Multibeam Energy Moments Of Multibeam Particle Velocity Distributions

High-resolution electron and ion velocity distributions, f(v), which consist of N effectively disjoint beams, have been measured by NASA's Magnetospheric Multiscale Mission and in reconnection simulations. Commonly used standard velocity moments assume a single mean-flow velocity for the entire distribution. This can lead to counterintuitive results for a multibeam f(v). An example is the standard thermal energy density moment (at a given space-time point) of a pair of equal and opposite cold particle beams. This standard moment is nonzero even though each beam has zero thermal energy density. By contrast, a multibeam moment of two or more cold beams at a given position and time has no thermal energy. A multibeam moment is obtained by taking a standard moment of each beam and then summing over beams. In this paper we will generalize these notions, explore their consequences, and apply them to an f(v) which is a sum of tri-Maxwellians. Both standard and multibeam energy moments have coherent and incoherent pieces. Examples of incoherent moments are the thermal energy density, the pressure, and the thermal energy flux (enthalpy flux plus heat flux). Corresponding coherent moments are the bulk kinetic energy density, the ram pressure, and the bulk kinetic energy flux. The difference between a standard incoherent moment and its multibeam counterpart will be defined as the "pseudothermal part" of the standard moment. The sum of a pair of corresponding coherent and incoherent moments is the undecomposed moment. Undecomposed standard moments are always equal to the corresponding undecomposed multibeam moments.