Theory of one-dimensional Vlasov-Maxwell equilibria:
with applications to collisionless current sheets and flux tubes \supervisorProfessor Thomas Neukirch \examiner \degreePh.D. \addresses \subjectMathematics \universityUniversity of St Andrews \departmentSchool of Mathematics and Statistics \groupSolar and Magnetospheric Theory \facultyFaculty of Science \pdfstringdefDisableCommands\pdfstringdefDisableCommands
Oliver Douglas Allanson
This thesis is submitted in partial fulfilment for the degree of PhD
at the University of St Andrews
July 5, 2019
Vlasov-Maxwell equilibria are characterised by the self-consistent descriptions of the steady-states of collisionless plasmas in particle phase-space, and balanced macroscopic forces. We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, as well as its application to current sheet and flux tube models.
The ‘inverse problem’ is that of determining a Vlasov-Maxwell equilibrium distribution function self-consistent with a given magnetic field. We develop the theory of inversion using expansions in Hermite polynomial functions of the canonical momenta. Sufficient conditions for the convergence of a Hermite expansion are found, given a pressure tensor. For large classes of DFs, we prove that non-negativity of the distribution function is contingent on the magnetisation of the plasma, and make conjectures for all classes.
The inverse problem is considered for nonlinear ‘force-free Harris sheets’. By applying the Hermite method, we construct new models that can describe sub-unity values of the plasma beta () for the first time. Whilst analytical convergence is proven for all , numerical convergence is attained for , and then after a ‘re-gauging’ process.
We consider the properties that a pressure tensor must satisfy to be consistent with ‘asymmetric Harris sheets’, and construct new examples. It is possible to analytically solve the inverse problem in some cases, but others must be tackled numerically. We present new exact Vlasov-Maxwell equilibria for asymmetric current sheets, which can be written as a sum of shifted Maxwellian distributions. This is ideal for implementations in particle-in-cell simulations.
We study the correspondence between the microscopic and macroscopic descriptions of equilibrium in cylindrical geometry, and then attempt to find Vlasov-Maxwell equilibria for the nonlinear force-free ‘Gold-Hoyle’ model. However, it is necessary to include a background field, which can be arbitrarily weak if desired. The equilibrium can be electrically non-neutral, depending on the bulk flows.
Prof Thomas Neukirch, University of St Andrews
Dr Andrew Wright, University of St Andrews
Prof Alexander Schekochihin, University of Oxford.
From the different research projects in my doctoral studies, the following papers have been published, or have been submitted:
O. Allanson, T. Neukirch, F. Wilson & S. Troscheit:
An exact collisionless equilibrium for the Force-Free Harris Sheet with low plasma beta, Physics of Plasmas, 22, 102116, 2015
O. Allanson, T. Neukirch, S. Troscheit & F. Wilson:
From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials, Journal of Plasma Physics, 82, 905820306, 2016
O. Allanson, F. Wilson & T. Neukirch:
Neutral and non-neutral collisionless plasma equilibria for twisted flux tubes: The Gold-Hoyle model in a background field, Physics of Plasmas 23, 092106, 2016
O. Allanson, F. Wilson, T. Neukirch, Y.-H. Liu, and J. D. B. Hodgson:
Exact Vlasov-Maxwell equilibria for asymmetric current sheets, Geophysical Research Letters, 44, 2017
O. Allanson, T. Neukirch and S. Troscheit:
The inverse problem for collisionless plasma equilibria, Invited paper for The IMA Journal of Applied Mathematics, submitted
There are people without whom this PhD would not have been possible, and there are those without whom it would not have been the same.
Those in the latter camp:
- Thomas Neukirch
Chief amongst them. Thank you Thomas for supporting me wholeheartedly at every stage, for making sure that I didn’t make a complete mess out of this, and for this adventure in Vlasov theory. One day we’ll convince everyone that it is so much more interesting than MHD.
To Sophie Dawe, Thomas Elsden and Cara Fraser (with an honourable mention for Jonathan Fraser, you were there often enough). Thank you for teaching me that you can never have too much of a good thing. And thank you for the impromptu - but regular - dancing to the house song. A particular thank you to Tom for the camaraderie of our days spent at Martyr’s Kirk; the gleeful mid-morning breaks spent shivering over bacon sandwiches and the Bialetti are crystallised in my memory.
- The YRM2016 team
Thomas Bourne, Zoë Sturrock, Sascha Troscheit, Cristina Evans, Daniel Bennett, and Fiona MacFarlane. That was all very fun wasn’t it, and extremely efficient!?
- Sascha Troscheit
I don’t know how you managed to get mentioned twice? Thank you for the innumerable distracting visits to my office, and for implicitly agreeing to make them useful, by engaging with me in my plasma physics problems. One day I’ll help you with slippery devil’s staircases.
