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Other physicists had already made the mathematical extension of Kelvin's theorem to compressible fluids, but it was not until Vilhelm Bjerknes' landmark 1898 paper that meteorology and oceanography began to adopt this insight.
This essay is primarily about the Bjerknes circulation theorem (Bjerknes 1898). This theorem, formulated by Vilhelm Bjerknes, was published in a paper of 1898, in the Proceedings of the Royal Swedish Academy of Sciences in Stockholm. The paper is in German, being one of the primary languages of scientific publications of that era. The title of the paper is "On a Fundamental Theorem of Hydrodynamics and Its Applications Particularly to the Mechanics of the Atmosphere and the World's Oceans." It has 35 pages, 36 sections, and 14 figures. The significance of this paper is twofold. It provided a key insight into the way circulation develops in geophysical fluids. But it also marked the beginning of the transition of Vilhelm Bjerknes' research from the field of electrostatics, electromagnetic theory, and pure hydrodynamics into that of atmospheric physics.
Prior to the introduction of this theorem the thinking on rotation in a fluid followed two distinct lines. The quantity we now call vorticity (curl of the velocity vector) had been the subject of fundamental studies by H. Helmholtz, although its recognition as a fluid property predates these studies by at least 80 years. In Helmholtz (1858) equations for the rate of change of vorticity had been derived for a homogeneous inviscid fluid. Helmholtz thereby deduced that for a constant density fluid a material line element (a "vortex filament") aligned with the vorticity vector will always remain so aligned. In essence if no vorticity currently exists, then none could be generated by conservative forces.
Lord Kelvin, in a paper in 1867 (Thomson 1867), had approached the problem of rotation in a different way by defining a quantity called circulation given by the following expression:
where the integral is around any closed curve in the fluid and dl is a line element vector pointing along the curve. Kelvin's theorem states that the circulation around a material circuit is constant for a homogeneous inviscid fluid. A material circuit is one that always consists of the same fluid parcels.
By using Stokes's theorem one can easily derive the...