Abstract
This Chapter is devoted to the derivation of basic equations that will lay the foundation for the more complicated applications specific to boundary physics. We start with fundamental concepts, gradually obtaining equations at different levels of accuracy and complexity. The plasma kinetic model is introduced first, including Boltzmann and Vlasov equations and remarks on how to construct a collisional operator that can capture the complexity of the long range interactions typical of charged particles. A fluid moment approach, based on well defined orderings of the variables, is then derived from the kinetic equation, allowing for terms that are important in the boundary plasma, such as sinks and sources of particles, momentum and energy. A number of different closures for these equations are discussed, so that the infinite hierarchy of moments can be broken in a rigorous way. In particular, we discuss the MHD and the drift ordering, which allow to specialise (by pruning) the general and complex fluid equations into more manageable ones, which still provide deep insight into the processes and phenomena that they can describe, but without additional and unnecessary costs. While this chapter is rather general and the concepts describes likely familiar to plasma physicists with sufficient experience (kinetic and fluid models, collision operators, moment equations, quasi-neutrality, ambipolarity, drifts are all discussed here), it is extremely important to understand the calculations in the following Chapters. A confident reader can skip it, but with the warning that some of the derivations might be unusual to a core physicist as they specialise to the boundary region of the plasma (although here we still focus on electron/ion plasmas, leaving detailed multi-species effect to later Chapters).
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Notes
- 1.
Note that i does not stand for ‘ion’ here! it is just a subscript that represents a generic particle.
- 2.
The right hand side of this definition is probably more intuitive. The factor 1∕2 is introduced to avoid double-counting [since Φ(x i, x j) = Φ(x j, x i)] and he condition i ≠ j takes into account the obvious restriction that a particle cannot interact with itself.
- 3.
Due to the limited amount of letters and symbols available, and the dislike of the author for excessive subscripting, we use the same letter, f or F, for both distribution functions and the forces acting on the particle. The two are concepts are made distinct and recognizable by the fact that the forces are in bold font, F, as all the vectors in this book, while the distribution functions are scalars and have normal font, F. This is not ideal, but necessary if we want to maintain an intuitive understanding of what the symbols mean.
- 4.
Throughout the book, we used different conventions to represent differential operators. Here we use ∂∕∂ x i to represent a gradient in a multidimensional space, but later on we will also use the symbol ∇ for the same purpose (although in general this latter notation is used more frequently in the book when we have a spatial gradient).
- 5.
The notation can be quite deceiving here: the bracket is multiplied by F, not applied to F. Indeed, \(\frac {\partial }{\partial {\mathbf {x}}_i}\cdot \dot {\mathbf {x}}_i \) is the spatial divergence of the vector \(\dot {\mathbf {x}}_i\).
- 6.
More rigorously, n α is the number of particles in the infinitesimal cube with edges x and x + d x.
- 7.
Bogoliubov, Born, Green, Kirkwood and Yvon.
- 8.
In reality, this is just an electromagnetic kinetic equation until we specify the form of the collision operator. The Boltzmann closure will be discussed in Sect. 2.3.
- 9.
As it is, the closed system is inconsistent, since (2.37) is derived under classical assumptions and is therefore Galilean invariant, while Maxwell’s equations are Lorentz invariant. This disagreement will be solved in Sect. 2.5 by removing the displacement current from (2.41)—hoping that this will not spoil the surprise.
- 10.
The average thermal energy (or thermal energy per particle) is proportional to the temperature, as we will see in (2.61). Here, and in the rest of the book unless otherwise stated, we will express the temperature in units of energy (Joules), which are obtained by multiplying the temperature expressed in Kelvin by the Boltzmann constant. The electrostatic energy is given by eϕ, see (2.5), with ϕ give, for example, by (2.9).
- 11.
We use here the small letter to represent the modulus of a vector, so that w = |w|. In the following, we will use this convention also for other vectors, whenever it does not cause confusion.
- 12.
As a matter of fact, Σαβ contains quite a lot of redundant information. Indeed, moving to the rest frame of the particles of species β, for example, already removes six variables from the argument of this differential cross section (\({\mathbf {v}}_{\beta }={\mathbf {v}}_\beta ^{\prime }=0\)). Finally, as we will see in Sect. 4.1, the conservation properties of the elastic collisions (momentum and energy) impose four constraints and reduce the free variables in the arguments of the differential cross section to two, as in the solid angle.
- 13.
The de Broglie length represents the length scale at which wave-like properties become important for a particle. The distance of closest approach is the distance at which an incoming particle, aimed head-on towards another, stops and reverses its direction (this happens when the Coulomb energy becomes identical to the initial kinetic energy of the moving particle).
- 14.
A Fokker-Plank equation typically represents the evolution of a quantity, usually a probability density function, under the effect of a drag and a diffusive force.
- 15.
In reality, mathematicians cannot yet prove that the H-theorem is rigorous since it is unclear if the solutions of the Boltzmann equation are smooth enough with sufficient generality.
