Neutral and non-neutral collisionless plasma equilibria for twisted flux tubes: The Gold-Hoyle model in a background field
NNeutral and non-neutral collisionless plasma equilibria for twisted flux tubes:The Gold-Hoyle model in a background field
O. Allanson, a) F. Wilson, and T. Neukirch School of Mathematics & Statistics, University of St Andrews, United Kingdom,KY16 9SS (Dated: 17 th August 2016)
We calculate exact one-dimensional collisionless plasma equilibria for a continuum of flux tube models, forwhich the total magnetic field is made up of the ‘force-free’ Gold-Hoyle magnetic flux tube embedded in auniform and anti-parallel background magnetic field. For a sufficiently weak background magnetic field, theaxial component of the total magnetic field reverses at some finite radius. The presence of the backgroundmagnetic field means that the total system is not exactly force-free, but by reducing its magnitude thedeparture from force-free can be made as small as desired. The distribution function for each species is afunction of the three constants of motion; namely the Hamiltonian and the canonical momenta in the axialand azimuthal directions. Poisson’s Equation and Amp`ere’s Law are solved exactly, and the solution allowseither electrically neutral or non-neutral configurations, depending on the values of the bulk ion and electronflows. These equilibria have possible applications in various solar, space and astrophysical contexts, as wellas in the laboratory.
I. INTRODUCTION
There has been significant recent work on Vlasov-Maxwell (VM) equilibria that are consistent with nonlin-ear force-free and ‘nearly force-free’ magnetic fieldsin Cartesian geometry. Therein, force-free refers to amagnetic field for which the associated current density isexactly parallel, which is the definition we shall also use, ∇ × B = µ j , j × B = . These works consider one-dimensional (1D) collisionlesscurrent sheets, with Refs. 1–8 specifically calculatingVM equilibrium distribution functions (DFs) that areself-consistent with a given specific magnetic field con-figuration. A natural question to consider is whether itis also possible to find self-consistent force-free (or nearlyforce-free) VM equilibria for other geometries, in partic-ular cylindrical geometry. In this paper we shall presentparticular VM equilibria for 1D magnetic fields whichare nearly force-free in cylindrical geometry, i.e. fluxtubes/ropesTwo of the archetypal field configurations in cylindri-cal geometry are the z -Pinch (with axial current and az-imuthal magnetic field), a classical example of which isthe Bennett Pinch ; and the θ -Pinch (azimuthal cur-rent and axial magnetic field). Consideration of ‘Vlasov-fluid’ models of z -Pinch equilibria was given in Ref. 11,with Ref. 12 calculating z -Pinch equilibria and an exten-sion with azimuthal ion-currents. Others have also con-structed kinetic models of the θ -pinch, see Refs. 13 and14 for examples. In the same year as Pfirsch , cylindricalkinetic equilibria with only azimuthal current were stud- a) Electronic mail: [email protected] ied in Ref. 16 . For examples of treatments of the stabil-ity of fluid and kinetic linear pinches, see Refs 15, 17, and18 respectively.Recently there have been studies on ‘tokamak-like’ VMequilibria with flows , starting from the VM equa-tion in cylindrical geometry and working towards Grad-Shafranov equations for the vector potential. We alsonote two Vlasov equilibrium DFs in the literature thatare close in style to the one that we shall present. Thefirst is described in a brief paper , with an equilibriumpresented for a cylindrical pinch. However, their distri-bution describes a different magnetic field and the DF ap-pears not to be positive over all phase space. The secondDF is a very recent paper that actually describes a mag-netic field much like the one that we discuss. Their DFis designed to model ‘ion-scale’ flux tubes in the Earth’smagnetosphere. Formally, their quasineutral model ap-proaches a nonlinear force-free configuration in the limitof a vanishing electron to ion mass ratio. In their model,current is carried exclusively by electrons and the non-negativity of the DF depends on a suitable choice of mi-croscopic parameters. Finally, we mention that in beamphysics, much work on constructing cylindrical VM equi-libria is done by looking for mono-energetic distributionswith conserved angular momentum, see Refs. 24–27 forsome examples.Magnetic flux tubes and flux ropes are prevelant inthe study of plasmas, with a wide variety of observedforms in nature and experiment, as well as uses and ap-plications in numerical experiments and theory. Someexamples of the environments and fields of study inwhich they