Jacobi Equation, Riemannian Curvature and the Motion of a Perfect Incompressible Fluid

Following some Arnol’d results relative to the geometry underlying the dynamics of a perfect incompressible fluid [3] (geodesics of left-invariant metrics on Lie groups), the linear differential equation relative to a Lagrangian stability analysis are established. This differential equation, called Jacobi equation, describes, for the same fluid element, the time evolution of the difference between the trajectory starting from reference initial position in a reference flow and the trajectory starting from a perturbed initial position in a perturbed flow. Links with the linear differential equation relative to the classical Eulerian stability analysis are given. The stability of the solution of the Jacobi equation can be investigated through the sign of the Riemannian curvature. we prove here that, for all flows, with the exception of the perfect eddy with constant vorticity (corresponding to a stationnary rotation around a fixed axis), there always exists small perturbations with strictly negative curvature. If one assumes that negative curvature implies instability with exponential divergence of the geodesic flow, the above result proves effectively the instability, from a Lagrangian viewpoint, of all the solutions of the Euler equation, with the exception of the perfect eddy. An estimation of the most negative part of the curvature provides an interesting interpretation of the Kolmogorov timescale √ ν/ε as the smallest time-constant relative to exponential divergence of fluid elements that are initially close. Running title: Jacobi equation and the motion of an incompressible fluid. ∗École des Mines de Paris, Centre Automatique et Systèmes, 60, Bd. Saint-Michel, 75006 Paris, FRANCE. Tel: (33) (1) 40 51 91 15. Fax: (33) (1) 43 54 18 93. Email: [email protected] .