From MAILER-DAEMON Wed Dec 13 12:30:56 2000 Date: 13 Dec 2000 12:30:56 -0500 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA X-IMAP: 0976728656 0000000000 Status: RO This text is part of the internal format of your mail folder, and is not a real message. It is created automatically by the mail system software. If deleted, important folder data will be lost, and it will be re-created with the data reset to initial values. From tamay@rsmas.miami.edu Wed Dec 13 16:52:45 2000 -0500 Status: R X-Status: X-Keywords: Return-Path: Delivered-To: tamay@rsmas.miami.edu Received: (qmail 23757 invoked by uid 7794); 13 Dec 2000 16:52:44 -0000 Received: from linda@panoramix.rsmas.miami.edu by umigw.miami.edu with scan4virus-0.51 (sweep: 1.11/3.38. . Clean. Processed in 0.389655 secs); 13/12/2000 11:52:44 Received: from panoramix.rsmas.miami.edu (129.171.98.20) by umigw.miami.edu with SMTP; 13 Dec 2000 16:52:44 -0000 Received: (from linda@localhost) by panoramix.rsmas.miami.edu (950413.SGI.8.6.12/950213.SGI.AUTOCF) id LAA02391 for tamay@rsmas.miami.edu; Wed, 13 Dec 2000 11:53:31 -0500 Date: Wed, 13 Dec 2000 11:53:31 -0500 From: linda@panoramix.rsmas.miami.edu (Linda Smith) Message-Id: <200012131653.LAA02391@panoramix.rsmas.miami.edu> Apparently-To: tamay@rsmas.miami.edu \documentclass[12pt]{article} \usepackage{psfig,natbib,amspaper} \oddsidemargin 0pt \flushbottom \parskip 10pt %\parindent 0pt \textwidth 16cm \topmargin -0.2cm \textheight 24cm \renewcommand{\baselinestretch}{1.2} \input com.tex \title{ \Large \bf DOME - Dynamics of Overflow Mixing and Entrainment} \begin{document} \renewcommand{\thepage}{$\,$} \date{} \maketitle \begin{center} \Large Working Notes For a Model Comparison Study Prepared by T. Ozgokmen, J. Stephens, and R. Hallberg \footnote{ {\em Contact details:} Tamay Ozgokmen, MPO, RSMAS, 4600 Rickenbacker Causeway, Miami, FL 33149-1098. Email: tamay@rsmas.miami.edu Tel: (001) 305 361 4053 } \large{http://www.gfdl.gov/$^\sim$jns/pubs/dome.html} \end{center} DOME is an international study to compare the performance of a number of state of the art ocean models in representing overflow processes, aimed at quantifying model performance and identifying areas for improvement in individual models. The original idea for some kind of comparison project to address overflow related issues was conceived in discussions between Robert Hallberg, Peter Killworth and James Stephens. This idea rapidly grew into what is now the DOME project. DOME has attracted a great deal of interest, at this stage mostly from the large-scale numerical modeling community. Comparisons are taking place within the following, quasi-chronological framework of tests: \begin{itemize} \item {\bf Phase 1:} An idealised dense overflow. \item {\bf Phase 2:} Mediterranean Outflow and Denmark Straits. \item {\bf Phase 3:} Quasi Global simulations. \end{itemize} \newpage \setcounter{page}{1} \renewcommand{\thepage}{\arabic{page}} \tableofcontents \newpage \section{MOTIVATION} The far-reaching significance of deep cold temperatures in the ocean was first realized by Count Rumford in 1797. Rumford, in analyzing ship-recorded temperatures obtained almost 50 years earlier, inferred a polar origin for the deep water masses and a corresponding meridional overturning circulation to carry deep cold waters equatorward and warm surface waters poleward. We now know that this circulation is driven by heat and/or freshwater loss at the ocean surface in a few special regions near to the poles, and refer to it as the \textit{thermohaline circulation}. This circulation transports a considerable amount of heat, and in so doing strongly moderates weather and climate. For this reason, in view of anthropogenically induced changes in atmospheric carbon dioxide and the resultant possibility of global warming, we need to understand how the oceans' role in storing and transporting heat will vary in response to climate change. The wealth of recent data from programs such as the World Ocean Circulation Experiment (WOCE) indicates clearly that ocean models are currently lacking in their representation of deep water mass properties and circulation. A significant part of this is likely due to the poor representation of intense mixing and bottom boundary processes occurring at overflows, where many deep and intermediate water masses are sourced. This is therefore one of the most important problems to be addressed in current ocean models. COMMENT: WE NEED A MUCH STRONGER MOTIVATIONAL STATEMENT... Entraining gravity currents and other processes related to dense water overflows from marginal seas are central to the formation of several major water masses. These processes must be accurately portrayed in order for numerical models to provide reliable insight into the dynamics and variability of the ocean. There is abundant evidence that ocean models of a resolution that is feasible for large-scale studies do not depict overflows realistically, without an explicit treatment of the relevant processes. This is especially clearly demonstrated in the DYNAMO model