The Physics of Twisted Magnetic Tubes Rising in a Stratified Medium: Two-dimensional Results

The physics of a twisted magnetic flux tube rising in a stratified medium is studied using a numerical magnetohydrodynamic (MHD) code. The problem considered is fully compressible (has no Boussinesq approximation), includes ohmic resistivity, and is two-dimensional, i.e., there is no variation of the variables in the direction of the tube axis. We study a high-plasma β-case with a small ratio of radius to external pressure scale height. The results obtained will therefore be of relevance to understanding the transport of magnetic flux across the solar convection zone.

We confirm that a sufficient twist of the field lines around the tube axis can suppress the conversion of the tube into two vortex rolls. For a tube with a relative density deficit on the order of 1/β (the classical Parker buoyancy) and a radius smaller than the pressure scale height (R²≪H²_p), the minimum amount of twist necessary corresponds to an average pitch angle on the order of sin^-1 [(R/H_p)^1/2]. The evolution of a tube with this degree of twist is studied in detail, including the initial transient phase, the internal torsional oscillations, and the asymptotic, quasi-stationary phase. During the initial phase, the outermost, weakly magnetized layers of the tube are torn off its main body and endowed with vorticity. They yield a trailing magnetized wake with two vortex rolls. The fraction of the total magnetic flux that is brought to the wake is a function of the initial degree of twist. In the weakly twisted case, most of the initial tube is turned into vortex rolls. With a strong initial twist, the tube rises with only a small deformation and no substantial loss of magnetic flux. The formation of the wake and the loss of flux from the main body of the tube are basically complete after the initial transient phase.

A sharp interface between the tube interior and the external flows is formed at the tube front and sides; this area has the characteristic features of a magnetic boundary layer. Its structure is determined as an equilibrium between ohmic diffusion and field advection through the external flows. It is the site of vorticity generation via the magnetic field during the whole tube evolution.

From the hydrodynamical point of view, this problem constitutes an intermediate case between the rise of air bubbles in water and the motion of a rigid cylinder in an external medium. As with bubbles, the tube is deformable and the outcome of the experiment (the shape of the rising object and the wake) depends on the value of the Weber number. Several structural features obtained in the present simulation are also observed in rising air bubbles, such as a central tail, and a skirt enveloping the wake. As in rigid cylinders, the boundary layer satisfies a no-slip condition (provided for in the tube by the magnetic field), and secondary rolls are formed at the lateral edges of the moving object.