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Multi-beam 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 Multi-Scale Mission (MMS) observatories and in reconnection simulations. Commonly used standard velocity moments generally assume a single mean-flow-velocity for the entire distribution, which can lead to counterintuitive results for a multibeam f(v). An example is the (false) standard thermal energy moment of a pair of equal and opposite cold particle beams, which is nonzero even though each beam has zero thermal energy. By contrast, a multibeam moment of two or more beams has no false 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 sum of tri-Maxwellians. Both standard and multibeam energy moments have coherent and incoherent forms. 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 false part of an incoherent moment is defined as the difference between the standard incoherent moment and the corresponding multibeam moment. The sum of a pair of corresponding coherent and incoherent moments will be called the undecomposed moment. Undecomposed moments are independent of whether the sum is standard or multibeam and therefore have advantages when studying moments of measured f(v).


Abstract
High resolution electron and ion velocity distributions, f(v), which consist of N effectively disjoint beams, have been measured by NASA's Magnetospheric Multi-Scale Mission (MMS) observatories and in reconnection simulations. Commonly used standard velocity moments generally assume a single mean-flow-velocity for the entire distribution, which can lead to counterintuitive results for a multibeam f(v). An example is the (false) standard thermal energy moment of a pair of equal and opposite cold particle beams, which is nonzero even though each beam has zero thermal energy. By contrast, a multibeam moment of two or more beams has no false 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 sum of tri-Maxwellians.
Both standard and multibeam energy moments have coherent and incoherent forms. 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 false part of an incoherent moment is defined as the difference between the standard incoherent moment and the corresponding multibeam moment. The sum of a pair of corresponding coherent and incoherent moments will be called the undecomposed moment. Undecomposed moments are independent of whether the sum is standard or multibeam and therefore have advantages when studying moments of measured f(v).

VELOCITY MOMENTS OF MULTIBEAM VELOCITY DISTRIBUTIONS, f(v)
In space plasma physics is it very common to discuss the properties of a plasma in terms of its bulk properties -density, velocity, pressure, etc. and to use a fluid theory framework to describe and predict the behavior of many space plasma phenomena (Cravens, T.E.; Bellan, P.; Kulsrud, R.M..). Two different kinds of space plasmas are commonly treated as fluids: collisional and collisionless.
• In collisional space plasmas, such as the lower solar corona and lower ionosphere, the collisional mean free path is short and the particle velocity distributions, f(v), are at or close to Maxwellian (e.g., thermal equilibrium). In this case the fluid equations for a given species can be derived from the Boltzmann equation for that species by the Chapman-Enskog procedure (Braginskii) and include collisional transport coefficients such as electron-ion equilibration rates and thermal conductivity.
• However, in Earth's upper ionosphere, magnetosphere and solar wind, the collisional mean free path is large, so the plasma is collisionless and often far from thermal equilibrium [Pashmann et al 1998]. The velocity distributions, can have nonthermal tails and may even be multipeaked, leading to effectivey disjoint f(v) [Burch, et al]. Standard collisionless fluid equations and energy transport equations can be derived for constituent species by taking velocity moments of the Vlasov eqn. for each species (Bellan, 2006;Aunai, et al, ;Goldman, et al, 2015). The moment equations are expressed in terms of (standard) velocity moments of the particle distribution, f(v). In collisionless space-plasma fluid equations there are no electron-ion collisions and there is no collisional conductivity. Energy transport can only occur by convection, described in terms of particle energy fluxes or by radiation (Poynting flux).
Other terms in the standard collisionless fluid equations resemble those in the collisional fluid equations even though f(v) is not generally Maxwellian. For a given f(v) there is one flow coherent velocity, u and one mean density, n; higher-order moments decompose into a sum of a cohererent part (involving u) and an incoherent ("thermal") corresponding to moments of velocity fluctuations, such as òd 3 vf(v)|v -u| 2 or òd 3 vf(v) (v -u) |v -u| 2 . Thus, the energy density of a given species is the sum of a bulk kinetic energy density, nmu 2 /2, and a "thermal" energy density moment, mòd 3 vf(v)|v -u| 2 /2; the energy density flux is a sum of a coherent bulk kinetic energy flux, unmu 2 /2, and a "thermal" energy flux moment (enthalpy plus heat flux).
The meaning of each of the decomposed moments is well understood and not controversial for contiguous and effectively single-peaked velocity distributions, f(v). However, for multibeam (effectively disjoint) f(v), standard velocity moments can give rise to misinterpretations such as false identifications of incoherent energy density and temperature. In this paper an alternate method is developed for taking velocity moments of an N-beam velocity distribution, f(v), of form f(v) = f1(v) + f2(v) + … + fN (v). Such multibeam moments are formed by taking standard coherent moments or standard thermal moments of each beam, fj(v) and then summing over beams j = 1 to N. This procedure eliminates false thermal parts of higher-order standard decomposed moments of f(v), such as the standard "thermal" energy density and related false temperatures. However, it is important to note that the sum of the coherent plus the "thermal" parts of a higher-order moment remains the same whether the velocity moments are standard or multibeam. Such a sum will be called an undecomposed higher-order moment. An example is the undecomposed particle energy density, nm<|v| 2 >/2, which can be decomposed by writing v = u + (v -u).
In practical space plasma applications f(v) is measured by electrostatic analyser instruments , and (standard) moments can be constructed on the ground or onboard (so reducing the telemetry) (Reme et al. Ann Geophys 2001;McFadden et al. 2007) The launch of the Fast Plasma Instrument (FPI) (Pollock et al., 2016) on board the Magnetosphere MultiScale (MMS) Mission [Burch, 2016] now enables the full 3D electron and ion distribution to be measured over unprecedented time-intervals of 30 ms and 150 ms cadence respectively (or 7.5 ms and 37.5 ms in certain operating modes [Rager, et al 2018]).
In many regions of interest, not least the electron dissipation region of magnetic reconnection (e.g., Burch, et al, 2015), the ion and electron velocity distributions, f(v), consist of a number of effectively disjoint pieces in velocity space, which we will refer to as beams. This calls into question the use of standard moments because experimentally the distribution is not contiguous; nor is it single peaked. In particular, difficulties of interpretation may arise, and the underlying physics may become obscured by using a description which does not adequately capture the phenomena being observed.
Consequently, for multibeam f(v) it may be more useful not to take moments in the standard way but rather to take standard decomposed moments beam by beam and then add them together. We call such sums multibeam moments. The purpose of this paper is to lay out a systematic framework for understanding multibeam moments. By way of introduction, we first examine a highly idealized theoretical example which also serves to illustrate difficulties of interpretation that may arise. Fig. 1 shows two equal and opposite cold electron beams, with velocities u0 and -u0 and equal densities n0. According to standard moment theory the effective velocity distribution, f(v) is one entity, with one flow velocity, u. In this example, u = 0, so the bulk kinetic energy moment, Ubulk is zero and the (single) density is n = 2n0. The incoherent energy density moment found from standard moment theory is mn(u0) 2 /2. This incoherent part of the energy density is often called the thermal energy density and written as Utherm = n"T". This yields an effective temperature, "T" = m(u0) 2 /2. A difficulty with such a standard moment is that the pair of cold moving beams appears to have a temperature, which we will call here a falsetemperature, but no bulk kinetic energy density. This is customarily remedied by simply considering f(v) to be a two-beam system, f(v) = f1(v) + f2(v). The density and velocity moments of f1 and f2 are {n1, u1} and {n2, u2}, where, n1 = n2 = n0, u1 = u0 and u2 = -u0. The energy moments of each beam are now Ubulk1,2 = n0m(u0) 2 /2 and Utherm1,2 = 0. For the system of two beams, (Ubulk) 2-beam = Ubulk1 + Ubulk2 = nm(u0) 2 /2, where n = n1 + n2 = 2n0 and (Utherm) 2-beam = 0 + 0, which is more intuitive than the results of the standard moment analysis, Ubulk = 0 and Utherm = nm(u0) 2 /2.