intercomparison project. Two models, based on geopotential and sigma coordinates, exhibited grossly excessive entrainment in the critical Denmark Strait overflow. A third, isopycnic coordinate, model had essentially no entrainment in that plume. Motivated by this evident deficiency, several methods for improving the representation of overflow processes in the various ocean model classes have been suggested in recent years. While all of these proposals show clear improvements in selected test cases over the models without any deliberate treatment of the overflow processes, it is not clear whether these improvements are robust, or whether they all converge to the true behavious of the ocean. DOME is an effort to systematically assess the various suggested parameterizations through a series of model intercomparisons. The tests to be performed will include a series of idealized cases spanning a reasonable range in parameter space, for which very high resolution simulations provide some insight into the desired behavior. In addition, comparison of isolated simulations of realistic overflows with each other and with observations will provide further insight into the behavior of the various approaches. Finally, the impact of the parameterizations on global simulations will be assessed. The objectives of DOME are threefold. First, the fundamental physics of overflow processes will be studied to attain a deeper understanding. Second, model intercomparison will illucidate the intrinsic strenghts and shortcomings of the various approaches. Third, optimal parameterizations for overflow processes will be identified or developed, including advice on the conditions under which each viable approach is most suitable. \newpage \section{PARTICIPANTS/INSTITUTES/MODELS INVOLVED} \subsection{Participants} \begin{table}[!ht] \center \vskip 5mm \begin{tabular}{|l|l|} \hline \hline \emph{Participant} & \emph{Email} \\ \hline James Stephens & (jns@gfdl.gov)\\ Robert Hallberg & (rwh@gfdl.gov)\\ Tal Ezer & (tne@gfdl.gov)\\ Peter Killworth & (peter.d.killworth@soc.soton.ac.uk)\\ Neil Edwards & (nre@soc.soton.ac.uk)\\ Steven Alderson & (sga@soc.soton.ac.uk)\\ George Nurser & (agn@soc.soton.ac.uk)\\ David Holland & (holland@cims.nyu.edu)\\ Dale Haidvogel & (dale@imcs.rutgers.edu)\\ Paul Schopf & (schopf@cola.iges.org)\\ Whit Anderson & (anderson@cola.iges.org)\\ Igor Polyakov & (igor@iarc.uaf.edu, ivp@gfdl.gov)\\ Claus B\"{o}ening & (cboening@ifm.uni-kiel.de)\\ Joachim Dengg & (jdengg@ifm.uni-kiel.de)\\ Alistair Adcroft & (adcroft@sea.mit.edu)\\ Sonya Legg & (slegg@whoi.edu)\\ \hline \hline \end{tabular} \end{table} \subsection{Institutes/Models} \subsubsection{Geophysical Fluid Dynamics Laboratory (GFDL)} \indent HIM\,1.0 - Bob Hallberg's isopycnal model \footnote{Hereafter referred to simply as HIM.} incorporating a Richardson number dependent entrainment scheme, which will be run by James Stephens \mbox{(jns@gfdl.gov)}.\\ \indent MOM\,3 - an up to date version of GFDL's z-coordinate model \footnote{Hereafter referred to simply as MOM.} with the combination of a fixed thickness bbl and partial cells. Igor Polyakov will be running MOM\,3.\\ \subsubsection{Princeton University} \indent POM - Sigma-coordinate ocean model which has a Mellor-Yamada 2.5 closure scheme. This, run at absurdly high resolution - maybe 1/20 degree with 60 layers, will give us an independent test (and is likely to converge at this high resolution) for the idealised testcase. Tal Ezer has taken on the task of configuring this model.\\ \subsubsection{Southampton Oceanography Centre} \indent MOMBLE and/or UNIFIED MODEL (Met Office) with a variable thickness bottom boundary layer as described in \cite{ke99}. These are both geopotential coordinate models. Peter Killworth, George Nurser, Steve Alderson and Neil Edwards are the people to speak to. \subsubsection{Courant Institute} \indent MICOM - David Holland is implementing a diapycnal mixing scheme in this pure isopycnal model along with a slab bottom boundary layer beneath MICOM to aid the representation of polar abyssal water masses where density gradients are very small. \subsubsection{Rutgers University} \indent SPEM, SCRUM or ROMS - Dale Haidvogel will participate in the tests and supply another model which, in addition to POM, may be used to provide ground truth. \subsubsection{Kiel University} \indent The model is basically MOM\,2 with a bottom boundary layer as described in \cite{bd97}. Claus B\"{o}ening and Joachim Dengg are configuring this model in a Denmark Straits scenario for Phase II (they will not be participating in Phase I). \subsubsection{Center For Ocean, Land, Atmosphere (COLA)} \indent Paul Schopf has a model (Poseidon) with a choice of grid (B or C) and which has a hybrid vertical coordinate. The usual configuration in the deep is quasi-isopycnal, although configurations with sigma or z coordinates can be made at run-time, and an iso-neutral tracking coordinate is currently in development. \subsubsection{MIT, WHOI} Alistair Adcroft and Sonya Legg are interested in participating, although do not as yet have an active role. Sonya has carried out nonrotating calculations of mixing and entrainment in straits and sills, and wishes to investigate the modification of this mixing and entrainment at lower Rossby numbers. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{IDEALISED TESTCASE} The idea here is to focus our attentions on the ability of the models to represent the intense mixing, descent and subsequent geostrophic adjustment of a dense plume from an embayment. We will therefore want to keep things simple in those aspects not directly concerned with the overflow dynamics and that may result in model to model differences. \subsection{Free parameters} In order to determine what parameters to perturb as part of this study, we are required to determine some non-dimensional numbers. In order to obtain these I have chosen the following dimensional free parameters as being appropriate for our system; a number of these are depicted in a schematic representation of a side view of the domain in Fig:\,\ref{fig:bbl_domain}. These are open to discussion. I have implicitly assumed for instance that the length of the domain and the horizontal dimensions of the embayment will not important, by making these dimensions reasonably large in the specification detailed in the next section. \begin{enumerate} \item The horizontal extent of the slope, $W$. \item The vertical extent of the slope, $D$. \item The vertical extent of the embayment,$H$. \item The Coriolis parameter, $f$. \item The reduced gravity associated with the density difference between the dense overflow water and the interior, $g_o$. \item The reduced gravity associated with the density difference between the surface and deepest water in the interior, $g_i$. \item The horizontal resolution, $\Delta x$. \item The vertical resolution, $\Delta z$ \item The amount of dense water injected in the embayment, T. For reasons pertaining the way we intend to force the model (see subsection entitled ``Forcing in the embayment'', we expect this to obey the following scaling for rotating-hydraulic control (\cite{wlk74}) \beq T = \frac{2}{3} \frac{g_o h_d^2}{f} \label{eq_transport} ~, \eeq where $h_d = h (H-h)/ H$ is the appropriate baroclinic two layer height scale and $h$ and $H-h$ are the respective upper and lower layer depths in the embayment. Since $T$ and $h$ are directly related through other, known parameters, in order to simplify the non-dimensional numbers we will use $h$, the vertical extent of the dense overflow water in the embayment as a proxy for $T$. \end{enumerate} Since we are dealing with only two units (length and time), it is possible to define seven non-dimensional numbers from these nine parameters, as detailed in the Buckingham Pi Theorem (\cite{e14}). I suggest we fix two of these parameters, $H/D=0.2$ (where $H=600$\,m and $D=3000$\,m) and $h/H = 0.5$ (where $h=300$\,m and we will choose the transport given the other parameters such that this is satisfied). This leaves five possible non-dimensional numbers, for which I suggest \begin{enumerate} \item $\mathbf{D/W}$ - This is simply the slope of the bathymetry. I suggest we try values of 0.005 and 0.01. \item $\mathbf{g_i/g_o}$ - This determines some measure of the stratification. It is zero if the interior is unstratified, and one if the interior surface to bottom density difference is the same as that of the dense water in the embayment. We should consider both of these cases, and in addition, an more highly structured vertical profile. \item $\mathbf{\Delta z/ h}$ - This parameter determines how well the overflow current is resolved in the vertical in the embayment. In isopycnal coordinate models, the corresponding parameter is $\mathbf{g_{lay}/g_o}$, where $g_{lay}$ is the reduced gravity difference between layers. This parameter is obviously explored by varying the vertical resolution. \item $\mathbf{\Delta x / \sqrt{g_o h}/ f}$ - This parameter determines how well an internal deformation radius based on $g_o$ and $h$ is resolved by the horizontal grid. The importance of this parameter can be examined by changing the horizontal resolution. \item $\mathbf{g_o D/ \lp f^2 W h \rp}$ - This parameter measures the ratio of the depth of the plume once on the slope and its depth in the embayment, $h$. If the plume thickness is constrained by rotation, a reasonable estimate for its thickness is $h_{plume} \approx 0.5 \Delta u / f$, while the density difference across the top of the plume may be assumed to be $\Delta u = g_o D / W f$. So the plume thickness might be proportional to {g_o D/ \lp f^2 W \rp}$. It is also possible that the plume thickness is constrained by stratification, in which case a reasonable estimate would be $h_{plume} \approx 20 \Delta u \sqrt{h_{plume} / g_o}$. In this limit, the plume thickness becomes $h_{plume} \approx 400\, g_o D^2/ \lp f^2 W^2 \rp$. As the stratification constrained limit applies when $D/W < 1/800$, it is assumed that the rotationally constrained limit is most appropriate. (See Killworth and Edwards for a discussion of these two limits.) Results are presented later of how important this parameter appears to be for MOM, where the bottom boundary layer thickness is fixed. Variations of this parameter can be examined by perturbing the density difference between the overflow and the ambient water. \end{enumerate} These non-dimensional numbers are not unique and it is possible to come up with others given the above dimensional scales. In addition, given different initial dimensional scales, one may obtain different non-dimensional numbers from these. I therefore invite discussion and comment in respect of the above choices. \subsection{Specification of Experiments} {\bf Domain Geometry.