Alfvén simple waves: Euler potentials and magnetic helicity

Abstract

The magnetic helicity characteristics of fully nonlinear, multi-dimensional Alfvén simple waves are investigated, by using relative helicity formulae and also by using an approach involving poloidal and toroidal decomposition of the magnetic field and magnetic vector potential. Different methods to calculate the magnetic vector potential are used, including the homotopy and Biot-Savart formulae. Two basic Alfvén modes are identified: (1) the plane one-dimensional Alfvén simple wave given in standard texts, in which the Alfvén wave propagates along the z-axis with wave phase φ = k0(z - λt), where k0 is the wave number and λ is the group velocity of the wave and (2) the generalized Barnes simple Alfvén wave in which the wave normal n moves in a circle in the xy-plane perpendicular to the mean field, which is directed along the z-axis. The plane Alfvén wave (1) is analogous to the slab Alfvén mode and the generalized Barnes solution (2) is analogous to the twodimensional mode in Alfvénic, incompressible turbulence. The helicity characteristics of these two basic Alfvén modes are distinct. The helicity characteristics of more general multi-dimensional simple Alfvén waves are also investigated. Applications to nonlinear Alfvénic fluctuations and structures observed in the solar wind are discussed. © 2010 The American Astronomical Society. All rights reserved.