Compound ion distributions, fi(v), have been measured by NASAs Magnetospheric Multi-Scale Mission (MMS) and have been found in reconnection simulations. A complex 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), forwhich there is only one mean flow velocity. For example, the standard thermal energy momentof 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 pseudo-thermal. By contrast, a multi-beam moment of two or more beams has no pseudo-thermal energy. We develop three different ways of decomposing into a sum and finding multi-beam moments for both a multi-beam fi(v) measured by MMS in the dayside magnetosphere during reconnection and a multi-beam fi(v) found in a PIC simulation of magnetotail reconnection. The three methods are: A visual method in which the velocity centroid of each beam is estimated and its density determined self-consistently; 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; 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 may provide significant insight into the properties of observed space plasmas.
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