Comparative study of a constant-alpha force-free field and its approximations in an ideal toroid

¹ Astronomical Institute, Academy of Sciences of the Czech Republic, Boční II 1401, 141 00 Praha 4, Czech Republic
e-mail: vandas@asu.cas.cz
² Lone Star College, Houston, TX77088 Texas, USA
e-mail: romash7@gmail.com

Received: 2 April 2015
Accepted: 19 June 2015

Abstract

Aims. Magnetic clouds in the solar wind are large, loop-like interplanetary flux ropes and may be locally approximated by a toroidal flux rope. We compare approximate constant-alpha force-free fields in an ideal toroid, used in magnetic cloud analysis, with the exact solution, and examine their validity for low aspect ratios, which can be found in magnetic clouds. The approximate toroidal solutions were originally derived under the assumption of large aspect ratios.

Methods. Three analytic simple approximate constant-alpha force-free solutions and the exact analytic solution are compared with respect to magnetic field profiles, magnetic field magnitude distributions, and magnetic helicity, with moderate (2–3) and very low (<2) aspect ratios.

Results. The Miller & Turner (1981, Phys. Fluids, 24, 363) field and its modification (to satisfy exact solenoidality) match the position of the magnetic axis in the toroidal flux rope well even for very low aspect ratios. The same can be said for the modified field and the position of the magnetic field maximum. When calculating helicity of the toroidal flux rope, the Miller & Turner field yields better results. A simple formula for magnetic helicity derived from the Miller & Turner solution is valid with a good accuracy even for very low aspect ratios.

Conclusions. The Miller & Turner solution is a reasonable substitute for the exact solution even for low aspect ratios (≈2).