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% $Header: /u/gcmpack/manual/s_examples/global_oce_latlon/climatalogical_ogcm.tex,v 1.19 2010/08/30 23:09:20 jmc Exp $ |
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% $Name: $ |
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\section[Global Ocean MITgcm Example]{Global Ocean Simulation at $4^\circ$ Resolution} |
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%\label{www:tutorials} |
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\label{sec:eg-global} |
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\begin{rawhtml} |
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<!-- CMIREDIR:eg-global: --> |
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\end{rawhtml} |
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\begin{center} |
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(in directory: {\it verification/tutorial\_global\_oce\_latlon/}) |
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\end{center} |
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\bodytext{bgcolor="#FFFFFFFF"} |
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%\begin{center} |
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%{\Large \bf Using MITgcm to Simulate Global Climatological Ocean Circulation |
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%At Four Degree Resolution with Asynchronous Time Stepping} |
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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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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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The files for this experiment are in the verification directory |
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under tutorial\_global\_oce\_latlon. |
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Twenty levels 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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\subsection{Overview} |
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%\label{www:tutorials} |
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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 DaSilva \cite{DaSilva94}. Climatological data |
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from Levitus \cite{Levitus94} is used to initialize the model hydrography. |
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Levitus seasonal climatology data is also used throughout the calculation |
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to provide additional air-sea fluxes. |
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These fluxes are combined with the DaSilva 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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\begin{eqnarray} |
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\label{eq:eg-global-global_forcing} |
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\label{eq:eg-global-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:eg-global-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:eg-global-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:eg-global-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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\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 in |
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meters. |
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It is used in conjunction with a reference density, $\rho_{0}$ |
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(here set to $999.8\,{\rm kg\,m^{-3}}$), a |
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reference salinity, $S_{0}$ (here set to 35~ppt), |
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and a specific heat capacity, $C_{p}$ (here set to |
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$4000~{\rm J}~^{\circ}{\rm C}^{-1}~{\rm kg}^{-1}$), to convert |
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input dataset values into time tendencies of |
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potential temperature (with units of $^{\circ}{\rm C}~{\rm s}^{-1}$), |
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salinity (with units ${\rm ppt}~s^{-1}$) and |
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velocity (with units ${\rm m}~{\rm s}^{-2}$). |
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The externally supplied forcing fields used in this |
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experiment are $\tau_{x}$, $\tau_{y}$, $\theta^{\ast}$, $S^{\ast}$, |
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$\cal{Q}$ and $\cal{E}-\cal{P}-\cal{R}$. The wind stress fields ($\tau_x$, $\tau_y$) |
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have units of ${\rm N}~{\rm m}^{-2}$. The temperature forcing fields |
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($\theta^{\ast}$ and $Q$) have units of $^{\circ}{\rm C}$ and ${\rm W}~{\rm m}^{-2}$ |
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respectively. The salinity forcing fields ($S^{\ast}$ and |
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$\cal{E}-\cal{P}-\cal{R}$) have units of ${\rm ppt}$ and ${\rm m}~{\rm s}^{-1}$ |
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respectively. The source files and procedures for ingesting this data into the |
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simulation are described in the experiment configuration discussion in section |
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\ref{sec:eg-global-clim_ocn_examp_exp_config}. |
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\subsection{Discrete Numerical Configuration} |
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%\label{www:tutorials} |
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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 on the sphere |
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$\Delta \phi=\Delta \lambda=4^{\circ}$, so |
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that there are ninety grid cells in the zonal and forty in the |
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meridional direction. The internal model coordinate variables |
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$x$ and $y$ are initialized according to |
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\begin{eqnarray} |
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x=r\cos(\phi),~\Delta x & = &r\cos(\Delta \phi) \\ |
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y=r\lambda,~\Delta y &= &r\Delta \lambda |
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\end{eqnarray} |
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Arctic polar regions are not |
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included in this experiment. Meridionally the model extends from |
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$80^{\circ}{\rm S}$ to $80^{\circ}{\rm N}$. |
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Vertically the model is configured with twenty layers with the |
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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, ${\tt k}$) to |
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give a total depth, $H$, of $-5450{\rm m}$. |