I would like to thank the STFC for allowing me to spend three and a half more years as a student, enabling me to fiddle around in plasma physics. I would also like to acknowledge grants from the National Science Foundation, the Vlasovia conference, and the Royal Astronomical Society for making possible my trips to the 2016 AGU Chapman conference in Dubrovnik, the 2016 Vlasovia meeting in Calabria, and the 2017 American Geophysical Union fall meeting in San Francisco, respectively.
I, Oliver Douglas Allanson, hereby certify that this thesis, which is approximately 38000 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for a higher degree.
I was admitted as a research student in September 2013 and as a candidate for the degree of Ph.D in September 2014; the higher study for which this is a record was carried out in the University of St Andrews between 2013 and 2017.
Signature of candidate:
I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of Ph.D in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree.
Signature of supervisor:
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Some important notation
|Magnetic vector potential|
|Electrostatic scalar potential|
|Particle distribution function (DF)|
|Particle number density|
|Electric charge density|
|Electric current density|
|Particle flow relative to the bulk|
|Thermal pressure tensor|
|Scalar thermal pressure|
|Particle Hamiltonian (energy)|
|Particle canonical momenta|
|Particle thermal velocity|
|Thermal Larmor radius|
|Macroscopic length scale||e.g. current sheet width|
Physical constants (SI units)
|Speed of light in a vacuum|
|Permittivity of free space|
|Permeability of free space|
|GEM||Geospace Environmental Modelling|
|RHS||Right Hand Side|
|LHS||Light Hand Side|
|IFT||Inverse Fourier Transform|
|FFHS||Force-Free Harris Sheet|
|MMS||Magnetospheric MultiScale mission|
|AHS||Asymmetric Harris Sheet|
|AH+G||Asymmetric Harris plus Guide|
|GH+B||Gold-Hoyle plus Background|
This thesis is dedicated to
Sophie Dawe’s love, and levity
Phil Michaels’ life, and spirit
my parents’ support, and tender care
my family present, passed and in-law
my friends for bringing me to life,
and cutting me down to size.
At quite uncertain times and places,
The atoms left their heavenly path,
And by fortuitous embraces,
Engendered all that being hath.
And though they seem to cling together,
And form "associations" here,
Yet, soon or late, they burst their tether,
And through the depths of space career.
Soon, all too soon, the chilly morning,
This flow of soul will crystallize,
Then those who Nonsense now are scorning,
May learn, too late, where wisdom lies.
James Clerk Maxwell
Molecular evolution (abridged)
Nature, 8, 205, page 473 (1873)
- 1 The hierarchy of plasma models
- 2 Collisions in plasmas
3 Collisionless plasma equilibria
- 3.1 The ‘forward’ and ‘inverse’ approaches
- 3.2 Motivating translationally invariant Vlasov-Maxwell (VM) equilibria
- 3.3 Magnetic Reconnection
- 3.4 Forward approach for one-dimensional (1D) VM equilibria
- 3.5 Inverse approach for 1D VM equilibria
- 3.6 Previous work on VM equilibria
- 4 Thesis motivation and outline
- 5 Preamble
- 6 Introduction
- 7 Formal solution by Hermite polynomials
- 8 Mathematical validity of the method
- 9 Non-negativity of the Hermite expansion
- 10 Illustrative case of the use of the method: correspondence with the Fourier transform method
- 11 Summary
- 12 Preamble
- 13 Introduction
- 14 Force-free current sheet VM equilibria
15 VM equilibria for the force-free Harris sheet:
- 15.1 Calculating the DF
- 15.2 Convergence and boundedness of the DF
- 15.3 Moments of the DF
- 15.4 Properties of the DF
- 16 ‘Re-gauged’ equilibrium DF for the FFHS
- 17 Summary
- 18 Preamble
- 19 Introduction
- 20 Exact VM equilibria for 1D asymmetric current sheets
- 21 The numerical/“” equilibrium DF
- 22 The analytical/“exponential” equilibrium DF
- 23 Discussion
- 24 Preamble
- 25 Introduction
26 General theory
- 26.1 Vlasov equation in time-independent orthogonal coordinates
- 26.2 Vlasov equation in cylindrical geometry
- 26.3 Equation of Motion in cylindrical geometry
- 26.4 The Gold-Hoyle (GH) magnetic field
- 26.5 Methods for calculating an equilibrium DF
- 26.6 GH flux tube plus background field (GH+B)
- 27 The equilibrium DF
28 Analysis of the equilibrium
- 28.1 Non-neutrality & the electric field
- 28.2 The equation of state and the plasma beta
- 28.3 Origin of terms in the equation of motion
- 28.4 Plots of the DF
- 29 Summary
- 30 Context
- 31 Broader theoretical questions
- A Species-independent integrals
Chapter \thechapter Introduction
Most important part of doing physics is the knowledge of approximation.
