Lagrangian Derivatives in Chaotic Flows: Asymptotic Behaviour and a Numerical Method

Lagrangian coordinates describe a local frame moving and deforming with a flow, where by flow we mean either a physical flow or the solution of a smooth dynamical system. The rôle of chaos in these coordinates is reflected by a mean exponential stretching of fluid elements along characteristic directions. Many partial differential equations, such as the kinematic advection–diffusion equation, are greatly simplified by a change to Lagrangian coordinates. To evaluate the terms in such equations, it is necessary to obtain the form of the Lagrangian derivatives of the frame. These derivatives are related to generalised Lyapunov exponents, which describe deformations of fluid elements beyond ellipsoidal. When the flow is chaotic, care must be taken in evaluating the derivatives because of the extreme separation of scale along the different characteristic directions. We show this can be accomplished by projection onto orthonormal frames. Two matrix decomposition methods are used to accomplish the projections, the first appropriate in finding the asymptotic behaviour of the derivatives analytically, the second better suited to numerical evaluation. As an illustration, we show that for the kinematic dynamo problem the induced current becomes perpendicular to the magnetic field. PACS numbers: 05.45.-a, 47.52.+j Submitted to: Nonlinearity