s_examples/plume_on_slope/plume_on_slope.tex

\section{Gravity Plume On a Continental Slope}
\label{www:tutorials}
\label{sect:eg-gravityplume}
\begin{rawhtml}
<!-- CMIREDIR:eg-gravityplume: -->
\end{rawhtml}

\begin{rawhtml}MITGCM_INSERT_FIGURE_BEGIN T-plume-on-slope\end{rawhtml}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/billows.eps}
\end{center}
\caption{Temperature after 23~hours of cooling. The cold dense water is
mixed with ambient water as it accelerates down the slope and hence
is warmed than the unmixed plume.
}
\label{fig:T-plume-on-slope}
\end{figure}
\begin{rawhtml}MITGCM_INSERT_FIGURE_END\end{rawhtml}

An important test of any ocean model is the ability to represent the
flow of dense fluid down a slope. One example of such a flow is a
non-rotating gravity plume on a continental slope, forced by a limited
area of surface cooling above a continental shelf. Because the flow is
non-rotating, a two dimensional model can be used in the across slope
direction. The experiment is non-hydrostatic and uses open-boundaries
to radiate transients at the deep water end.  (Dense flow down a slope
can also be forced by a dense inflow prescribed on the continental
shelf; this configuration is being implemented by the DOME (Dynamics
of Overflow Mixing and Entrainment) collaboration to compare solutions
in different models).

The fluid is initially unstratified.  The surface buoyancy loss $B_0$
(dimensions of L$^2$T$^{-3}$) over a cross-shelf distance $R$ causes
vertical convective mixing and modifies the density of the fluid by an
amount
\begin{equation}
\Delta \rho = \frac{B_0 \rho_0 t}{g H}
\end{equation}
where $H$ is the depth of the shelf, $g$ is the acceleration due to
gravity, $t$ is time since onset of cooling and $\rho_0$ is the
reference density. Dense fluid slumps under gravity, with a flow speed
close to the gravity wave speed:
\begin{equation}
U
\sim \sqrt{g' H}
\sim \sqrt{ \frac{g \Delta \rho H}{\rho_0} }
\sim \sqrt{B_0 t}
\end{equation}
A steady state is rapidly established in which the buoyancy flux out of 
the cooling region is balanced by the surface buoyancy loss. 
Then 
\begin{equation}
U \sim (B_0 R)^{1/3} \mbox{  ;  } \Delta \rho \sim \frac{\rho_0}{g H} (B_0 R)^{2/3} 
\end{equation}
The Froude number of the flow on the shelf is close to unity (but in 
practice slightly less than unity, giving subcritical flow). 
When the flow reaches the slope, it accelerates, so that it may become 
supercritical (provided the slope angle $ \alpha $ is steep enough). 
In this case, a hydraulic control is established at 
the shelf break. On the slope, where the Froude number is greater 
than one, and gradient Richardson number 
(defined as $Ri \sim g' h^*/U^2$ where $h^*$ is the thickness of the 
interface between dense and ambient fluid) is reduced 
below 1/4, Kelvin-Helmholtz instability is possible, and leads to 
entrainment of ambient fluid into the plume, modifying the 
density, and hence the acceleration down the slope. 
Kelvin-Helmholtz instability is suppressed at low Reynolds and 
Peclet numbers given by 
\begin{equation}
Re \sim \frac{U h}{ \nu} \sim \frac{(B_0 R)^{1/3} h}{\nu} \mbox{  ;  } Pe = Re Pr
\end{equation}
where $h$ is the depth of the dense fluid on the slope. 
Hence this experiment is carried out in the high Re, Pe regime. 
A further constraint is that the convective heat flux must be much greater 
than the diffusive heat flux (Nusselt number $>> 1$). 
Then 
\begin{equation}
Nu = \frac{U h^* }{\kappa} >> 1
\end{equation}
Finally, since we have assumed that the convective mixing on the shelf
occurs in a much shorter time than the horizontal equilibration, 
this implies $H/R << 1$.  

