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% $Header: /u/gcmpack/manual/part3/case_studies/held_suarez_cs/held_suarez_cs.tex,v 1.4 2005/08/09 18:46:39 jmc Exp $ |
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% $Name: $ |
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\section[Held-Suarez Atmosphere MITgcm Example]{Held-Suarez atmospheric simulation |
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on cube-sphere grid with 32 square cube faces.} |
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\label{www:tutorials} |
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\label{sect:eg-hs} |
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\begin{rawhtml} |
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<!-- CMIREDIR:eg-hs: --> |
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\end{rawhtml} |
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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 illustrates the use of the MITgcm as an Atmospheric GCM, |
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using simple \cite{held-suar:94} forcing |
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to simulate Atmospheric Dynamics on global scale. |
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The set-up use the rescaled pressure coordinate ($p^*$)\cite[]{adcroft:04a} |
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in the vertical direction, with 20 equaly-spaced levels, and |
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the conformal cube-sphere grid (C32) \cite[]{adcroft:04b}. |
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\subsection{Overview} |
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\label{www:tutorials} |
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This example demonstrates using the MITgcm to simulate |
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the planetary atmospheric circulation, with flat orography |
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and simplified forcing. |
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In particular, only dry air processes are considered and |
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radiation effects are represented by a simple newtownien cooling, |
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Thus this example does not rely on any particular atmospheric |
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physics package. |
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This kind of simplified atmospheric simulation has been widely |
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used in GFD-type experiments and in intercomparison projects of |
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AGCM dynamical cores \cite[]{held-suar:94}. |
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The horizontal grid is obtain from the projection of a uniform gridded cube |
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to the sphere. Each of the 6 faces has the same resolution, with |
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$32 \times 32$ grid points. The equator line coincide with a grid line |
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and crosses, right in the midle, 4 of the 6 faces, leaving 2 faces |
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for the Northern and Southern polar regions. |
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This curvilinear grid requires the use of the 2nd generation exchange |
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topology ({\it pkg/exch2}) to connect tile and face edges, |
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but without any limitation on the number of processors. |
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The use of the $p^*$ coordinate with 20 equally spaced levels |
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($20 \times 50\,{\rm mb}$, from $p^*=1000,{\rm mb}$ to $0$ at the |
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top of the atmosphere) follows the choice of \cite{held-suar:94}. |
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Note that without topography, the $p^*$ coordinate and the normalized |
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pressure coordinate ($\sigma_p$) coincide exactly. |
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No viscosity and zero diffusion are used here, but |
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a $8^th$ order \cite{Shapiro_70} filter is applied to both momentum and |
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potential temperature, to remove selectively grid scale noise. |
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Apart from the horizontal grid, this experiment is made very similar to |
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the grid-point model case used in \cite{held-suar:94} study. |
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At this resolution, the configuration can be integrated forward |
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for many years on a single processor desktop computer. |
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\\ |
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\subsection{Forcing} |
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\label{www:tutorials} |
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The model is forced by relaxation to a radiative equilibrium temperature from |
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\cite{held-suar:94}. |
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A linear frictional drag (Rayleigh damping) is applied in the lower |
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part of the atmosphere and account from surface friction and momentum |
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dissipation in the boundary layer. |
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Altogether, this yields the following forcing |
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\cite[from][]{held-suar:94} that is applied to the fluid: |
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\begin{eqnarray} |
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\label{EQ:eg-hs-global_forcing} |
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\label{EQ:eg-hs-global_forcing_fv} |
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\vec{{\cal F}_\mathbf{v}} & = & -k_\mathbf{v}(p)\vec{\mathbf{v}}_h |
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\\ |
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\label{EQ:eg-hs-global_forcing_ft} |
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{\cal F}_{\theta} & = & -k_{\theta}(\varphi,p)[\theta-\theta_{eq}(\varphi,p)] |
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\end{eqnarray} |
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\noindent where $\vec{\cal F}_\mathbf{v}$, ${\cal F}_{\theta}$, |
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are the forcing terms in the zonal and meridional |
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momentum and in the potential temperature equations respectively. |
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The term $k_\mathbf{v}$ in equation (\ref{EQ:eg-hs-global_forcing_fv}) applies a |
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Rayleigh damping that is active within the planetary boundary layer. |
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It is defined so as to decay as pressure decreases according to |
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\begin{eqnarray*} |