FALSE THERMAL ENERGY MOMENT OF A PAIR OF COLD BEAMS
We summarize the results for the two different ways of taking energy moments of f(v) = f1(v) + f2(v) in the table below: The bottom (shaded) row demonstrates an important general result: that the sum of the bulk and thermal energy terms has the same value, U = nm(u0) 2 /2, for both the standard and the multibeam methods of taking moments. In this paper, the generalized U will be referred to as the undecomposed energy moment and the bulk and thermal energies as the decomposed energy moments. It is clear from Table 1 that the values of the decomposed moments differ, depending on the method used for taking moments. Since U is independent of the approach used, it can be employed reliably to quantify the properties of the plasma without ambiguity. This offers a path forward in studying the properties of complex distribution functions such as those observed by MMS.

ORGANIZATION AND OUTLINE OF THIS PAPER
In the present paper the concepts discussed above in Fig. 1 and Table 1 for two cold beams will be generalized to N-beams and the full set of multibeam pressure, energy and energy flux moments. Standard and multibeam moments then will be evaluated and compared for an analytic particle velocity distribution, f(v), consisting of a sum of N tri-Maxwellian beams.
The content of this paper is indicated below in terms of questions to be addressed: 1. Examples of measured and simulated f(v) which are effectively disjoint.
• What are examples of effectively disjoint velocity distributions, f(v) º f(v, r0, t0), found at certain locations and times during magnetic reconnection in the magnetopause and found in PIC simulations of reconnection in the magnetotail?
• What is a possible physical origin of such beams at {r0, t0} in terms of their Lagrangian trajectories at other points in space and time?
2. Standard energy transport theory • What are the various energy moments and transport equations associated with a given f(r,v,t)?
• How are transport equations and transport coefficients derived from kinetic theory (i.e., the Vlasov equation) in a collisionless plasma?

Multibeam moments
• What is the difference between undecomposed energy density, pressure and energy flux moments and decomposed energy density moments (bulk, thermal), pressure moments (RAM, thermal) and energy flux moments (bulk, enthalpy, heat flux)?
• What is the relationship between standard and multibeam decomposed moments?
• How are false-thermal parts of standard decomposed moments determined?
• What are the standard and multibeam moments of a set of N tri-Maxwellian beams?

MULTIPLE ION BEAMS NEAR X-LINE DURING DAYSIDE RECONNECTION
In a pioneering paper , high resolution data from the Fast Plasma Instrument (FPI) onboard MMS satellites was used to study electron and ion velocity distributions in the electron diffusion region of Earth's magnetopause during magnetic reconnection. The left image in Figure 2 shows the magnetic field and standard particle and field energy fluxes as functions of time along the trajectory of MMS4. The reconnection x-line is crossed at time 1307, as indicated by the vertical dashed line. The right image (here constructed from the data) shows 3D contour maps of the ion distribution, f(v), measured over a time interval of 150ms a few seconds before and a few seconds after an x-line crossing. Details are given in the Figure caption. It is clear that both ion distributions, f(v), are far from equilibrium and disjoint. Each f(v) is therefore an ion multibeam distribution amenable to the multibeam moment treatment described in this paper. Such a treatment will generally give ion energy moments different from the ones on the left in Figure (2).  . Left panel: MMS4-measured particle energy and Poynting fluxes showing x-point around time 1307 (vertical black dashed line). The two vertical orange dotted lines labelled MB are at times ∼3s before 1307 and ∼6s after 1307. Multiple-beam-like features are found in the ion velocity distribution, f(v), at both of those times. The right two panels show f(v) determined by the Fast Plasma Instrument (FPI) at the later time from two different perspectives. In those panels the black arrow is in the direction of B and the blue, green and red arrows show the directions of the magnetopause GSE coordinates in x, y, z. The tip of each colored arrow is at a velocity corresponding to an energy of 1.6 keV. The values of f(v) on successive surface contours differ by factors of 5.