feature include solar ; solar wind ;planetary magnetospheres and magnetopauses ; as-trophysical plasmas ; tokamak , laboratory pinchexperiments and the basic study of energy release inmagnetised plasmas , to give a small selection of refer-ences.One application of flux tubes is in the study of solar a r X i v : . [ phy s i c s . p l a s m - ph ] S e p active regions and the onset of solar flares and coro-nal mass ejections . A classic magnetohydrodynamic(MHD) model for magnetic flux tubes was first presentedby T. Gold and F. Hoyle (GH) , initially intended foruse in the study of solar flares. The GH model is aninfinite, straight, 1D and nonlinear force-free magneticflux tube with constant ‘twist’ . Mathematically, theGH magnetic field could be regarded as the cylindricalanalogue of the Force-Free Harris sheet (a planar cur-rent sheet model), as the Bennett Pinch might be tothe ‘original’ Harris Sheet .It is typical to consider solar, space and astrophysi-cal flux tubes within the framework of MHD, e.g. seeRef. 49. However, many of these plasmas can be weaklycollisional or collisionless, with values of the collisionalfree path large against any fluid scale , making a de-scription using collisionless kinetic theory necessary. Itis the intention of this paper to study the GH flux tubemodel beyond the MHD description, since - apart fromthe very recent work in Ref. 23 - we see no attemptin the literature of a microscopic description of the GHfield. Other than any interesting theoretical advances,a possible application of the results of this study couldbe to implement the obtained model in kinetic (particle)numerical simulations.In Cartesian geometry, the work in Refs. 1–8 usedthe method proposed by Channell to tackle the VMinverse problem, i.e. to determine self-consistent equilib-rium DFs for a given magnetic field configuration. Chan-nell described the extension of his work to cylindrical ge-ometry as ‘not possible in a straightforward manner.’ Asexplained in Ref. 20 (in which cylindrical coordinates areused to model a torus), this is due in part to the ‘toroidic-ity’ of the problem, i.e. the 1 /r factor in the equations.As we shall see in this paper, another potential com-plication is the need to allow – at least in principle – anon-zero charge density. The work in this paper does notpresent a generalised method for the VM inverse prob-lem in cylindrical geometry, but instead some particularsolutions for a specific given magnetic field.The paper is structured as follows. In Section 2 wefirst review the theory of the equation of motion consis-tent with a collisionless DF in cylindrical geometry, anddiscuss the question of the possibility of 1D force-freeequilibria. Then we introduce the magnetic field to beused. We note that whilst the work in this paper is ap-plied to a particular magnetic field from Subsection 2.1onwards, the steps taken to calculate the equilibrium DFseem as though they could be adaptable to other cases.In Section 3 we present the form of the DF that gives therequired macroscopic equilibrium, and proceed to ‘fix’the parameters of the DF by explicitly solving Amp`ere’sLaw and Poisson’s Equation. Note that whilst we chooseto consider a two-species plasma of ions and electrons, wesee no obvious reason preventing the work in this paperbeing used to describe plasmas with a different composi-tion. In Section 4. we present a preliminary analysis ofthe physical properties of the equilibrium. Particularly technical calculations are in the Appendices. Appendix Acontains the zeroth and first order moment calculations,used to find the number densities and bulk flows directly,and in turn the charge and current densities. AppendixB contains the mathematical details of the existence andlocation of multiple maxima of the DF in velocity-space. II. GENERAL THEORYA. The Vlasov equation and the equation of motion
A collisionless equilibrium is characterised by the 1-particle distribution function, f s , a solution of the steady-state Vlasov equation (e.g. see Ref. 52). The Vlasovequation in cylindrical coordinates is ∂f s ∂t + v i ∂f s ∂x i + q s m s (cid:0) E i + ε ijk v j B k (cid:1) ∂f s ∂v i + (cid:20) v θ r ∂f s ∂v r − v r v θ r ∂f s ∂v θ (cid:21) = 0 , (1)see for example Refs. 16, 19, and 53. Here i, j and k areused as ‘spatial’ indices running over { , , } , and s isused as the particle species index. Individual particle po-sitions and velocities are given by ( x , x , x ) = ( r, θ, z )and ( v , v , v ) = ( v r , v θ , v z ) respectively, for r the hor-izontal distance from the z axis, and θ the azimuthalangle. The totally antisymmetric unit tensor of rank 3(the Levi-Civita tensor) is ε ijk , and the Einstein sum-mation convention is applied (such that repeated indicesare summed over, with subscript and superscript indicesused to describe co- and contravariant components re-spectively. The mass and charge of particle species s are m s and q s respectively. The electric and magnetic fieldsare defined as E = −∇ φ and B = ∇ × A , for φ the scalarpotential and vector potential A .The ‘fluid’ equation of motion of a particular species s is found by taking first-order velocity moments of theVlasov equation. After a routine but laborious moment-taking calculation, we see that - in equilibrium ( ∂/∂t =0), assuming a one-dimensional configuration with onlyradial dependence ( ∂/∂θ = ∂/∂z = 0), and letting f s bean even function of the radial velocity v r - force balancefor species s is maintained according to( ∇ · P s ) r = ( j s × B ) r + σ s E + ρ s r u θs . (2)The pressure tensor for species s is a rank-2 tensor andis defined by P ij,s = (cid:90) w is w js f s d v, where v i = u is + w is , for u is the bulk velocity of species s and v i the individual particle velocity. Note that theassumption of f s to be an even function of v r automat-ically implies that u rs = P rθ = P zr = 0. Equation (2)can be summed over species to give( ∇ · P ) r = ( j × B ) r + σ E + 1 r F c , (3)where F c = ρ i u θi + ρ e u θe is the force density associated with the rotating bulk flowsof the ions and electrons. Equation (3) is a cylindricalanalogue of the force balance equation in Cartesian ge-ometry (e.g. see 54). There are ‘extra inertial terms’ ascompared to the case of Cartesian geometry. From thepoint of view of a particular magnetic field B (which isthe point we take by specifying a particular macroscopicequilibrium), we see that equilibrium is maintained bya combination of density/pressure variations as in thecase of Cartesian geometry, but with additional contri-butions from centrifugal forces and as an inevitable resultof the resultant charge separation, an electric field. Thisclearly demonstrates that ‘sourcing’ an exactly force-freemacroscopic equilibrium with an equilibrium DF in a 1Dcylindrical geometry is inherently a more difficult taskthan in the Cartesian case. The presence of ‘extra’ posi-tive definite inertial forces and, almost inevitably, forcesassociated with charge separation raises the question ofwhether exactly force-free equilibria are possible at all inthis paradigm.Before proceeding, we comment that given certainmacroscopic constraints on the electromagnetic fields orfluid quantities - such as the force-free condition, or a spe-cific given magnetic field (for example) - it is not a priori known how to calculate a self-consistent Vlasov equilib-rium, or if one even exists within the framework of theassumptions made. Hence one has to proceed more orless on a case by case basis, with the intention of achiev-ing consistency with the required macroscopic conditions,upon taking moments of the DF. B. Methods for calculating an equilibrium DF
In Refs. 2 and 51 for example, a method used to cal-culate a DF, given a prescribed 1D magnetic field wasInverse Fourier Transforms (IFT). A distribution func-tion of the form f s ∝ e − β s H s g s ( p xs , p ys ) , (4)was used, with H s , p xs and p ys the conserved particleHamiltonian and canonical momenta in the x and y di-rections, and g s an unknown function, to be determined.Since our problem is one of a 1D equilibrium with varia-tion in the radial direction, the three constants of motionare the Hamiltonian, and the canonical momenta in the θ and z directions: H s = m s (cid:0) v r + v θ + v z (cid:1) + q s φ,p θs = r ( m s v θ + q s A θ ) , p zs = m s v z + q s A z . (5) A function of a subset of the constants of motion is au-tomatically a solution of the VM equation (e.g. see Ref.52). One can try to calculate an equilibrium distributionfor the Gold-Hoyle force-free flux tube without a back-ground field by a similar method, assuming a DF of theform f s ∝ e − β s H s g s ( p θs , p zs ) . (6)By exploiting the convolution in the definition of the cur-rent density, j ( A , r ) = (cid:88) s q s (cid:90) v f s ( H s , p θs , p zs ) d v, = r (cid:88) s q s m s (cid:90) ( p s − q s A ) f s ( H s , r p θs , p zs ) d p s , Amp`ere’s law can be solved by IFT, with the quantity p s defined by p rs = p rs , p θs = p θs r , p zs = p zs . Notice how when written in this integral form, j is notonly a function of A , but - in contrast with the Carte-sian case - also of the relevant spatial co-ordinate, r . Inthe case of zero scalar potential, the result of the cal-culation is to give a distribution function that is not asolution of the Vlasov equation as it is not a function ofthe constants of motion only. In essence, an additionalexp( − r ) factor is required