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\label{EQ:eg-hs-define_kv} |
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k_\mathbf{v} & = & k_{f}~\max[0,~(p^*/P^{0}_{s}-\sigma_{b})/(1-\sigma_{b})] |
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\\ |
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\sigma_{b} & = & 0.7 ~~{\rm and}~~ |
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k_{f} = 1/86400 ~{\rm s}^{-1} |
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\end{eqnarray*} |
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where $p^*$ is the pressure level of the cell center |
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and $P^{0}_{s}$ is the pressure at the base of the atmospheric column, |
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which is constant and uniform here ($= 10^5 {\rm Pa}$), in the absence |
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of topography. |
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The Equilibrium temperature $\theta_{eq}$ and relaxation time scale $k_{\theta}$ |
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are set to: |
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\begin{eqnarray} |
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\label{EQ:eg-hs-define_Teq} |
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\theta_{eq}(\varphi,p^*) & = & \max \{ 200.K (P^{0}_{s}/p^*)^\kappa,\\ |
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\nonumber |
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& & \hspace{8mm} 315.K - \Delta T_y~\sin^2(\varphi) |
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- \Delta \theta_z \cos^2(\varphi) \log(p^*/P^{0}_s) \} |
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\\ |
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\label{EQ:eg-hs-define_kT} |
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k_{\theta}(\varphi,p^*) & = & |
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k_a + (k_s -k_a)~\cos^4(\varphi)~\max[0,(p^*/P^{0}_{s}-\sigma_{b})/(1-\sigma_{b})] |
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\end{eqnarray} |
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with: |
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\begin{eqnarray*} |
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\Delta T_y = 60.K & k_a = 1/(40 \cdot 86400) ~{\rm s}^{-1}\\ |
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\Delta \theta_z = 10.K & k_s = 1/(4 \cdot 86400) ~{\rm s}^{-1} |
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\end{eqnarray*} |
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Initial conditions correspond to a resting state with horizontally uniform |
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stratified fluid. The initial temperature profile is simply the |
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horizontally average of the radiative equilibrium temperature. |
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\subsection{Set-up description} |
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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, using non-boussinesq |
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$p^*$ coordinate. |
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The vertical resolution is uniform, $\Delta p^* = 50.10^2 Pa$, |
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with 20 levels, from $p^*=10^5 Pa$ to $0$ at the top. |
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The domain is discretised using C32 cube-sphere grid \cite[]{adcroft:04b} |
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that cover the whole sphere with a relatively uniform grid-spacing. |
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The resolution at the equator or along the Greenwitch meridian |
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is similar to the $128 \times 64$ equaly spaced longitude-latitude grid, |
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but requires $25\%$ less grid points. |
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Grid spacing and grid-point location are not computed by the model but |
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read from files. |
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The vector-invariant form of the momentum equation (see section |
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\ref{sect:vect-inv_momentum_equations}) is used so that no explicit |
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metrics are necessary. |
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Applying the vector-invariant discretization to the |
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atmospheric equations \ref{eq:atmos-prime}, and adding the |
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forcing term |
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(\ref{EQ:eg-hs-global_forcing_fv}, \ref{EQ:eg-hs-global_forcing_ft}) |
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on the right-hand-side, |
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leads to the set of equations that are solved in this configuration: |
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%the The set of equations solved here is der |
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%Wind-stress forcing is 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:eg-hs-global_forcing_fv}) and (\ref{EQ:eg-hs-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:eg-hs-global_forcing_ft}) and (\ref{EQ:eg-hs-global_forcing_fs}). |
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\begin{eqnarray} |
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\label{EQ:eg-hs-model_equations} |
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\frac{\partial \vec{\mathbf{v}}_h}{\partial t} |
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+(f + \zeta)\hat{\mathbf{k}} \times \vec{\mathbf{v}}_h |
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%+\mathbf{\nabla }_{p} ({\rm KE}) |
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+\mathbf{\nabla }_{p} (\mbox{\sc ke}) |
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+ \omega \frac{\partial \vec{\mathbf{v}}_h }{\partial p} |
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+\mathbf{\nabla }_p \Phi ^{\prime } |
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&=& |
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% \vec{{\cal F}_v} = |
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-k_\mathbf{v}\vec{\mathbf{v}}_h |
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\\ |
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\frac{\partial \Phi ^{\prime }}{\partial p} |
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+\frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&0 |
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\\ |
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\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_h+\frac{\partial \omega }{ |
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\partial p} &=&0 |
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\\ |
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\frac{\partial \theta }{\partial t} |
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+ \mathbf{\nabla }_{p}\cdot (\theta \vec{\mathbf{v}}_h) |
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+ \frac{\partial (\theta \omega)}{\partial p} |
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%= \frac{\mathcal{Q}}{\Pi } |
