FAQ-Kinematics

Consider the three flow fields that vary as a function of time t and location (x,y) given by

u = (D/2)x —> x = X exp(Dt/2)
v = (D/2)y —> y=Yexp(Dt/2)

u = λx —> x = Xexp(λt)
v = λy —> y = Yexp(-λt)

u = Ωy —> x = Xcos(Ωt)+ΩYsin(Ωt)
v = -Ωx —> y = -ΩXsin(Ωt)+Ycos(Ωt)

1. Calculate the vorticity, divergence, and deformation (rate) for these flows.
2. Pick values for D, λ, and Ω typical of oceanic conditions and plot the evolution of the unit square under these three flows. Recall that the unit square has vertices at (1,1), (-1,1), (-1,-1), and (1,-1).
3. Graph the area and perimeter of the unit square for these three flows as a function of time. Extra credit may be available if you show the analytic formulae.
4. If you got the answers above correctly then you have learned a sophisticated perspective on the unique physical attributes of D, λ, and Ω. Write a short discussion of their implications for non-advective flux between the unit square and its environment.
5. These flow representations can be inserted into the equations of motion to develop diagnostic relations for the pressure gradient, divided by a reference density. Graph the pressure fields responsible for each of these flows.
6. Lagrangian observations of the hydrosphere are now routine. A simple kinematic model of eddies is a translating vortex given by:

u+Ωy=-U —> x= -Ut+Xcos(Ωt)-Ysin(Ωt)
-Ωx+v=UΩt —> y= Xsin(Ωt)+Ycos(Ωt)

Taking typical values from the literature for the size, translation velocity U, and vorticity Ω of an open ocean anticyclone translating to the west, as they typically do in the northern hemisphere, plot trajectories for three drifters deployed on the north-south radius of an anticyclone. That is, the initial north-south positions for the three drifters are X = 0. For the east-west positions, take ΩY < U, ΩY = U, ΩY > U.
7. The first and third trajectories plotted in 6 are quite different, although they are in the same anticyclone. This difference has caused some confusion in the literature about interpretations of Lagrangian data. The wavy trajectory could be interpreted as arising from a drifter being advected by a meandering jet, such as the Gulf Stream. What additional information would you need to decide whether that type of trajectory was from a drifter in an anticyclonic ring or a meandering jet?

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