TAIL RECONNECTION SIMULATION SHOWS MULTIPLE ION BEAMS
Multibeam (disjoint) ion distributions, f(v), have also been found in simulations of magnetic reconnection in Earth's magnetotail. An example can be found in Eastwood, et al, in which PIC simulations are carried out in support of THEMIS observations of magnetic reconnection on February 27, 2009 where complex ion distributions were observed. The THEMIS magnetic field data were used to establish appropriate comparison cuts through a particle-in-cell simulation of reconnection, and very good agreement was found between the observed and simulated ion distributions on both sides of the dipolarization front.
The principal feature of interest in the simulation is the dynamics of a pair of counterpropagating ion beams found in the dipolarization front about five ion inertial lengths in front of the unperturbed plasma just below the neutral sheet at time t = 30. They are visible in the 2D reduced distribution in the v x -v y (reconnection) plane shown in Figure 3a. This reduced distribution is integrated over v z . The two beams are represented by two small cubes in velocity space labeled A and B. The other two orthogonal reduced distribution functions are shown in Figures 3b and 3c.
Figures 3d-3g show the real space self-consistent trajectories of the bunches of ions that pass through these two phase-space cubes at t = 30 over the time interval 23 < t < 35. Trajectories are of particular interest because they illustrate how multibeam particle distributions, f(v), might arise.
The trajectories are superposed on plots of E zSIM (approximately the reconnection electric field) and E ySIM (approximately the Hall electric field), respectively. The fields and circled groups of ions in Figures 3d and 3e correspond to the start time for the trajectories (t = 23.28). The fields and circled bunches of ions in Figures 3f and 3g correspond to the time when the trajectories cross, thereby giving rise to the disjoint reduced distribution function show in Figures 3a-c. The ions composing the two beams are clearly distinguishable, with beam A moving primarily downward and to the right, while beam B moves primarily upward and to the right (as labeled in Figure 3d). At the start time, the ion bunches are widely separated in space, with each bunch near a different separatrix. The ion bunch trajectories later in time are determined by the electric and magnetic forces. The ions are not frozen-in (this is the ion-diffusion region). For details concerning the forces and their influence on the ion trajectories refer to Eastwood, et al.

REVIEW OF STANDARD ENERGY-TRANSPORT THEORY
There are a number of different ways to take energy moments and derive corresponding energy transport equations for a given particle species. A common strategy, the so-called standard approach, is based on the assumption that there is a single mean-flow-velocity, u of a distribution f(v). For a given particle species the well-known equations (Bellan) for the transport of the bulk coherent energy density, Ubulk, and the "thermal" (incoherent) energy density, Uthermal are given by (1) In Eqn. (1a), u is the flow-velocity of the species, Ubulk = mnu 2 /2, Qbulk is the bulk kinetic energy density flux, uUbulk, J·E is the work performed by the electric field, E against the current, Je or Ji. The term -u·Ñ·P is the work performed by the pressure force, -Ñ· P, on the flow, u.
In Eqn. (1b), Qtherm is the thermal (incoherent) energy flux, Qtherm = Qheatflux + Qenthalpy, consisting of a part, Qheatflux, which is invariant under velocity frame transformations and a part, Qenthalpy, proportional to u. The term +u·Ñ· P is the work done by u against the pressure force. Hence, the work associated with pressure governs the transfer of energy between coherent and incoherent energy densities.
Another energy transport equation results from adding Eqn. (1a) to Eqn. (1b). The work terms associated with pressure forces then drop out entirely, resulting in: Eqn. (2) describes particle energy transport (for ions or electrons). We shall refer to Eqn.
(2) as the undecomposed particle energy transport equation, and U as the undecomposed particle energy density. Ubulk and Uthermal can be understood as resulting from the decomposition of U into a coherent and an incoherent part. In a similar manner, Q is the undecomposed energy flux whose decomposed pieces, Qbulk and Qthermal are its coherent and incoherent parts.
At first glance energy transport Eqn.
(2) appears less useful than energy transport Eqns.
(1) because U and Q do not distinguish between coherent and incoherent (thermal) energies and fluxes. However, energy transport in terms of U and Q can be very useful because U and Q are independent of whether their decomposed parts are standard (single-beam) moments or multibeam moments of a given velocity distributions, f(v). Note also that U and Q enter into the electromagnetic energy transport equation in the same way as the electromagnetic energy density and Poynitng flux when -(Je+Ji)·E is eliminated by using Eqn. (2a) for each species: (3) In Eqn. (3) the electromagnetic energy, UEM ∝ E 2 +B 2 , and Poynting flux, S ∝ ExB, are on the same footing as Ue,I and Qe,i. From Eqn. (3) it is evident that when magnetic energy is lost locally it can transform into local particle energy densities, Ue,i, and/or it can be transported remotely by radiation (Poynting flux) and convection (particle fluxes, Qe and Qi).
At the level of description of Eqn (3) one cannot determine whether lost magnetic energy goes mainly into local particle heating, into particle acceleration or is spatially transported by radiation or coherent or incoherent particle convection. To determine in standard transport theory whether coherent or incoherent processes dominate changes in particle energy (or energy flux) one must employ the standard decomposed energy transport Eqns. (1).

