OC-513 GEOPHYSICAL FLUID DYNAMICS - II

SPRING 2007      Tues/Thurs 9.30 - 11.00 sort of

P.B. RHINES

• Week 9/10:
  

  Solutions from problem set 3 are here.

  ***TAKEHOME QUIZ POSTPONED****see email from Peter R.***

  Lecture slides on the MOC posted here as powerpoint file.
  The final quiz will be handed out Tuesday, 5 June at 3.00 pm at OSB 310, to be returned Wednesday 6 June at 10.00 am. It is open book, and will consist of problems and discussion questions typical of the material on Rossby waves, potential vorticity and general circulation.

  A copy of the final quiz from a previous year (2004) is here. Also another quiz from that same year here. Neither quiz is quite appropriate for this year's class in the sense that our emphasis has been more on theory and observations of oceanic Rossby waves and general circulation (particularly the Reid and Keffer papers for observations, the Stommel-Arons-Faller, Hallberg et al. and Vallis and class note handouts for PV and modeling). The 2004 first quiz asks you to derive something we in fact derived in class, to consider maps of Ekman transport which you did as homework, and refers to a standing Rossby wave problem that we did not derive in class...but, at least they show you the scope of typical quizzes.

  
  
  • Week 7/8:

    Bill Schmitz' conveyor belts: click image
    

    Notes on PV in the Pacific, and the depth penetration of the wind-driven gyre circulations here.
    

    An exercise in thermal wind and PV dynamics: Stratified f-plane flow over bumps and valleys...deep level in black contours, upper level in green contours, topographic contours in green/blue. Note that the horizontal velocity vector rotates with depth, following the topography down deep but going more directly over it at upper levels. This 'velocity spiral' seen when you plot points (u,v) for all depths on one plot, tells us the vertical velocity induced by the topography.
    

    Stommel's 'cooling spiral' (from Proc Nat Acad Sci, 1979 paper on the warming of western Europe by the ocean circulation). Here the (u,v) plot spirals due to cooling of the water by the atmosphere in winter.

    Reading:

    • Stommel, Arons & Faller, Some examples of stationary flow patterns in bounded basins, Tellus, 1958 (see Collected Papers, H.Stommel, Hogg & Huang Eds., Amer. Met. Soc. 1995)
    • Stommel, The abyssal circulation, Deep-Sea Res. 1958.
    • Hallberg, R. and P.B.Rhines, 1996, Buoyancy driven circulations in an ocean basin with isopycnals intersecting the sloping boundary, J.Phys.Oceanogr.
    • Lecture notes on MOCs

    The previous week's papers describe aspects of the wind-driven circulation particularly, both from a GFD standpoint and from careful compilation of hydrographic and tracer observations at sea. This week we are looking at the merdional overturning circulation (MOC), which interacts with the wind-driven gyres yet we can introduce some ideas imagining an ocean driven only by buoyancy forcing...hot, cold, evaporation, precipitation plus cryosphere (ice-) effects. The Stommel, Arons, Faller model looks at the GFD of the ocean beneath the thermocline, imposing a uniform upward velocity and a small region of source water input. This use of the generalized Sverdrup equation (which we have identified with a particularly simple, almost geometric rule about rotationally induced stiffness of the fluid) led Stommel and collaborators to a complete, world-wide scheme of deep ocean circulation. The missing element, which still plagues us today, is the missing mixing with is required to allow water to warm up and rise back to the ocean surface...the 'dark mixing' problem, in analogy with modern cosmology's 'dark energy' and 'dark matter'.

    We are standing back and looking at the global climate system as a 'heat and fresh-water engine', driven by solar radiation, with exchange of fresh water between air and sea, and poleward transport of thermal energy and fresh water required by the vertical exchange and radiation processes.
    

  • Week 6:
    Breathing biosphere...wmv file
    Breathing biosphere...mpg file (too fast..slow it down)

    Reading:

    • Joe Reid paper: On the total geostrophic circulation of the Pacific Ocean: flow patterns, tracers, and transports, Prog.Oceanography 1997 (available as e-journal online).
    • Keffer, The ventilation of the world's oceans: maps of the potential vorticity field. J.Phys.Oceanogr. 1985.
    • Holland, Rhines & Keffer, Dynamics of the oceanic general circulation: the potential vorticity field. Nature, 1984.
    • Vallis, G., excerpt on deriving the 3-dimensional PV equation for mesoscale motion, from his new textbook, Atmospheric and Oceanic Fluid Dynamics, Cambridge, 2006.
    
    
    These papers will lead us to a discussion of the general circulation from two complemetary viewpoints: hydrography and PV dynamics. Included in hydrography are nutrients, trace chemicals, thermal energy and fresh-water. Thus we interface with thermodynamics as well as wind-stress dynamics, and with biological cycles and ecosystem distributions.