1 The hierarchy of plasma models
More than of the known matter in the Universe is in the plasma state (Baumjohann and Treumann, 1997), by far the most significant material constituent of stellar, interplanetary, interstellar and intergalactic media. Not only is a deep understanding of plasmas then clearly necessary to understand the physics of our universe, but plasmas are also of real interest to us on Earth. Nuclear fusion experiments - and in principle, future power stations - necessarily exploit the plasma state to work, either using high-temperature plasmas confined by strong magnetic fields, or plasmas formed by the laser ablation of a solid fuel target.
Plasmas are often known as the ‘fourth’ state of matter, lying after the ‘third’, and more familiar gaseous state. At a temperature above K, most matter exists in an ionised state, however plasmas can exist at much lower temperatures should ionisation mechanisms exist, and if the density is sufficiently low (Krall and Trivelpiece, 1973). Figures 0(a) and 0(b) display some examples from the rich array of plasma environments in temperature-density scatter plots; from the relatively cool and diffuse plasmas of interstellar space, to the incredibly dense and hot plasmas of stellar and laboratory fusion. Since there is such variety in the physical conditions able to sustain plasmas, the ‘plasma state’ may best describe collective behaviours, the characteristics that persist despite the range of physical conditions that can sustain plasmas (we see from Figure 0(b) that even the free electrons in metals can be considered, or modelled, as a plasma). Matter is in a plasma state when the degree of ionisation is sufficiently high that the dynamical behaviour of the particles is dominated by electromagnetic forces (Fitzpatrick, 2014), and this can even be the case for ionisation levels as low as a fraction of a percent (Peratt, 1996).
Whilst many of these plasmas possess some shared tendencies and behaviours, it is not possible to capture all the detailed physics of the entire variety of plasma processes with one particular mathematical toolkit or model. Not only may some models fail to capture certain aspects of the physics by virtue of the approximations made, but they may be inefficient, or in fact insoluble when applied in practice. Hence, plasma physics is a discipline with a rich variety of perspectives and methods. Within each of these paradigms we make certain approximations and ordering assumptions, in order to capture the essence of the problem at hand.
1.1 Single particle motion
Taking the viewpoint of particulate matter as the fundamental approach, then a ‘full’ description of plasmas is found by solving the (Lorentz) equation of motion of each individual particle, written in classical form as
with the force, , on a test particle of species , of charge , at position , and with velocity , when under the influence of electric and magnetic fields, and . One can in principle integrate in time to calculate the trajectory of the particle for all future times (e.g. see Vekstein et al. (2002)),
for some initial condition. However, in all but the simplest electromagnetic field geometries these integrals may not even be able to be written down, and/or one might have to resort to numerical methods to calculate the trajectory. One more complication is the effect of the charged particles on the electromagnetic fields, and , and this shall be discussed in Section 1.2.
If a plasma is sufficiently magnetised it has small parameters
for and the characteristic values of the Larmor radius and gyrofrequency of individual particle gyromotion respectively, and and the characteristic length and time scales upon which the electromagnetic fields vary. In such a case there is a well understood treatment for particle orbits, namely Guiding Centre theory (e.g. see Northrop (1961); Littlejohn (1983); Cary and Brizard (2009)). Guiding centre theory models particle motion as a superposition of rapid gyromotion and a comparitively slow secular drift (e.g. see Morozov and Solov’ev (1966)). This gyromotion is depicted in Figure 2, reproduced from Northrop (1963); in which the notation and are used for the gyroradius ‘vector’ and magnitude respectively (in contrast to the use of herein); is the particle position; and is the guiding center position, such that .
The local gyromotion is governed by the conservation (to lowest order) of the magnetic moment,
for the mass of a particle, and the square magnitude of the particle velocity normal to the local magnetic field. This theory is very useful for heuristic understanding of individual particle motion, and for the study of ‘test particles’ embedded in a system of interest (e.g. see Threlfall et al. (2015); Borissov et al. (2016)), however not for ‘building up’ a theory that models the evolution of the particles and electromagnetic fields self-consistently. In a situation in which many particles are present, the self-consistent modelling of all of the particles would in practice require knowledge of the individual particle interactions via the electromagnetic fields of mixed origin (microscopic/self-generated and macroscopic/external fields), and in principle collisions, which is mathematically unwieldy. However, we note here that it is possible - whilst unconventional - to use -body particle dynamics to study collective effects in plasma physics (e.g. see Pines and Bohm (1952); Escande et al. (2016)), including the recent work of Dominique Escande and collaborators, who have taken an N-body approach to ‘re-deriving’ physical phenomena, such as Debye shielding and Landau Damping (see Figure 3 for a representation of how their work ‘sidesteps’ the more traditional routes).
