s_examples/plume_on_slope/plume_on_slope.tex

\section{Gravity Plume On a Continental Slope}
%\label{www:tutorials}
\label{sec:eg-gravityplume}
\begin{rawhtml}
<!-- CMIREDIR:eg-gravityplume: -->
\end{rawhtml}
\begin{center}
(in directory: {\it verification/tutorial\_plume\_on\_slope/})
\end{center}

\begin{rawhtml}MITGCM_INSERT_FIGURE_BEGIN T-plume-on-slope\end{rawhtml}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth,height=.3\textheight]{s_examples/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 files for this experiment can be found in
the verification directory under tutorial\_plume\_on\_slope.

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{sec:plume-generating}. Other model parameters are
specified in file {\tt data} and {\tt data.obcs} and detailed in
section~\ref{sec:plume-params}.

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

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth,height=.3\textheight]{s_examples/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]{s_examples/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]{s_examples/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
$0-0.01$~$^{\circ}\mathrm{K}$.

\subsection{Code configuration}
%\label{www:tutorials}
\label{sec: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 by adding line {\bf obcs} to 
package configuration file
{\tt code/packages.conf} and activated via {\bf useOBCS=.TRUE,} in
namelist {\em PACKAGES} of {\tt input/data.pkg}.
\end{itemize}

