Magnetic Structure in Steady Integrable Flows

Abstract

From the examples of Chaps. 2 and 3 we have seen that fast dynamo activity is accompanied by the emergence of intense, small-scale magnetic structure. The most direct connection of this structure to the geometry of the flow is through the stretching of vectors as measured by the Liapunov exponent of the flow. There are, however, other ways to produce these effects in flows having zero Liapunov exponent. In the present chapter we shall consider processes of this kind for several simple steady flow fields. These examples will share many features with a classical problem of fluid mechanics, namely the formation and structure of boundary layers in viscous fluid flows at large Reynolds number Re ≡ U L/v, with v the kinematic viscosity (see, e.g., Prandtl 1952, Batchelor 1967). Boundary-layer theory originated in Prandtl’s observations of the flow adjacent to a rigid wall where the fluid adheres. The flow is arrested in a thin layer of thickness O(Re^−1/2). Mathematically, the Prandtl boundary-layer theory has been recognized to be a singular perturbation of the inviscid or Euler limit (see, e.g., Van Dyke 1975). In such a perturbation theory, the ideal fluid with zero viscosity is analogous to our perfectly-conducting limit. In the thin boundary layer, viscous forces are in balance with inertial forces and the ideal or Euler equations are not valid limits of the Navier-Stokes equations. Solutions of the Navier-Stokes equations which are valid in the limit of large Reynolds number, uniformly over the flow domain, must in general contain such boundary-layer structures.