} For simplicity, the f-plane approximation is made, and it is expected that a uniform spacing Cartesian grid is used in the horizontal. The domain is zonally reentrant and 2000 km long by 650 km in the cross-slope direction. The maximum depth is 3600 m, and forcing occurs in a flat-bottomed embayment at the top of the slope. The embayment is 50 km square. If the 1% slope is used, and the southwest corner of the domain is specified as the origin, the bottom depth D can be defined as follows: if (y<300 km) D = 3600 m, otherwise if (y>300 km) and (y<600 km) D = 3600 m - 10 m/km * (y-300 km), otherwise (if (x>1800 km) and (x<1850 km) D = 600 m, and D = 0 m otherwise). The geometry and bathymetry for a slope of 0.01 is depicted in Fig:\,\ref{fig:depths}. When the 0.5% slope is used, the corresponding specification is: if (y<600 km) D = 3600 m - 5 m/km * y, otherwise (if (x>1800 km) and (x<1850 km) D = 600 m, and D = 0 m otherwise). This long domain helps minimize the impact of the zonally reentrant boundary condition. {\bf Forcing.} The only forcing should be due to an injection of dense water into the embayment. There should be no surface wind stress or surface buoyancy forcing. In general, the domain should be allows to fill with water, but a broadly distributed surface sink is permissible to balance the inflow. The parameters suggested below give an inflow of 3 Sv. In coarse-resolution versions of these tests, the embayment may have to be enlarged, in which case the other parameters should be adjusted to retain the 3 Sv inflow. {\bf Base-case Parameters:} \begin{itemize} \item Slope: 0.01 (1%) \item Surface Density (also Rho$_0$ in Boussinesq models): 1030 kg m$^{-3}$. \item Ambient stratification: None \item Coriolis Parameter: $f = 1 \times 10^{-4}\, \mathrm{s}^{-1}$ \item Gravitational Constant: 9.81 m s$^{-2}$. \item Inflow Density: 1032 kg m$^{-3}$. \item Inflow Thickness: 300 m. \item Inflow Velocity: 0.2 m s-1. \item Lateral Viscosity: As needed by the model at a given resolution. \item Lateral Diffusivity: As little as can be tolerated by the model at a given resolution. \item Interior vertical viscosity: This may be a part of the parameterization of the overflow physics and is left unspecified. \item Background Diapycnal Diffusivity: $1 \times 10^{-5} \mathrm{m}^{2} \mathrm{s}^{-1}$, or a larger value if required by the model. \item Bottom Drag: A quadratic drag law, with c$_D$ = 0.002? and an assumed background velocity of 0.05 m s-1. That is $\tau_{bottom} = c$_D$ * \sqrt(u^2 + v^2 + 0.0025 m^2 s^-1)$. This also may be a part of the parameterization, and this suggestion may be revisited. \item Horizontal resolution: This crucial parameter is left to the discression of the participants, but it is expected that a range (and certainly at least 2) resolutions will be examined with each model. Common resolutions will be 50 km, 25 km, 12.5 km, 5 km. \item Vertical resolution: The number of vertical degrees of freedom is another crucial parameter, for which it is expected that a range of values will be examined with each model. With the specified domain and forcing, regularly spaced Z-coordinate models will work best with 12*n levels, where n = 1, 2, 3, etc. Comparable numbers of vertical degrees of freedom should be considered with other models. \end{itemize} {\bf Suggested Cases:} All of these perturbation cases are same as the base-case, except for the listed changes. \begin{enumerate} \item The base-case, as describe above. \item Inflow density changed to 1031 kg m$^{-3}$. \item Inflow density changed to 1030.2 kg m$^{-3}$. \item Coriolis parameter set to 0 s$^{-1}$. \item Slope set to 0.005 (0.5%). \item Linear ambient stratification imposed, with the density profile given by $\rho_{ambient} = 1030 kg m$^{-3}$ - z * (2 kg m$^{-3}$) / (3600 m)$, where z is the distance from the surface, negative downward. \item Ambient stratification contains a thermcline. (Details to be specified later). \end{enumerate} It is assumed that this list of cases will be run at a wide range of horizontal and vertical resolutions. For this reason, the list of cases has been deliberately limited to a few, hopefully representative, cases. There are several other considerations which must be spelled out: \begin{itemize} \item {\bf Linear equation of state.