  Complex ion distributions, fi(v), have been measured by NASA's 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.
