Torsional Alfvén waves in a dipolar magnetic field: experiments and simulations
Résumé
The discovery of torsional Alfv\'en waves (geostrophic Alfv\'en waves) in the Earth's core \citep{gillet_fast_2010} calls for a better understanding of their properties.
We present the first experimental observations of torsional Alfv\'en waves, performed in the \dts set-up.
In this set-up, 50L of liquid sodium (magnetic Prandtl number $Pm = 7.4 \times 10^{-6}$) 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 vertically aligned 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 and short rotation (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.
The waves produce an azimuthal magnetic field, which we measure 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, and because of the radial decay of the magnetic field intensity, the dynamical regimes we achieve are characterized by dimensionless numbers in the following ranges: Lundquist number $0.5 < Lu < 16$, Lehnert number $0.01 < Le < 0.3$, Rossby number $Ro \sim 0.1$.
We observe that the magnetic signal propagates away from the inner sphere, 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 Alfv\'en wave.
To complement these observations, we ran numerical simulations, using the XSHELLS pseudo-spectral 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 Alfv\'en waves, and the excitation of inertial modes for abrupt jerks of the inner sphere.
In the absence of rotation, inertial effects become dominant both in the experiments and in the simulations.
The resulting non-linear regimes reveal the formation of an equatorial sheet with a mushroom-shape cross-section.
We establish scaling laws for the magnetic and kinetic energies of Alfv\'en waves with and without rotation.
In both cases, we find that the magnetic energy $E_M$ 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 short rotation, increases linearly with the jerk duration.
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