in the DF to counter exp( r )terms that manifest by completing the square in the inte-gration. The physical cause here would appear to be theinertial forces associated with the rotational bulk flow.If one assumes a non-zero scalar potential, then itseems impossible to satisfy Amp`ere’s Law. The physicalcause seems to be that, in the case of force-free fields, onewould require a ‘different’ electrostatic potential to bal-ance the inertial forces for the ions and electrons, whichis of course nonsensical. Thus, our investigation seemsto suggest that it is not possible to calculate a DF of theform of equation (6) for the exact GH field. C. The magnetic field:A Gold-Hoyle flux tube plus a background field
To make progress, we introduce a background field inthe negative z direction. The mathematical motivationfor this change is to balance the ‘exp( r ) problem’. Phys-ically, it seems that the background field introduces anextra term (whose sign depends on species) into the force-balance, to allow for both the ion and electrons to be inforce balance simultaneously, given one unique expressionfor the scalar potential.The vector potential, magnetic field and current den-sity used in this paper are as follows (GH+B): A (˜ r ) = B τ (cid:18) , r ln (cid:0) r (cid:1) − k ˜ r, − ln (cid:0) r (cid:1)(cid:19) , = A GH − (cid:0) , B kτ − ˜ r, (cid:1) . (7) B (˜ r ) = B (cid:18) , ˜ r r ,
11 + ˜ r − k (cid:19) , = B GH − (0 , , kB ) . (8) j (˜ r ) = 2 τ B µ (cid:18) , ˜ r (1 + ˜ r ) , r ) (cid:19) , = j GH . (9)The magnetic permeability in vacuo is given by µ andthe characteristic magnetic field strength by B . Theconstant τ has units of inverse length, and we use 1 /τ to represent the characteristic length scale of the system(˜ r = τ r ) (see Table 1 for a concise list of the dimen-sionless quantities used in this paper, all denoted witha tilde, ˜ ). The dimensionless constant k > z direction,and as a result there are now two different interpreta-tions to be made. We could either consider the systemas a GH flux tube of uniform twist embedded in an un-twisted uniform background field, or consider the wholeGH+B magnetic field as a non-uniformly twisted fluxtube. We note that flux tubes embedded in an axiallydirected background field have recently been observedduring reconnection events in the Earth’s magnetotail,by the Cluster spacecraft .In the first interpretation, τ is a direct measure of the‘twist’ of the embedded flux tube (see Ref. 46), withthe number of turns per unit length (in z ) along a fieldline given by τ / (2 π ) . In the second interpretation, wesee that the system is not uniformly twisted, with the z distance traversed when following a field line (e.g. 56)given by (cid:90) rB z B θ dθ = 1 τ (cid:0) − k (1 + ˜ r ) (cid:1) (cid:90) dθ. The fact that this depends on r demonstrates that thesystem as a whole has non-uniform twist. The number ofturns per unit length in z of the GH+B field: the ‘twist’is given by (cid:32)(cid:90) θ =2 πθ =0 rB z B θ dθ (cid:33) − = τ π (cid:0)(cid:0) − k (1 + ˜ r ) (cid:1)(cid:1) − , and is plotted in Figure 1 for three values of k . Since k < / k ≥ / z direction as we wind round the GH+B flux tube in theanti-clockwise direction.The magnetic field is plotted in Figures 2a-2b for twovalues of k . The k = 0 . B z field direction and as such is akin to a Reversed FieldPinch (e.g. see Ref. 57 for a laboratory interpretation):this configuration may be of use in the study of astro-physical jets, see Ref. 36 for example. The value k = 1 / B z at ˜ r = 0, and as such is the value thatdistinguishes the two different classes of field configura-tion, namely unidirectional ( k ≥ /
2) or including fieldreversal ( k < / r for which the ˜ B z fieldreverses is plotted in Figure 2c. The magnitude of theGH+B magnetic field is plotted in Figure 3 for three val-ues of k . For all values of k , | ˜ B | → k for large ˜ r , i.e. toa potential field.The primary task of this paper is to calculate self-consistent collisionless equilibrium distribution functionsfor the GH+B field. This problem essentially reducesto solving Amp`ere’s Law such that equation (1) is satis-fied. We assume nothing about the electric field however,and in fact use that degree of freedom to solve Amp`ere’sLaw. The resultant form of the scalar potential is thensubstituted into Poisson’s equation, to establish the finalrelationships between the microscopic and macroscopicparameters of the equilibrium. III. THE EQUILIBRIUM DISTRIBUTION FUNCTION