- 16.
In 1948, C. Shannon introduced the concept of information entropy, and represents the level of information (or conversely, uncertainty) associated with a random variable. Given a discrete random variable X that can take the values x 1, x 2, …, x n with probability P(x 1), P(x 2) and so on, Shannon’s entropy is defined as \(H(X)= -\sum _{i=1}^nP(x_i)\log P(x_i)\), which is reminiscent of (2.57).
- 17.
We are also assuming that the system is spatially homogeneous or, at least, that the gradients of its macroscopic properties are sufficiently weak.
- 18.
At the moment, we neglect the electric energy, the contribution of which is important for charged particles and will be discussed in Sect. 3.4.4.
- 19.
In the common practice, the terminology around the Maxwell-Boltzmann distribution (our Maxwellian) is sometimes ambiguous. Strictly speaking, this distribution describes how the speed (or the modulus of the velocity) is distributed, not the velocities themselves. The Maxwell-Boltzmann distribution has therefore a positive definite support and a functional form corresponding to a χ distribution. It is therefore evident that we are calling a Maxwellian something that is not. However, if the speed of the particles is Maxwellian, it necessarily follows the velocity distribution has the form (2.60), so that we can turn a blind eye and carry on with this imprecise terminology.
- 20.
The relative velocity makes use of the reduced mass, as this gives the representation of a two body interaction in the frame of reference of one of the bodies.
- 21.
It immediately follows that, for a low Z single ion plasma in thermal equilibrium, τ c,ee ∼ τ c,ei ≪ τ c,ii ≪ τ c,ie, as (2.62) confirms.
- 22.
Not to be confused with the BBGKY formalism!
- 23.
The heat flux density is the rate of heat transfer per unit area normal to the direction of heat transfer and is often simply called heat flux. As a matter of fact, often we omit the word ‘density’ when dealing with the quantities in (2.72). However, this equation deals with the evolution of the energy per unit volume!
- 24.
We will show in Sect. 2.5 that the electric energy is negligible compared to the magnetic energy if the plasma dynamics is not relativistic.
- 25.
This does not mean that relativistic corrections can be always discarded in tokamaks. The acceleration of the runaway electrons is a classical example.
- 26.
In reality, Maxwell’s equations can have a few Galilean limits. We will focus on the one relevant for tokamak research, where the magnetic field dominates over the electric field.
- 27.
In our derivations we will closely follow the 2013 European Journal of physics paper by Manfredi [1] on the non-relativistic limit of Maxwell’s equations.
- 28.
In order to arrive to this conclusion, we need to define a reference electrostatic potential, \(\overline {\phi } = \overline {E}\overline {L}\), and a reference temperature of the plasma associated with the charge and the current, \(\overline {T}\). Then we take \(e\overline {\phi } = \overline {T}\), corresponding to an instantaneous and dissipationless response of the plasma. Noticing that \(\lambda _D = \sqrt {\overline {T}\epsilon _0/(e\overline {\rho })}\), as will be discussed in Chap. 3, we find the expression for a.
- 29.
Electromagnetic radiation cannot be recovered in this approximation, since waves are associated with fields that satisfy |E|∼ c|B|. In addition, we are kind of going in a circle here: in order for ∇⋅E 0 to vanish, we are implicitly assuming that the zeroth and first order of the electric charge vanish because we are assuming small charge separation. This leads to the fact that the magnetic field is much larger than the electric field. We could have taken also the opposite path, starting by requesting that the magnetic field is dominant over the electric field and obtained that the electric charge must be second order.
- 30.
Here and in the following, we still define collisionality in a loose sense. Substantially, we refer to high collisionality when the typical length scale of the problem we are treating is of the order of or shorter than the mean free path between collisions. The term ‘collisionality’, however, is part of the boundary physics jargon and has a precise definition that will be formally given in Sect. 5.5.
- 31.
There are a number of complicated passages in this derivation, which we avoid for the sake of brevity. While mathematically rewarding, they do not add to the philosophy of the derivation.
- 32.
As a technical side note, it is useful to know that finding a distribution function by inverting a collision operator is commonly called a Spitzer problem, after the great plasma physicist that was among the first to tackle this issue.
- 33.
Sometimes the viscosity is expressed as a kinematic viscosity, which is simply ν α = μα∕(m α n α).
- 34.
- 35.
Their importance is a reflection of the fact that the plasma is intrinsically anisotropic.
- 36.
At T e ∼ 2500 eV, which are relevant temperatures for present-day experiments, the plasma has a comparable resistivity to the best conducting metals at room temperature (like copper, silver). It reduces by one order or magnitude in very hot cores of future reactors, where T e ∼ 10, 000 eV. This is due to the strong inverse temperature dependence, and it also implies that resistivity goes up significantly in the SOL, where it reaches values comparable to amorphous carbon (four orders of magnitude higher than copper) in detached conditions.