1.2 Kinetic theory
To move forward we require a mean-field/statistical formalism that allows for a self-consistent set of evolution equations, involving the quantities that both describe the particles and electromagnetic fields. The electromagnetic fields are governed by Maxwell’s equations, and given in free space as
for and the charge and current densities respectively (e.g. see Griffiths (2013)). The electric permittivity and magnetic permeability in vacuo are given by and respectively, and they are related by , for the speed of light in free space. The electric and magnetic fields are defined as derivatives of the electrostatic scalar potential, , and the magnetic vector potential, , according to
The potential functions are themselves ‘sourced’ by and , respectively,
for the retarded time (Griffiths, 2013). The charge and current densities can be calculated by taking moments of the 1-particle distribution functions (DF), for particle species (e.g. see Krall and Trivelpiece (1973); Schindler (2007)), over velocity space
with and the number density and bulk velocity of particle species respectively. Hereafter we us the notation and to imply triple integration over all position and velocity space respectively,
unless otherwise stated. The DF, , represents the number density of particles in a microscopic volume of six-dimensional phase-space at a particular time, such that
Note that one can instead use the Klimontovich-Dupree description to exactly describe the particles using Dirac-Delta functions in phase space, but this approach is really only useful for formal considerations (Krall and Trivelpiece, 1973).
Now we are in a position to imagine the ‘machine’ behind nature’s self-consistent evolution of the particles and fields in the plasma, in the following way:
- Statistical description:
- Source terms:
is found by integrating the Lorentz force, , for some infinitesimal time (Equation 1)
is found by integrating
- Statistical description:
is found by … and so the cycle continues.
To put these ideas on a firm mathematical footing, we need to understand the evolution of in phase space, . The DF evolves according to an equation typically known as the Boltzmann equation,
with the right-hand side (RHS) of the equation describing the evolution of the DF according to ‘collisions’ (e.g. binary Coulomb collsions, see Fitzpatrick (2014)). Properly, this equation is specifically named after the form of collision operator assumed, e.g. Boltzmann, Fokker-Planck or Lenard-Balescu (Schindler, 2007). If the collision operator chosen is a function of alone, then the Boltzmann equation and Maxwell’s equations form a closed set, and the plasma is said to be in a kinetic regime Schindler (2007). In its general form, the Boltzmann equation can be obtained by integrating the Liouville equation for the N-particle DF in dimensional phase-space,
over the positions and velocities of all but one particle (Krall and Trivelpiece, 1973) (made possible by the fact that particles of a particular species are identical (Tong, 2012)). This also involves some assumptions made about the weak nature of the particle coupling in the plasma, characterised by
for the small parameter , i.e. a weakly coupled plasma (Schindler, 2007; Krall and Trivelpiece, 1973). Here, is the plasma parameter, equal to the number of electrons in the Debye sphere, a sphere of radius beyond which charge density inhomogeneities are shielded (Krall and Trivelpiece, 1973; Fitzpatrick, 2014). The small parameter is used as the ordering parameter in an infinite hierarchy of statistical equations - the so called BBGKY hierarchy - for which closure is achieved by neglecting terms of the desired order in (Krall and Trivelpiece, 1973). The standard collisional framework is achieved by neglecting terms of order and above.
It is a feature common to many weakly coupled plasmas that typical spatial variations, , are much larger than a quantity known as the Debye radius, ,
for Boltzmann’s constant, the electron temperature, and the fundamental charge. In such a situation the plasma is considered to be quasineutral (Schindler, 2007), typically taken to mean that
Note that this is in an asymptotic sense, and formally does not imply that vanishes, see e.g. Freidberg (1987); Schindler (2007); Harrison and Neukirch (2009a). To see how this works, first notice that if one normalises Poisson’s equation by
for characteristic values , and of the scalar potential, length scales and number densities, then one obtains
for , and . In the quasineutral limit the parameter is vanishingly small. If one then makes an expansion of small parameters
then one sees that formally, for ,
As such, letting is an approximation to the quasineutral limit, valid to first order.