Although the IFT method did not yield a self-consistent equilibrium DF for the GH field without abackground field, the outcome of the calculation can stillbe used as an indication of possible forms for the DF forthe GH+B field. Using trial and error we arrived at thedistribution function f s = n s ( √ πv th,s ) × (cid:104) e − ( ˜ H s − ˜ ω s ˜ p θs − ˜ U zs ˜ p zs ) + C s e − ( ˜ H s − ˜ V zs ˜ p zs ) (cid:105) , (10)which is a superposition of two terms that are consis-tent macroscopically with a ‘Rigid-Rotor’, see Ref. forexample. A Rigid-Rotor is microscopically described bya DF of the form F ( H − ωp θ − V p z ) (with V = 0 inthe second term of the DF in equation (10)). Each F ( H − ωp θ − V p z ) term corresponds to an average macro-scopic motion of rigid rotation with angular frequency ω ,and rectilinear motion with velocity V .The dimensionless constants ˜ ω s , ˜ U zs , ˜ V zs and C s are yet to be determined, with C s > β s = 1 / ( k B T s ) and v th,s is the thermal velocity ofspecies s . The ratio of the thermal Larmor radius, r L = m s v th,s / ( e | B | ) (for e = | q s | ) to the macroscopiclength scale of the system L (= 1 /τ ), is given by δ s ( r ) = r L L = m s v th,s τeB ( r ) , typically known as the ‘magnetisation parameter’ . Inour system, the magnitude of the magnetic field andhence δ s itself is spatially variable. For the purposes ofthe calculations in this paper however, we set m s v th,s τeB = δ s = const . as a characteristic value (see Table II for a concise listof the micro and macroscopic parameters of the equilib-rium). A. Maxwell’s equations: Fixing the parameters of the DF
By insisting on a specific magnetic field configuration(the GH+B field) we have made a statement on themacroscopic physics. In searching for the equilibriumDF, we are trying to understand the microscopic physics.In this sense we are tackling an ‘inverse problem’. Oncean assumption on the form of the DF is made then –should the assumed form be able to reproduce the cor-rect moments – this inverse problem reduces to establish-ing the relationships between the microscopic and macro-scopic parameters of the equilibrium. In this section we‘fix’ the free parameters of the DF in equation (10), such that Maxwell’s equations are satisfied; ∇ · E = 1 ε (cid:88) s q s (cid:90) f s d v, (11) ∇ × B = µ (cid:88) s q s (cid:90) v f s d v. (12)Note that the solenoidal constraint and Faraday’s law areautomatically satisfied for the GH+B field in equilibrium,since B = ∇ × A implies that ∇ · B = 0 and E = −∇ φ implies that ∇ × E = = − ∂ B ∂t .
1. Amp`ere’s Law
In Appendix A we have calculated the j z current den-sity, found by summing first order moments in v z of theDF. We now substitute in the macroscopic expressionsfor j z (˜ r ), A θ (˜ r ) and A z (˜ r ) from (9) and (7) into the ex-pression for the j z current density of equation (A4). Af-ter this substitution, we can calculate a φ ( r ) that makesthe system consistent. The substitution of the knownexpressions for j z , A z and A θ gives j z (˜ r ) = 2 τ B µ r ) = (cid:88) s n s q s v th,s e − q s β s φ × (cid:18) ˜ U zs e ( ˜ U zs +˜ r ˜ ω s ) / − sgn( q s )˜ ω s ˜ r k/δ s (cid:0) r (cid:1) sgn( q s )(˜ ω s − ˜ U zs ) / (2 δ s ) + ˜ V zs C s e ˜ V zs / (cid:0) r (cid:1) − sgn( q s ) ˜ V zs / (2 δ s ) (cid:19) (13)In order to satisfy the above equality we can construct a solution by introducing a ‘separation constant’ γ (cid:54) = 0 , r ) which makes the left-hand side constant, whilst the right-hand side is asum of two terms, one depending on ion parameters and the second depending on electron parameters. Then we candefine γ by 2 τ B µ = 2 τ B µ (1 − γ ) + 2 τ B µ γ , (14)associating the ‘ion term’ with the first term on the right-hand side of (14), and the ‘electron term’ with the secondterm on the right-hand side of (14). After some algebra we can rearrange these two associations to give two expressionsfor the scalar potential, one in terms of the ion parameters, and one in terms of the electron parameters: φ ( r ) = 1 q i β i ln (cid:26) µ n i q i v th,i τ B (1 − γ ) (cid:20) ˜ U zi e ( ˜ U zi +˜ r ˜ ω i ) / − ˜ ω i ˜ r k/δ i (cid:0) r (cid:1) ω i − ˜ U zi ) / (2 δ i ) + ˜ V zi C i e ˜ V zi / (cid:0) r (cid:1) − ˜ V zi / (2 δ i ) (cid:21)(cid:27) φ ( r ) = 1 q e β e ln (cid:26) µ n e q e v th,e τ B γ (cid:20) ˜ U ze e ( ˜ U ze +˜ r ˜ ω e ) / ω e ˜ r k/δ e (cid:0) r (cid:1) − (˜ ω e − ˜ U ze ) / (2 δ e ) + ˜ V ze C e e ˜ V ze / (cid:0) r (cid:1)
