Abstract
This chapter is aimed at helping to understand the magnetospheric plasma as a self-organizing entity with self-generated electromagnetic fields (whereas in all previous chapters the magnetic and electric fields were given, of sources external to the particle population). It starts with an introduction to collisionless plasma physics exclusively based on the understanding of adiabatic motion of individual particles. The concepts of particle (kinetic) and guiding center fluids are introduced as yet another example of “physics as the art of modeling”. The corresponding distribution functions and their relations to macroscopic quantities are examined for the hypothetical case of identical particle ensembles, linking magnetization density with perpendicular pressure in a guiding center fluid. On the basis of very simple examples (throughout the text we call these “kindergarten examples”) the physical meanings of equivalent and convective current densities and their return circuits in a guiding center fluid are analyzed in detail, with emphasis on their origin in geometric aspects of cyclotron motion. Different classes of current densities are defined in general terms and their role in the generation of magnetic stresses in a particle ensemble is thoroughly examined. We finally turn to quasi-neutral mixtures of positive and negative particles, introducing the so-called center of mass fluid, discussing its properties and related equations. The concept of quasi-neutrality is examined and the plasma parameter known as Debye length is introduced; the reason why the magnetic field does not appear in the Debye length is discussed explicitly. All this leads to the plasma momentum and magnetohydrodynamic equations. The chapter concludes with the introduction of collisions and the formulation of the so-called generalized Ohm equation; the physical meaning of its terms is discussed as well its link to Maxwells equations and the “chicken-and-egg” problem of whether currents drive fields or fields drive currents in a plasma. Several simplifications for some special situations are discussed, introducing concepts like Hall conductivity, magnetic field diffusion, frozen-in magnetic field lines and Alfvén waves.
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Notes
- 1.
This sounds a bit pompous. The maximum total energy flow in the astronomically-sized magnetosphere can be estimated at barely ∼ 20,000 MW, its total plasma mass a mere 20 t.
- 2.
A brief detour into Foundations of Physics is in order here. In the Preface we already stated that “physics is the art of modeling”, and in Sect. 1.1 we introduced the model of a “guiding center particle”. A fluid (any fluid!) is also a model—the model of a system in which a huge, mathematically unmanageable, number of physically real particles (molecules, atoms, electrons, nucleons, quarks, gluons, sand grains, etc., depending on the ensemble in question) has been replaced in our mental image and in the quantitative description by a virtual continuum (see also Sect. A.1, page 160). We speak of and quantitatively describe “parcels” of fluid and imagine how they are deformed as they move, and guided by what our physiological senses experience when exposed to liquids or flowing gases, we introduce macroscopic variables which can be used for practical purposes, like density, bulk velocity, pressure, temperature, internal energy, entropy, etc. Statistical mechanics and, as a corollary, plasma physics were developed to link approximate but intuitive macroscopic continuum descriptions of matter with their physically real microscopic structures that can only be revealed through the use of scientific instruments. In our case, distribution functions and the differential equations which they obey establish such a link. The main aim of any fluid description is to formulate physical-mathematical relationships between the macroscopic variables so as to provide a “coarse-grained” quantitative description of the dynamic state of the ensemble—regardless of the unknowable detailed state (position, velocity) of each elementary constituent.
- 3.
From now on, all macro-variables in the guiding center fluid will carry the subindex g, whereas homologous variables in the kinetic particle fluid model will not be subindexed.
- 4.
- 5.
Remember that this is the usual explanation given in E & M texts to justify the appearance of an equivalent \({\boldsymbol \nabla }\times {\boldsymbol M}\) current in magnetized materials (although in ferromagnetism the magnetization is not due to “little current loops” in atoms but due to the intrinsic quantum magnetic moment (spin) of electrons). Since in an ensemble of trapped particles the magnetic moment associated to a guiding center particle is always directed antiparallel to B, plasmas behave like a diamagnetic gas—as we already had anticipated.
- 6.
Never mind that in these kindergarten examples we have considered only one class of particles—the results about currents thus far are independent of the electric charge of the particles involved.
- 7.
This dropout, predicted by theoreticians in the early days of magnetospheric physics, caused confusion among experimentalists studying ring current data, who from the beginning assumed this West-East current to be due to the convective E-W and W-E drift of trapped protons and electrons, respectively. However, the ring current is the superposition of a E-W convection drift current with an equivalent diamagnetic current (5.11), the latter with an W-E inner ring (radially outward directed density or pressure gradient in (4.24)) and a E-W outer ring where the density gradient is reversed.
- 8.
A question still subsists: How can two different solutions, either \({\boldsymbol V }_{g}\) or \({\boldsymbol V }\), be obtained for the two different fluid models from one and the same equation? The answer is that \({\boldsymbol V }_{g}\) or \({\boldsymbol V }\) sit inside \({\boldsymbol J}\), which in the case of the magnetized guiding center fluid model also contains \({\boldsymbol \nabla }\times {\boldsymbol M}\), with \({\boldsymbol M}\) in turn being a function of \({\boldsymbol B}\) and p ⊥ .
- 9.
Of course, we can show only a subgroup of particles of each class in the central element of volume at time t; it may be crossed by many other particles coming from other pairs of pre-t parcels.
- 10.
In most plasma physics books, the Debye length and the plasma frequency are introduced at the very beginning, without any mention of the magnetic field (and often assuming a Maxwellian distribution). This sometimes confuses the student, especially if the book mainly deals with magnetized plasmas.
- 11.
It is assumed that the actual momentum transfer can vary with equal probability distribution between 0 and a maximum of \(2n_{e}m_{e}\nu _{\mathit{coll}}\).
- 12.
- 13.
Radio propagation engineers define the Hall coefficient and Hall conductivity with opposite sign.
References
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Roederer, J., Zhang, H. (2014). Collisionless Plasmas. In: Dynamics of Magnetically Trapped Particles. Astrophysics and Space Science Library, vol 403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41530-2_5
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