Alfvén waves at low magnetic Reynolds number: transitions between diffusion, dispersive Alfvén waves and nonlinear propagation

We seek the conditions in which Alfvén waves (AW) can be produced in laboratory-scale liquid metal experiments, i.e. at low magnetic Reynolds Number ( $Rm$ ). Alfvén waves are incompressible waves propagating along magnetic fields typically found in geophysical and astrophysical systems. Despite the high values of $Rm$ in these flows, AW can undergo high dissipation in thin regions, for example in the solar corona where anomalous heating occurs (Davila, Astrophys. J. , vol. 317, 1987, p. 514; Singh & Subramanian, Sol. Phys. , vol. 243, 2007, pp. 163–169). Understanding how AW dissipate energy and studying their nonlinear regime in controlled laboratory conditions may thus offer a convenient alternative to observations to understand these mechanisms at a fundamental level. Until now, however, only linear waves have been experimentally produced in liquid metals because of the large magnetic dissipation they undergo when $Rm\ll 1$ and the conditions of their existence at low $Rm$ are not understood. To address these questions, we force AW with an alternating electric current in a liquid metal in a transverse magnetic field. We provide the first mathematical derivation of a wave-bearing extension of the usual low- $Rm$ magnetohydrodynamics (MHD) approximation to identify two linear regimes: the purely diffusive regime exists when $N_{\omega }$ , the ratio of the oscillation period to the time scale of diffusive two-dimensionalisation by the Lorentz force, is small; the propagative regime is governed by the ratio of the forcing period to the AW propagation time scale, which we call the Jameson number $Ja$ after (Jameson, J. Fluid Mech. , vol. 19, issue 4, 1964, pp. 513–527). In this regime, AW are dissipative and dispersive as they propagate more slowly where transverse velocity gradients are higher. Both regimes are recovered in the FlowCube experiment (Pothérat & Klein, J. Fluid Mech. , vol. 761, 2014, pp. 168–205), in excellent agreement with the model up to $Ja \lesssim 0.85$ but near the $Ja=1$ resonance, high amplitude waves become clearly nonlinear. Hence, in electrically driving AW, we identified the purely diffusive MHD regime, the regime where linear, dispersive AW propagate, and the regime of nonlinear propagation.