16 |
\section{A Rotating Tank in Cylindrical Coordinates} |
\section{A Rotating Tank in Cylindrical Coordinates} |
17 |
\label{sect:eg-tank} |
\label{sect:eg-tank} |
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\label{www:tutorials} |
\label{www:tutorials} |
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\begin{rawhtml} |
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<!-- CMIREDIR:eg-tank: --> |
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\end{rawhtml} |
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This section illustrates an example of MITgcm simulating a laboratory |
This section illustrates an example of MITgcm simulating a laboratory |
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experiment on much smaller scales than those common to geophysical |
experiment on much smaller scales than those commonly considered in |
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geophysical |
26 |
fluid dynamics. |
fluid dynamics. |
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28 |
\subsection{Overview} |
\subsection{Overview} |
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\label{www:tutorials} |
\label{www:tutorials} |
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This example configuration demonstrates using the MITgcm to simulate a |
32 |
This example experiment demonstrates using the MITgcm to simulate |
laboratory demonstration using a differentially heated rotating |
33 |
a laboratory experiment with a rotating tank of water with an ice |
annulus of water. The simulation is configured for a laboratory scale |
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bucket in the center. The simulation is configured for a laboratory |
on a $3^{\circ}$ $\times$ 1cm cyclindrical grid with twenty-nine |
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scale on a |
vertical levels of 0.5cm each. This is a typical laboratory setup for |
36 |
$3^{\circ}$ $\times$ 20cm |
illustration principles of GFD, as well as for a laboratory data |
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cyclindrical grid with twenty-nine vertical |
assimilation project. |
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levels. |
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\\ |
\\ |
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example illustration from GFD lab here |
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\\ |
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This example experiment demonstrates using the MITgcm to simulate |
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a Barotropic, wind-forced, ocean gyre circulation. The experiment |
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is a numerical rendition of the gyre circulation problem similar |
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to the problems described analytically by Stommel in 1966 |
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\cite{Stommel66} and numerically in Holland et. al \cite{Holland75}. |
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In this experiment the model |
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is configured to represent a rectangular enclosed box of fluid, |
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$1200 \times 1200 $~km in lateral extent. The fluid is $5$~km deep and is forced |
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by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally |
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in the ``north-south'' direction. Topologically the grid is Cartesian and |
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the coriolis parameter $f$ is defined according to a mid-latitude beta-plane |
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equation |
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\begin{equation} |
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\label{EQ:eg-baro-fcori} |
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f(y) = f_{0}+\beta y |
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\end{equation} |
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\noindent where $y$ is the distance along the ``north-south'' axis of the |
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simulated domain. For this experiment $f_{0}$ is set to $10^{-4}s^{-1}$ in |
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(\ref{EQ:eg-baro-fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$. |
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\\ |
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\\ |
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The sinusoidal wind-stress variations are defined according to |
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\begin{equation} |
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\label{EQ:eg-baro-taux} |
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\tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y}) |
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\end{equation} |
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45 |
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\noindent where $L_{y}$ is the lateral domain extent ($1200$~km) and |
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$\tau_0$ is set to $0.1N m^{-2}$. |
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\\ |
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\\ |
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Figure \ref{FIG:eg-baro-simulation_config} |
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summarizes the configuration simulated. |
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%% === eh3 === |
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\begin{figure} |
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%% \begin{center} |
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%% \resizebox{7.5in}{5.5in}{ |
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%% \includegraphics*[0.2in,0.7in][10.5in,10.5in] |
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%% {part3/case_studies/barotropic_gyre/simulation_config.eps} } |
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%% \end{center} |
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\centerline{ |
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\scalefig{.95} |