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&=& -k_{\theta}[\theta-\theta_{eq}] |
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\end{eqnarray} |
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%\begin{equation} |
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%\partial_t \vec{v} + ( 2\vec{\Omega} + \vec{\zeta}) \wedge \vec{v} |
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%- b \hat{r} |
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%+ \vec{\nabla} B = \vec{\nabla} \cdot \vec{\bf \tau} |
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%\end{equation} |
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%{\cal F}_{\theta} & = & -k_{\theta}(\varphi,p)[\theta-\theta_{eq}(\varphi,p)] |
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|
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\noindent where $\vec{\mathbf{v}}_h$ and $\omega = \frac{Dp}{Dt}$ |
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are the horizontal velocity vector and the vertical velocity in pressure coordinate, |
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$\zeta$ is the relative vorticity and $f$ the Coriolis parameter, |
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$\hat{\mathbf{k}}$ is the vertical unity vector, |
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{\sc ke} is the kinetic energy, $\Phi$ is the geopotential |
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and $\Pi$ the Exner function |
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($\Pi = C_p (p/p_c)^\kappa ~{\rm with}~ p_c = 10^5 Pa$). |
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Variables marked with ' corresponds to anomaly from |
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the resting, uniformly stratified state. |
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As described in MITgcm Numerical Solution Procedure \ref{chap:discretization}, |
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the continuity equation is integrated vertically, to give a prognostic |
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equation for the surface pressure $p_s$: |
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\begin{equation} |
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\frac{\partial p_s}{\partial t} + \nabla_{h}\cdot \int_{0}^{p_s} \vec{\mathbf{v}}_h dp |
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= 0 |
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\end{equation} |
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The implicit free surface form of the pressure equation described in |
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\cite{marshall:97a} is employed to solve for $p_s$; |
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Integrating vertically the hydrostatic balance |
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gives the geopotential $\Phi'$ and allow to step forward the momentum equation |
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\ref{EQ:eg-hs-model_equations}. |
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The potential temperature, $\theta$, is stepped forward using the |
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new velocity field ({\it staggered time-step}, section |
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\ref{sect:adams-bashforth-staggered}). |
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\\ |
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\subsubsection{Numerical Stability Criteria} |
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\label{www:tutorials} |
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\noindent The numerical stability for inertial oscillations |
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\cite{adcroft:95} |
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\begin{eqnarray} |
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\label{EQ:eg-hs-inertial_stability} |
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S_{i} = f^{2} {\Delta t}^2 |
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\end{eqnarray} |
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\noindent evaluates to $4.\times10^{-3}$ at the poles, |
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for $f=2\Omega\sin(\pi / 2) =1.45\times10^{-4}~{\rm s}^{-1}$, |
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which is well below the $S_{i} < 1$ upper limit for stability. |
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\\ |
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\noindent The advective CFL \cite{adcroft:95} |
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for a extreme maximum horizontal flow speed of $ | \vec{u} | = 90. {\rm m/s}$~ |
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and the smallest horizontal grid spacing $ \Delta x = 1.1\times10^5 {\rm m}$~: |
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\begin{eqnarray} |
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\label{EQ:eg-hs-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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\noindent evaluates to $0.37$, which is close to the stability |
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limit of 0.5. |
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\\ |
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\noindent The stability parameter for internal gravity waves propagating |
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with a maximum speed of $c_{g}=100~{\rm m/s}$ |
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\cite{adcroft:95} |
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\begin{eqnarray} |
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\label{EQ:eg-hs-gfl_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 $4 \times 10^{-1}$. 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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\label{www:tutorials} |
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\label{SEC:eg-hs_examp_exp_config} |
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The model configuration for this experiment resides under the |
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directory {\it verification/tutorial\_held\_suarez\_cs}. 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/data.shap}, |
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\item {\it code/packages.conf}, |
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\item {\it code/CPP\_OPTIONS.h}, |
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\item {\it code/SIZE.h}, |
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\item {\it code/DIAGNOSTICS\_SIZE.h}, |
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\item {\it code/external\_forcing.F}, |
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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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\subsubsection{File {\it input/data}} |
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\label{www:tutorials} |
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\input{part3/case_studies/held_suarez_cs/inp_data} |
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\subsubsection{File {\it input/data.pkg}} |
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\label{www:tutorials} |
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\input{part3/case_studies/held_suarez_cs/inp_data.pkg} |
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\subsubsection{File {\it input/eedata}} |
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\label{www:tutorials} |