} A nonlinear equation of state will complicate the specification of the problem and comparison of results. Initially, a linear equation of state (perhaps with density as the state variable) should be used. In this specification, densities will be given, rather than temperatures and salinities. Effects of a nonlinear equation of state might be revisited later. \item {\bf Integration time.} Based on preliminary simulations, 60 days might be a sensible duration for the simulations. It may be advisable to intensively analyze the results at 30 days and 60 days, with output saved every 5 days. If on-line time means are available, these would be valuable diagnostic quantities, especially time-averaged cross-slope volume fluxes by density class. If on-line time means are not available, a higher frequency of output may be necessary. \item{\bf Tracers.} See the following subsection, in which the diagnostic use of passive tracers is discussed. \end{itemize} \subsection{Tracers} Judicious use of passive tracers will allow us to diagnose mixing. In HIM, I am prescribing a number of passive tracers, 11 in total, which sounds like a lot (but maybe we need more), in the following way: (Source and/or initial concentrations for each passive tracer are set to one). One tracer is injected into source water in the embayment. There is none of this tracer at time $t=0$. Five tracers are placed in vertical bands 0.5 degrees wide in latitude and along the entire longitudinal extent of the domain. For a slope of 0.01, reasonable latitude bands for each tracer are 350-400 km, 400-450 km, 450-500 km, 500-550 km and 550-600 km from the southern wall. These tracers extend the entire depth of the fluid. Five tracers are placed in horizontal bands 300m in thickness (since this is the largest unit of resting layer depth we consider). The thickness bands are 600-900\,m, 1200-1500\,m, 1800-2100\,m, 2400-2700\,m and 3000-3300\,m, and extend horizontally wherever fluid exists. \subsection{Forcing in the embayment} We can only really make a meaningful comparison between models if the overflow properties and flow rate and integrated transport of dense water are quite similar. We have choices as whether to use surface cooling, relaxation, or straightforward injection of mass to generate a source of dense water. Injection appears to be the preferred method, beacuse it ensures the same net transport of dense water \textit{and} also buoyancy (the problems with surface cooling and relaxation are documented below). In HIM I simply add mass uniformly to the densest layer in the northern 1 degree of the embayment at each timestep, such that the desired volume flow rate input of dense water occurs over time. I remove the effect of the mass input uniformly from the surface layer in the interior of the domain south of the embayment entrance. This injection method works well - I obtain very similar transports and water mass properties for different resolutions. An initial problem whereby about 25\,\% of the input dense water is mixed away and exits in lower density classes was trivially fixed by turning off the diapycnal mixing scheme in the embayment. In geopotential/sigma models an equivalent effect could be achieved by injecting mass at the northern boundary of the domain in the densest layer(s), but this may take some work. It is however, the only sensible way to proceed. \subsubsection{Problems with Surface Cooling} With cooling there is no guarantee that the outflow will occur at the same density in all cases, but the net export of buoyancy will be the same, and that only at steady state. Lighter water flows out first, followed by progressively denser water. It really isn't a very satisfactory way to proceed. \subsubsection{Problems with Relaxation} With relaxation of layer interfaces to a specified thickness, there is no guarantee that the steady state net export of buoyancy will be the same, but the outflow is certain to occur with the specified buoyancy. The issue here is one of transport - differences in the representation of boundary conditions and Kelvin wave bores due both to the horizontal and vertical resolution and also the model grid, alter the strength of the outflow to too great a degree. I have numbers, obtained from a comparison of MOM and HIM, if people are interested. %I find the following transports (in Sv) of the densest water over the sill %for a stratified case with a density difference of $2 \mathrm{kg}\,\mathrm{m}^-3$: % 1/4 degree, 12 layers 1/8 degree 48 layers %HIM 30 days 3.0 3.2 % 60 days 2.9 3.3 %MOM 30 days 2.5 3-3.5 (*) % 2.3 3-3.5 %* - There is a problem computing this in MOM. %I have not tried an unstratified scenario, but I beleive %this will only cause significant differences in the interior. The time integrated transport over the sill is important too, and in MOM the overflow builds up noticably slower than HIM, which may well be due to boundary wave differences (no slip B-Grid Kelvin waves in MOM are slow). %For the 1/4 degree, 12 layer case above, transports (in Sv) %of the DENSEST water over the sill increase with time as follows: % HIM MOM % 4 days 1.8 0 % 8 days 2.6 0.3 % 10 days 2.6 1.1 % 12 days 2.7 2.0 % 20 days 2.8 2.4 % 30 days 3.0 2.5 %At high resolution (1/8 with 48 layers) in HIM things are little %different % 4 days 1.8 % 6 days 2.9 % 8 days 3.2 % 10 days 3.2 %The way I suggest we do this is as follows: %In order to obtain similar properties and transports for the overflows %with each model, the strong relaxation in the sponge zone has to be %equivalent. This is most easily accomplished by relaxing to a two layer %configuration in the sponge zone, where each of the layers %has equal depth. All the internal interfaces in the isopycnal model %can be relaxed to a single depth in a portion of the embayment %and a vertical temperature profile can be prescribed in the z-coordinate %model such that there is an abrupt and equal change in density at the %same depth as the convergence of the isopycnal interfaces. %Choosing the density difference between the two layers in the sponge %to be say 2 kg m-3 (like the Med overflow), the transport scaling %suggests that for an overflow of order 2 Sv one would want h~150 m. %Bearing in mind that the appropriate baroclinic two layer height scale, %h, is (h1 h2)/ (h1+ h2), where h1 and h2 are the upper and lower %layer depths respectively, %we require h1=h2=300\,m. What I have done is to make the embayment %600\,m deep and the maximum depth 3600\,m, which I suggest is nice %because it requires a vertical resolution of 12 n layers where we %can consider n= 1,2,3 ........ and we will always be forcing the %same way between the models and between these different vertical %resolutions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{DIAGNOSTICS} In order to most easily compare between models, we need to produce quantitative indices of model performance. This will enable open participation in DOME now and in future years by the whole scientific community. \subsection{Naming convention} In order to enable such indices to be collated we also need some sort of naming convention. I suggest that we use the following approach, whereby the value of each of the free parameters is listed in the name of the output file in the following order, separated by underscores and followed by an appropriate filename extension. \begin{itemize} \item Performance Index, e.g ADF for alongslope density flux. The names and specifications of each of the performance indices, and how they should be output to file, can be decided at a later date. However, I give a suggested example below under the subsection entitled ``Alongslope flux of fluid by density class''. \item Model name, e.g. HIM. \item Horizontal resolution in degrees, e.g. 0.25. \item Number of layers, e.g. 12. \item Slope, as a fraction, e.g. 0.01. \item Stratification in the interior. This is the density difference between the surface water in the domain, and the deep water, in units of kg\,m$^{-3}$, e.g. 2. \item The density difference between the lightest water in the domain, and the dense water injected, in units of kg\,m$^{-3}$, e.g. 2. \end{itemize} With the examples above and an ``nc'' extension if it is netcdf, the filename will be called:\\ \mbox{ADF\_HIM\_0.25\_12\_0.01\_2\_2.nc} The kind of things I suggest we should be looking at for performance indices are: \subsection{Alongslope flux of fluid by density class} We could compute this quantity at specific longitudes for fluid on or above the topographic slope. In an isopycnal model the total (mean + eddy) alongslope flux of density in layer n per unit width of current, $F_{\rho}^n$, is defined \beq F_{\rho}^n = \rho^n \overline{\lp h u \rp^n} \label{rhoflux_iso}~, \eeq where the overbar is a time average. It is important to compute $\lp h u \rp^n$ at each timestep and then conduct the averaging, since this implicity includes the contribution from eddies\footnote{$\overline{h u} = \overline{h} \overline{u} + \overline{h' u'}$} by taking time averages of the layer thickness fluxes. One may also separately compute $\overline{h}$ and $\overline{u}$ to enable the eddy bolus transport $\overline{h' u'}$ to be computed. In geopotential/sigma models the alongslope flux of density in layer n per unit width of current, $F_{\rho}^n$, is defined \beq F_{\rho}^n = \overline{\lp \rho u \Delta z \rp^n} + \kappa_x \overline{\lp \frac{\partial \rho}{\partial x} \Delta z \rp^n} ~, \eeq where $\partial \rho /\partial x$ in sigma coordinates is related to the same quantity in geopotential coordinates by \beq \left . \frac{\partial \rho}{\partial x} \right|_s = \left . \frac{\partial \rho}{\partial x} \right|_z + \frac{\partial \rho}{\partial x} \frac{\partial z}{\partial x} ~, \eeq where the subscripts $s$ and $z$ refer to sigma and geopotential coordinates respectively. To show the kind of thing we might expect, Fig:\,\ref{fig:ADF_HIM} shows the total alongslope transport in three different density classes at 15\,W in a particular configuration of HIM and as a function of vertical resolution, at (a) $30-35$ days as an average, and (b) $60-65$ days as an average. The values were computed using Eq:\,(\ref{rhoflux_iso}) for fluid on or above the topographic slope. What we may want to do is plot a figure similar to this, but as a function of of model type for a particular set of non-dimensional parameters. I am currently thinking of other diagnostics to use. \subsection{Picture figures} Inevitably we will also need picture type figures. I do not think data format need be an issue for producing figures, but at some stage we will need a specification of what needs to be produced and how. Obvioulsy it would be nice if everybody used the same plotting tool, although I appreciate this could cause difficulties. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{RESULTS FROM A PRELIMINARY COMPARISON} This section details preliminary results from a comparison between MOM and HIM at 1/4 degree resolution and with 12 layers. High resolution sigma-coordinate model runs will ultimately provide the ground truth. The models are setup as detailed in the previous section, although the forcing is via relaxation for initial simplicity and the embayment is wider than specified. There is a density anomaly of 2 kg\,m$^{-3}$ in the relaxation zone at then northern end of the embayment. I have imposed a linear basic state stratification in the interior encompassing this same 2 kg\,m$^{-3}$ density difference between the surface and the maximum depth of the basin (3600\,m). With this configuration, any dense water from the overflow that is not mixed away should just be able to descend to its neutral buoyancy layer at the bottom of the slope. The slope is a uniform 0.01. I have used a biharmonic Smagorinsky lateral eddy viscosity in both models. \subsection{HIM} \indent Fig:\,\ref{fig:bot_density_strat_60}\,(a) clearly shows the deepest water in the plume spreading entirely \textit{along} the slope, having reached neutral buoyancy at a little deeper than 2500\,m. A significant volume of dense water is also visible along the boundary at the top of the slope. An explanation for this may be the following: The representation of Kelvin wave speed on a C-Grid is particularly good at all ratios of grid spacing/deformation radius, $\Delta$, (\cite{hdw83}). However, since the internal deformation radius at the top of the slope is only approximately 5\,km, versus 25\,km for the grid spacing, the transport is $\Delta \sim$5 times too efficient in this case and we see far too much dense water carried along in the Kelvin wave. The density section show in Fig:\,\ref{fig:density_sec_strat_60}\,(a) reveals a double peaked structure associated with the plume and the Kelvin wave.\\ \subsection{MOM} \indent After 60 days Fig:\,\ref{fig:bot_density_strat_60}\,(b) shows a plume, although its descent downslope is certainly less than in HIM (see (a)) The density section shown in Fig:\,\ref{fig:density_sec_strat_60}\,(b) reveals that much of the dense water has been mixed away vertically. My interpretation of the behaviour we see in the MOM simulations is the following: We are prescribing a 300\,m thick layer of dense water in the relaxation zone of the embayment, and the fixed thickness bottom boundary layer in MOM is set to 100\,m. Therefore when dense water executes its egress from the embayment, only 1/3 of it is part of the bottom boundary layer. The other 2/3 is subject to convective overturning, since the grid aspect ratio is about the same as the slope. This is the very problem bottom boundary layers were designed to avoid and in MOM is unavoidable since setting the bbl thickness to 300\,m would prevent the dense water from descending the slope via the action of friction. I expect that Neil and Peter's bbl will fix this problem in part, although not completely since their bbl thickness is unable to exceed 1/2 the layer thickness. There is virtually no Kelvin wave problem in MOM, (evidenced by the lack of dense water at the top of the slope at the western end of the domain), since with no slip boundary conditions and $\Delta \sim$5, Kelvin waves travel much more slowly on a B-Grid than in the continuum limit (\cite{hdw83}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{MEDITERRANEAN OUTFLOW} The mixing is more intense here than for other overflows, and one has to mix just enough (and no more) to get the water to the right level in the interior. I have run some simulations of the Mediterranean Outflow using HIM; this configuration will serve here as a first suggestion of what we need.\\ \indent The domain is 32:42\,N, 14:2\,W and includes an explicit Mediterranean exchange, although our Mediterranean basin is simply a channel 300\,m deep and 1 degree wide. The Atlantic bathymetry is taken from \cite{rhd95} and smoothed once; the Strait of Gibraltar is a zonal channel two grid boxes wide and 200\,m deep whatever the resolution. The initial stratification on the Atlantic side (without the Med'n tongue) and the Med'n side is taken from \cite{l82}. We relax to these observations at the western side of the Atlantic (to kill Kelvin waves travelling around the solid walls of the domain) and at the eastern end of the Mediterranean basin to keep the properties of the overflow constant. There is no other forcing.