrete Numerical Configuration} |
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|
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|
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The model is configured in hydrostatic form. The domain is discretised with |
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a uniform grid spacing in latitude and longitude of |
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$\Delta x=\Delta y=4^{\circ}$, so |
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that there are ninety grid cells in the $x$ and forty in the |
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$y$ direction (Arctic polar regions are not |
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included in this experiment). Vertically the |
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model is configured with twenty layers with the following thicknesses |
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$\Delta z_{1} = 50\,{\rm m},\, |
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\Delta z_{2} = 50\,{\rm m},\, |
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\Delta z_{3} = 55\,{\rm m},\, |
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\Delta z_{4} = 60\,{\rm m},\, |
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\Delta z_{5} = 65\,{\rm m},\, |
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$ |
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$ |
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\Delta z_{6}~=~70\,{\rm m},\, |
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\Delta z_{7}~=~80\,{\rm m},\, |
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\Delta z_{8}~=95\,{\rm m},\, |
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\Delta z_{9}=120\,{\rm m},\, |
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\Delta z_{10}=155\,{\rm m},\, |
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$ |
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$ |
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\Delta z_{11}=200\,{\rm m},\, |
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\Delta z_{12}=260\,{\rm m},\, |
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\Delta z_{13}=320\,{\rm m},\, |
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\Delta z_{14}=400\,{\rm m},\, |
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\Delta z_{15}=480\,{\rm m},\, |
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$ |
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$ |
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\Delta z_{16}=570\,{\rm m},\, |
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\Delta z_{17}=655\,{\rm m},\, |
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\Delta z_{18}=725\,{\rm m},\, |
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\Delta z_{19}=775\,{\rm m},\, |
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\Delta z_{20}=815\,{\rm m} |
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$ (here the numeric subscript indicates the model level index number, ${\tt k}$). |
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The implicit free surface form of the pressure equation described in Marshall et. al |
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\cite{Marshall97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous |
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dissipation. Thermal and haline diffusion is also represented by a Laplacian operator. |
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\\ |
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|
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Wind-stress momentum inputs are added to the momentum equations for both |
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the zonal flow, $u$ and the meridional flow $v$, according to equations |
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(\ref{EQ:global_forcing_fu}) and (\ref{EQ:global_forcing_fv}). |
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Thermodynamic forcing inputs are added to the equations for |
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potential temperature, $\theta$, and salinity, $S$, according to equations |
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(\ref{EQ:global_forcing_ft}) and (\ref{EQ:global_forcing_fs}). |
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This produces a set of equations solved in this configuration as follows: |
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% {\fracktur} |
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|
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|
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\begin{eqnarray} |
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\label{EQ:model_equations} |
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\frac{Du}{Dt} - fv + |
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\frac{1}{\rho}\frac{\partial p^{'}}{\partial x} - |
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A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}} |
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& = & |
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{\cal F}_{u} |
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\\ |
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\frac{Dv}{Dt} + fu + |
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\frac{1}{\rho}\frac{\partial p^{'}}{\partial y} - |
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A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}} |
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& = & |
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{\cal F}_{v} |
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\\ |
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\frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u} |
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&=& |
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0 |
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\\ |
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\frac{D\theta}{Dt} - |
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K_{h}\nabla_{h}^2\theta - \Gamma(K_{z})\frac{\partial^{2}\theta}{\partial z^{2}} |
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& = & |
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{\cal F}_{\theta} |
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\\ |
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\frac{D s}{Dt} - |
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K_{h}\nabla_{h}^2 s - \Gamma(K_{z})\frac{\partial^{2} s}{\partial z^{2}} |
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& = & |
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{\cal F}_{s} |
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\\ |
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g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz & = & p^{'} |
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\\ |
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\end{eqnarray} |
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|
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\noindent where $u$ and $v$ are the $x$ and $y$ components of the |
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flow vector $\vec{u}$. The suffices ${s},{i}$ indicate surface and |
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interior model levels respectively. As described in |
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MITgcm Numerical Solution Procedure \ref{chap:discretization}, the time |
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evolution of potential temperature, $\theta$, equation is solved prognostically. |
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The total pressure, $p$, is diagnosed by summing pressure due to surface |
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elevation $\eta$ and the hydrostatic pressure. |
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\\ |
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|
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\subsubsection{Numerical Stability Criteria} |
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|
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The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$. |
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This value is chosen to yield a Munk layer width \cite{adcroft:95}, |
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|
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\begin{eqnarray} |
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\label{EQ:munk_layer} |
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M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}} |
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\end{eqnarray} |
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|
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\noindent of $\approx 100$km. This is greater than the model |
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resolution in mid-latitudes $\Delta x$, ensuring that the frictional |
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boundary layer is well resolved. |
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\\ |
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|