Hence to summarize the important nondimensional parameters, and 
the limits we are considering:
\begin{equation}
\frac{H}{R} << 1 \mbox{ ; } Re >> 1 \mbox{  ; } Pe >> 1 \mbox{  ; } Nu >> 1
\mbox{  ;  } \mbox{  ; } Ri < 1/4 
\end{equation}
In addition we are assuming that the slope is steep enough to provide 
sufficient acceleration  to the gravity plume, but nonetheless much less 
that $1:1$, since many Kelvin-Helmholtz billows appear on the slope, 
implying horizontal lengthscale of the slope $>>$ the depth of the 
dense fluid. 

\subsection{Configuration}
\label{www:tutorials}

The topography, spatial grid, forcing and initial conditions are all
specified in binary data files generated using a {\em Matlab} script
called {\tt gendata.m} and detailed in
section~\ref{sect:plume-generating}. Other model parameters are
specified in file {\tt data} and {\tt data.obcs} and detailed in
section~\ref{sect:plume-params}.

\subsection{Binary input data}
\label{www:tutorials}
\label{sect:plume-generating}

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/dx.eps}
\end{center}
\caption{Horizontal grid spacing, $\Delta x$, in the across-slope
direction for the gravity plume experiment.}
\label{fig:dx-plume-on-slope}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/Depth.eps}
\end{center}
\caption{Topography, $h(x)$, used for the gravity plume experiment.}
\label{fig:depth-plume-on-slope}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/Qsurf.eps}
\end{center}
\caption{Upward surface heat flux, $Q(x)$, used as forcing in the
gravity plume experiment.}
\label{fig:Q-plume-on-slope}
\end{figure}

The domain is $200$~m deep and $6.4$~km across. Uniform resolution of
$60\times3^1/_3$~m is used in the vertical and variable resolution of
the form shown in Fig.~\ref{fig:dx-plume-on-slope} with $320$ points
is usedin the horizontal. The formula for $\Delta x$ is:
\begin{displaymath}
\Delta x(i) = \Delta x_1 + ( \Delta x_2 - \Delta x_1 )
( 1 + \tanh{\left(\frac{i-i_s}{w}\right)} ) /2
\end{displaymath}
where
\begin{eqnarray*}
Nx & = & 320 \\
Lx & = & 6400 \;\; \mbox{(m)} \\
\Delta x_1 & = & \frac{2}{3} \frac{Lx}{Nx} \;\; \mbox{(m)} \\
\Delta x_2 & = & \frac{Lx/2}{Nx-Lx/2 \Delta x_1} \;\; \mbox{(m)} \\
i_s & = & Lx/( 2 \Delta x_1 ) \\
w & = & 40 
\end{eqnarray*}
Here, $\Delta x_1$ is the resolution on the shelf, $\Delta x_2$ is the
resolution in deep water and $Nx$ is the number of points in the
horizontal.

The topography, shown in Fig.~\ref{fig:depth-plume-on-slope}, is given
by:
\begin{displaymath}
H(x) = -H_o + (H_o - h_s) ( 1 + \tanh{\left(\frac{x-x_s}{L_s}\right)} ) / 2
\end{displaymath}
where
\begin{eqnarray*}
H_o & = & 200 \;\; \mbox{(m)} \\
h_s & = & 40 \;\; \mbox{(m)} \\
x_s & = & 1500 + Lx/2 \;\; \mbox{(m)} \\
L_s & = & \frac{(H_o - h_s)}{2 s} \;\; \mbox{(m)} \\
s & = & 0.15
\end{eqnarray*}
Here, $s$ is the maximum slope, $H_o$ is the maximum depth, $h_s$ is
the shelf depth, $x_s$ is the lateral position of the shelf-break and
$L_s$ is the length-scale of the slope.

