GFD-1 LAB GEOSTROPHIC ADJUSTMENT AND GEOSTROPHIC FLOW - I
(images from 26 Jan 2001 lab)




A rapidly spinning, dense fluid has a lower density (orange) fluid poured onto it, at the center. This forms a buoyant lens whose lower interface is sloping relative to equipotential surfaces (the equipotentials are identical paraboloids, like the free surface when the fluid is totally spun up). {Note that the free surface of the fluid is just above the colored region. This is a strong optical lens, making secondary images above.} This symmetric eddy will continue to spin, relative to the underlying fluid, for many hours, until its available potential energy is used up. It is remarkable that its kinetic energy can live for so long, given the small amount of APE. When it is spun down, the orange layer thickness will be uniform (measured along the rotation axis). Its center of mass will have moved downward by a few mm. This is happening in the distorted coordinate system of these equipotential surfaces; ironically, to reduce its potential energy much of the orange buoyant fluid will have to climb up (or be 'flung up') the paraboloid!

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Thermal Wind Balance

Shown above is a cylindrical annulus of fluid, made by placing a small cylinder inside a larger one. There is ice inside the inner cylinder in this slowly rotating experiment (roughly 1.5 radian/sec. or 4 sec. period). It causes the water near the inner wall to cool and sink. This drives outflow along the bottom, and a compensating inflow above. The Coriolis force on this radial (or, 'meridional' if this were the Earth) velocity causes a leftward zonal velocity at depth, and rightward (cyclonic) zonal velocity at the surface. The roughly linear profile of zonal velocity is evident in the dye traces, which were injected as vertical lines. Actually it is not symmetrical above and below, because of frictional drag at the bottom (with little at the free surface above). Thus there is a bias toward cyclonic 'eastward' or rightward flow, which is stronger than the 'westward' deep flow. The later image shows that the zonal flow and meridionala flow have some instability, distorting the simple tipping of the dye lines..but of course the combination of meridional overturning and zonal sheared flow will themselves make a complex Lagrangian flow pattern.

This nearly symmetric flow resembles the linear shear described by the basic 'slumping due to gravity' of a horizontally stratified fluid, the short mathematical solution given in lectures.

We did not make precise measurements, but a radial temperature difference over the 5 cm gap was about 2C, and vertical temperature difference over the 10 cm layer was about 6C. Temperature ranged between 5C and 12C. Thus the isotherms were sloping upward/inward with a slope of roughly 1 in 3. According to the thermal wind equation, f du/dz = (g/ro)alpha dT/dr; what du/dz is predicted, and what did we see? Alpha is tabulated in Gill, Appendix 3.

This symmetric flow varies only in the vertical and radial ('meridional', north-south) directions, and not in the azimuthal ('zonal', east-west) direction. If we make the gap containing fluid wider, or rotate faster, or cool the fluid less intensely the flow becomes fully 3-dimensional: waves, eddies and jet-streams form which vary in all 3 directions. The 'annulus experiment', very like this, was the first concrete demonstration of the fundamental dynamics of baroclinic instability, synoptic scale eddies and jet streams in the atmosphere or Southern Ocean. It was pioneered by Dave Fultz, Univ. of Chicago, Raymond Hide of MIT, and earlier by C.G. Rossby at Chicago. You can see the view from above of this annulus experiment animated on the mast-head for this web-page.
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View from above of a 2-layer geostrophic adjustment experiment in a channel.
Geostrophic adjustment of a thin layer of green fresh water (5 cm deep), riding atop a deep layer (10 cm deep) of clear (undyed) 0.5% salinity water; view from above. Rotation of the table is 0.5 radians/sec. The channel is 100 cm long and 25 cm wide.In the green region, an extra thickness of upper layer, fresh water is added, so that the interface is lower and the surface higher than in the clear fluid. The green fluid is initially at rest in the rotating system; the barrier is raised and the hydrostatic imbalance causes the green fluid to 'slump' outward. First of all the slumping is arrested as the velocity is turned to the right, and runs along the green front ('downward' in the images). Then this jet hits the wall and acts as a source of a fast Kelvin wave which proceeds down the boundary at the lower edge of the images. It can be seen moving the blue dye lines all the way round the tank.

The important scale at this stage is the Rossby deformation radius, lambda=Co/f ~ 9 cm where Co ~ 4.5 cm/sec is the speed of long gravity waves ... internal waves because the free surface is almost unmoved by these internal flows. Note correction due to insightful student question, 6ii2001: Lambda is the length scale of the interface field after adjustment...the distance traveled by a long gravity wave in the 1/f adjustment time. There is another length scale connected to the amplitude of the flow which is set by the amount of green fluid added to the upper layer at t=0. This is the distance that fluid particles move down channel during the adjustment. This is seen here as the distance between the edge of the green fluid in image 2 and its initial position in image 1, before the barrier was lifted. For a nearly linear, small amplitude problem this particle displacement (simply equal to v/f where v is the cross channel geostrophic velocity after adjustment) is ~ (delta eta/H)lambda where delta eta is the change in interface height due to the added green fluid, and H is the upper layer's mean thickness...See review problem #2. Indeed, that distance is significantly less than 9 cm. (9 cm is a bit more than 1/3 of the channel width).

Lamda also should be the e-folding width of the Kelvin wave (the distance from the wall occupied by the wave). This looks about right. Tracking the wave right round the channel we find its speed to be 4.0 cm/sec, compared with the predicted 4.5. The first 3 images contain all the information you will want to think about, based on our linear discussion of geostrophic adjustment. All the rest of the evolution shows how nonliner, nearly geostrophic flows continue to have interesting life cycles.

Following on more slowly behind the Kelvin wave is a jet of green fluid itself, a current following a path set up by a wave. By the 5th image (about 30 sec.) the Kelvin wave has circulated round the entire channel and bumps into the green fluid again. It sets up a cyclonic gyre thta erodes the green front. By this time essentially everything in view is nearly geostrophic.
The overall anticyclonic (clockwise) vorticity of the green fluid left behind is caused by the 'slumping' and reduction in layer thickness, roughly conserving potential vorticity (giving the SAME result as the argument about Coriolis force acting on the horizontally diverging, outflowing fluid). A cyclonic green eddy also appears from instability of the front, and the boundary current itself is unstable, eventually filling the channel with roundish eddies. This is an example of the strong tendency for geophysical flows to break into synoptic scale eddies.
This experiment proceeds through several stages: linear wave propagation and geostrophic adjustment, development of the green wall jet following the waves, nearly horizontal circulation with its own quasi-two-dimensional evolution and instability. The fluid is trying to come to rest, with upper layer of uniform thickness, but it takes a very long time to run down the geostrophic eddies and circulations.