KINETIC THEORY OF STANDARD ENERGY TRANSPORT
Further insight into the meaning of Eqn.
(2) is gained by deriving it from kinetic theory (Aunai, et al; Bellan; Goldman, et al). In kinetic theory, undecomposed energy transport is described by taking a kinetic energy velocity moment of the Vlasov equation for either species: The result is Eqn. (2) with the undecomposed U and Q now explicitly defined as velocity moments of f(v), In the usual standard approach employing the mean-flow-velocity decomposition of U and Q moments, one substitutes v = u + dv into mv 2 /2 and mvv 2 /2 in the velocity integrals in Eqns (4a) and (4b). Integrals whose integrands are linear in dv = (v-u) vanish -hence the term "incoherent" for moments proportional to (dv). 2 This kind of decomposition is most meaningful for a single beam (f(v) single-peaked and contiguous). The mean-flow-velocity decomposition of U and Q moments is The three fluxes, Q, in Eqn. (5b-d) correspond to standard energy transport by convection of coherent and incoherent particle energies.
The enthalpy flux is second-order in dv so it may be defined in terms of the pressure tensor, P. In dimensional (as opposed to dimensionless) units, The moments in Eqn. (5) enter into the two independent energy transport Eqns. (1a,b) for Ubulk and Utherm. They can be derived from the Vlasov Eqn (Bellan; Goldman, et al) by using • the transport Eqn (2a) for U = Ubulk + Utherm (as found above from the Vlasov Eqn.) and • a transport equation for Ubulk obtained by dotting u into the momentum eqn (first moment of the Vlasov eqn.).
The distinction between undecomposed and decomposed moments may be applied to the pressure tensor moment as well as to the energy and energy flux moments. The stress tensor, T, is the undecomposed moment which decomposes into a RAM pressure tensor (dyadic) and a "thermal" pressure, when the mean velocity, u is introduced: Note that the pressure tensor, , is frame-invariant (like Utherm and Qheatflux), whereas T and PRAM are not. In dimensional units, 6a) Tr( )/2 = Utherm This helps explain the enthalpy flux in Eqn. (5e), since the first term on the right is now uUtherm. The undecomposed stress tensor enters directly into the force-momentum equation, which is the first moment of the Vlasov equation and may be written in the following form: When T is decomposed the "thermal" pressure force appears explicitly and the RAM pressure leads to the convective derivative, u·Ñu.
A space-time transport eqn for T(r,t) is generated from kinetic theory by multiplying the Vlasov eqn by mvv and integrating over v. However, space-time transport will not be studied in this paper.

STANDARD MOMENTS OF ONE TRI-MAXWELLIAN BEAM, f(v)
To give explicit examples of standard and multibeam energy moments, we first consider a single tri-Maxwellian distribution function, f(v), of form, Here, f(v) is normalized to the density, n, and centered at the velocity, u = {ux, uy, uz}. The three thermal velocities are wj = √(kBTj/ms), where j = (x, y, z), s = e or i, and Tj is the temperature associated with direction i. We define a thermal velocity vector, w = {wx, wy, wz}.
All of the standard moment integrals in Eqns. (5) and (6) can be performed analytically, with the following results: Note that the pressure tensor is diagonal in this coordinate system. Such a coordinate system may be found for a non-diagonal pressure tensor in terms of its eigenvalues and eigenvectors.
The enthalpy flux vector has been calculated in terms of the pressure, as in Eqn. (5e), The standard heat flux, Qheatflux, is zero for a tri-Maxwellian beam of the form in Eqn. (7) due to reflection symmetries which makes the heat flux integral vanish.

Multibeam moments
Consider a 3-D velocity distribution, f(v), consisting of N pieces, which will be referred to as beams: In the standard method for taking moments, a single mean flow velocity, u, is used, as in Eqns. (5) and (6) for the entire assembly, f(v) of beams. Such an approach is used in the typical calculation of on-board moments on a satellite, such as the Fast Plasma Instruments on board the MMS satellites as well as instruments on Cluster, THEMIS, etc.. It is usually applied without regard to whether or not the measured f(v) consists of one or more than one effectively disjoint "beams".
In the multibeam decomposition the moments of each beam are taken first (using flow velocities uj of each beam, j). One then sums the moments over all beams. In general, the multibeam and standard decomposition methods give different results for the decomposed moments.
By contrast, the undecomposed moments, U, Q, and T, which are independent of flow velocities, are the same whether one uses f(v) or its equivalent, f1(v) + f2(v) + ... + fN(v): The sum in Eqn. (10a) is a sum over the N beams of undecomposed beam energy densities, Uj; the sum in Eqn. (10b) is a sum over beams of undecomposed beam energy fluxes, Qj; and the sum in Eqn. (10c) is a sum over beams of undecomposed individual beam stress tensors. It should be clear from Eqns. (10) that the sum over undecomposed moments of each beam gives the same result as the undecomposed moments of f(v). The undecomposed moments are therefore a robust and unambiguous characterization of the plasma, whether it consists of one or more beams.
The "standard" decomposition of U, Q and T based on the mean flow velocity u, is correctly given by Eqns. (5) and (6). The standard decomposed moments Ubulk, Utherm, Qbulk, Qheatflux, and Qenthalpy, TRAM and T are found, for example, in the moments file of the Fast Plasma Device (FPI) on each of the MMS spacecraft.
By contrast, in the alternative, multibeam decomposition one first decomposes the moments of each beam using that beam's mean velocity, uj and then substitutes, v = uj + (vj -uj) in mv 2 /2 and mvv 2 /2 in the velocity integrals for Uj and Qj and Tj in Eqn. (10).
A key point in this paper is that each multibeam decomposed moment of the assembly of beams is given by the sum over beams, j, of the corresponding standard moment of each beam j, In general, none of the multibeam decomposed moments is equal to the corresponding standard decomposed moment.
Based on Eqns. (10) and (11) the undecomposed moments U, Q and T may therefore be decomposed into either a sum of standard moments or a sum of multibeam moments: The thermal energy density fluxes Qtherm and (Q MB )therm have been defined in Eqn (12d) as sums of enthalpy flux and heat flux. Section (4.4) will show how Eqns. (12) determine the false-thermal parts of standard moments. First, we consider lower-order moments.