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\epsfbox{part3/case_studies/barotropic_gyre/simulation_config.eps} |
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} |
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\caption{Schematic of simulation domain and wind-stress forcing function |
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for barotropic gyre numerical experiment. The domain is enclosed bu solid |
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walls at $x=$~0,1200km and at $y=$~0,1200km.} |
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\label{FIG:eg-baro-simulation_config} |
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\end{figure} |
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\subsection{Equations Solved} |
\subsection{Equations Solved} |
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\label{www:tutorials} |
\label{www:tutorials} |
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The model is configured in hydrostatic form. The implicit free surface form of the |
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pressure equation described in Marshall et. al \cite{marshall:97a} is |
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employed. |
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A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous |
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dissipation. The wind-stress momentum input is added to the momentum equation |
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for the ``zonal flow'', $u$. Other terms in the model |
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are explicitly switched off for this experiment configuration (see section |
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\ref{SEC:code_config} ), yielding an active set of equations solved in this |
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configuration as follows |
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\begin{eqnarray} |
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\label{EQ:eg-baro-model_equations} |
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\frac{Du}{Dt} - fv + |
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g\frac{\partial \eta}{\partial x} - |
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A_{h}\nabla_{h}^2u |
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& = & |
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\frac{\tau_{x}}{\rho_{0}\Delta z} |
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\\ |
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\frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} - |
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A_{h}\nabla_{h}^2v |
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& = & |
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0 |
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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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\end{eqnarray} |
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\noindent where $u$ and $v$ and the $x$ and $y$ components of the |
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flow vector $\vec{u}$. |
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\\ |
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49 |
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50 |
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\subsection{Discrete Numerical Configuration} |
\subsection{Discrete Numerical Configuration} |
52 |
\label{www:tutorials} |
\label{www:tutorials} |
53 |
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54 |
The domain is discretised with |
The domain is discretised with a uniform cylindrical grid spacing in |
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a uniform grid spacing in the horizontal set to |
the horizontal set to $\Delta a=1$~cm and $\Delta \phi=3^{\circ}$, so |
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$\Delta x=\Delta y=20$~km, so |
that there are 120 grid cells in the azimuthal direction and |
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that there are sixty grid cells in the $x$ and $y$ directions. Vertically the |
thirty-one grid cells in the radial, representing a tank 62cm in |
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model is configured with a single layer with depth, $\Delta z$, of $5000$~m. |
diameter. The bathymetry file sets the depth=0 in the nine lowest |
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radial rows to represent the central of the annulus. Vertically the |
60 |
\subsubsection{Numerical Stability Criteria} |
model is configured with twenty-nine layers of uniform 0.5cm |
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\label{www:tutorials} |
thickness. |
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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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\begin{eqnarray} |
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\label{EQ:eg-baro-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 $\approx 100$km. This is greater than the model |
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resolution $\Delta x$, ensuring that the frictional boundary |
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layer is well resolved. |
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\\ |
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\noindent The model is stepped forward with a |
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time step $\delta t=1200$secs. With this time step the stability |
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parameter to the horizontal Laplacian friction \cite{adcroft:95} |
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\begin{eqnarray} |
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\label{EQ:eg-baro-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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\noindent evaluates to 0.012, which is well below the 0.3 upper limit |
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for stability. |
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62 |
\\ |
\\ |
63 |
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something about heat flux |