It should also be mentioned that quasineutrality implies that the characteristic frequencies are much less than the (electron) plasma frequency,
Quoting Freidberg (1987) directly: “For any low-frequency macroscopic charge separation that tends to develop, the electrons have more than an adequate time to respond, thus creating an electric field which maintains the plasma in local quasineutrality”. The assumption of quasineutrality is consistent with neglecting the displacement current in Maxwell’s equations (Schindler, 2007). These ordering assumptions give the quasineutral ‘low-frequency/pre-Maxwell’ equations that are commonly used in plasma physics
In practice, Gauß’ Law is often not considered as a ‘core equation’ in plasma physics, and is implicitly ‘replaced’ by . Faraday’s law is also often ‘reformulated’ by eliminating the electric field using some version of Ohm’s law (e.g. see Schindler (2007); Kulsrud (1983); Freidberg (1987); Krall and Trivelpiece (1973); Fitzpatrick (2014)).
1.4 Fluid Models
Fluid models are the next step in the hierarchy after kinetic models, and are characterised by variables that depend only on space and time. Hence, the fluid equations are calculated by integrating over velocity space: taking velocity space moments of the kinetic equation at hand (Schindler, 2007). This process was laid down in the seminal work of Braginskii (1965), giving the collisional transport (or Braginskii) equations
|Electron mass transport|
|Electron mom. transport|
|Electron energy transport|
for electrons, and
|Ion mass transport|
|Ion mom. transport|
|Ion energy transport|
for ions, using the notation from Fitzpatrick (2014). In these equations defines the mass density, the scalar pressure for species , defined by the trace of the pressure tensor of species
for the velocity of a particle relative to the bulk flow, and for which
is the stress/generalised viscosity tensor. The vector ,
is the heat flux density. Finally, and are found by taking the momentum- and energy- moments of the collision operator (the RHS of the Boltzmann equation), and represent the collisional friction force, and collisional energy change, respectively.
These are the two-fluid equations. They describe the spatio-temporal evolution of the moments of the ion and electron DFs resepctively, and these are coupled by the EM fields. In their current form they are not closed: there are more unknowns than equations (Freidberg, 1987). It is not the purpose of this introduction to explore the subtle details of fluid closure, two-fluid, single fluid and magnetohydrodynamic (MHD) theories. For details on these topics see Schindler (2007); Kulsrud (1983); Freidberg (1987); Krall and Trivelpiece (1973); Fitzpatrick (2014).
2 Collisions in plasmas
2.1 Collisional plasmas
The collisionality of a plasma species is characterised in time and space by two quantities (Fitzpatrick, 2014): the collision rate/frequency, ; and the mean free path , such that
That is to say that the total collision rate for a species is made up of the collision rates with all species (including its own), the mean free path measures the typical distance a particle travels between collisions, and that in the case of an isothermal plasma the collision rate for electrons is much greater than that for ions. The thermal velocity, , gives the energy of random particle motion , such that in thermal equilibrium (Schindler, 2007). We note here that a collision is classified as a scattering event, and as such a particle may have numerous ‘small-angle’ scattering (i.e. ) events before a successful ‘collision’ (Fitzpatrick, 2014).
A collision dominated plasma is one for which the mean free path is much smaller than typical plasma length scales,
with the opposite limit indicating a collisionless plasma. The collisional frequency typically has magnitude
(Fitzpatrick, 2014) and as such
That is to say that weakly coupled plasmas are those for which collisions are not able to prevent plasma oscillations from regulating charge separation. In the case of a sufficiently collisional plasma characterised by
for which is a order velocity moment of the DF, then the plasma is in a local thermal equilibrium (e.g. see Cowley (2003/4)), characterised by a temperature , and the DF can be written as a Maxwellian of the form
to lowest order. This DF describes a plasma species with local number density and local bulk velocity . The DF in Equation (15) is clearly not an equilibrium solution, since the number density, bulk flow and temperature explicitly depend on time. Given sufficient time, Boltzmann’s H-Theorem implies that collisions will always attempt to drive a system towards thermal equilibrium (e.g. see Grad (1949); Brush (2003)), defined by a DF of the form
2.2 Collisionless plasmas
The statement that collisionless plasmas are those for which is rather truistic, and not particularly helpful in physical terms. Using the definition of the plasma parameter (Fitzpatrick, 2014),
we see that the collision frequency behaves like
Hence, dense and low temperature plasmas are more likely to be collisional, whereas diffuse and high temperature plasmas tend to be collisionless. In such situations, it is reasonable to neglect the RHS of the Boltzmann equation (Equation (12)), giving the Vlasov equation (Vlasov, 1968),
In closed form this equation can be written, using Hamilton’s equations (Tong, 2012), as
Here, the Hamiltonian is given by , the canonical momenta by , and the brackets are Poisson brackets, whose definition can be inferred from above. We can go from using velocity variables in the first line, to momentum variables in the second since . The Vlasov equation essentially states that the DF is conserved along a particle trajectory in phase-space (Schindler, 2007), since the characteristics of the Vlasov equation are the single particle equations of motion,
The solutions of this equation are in principle completely reversible in time, and hence entropy conserving (Krall and Trivelpiece, 1973).
