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. See my earlier post on collisional and collisionless plasmas, e.g.

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. 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.

As reported in * EOS: Earth & Space Science*, (March, p. 44) M.V. Goldman

*et al*- in a new paper ('

**Multibeam Energy Moments of Multibeam Particle Velocity Distributions'**) appearing in

*Journal of Geophysical Research: Space Physics:*

https://doi.org/10.1029/2020JA028340

have presented a novel method to treat space plasmas using multibeam velocity components for a measured plasma distribution. The approach allows a better understanding of how much of the system's fluid energy is kinetic and how uch is thermal (ie. associated with velocity fluctuations about the primary beam flow velocity, **u.** The authors suggest that a multibeam approach offers clear advantages when interpreting energy transport in complex plasmas, although they note that the approach is based on assumptions, such as the number of beams into which a given distribution should be decomposed. In this post I examine the multibeam concept as laid out in their paper with the view to better grasping the importance of their work.

**u**and one mean number density, n. An illustration of a dual beam system from a previous blog post is shown below:

^{3}

**v**f(

**v**)|

**v**–

**u**|

^{2/}/2 or: ò d

^{3}

**v**f(

**v**) (

**v**–

**u**) |

**v**–

**u**|

^{2}.

^{2/}/2, and a thermal energy density moment, m ò d

^{3}

**v**f(

**v**)|

**v**–

**u**|

^{2/}/2 ; the energy density flux is a sum of a coherent bulk kinetic energy flux, unmu2 /2, and a “thermal” energy flux moment (enthalpy plus heat flux).

^{2}>/2, which can be decomposed by writing:

**v**=

**u**+ (

**v**–

**u**) In practical space plasma applications f(v) is measured by electrostatic analyzer instruments

*beam by beam*and

*add them together. The authors call such sums*

__then__**.**

*multibeam moments*_{ o}and -u

_{o }and equal densities n

_{ o}. 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, U (bulk) is zero and the (single) density is n = 2n

_{ o}.

_{ o})

^{2}/2. This incoherent part of the energy density is often called the thermal energy density and written as U(therm) = nT'. This yields an effective temperature, T ' = m(u

_{ o})

^{2}/2. The authors provide the following cautionary note:

*false*temperature, T' 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)."

_{ o}+ f

_{ b}.)

_{ o }, u1 = u

_{ o}and u2 = - u

_{ o}. The energy moments of each beam are now U(bulk)1,2 = n

_{ o }m(u

_{ o})

^{2}/2 and U(therm)1,2 = 0.

_{ }m(u

_{ o})

^{2}/2, where n = n1 + n2 = 2n

_{ o}and (Utherm) 2-beam = 0 + 0, which is more intuitive than the results of the standard moment analysis, with respective velocities: U(bulk) = 0 and U(therm) = nm(u

_{ o})

^{2}/2.

*et al*summarize their results for the two different ways of taking energy moments of f(v) = f1(v) + f2(v) in their table below:

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