\\ \indent We are able to get some nice plume cross sections of velocity and salinity in comparison with observations, although not at exactly the right depths. We are hoping that including a non-linear eqn of state will clear this up; this hasn't been done yet but I think we would want it for the comparison.\\ \indent Rotational hydraulic control seems to do a good job of controlling the exchange through the model Strait of Gibraltar (SOG) in the runs I have done. However, it might be a better idea to prescribe the overflow and separate the issue of the overflow dynamics from the channel dynamics. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{DENMARK STRAITS OVERFLOW} >From a climatic perspective, the Denmark Straits overflow is the most important North atlantic overflow. Setting up a testcase will require some thought ....................... As stated earlier, Claus B\"{o}ening and Joachim Dengg are giving this some thought, so hold tight for an update. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{QUASI-GLOBAL SIMULATIONS} These are still in the planning stage but I have had some suggestions thus far from Peter Killworth on behalf of the Southampton participants (I believe George Nursrer is setting up a 1 degree model as I write this): ............................................... We suggest a 1 deg resolution if folk can afford it (we don't know if we can yet!), stopping at $\pm$ 75 north, no ice, surface forcing either original Haney '71 or climatology + relaxation. We aren't sure what to do about knobs/whistles such as parameterisations. Certainly momble doesn't have those, and we haven't thought through the interaction with the bottom layer anyway. It seems to us that if not all models have a feature, comparison will be difficult (DYNAMO tendency to lowest common denominator showing up here again). Conversely, while we want to show improvements or lack thereof by using a bbl, we don't want a thoroughly unrealistic control run... thoughts? I have a few thoughts on this too . Peter has some very good points and I do think comparison wil be difficult. For example, GFDL are currently setting up a 1 degree global version of MOM with enhanced equatorial resolution of 1/3 degree for ENSO studies. This is a big investment and poeple aren't going to want to redo the thing stopping at 70\,N and without their best parameterisations just to obtain standardisation with another model. Again, your opinions are needed .................................... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \def\refname{References} \bibliographystyle{natbib} \bibliography{skeleton} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \begin{figure}[!ht] \vspace{2cm} \center \mbox{\psfig{figure=figs/bbl_domain.epsi,height=8cm,angle=-90}} \caption{Schematic depicting cross-section of domain and important chosen length scales for determining non-dimensional parameters; the horizontal extent of the slope, $W$, the vertical extent of the slope, $D$, the vertical extent of the embayment, $H$, and the vertical extent of the dense overflow water in the embayment, $h$. \label{fig:bbl_domain} } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \begin{figure}[!ht] \vspace{2cm} \center \mbox{\psfig{figure=figs/depth_col.epsi,height=10cm,angle=90}} \caption{Bathymetry for idealised testcase with uniform slope of 0.01. Sill depth is 600\,m. Maximum depth in basin is 3600\,m. \label{fig:depths} } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \begin{figure}[ht] \center \raisebox{9cm}{(a)~}\mbox{\psfig{figure=figs/ADF_HIM_0.25_x_0.01_2_2_30:35.ps,height=9cm,angle=-90}} \vskip 2cm \raisebox{9cm}{(b)~}\mbox{\psfig{figure=figs/ADF_HIM_0.25_x_0.01_2_2_60:65.ps,height=9cm,angle=-90}} \caption{ Total (mean + eddy) transport (Sv) as a function of density class (kg\,m$^{-3}$) at 15\,W after (a) a $30-35$ day time average. (b) a $60-65$ day time average. The horizontal resolution is 0.25 degrees, with a linear initial stratification and a slope of 0.01. Solid - 12 layers. Short dashes - 24 layers. Long dashes - 48 layers. \label{fig:ADF_HIM}} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \begin{figure}[!ht] \center \mbox{\psfig{figure=figs/bot_density_strat_60.epsi,height=18cm}} \caption{Bottom density anomalies (kg\,m$^{-3}$) after 60 days from a) HIM and b) MOM. The horizontal resolution is 0.25 degrees with 12 vertical layers, a linear initial stratification and a slope of 0.01. In HIM, bottom density is averaged over the bottom 100\,m to assist comparison with MOM. \label{fig:bot_density_strat_60} } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \begin{figure}[!ht] \center \mbox{\psfig{figure=figs/density_sec_strat_60.epsi,height=18cm}} \caption{Cross sections of density anomaly (kg\,m$^{-3}$) at 15\,W after 60 days from a) HIM and b) MOM. The horizontal resolution is 0.25 degrees with 12 vertical layers, a linear initial stratification and a slope of 0.01. \label{fig:density_sec_strat_60} } \end{figure} \end{document}