2+ ˜ V ze / (2 δ e ) (cid:21)(cid:27) The two values of the scalar potential above must be made identical by a suitable choice of relationships betweenthe ion and electron parameters. Given enough freedom in parameter space, we could say that the z component ofAmp`ere’s Law is implicitly solved the above equations, in that one just needs to choose a consistent set of parameters.However, we seek a solution in an explicit sense.In order to make progress we non-dimensionalise the above equations by multiplying both sides by eβ r with β r = β i β e β e + β i . Once this is done we can write the scalar potential in the form eβ r φ ( r ) = ln (cid:110) [ion terms] eβrqiβi (cid:111) , (15) eβ r φ ( r ) = ln (cid:110) [electron terms] eβrqeβe (cid:111) . (16)Specifically, equations (15) and (16) require the equality of the arguments of the logarithm to hold in order for ameaningful solution to be obtained for the scalar potential. A first step towards this is made by requiring consistentpowers of the 1 + ˜ r ‘profile’ in the right-hand side of the above expression to allow factorisation. Hence(˜ ω i − ˜ U zi ) / (2 δ i ) = − ˜ V zi / (2 δ i ) , − (˜ ω e − ˜ U ze ) / (2 δ e ) = ˜ V ze / (2 δ e ) , = ⇒ ˜ ω i = ˜ U zi − ˜ V zi , ˜ ω e = ˜ U ze − ˜ V ze , (17)and hence the rigid-rotation, ˜ ω s , is fixed by the difference of the rectilinear motion, ˜ U zs − ˜ V zs . On top of this, werequire that the power of the 1 + ˜ r ‘profile’ on the right-hand side is the same for both the ions and electrons, thus eβ r q i β i (cid:16) − ˜ V zi / (2 δ i ) (cid:17) = E = eβ r q e β e (cid:16) V ze / (2 δ e ) (cid:17) . (18)This condition seems to be a statement on an average potential energy associated with the particles. Once more toallow factorisation of the 1 + ˜ r ‘profile’, we insist that net exp( r ) terms cancel, i.e.˜ ω i kδ i > , ˜ ω e − kδ e < . (19)The physical meaning of this condition seems to be that the frequencies of the rigid rotor for each species are matchedaccording to the relevant magnetisation, and the background field magnitude. The remaining task is to ensure equalityof the ‘coefficients’ (cid:40) δ i (1 − γ ) n i m i v th,i B / (2 µ ) (cid:104) ˜ U zi e ˜ U zi / + ˜ V zi C i e ˜ V zi / (cid:105)(cid:41) eβrqiβi = D = (cid:40) − δ e γ n e m e v th,e B / (2 µ ) (cid:104) ˜ U ze e ˜ U ze / + ˜ V ze C e e ˜ V ze / (cid:105)(cid:41) eβrqeβe (20)These seem to be conditions on the ratios of the energy densities associated with the bulk rectilinear motion andthe magnetic field respectively. Thus far we have 8 constraints and 12 unknowns ( ˜ U zs , ˜ V zs , ˜ ω s , C s , n s , β s ) given fixedcharacteristic macroscopic parameters of the equilibrium B , τ , and k . We can now write down an expression for φ that explicitly solves the z component of Amp`ere’s law; φ (˜ r ) = 1 eβ r E ln (cid:0) r (cid:1) + φ (0) , (21)with φ (0) = 1 eβ r ln D . Clearly, we require that D > γ could, in principle,affect the sign of D . It is seen from (20) that positivity of D implies that11 − γ (cid:104) ˜ U zi e ˜ U zi / + ˜ V zi C i e ˜ V zi / (cid:105) > , (22)1 γ (cid:104) ˜ U ze e ˜ U ze / + ˜ V ze C e e ˜ V ze / (cid:105) < . (23)By rearranging the above inequalities to make C s the subject, it can be seen after some algebra that positivity of D and C s is guaranteed when γ > , sgn( ˜ U zs ) = − sgn( ˜ V zs ) . Note that these conditions are sufficient, but not necessary, i.e. it is possible to have D > C s > γ (cid:54) = 0 ,
1, and even for sgn( ˜ U zs ) = sgn( ˜ V zs ) in the case of γ < j z component, and it is premature to consider all components of Amp`ere’sLaw satisfied. Let us move on to consider the θ component. In a process similar to that above, we substitute in themacroscopic expressions for j θ (˜ r ), A θ (˜ r ) and A z (˜ r ) for the GH+B field into the expression for the j θ current densityof equation (A6) in Appendix A. After this substitution, we can once more calculate the φ that makes the systemconsistent. The substitution gives j θ = 2 τ B µ = (cid:88) s n s q s v th,s ˜ ω s e − q s β s φ e ( ˜ U zs +˜ r ˜ ω s ) / − sgn( q s ) ˜ ω s ˜ r k/δ s (cid:0) r (cid:1) q s )(˜ ω s − ˜ U zs ) / (2 δ s ) (24)Using the parameter relations as above, we determine that the scalar potential is again given in the form of (21), φ (˜ r ) = 1 eβ r E ln (cid:0) r (cid:1) + φ (0) . Hence, this form of the scalar potential is consistent provided (cid:20) − γ δ i n i m i v th,i ω i /τB / (2 µ ) e ˜ U zi / (cid:21) eβrqiβi = D = (cid:20) − γ δ e n e m e v th,e ω e /τB / (2 µ ) e ˜ U ze / (cid:21) eβrqeβe (25)for γ (cid:54) = 1 another separation constant. These seem to be conditions on the ratios of the energy densities associatedwith the bulk rotation and the magnetic field respectively. This has added two more constraints.Once again we must ensure that D >
0. Since ω e <
0, the right-hand side of the above equation implies that γ > D >
0. Whilst the left-hand side implies that γ < D since ω i >
0. Hence we cansay that for positivity 0 < γ < . We can now consider Amp`ere’s Law satsified, given a φ that solves Poisson’s equation. As a result, the problem ofconsistency is now shifted to solving Poisson’s Equation, where the remaining degrees of freedom lie.
