Evaluation of magnetic helicity density in the wave number domain using multi-point measurements in space

We develop an estimator for the magnetic helicity density, a measure of the spiral geometry of magnetic field lines, in the wave number domain as a wave diagnostic tool based on multi-point measurements in space. The estimator is numerically tested with a synthetic data set and then applied to an observation of magnetic field fluctuations in the Earth foreshock region provided by the four-point measurements of the Cluster spacecraft. The energy and the magnetic helicity density are determined in the frequency and the wave number domain, which allows us to identify the wave properties in the plasma rest frame correcting for the Doppler shift. In the analyzed time interval, dominant wave components have parallel propagation to the mean magnetic field, away from the shock at about Alfvén speed and a left-hand spatial rotation sense of helicity with respect to the propagation direction, which means a right-hand temporal rotation sense of polarization. These wave properties are well explained by the right-hand resonant beam instability as the driving mechanism in the foreshock. Cluster observations allow therefore detailed comparisons with various theories of waves and instabilities