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The implicit free surface form of the pressure equation described in Marshall et. al |
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\cite{marshall:97a} 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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Wind-stress forcing is added to the momentum equations in (\ref{eq:eg-global-model_equations}) |
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for both the zonal flow, $u$ and the meridional flow $v$, according to equations |
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(\ref{eq:eg-global-global_forcing_fu}) and (\ref{eq:eg-global-global_forcing_fv}). |
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Thermodynamic forcing inputs are added to the equations |
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in (\ref{eq:eg-global-model_equations}) for |
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potential temperature, $\theta$, and salinity, $S$, according to equations |
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(\ref{eq:eg-global-global_forcing_ft}) and (\ref{eq:eg-global-global_forcing_fs}). |
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This produces a set of equations solved in this configuration as follows: |
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\begin{eqnarray} |
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\label{eq:eg-global-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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\nabla_{h}\cdot A_{h}\nabla_{h}u - |
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\frac{\partial}{\partial z}A_{z}\frac{\partial u}{\partial z} |
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& = & |
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\begin{cases} |
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{\cal F}_u & \text{(surface)} \\ |
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0 & \text{(interior)} |
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\end{cases} |
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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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\nabla_{h}\cdot A_{h}\nabla_{h}v - |
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\frac{\partial}{\partial z}A_{z}\frac{\partial v}{\partial z} |
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& = & |
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\begin{cases} |
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{\cal F}_v & \text{(surface)} \\ |
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0 & \text{(interior)} |
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\end{cases} |
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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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\nabla_{h}\cdot K_{h}\nabla_{h}\theta |
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- \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial\theta}{\partial z} |
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& = & |
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\begin{cases} |
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{\cal F}_\theta & \text{(surface)} \\ |
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0 & \text{(interior)} |
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\end{cases} |
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\\ |
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\frac{D s}{Dt} - |
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\nabla_{h}\cdot K_{h}\nabla_{h}s |
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- \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial s}{\partial z} |
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& = & |
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\begin{cases} |
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{\cal F}_s & \text{(surface)} \\ |
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0 & \text{(interior)} |
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\end{cases} |
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\\ |
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g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz & = & p^{'} |
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\end{eqnarray} |
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\noindent where $u=\frac{Dx}{Dt}=r \cos(\phi)\frac{D \lambda}{Dt}$ and |
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$v=\frac{Dy}{Dt}=r \frac{D \phi}{Dt}$ |
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are the zonal and meridional components of the |
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flow vector, $\vec{u}$, on the sphere. 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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\subsubsection{Numerical Stability Criteria} |
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%\label{www:tutorials} |
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|
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The Laplacian dissipation coefficient, $A_{h}$, is set to $5 \times 10^5 m s^{-1}$. |
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This value is chosen to yield a Munk layer width \cite{adcroft:95}, |
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\begin{eqnarray} |
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\label{eq:eg-global-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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\noindent of $\approx 600$km. This is greater than the model |
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resolution in low-latitudes, $\Delta x \approx 400{\rm km}$, ensuring that the frictional |
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boundary layer is adequately resolved. |
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\\ |
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\noindent The model is stepped forward with a |
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time step $\delta t_{\theta}=30~{\rm hours}$ for thermodynamic variables and |
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$\delta t_{v}=40~{\rm minutes}$ for momentum terms. With this time step, the stability |
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parameter to the horizontal Laplacian friction \cite{adcroft:95} |
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\begin{eqnarray} |
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\label{eq:eg-global-laplacian_stability} |
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&& S_{l} = 4 \frac{A_{h} \delta t_{v}}{{\Delta x}^2} |
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\end{eqnarray} |
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\noindent evaluates to 0.16 at a latitude of $\phi=80^{\circ}$, which is below the |
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0.3 upper limit for stability. The zonal grid spacing $\Delta x$ is smallest at |
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$\phi=80^{\circ}$ where $\Delta x=r\cos(\phi)\Delta \phi\approx 77{\rm km}$. |
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\\ |
