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adcroft |
1.2 |
\section{Example: Gravity plume on a continental slope} |
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adcroft |
1.1 |
\begin{figure} |
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\begin{center} |
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\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/dx.eps} |
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\end{center} |
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\caption{Horizontal grid spacing, $\Delta x$, in the across-slope |
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direction for the gravity plume experiment.} |
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\label{fig:dx-plume-on-slope} |
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\end{figure} |
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\begin{figure} |
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\begin{center} |
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\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/Depth.eps} |
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\end{center} |
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\caption{Topography, $h(x)$, used for the gravity plume experiment.} |
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\label{fig:depth-plume-on-slope} |
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\end{figure} |
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\begin{figure} |
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\begin{center} |
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\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/Qsurf.eps} |
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\end{center} |
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\caption{Upward surface heat flux, $Q(x)$, used as forcing in the |
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gravity plume experiment.} |
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\label{fig:Q-plume-on-slope} |
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\end{figure} |
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\begin{figure} |
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\begin{center} |
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\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/plume_on_slope/billows.eps} |
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\end{center} |
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\caption{Temperature after 23~hours of cooling. The cold dense water is |
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mixed with ambient water as it accelerates down the slope and hence |
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is warmed than the unmixed plume. |
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} |
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\label{fig:T-plume-on-slope} |
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\end{figure} |
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An important test of any ocean model is the ability to represent the |
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flow of dense fluid down a slope. One example of such a flow is a |
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non-rotating gravity plume on a continental slope, forced by a limited |
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area of surface cooling above a continental shelf. Because the flow is |
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non-rotating, a two dimensional model can be used in the across slope |
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direction. The experiment is non-hydrostatic and uses open-boundaries |
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to radiate transients at the deep water end. (Dense flow down a slope |
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can also be forced by a dense inflow prescribed on the continental |
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shelf; this configuration is being implemented by the DOME (Dynamics |
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of Overflow Mixing and Entrainment) collaboration to compare solutions |
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in different models). |
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The fluid is initially unstratified. The surface buoyancy loss $B_0$ |
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(dimensions of L$^2$T$^{-3}$) over a cross-shelf distance $R$ causes |
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vertical convective mixing and modifies the density of the fluid by an |
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amount |
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\begin{equation} |
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\Delta \rho = \frac{B_0 \rho_0 t}{g H} |
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\end{equation} |
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where $H$ is the depth of the shelf, $g$ is the acceleration due to |
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gravity, $t$ is time since onset of cooling and $\rho_0$ is the |
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reference density. Dense fluid slumps under gravity, with a flow speed |
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close to the gravity wave speed: |
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\begin{equation} |
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U |
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\sim \sqrt{g' H} |
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\sim \sqrt{ \frac{g \Delta \rho H}{\rho_0} } |
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\sim \sqrt{B_0 t} |
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\end{equation} |
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A steady state is rapidly established in which the buoyancy flux out of |
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the cooling region is balanced by the surface buoyancy loss. |
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Then |
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\begin{equation} |
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U \sim (B_0 R)^{1/3} \mbox{ ; } \Delta \rho \sim \frac{\rho_0}{g H} (B_0 R)^{2/3} |
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\end{equation} |
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The Froude number of the flow on the shelf is close to unity (but in |
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practice slightly less than unity, giving subcritical flow). |
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When the flow reaches the slope, it accelerates, so that it may become |
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supercritical (provided the slope angle $ \alpha $ is steep enough). |
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In this case, a hydraulic control is established at |
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the shelf break. On the slope, where the Froude number is greater |
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than one, and gradient Richardson number |
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(defined as $Ri \sim g' h^*/U^2$ where $h^*$ is the thickness of the |
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interface between dense and ambient fluid) is reduced |
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below 1/4, Kelvin-Helmholtz instability is possible, and leads to |
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entrainment of ambient fluid into the plume, modifying the |
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density, and hence the acceleration down the slope. |
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Kelvin-Helmholtz instability is suppressed at low Reynolds and |