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\noindent The model is stepped forward with a |
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time step $\delta t=1200$secs. With this time step the stability |
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parameter to the horizontal Laplacian friction \cite{adcroft:95} |
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|
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\begin{eqnarray} |
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\label{EQ:laplacian_stability} |
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S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2} |
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\end{eqnarray} |
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|
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\noindent evaluates to 0.012, which is well below the 0.3 upper limit |
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for stability. |
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\\ |
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|
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\noindent The vertical dissipation coefficient, $A_{z}$, is set to |
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$1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit |
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|
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\begin{eqnarray} |
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\label{EQ:laplacian_stability_z} |
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S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2} |
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\end{eqnarray} |
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|
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\noindent evaluates to $4.8 \times 10^{-5}$ which is again well below |
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the upper limit. |
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The values of $A_{h}$ and $A_{z}$ are also used for the horizontal ($K_{h}$) |
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and vertical ($K_{z}$) diffusion coefficients for temperature respectively. |
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\\ |
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|
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\noindent The numerical stability for inertial oscillations |
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\cite{adcroft:95} |
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|
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\begin{eqnarray} |
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\label{EQ:inertial_stability} |
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S_{i} = f^{2} {\delta t}^2 |
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\end{eqnarray} |
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|
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\noindent evaluates to $0.0144$, which is well below the $0.5$ upper |
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limit for stability. |
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\\ |
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|
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\noindent The advective CFL \cite{adcroft:95} for a extreme maximum |
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horizontal flow |
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speed of $ | \vec{u} | = 2 ms^{-1}$ |
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|
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\begin{eqnarray} |
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\label{EQ:cfl_stability} |
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S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x} |
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\end{eqnarray} |
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|
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\noindent evaluates to $5 \times 10^{-2}$. This is well below the stability |
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limit of 0.5. |
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\\ |
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|
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\noindent The stability parameter for internal gravity waves |
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\cite{adcroft:95} |
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|
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\begin{eqnarray} |
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\label{EQ:cfl_stability} |
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S_{c} = \frac{c_{g} \delta t}{ \Delta x} |
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\end{eqnarray} |
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|
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\noindent evaluates to $5 \times 10^{-2}$. This is well below the linear |
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stability limit of 0.25. |
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|
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\subsection{Code Configuration} |
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\label{SEC:code_config} |
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|
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The model configuration for this experiment resides under the |
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directory {\it verification/exp1/}. The experiment files |
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\begin{itemize} |
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\item {\it input/data} |
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\item {\it input/data.pkg} |
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\item {\it input/eedata}, |
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\item {\it input/windx.sin\_y}, |
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\item {\it input/topog.box}, |
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\item {\it code/CPP\_EEOPTIONS.h} |
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\item {\it code/CPP\_OPTIONS.h}, |
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\item {\it code/SIZE.h}. |
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\end{itemize} |
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contain the code customizations and parameter settings for this |
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experiments. Below we describe the customizations |
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to these files associated with this experiment. |
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|
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\subsubsection{File {\it input/data}} |
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|
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This file, reproduced completely below, specifies the main parameters |
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for the experiment. The parameters that are significant for this configuration |
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are |
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|
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\begin{itemize} |
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|
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\item Line 4, |
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\begin{verbatim} tRef=20.,10.,8.,6., \end{verbatim} |
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this line sets |
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the initial and reference values of potential temperature at each model |
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level in units of $^{\circ}$C. |
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The entries are ordered from surface to depth. For each |
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depth level the initial and reference profiles will be uniform in |
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$x$ and $y$. |
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|
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\fbox{ |
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\begin{minipage}{5.0in} |
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{\it S/R INI\_THETA}({\it ini\_theta.F}) |
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\end{minipage} |
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} |
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|
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|
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\item Line 6, |
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\begin{verbatim} viscAz=1.E-2, \end{verbatim} |
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this line sets the vertical Laplacian dissipation coefficient to |
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$1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions |
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for this operator are specified later. This variable is copied into |
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model general vertical coordinate variable {\bf viscAr}. |
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|
