| dc.description.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). | en_GB |