2. Poisson’s Equation
The final step in ‘self-consistency’ is to solve Poisson’s Equation. Frequently in such equilibrium studies, this stepis replaced by satisfying quasineutrality and in essence solving a first order approximation of Poisson’s equation, seefor example Refs. 1, 20, and 52. Here we solve Poisson’s equation exactly, i.e. to all orders. Poisson’s equation incylindrical coordinates with only radial dependence gives ∇ · E = − r ∂∂r (cid:18) r ∂φ∂r (cid:19) = σε . (26)The electric field is calculated as E = −∇ φ , giving E r = − ∂ r φ = − τ E eβ r ˜ r (1 + ˜ r ) . We can now take the divergence of the electric field ∇ · E = τ ˜ r − ∂ ˜ r (˜ rE r ) and so ∇ · E = − τ E eβ r r ) = ⇒ σ = − ε τ E eβ r r ) . (27)This gives a non-zero net charge per unit length (in z ) of Q = (cid:90) θ =2 πθ =0 (cid:90) r = ∞ r =0 σ r dr dθ = − πε E eβ r . (28)The charge density derived in equation (27) must equal the charge density calculated by taking the zeroth momentof the DF. The expression for the charge density calculated in (A2) gives σ = (cid:88) s q s n s = (cid:88) s n s q s e − q s β s φ (cid:16) e ( ˜ U zs +˜ r ˜ ω s ) / e ˜ U zs ˜ A zs e ˜ ω s ˜ r ˜ A θs + C s e ( ˜ U zs − ˜ ω s ) / e ( ˜ U zs − ˜ ω s ) ˜ A zs (cid:17) , = (cid:88) s n s q s e − q s β s φ (cid:0) r (cid:1) sgn( q s )(˜ ω s − ˜ U zs ) / (2 δ s ) (cid:16) e ˜ U zs / + C s e ( ˜ U zs − ˜ ω s ) / (cid:17) , = 1(1 + ˜ r ) (cid:88) s n s q s D − qsβseβr (cid:16) e ˜ U zs / + C s e ( ˜ U zs − ˜ ω s ) / (cid:17) . (29)The second equality is found by substituting the form of the vector potential from equation (7), and the final equalityis reached by using the conditions derived in equations (17) - (21).We can now match equations (27) and (29) to get σ = − ε τ E eβ r = (cid:88) s n s q s D − qsβseβr (cid:16) e ˜ U zs / + C s e ˜ V zs / (cid:17) . (30)We now have 12 physical parameters ( ˜ U zs , ˜ V zs , ˜ ω s , C s , n s , β s ) with 11 constraints (17-20), (25) & (30). For example,if one picks B , τ , k and one microscopic parameter, say β i , then the remaining parameters of the equilibrium,( ˜ U zs , ˜ V zs , ˜ ω s , C s , n s , β e ), are now determined. One could of course choose the values of a different set of parameters,and determine those that remain by using the constraints derived. Note that whilst the constants γ (cid:54) = 0 , < γ < IV. ANALYSIS OF THE EQUILIBRIUMA. Non-neutrality & the electric field
It is seen from equations (27) and (28) that basic elec-trostatic properties of the equilibrium described by f s are encoded in E . The equilibrium is electrically neutralonly when E = 0, and non-neutral otherwise. Specifi-cally, there is net negative charge when E >
0, and netpositive charge when E <
0. This net charge is finite inthe ( r, θ ) plane and given by Q in equation (28).Physically, the sign of E seems to be related to therespective magnitudes of the bulk rotation frequencies,˜ ω s . From equations (17) and (18) we see that E > ω i > ω (cid:63)i = ˜ U zi − δ i , | ˜ ω e | < ω (cid:63)e = − ˜ U ze − δ e , and E < ω i < ω (cid:63)i = ˜ U zi − δ i , | ˜ ω e | > ω (cid:63)e = − ˜ U ze − δ e . Hence, E > E corresponds to an electric field directed radially ‘in-wards’. This seems to make sense physically, by thefollowing argument. A ‘larger’ (˜ ω i > ω (cid:63)i ) bulk ion ro-tation freqency gives a ‘larger’ centrifugal force, and a‘smaller’ ( | ˜ ω e | < ω (cid:63)e ) bulk electron rotation frequencygives a ‘smaller’ centrifugal force. For a dynamic inter-pretation, at a fixed r , the ions are forced to a slightlylarger radius than the electrons, i.e. a charge separationmanifests on small scales. This charge separation resultsin an inward electric field, E r <