3 Collisionless plasma equilibria
A Vlasov equilibrium is obtained when the DF satisfies
This statement does not mean that there are no macroscopic particle flows or currents; density, pressure or temperature gradients; or even heat fluxes, for example. That is to say that the moments of the DF can still have gradients in space. Rather, it is an equilibrium in the sense of a particle distribution. This means that the value of the DF at each individual point in phase-space is independent of time.
It is a standard result in classical mechanics that constants of motion, , (that do not depend explicitly on time) are in ‘involution’ with/commute with the Hamiltonian (Tong, 2004),
Using this result, and the linearity of the Poisson bracket, we see that any function of the constants of motion is a Vlasov equilibrium DF, since
We can also show that the reverse is true, namely that any Vlasov equilibrium DF is a function of the constants of motion. First consider a Vlasov equilibrium DF for arbitrary linearly independent functions . Then by linearity of the Poisson Bracket,
This sum must be zero for an equilibrium, and since the are linearly independent, that implies that each of the Poisson brackets must be zero independently. Hence the must be constants of motion and so
It is clear that a Vlasov equilibrium DF also satisfies the time-dependent Vlasov equation itself Schindler (2007), since
Using this fact, one can construct time-dependent solutions for ‘nonlinear’ propagating structures to the Vlasov equation by using a frame transformation (Schamel, 1979). Then one can solve for Vlasov equilibria in the wave frame, e.g. the famous BGK modes (Bernstein et al., 1957) and Schamel’s theory (Schamel, 1986), amongst other examples, e.g. see Abraham-Shrauner (1968); Ng and Bhattacharjee (2005); Vasko et al. (2016); Hutchinson (2017).
3.1 The ‘forward’ and ‘inverse’ approaches
As described above, one can easily construct equilibrium solutions of the Vlasov equation provided that at least one constant of motion has been identified. Any differentiable function of the constants of motion is an equilibrium solution of the Vlasov equation (Schindler, 2007), and is physically meaningful provided all velocity moments exist,
and the function is non-negative over all phase-space,
Whilst such a function may well satisfy these mathematical/microscopic conditions, the next question to ask is of the macroscopic electromagnetic fields that are consistent with such a function. Through Equations (10) and (11), we see that the distribution of particles in phase-space determines the charge and current densities respectively, in configuration-space. These charge and current densities are consistent with certain electric and magnetic fields through Maxwell’s equations (Equations (2) - (3)). Hence, a full understanding of the macroscopic and microscopic physics of a plasma necessitates a self-consistent ‘solution’ of the Vlasov-Maxwell (VM) system.
From these considerations, it should be clear that there are two possible routes to follow, in the absence of a comprehensive self-consistent theory, namely
The forward approach is the one that is most frequently seen in the literature. This is partly due, mathematically, to the fact that this involves solving differential equations, as opposed to the often less tractable inversion of integral equations in the case of the inverse approach. But also, as argued in Section 2.1, it is reasonable on physical grounds to assume that - for sufficiently collisional (Cowley, 2003/4) and ‘not-too-turbulent’ plasmas (Alpers, 1969) - that the DF is (locally) Maxwellian, and then to proceed with the forwards approach from thereon.
In the case of collisionless plasmas, there are an infinite class of equilibrium solutions in principle, and hence the forwards approach would have to be predicated on some prior knowledge of the DF. In-situ observations of DFs have only recently become available with spatio-temporal resolution on kinetic scales, for example the NASA Multiscale Magnetospheric (MMS) mission (Hesse et al., 2016), and the ESA candidate mission: Turbulent Heating ObserveR (THOR) (Vaivads et al., 2016).
Due to the ubiquitous nature and reasonable validity of the MHD approach in many environments, and the relative wealth and long history of magnetic field measurements, the equilibrium structures and dynamics of electromagnetic fields are better understood and more often used as the fundamental basis, or object, of plasma physics discussions and theory. Hence, it is of use, and necessity, to consider the inverse approach.
3.2 Motivating translationally invariant Vlasov-Maxwell (VM) equilibria
3.2.1 Current sheets
In a planar geometry, localised electric currents in a plasma are known as current sheets: frequently considered to be the initial state of wave processes (Fruit et al., 2002), instabilities (Schindler, 2007), reconnection (Yamada et al., 2010) and various dynamical phenomena in laboratory (Beidler and Cassak, 2011), space (Zelenyi et al., 2011) and astrophysical (DeVore et al., 2015) plasmas. The formation of current sheets is ubiquitous in plasmas. They can form between plasmas of different origins that encounter each other, such as at Earth’s magnetopause between the magnetosheath plasmas and magnetospheric plasmas (e.g. see Dungey (1961); Phan and Paschmann (1996)); or they can develop spontaneously in magnetic fields that are subjected to random external driving (e.g. see Parker (1994)), such as in the solar corona.
