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Compound ion distributions, fi(v), have been measured by NASA'sMagnetospheric Multi-Scale Mission (MMS) and have been found in reconnectionsimulations. A complex distribution, fi(v), consisting, for example, ofessentially disjoint pieces will be called a multi-beam distribution andmodeled as a sum of "beams," fi(v) = f1(v) + ... +fN(v). Velocity moments offi(v) are taken beam by beam and summed. Such multi-beam moments of fi(v) haveadvantages over the customary standard velocity moments of fi(v), forwhichthere is only one mean flow velocity. For example, the standard thermal energymomentof a pair of equal and opposite cold particle beams is non-zero eventhough each beam has zero thermal energy. We therefore call this thermal energypseudo-thermal. By contrast, a multi-beam moment of two or more beams has nopseudo-thermal energy. We develop three different ways of decomposing into asum and finding multi-beam moments for both a multi-beam fi(v) measured by MMSin the dayside magnetosphere during reconnection and a multi-beam fi(v) foundin a PIC simulation of magnetotail reconnection. The three methods are: Avisual method in which the velocity centroid of each beam is estimated and itsdensity determined self-consistently; A k-means method in which particles in aparticle-representation of fi(v) are sorted into a minimum energy configurationof N (= k) clusters; A nonlinear least squares method based on a fit to a sumof N kappa functions.  Multi-beam energy moments are calculated and compared with standard momentsfor the thermal energy density, pressure tensor, thermal energy flux (heat plusenthalpy fluxes), bulk kinetic energy density, RAM pressure and bulk kineticenergy flux. Applying this new formalism to real data demonstrates in detailhow multi-beam techniques may provide significant insight into the propertiesof observed space plasmas.

Multi-Beam Energy Moments of Compound Measured Ion Velocity Distributions