The forcing is through heat loss over the shelf, shown in
Fig.~\ref{fig:Q-plume-on-slope} and takes the form of a fixed flux
with profile:
\begin{displaymath}
Q(x) = Q_o ( 1 + \tanh{\left(\frac{x - x_q}{L_q}\right)} ) / 2
\end{displaymath}
where
\begin{eqnarray*}
Q_o & = & 200 \;\; \mbox{(W m$^{-2}$)} \\
x_q & = & 2500 + Lx/2 \;\; \mbox{(m)} \\
L_q & = & 100 \;\; \mbox{(m)}
\end{eqnarray*}
Here, $Q_o$, is the maximum heat flux, $x_q$ is the position of the
cut-off and $L_q$ is the width of the cut-off.

The initial tempeture field is unstratified but with random
perturbations, to induce convection early on in the run. The random
perturbation are calculated in computational space and because of the
variable resolution introduce some spatial correlations but this does
not matter for this experiment. The perturbations have range
$0-0.01$~$^\circ$K.

\subsection{Code configuration}
\label{www:tutorials}
\label{sect:plume-config}

The computational domain (number of points) is specified in {\tt
code/SIZE.h} and is configured as a single tile of dimensions
$320\times1\times60$. There are no experiment specific source files.

Optional code required to for this experiment are the non-hydrostatic
algorithm and open-boundaries:
\begin{itemize}
\item Non-hydrostatic terms and algorithm are enabled with {\bf
\#define ALLOW\_NONHYDROSTATIC} in {\tt code/CPP\_OPTIONS.h} and
activated with {\bf nonHydrostatic=.TRUE.,} in namelist {\em PARM01}
of {\tt input/data}.
\item Open boundaries are enabled with {\bf \#define ALLOW\_OBCS} in
{\tt code/CPP\_OPTIONS.h} and activated with {\bf use\_OBCS=.TRUE,} in
namelist {\em PACKAGES} of {\tt input/data.pkg}.
\end{itemize}

\subsection{Model parameters}
\label{www:tutorials}
\label{sect:plume-params}

\begin{table}
\begin{center}
\begin{tabular}{lll}
$g$ & $9.81$ m s$^{-2}$ & acceleration due to gravity \\
$\rho_o$ & $999.8$ kg m$^{-3}$ & reference density \\
$\alpha$ & $2 \times 10^{-4}$ K$^{-1}$ & expansion coefficient \\
$A_h$ & $1 \times 10^{-2}$ m$^2$s$^{-1}$ & horizontal viscosity \\
$A_v$ & $1 \times 10^{-3}$ m$^2$s$^{-1}$ & vertical viscosity \\
$\kappa_h$ & $0$ m$^2$s$^{-1}$ & (explicit) horizontal diffusion \\
$\kappa_v$ & $0$ m$^2$s$^{-1}$ & (explicit) vertical diffusion \\
\\
$\Delta t$ & $20$ s & time step \\
$\Delta z$ & $3.3\dot{3}$ m & vertical grid spacing \\
$\Delta x$ & $13.\dot{3}-39.5$ m & horizontal grid spacing
\end{tabular}
\end{center}
\caption{Model parameters used in the gravity plume experiment.}
\label{table:plume-on-slope}
\end{table}

The model parameters (Table~\ref{table:plume-on-slope}) are specified
in {\tt input/data} and if not assume the default values defined in
{\tt model/src/set\_defaults.F}. A linear equation of state is used,
{\bf eosType='LINEAR'}, but only temperature is active, {\bf
sBeta=0.E-4}. For the given heat flux, $Q_o$, the buoyancy forcing is
$B_o = \frac{g \alpha Q}{\rho_o c_p} \sim
10^{-7}$~m$^2$s$^{-3}$. Using $R=10^3$~m, the shelf width, then this
gives a velocity scale of $U\sim 5 \times 10^{-2}$~m~s$^-1$ for the
initial front but will accelerate by an order of magnitude over the
slope. The temperature anomaly will be of order $\Delta \theta \sim 3
\times 10^{-2}$~K.  The viscosity is constant and gives a Reynolds
number of $100$, using $h=20$~m for the initial front and will be an
order magnitude bigger over the slope. There is no explicit diffusion
but a non-linear advection scheme is used for temperature which adds
enough diffusion so as to keep the model stable. The time-step is set
to $20$~s and gives Courant number order one when the flow reaches the
bottom of the slope.