As to be introduced in Section 3.3, localised electric currents are an important ingredient for magnetic reconnection: acting as a signature of sheared magnetic fields, and reconnection electric fields (e.g. see Biskamp (2000); Hesse et al. (2011)). As per Poynting’s theorem (Poynting, 1884), with , and neglecting electric field energy,
intense current sheets are ideal locations for magnetic energy conversion and dissipation (Birn and Hesse, 2010; Zenitani et al., 2011). The dominant mechanisms that release the free energy include magnetic reconnection, and various plasma instabilities.
The currents themselves are usually considered synonymous with a stressed and/or anti-parallel magnetic field configuration, since in a quasineutral plasma (or a plasma in equilibrium), the current density is given by
with the current sheet ‘width’, normalising ; the asymptotic values of the magnetic field, normalising ; and normalising the current density and scalar pressure respectively. The maximum shear of is localised in the region , and this is where we see the maximum values of the current density: the current sheet itself. A Vlasov equilibrium DF self-consistent with the Harris sheet is given by
with ; a constant with dimensions of spatial number density (and not necessarily representing the number density itself); and with a bulk flow parameter, that in this case coincides with the bulk flow itself, i.e. . Note that one can derive other equilibrium DFs for the Harris sheet, e.g. the Kappa () DF (Fu and Hau, 2005).
3.2.2 Harris-type distribution functions (DFs)
If we were to ‘generalise’ the DF in Equation (25) to one that supports two current density components (and hence a DF self-consistent with a different magnetic field), then we have
One particularly nice feature of a DF that is a function of ,
is that the bulk flows are directly related to the flow parameters, i.e. and . This is seen by the following argument. If we define for
If we now consider the first-order moment of by , the result must be zero since only depends on , through . Consequently
and hence .
3.2.3 Other applications
Current sheets are by no means the only application of the work on translationally invariant VM equilibria in this thesis. As indicated in Section 3.6, translationally invariant VM equilibria are of use for numerous other applications in plasma physics. Examples include nonlinear waves (e.g. see Bernstein et al. (1957); Ng et al. (2012)); electron holes, ion holes and double layers (e.g. see Schamel (1986)); and colllisionless shock fronts (e.g. see Montgomery and Joyce (1969); Burgess and Scholer (2015)).
3.3 Magnetic Reconnection
Magnetic reconnection is a ubiquitous phenomenon in solar, space, astrophysical and laboratory plasmas, and now considered to be “among the most fundamental unifying concepts in astrophysics, comparable in scope and importance to the role of natural selection in biology.” (Moore et al., 2015): see authoritative discussions of ‘classical’ reconnection in Schindler (2007); Priest and Forbes (2000); Biskamp (2000); Hesse et al. (2011); on modern theories of ‘fast’ reconnection and ‘turbulent/stochastic reconnection’ in Lazarian et al. (2015); Loureiro and Uzdensky (2016); and ‘fractal reconnection’ in Shibata and Tanuma (2001). The literature on the topic is vast and there are many complex concepts to consider regarding the precise mathematical definition (e.g. see Hesse and Schindler (1988); Priest (2014)) of reconnection and its physical behaviour in different dimensions and plasma environments. The phenomenon also appears in physical environments as numerous as the number of plasma environments themselves, e.g. solar corona, planetary and pulsar magnetospheres, magnetic dynamos, gamma-ray bursts, geomagnetic storms and sawtooth crashes in tokamaks. However, there are common features that are agreed upon:
There is a change in the topology of the magnetic field, caused by processes in non-ideal () regions of plasma with strong localised electric currents and parallel electric fields.
- Diffusion region:
Ideal MHD breaks down within the diffusion region, kinetic physics is dominant, and the plasma decouples from the magnetic field, enabling stored magnetic energy to be released to the physical medium.
