Generating 3D Magnetic Fields of Constant Magnitude for Solar Wind Applications

Large-amplitude Alfvén waves can still be exact solutions to the Magnetohydrodynamic (MHD) equations if the wave is circularly, arc, or spherically polarized, thus keeping the total magnitude of the magnetic field, |B|, constant. Even at large amplitude, the strength of the total magnetic field remains nearly constant for Alfvénic fluctuations in the solar wind, (|B|/B_0 << 1) implying that the magnetic field is arc polarized. Unfortunately, simulating a fully 3D magnetic field that both preserves |B| = const and the divergence of B is 0 is nontrivial. Analytically, it seems impossible, and only a few studies have used numerical methods to study Alfvénic turbulence. Here, we discuss a new numerical algorithm for generating a 3D magnetic field that satisfies these conditions and its potential applications for understanding how Alfvén waves evolve in the solar wind. The algorithm starts with a general form of B that ensures a constant magnitude, and then imposes the divergence of B condition and, as long as appropriate initial conditions are chosen, numerically solves the field for all space in a 3D box. This numerical method can successfully find B configurations where the divergence is much smaller than the spatial resolution of the box.