\subsection{Model parameters}
%\label{www:tutorials}
\label{sec: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	\section{Gravity Plume On a Continental Slope}
2	%\label{www:tutorials}
3	\label{sec:eg-gravityplume}
4	\begin{rawhtml}
5	<!-- CMIREDIR:eg-gravityplume: -->
6	\end{rawhtml}
7	\begin{center}
8	(in directory: {\it verification/tutorial\_plume\_on\_slope/})
9	\end{center}
10
11	\begin{rawhtml}MITGCM_INSERT_FIGURE_BEGIN T-plume-on-slope\end{rawhtml}
12	\begin{figure}
13	\begin{center}
14	\includegraphics[width=\textwidth,height=.3\textheight]{s_examples/plume_on_slope/billows.eps}
15	\end{center}
16	\caption{Temperature after 23~hours of cooling. The cold dense water is
17	mixed with ambient water as it accelerates down the slope and hence
18	is warmed than the unmixed plume.
19	}
20	\label{fig:T-plume-on-slope}
21	\end{figure}
22	\begin{rawhtml}MITGCM_INSERT_FIGURE_END\end{rawhtml}
23
24	An important test of any ocean model is the ability to represent the
25	flow of dense fluid down a slope. One example of such a flow is a
26	non-rotating gravity plume on a continental slope, forced by a limited
27	area of surface cooling above a continental shelf. Because the flow is
28	non-rotating, a two dimensional model can be used in the across slope
29	direction. The experiment is non-hydrostatic and uses open-boundaries
30	to radiate transients at the deep water end. (Dense flow down a slope
31	can also be forced by a dense inflow prescribed on the continental
32	shelf; this configuration is being implemented by the DOME (Dynamics
33	of Overflow Mixing and Entrainment) collaboration to compare solutions
34	in different models). The files for this experiment can be found in
35	the verification directory under tutorial\_plume\_on\_slope.
36
37	The fluid is initially unstratified. The surface buoyancy loss $B_0$
38	(dimensions of L$^2$T$^{-3}$) over a cross-shelf distance $R$ causes
39	vertical convective mixing and modifies the density of the fluid by an
40	amount
41	\begin{equation}
42	\Delta \rho = \frac{B_0 \rho_0 t}{g H}
43	\end{equation}
44	where $H$ is the depth of the shelf, $g$ is the acceleration due to
45	gravity, $t$ is time since onset of cooling and $\rho_0$ is the
46	reference density. Dense fluid slumps under gravity, with a flow speed
47	close to the gravity wave speed:
48	\begin{equation}
49	U
50	\sim \sqrt{g' H}
51	\sim \sqrt{ \frac{g \Delta \rho H}{\rho_0} }
52	\sim \sqrt{B_0 t}
53	\end{equation}
54	A steady state is rapidly established in which the buoyancy flux out of
55	the cooling region is balanced by the surface buoyancy loss.
56	Then
57	\begin{equation}
58	U \sim (B_0 R)^{1/3} \mbox{ ; } \Delta \rho \sim \frac{\rho_0}{g H} (B_0 R)^{2/3}
59	\end{equation}
60	The Froude number of the flow on the shelf is close to unity (but in
61	practice slightly less than unity, giving subcritical flow).
62	When the flow reaches the slope, it accelerates, so that it may become
63	supercritical (provided the slope angle $ \alpha $ is steep enough).
64	In this case, a hydraulic control is established at
65	the shelf break. On the slope, where the Froude number is greater
66	than one, and gradient Richardson number
67	(defined as $Ri \sim g' h^/U^2$ where $h^$ is the thickness of the
68	interface between dense and ambient fluid) is reduced
69	below 1/4, Kelvin-Helmholtz instability is possible, and leads to
70	entrainment of ambient fluid into the plume, modifying the
71	density, and hence the acceleration down the slope.
72	Kelvin-Helmholtz instability is suppressed at low Reynolds and
73	Peclet numbers given by
74	\begin{equation}
75	Re \sim \frac{U h}{ \nu} \sim \frac{(B_0 R)^{1/3} h}{\nu} \mbox{ ; } Pe = Re Pr
76	\end{equation}
77	where $h$ is the depth of the dense fluid on the slope.
78	Hence this experiment is carried out in the high Re, Pe regime.
79	A further constraint is that the convective heat flux must be much greater
80	than the diffusive heat flux (Nusselt number $>> 1$).
81	Then
82	\begin{equation}
83	Nu = \frac{U h^* }{\kappa} >> 1
84	\end{equation}
85	Finally, since we have assumed that the convective mixing on the shelf
