Lagrangian instability of barotropic Rossby waves

The Lagrangian laws for barotropic conservation of volume and barotropic conservation of the vertical component of the Cauchy vector lead to the theorem for perturbation kinetic energy and perturbation enstrophy derived by Fjørtoft [On the changes in the spectral distribution of kinetic energy for twodimensional, nondivergent flow. Tellus 1953, 5] from the Eulerian laws for conservation of volume and the vertical component of vorticity. The Lagrangian laws, like the Eulerian laws, admit an exact solution in the form of a monochromatic Rossby wave. Linearising the Lagrangian laws about the Rossby wave by the inclusion of small disturbances in the particle paths leads to single equation which differs significantly from the Eulerian linearised vorticity equation derived by Gill [The stability of planetary waves on an infinite beta-plane. Geophysic. Astrophys. Fluid Dyn. 1974, 6, 29–47.]. However, Floquet series expansions of both equations yield almost identical Floquet matrices. At the maximal truncation to a triplet state the resulting Floquet equations yields identical stability criteria and growth rates for the Floquet waves, both in the strong interaction limit and in the weak interaction limit.