Lagrangian Barotropic and Baroclinic Instability

Abstract The standard geophysical fluid dynamics of barotropic and baroclinic instability is rederived here in a Lagrangian representation. The labels are the initial positions of a fluid particle using either in situ density coordinates or potential temperature coordinates. In all cases, small disturbances obey the Rayleigh stability equation with the Lagrangian coordinates simply replacing the Eulerian coordinates. The results are not trivial, owing to the Lagrangian representation of the gradients of the Montgomery potentials. If the mean vertical shear in a baroclinic flow is uniform, then the Eady dispersion relation obtains. The geometrical relationship between the mean shear and the vertical phase tilt for growing disturbances is the same in the label space as in the Eulerian representation. Significance Statement Advances in the technology of autonomous underwater vehicles and inertial navigation are capable of revolutionizing the collection of ocean data by an observer following the currents. The fluid dynamics which generates the dominant current patterns is reconstructed here in a mathematical representation also following the currents. This new formulation enables the application of fluid dynamics to the analysis of future ocean data.