86	occurs in a much shorter time than the horizontal equilibration,
87	this implies $H/R << 1$.
88
89	Hence to summarize the important nondimensional parameters, and
90	the limits we are considering:
91	\begin{equation}
92	\frac{H}{R} << 1 \mbox{ ; } Re >> 1 \mbox{ ; } Pe >> 1 \mbox{ ; } Nu >> 1
93	\mbox{ ; } \mbox{ ; } Ri < 1/4
94	\end{equation}
95	In addition we are assuming that the slope is steep enough to provide
96	sufficient acceleration to the gravity plume, but nonetheless much less
97	that $1:1$, since many Kelvin-Helmholtz billows appear on the slope,
98	implying horizontal lengthscale of the slope $>>$ the depth of the
99	dense fluid.
100
101	\subsection{Configuration}
102	%\label{www:tutorials}
103
104	The topography, spatial grid, forcing and initial conditions are all
105	specified in binary data files generated using a {\em Matlab} script
106	called {\tt gendata.m} and detailed in
107	section~\ref{sec:plume-generating}. Other model parameters are
108	specified in file {\tt data} and {\tt data.obcs} and detailed in
109	section~\ref{sec:plume-params}.
110
111	\subsection{Binary input data}
112	%\label{www:tutorials}
113	\label{sec:plume-generating}
114
115	\begin{figure}
116	\begin{center}
117	\includegraphics[width=\textwidth,height=.3\textheight]{s_examples/plume_on_slope/dx.eps}
118	\end{center}
119	\caption{Horizontal grid spacing, $\Delta x$, in the across-slope
120	direction for the gravity plume experiment.}
121	\label{fig:dx-plume-on-slope}
122	\end{figure}
123
124	\begin{figure}
125	\begin{center}
126	\includegraphics[width=\textwidth,height=.3\textheight]{s_examples/plume_on_slope/Depth.eps}
127	\end{center}
128	\caption{Topography, $h(x)$, used for the gravity plume experiment.}
129	\label{fig:depth-plume-on-slope}
130	\end{figure}
131
132	\begin{figure}
133	\begin{center}
134	\includegraphics[width=\textwidth,height=.3\textheight]{s_examples/plume_on_slope/Qsurf.eps}
135	\end{center}
136	\caption{Upward surface heat flux, $Q(x)$, used as forcing in the
137	gravity plume experiment.}
138	\label{fig:Q-plume-on-slope}
139	\end{figure}
140
141	The domain is $200$~m deep and $6.4$~km across. Uniform resolution of
142	$60\times3^1/_3$~m is used in the vertical and variable resolution of
143	the form shown in Fig.~\ref{fig:dx-plume-on-slope} with $320$ points
144	is usedin the horizontal. The formula for $\Delta x$ is:
145	\begin{displaymath}
146	\Delta x(i) = \Delta x_1 + ( \Delta x_2 - \Delta x_1 )
147	( 1 + \tanh{\left(\frac{i-i_s}{w}\right)} ) /2
148	\end{displaymath}
149	where
150	\begin{eqnarray*}
151	Nx & = & 320 \\
152	Lx & = & 6400 \;\; \mbox{(m)} \\
153	\Delta x_1 & = & \frac{2}{3} \frac{Lx}{Nx} \;\; \mbox{(m)} \\
154	\Delta x_2 & = & \frac{Lx/2}{Nx-Lx/2 \Delta x_1} \;\; \mbox{(m)} \\
155	i_s & = & Lx/( 2 \Delta x_1 ) \\
156	w & = & 40
157	\end{eqnarray*}
158	Here, $\Delta x_1$ is the resolution on the shelf, $\Delta x_2$ is the
159	resolution in deep water and $Nx$ is the number of points in the
160	horizontal.
161
162	The topography, shown in Fig.~\ref{fig:depth-plume-on-slope}, is given
163	by:
164	\begin{displaymath}
165	H(x) = -H_o + (H_o - h_s) ( 1 + \tanh{\left(\frac{x-x_s}{L_s}\right)} ) / 2
166	\end{displaymath}
167	where
168	\begin{eqnarray*}
169	H_o & = & 200 \;\; \mbox{(m)} \\
170	h_s & = & 40 \;\; \mbox{(m)} \\
171	x_s & = & 1500 + Lx/2 \;\; \mbox{(m)} \\
172	L_s & = & \frac{(H_o - h_s)}{2 s} \;\; \mbox{(m)} \\
173	s & = & 0.15
174	\end{eqnarray*}
175	Here, $s$ is the maximum slope, $H_o$ is the maximum depth, $h_s$ is
176	the shelf depth, $x_s$ is the lateral position of the shelf-break and
177	$L_s$ is the length-scale of the slope.
178
179	The forcing is through heat loss over the shelf, shown in
180	Fig.~\ref{fig:Q-plume-on-slope} and takes the form of a fixed flux
181	with profile:
182	\begin{displaymath}
183	Q(x) = Q_o ( 1 + \tanh{\left(\frac{x - x_q}{L_q}\right)} ) / 2
184	\end{displaymath}
185	where
186	\begin{eqnarray*}
187	Q_o & = & 200 \;\; \mbox{(W m$^{-2}$)} \\
188	x_q & = & 2500 + Lx/2 \;\; \mbox{(m)} \\
189	L_q & = & 100 \;\; \mbox{(m)}
190	\end{eqnarray*}