CONSERVATION OF PARTICLES AND OF PARTICLE FLUX
The zero-order-moment of f(v) = f1 (v) + f2 (v) + ... + fN (v), yields conservation of particles, Here, nj is the density of beam j, n is the mean density of the ensemble of beams and hj = nj/n is the fractional density of each beam. The first-order moment of f(v) yields conservation of particle flux. When divided by n, this becomes where uj is the velocity of the j-th beam and u is the mean velocity of the ensemble of beams. Each beam velocity, uj, in the weighted mean, u, is multiplied by the fractional beam density, hj.
Eqns. (13) illustrate that the beam velocities, uj, and the beam densities, nj, are not completely independent of each other. Their relationship depends on the standard density, n and flow velocity, u. We next consider a sum of tri-Maxwellian beams.

MULTIBEAM MOMENTS OF A SUM OF TRI-MAXWELLIANS
Consider an analytic velocity distribution consisting of a sum of N electron or N ion beams, f(v) = f1(v)+ f2(v)+... fN(v), in which each beam, fj(v), is tri-Maxwellian: ,...,N Here nj is the density of beam j, uj is its mean velocity, and mwjx 2 = Tjx, mwjy 2 = Tjy, mwjz 2 = Tjz are the three temperatures of beam j. We have taken all of the tri-Maxwellians to be co-aligned, with temperatures along the x, y and z axes. Consequently, all beam pressure tensor moments will here be diagonal. This of course, is not the most general possible sum of tri-Maxwellians but it will suffice to illustrate the concepts considered in this paper. More commony treated is the even less general limit of a sum of spherically symmetric Maxwellians, each of temperature Tj, arrived at by setting wjx = wjy = wjz = √(Tj/m).
As in Eqns. (11), the decomposed multibeam moments of any multibeam f(v) are given simply by the sum over beams, j, of the corresponding standard moments of each beam j. Since each beam is a tri-Maxwellian and the standard moments of a tri-Maxwellian are given by Eqns. (8), the decomposed multibeam moments of the sum of tri-Maxwellians can be expressed as, Here the integer j labels beams while the integers l and s label vector and tensor components in the three Cartesian coordinates, {x, y, z}. The multibeam heat flux is zero because of velocity reflection symmetry in each tri-Maxwellian beam. We have defined a thermal vector, wj º {wjx, wjy, wjz}. In the spherically symmetric limit, |wj| 2 = 3Tj/m. In the cold beam limit, wj ® 0, for all beams so the ensemble thermal energy, enthalpy flux and pressure go to zero.

HOW TO FIND STANDARD MOMENTS FROM MULTIBEAM MOMENTS
The decomposed moments obtained using the standard (e.g., FPI) mean-flow-velocity, u, and associated density, n, are given by the f(v) integrals in Eqns. (5) and (6).

STANDARD MOMENTS OF AN ASSEMBLY OF TRI-MAXWELLIANS
For the assembly of N co-aligned tri-Maxwellians, (Utherm) MB and P MB are known, so the standard thermal moments for the assembly are: δu jx δu jy δu jx δu jz δu jx δu jy w jy 2 + δu jy 2 δu jz δu jy δu jx δu jz δu jz δu jy w jz 2 + δu jz There are both true and false thermal parts contained in the standard thermal energy density, heat flux and enthalpy flux of the assembly of N co-alignd tri-Maxwellian beams Although the multibeam heat flux, (Qheatflux) MB of the assembly is zero, owing to the symmetry of each tri-Maxwellian beam, the standard heat flux of the assembly, Qheatflux, does not vanish, but is composed entirely of false thermal heat flux. Note that the dependence of Qheatflux on Pj means that the false thermal heatflux contains thermal energy |wjl| 2 , where the index l is the Cartesian coordinate, l = x, y or z. However, in the cold beam limit wjl ® 0, Qheatflux does not go to zero.

MOMENTS OF EQUAL AND OPPOSITE MAXWELLIAN DISTRIBUTIONS
The above case is a generalization of the "false-thermal" standard energy density moment of two equal and opposite cold beams described in the introduction. For the two equal and opposite cold beams in Fig. (1), the mean flow velocity, u = 0, and, from Eqns. (15) and (16), Thus, the standard (FPI) moments identify the energy of an assembly of cold beams as totally "thermal," whereas multibeam moments correctly identify the system energy as composed of bulk kinetic energy only. This is what is termed here as a false thermal component.
In general, Uthermt will be a mixture of real and false thermal energy densities. If the two cold beams in Fig. 1 and Table 1 are replaced by two Maxwellian beams, each with isotropic temperature, T = mw 2 , Eqns. (17) still hold, except for the "thermal" energy densities expressions which are changed to: Table (2), below, compares the standard energy density moments to the multibeam energy density moments for this example of two beams.