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Peclet numbers given by |
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\begin{equation} |
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Re \sim \frac{U h}{ \nu} \sim \frac{(B_0 R)^{1/3} h}{\nu} \mbox{ ; } Pe = Re Pr |
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\end{equation} |
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where $h$ is the depth of the dense fluid on the slope. |
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Hence this experiment is carried out in the high Re, Pe regime. |
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A further constraint is that the convective heat flux must be much greater |
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than the diffusive heat flux (Nusselt number $>> 1$). |
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Then |
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\begin{equation} |
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Nu = \frac{U h^* }{\kappa} >> 1 |
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\end{equation} |
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Finally, since we have assumed that the convective mixing on the shelf |
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occurs in a much shorter time than the horizontal equilibration, |
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this implies $H/R << 1$. |
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Hence to summarize the important nondimensional parameters, and |
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the limits we are considering: |
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\begin{equation} |
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\frac{H}{R} << 1 \mbox{ ; } Re >> 1 \mbox{ ; } Pe >> 1 \mbox{ ; } Nu >> 1 |
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\mbox{ ; } \mbox{ ; } Ri < 1/4 |
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\end{equation} |
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In addition we are assuming that the slope is steep enough to provide |
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sufficient acceleration to the gravity plume, but nonetheless much less |
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that $1:1$, since many Kelvin-Helmholtz billows appear on the slope, |
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implying horizontal lengthscale of the slope $>>$ the depth of the |
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dense fluid. |
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\subsection{Configuration} |
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The topography, spatial grid, forcing and initial conditions are all |
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specified in binary data files generated using a {\em Matlab} script |
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called {\tt gendata.m} and detailed in |
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section~\ref{sect:plume-generating}. Other model parameters are |
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specified in file {\tt data} and {\tt data.obcs} and detailed in |
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section~\ref{sect:plume-params}. |
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\subsubsection{Binary input data} |
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\label{sect:plume-generating} |
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The domain is $200$~m deep and $6.4$~km across. Uniform resolution of |
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$60\times3^1/_3$~m is used in the vertical and variable resolution of |
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the form shown in Fig.~\ref{fig:dx-plume-on-slope} with $320$ points |
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is usedin the horizontal. The formula for $\Delta x$ is: |
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\begin{displaymath} |
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\Delta x(i) = \Delta x_1 + ( \Delta x_2 - \Delta x_1 ) |
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( 1 + \tanh{\left(\frac{i-i_s}{w}\right)} ) /2 |
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\end{displaymath} |
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where |
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\begin{eqnarray*} |
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Nx & = & 320 \\ |
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Lx & = & 6400 \;\; \mbox{(m)} \\ |
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\Delta x_1 & = & \frac{2}{3} \frac{Lx}{Nx} \;\; \mbox{(m)} \\ |
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\Delta x_2 & = & \frac{Lx/2}{Nx-Lx/2 \Delta x_1} \;\; \mbox{(m)} \\ |
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i_s & = & Lx/( 2 \Delta x_1 ) \\ |
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w & = & 40 |
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\end{eqnarray*} |
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Here, $\Delta x_1$ is the resolution on the shelf, $\Delta x_2$ is the |
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resolution in deep water and $Nx$ is the number of points in the |
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horizontal. |
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The topography, shown in Fig.~\ref{fig:depth-plume-on-slope}, is given |
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by: |
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\begin{displaymath} |
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H(x) = -H_o + (H_o - h_s) ( 1 + \tanh{\left(\frac{x-x_s}{L_s}\right)} ) / 2 |
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\end{displaymath} |
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where |
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\begin{eqnarray*} |
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H_o & = & 200 \;\; \mbox{(m)} \\ |
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h_s & = & 40 \;\; \mbox{(m)} \\ |
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x_s & = & 1500 + Lx/2 \;\; \mbox{(m)} \\ |
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L_s & = & \frac{(H_o - h_s)}{2 s} \;\; \mbox{(m)} \\ |
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s & = & 0.15 |
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\end{eqnarray*} |
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Here, $s$ is the maximum slope, $H_o$ is the maximum depth, $h_s$ is |
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the shelf depth, $x_s$ is the lateral position of the shelf-break and |
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$L_s$ is the length-scale of the slope. |
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The forcing is through heat loss over the shelf, shown in |
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Fig.~\ref{fig:Q-plume-on-slope} and takes the form of a fixed flux |
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with profile: |
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\begin{displaymath} |
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Q(x) = Q_o ( 1 + \tanh{\left(\frac{x - x_q}{L_q}\right)} ) / 2 |
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\end{displaymath} |
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where |
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\begin{eqnarray*} |
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Q_o & = & 200 \;\; \mbox{(W m$^{-2}$)} \\ |
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x_q & = & 2500 + Lx/2 \;\; \mbox{(m)} \\ |
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L_q & = & 100 \;\; \mbox{(m)} |
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\end{eqnarray*} |
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Here, $Q_o$, is the maximum heat flux, $x_q$ is the position of the |