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\fbox{ |
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\begin{minipage}{5.0in} |
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{\it S/R CALC\_DIFFUSIVITY}({\it calc\_diffusivity.F}) |
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\end{minipage} |
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} |
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|
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\item Line 7, |
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\begin{verbatim} |
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viscAh=4.E2, |
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\end{verbatim} |
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this line sets the horizontal Laplacian frictional dissipation coefficient to |
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$1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions |
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for this operator are specified later. |
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|
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\item Lines 8, |
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\begin{verbatim} |
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no_slip_sides=.FALSE. |
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\end{verbatim} |
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this line selects a free-slip lateral boundary condition for |
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the horizontal Laplacian friction operator |
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e.g. $\frac{\partial u}{\partial y}$=0 along boundaries in $y$ and |
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$\frac{\partial v}{\partial x}$=0 along boundaries in $x$. |
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|
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\item Lines 9, |
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\begin{verbatim} |
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no_slip_bottom=.TRUE. |
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\end{verbatim} |
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this line selects a no-slip boundary condition for bottom |
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boundary condition in the vertical Laplacian friction operator |
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e.g. $u=v=0$ at $z=-H$, where $H$ is the local depth of the domain. |
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|
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\item Line 10, |
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\begin{verbatim} |
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diffKhT=4.E2, |
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\end{verbatim} |
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this line sets the horizontal diffusion coefficient for temperature |
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to $400\,{\rm m^{2}s^{-1}}$. The boundary condition on this |
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operator is $\frac{\partial}{\partial x}=\frac{\partial}{\partial y}=0$ at |
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all boundaries. |
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|
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\item Line 11, |
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\begin{verbatim} |
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diffKzT=1.E-2, |
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\end{verbatim} |
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this line sets the vertical diffusion coefficient for temperature |
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to $10^{-2}\,{\rm m^{2}s^{-1}}$. The boundary condition on this |
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operator is $\frac{\partial}{\partial z}$ = 0 on all boundaries. |
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|
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\item Line 13, |
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\begin{verbatim} |
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tAlpha=2.E-4, |
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\end{verbatim} |
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This line sets the thermal expansion coefficient for the fluid |
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to $2 \times 10^{-4}\,{\rm degrees}^{-1}$ |
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|
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\fbox{ |
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\begin{minipage}{5.0in} |
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{\it S/R FIND\_RHO}({\it find\_rho.F}) |
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\end{minipage} |
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} |
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|
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\item Line 18, |
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\begin{verbatim} |
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eosType='LINEAR' |
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\end{verbatim} |
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This line selects the linear form of the equation of state. |
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|
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\fbox{ |
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\begin{minipage}{5.0in} |
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{\it S/R FIND\_RHO}({\it find\_rho.F}) |
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\end{minipage} |
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} |
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|
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|
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|
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\item Line 40, |
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\begin{verbatim} |
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usingSphericalPolarGrid=.TRUE., |
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\end{verbatim} |
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This line requests that the simulation be performed in a |
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spherical polar coordinate system. It affects the interpretation of |
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grid input parameters, for example {\bf delX} and {\bf delY} and |
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causes the grid generation routines to initialize an internal grid based |
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on spherical polar geometry. |
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|
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\fbox{ |
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\begin{minipage}{5.0in} |
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{\it S/R INI\_SPEHRICAL\_POLAR\_GRID}({\it ini\_spherical\_polar\_grid.F}) |
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\end{minipage} |
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} |
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|
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\item Line 41, |
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\begin{verbatim} |
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phiMin=0., |
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\end{verbatim} |
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This line sets the southern boundary of the modeled |
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domain to $0^{\circ}$ latitude. This value affects both the |
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generation of the locally orthogonal grid that the model |
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uses internally and affects the initialization of the coriolis force. |
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Note - it is not required to set |
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a longitude boundary, since the absolute longitude does |
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not alter the kernel equation discretisation. |
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|
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\item Line 42, |
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\begin{verbatim} |
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delX=60*1., |
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\end{verbatim} |
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This line sets the horizontal grid spacing between each y-coordinate line |
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in the discrete grid to $1^{\circ}$ in longitude. |
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|
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\item Line 43, |
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\begin{verbatim} |
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delY=60*1., |
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\end{verbatim} |
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This line sets the horizontal grid spacing between each y-coordinate line |