FALSE-THERMAL PARTS OF DIMENSIONLESS STANDARD MOMENTS
Thus, standard thermal energy moments may contain false-thermal parts, whereas the corresponding multibeam thermal energy moments are entirely true-thermal.
In general, false-thermal parts of standard moments are obtained by subtracting from each standard moment the corresponding multibeam moment. Hence, the false-thermal moments, DU, DP, and DQ are defined as: Since Eqn.(12a), for example, may be rewritten as , Eqns. (12) allow us to replace the difference moments in Eqn. (18) by differences of the much simpler bulk moments: First consider energy density moments. Define dimensionless energy units by dividing all energy densities by a normalization standard bulk energy density, Unorm = mnun 2 /2. We allow for the possibility that the normalization flow velocity moment, un may (or may not) differ from the standard flow velocity moment, u, of the given f(v).
The standard and multibeam bulk energy density moments are independent of the exact shape of f(v) = f1(v) + ... + f N(v) and of the shape of the individual beam distributions, fj(v). These moments depend only on densities and mean velocities of f(v), (n, u), and of fj(v), (nj, uj), so that the false-thermal energy density moment is Two equivalent alternate expressions for the dimensionless DU are (20b) (20c) Eqn. (20b) has N terms. It is equivalent to Eqn. (20a) as one can see by explicitly squaring (u-uj) 2 in each term and summing. In this form, DU is manifestly frame-independent, since the beam velocities, uj are all measured from the mean flow velocity, u. The right side of Eqn. (20c) depends on differences of beam velocities, (ui-uj) so it too is explicitly frameindependent. In this format there are N(N-1)/2 terms.
Any of the Eqns. (20) may be used to calculate the false-thermal energy density, DU, in terms of zero-order and first-order moments of f(v) and of zero-order and first-order moments of each of the beams, fj(v). However, there is another possible use of DU which will be demonstrated in a later paper dealing with methods for explicit multibeam moments of measured f(v). From Eqn. (19a), the multibeam thermal energy density (U therm) MB is, Thus, for a given (i.e., measured) f(v), and a known (i.e., measured) U therm the decomposed multibean energy density, (U therm), MB can be found by sutracting DU. This is usually easier than calculating (U therm) MB by summing all of the beam thermal energy densities (as in Eqn. (11a)) Consistent with the treatment leading to the multibeam energy density, we next treat the vector, false energy flux, DQ. Here, all energy fluxes are rendered dimensionless by dividing by mnun 3 /2: By the same reasoning that led to the "thermal" multibeam energy moment in Eqn. (21), we can express the dimensionless "thermal" multibeam energy flux vector moment as, Hence, if we know the standard moment Qtherm for f(v), we can find (Qtherm) MB by subtracting the "false thermal" part, DQ. Adding Eqns. (22) and (23) verifies that either way the moments are taken (standard method or multibeam method) one obtains the same undecomposed Q. The same is true of the undecomposed energy density, U, as is evident when DU is eliminated from Eqns. (18a) and (21).
One can use a similar strategy to find the multibeam pressure tensor, P MB , from the known standard pressure tensor, P by introducing the RAM pressure tensor moment, PRAM and stress tensor moment, T, as in Eqn. (12). Here, T = P + PRAM, has the same value for standard or multibeam pressure moments. Dimensionless pressures are obtained by dividing all pressures by nmun 2 . If DP is defined as , then, by the same arguments as above, Note that the multibeam pressure moment tensor, P MB , contains the scalar multibeam energy density, , since, in dimensionless units, the trace of P MB , Tr[P MB ] = .
When u, P, and Q are known standard moments of a known or measured f(v), one need only find the density and velocity (nj, uj) of each port Equation, fj(v) in order to evaluate DP and DQ and, from them, P MB , (Utherm) MB and (Qitherm) MB . (See Table 3). This is a more effective strategy than calculating a sum of higher order beam thermal moments, as in the definitions of P MB , (Utherm) MB and (Qitherm) MB given in Eqns. (11).

MULTIBEAM ENERGY TRANSPORT EQUATIONS
The multibeam energy moments satisfy their own multibeam space-time energy transport equations, analogous to the normal energy transport equations, Eqns. (1): Here J is the system current equal to the sum over all N beam currents. The transfer of energy between multibeam bulk energy and multibeam thermal energy is governed by the sum over beams of the work done by each beam's pressure against that beam's flow velocity. Note that the multibeam pressure which appears in the multibeam force-momentum eqn. does not appear in the multibeam energy transport equations because the work done by the pressure in the multibeam energy transport equation must be computed beam by beam and then summed.