    One goal will be to compare and contrast the Pacific and Atlantic oceans. There is much similar and much different between them. They play very different roles in the global overturning circulation and in global climate.

    This week's dynamics includes the development of potential density and improved QG PV equation, discussion of the 'dynamical thickness' term in 1, 1 1/2, 2 layers and with continous stratification, the non-dispersive baroclinic Rossby wave case L >> &lambda (&lambda is the Rossby radius, Co/f).

    The group project, running baroclinic Rossby wave/eddy simulations with HIM can develop with tracer and PV fields, following single eddies and clusters of eddies at a variety of amplitudes. A simple barotropic example of this kind of run can be see at Prof. Hakim's website for the 'GFD2' of Atmospheric Sciences: here.

  
  
• Week 5:

  Derive stratified QG PV equation, look at barotropic f/h contours of the ocean. Topographic/planetary Sverdrup equation. Free flow pathways. The time-dependent spin-up of the wind-driven circulation from rest, in the 1 1/2 layer model. Comparison with Anderson & Gill : the Rossby wave establishment of the Sverdrup flow, plus the western boundary layer development. The PV crisis in the one-layer barotropic wind-driven circulation model (which is very visible in numerical simulations of the barotropic spin-up).
  

• Week 4:
  • Look at the global ocean surface altimetry animation: click here.
  • Term Project discussion, 23iv07
  • QuikSCAT wind stress data. full URL from A.R..
  • Chelton,Schlax,Freilich Rossby wave altimetry paper. Color figures: 1, 2, 3.
    
  • surface density data. I suggested that it would be interesting to look at the Ekman flux across 'outcrops', the curves of constant density at the ocean surface. World Ocean Atlas data is readily obtained at http://www.nodc.noaa.gov/cgi-bin/OC5/WOA05/woa05.pl
    A map of (Ekman flux) dot grad (surface density) would be very interesting, using wintertime data.
    
  • wind stress data for PS2: I have put Matlab data files .mat for monthly mean QuikSCAT data from IFREMER in the directory on tsunami:
    /www/htdocs/courses/oc513/quikscat
    See if you can access it. The MODIS (Aqua, Terra satellites website of scatterometer winds (as well SST...) is also very interesting with good browsing. Their HDF data files can be read into Matlab, and I will work on this. For example: podaac at NASA.

    Part of the goal of this assignment is for you to look at the space time structure surface winds. You can do this with weather center data, derived from sparse balloon radiosondes plus satellite radiation sensing (that gives weighted integrals of the vertical temperature profiles in the atmosphere). Or, you can over the ocean look at scatterometer winds as described in class: radar backscatter from capillary ruffles on the sea surface which miraculously can be interpreted as vector wind speed near the sea surface. Here are some links. First browse images of daily winds at the Ifremer France site (the French are very good with satellite data): www.ifremer.fr/cersat/en. Choose the 'windstress' variable and click through the images as if it were a movie. This QuikSCAT data began in 1999 and continues to the present (though there is danger that NASA will not launch a replacement satellite to continue).The data is 50 km resolution which is much better than the weather center data. It is updated to about 2 weeks ago.

    Next switch from daily to monthly resolution. Think about the timescale of topographic Rossby waves and baroclinic Rossby waves as you look the time content of the windstress. Look at other variables: 'zonal wind stress' also, as this a big contributor to wind-stress curl. You will see how the storm tracks in N Atlantic and Pacific affect the stress, and how different the Southern Ocean is. Windstress curl is included, so you can directly see its pattern structure and visualize the red upwelling and blue downward Ekman pumping regions.

    The Ifremer 0.5 degree data can be downloaded from http://www.ifremer.fr/cersat/en/data/download/download.htm#gridded in ascii or NetCDF format. NetCDF can be read into Matlab; normally I have had to install the netcdf toolbox to do this but I'm not sure if that is still necessary...does anyone know if Matlab commands like cdfread are native commands, or part of the Toolbox?

    Getting the data. This can be done directly from the IFREMER website

    As soon as I get a good set of QuikSCAT vector stress data I will put in on this site, as .mat Matlab data files.

    The weather center version of wind stress, on a much coarser grid, can be found at www.cdc.noaa.gov, which is the Boulder, Colorado climate group of NOAA. They are good because there are smart people who can answer questions.
    
    

  • Lecture on time-development of the wind-driven circulation: basic barotropic models.
  • Lecture on wind-driven circulation: basic barotropic model.
  • Lecture notes on water masses, f/h contours, some circulation basics (like topographic-Sverdrup circulation driven by wind).
  
  
  
• Week 2/3: Lecture on Rossby waves (called chapter 14) in .pdf form, including color figures.
  
  • Math review notes: these are from a previous year's GFD-1 class, but you might find them helpful.
    
    
  • Problem Set 1 here.
    