\subsection{Build and run the model}
\label{www:tutorials}

Build the model per usual. For example:
\begin{verbatim}
% cd verification/plume_on_slope
% mkdir build
% cd build
% ../../../tools/genmake -mods=../code -disable=gmredi,kpp,zonal_filt
  ,shap_filt
% make depend
% make
\end{verbatim}

When compilation is complete, run the model as usual, for example:
\begin{verbatim}
% cd ../
% mkdir run
% cp input/* build/mitgcmuv run/
% cd run
% ./mitgcmuv > output.txt
\end{verbatim}
1	cnh	1.4	\section{Gravity Plume On a Continental Slope}
2	adcroft	1.5	\label{www:tutorials}
3	cnh	1.4	\label{sect:eg-gravityplume}
4	edhill	1.6	\begin{rawhtml}
5			<!-- CMIREDIR:eg-gravityplume: -->
6			\end{rawhtml}
7	adcroft	1.2
8	cnh	1.4	\begin{rawhtml}MITGCM_INSERT_FIGURE_BEGIN T-plume-on-slope\end{rawhtml}
9	adcroft	1.1	\begin{figure}
10			\begin{center}
11			\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/billows.eps}
12			\end{center}
13			\caption{Temperature after 23~hours of cooling. The cold dense water is
14			mixed with ambient water as it accelerates down the slope and hence
15			is warmed than the unmixed plume.
16			}
17			\label{fig:T-plume-on-slope}
18			\end{figure}
19	cnh	1.4	\begin{rawhtml}MITGCM_INSERT_FIGURE_END\end{rawhtml}
20	adcroft	1.1
21			An important test of any ocean model is the ability to represent the
22			flow of dense fluid down a slope. One example of such a flow is a
23			non-rotating gravity plume on a continental slope, forced by a limited
24			area of surface cooling above a continental shelf. Because the flow is
25			non-rotating, a two dimensional model can be used in the across slope
26			direction. The experiment is non-hydrostatic and uses open-boundaries
27			to radiate transients at the deep water end. (Dense flow down a slope
28			can also be forced by a dense inflow prescribed on the continental
29			shelf; this configuration is being implemented by the DOME (Dynamics
30			of Overflow Mixing and Entrainment) collaboration to compare solutions
31			in different models).
32
33			The fluid is initially unstratified. The surface buoyancy loss $B_0$
34			(dimensions of L$^2$T$^{-3}$) over a cross-shelf distance $R$ causes
35			vertical convective mixing and modifies the density of the fluid by an
36			amount
37			\begin{equation}
38			\Delta \rho = \frac{B_0 \rho_0 t}{g H}
39			\end{equation}
40			where $H$ is the depth of the shelf, $g$ is the acceleration due to
41			gravity, $t$ is time since onset of cooling and $\rho_0$ is the
42			reference density. Dense fluid slumps under gravity, with a flow speed
43			close to the gravity wave speed:
44			\begin{equation}
45			U
46			\sim \sqrt{g' H}
47			\sim \sqrt{ \frac{g \Delta \rho H}{\rho_0} }
48			\sim \sqrt{B_0 t}
49			\end{equation}
50			A steady state is rapidly established in which the buoyancy flux out of
51			the cooling region is balanced by the surface buoyancy loss.
52			Then
53			\begin{equation}
54			U \sim (B_0 R)^{1/3} \mbox{ ; } \Delta \rho \sim \frac{\rho_0}{g H} (B_0 R)^{2/3}
55			\end{equation}
56			The Froude number of the flow on the shelf is close to unity (but in
57			practice slightly less than unity, giving subcritical flow).
58			When the flow reaches the slope, it accelerates, so that it may become