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in the discrete grid to $1^{\circ}$ in latitude. |
| 420 |
|
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\item Line 44, |
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\begin{verbatim} |
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delZ=500.,500.,500.,500., |
| 424 |
\end{verbatim} |
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This line sets the vertical grid spacing between each z-coordinate line |
| 426 |
in the discrete grid to $500\,{\rm m}$, so that the total model depth |
| 427 |
is $2\,{\rm km}$. The variable {\bf delZ} is copied into the internal |
| 428 |
model coordinate variable {\bf delR} |
| 429 |
|
| 430 |
\fbox{ |
| 431 |
\begin{minipage}{5.0in} |
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{\it S/R INI\_VERTICAL\_GRID}({\it ini\_vertical\_grid.F}) |
| 433 |
\end{minipage} |
| 434 |
} |
| 435 |
|
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\item Line 47, |
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\begin{verbatim} |
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bathyFile='topog.box' |
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\end{verbatim} |
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This line specifies the name of the file from which the domain |
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bathymetry is read. This file is a two-dimensional ($x,y$) map of |
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depths. This file is assumed to contain 64-bit binary numbers |
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giving the depth of the model at each grid cell, ordered with the x |
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coordinate varying fastest. The points are ordered from low coordinate |
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to high coordinate for both axes. The units and orientation of the |
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depths in this file are the same as used in the MITgcm code. In this |
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experiment, a depth of $0m$ indicates a solid wall and a depth |
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of $-2000m$ indicates open ocean. The matlab program |
| 449 |
{\it input/gendata.m} shows an example of how to generate a |
| 450 |
bathymetry file. |
| 451 |
|
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|
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\item Line 50, |
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\begin{verbatim} |
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zonalWindFile='windx.sin_y' |
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\end{verbatim} |
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This line specifies the name of the file from which the x-direction |
| 458 |
surface wind stress is read. This file is also a two-dimensional |
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($x,y$) map and is enumerated and formatted in the same manner as the |
| 460 |
bathymetry file. The matlab program {\it input/gendata.m} includes example |
| 461 |
code to generate a valid |
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{\bf zonalWindFile} |
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file. |
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|
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\end{itemize} |
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|
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\noindent other lines in the file {\it input/data} are standard values |
| 468 |
that are described in the MITgcm Getting Started and MITgcm Parameters |
| 469 |
notes. |
| 470 |
|
| 471 |
\begin{small} |
| 472 |
% \input{part3/case_studies/carbon_outgassing_sensitivity/input/data} |
| 473 |
\end{small} |
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|
| 475 |
\subsubsection{File {\it input/data.pkg}} |
| 476 |
|
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This file uses standard default values and does not contain |
| 478 |
customizations for this experiment. |
| 479 |
|
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\subsubsection{File {\it input/eedata}} |
| 481 |
|
| 482 |
This file uses standard default values and does not contain |
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customizations for this experiment. |
| 484 |
|
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\subsubsection{File {\it input/windx.sin\_y}} |
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|
| 487 |
The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$) |
| 488 |
map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$. |
| 489 |
Although $\tau_{x}$ is only a function of $y$n in this experiment |
| 490 |
this file must still define a complete two-dimensional map in order |
| 491 |
to be compatible with the standard code for loading forcing fields |
| 492 |
in MITgcm. The included matlab program {\it input/gendata.m} gives a complete |
| 493 |
code for creating the {\it input/windx.sin\_y} file. |
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|
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\subsubsection{File {\it input/topog.box}} |
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|
| 497 |
|
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The {\it input/topog.box} file specifies a two-dimensional ($x,y$) |
| 499 |
map of depth values. For this experiment values are either |
| 500 |
$0m$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep |
| 501 |
ocean. The file contains a raw binary stream of data that is enumerated |
| 502 |
in the same way as standard MITgcm two-dimensional, horizontal arrays. |
| 503 |
The included matlab program {\it input/gendata.m} gives a complete |
| 504 |
code for creating the {\it input/topog.box} file. |
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|
| 506 |
\subsubsection{File {\it code/SIZE.h}} |
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|
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Two lines are customized in this file for the current experiment |
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|
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\begin{itemize} |
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|
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\item Line 39, |
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\begin{verbatim} sNx=60, \end{verbatim} this line sets |
| 514 |
the lateral domain extent in grid points for the |
| 515 |
axis aligned with the x-coordinate. |
| 516 |
|
| 517 |
\item Line 40, |
| 518 |
\begin{verbatim} sNy=60, \end{verbatim} this line sets |
| 519 |
the lateral domain extent in grid points for the |
| 520 |
axis aligned with the y-coordinate. |
| 521 |
|
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\item Line 49, |
| 523 |
\begin{verbatim} Nr=4, \end{verbatim} this line sets |
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the vertical domain extent in grid points. |
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|
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\end{itemize} |
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|
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\begin{small} |
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% \include{code/SIZE.h} |
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\end{small} |
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|
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\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
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|
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This file uses standard default values and does not contain |
| 535 |
customizations for this experiment. |
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|
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|
| 538 |
\subsubsection{File {\it code/CPP\_EEOPTIONS.h}} |
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|
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This file uses standard default values and does not contain |
| 541 |
customizations for this experiment. |
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|
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\subsubsection{Other Files } |
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|
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Other files relevant to this experiment are |
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\begin{itemize} |
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\item {\it model/src/ini\_cori.F}. This file initializes the model |
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coriolis variables {\bf fCorU}. |
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\item {\it model/src/ini\_spherical\_polar\_grid.F} |
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\item {\it model/src/ini\_parms.F}, |
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\item {\it input/windx.sin\_y}, |
| 552 |
\end{itemize} |
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contain the code customizations and parameter settings for this |
| 554 |
experiments. Below we describe the customizations |
| 555 |
to these files associated with this experiment. |
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