  Compound 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.