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This file uses standard default values except line 6: |
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\begin{verbatim} |
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useCubedSphereExchange=.TRUE., |
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\end{verbatim} |
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This line selects the cubed-sphere specific exchanges to |
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to connect tiles and faces edges. |
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\subsubsection{File {\it input/data.shap}} |
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\label{www:tutorials} |
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\input{part3/case_studies/held_suarez_cs/inp_data.shap} |
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\subsubsection{File {\it code/SIZE.h}} |
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\label{www:tutorials} |
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Four lines are customized in this file for the current experiment |
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\begin{itemize} |
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\item Line 39, |
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\begin{verbatim} sNx=32, \end{verbatim} |
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sets the lateral domain extent in grid points along the x-direction, |
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for 1 face. |
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\item Line 40, |
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\begin{verbatim} sNy=32, \end{verbatim} |
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sets the lateral domain extent in grid points along the y-direction, |
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for 1 face. |
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\item Line 43, |
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\begin{verbatim} nSx=6, \end{verbatim} |
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sets the number of tiles in the x-direction, for the model domain |
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decomposition. In this simple case (one processor and 1 tile per |
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face), this number corresponds to the total number of faces. |
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|
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\item Line 49, |
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\begin{verbatim} Nr=20, \end{verbatim} |
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sets the vertical domain extent in grid points. |
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|
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\end{itemize} |
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%\begin{small} |
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%\input{part3/case_studies/held_suarez_cs/code/SIZE.h} |
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%\end{small} |
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|
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\subsubsection{File {\it code/packages.conf}} |
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\label{www:tutorials} |
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\input{part3/case_studies/held_suarez_cs/cod_packages.conf} |
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|
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\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
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\label{www:tutorials} |
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This file uses standard default except for Line 40\\ |
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({\it diff CPP\_OPTIONS.h ../../../model/inc}): |
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\begin{verbatim} |
| 357 |
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#define NONLIN_FRSURF |
| 358 |
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\end{verbatim} |
| 359 |
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This line allow to use the non-linear free-surface part of the code, |
| 360 |
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which is required for the $p^*$ coordinate formulation. |
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|
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\subsubsection{Other Files } |
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\label{www:tutorials} |
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Other files relevant to this experiment are |
| 366 |
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\begin{itemize} |
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\item {\it code/external\_forcing.F} |
| 368 |
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\item {\it input/grid\_cs32.face00[n].bin}, with $n=1,2,3,4,5,6$ |
| 369 |
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\end{itemize} |
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contain the code customisations and binary input files for this |
| 371 |
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experiments. Below we describe the customisations |
| 372 |
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to these files associated with this experiment.\\ |
| 373 |
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| 374 |
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The file {\it code/external\_forcing.F} contains 4 subroutines |
| 375 |
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that calculate the forcing terms (Right-Hand side term) in the |
| 376 |
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momentum equation (\ref{EQ:eg-hs-global_forcing_fv}, |
| 377 |
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{\it S/R EXTERNAL\_FORCING\_U} and {\it EXTERNAL\_FORCING\_V}) |
| 378 |
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and in the potential temperature equation |
| 379 |
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(\ref{EQ:eg-hs-global_forcing_ft}, {\it S/R EXTERNAL\_FORCING\_T}). |
| 380 |
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The water-vapour forcing subroutine ({\it S/R EXTERNAL\_FORCING\_S}) |
| 381 |
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is left empty for this experiment.\\ |
| 382 |
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|
| 383 |
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The grid-files {\it input/grid\_cs32.face00[n].bin}, with $n=1,2,3,4,5,6$, |
| 384 |
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are binary files (direct-access, big-endian 64.bits real) that |
| 385 |
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contains all the cubed-sphere grid lengths, areas and grid-point |
| 386 |
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positions, with one file per face. |
| 387 |
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Each file contains 18 2-D arrays (dimension $33 \times 33$) that correspond |
| 388 |
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1.3 |
to the model variables: |
| 389 |
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{\it |
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XC YC DXF DYF RA XG YG DXV DYU RAZ DXC DYC RAW RAS DXG DYG AngleCS AngleSN |
| 391 |
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} |
| 392 |
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(see {\it GRID.h} file) |
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