59			supercritical (provided the slope angle $ \alpha $ is steep enough).
60			In this case, a hydraulic control is established at
61			the shelf break. On the slope, where the Froude number is greater
62			than one, and gradient Richardson number
63			(defined as $Ri \sim g' h^/U^2$ where $h^$ is the thickness of the
64			interface between dense and ambient fluid) is reduced
65			below 1/4, Kelvin-Helmholtz instability is possible, and leads to
66			entrainment of ambient fluid into the plume, modifying the
67			density, and hence the acceleration down the slope.
68			Kelvin-Helmholtz instability is suppressed at low Reynolds and
69			Peclet numbers given by
70			\begin{equation}
71			Re \sim \frac{U h}{ \nu} \sim \frac{(B_0 R)^{1/3} h}{\nu} \mbox{ ; } Pe = Re Pr
72			\end{equation}
73			where $h$ is the depth of the dense fluid on the slope.
74			Hence this experiment is carried out in the high Re, Pe regime.
75			A further constraint is that the convective heat flux must be much greater
76			than the diffusive heat flux (Nusselt number $>> 1$).
77			Then
78			\begin{equation}
79			Nu = \frac{U h^* }{\kappa} >> 1
80			\end{equation}
81			Finally, since we have assumed that the convective mixing on the shelf
82			occurs in a much shorter time than the horizontal equilibration,
83			this implies $H/R << 1$.
84
85			Hence to summarize the important nondimensional parameters, and
86			the limits we are considering:
87			\begin{equation}
88			\frac{H}{R} << 1 \mbox{ ; } Re >> 1 \mbox{ ; } Pe >> 1 \mbox{ ; } Nu >> 1
89			\mbox{ ; } \mbox{ ; } Ri < 1/4
90			\end{equation}
91			In addition we are assuming that the slope is steep enough to provide
92			sufficient acceleration to the gravity plume, but nonetheless much less
93			that $1:1$, since many Kelvin-Helmholtz billows appear on the slope,
94			implying horizontal lengthscale of the slope $>>$ the depth of the
95			dense fluid.
96
97			\subsection{Configuration}
98	adcroft	1.5	\label{www:tutorials}
99	adcroft	1.1
100			The topography, spatial grid, forcing and initial conditions are all
101			specified in binary data files generated using a {\em Matlab} script
102			called {\tt gendata.m} and detailed in
103			section~\ref{sect:plume-generating}. Other model parameters are
104			specified in file {\tt data} and {\tt data.obcs} and detailed in
105			section~\ref{sect:plume-params}.
106
107	adcroft	1.3	\subsection{Binary input data}
108	adcroft	1.5	\label{www:tutorials}
109	adcroft	1.1	\label{sect:plume-generating}
110
111	adcroft	1.3	\begin{figure}
112			\begin{center}
113			\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/dx.eps}
114			\end{center}
115			\caption{Horizontal grid spacing, $\Delta x$, in the across-slope
116			direction for the gravity plume experiment.}
117			\label{fig:dx-plume-on-slope}
118			\end{figure}
119
120			\begin{figure}
121			\begin{center}
122			\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/Depth.eps}
123			\end{center}
124			\caption{Topography, $h(x)$, used for the gravity plume experiment.}
125			\label{fig:depth-plume-on-slope}
126			\end{figure}
127
128			\begin{figure}
129			\begin{center}
130			\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/Qsurf.eps}
131			\end{center}
132			\caption{Upward surface heat flux, $Q(x)$, used as forcing in the
