Three-dimensional magnetohydrodynamic turbulence in cylindrical geometry

Abstract

Ideal magnetohydrodynamic turbulence is treated using more realistic boundary conditions than rectangular periodic boundary conditions. The dynamical equations of incompressible magnetohydrodynamics and the associated fields are expanded in a set of vector eigenfunctions of the curl. The individual eigenfunctions represent force-free fields, but superpositions of them do not. Three integral invariants have simple quadratic expressions in the expansion coefficients: the total energy, the magnetic helicity, and the cross helicity. The invariants remain temporally constant in the face of a truncation at a large but finite number of coefficients. Boundary conditions imposed are those for a rigid, perfectly-conducting cylindrical boundary, with an arbitrary periodicity length parallel to the axis. Canonical distributions are constructed from the invariants. Mean-square turbulent velocity fields 〈v2〉 have finite values for virtually all initial conditions, including quiescient ones. The stability problem can be reformulated as a search for values of the integral invariants which will minimize 〈v2〉. This leads to a principle of extremal helicity, which requires a magnetic configuration which will minimize 〈v2〉 for a given total energy. The development of helical macroscopic structures in the cylinder as a function of increasing ratio of axial current to axial magnetic flux is predicted.

Power spectral signatures of interplanetary corotating and transient flows

“…This has been hypothesized to be a direct consequence of the conservation of the three invariants of homogeneous incompressible MHD (total energy, magnetic helicity and cross helicity) in forced dissipative situations by Frisch et al (1975) and in selective decay situations by Montgomery et al (1978) and Matthaeus and Montgomery (1980).…”

“…At lower frequencies, the spectra apparently break to a flatter slope of -1. Although a slope of -1 is expected to characterize fully developed (isotropic) MHD turbulence at scales longer than the correlation length (Frisch et al, 1975;Montgomery et al, 1978;Matthaeus and Montgomery, 1981), it is unlikely that interplanetary turbulence is the origin of this feature in these spectra. One reason is that the Cl spectrum is not accompanied by a significant amount of magnetic helicity, nor is the helicity correlation length large.…”

Observation of a Relaxed Plasma State in a Quasi-Infinite Cylinder

“…Taylor stated that provided a finite resistivity, magnetic reconnection approximately conserves global helicity and provides for the dissipation mechanism for reducing magnetic energy. [1,3] The selective decay hypothesis posits the energy selectively decays relative to magnetic helicity because the energy spectra peaks at higher wavenumbers, where dissipation is higher [4,5]. In the laboratory, experiments which rely on plasma relaxation are often confined with the help of flux conserving boundaries.…”

Efficiency and scaling of current drive and refuelling by spheromak injection into a tokamak

“…Frisch et al [14] have suggested that threedimensional turbulent MHD flows can lead to an inverse cascade of magnetic helicity generating large scale magnetic structure. Montgomery et al [15] have discussed in detail the role of the ratio of helicity to energy, K/W, in the relaxation of turbulent flows.…”