Multi-beam energy moments of measured compound ion velocity distributions

Abstract

Compound ion distributions, fi(v), have been measured with high-time resolution by NASA's Magnetospheric Multi-Scale Mission (MMS) and have been found in reconnection simulations. A compound distribution, fi(v), consisting, for example, of essentially disjoint pieces will be called a multi-beam distribution and modeled as a sum of “beams,” fi(v) = f1(v) + ⋯ + fN(v). Velocity moments of fi(v) are taken beam by beam and summed. Such multi-beam moments of fi(v) have advantages over the customary standard velocity moments of fi(v), for which there is only one mean flow velocity. For example, the standard thermal energy moment of a pair of equal and opposite cold particle beams is non-zero even though each beam has zero thermal energy. We therefore call this thermal energy pseudothermal. By contrast, a multi-beam moment of two or more beams has no pseudothermal energy. We develop three different ways of approximating a compound ion velocity distribution, fi(v), as a sum of beams and finding multi-beam moments for both a compound fi(v) measured by MMS in the dayside magnetosphere during reconnection and a compound fi(v) found in a particle-in-cell simulation of magnetotail reconnection. The three methods are (i) a visual method in which the velocity centroid of each beam is estimated and the beam densities are determined self-consistently, (ii) a k-means method in which particles in a particle representation of fi(v) are sorted into a minimum energy configuration of N (= k) clusters, and (iii) a nonlinear least squares method based on a fit to a sum of N kappa functions. Multi-beam energy moments are calculated and compared with standard moments for the thermal energy density, pressure tensor, thermal energy flux (heat plus enthalpy fluxes), bulk kinetic energy density, ram pressure, and bulk kinetic energy flux. Applying this new formalism to real data demonstrates in detail how multi-beam techniques provide new insights into the energetics of observed space plasmas.