133			gravity plume experiment.}
134			\label{fig:Q-plume-on-slope}
135			\end{figure}
136
137	adcroft	1.1	The domain is $200$~m deep and $6.4$~km across. Uniform resolution of
138			$60\times3^1/_3$~m is used in the vertical and variable resolution of
139			the form shown in Fig.~\ref{fig:dx-plume-on-slope} with $320$ points
140			is usedin the horizontal. The formula for $\Delta x$ is:
141			\begin{displaymath}
142			\Delta x(i) = \Delta x_1 + ( \Delta x_2 - \Delta x_1 )
143			( 1 + \tanh{\left(\frac{i-i_s}{w}\right)} ) /2
144			\end{displaymath}
145			where
146			\begin{eqnarray*}
147			Nx & = & 320 \\
148			Lx & = & 6400 \;\; \mbox{(m)} \\
149			\Delta x_1 & = & \frac{2}{3} \frac{Lx}{Nx} \;\; \mbox{(m)} \\
150			\Delta x_2 & = & \frac{Lx/2}{Nx-Lx/2 \Delta x_1} \;\; \mbox{(m)} \\
151			i_s & = & Lx/( 2 \Delta x_1 ) \\
152			w & = & 40
153			\end{eqnarray*}
154			Here, $\Delta x_1$ is the resolution on the shelf, $\Delta x_2$ is the
155			resolution in deep water and $Nx$ is the number of points in the
156			horizontal.
157
158			The topography, shown in Fig.~\ref{fig:depth-plume-on-slope}, is given
159			by:
160			\begin{displaymath}
161			H(x) = -H_o + (H_o - h_s) ( 1 + \tanh{\left(\frac{x-x_s}{L_s}\right)} ) / 2
162			\end{displaymath}
163			where
164			\begin{eqnarray*}
165			H_o & = & 200 \;\; \mbox{(m)} \\
166			h_s & = & 40 \;\; \mbox{(m)} \\
167			x_s & = & 1500 + Lx/2 \;\; \mbox{(m)} \\
168			L_s & = & \frac{(H_o - h_s)}{2 s} \;\; \mbox{(m)} \\
169			s & = & 0.15
170			\end{eqnarray*}
171			Here, $s$ is the maximum slope, $H_o$ is the maximum depth, $h_s$ is
172			the shelf depth, $x_s$ is the lateral position of the shelf-break and
173			$L_s$ is the length-scale of the slope.
174
175			The forcing is through heat loss over the shelf, shown in
176			Fig.~\ref{fig:Q-plume-on-slope} and takes the form of a fixed flux
177			with profile:
178			\begin{displaymath}
179			Q(x) = Q_o ( 1 + \tanh{\left(\frac{x - x_q}{L_q}\right)} ) / 2
180			\end{displaymath}
181			where
182			\begin{eqnarray*}
183			Q_o & = & 200 \;\; \mbox{(W m$^{-2}$)} \\
184			x_q & = & 2500 + Lx/2 \;\; \mbox{(m)} \\
185			L_q & = & 100 \;\; \mbox{(m)}
186			\end{eqnarray*}
187			Here, $Q_o$, is the maximum heat flux, $x_q$ is the position of the
188			cut-off and $L_q$ is the width of the cut-off.
189
190			The initial tempeture field is unstratified but with random
191			perturbations, to induce convection early on in the run. The random
192			perturbation are calculated in computational space and because of the
193			variable resolution introduce some spatial correlations but this does
194			not matter for this experiment. The perturbations have range
195			$0-0.01$~$^\circ$K.
196
197	adcroft	1.3	\subsection{Code configuration}
198	adcroft	1.5	\label{www:tutorials}
199	adcroft	1.1	\label{sect:plume-config}
200
201			The computational domain (number of points) is specified in {\tt
202			code/SIZE.h} and is configured as a single tile of dimensions
203			$320\times1\times60$. There are no experiment specific source files.
204
205			Optional code required to for this experiment are the non-hydrostatic
206			algorithm and open-boundaries:
207			\begin{itemize}
208			\item Non-hydrostatic terms and algorithm are enabled with {\bf
