Torsional Alfvén waves in a dipolar magnetic field: experiments and simulations

The discovery of torsional Alfven waves (geostrophic Alfven waves) in the Earth's core (Gillet et al. 2010) calls for a better understanding of their properties. We present the first experimental observations of torsional Alfven waves, performed in the DTS-Omega set-up. In this set-up, 50 L of liquid sodium are confined between an inner sphere (r(i) = 74 mm) and an outer shell (r(o) = 210 mm). The inner sphere houses a permanent magnet, imposing a dipolar magnetic field (B-max = 345 mT). Both the inner sphere and the outer shell can rotate around the vertical axis. Alfven waves are triggered by a sudden jerk of the inner sphere. We study the propagation of these waves when the fluid is initially at rest, and when it spins at a rotation rate up to 15 Hz. We measure the azimuthal magnetic field of the wave at different radii inside the fluid with magnetometers installed in a sleeve. We also record the electric potential signature on the outer shell at several latitudes. Besides, we probe the associated azimuthal velocity field using ultrasound Doppler velocimetry. With a 15 Hz rotation rate, the dynamical regimes we achieve are characterized by dimensionless numbers in the following ranges: Lundquist number 0.5 < Lu < 12, Lehnert number 0.01 < Le < 0.26, Rossby number Ro similar to 0.1. We observe that the magnetic signal propagates away from the inner sphere, strongly damped by magnetic diffusion. Rotation affects the magnetic signature in a subtle way. Its effect is more pronounced on the surface electric potentials, which are sensitive to the actual fluid velocity of the wave. The ultrasound Doppler probes provide the first experimental measurement of the fluid velocity of an Alfven wave. To complement these observations, we ran numerical simulations, using the XSHELLS pseudospectral code with parameters as close as possible to the experimental ones. The synthetic magnetic and electric signals match our measurements. The meridional snapshots of the synthetic azimuthal velocity field reveal the formation of geostrophic cylinders expected for torsional Alfven waves. We establish scaling laws for the magnetic and kinetic energies of Alfven waves with and without rotation. In both cases, we find that the magnetic energy EM saturates at a level proportional to Rm(jerk)(2), where Rm(jerk) = U(jerk)r(o)/eta is the magnetic Reynolds number built with the maximum azimuthal velocity of the inner sphere during the jerk. The E-K(max)/E-M(max) ratio (where E-K(max) is the maximum kinetic energy), close to 1 for very quick jerks, increases linearly with the jerk duration.