Hence, magnetic reconnection explicitly couples (via the transmission of energy) the macroscopic ideal MHD picture of relatively slow-evolving and large scale neutral, conducting fluids to the small-scale, short-timescale and non-neutral kinetic plasma physics. Reconnection can of course occur in many different ways. It could occur in one of following ways
One physical phenomenon out of many (and not necessarily dominant), occurring in a dynamical plasma, e.g. small scale reconnection in a turbulent plasma (e.g. Lazarian and Vishniac (1999));
3.3.1 Approximate equilibria in particle-in-cell (PIC) simulations
Magnetic reconnection processes can critically depend on a variety of length and time scales, for example on lengths of the order of the Larmor orbits and below that of the mean free path (e.g. see Biskamp (2000); Birn and Priest (2007)). In such situations a collisionless kinetic theory could be necessary to capture all of the relevant physics, and as such an understanding of the differences between using MHD, two-fluid, hybrid, Vlasov and other approaches is of paramount importance, for example see Birn et al. (2001, 2005) for discussions of this problem in the context of one-dimensional (1D) current sheets: the ‘Geospace Environmnetal Modelling (GEM)’ and ‘Newton’ challenges.
In the absence of an exact collisionless kinetic equilibrium solution, one has to use non-equilibrium DFs to start kinetic simulations, without knowing how far from the true equilibrium DF they are. In such cases, non-equilibrium drifting Maxwellian distributions are frequently used (see Swisdak et al. (2003); Hesse et al. (2005); Pritchett (2008); Malakit et al. (2010); Aunai et al. (2013b); Hesse et al. (2013); Guo et al. (2014); Hesse et al. (2014); Liu and Hesse (2016) for examples),
with a characteristic value of the thermal velocity, the number density, and the bulk velocity of species . These DFs can reproduce the same moments (and , typically with ) necessary for a fluid equilibrium, maintained by the gradient of a scalar pressure,
However, the DF, , in Equation (26) is not an exact solution of the Vlasov equation and hence does not describe a kinetic equilibrium. The macroscopic force balance self-consistent with a quasineutral Vlasov/kinetic equilibrium is maintained by the divergence of a rank-2 pressure tensor, (e.g. see Channell (1976); Mynick et al. (1979); Schindler (2007)), according to
As explained in Aunai et al. (2013b) on the subject of PIC simulations, the fluid equilibrium characterised by a drifting Maxwellian can evolve to a quasi-steady state “with an internal structure very different from the prescribed one”, and as demonstrated in Pritchett (2008), undesired electric fields, “coherent bulk oscillations”, and other perturbations may form, in nature’s attempt to maintain force-balance. Figure 6 is taken from Pritchett (2008), and demonstrates this phenomenon. Each of the panels relates, in principle, to a 1D MHD equilibrium characterised by , in which the PIC simulation is intialised with a DF of the form of that in Equation (26). Panel (a) demonstrates how the initial condition is self-consistent with a magnetic field profile and number density that are very close to those prescribed by the fluid equilibrium. However, panel (b) shows an electric field that forms due to the non-equilbrium initial state, and panel (c) demonstrates the resultant disparity between the exact/‘fluid’ current density (black), and that derived from the PIC simulation (red).
The knowledge of exact VM equilibria thus provides the chance to initialise PIC simulations in full confidence, with the intended macroscopic quantities reproduced. Exact VM equilibria would also permit analytical and numerical studies of the linear phase of collisionless instabilities (Gary, 2005), such as the tearing mode (e.g. see Drake and Lee (1977); Quest and Coroniti (1981a)). This sort of exact analysis is formally out of reach without an exact initial condition since - as discussed by e.g. Pritchett (2008); Aunai et al. (2013b) - a non-exact Vlasov solution creates perturbations itself, by virtue of not being an equilibrium.
Of course, one could make an argument on the basis of ordering arguments that a non-exact equilibrium DF such as that in Equation (26) allows the study of the nonlinear (and perhaps the linear) phase dynamics of plasma instabilities, such as the tearing mode. This sort of argument would be based on the assumption that a drifting Maxwellian such as that in Equation (26) is sufficiently close to a VM equilibrium so as not to significantly affect the physical processes. However, it is generally unclear how far such an initial condition is from exact equilibrium.
3.4 Forward approach for one-dimensional (1D) VM equilibria
To give context and to demonstrate the contrast, I will briefly introduce the ‘forward approach’ in VM equilibria, as used and discussed in e.g. Grad (1961); Harris (1962); Sestero (1967); Lee and Kan (1979a); Schindler (2007). In these - and other - works, a self-consistent solution to the VM system is found first by specifying the equilibrium DF as a function of the constants of motion. For example, a 1D system with , has the Hamiltonian, and two canonical momenta as the constants of motion,
These quantities are constants of motion in the sense that for an individual particle trajectory (the characteristics of the Vlasov equation) parameterised by ,
where the is in fact an operator involving derivatives over phase-space,
Using these relationships, it is now clear how one can justify writing the equilibrium DF as a function of the constants of motion
and a solution of Vlasov’s equation. Note how the second equality above demonstrates that the non-uniqueness of the correspondences,
In order to now satisfy the equilibrium Maxwell equations, scalar and vector potentials must be found that satisfy the following,