  • Download syllabus for 2007.
    
    
    
    

  Stream function and pressure for 2d flow round a circular cylinder with and without rotation (the streamlines are unaffected by rotation in 2D if the boundary conditions are unaffected by rotation). The streamfunction for the flow corresponds to a constant u-velocity plus the field of a dipole source-sink pair which 'blow out' the streamlines and mimick a cylinder. Using polar coordinates (r,ϑ),

Rossby wave Green function viewed from southwest

Eastward flow past a cylindrical mountain: semicircular lee-Rossby wavecrests

Potential vorticity (PV) of a single layer of fluid, (f+ζ)/h, is of interest for the barotropic mode, involving the depth-averaged velocity. This is seen in the f/h contours ('geostrophic contours' or 'isostrophes') in the exercise described a few figures below here.

The Ertel-Rossby PV describes the internal, baroclinic dynamics of a stratifed fluid, and is proportional to (f+ζ)/h where now h is the vertical thickness of layers between surfaces of constant potential density. In absence of forcing effects or dissipation or time-dependence (on the isopycnal layer of interest) the fluid circulation should follow constant-PV contours. We set out to make a 'world atlas' of PV maps when this idea came clear (it was motivated by a theoretical solution for the general circulation, Rhines and Young J. Fluid Mech. 1982; J. Marine Res. 1982). The first maps were in McDowell and Rhines, JPO 1982. A 'world Atlas' is Keffer's paper from JPO 1985, here.

For the North Pacific see also Talley (JPO 1988). Other papers fill out the many sources of PV in the ocean circulation, including western boundary currents injecting PV into the interior ocean (Hallberg and Rhines 2000 in Deveopments in Geophysical Turbulence, Kerr and Kimura Eds., Kluwer), mixing at boundaries and injection from the mixed layer (Williams, JPO 1991), and abyssal flows into deep basins (Rossenov, Willimas and O'Dwyer JPO 2002). In subtropical gyres Ekman pumping from the upper-ocean mixed layer directly injects mixed layer fluid into the deep geostrophic flow, the fluid then circumnavigating the subtropical gyre (Luyten, Pedlosky and Stommel, JPO 1983).

For a complete set of figures for the western and central sections click here.

30 iii 2004: f/h contour exercise

A global topographic data set is available as a Matlab file here. This is a binary file that should load into Matlab directly ("load etopo20"). [To download the file use the 'save target as' option in Netscape or something similar in Explorer.]

The variables are latt, lonn and hh (latitude in degrees, longitude, height in meters relative to mean sea level). This is about 20 nautical mile (20 minutes of arc) resolution...37.6 km resolution. The original file has 10 times better resolution..3.76 km but it is a bit big to post here. Try exploring the world with plot statements (plotting the topography along latitude circles, for example) and look at the character and height of the Rocky Mountains, the Himalayan Plateau, Antarctica, the Andes Mountains, the deep ocean basins, the shape of Greenlands ice cap...just as line plots. Then try contouring or using pcolor(lonn,latt,ha), shading interp to give a shaded color map. Matlab has other graphical devices like 'mesh' to try.

The f/h contours can then be constructed (taking care to avoid zeros in the denominator. h is the thickness, so set it h = -hh in the ocean and to h = (8000 - hh) in the atmosphere: this uses the 8000m scale height of the atmosphere to set the upper surface of our fluid. Then contour f/h and look at its geography. Then mask out the regions where the thickness hh is very small.

 

Above (click on figure to enlarge) f/h geostrophic contours centered on the Pacific. Note the mainly closed islands of f/h for the major mountain massifs, Himalayan Plateau, Rockies and Greenland. The Pacific Ocean shows f/h running east-west, yet the gradual slope of the basin pinches them together in the eastern Pacific. Lots of small islands of f/h exist, and f/h contours always bend toward the Equator as they rise up continental slopes.

Above, the North Atlantic and Greenland f/h. Over the Mid-Atlantic Ridge the geostrophic contours warp southward, yet general f/h increases poleward: both from f increasing and from the general shoaling of the depth.

Above, the topography of Greenland viewed from the west: it is 3000m high with sharp edges north and south; it is the only high topography in the far north. At left is the Arctic Ocean basin, and in the right foreground are the lower mountains of Baffin Island.


Above, directly measured currents at mid-depth in the ocean. This level (1750m) is below the strongest currents of the Gulf Stream and wind-driven gyres yet the Antarctic Circumpolar Current is still very strong.

Synoptic scale (mesoscale) eddies in the boundary current along the east coast of Australia, as seen by color variations associated with biological activity (Seawifs satellite, false color rendering), 22 Nov 1997). Many important branches of the general circulation are obscured from view by eddies that dominate the kinetic-energy spectrum: in both atmosphere and ocean "climate/general circulation is what you expect but weather is what you get".