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cut-off and $L_q$ is the width of the cut-off. |
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The initial tempeture field is unstratified but with random |
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perturbations, to induce convection early on in the run. The random |
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perturbation are calculated in computational space and because of the |
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variable resolution introduce some spatial correlations but this does |
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not matter for this experiment. The perturbations have range |
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$0-0.01$~$^\circ$K. |
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\subsubsection{Code configuration} |
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\label{sect:plume-config} |
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The computational domain (number of points) is specified in {\tt |
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code/SIZE.h} and is configured as a single tile of dimensions |
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$320\times1\times60$. There are no experiment specific source files. |
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Optional code required to for this experiment are the non-hydrostatic |
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algorithm and open-boundaries: |
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\begin{itemize} |
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\item Non-hydrostatic terms and algorithm are enabled with {\bf |
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\#define ALLOW\_NONHYDROSTATIC} in {\tt code/CPP\_OPTIONS.h} and |
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activated with {\bf nonHydrostatic=.TRUE.,} in namelist {\em PARM01} |
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of {\tt input/data}. |
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\item Open boundaries are enabled with {\bf \#define ALLOW\_OBCS} in |
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{\tt code/CPP\_OPTIONS.h} and activated with {\bf use\_OBCS=.TRUE,} in |
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namelist {\em PACKAGES} of {\tt input/data.pkg}. |
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\end{itemize} |
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\subsubsection{Model parameters} |
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\label{sect:plume-params} |
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\begin{table} |
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\begin{center} |
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\begin{tabular}{lll} |
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$g$ & $9.81$ m s$^{-2}$ & acceleration due to gravity \\ |
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$\rho_o$ & $999.8$ kg m$^{-3}$ & reference density \\ |
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$\alpha$ & $2 \times 10^{-4}$ K$^{-1}$ & expansion coefficient \\ |
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$A_h$ & $1 \times 10^{-2}$ m$^2$s$^{-1}$ & horizontal viscosity \\ |
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$A_v$ & $1 \times 10^{-3}$ m$^2$s$^{-1}$ & vertical viscosity \\ |
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$\kappa_h$ & $0$ m$^2$s$^{-1}$ & (explicit) horizontal diffusion \\ |
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$\kappa_v$ & $0$ m$^2$s$^{-1}$ & (explicit) vertical diffusion \\ |
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\\ |
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$\Delta t$ & $20$ s & time step \\ |
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$\Delta z$ & $3.3\dot{3}$ m & vertical grid spacing \\ |
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$\Delta x$ & $13.\dot{3}-39.5$ m & horizontal grid spacing |
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\end{tabular} |
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\end{center} |
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\caption{Model parameters used in the gravity plume experiment.} |
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\label{table:plume-on-slope} |
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\end{table} |
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The model parameters (Table~\ref{table:plume-on-slope}) are specified |
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in {\tt input/data} and if not assume the default values defined in |
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{\tt model/src/set\_defaults.F}. A linear equation of state is used, |
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{\bf eosType='LINEAR'}, but only temperature is active, {\bf |
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sBeta=0.E-4}. For the given heat flux, $Q_o$, the buoyancy forcing is |
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$B_o = \frac{g \alpha Q}{\rho_o c_p} \sim |
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10^{-7}$~m$^2$s$^{-3}$. Using $R=10^3$~m, the shelf width, then this |
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gives a velocity scale of $U\sim 5 \times 10^{-2}$~m~s$^-1$ for the |
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initial front but will accelerate by an order of magnitude over the |
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slope. The temperature anomaly will be of order $\Delta \theta \sim 3 |
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\times 10^{-2}$~K. The viscosity is constant and gives a Reynolds |
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number of $100$, using $h=20$~m for the initial front and will be an |
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order magnitude bigger over the slope. There is no explicit diffusion |
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but a non-linear advection scheme is used for temperature which adds |
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enough diffusion so as to keep the model stable. The time-step is set |
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to $20$~s and gives Courant number order one when the flow reaches the |
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bottom of the slope. |
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\subsubsection{Build and run the model} |
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Build the model per usual. For example: |
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\begin{verbatim} |
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% cd verification/plume_on_slope |
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% mkdir build |
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% cd build |
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% ../../../tools/genmake -mods=../code -disable=gmredi,kpp,zonal_filt |
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,shap_filt |
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% make depend |
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% make |
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\end{verbatim} |
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When compilation is complete, run the model as usual, for example: |
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\begin{verbatim} |
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% cd ../ |
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% mkdir run |
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% cp input/* build/mitgcmuv run/ |
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% cd run |
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% ./mitgcmuv > output.txt |
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\end{verbatim} |
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