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\noindent The vertical dissipation coefficient, $A_{z}$, is set to |
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$1\times10^{-3} {\rm m}^2{\rm s}^{-1}$. The associated stability limit |
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\begin{eqnarray} |
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\label{eq:eg-global-laplacian_stability_z} |
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S_{l} = 4 \frac{A_{z} \delta t_{v}}{{\Delta z}^2} |
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\end{eqnarray} |
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\noindent evaluates to $0.015$ for the smallest model |
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level spacing ($\Delta z_{1}=50{\rm m}$) which is again well below |
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the upper stability limit. |
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\\ |
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The values of the horizontal ($K_{h}$) and vertical ($K_{z}$) diffusion coefficients |
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for both temperature and salinity are set to $1 \times 10^{3}~{\rm m}^{2}{\rm s}^{-1}$ |
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and $3 \times 10^{-5}~{\rm m}^{2}{\rm s}^{-1}$ respectively. The stability limit |
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related to $K_{h}$ will be at $\phi=80^{\circ}$ where $\Delta x \approx 77 {\rm km}$. |
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Here the stability parameter |
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\begin{eqnarray} |
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\label{eq:eg-global-laplacian_stability_xtheta} |
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S_{l} = \frac{4 K_{h} \delta t_{\theta}}{{\Delta x}^2} |
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\end{eqnarray} |
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evaluates to $0.07$, well below the stability limit of $S_{l} \approx 0.5$. The |
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stability parameter related to $K_{z}$ |
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\begin{eqnarray} |
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\label{eq:eg-global-laplacian_stability_ztheta} |
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S_{l} = \frac{4 K_{z} \delta t_{\theta}}{{\Delta z}^2} |
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\end{eqnarray} |
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evaluates to $0.005$ for $\min(\Delta z)=50{\rm m}$, well below the stability limit |
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of $S_{l} \approx 0.5$. |
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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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1.19 |
\label{eq:eg-global-inertial_stability} |
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S_{i} = f^{2} {\delta t_v}^2 |
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\end{eqnarray} |
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\noindent evaluates to $0.24$ for $f=2\omega\sin(80^{\circ})=1.43\times10^{-4}~{\rm s}^{-1}$, which is close to |
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the $S_{i} < 1$ upper 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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1.1 |
horizontal flow |
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speed of $ | \vec{u} | = 2 ms^{-1}$ |
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\begin{eqnarray} |
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jmc |
1.19 |
\label{eq:eg-global-cfl_stability} |
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S_{a} = \frac{| \vec{u} | \delta t_{v}}{ \Delta x} |
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\end{eqnarray} |
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\noindent evaluates to $6 \times 10^{-2}$. 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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cnh |
1.3 |
\noindent The stability parameter for internal gravity waves propagating |
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1.1 |
with a maximum speed of $c_{g}=10~{\rm ms}^{-1}$ |
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adcroft |
1.4 |
\cite{adcroft:95} |
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1.1 |
|
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\begin{eqnarray} |
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jmc |
1.19 |
\label{eq:eg-global-gfl_stability} |
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S_{c} = \frac{c_{g} \delta t_{v}}{ \Delta x} |
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\end{eqnarray} |
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\noindent evaluates to $3 \times 10^{-1}$. This is close to the linear |
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stability limit of 0.5. |
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\subsection{Experiment Configuration} |
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jmc |
1.19 |
%\label{www:tutorials} |
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\label{sec:eg-global-clim_ocn_examp_exp_config} |
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adcroft |
1.1 |
|
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The model configuration for this experiment resides under the |
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directory {\it tutorial\_examples/global\_ocean\_circulation/}. |
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The experiment files |
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|
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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.bin}, |
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\item {\it input/windy.bin}, |
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\item {\it input/salt.bin}, |
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\item {\it input/theta.bin}, |
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\item {\it input/SSS.bin}, |
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\item {\it input/SST.bin}, |
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\item {\it input/topog.bin}, |
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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 these |
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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{Driving Datasets} |
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%\label{www:tutorials} |
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|
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1.19 |
Figures ({\it --- missing figures ---}) |
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%(\ref{fig:sim_config_tclim}-\ref{fig:sim_config_empmr}) |
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show the relaxation temperature ($\theta^{\ast}$) and salinity ($S^{\ast}$) |
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fields, the wind stress components ($\tau_x$ and $\tau_y$), the heat flux ($Q$) |
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and the net fresh water flux (${\cal E} - {\cal P} - {\cal R}$) used |
