GFD-1 LAB #5 EKMAN LAYERS and SPIN-UP
P.B. Rhines, E.G. Lindahl Feb. 9 2001

Because vertical motion is small near a horizontal boundary, the pressure field of the interior flow is felt right through the viscous boundary layer. If that interior flow is in geostrophic balance, the slower velocity in the boundary layer has a weaker Coriolis force, and hence cannot resist the pressure-gradient force. The result is a flow partially directed 'down the pressure gradient' in the boundary layer.

Fluid reaching the center rises in the tornado vortex to the outflowing sink. A single fluid parcel follows a helical spiral upward. Thus the ribbon of dye forms such a spiral, and you see a cross-section of it as a 'fir tree'. The limbs of the tree are quite good indicators of the orientation of the principal shear. Notice the two regions of opposing slope on the 'limbs'...what does this indicate?

A cylinder of homogeneous is spinning on a rotating table, cyclonically. In the first set of images we created a symmetrical swirling flow (zonal or azimuthal velocity) by sucking fluid out at the top, center, and injecting it back in at the outer wall of the cylinder. The radially inward flow from source to sink creates a cyclonic interior vortex: initially the inflow is more or less uniform in z, but then the lower boundary layer becomes active and begins to pump fluid inward, close to the bottom boundary (as seen by the inward spiral of the bottom dye streaks). This boundary layer 'secondary flow' or Ekman flux reduces the need for an interior inflow. In fact, inward flow in the interior cannot occur at all in the steady state (presuming rapid rotation..small Rossby number). For, if it did, the swirl velocity would be enormous, and Ekman suction would bring the inflow down into the boundary layer.

A simple inviscid calculation shows that without the Ekman layers the angular momentum (measured by a non-rotating observer) of inward moving rings of fluid would be conserved. If omega is the table rotation rate,
(r*u + omega*r^2) = C,
a constant following the ring, so that
u = C/r - omega*r
The zonal velocity is a combination of a solid body rotation plus a 1/r point vortex (which has zero vorticity outside of the tornado). C is determined by the injection process at the rim. The Ekman layers reduce this very strong swirl flow to a value ~ Q/r(2 pi delta) where Q is the volume flux of the sink, and delta is the Ekman layer thickness.

Time sequence of dye tracing out the inward/upward trajectory in a tornado vortex


Below you can see a fine dye line showing the vertical velocity/swirl structure for the entire tornado. Note the swarm of particles hovering near the sink, at top.


The North polar ice-cap of Mars: reminiscent of the swirling inflow pattern due to Ekman transport beneath a cyclonic vortex.



Meridional circulation driven by surface stress in a very viscous fluid

Here the fluid is rotating but with a 'lid' that is fixed in the lab frame of reference (it is a beaker visible in the image), hence exerting a strong anticyclonic stress on the fluid. The lid exerts a primarily azimuthal (zonal) stress, and creates a zonal velocity changes in the fluid, but what we see here is the radial/vertical or 'meridional' overturning circulation that helps to transfer the stress to the bottom of the fluid. Because of the high viscosity, the 'Ekman' layers pretty much fill the full depth. If you switch off the rotating table the flow pattern is perfectly captured...frozen...to look at.

The lower image occurs when some buoyancy is added to the top of the fluid so that the meridional circulation splits into two (actually three) cells. Some chaos is added by slightly breaking the symmetry, with the lid off center.

The radical vortex with a free surface here levitates two ping-pong balls. Note the cylindrical curtains of dye, where boundary layer suction is very active on the tops/bottoms of the two balls, and meridional recirculation is stimulated.



Stratified fluid causes major changes in spin-up. Here, below, we have an upper boundary that spins relative to the rest of the container. Ekman suction still is operating, but it must fight against heavy stratification. Initially just two layers of different density, the circulation driven by the upper rotating lid causes mixing at the interface.