- 37.
MHD is commonly used to study equilibria, which is apparently a paradox, considering that it is well-suited to describe just fast phenomena. In reality, this is correct because the theory allows to investigate the possibility of perturbations as well as the perturbations themselves!
- 38.
For a complete and exhaustive account, Friedberg’s [5] book (2014) is always a good suggestion.
- 39.
- 40.
The parallel and perpendicular length scale can be interpreted as associated either to the equilibrium or to perturbations, as long as the perturbations are sufficiently large with respect to the ion Larmor radius. For example, plasma filaments in the boundary region of the plasma are perturbations, but their characteristic perpendicular size is 10–20 times ρ i, as we will see in Chap. 7.
- 41.
Remember that we defined ∇∥ as a vector in the direction of the magnetic field. Hence our slightly inelegant notation for the parallel gradients, which are represented by b ⋅∇ or b ⋅ ∂∕∂ x.
- 42.
In reality, using (5.13), we find that the perpendicular current term is negligible when β ≪ q𝜖, where q is a parameter that tends to be larger than 2 for reasons of plasma stability, and typically between 3 and 8.
- 43.
Here ρ is the electric charge defined in (2.42), not the Larmor radius.
- 44.
The requirement that diamagnetic and parallel currents play comparable roles in the charge conservation equation implies that J ∥∼ J ⊥∼ (ρ i∕L ⊥)env t,i. Note that even if the parallel ion and electron velocity are almost sonic, the parallel current, which represents their difference, can and actually should remain small.
- 45.
While this scaling is in contradiction with (2.136), the vorticity equation is sometimes used in this regime, providing reasonable results.
- 46.
In 1971, Hinton and Horton [7] wrote a famous and pioneering article introducing for the first time many important concepts associated with the drift approximation.
- 47.
- 48.
Note that ∂ b∕∂ x is the gradient of the b vector and that u 0,e ⋅ ∂ b∕∂ x = (u 0,e ⋅ ∂∕∂ x)b is just a vector identity. We used here this alternative notation to draw the attention of the reader to this property of the convective derivatives. Remember also that, in general, u 0,e ⋅ ∂ b∕∂ x ≠ ∂ b∕∂ x ⋅u 0,e, as the dot product between vectors and tensors is not commutative. More details in Appendix A.1.
- 49.
This terminology is often misleading for people unfamiliar with it, as it is not strictly related to the commonly used thermodynamic concept of adiabatic behaviour.
- 50.
Some well known codes and papers adopt a different basis and notation, with y being the radial direction. This changes a number of things, including the signs of the drifts.
- 51.
To avoid confusion in the notation, it is useful to specify that we use x to represent the Cartesian coordinate system and x(x), y(x) and z(x) for our generic curvilinear coordinates. Note that having taken a normalized basis, all vectors and tensors are expressed with physical components (not covariant, nor contravariant)!
- 52.
This form is widely used, but it relies on a small β approximation, which allows neglecting the second term in the first equality of (2.144).
- 53.
This term includes friction coming from a number of interactions that will be discussed in Chap. 4 such as elastic collisions, (4.22) and ionization, (4.44), plus many others that involve molecules and impurities. Specifically, using the definition of R below (2.71) and the expression in Chap. 4, one can prove that for the electrons R o ≈ m e n e(u n −u e)∕τ, where τ is the shortest between the characteristic ionization and elastic time. Since the friction is proportional to m e, it is often neglected.
References
Manfredi, G.: Non-relativistic limits of Maxwell’s equations. Eur. J. Phys. 34(4), 859 (2013)
Braginskii, S.I.: Transport processes in a plasma. Rev. Plasma Phys. 1 (1965)
Mikhailovskii, A.B., Tsypin, V.S.: Transport equations in Plasmas in a curvilinear magnetic field. Beitr. Plasmaphys. 24(4), 335 (1984)
Simakov, A.N., Catto, P.J.: Drift-ordered fluid equations for field-aligned modes in low-β 2135 collisional plasma with equilibrium pressure pedestals. Phys. Plasmas 10, 4744 (2003)
Freidberg, J.P.: Ideal MHD. Cambridge University Press, Cambridge (2014)
Hazeltine, R.D., Kotschenreuther, M., Morrison, P.J.: A four-field model for tokamak plasma dynamics. Phys. Fluids 28(8), 2466 (1985)
Hinton, F.L., Horton Jr, C.W.: Amplitude limitation of a collisional drift wave instability. Phys. Fluids 14(1), 116 (1971)
Strauss, H.R.: Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19(1), 134 (1976)
Strauss, H.R.: Dynamics of high β tokamaks. Phys. Fluids 20(8), 1354 (1977)
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Militello, F. (2022). Plasma Equations. In: Boundary Plasma Physics. Springer Series on Atomic, Optical, and Plasma Physics, vol 123. Springer, Cham. https://doi.org/10.1007/978-3-031-17339-4_2
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