191	Here, $Q_o$, is the maximum heat flux, $x_q$ is the position of the
192	cut-off and $L_q$ is the width of the cut-off.
193
194	The initial tempeture field is unstratified but with random
195	perturbations, to induce convection early on in the run. The random
196	perturbation are calculated in computational space and because of the
197	variable resolution introduce some spatial correlations but this does
198	not matter for this experiment. The perturbations have range
199	$0-0.01$~$^{\circ}\mathrm{K}$.
200
201	\subsection{Code configuration}
202	%\label{www:tutorials}
203	\label{sec:plume-config}
204
205	The computational domain (number of points) is specified in {\tt
206	code/SIZE.h} and is configured as a single tile of dimensions
207	$320\times1\times60$. There are no experiment specific source files.
208
209	Optional code required to for this experiment are the non-hydrostatic
210	algorithm and open-boundaries:
211	\begin{itemize}
212	\item Non-hydrostatic terms and algorithm are enabled with {\bf
213	\#define ALLOW\_NONHYDROSTATIC} in {\tt code/CPP\_OPTIONS.h} and
214	activated with {\bf nonHydrostatic=.TRUE.,} in namelist {\em PARM01}
215	of {\tt input/data}.
216	\item Open boundaries are enabled by adding line {\bf obcs} to
217	package configuration file
218	{\tt code/packages.conf} and activated via {\bf useOBCS=.TRUE,} in
219	namelist {\em PACKAGES} of {\tt input/data.pkg}.
220	\end{itemize}
221
222	\subsection{Model parameters}
223	%\label{www:tutorials}
224	\label{sec:plume-params}
225
226	\begin{table}
227	\begin{center}
228	\begin{tabular}{lll}
229	$g$ & $9.81$ m s$^{-2}$ & acceleration due to gravity \\
230	$\rho_o$ & $999.8$ kg m$^{-3}$ & reference density \\
231	$\alpha$ & $2 \times 10^{-4}$ K$^{-1}$ & expansion coefficient \\
232	$A_h$ & $1 \times 10^{-2}$ m$^2$s$^{-1}$ & horizontal viscosity \\
233	$A_v$ & $1 \times 10^{-3}$ m$^2$s$^{-1}$ & vertical viscosity \\
234	$\kappa_h$ & $0$ m$^2$s$^{-1}$ & (explicit) horizontal diffusion \\
235	$\kappa_v$ & $0$ m$^2$s$^{-1}$ & (explicit) vertical diffusion \\
236	\\
237	$\Delta t$ & $20$ s & time step \\
238	$\Delta z$ & $3.3\dot{3}$ m & vertical grid spacing \\
239	$\Delta x$ & $13.\dot{3}-39.5$ m & horizontal grid spacing
240	\end{tabular}
241	\end{center}
242	\caption{Model parameters used in the gravity plume experiment.}
243	\label{table:plume-on-slope}
244	\end{table}
245
246	The model parameters (Table~\ref{table:plume-on-slope}) are specified
247	in {\tt input/data} and if not assume the default values defined in
248	{\tt model/src/set\_defaults.F}. A linear equation of state is used,
249	{\bf eosType='LINEAR'}, but only temperature is active, {\bf
250	sBeta=0.E-4}. For the given heat flux, $Q_o$, the buoyancy forcing is
251	$B_o = \frac{g \alpha Q}{\rho_o c_p} \sim
252	10^{-7}$~m$^2$s$^{-3}$. Using $R=10^3$~m, the shelf width, then this
253	gives a velocity scale of $U\sim 5 \times 10^{-2}$~m~s$^-1$ for the
254	initial front but will accelerate by an order of magnitude over the
255	slope. The temperature anomaly will be of order $\Delta \theta \sim 3
256	\times 10^{-2}$~K. The viscosity is constant and gives a Reynolds
257	number of $100$, using $h=20$~m for the initial front and will be an
258	order magnitude bigger over the slope. There is no explicit diffusion
259	but a non-linear advection scheme is used for temperature which adds
260	enough diffusion so as to keep the model stable. The time-step is set
261	to $20$~s and gives Courant number order one when the flow reaches the
262	bottom of the slope.
263
264	\subsection{Build and run the model}
265	%\label{www:tutorials}
266
267	Build the model per usual. For example:
268	\begin{verbatim}
269	% cd verification/plume_on_slope
270	% mkdir build
271	% cd build
272	% ../../../tools/genmake -mods=../code -disable=gmredi,kpp,zonal_filt
273	,shap_filt
274	% make depend
275	% make
276	\end{verbatim}
277
278	When compilation is complete, run the model as usual, for example:
279	\begin{verbatim}
280	% cd ../
281	% mkdir run
282	% cp input/* build/mitgcmuv run/
283	% cd run
284	% ./mitgcmuv > output.txt
285	\end{verbatim}