Constructing a class of topological solitons in magnetohydrodynamics

We present a class of topological plasma configurations characterized by their toroidal and poloidal winding numbers, $n_t$ and $n_p$, respectively. The special case of $n_t=1$ and $n_p=1$ corresponds to the Kamchatnov-Hopf soliton, a magnetic field configuration everywhere tangent to the fibers of a Hopf fibration so that the field lines are circular, linked exactly once, and form the surfaces of nested tori. We show that for $n_t\in {\mathbb{Z}}^{+}$ and $n_p=1$, these configurations represent stable, localized solutions to the magnetohydrodynamic equations for an ideal incompressible fluid with infinite conductivity. Furthermore, we extend our stability analysis by considering a plasma with finite conductivity, and we estimate the soliton lifetime in such a medium as a function of the toroidal winding number.

Figure 1

One lobe of the field configuration for $n_p=1$ and $n_t=2$. (a) A single, closed core magnetic field line. (b) The core field line is surrounded by nested toroidal surfaces, shown in cross section. (c) A complete magnetic surface filled entirely by one field line.

Figure 2

Topological solitons in MHD with $n_p=1$ and (a) $n_t=2$, (b) $n_t=3$, and (c) $n_t=4$. A single magnetic field line fills out each of the linked, toroidal surfaces.

Figure 3

The magnetic surfaces for $n_t=1$ and (a) $n_p=2$, (b) $n_p=3$, and (c) $n_p=4$. Solutions with $n_p\ne 1$ have zero angular momentum and are therefore not stable solitons. The magnetic field lines in each lobe wind in opposite directions, represented by the red and blue surfaces.