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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-baro-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 $0.0144$, which is well below the $0.5$ upper |
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limit for stability. |
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\\ |
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\noindent The advective CFL \cite{adcroft:95} for an extreme maximum |
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horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$ |
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\begin{eqnarray} |
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\label{EQ:eg-baro-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 0.12. This is approaching the stability limit |
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of 0.5 and limits $\delta t$ to $1200s$. |
|
64 |
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65 |
\subsection{Code Configuration} |
\subsection{Code Configuration} |
66 |
\label{www:tutorials} |
\label{www:tutorials} |
92 |
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93 |
\begin{itemize} |
\begin{itemize} |
94 |
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95 |
\item Line X, \begin{verbatim} viscAh=5.0E-6, \end{verbatim} this line sets |
\item Lines 9-10, \begin{verbatim} |
96 |
the Laplacian friction coefficient to $0.000006 m^2s^{-1}$, which is ususally |
viscAh=5.0E-6, |
97 |
low because of the small scale, presumably.... qqq |
viscAz=5.0E-6, |
98 |
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\end{verbatim} |
99 |
\item Line X, \begin{verbatim}f0=0.5 , \end{verbatim} this line sets the |
|
100 |
coriolis term, and represents a tank spinning at qqq |
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101 |
\item Line 10, \begin{verbatim} beta=1.E-11, \end{verbatim} this line sets |
These lines set the Laplacian friction coefficient in the horizontal |
102 |
$\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$ |
and vertical, respectively. Note that they are several orders of |
103 |
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magnitude smaller than the other examples due to the small scale of |
104 |
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this example. |
105 |
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106 |
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\item Lines 13-16, \begin{verbatim} |
107 |
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diffKhT=2.5E-6, |
108 |
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diffKzT=2.5E-6, |
109 |
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diffKhS=1.0E-6, |
110 |
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diffKzS=1.0E-6, |
111 |
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112 |
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\end{verbatim} |
113 |
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114 |
\item Lines 15 and 16 |
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115 |
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These lines set horizontal and vertical diffusion coefficients for |
116 |
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temperature and salinity. Similarly to the friction coefficients, the |
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values are a couple of orders of magnitude less than most |
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configurations. |
119 |
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120 |
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\item Line 17, \begin{verbatim}f0=0.5 , \end{verbatim} this line sets the |
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coriolis term, and represents a tank spinning at about 2.4 rpm. |
123 |
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\item Lines 23 and 24 |
125 |
\begin{verbatim} |
\begin{verbatim} |
126 |
rigidLid=.TRUE., |
rigidLid=.TRUE., |
127 |
implicitFreeSurface=.FALSE., |
implicitFreeSurface=.FALSE., |
128 |
\end{verbatim} |
\end{verbatim} |
129 |
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130 |
these lines do the opposite of the following: |
These lines activate the rigid lid formulation of the surface |
131 |
suppress the rigid lid formulation of the surface |
pressure inverter and suppress the implicit free surface form |
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pressure inverter and activate the implicit free surface form |
|
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of the pressure inverter. |
of the pressure inverter. |
133 |
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134 |
\item Line 27, |
\item Line 40, |
135 |
\begin{verbatim} |
\begin{verbatim} |
136 |
startTime=0, |
nIter=0, |
137 |
\end{verbatim} |
\end{verbatim} |
138 |
this line indicates that the experiment should start from $t=0$ |
This line indicates that the experiment should start from $t=0$ and |
139 |
and implicitly suppresses searching for checkpoint files associated |
implicitly suppresses searching for checkpoint files associated with |
140 |
with restarting an numerical integration from a previously saved state. |
restarting an numerical integration from a previously saved state. |
141 |
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Instead, the file thetaPol.bin will be loaded to initialized the |
142 |
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temperature fields as indicated below, and other variables will be |
143 |
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initialized to their defaults. |
144 |
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145 |
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146 |
\item Line 30, |
\item Line 43, |
147 |
\begin{verbatim} |
\begin{verbatim} |
148 |
deltaT=0.1, |
deltaT=0.1, |
149 |
\end{verbatim} |
\end{verbatim} |
150 |
This line sets the integration timestep to $0.1s$. This is an unsually |
This line sets the integration timestep to $0.1s$. This is an |
151 |
small value among the examples due to the small physical scale of the |
unsually small value among the examples due to the small physical |
152 |
experiment. |
scale of the experiment. Using the ensemble Kalman filter to produce |
153 |
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input fields can necessitate even shorter timesteps. |