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in equations |
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(\ref{eq:eg-global-global_forcing_fu}-\ref{eq:eg-global-global_forcing_fs}). |
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The figures also indicate the lateral extent and coastline used in the |
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experiment. Figure ({\it --- missing figure --- }) %ref{fig:model_bathymetry}) |
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shows the depth contours of the model domain. |
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|
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\subsubsection{File {\it input/data}} |
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jmc |
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%\label{www:tutorials} |
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|
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jmc |
1.20 |
\input{s_examples/global_oce_latlon/inp_data} |
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1.1 |
|
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\subsubsection{File {\it input/data.pkg}} |
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jmc |
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%\label{www:tutorials} |
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adcroft |
1.1 |
|
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This file uses standard default values and does not contain |
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customisations for this experiment. |
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|
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\subsubsection{File {\it input/eedata}} |
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jmc |
1.19 |
%\label{www:tutorials} |
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1.1 |
|
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This file uses standard default values and does not contain |
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customisations for this experiment. |
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|
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\subsubsection{File {\it input/windx.sin\_y}} |
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jmc |
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%\label{www:tutorials} |
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|
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The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$) |
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map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$. |
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Although $\tau_{x}$ is only a function of $y$n in this experiment |
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this file must still define a complete two-dimensional map in order |
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to be compatible with the standard code for loading forcing fields |
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in MITgcm. The included matlab program {\it input/gendata.m} gives a complete |
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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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jmc |
1.19 |
%\label{www:tutorials} |
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adcroft |
1.1 |
|
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|
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The {\it input/topog.box} file specifies a two-dimensional ($x,y$) |
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map of depth values. For this experiment values are either |
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$0m$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep |
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ocean. The file contains a raw binary stream of data that is enumerated |
390 |
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in the same way as standard MITgcm two-dimensional, horizontal arrays. |
391 |
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The included matlab program {\it input/gendata.m} gives a complete |
392 |
|
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code for creating the {\it input/topog.box} file. |
393 |
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|
394 |
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\subsubsection{File {\it code/SIZE.h}} |
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jmc |
1.19 |
%\label{www:tutorials} |
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adcroft |
1.1 |
|
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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, |
402 |
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\begin{verbatim} sNx=60, \end{verbatim} this line sets |
403 |
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the lateral domain extent in grid points for the |
404 |
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axis aligned with the x-coordinate. |
405 |
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|
406 |
|
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\item Line 40, |
407 |
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\begin{verbatim} sNy=60, \end{verbatim} this line sets |
408 |
|
|
the lateral domain extent in grid points for the |
409 |
|
|
axis aligned with the y-coordinate. |
410 |
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|
411 |
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\item Line 49, |
412 |
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\begin{verbatim} Nr=4, \end{verbatim} this line sets |
413 |
|
|
the vertical domain extent in grid points. |
414 |
|
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|
415 |
|
|
\end{itemize} |
416 |
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|
417 |
|
|
\begin{small} |
418 |
jmc |
1.18 |
\input{s_examples/global_oce_latlon/code/SIZE.h} |
419 |
adcroft |
1.1 |
\end{small} |
420 |
|
|
|
421 |
|
|
\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
422 |
jmc |
1.19 |
%\label{www:tutorials} |
423 |
adcroft |
1.1 |
|
424 |
|
|
This file uses standard default values and does not contain |
425 |
|
|
customisations for this experiment. |
426 |
|
|
|
427 |
|
|
|
428 |
|
|
\subsubsection{File {\it code/CPP\_EEOPTIONS.h}} |
429 |
jmc |
1.19 |
%\label{www:tutorials} |
430 |
adcroft |
1.1 |
|
431 |
|
|
This file uses standard default values and does not contain |
432 |
|
|
customisations for this experiment. |
433 |
|
|
|
434 |
|
|
\subsubsection{Other Files } |
435 |
jmc |
1.19 |
%\label{www:tutorials} |
436 |
adcroft |
1.1 |
|
437 |
|
|
Other files relevant to this experiment are |
438 |
|
|
\begin{itemize} |
439 |
|
|
\item {\it model/src/ini\_cori.F}. This file initializes the model |
440 |
|
|
coriolis variables {\bf fCorU}. |
441 |
|
|
\item {\it model/src/ini\_spherical\_polar\_grid.F} |
442 |
|
|
\item {\it model/src/ini\_parms.F}, |
443 |
|
|
\item {\it input/windx.sin\_y}, |
444 |
|
|
\end{itemize} |
445 |
|
|
contain the code customisations and parameter settings for this |
446 |
cnh |
1.3 |
experiments. Below we describe the customisations |
447 |
adcroft |
1.1 |
to these files associated with this experiment. |