Abstract
The Rossby adjustment of an initially circular column of water, the so-called collapse of a cylinder, continues to be a widely used method for forming lenslike eddies in the laboratory. Here, we consider the structure of an eddy so formed as well as some ramifications of that formation. We demonstrate that the calculation of the eddy structure can be reduced to the extraction of the roots of two nonlinear, coupled algebraic equations. Analytical solutions in the limit of the collapse of a needle are given and roots are obtained numerically otherwise. It is concluded that in the collapse of a cylinder initially spanning the entire column of water, the eddy always maintains contact with both surfaces. (This is not the case in the seemingly equivalent two-dimensional case with no variation in one Cartesian direction.) In the event the initial cold column is separated only slightly from the surface, the above solution acts as the lowest order solution in a regular perturbation sequence.
Next, these “collapse eddy” solutions, which possess motions in both layers and finite energies, are used to examine lens merger. Two collapse eddies of equal volume jointly possess less energy than one collapse eddy of twice the volume. However, we argue that two collapse eddies of equal volume can have more energy than the circularly symmetric end-state eddy formed from them if the two initial eddies “mix.” We also offer evidence that the energy budgets may be balanced exactly if the end-state eddy is slightly asymmetric. Comparisons with some previous laboratory experiments are made.