154 |
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|
155 |
\item Line 39, |
\item Line 56, |
156 |
\begin{verbatim} |
\begin{verbatim} |
157 |
usingCylindricalGrid=.TRUE., |
usingCylindricalGrid=.TRUE., |
158 |
\end{verbatim} |
\end{verbatim} |
159 |
This line requests that the simulation be performed in a |
This line requests that the simulation be performed in a |
160 |
cylindrical coordinate system. |
cylindrical coordinate system. |
161 |
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|
162 |
\item Line qqq, |
\item Line 57, |
163 |
\begin{verbatim} |
\begin{verbatim} |
164 |
dXspacing=3, |
dXspacing=3, |
165 |
\end{verbatim} |
\end{verbatim} |
166 |
This line sets the azimuthal grid spacing between each x-coordinate line |
This line sets the azimuthal grid spacing between each $x$-coordinate line |
167 |
in the discrete grid. The syntax indicates that the discrete grid |
in the discrete grid. The syntax indicates that the discrete grid |
168 |
should be comprise of $120$ grid lines each separated by $3^{\circ}$. |
should be comprised of $120$ grid lines each separated by $3^{\circ}$. |
169 |
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|
170 |
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|
171 |
\item Line qqq, |
\item Line 58, |
172 |
\begin{verbatim} |
\begin{verbatim} |
173 |
dYspacing=0.01, |
dYspacing=0.01, |
174 |
\end{verbatim} |
\end{verbatim} |
|
This line sets the radial grid spacing between each $\rho$-coordinate line |
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in the discrete grid to $1cm$. |
|
175 |
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|
176 |
\item Line 43, |
This line sets the radial cylindrical grid spacing between each |
177 |
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$a$-coordinate line in the discrete grid to $1cm$. |
178 |
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|
179 |
|
\item Line 59, |
180 |
\begin{verbatim} |
\begin{verbatim} |
181 |
delZ=29*0.005, |
delZ=29*0.005, |
182 |
\end{verbatim} |
\end{verbatim} |
|
This line sets the vertical grid spacing between each z-coordinate line |
|
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in the discrete grid to $5000m$ ($5$~km). |
|
183 |
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|
184 |
\item Line 46, |
This line sets the vertical grid spacing between each of 29 |
185 |
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z-coordinate lines in the discrete grid to $0.005m$ ($5$~mm). |
186 |
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|
187 |
|
\item Line 64, |
188 |
\begin{verbatim} |
\begin{verbatim} |
189 |
bathyFile='bathyPol.bin', |
bathyFile='bathyPol.bin', |
190 |
\end{verbatim} |
\end{verbatim} |
191 |
This line specifies the name of the file from which the domain |
This line specifies the name of the file from which the domain |
192 |
``bathymetry'' (tank depth) is read. This file is a two-dimensional |
``bathymetry'' (tank depth) is read. This file is a two-dimensional |
193 |
($x,y$) map of |
($a,\phi$) map of |
194 |
depths. This file is assumed to contain 64-bit binary numbers |
depths. This file is assumed to contain 64-bit binary numbers |
195 |
giving the depth of the model at each grid cell, ordered with the $x$ |
giving the depth of the model at each grid cell, ordered with the $\phi$ |
196 |
coordinate varying fastest. The points are ordered from low coordinate |
coordinate varying fastest. The points are ordered from low coordinate |
197 |
to high coordinate for both axes. The units and orientation of the |
to high coordinate for both axes. The units and orientation of the |
198 |
depths in this file are the same as used in the MITgcm code. In this |
depths in this file are the same as used in the MITgcm code. In this |
200 |
and a depth |
and a depth |
201 |
f $-0.145m$ indicates the tank itself. |
f $-0.145m$ indicates the tank itself. |
202 |
|
|
203 |
\item Line 49, |
\item Line 65, |
204 |
\begin{verbatim} |
\begin{verbatim} |
205 |
hydrogThetaFile='thetaPol.bin', |
hydrogThetaFile='thetaPol.bin', |
206 |
\end{verbatim} |
\end{verbatim} |
207 |
This line specifies the name of the file from which the initial values |
This line specifies the name of the file from which the initial values |
208 |
of $\theta$ |
of temperature |
209 |
are read. This file is a three-dimensional |
are read. This file is a three-dimensional |
210 |
($x,y,z$) map and is enumerated and formatted in the same manner as the |
($x,y,z$) map and is enumerated and formatted in the same manner as the |
211 |
bathymetry file. |
bathymetry file. |
212 |
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|
213 |
\item Line qqq |
\item Lines 66 and 67 |
214 |
\begin{verbatim} |
\begin{verbatim} |
215 |
tCyl = 0 |
tCylIn = 0 |
216 |
|
tCylOut = 20 |
217 |
\end{verbatim} |
\end{verbatim} |
218 |
This line specifies the temperature in degrees Celsius of the interior |
These line specify the temperatures in degrees Celsius of the interior |
219 |
wall of the tank -- usually a bucket of ice water. |
and exterior walls of the tank -- typically taken to be icewater on |
220 |
|
the inside and room temperature on the outside. |
221 |
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222 |
|
|
223 |
\end{itemize} |
\end{itemize} |
224 |
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|
225 |
\noindent other lines in the file {\it input/data} are standard values |
\noindent Other lines in the file {\it input/data} are standard values |
226 |
that are described in the MITgcm Getting Started and MITgcm Parameters |
that are described in the MITgcm Getting Started and MITgcm Parameters |
227 |
notes. |
notes. |
228 |
|
|
246 |
\label{www:tutorials} |
\label{www:tutorials} |
247 |
|
|
248 |
The {\it input/thetaPol.bin} file specifies a three-dimensional ($x,y,z$) |
The {\it input/thetaPol.bin} file specifies a three-dimensional ($x,y,z$) |
249 |
map of initial values of $\theta$ in degrees Celsius. |
map of initial values of $\theta$ in degrees Celsius. This particular |
250 |
|
experiment is set to random values x around 20C to provide initial |
251 |
|
perturbations. |
252 |
|
|
253 |
\subsubsection{File {\it input/bathyPol.bin}} |
\subsubsection{File {\it input/bathyPol.bin}} |
254 |
\label{www:tutorials} |
\label{www:tutorials} |