MULTIBEAM VELOCITY DISTRIBUTIONS AND FALSE THERMAL MOMENTS
In this paper we have addressed issues that arise when taking energy moments of effectively disjoint particle velocity distributions, f(v), which can be expressed as a sum of N beams, The standard moments of any f(v) are based on the assumption that there is only one flow velocity, u, associated with f(v). Examples of standard coherent moments are the bulk kinetic energy density, Ubulk = nmu 2 /2, of the system and the bulk kinetic energy density flux vector, Qbulk = numu 2 /2 of the system. Moments which depend on correlations of fluctuations (v -u) are said to be incoherent or "thermal". Examples are the standard thermal energy density, Uthermal = ∫d 3 vf(v) m|v -u| 2 /2, and the standard thermal energy flux vector, Qthermal, defined, in our convention, to be the sum of the enthalpy and heat flux vectors. There are separate spacetime transport equations for the coherent and the incoherent energy moments (Eqns. (1)). In this paper we do not address the space-time evolution of energy moments of electrons or ions but rather consider their properties averaged over short time periods and small spatial regions.
Standard moments of multibeam velocity distributions f(v) = f1(v) + … + fN(v) tend to inflate Uthermal and change the direction and magnitude of Qthermal by amounts we call the false parts of these standard thermal moments, DUthermal and DQthermal. A simple example is the false thermal energy, DUthermal, of a pair of cold beams. In this case DUthermal arises from the standard-moment misidentification of their center-of-mass-frame kinetic energy as thermal energy.
There is an alternate way of taking moments well-suited for multibeam velocity distributions of form f(v) = f1(v) + … + fN(v), which we call multibeam moments. Multibeam thermal energy density moments, do not have false parts. They are calculated by taking a standard thermal moment of each beam and summing over beams. For example, the multibeam thermal energy density moment is given by, The false part of a thermal moment is the difference between the standard and multibeam thermal moments. For example, . Coherent multibeam moments are calculated in the same way. For example, the multibeam kinetic energy flux is given by, The time-evolution of the multibeam coherent and thermal energy densities is governed by a pair of space-time energy equations (Eqns. (26)) similar to those for the normal bulk and thermal energy densities. Energy transfer between coherent and thermal multibeam energy densities is now governed by the sum over beams of the work done by the multibeam pressure density force of each beam on or against the beam's flow velocity, uj.
To illustrate the advantages of taking multibeam energy moments the method was applied to a system of N co-axial tri-Maxwellians. Due to symmetry, none of the N tri-Maxwellians had a standard heat flux moment. However, the standard heat flux moment of the sum of beams, f(v) = f1 + … + fN was non-zero. Hence all of the standard heat flux of f(v) is false and the multibeam heat flux momert of f(v) is zero.
For the special case of a pair of equal and opposite cold beams the normal thermal energy density is nonzero, while the multibeam thermal energy density is zero.

UNDECOMPOSED ENERGY MOMENTS
There is still another way of taking energy moments of f(v), this time with no reference to either a single beam flow velocity or to multiple beam flow velocities and hence no distinction between coherence or incoherence of energy moments.
An example is the overall energy moment, U = òd 3 vf(v)mv 2 /2. If v 2 is replaced by [(v-u) + u] 2 , U decomposes into a sum of the normal coherent (bulk kinetic) energy density and incoherent (thermal) energy density, U = Ubulk + Uthermal. For that reason, we have referred to U as the undecomposed energy density. For a system of N beams the substitution of f = f1 + … + fN in the integral for U, followed by a replacement of v 2 by [(v-uj) + uj] 2 in each beam integral, j, leads to the decomposition of U into a sum of coherent and incoherent multibeam energy densities, .
Therefore, regardless of whether f(v) consists of one or more beams, the undecomposed energy moments are meaningful and invariant under which method of decomposition is employed. This is true for the undecomposed energy flux vector, Q, as well as for U. The space-time energy transport equation for U and Q is given by Eqn. (2a).
Analysis of measured velocity distributions, f(v), using moments such as U and Q has the advantage that undecomposed energy moments are well-defined and meaningful for singlepeaked, multi-peaked, disjoint or arbitrarilly-shaped f(v), far from thermal equilibrium. The disadvantage is that the energy and energy flux cannot be interpreted as "coherent" or "incoherent" (i.e., bulk or thermal).
One interesting and important consequence of the invariance of U and Q is that false thermal parts of normal thermal moments which are absent in multibeam thermal moments appear in the corresponding multibeam bulk moments. As one example, false normal thermal energy density in an assembly of beams will appear as an increase in (multibeam) bulk kinetic energy density. Equivalently, false temperature should be reinterpreted as increased kinetic energy per particle in the assembly of beams. In another example, false normal heat flux in a distribution consisting of a sum of beams (e.g., N tri-Maxwellians) will show up as extra enthalpy flux and kinetic energy flux in the collection of beams. Since the fluxes are vectors the more precise statement is .
We have shown that the invariance of undecomposed energy moments leads to a simplification in the determination of the false thermal parts of the normal thermal energy and normal energy flux moments. For example, we may use the invariance of U and Q to rewrite the false normal thermal energy density and the false normal thermal energy flux in terms of . Hence, the false parts of these normal thermal moments can be expressed entirely in terms of coherent beam velocities and beam densities. This will form the basis of the so-called "visual method" for finding false thermal moments of measured f(v) to be be described in a future paper.

PRESSURE MOMENTS
In addition to U and Q, another undecomposed moment we have treated is the particle stress tensor, T = òd 3 vf(v)m vv, whose divergence appears in the particle momentum equation (Eqn. (6b)). T decomposes into the normal "thermal" pressure tensor moment, P and the RAM pressure tensor, P RAM = muu when both v's in the dyadic vv are replaced by v = (v -u) + u. It also decomposes into the multibeam pressure tensor, , and the RAM pressure tensor, , when f(v) is replaced by (f1(v) + … + fN(v)) and v is expressed as v = (v -u j ) + u j in the RAM pressure of each of the N beams. Once again, there wil be a false (tensor) part of the normal thermal pressure tensor moment, P, in a multibeam f(v), given by the difference, DP = P -. False pressure will show up in the multibeam RAM pressure tensor due to the invariance of T.