209			\#define ALLOW\_NONHYDROSTATIC} in {\tt code/CPP\_OPTIONS.h} and
210			activated with {\bf nonHydrostatic=.TRUE.,} in namelist {\em PARM01}
211			of {\tt input/data}.
212			\item Open boundaries are enabled with {\bf \#define ALLOW\_OBCS} in
213			{\tt code/CPP\_OPTIONS.h} and activated with {\bf use\_OBCS=.TRUE,} in
214			namelist {\em PACKAGES} of {\tt input/data.pkg}.
215			\end{itemize}
216
217	adcroft	1.3	\subsection{Model parameters}
218	adcroft	1.5	\label{www:tutorials}
219	adcroft	1.1	\label{sect:plume-params}
220
221			\begin{table}
222			\begin{center}
223			\begin{tabular}{lll}
224			$g$ & $9.81$ m s$^{-2}$ & acceleration due to gravity \\
225			$\rho_o$ & $999.8$ kg m$^{-3}$ & reference density \\
226			$\alpha$ & $2 \times 10^{-4}$ K$^{-1}$ & expansion coefficient \\
227			$A_h$ & $1 \times 10^{-2}$ m$^2$s$^{-1}$ & horizontal viscosity \\
228			$A_v$ & $1 \times 10^{-3}$ m$^2$s$^{-1}$ & vertical viscosity \\
229			$\kappa_h$ & $0$ m$^2$s$^{-1}$ & (explicit) horizontal diffusion \\
230			$\kappa_v$ & $0$ m$^2$s$^{-1}$ & (explicit) vertical diffusion \\
231			\\
232			$\Delta t$ & $20$ s & time step \\
233			$\Delta z$ & $3.3\dot{3}$ m & vertical grid spacing \\
234			$\Delta x$ & $13.\dot{3}-39.5$ m & horizontal grid spacing
235			\end{tabular}
236			\end{center}
237			\caption{Model parameters used in the gravity plume experiment.}
238			\label{table:plume-on-slope}
239			\end{table}
240
241			The model parameters (Table~\ref{table:plume-on-slope}) are specified
242			in {\tt input/data} and if not assume the default values defined in
243			{\tt model/src/set\_defaults.F}. A linear equation of state is used,
244			{\bf eosType='LINEAR'}, but only temperature is active, {\bf
245			sBeta=0.E-4}. For the given heat flux, $Q_o$, the buoyancy forcing is
246			$B_o = \frac{g \alpha Q}{\rho_o c_p} \sim
247			10^{-7}$~m$^2$s$^{-3}$. Using $R=10^3$~m, the shelf width, then this
248			gives a velocity scale of $U\sim 5 \times 10^{-2}$~m~s$^-1$ for the
249			initial front but will accelerate by an order of magnitude over the
250			slope. The temperature anomaly will be of order $\Delta \theta \sim 3
251			\times 10^{-2}$~K. The viscosity is constant and gives a Reynolds
252			number of $100$, using $h=20$~m for the initial front and will be an
253			order magnitude bigger over the slope. There is no explicit diffusion
254			but a non-linear advection scheme is used for temperature which adds
255			enough diffusion so as to keep the model stable. The time-step is set
256			to $20$~s and gives Courant number order one when the flow reaches the
257			bottom of the slope.
258
259	adcroft	1.3	\subsection{Build and run the model}
260	adcroft	1.5	\label{www:tutorials}
261	adcroft	1.1
262			Build the model per usual. For example:
263			\begin{verbatim}
264			% cd verification/plume_on_slope
265			% mkdir build
266			% cd build
267			% ../../../tools/genmake -mods=../code -disable=gmredi,kpp,zonal_filt
268			,shap_filt
269			% make depend
270			% make
271			\end{verbatim}
272
273			When compilation is complete, run the model as usual, for example:
274			\begin{verbatim}
275			% cd ../
276			% mkdir run
277			% cp input/* build/mitgcmuv run/
278			% cd run
279			% ./mitgcmuv > output.txt
280			\end{verbatim}