Eulerian and Lagrangian means in rotating, magnetohydrodynamic flows II. Braginsky’s nearly axisymmetric dynamo

Abstract

The Hybrid Euler–Lagrange (HEL) approach has been usefully applied to weakly dissipative systems characterised by waves riding on mean flow. Soward (Phil. Trans. R. Soc. Lond. A 1972, 272, 431) showed how the HEL-formulation could elucidate remarkable features of the nearly axisymmetric large magnetic Reynolds number dynamo of Braginsky (JETP 1964, 47, 1084). Since Braginsky’s treatment of the nearly axisymmetric dynamo relies on azimuthal averages, those can only be taken when the azimuth is a coordinate direction. In that respect, the unified derivation and presentation of the HEL-equations governing rotating magnetohydrodynamic convective flows, as later reviewed and extended by Roberts and Soward (Geophys. Astrophys. Fluid Dyn. 2006, 100, 457), suffer the shortcoming that it was developed relative to rectangular Cartesian coordinates. Here we undertake those modifications needed to transform the rectangular Cartesian coordinate formulation into cylindrical polar coordinates. Being a Lagrangian description, application of the HEL-method means that the variables used, dependent on coordinates x, do not describe conditions at the position P: x but on conditions elsewhere at some displaced position PL: xL(x, t) = x + ξ (x, t), generally dependent on time t. To address this issue Soward and Roberts (J. Fluid Mech. 2010, 661, 45) invoked an idea pioneered by Moffatt (J. Fluid Mech. 1986, 166, 359), whereby the point x is dragged to xL(x, t) by a “fictitious steady flow” η(x, t) in a unit of “fictitious time”. This is the “Lie dragging” technique of general tensor calculus, which we apply here to the HEL-equations governing Braginsky’s nearly axisymmetric dynamo. We consider the “effective-variables” introduced by Braginsky, appropriate for small displacement ξ , and show that η, rather than ξ , is their natural expansion variable. As well as revisiting Braginsky’s kinematic dynamo, we reassess the hydromagnetic extensions of Tough and Roberts (Phys. Earth Planet. Inter. 1968, 1, 288).