SIGNIFICANCE
Recent high-resolution simulations and measurements of electron and ion velocity distributions, f(v), during magnetic reconnection have revealed that f(v) can be very far from thermal equilibrium, quite complex, and often effectively disjoint. Hence the interpretation of particle energetics sometimes requires new tools although we are still wedded to using long-standing familiar ones such as fluid concepts and fluid models. Kinetic theory modeling of processes in collisionless plasmas is increasingly used to understand space physics measurements, especially through particle-in-cell simulations. Vlasov equations contain particle and field physics which is not present in fluid equations. Velocity moments of the Vlasov equation are often the basis for deriving fluid variables and equations.
In this paper we retain this framework but introduce a multibeam method of taking velocity moments which is sometimes more appropriate than the "normal" method of taking velocity moments for interpreting distributions, f(v), which are far from equilibrium and effectively disjoint. In the normal method of taking moments it is assumed that there is only one overall flow velocity associated with f(v) and the energy moments of f(v) are either coherent (bulk kinetic energy, bulk energy flux, RAM pressure, etc.) or incoherent (pressure, thermal energy density, enthalpy flux, heat flux, etc.). This can lead to counterintuitive results, such as the false normal thermal energy in a system, f(v) of two cold equal and opposite cold electron beams. The remedy is well known -consider each electron beam to be a separate species. When the system thermal energy is calculated for each species separately and then summed, the system energy is found to be entirely kinetic rather than entirely thermal.
The underlying approximation of the multibeam method is to express f(v) as a sum of distributions, which we call beams, so that for electrons or for ions, f(v) = f 1 (v) + … + f N (v). We can think of each beam as a separate subspecies within the N electrons or the N ions. Each beam has a centroid velocity, u j , a density, n j , etc. Multibeam energy moments of f(v) are constructed by taking normal energy moments of each beam and then summing over beams. Thus, multibeam energy moments can again be either coherent (multibeam bulk kinetic energy, multibeam bulk energy flux, multibeam RAM pressure, etc.) or incoherent (multibeam pressure, multibeam thermal energy density, multibeam enthalpy flux, multibeam heat flux, etc.). Multibeam incoherent moments do not contain false thermal parts. However, any of the normal incoherent energy moments can contain false thermal parts which actually should be reassigned (whether scalar vector or tensor) to a corresponding multibeam coherent moment.
This has practical significance for normal fluid interpretations of multibeam f(v) measured by spacecraft during magnetospheric reconnection or found in PIC simulations. Some hypothetical scenarios illustrating the differences in interpretation when taking multibeam moments of a multibeam electron distribution, f(v), rather than taking normal moments are: o If an elevated normal electron temperature T = (U thermal /n) measured by an MMS spacecraft over a 30 ms time interval is interpreted as due to a heating process and the electron velocity distribution, f(v) measured by the FPI instrument is found to consist of disjoint electron beams, the multibeam electron temperature will be lower, and the multibeam bulk kinetic energy per particle higher by the same amount. Although the normal temperature suggests electron heating has occurred, the multibeam temperature reveals that some of the higher electron energy is coherent rather than incoherent, arising from electron acceleration to a higher bulk kinetic energy. The relative magnitudes of thermal energy density vs bulk kinetic energy remain to be studied for specific measurements.
o Suppose a measured electron distribution, f(v), consists of a (high-energy) crescentshaped beam and a broad background with its own small flow velocity. The magnitude and direction of the (measured) normal heat flux moment of f(v) may differ from the magnitude and direction of the multibeam heat flux. In other words, the normal electron heat flux vector may have a false part which multibeam energy flux moments reveal to actually be carried by the electron enthalpy flux vector and the bulk electron kinetic energy flux vector.
It is worth restating and discussing the assumptions and limitations of the approach developed here for taking energy moments of non-equilibrium particle distributions, f(v). A number of these issues will be addressed by work currently in progress.
The multibeam moment method described here applies to non-equilibrium velocity distributions, f(v), essentially at one time and place, which can be approximated as a sum of sub-distributions, f 1 (v) + ... + f N , referred to as "beams." While an analytic example has been given of a sum of tri-Maxwellians, it has not been demonstrated in this paper when or if an arbitrary measured non-equilibrium f(v) can be expressed in this manner. One strategy would be to approximate f(v) as a sum of parametrized analytic sub-distributions (not necessarily tri-Maxwellians) and to determine the parameters by a least-squares fit to the measured f(v). There are preliminary indications that this method can be useful and can be adapted to treat high-energy distributed velocity-space "clouds" or "haloes" as particular subdistributions (beams) in the sum.
Aside from the question of uniqueness of a beam decomposition for a given N it is not always clear how many beams, N, should be included in the sum. In the theoretical limit in which N equals the number of particles in f(v), one enters the microscopic kinetic regime, where there is not even a concept of fluid variables such as temperature. However, published work on machine learning suggests a single digit-number of beams, N, is reasonable (DuPuis, et al, 2020).
The least squares method and other strategies for expressing ion distributions measured during magnetic reconnection in the magnetosphere (e.g., Fig. (2)) have been presented in papers at conferences, and is about to be submitted for publication.
Although multibeam space-time energy transport equations have been developed (Eqns. (26)), the present paper does not consider particle energy transport in space and time. The multibeam fluid moments taken here are of an Eulerian multibeam distribution, f(v), at one space-time point (or an average over at most a small range of such points). The study of processes such as multibeam particle energization and space-time energy transfer are beyond the scope of the moment methods discussed here. Nevertheless, undecomposed energy moments discussed in this paper may be useful in following spacecraft-measured Eulerian f(v) in time along the spacecraft orbit, especially when multiple beams appear to merge and split and when the distinction between coherent and incoherent moments is not essential.
Considerable insight is gained into the origins and energization of beams from Lagrangian studies based on kinetic theory (i.e., simulations), such as the particle trajectories traced in Fig. (3). For example, a pair of beams may be two-stream unstable only so long as the spacetime particle trajectories remain crossed at the same spatial point. Even if instability does occur, the spacecraft must remain at that crossing point long enough for it to be observed.
In conclusion, care must be taken when applying fluid concepts to highly non-Maxwellian particle velocity distributions, especially when they are disjoint.