0. An equally valid in-terpretation is to say that for an equilibrium to exist, anelectric field must exist to counteract the differences inthe centrifugal forces associated with the bulk ion andelectron rotational flows.In a similar manner, E < ω i < ω (cid:63)i ) bulk ion rotation frequencies,and ‘sufficiently large’ ( | ˜ ω e | > ω (cid:63)e ) bulk electron rotationfrequencies. A negative E corresponds to an electric field directed radially ‘outwards’. We can then interpret theseresult physically, in a manner like that above.Finally, we can interpret the neutral case, E = 0, as theintermediary between the two circumstances consideredabove. That is to say that the equilibrium is neutral whenthe bulk rotation flows are just matched accordingly, suchthat there is no charge separation and hence no electricfield. B. The equation of state and the plasma beta
For certain considerations, e.g. the solar corona, itwould be advantageous if the DF had the capacity to de-scribe plasmas with sub-unity values of the plasma beta:the ratio of the thermal energy density to the magneticenergy density β pl (˜ r ) = 2 µ k B B (cid:88) s n s T s . (31)For our configuration, the number density is seen to beproportional to the rr component of the pressure tensor, P rr,s = n s k B T s . This is demonstrated by the followingcalculation. In order to calculate P rr , we must considerthe integral P rr = (cid:88) s m s (cid:90) ∞−∞ w rs w rs f s d v. (32)However, we do not have to consider a bulk velocity inthe r direction here ( u rs = 0), since f s is an even functionof v r . Using the fact that (cid:90) ∞−∞ v r e − v r / (2 v th,s ) dv r = v th,s (cid:90) ∞−∞ e − v r / (2 v th,s ) dv r , and by consideration of equations (32) and the numberdensity, we see that P rr,s = m s v th,s n s , (33)that is to say that k B T s = m s v th,s . Note that if n i = n e := n and hence E = 0 (neutrality), then we have anequation of state given by P rr = β e + β i β e β i n. This resembles expressions found in the Cartesian case,in Refs. 3, 7, and 51 for example. Incidentally, we can usethe connection between n s and P rr to give an expressionfor the β pl that is perhaps more typically seen, β pl (˜ r ) = 2 µ B (cid:88) s P rr,s . The square magnitude of the magnetic field (equation(8)) is given by B = B (1 + ˜ r ) (cid:0) − k + 4 k (1 + ˜ r ) (cid:1) . Using the number density from equation (A1) in the def-inition of the plasma beta from equation (31), as well asthe equilibrium conditions (17) - (21) gives β pl (˜ r ) = 2 µ B (1 + ˜ r ) (1 − k + 4 k (1 + ˜ r )) × (cid:88) s n s β s D − qsβseβr (cid:16) e ˜ U zs / + C s e ˜ V zs / (cid:17) . (34)It is not immediately obvious from the above equationwhat values β pl can have. However it is readily seen thatas ˜ r → ∞ then β pl →
0, essentially since the numberdensity is vanishing at large radii. On the central axis ofthe tube we see that β pl (0) = 2 µ B (1 − k + 4 k ) × (cid:88) s n s β s D − qsβseβr (cid:16) e ˜ U zs / + C s e ˜ V zs / (cid:17) , (35)suggesting that for a suitable choice of parameters, itshould be possible to attain any value of β pl on the axis. C. Plots of the DF
A characteristic that one immediately looks for ina new DF is the existence of multiple maxima in ve-locity space, which are a direct indication of non-thermalisation, relevant for the existence of micro-instabilities (e.g. see ). Using an analysis very similarto that in , we can derive - for a given value of ˜ ω s - condi-tions on ˜ r and either ˜ v z or ˜ v θ , for the existence of multiplemaxima in the ˜ v θ or ˜ v z direction respectively. We presentthese calculations in Appendices B 1 and B 2. The mostreadily understood results are that multiple maxima inthe ˜ v θ direction can only occur for ˜ r > / | ˜ ω s | , and in the˜ v z direction for | ˜ ω s | >
2. Given these necessary condi-tions, one can then calculate that multiple maxima of f s will occur in the ˜ v θ direction for ˜ v z bounded above andbelow, and vice versa.In Figures (4-7) we present plots of the DFs over arange of parameter values. Figures (4) and (5) show theion DFs for k = 0 . k = 1 respectively, for all com-binations of ˜ ω i = 1 ,
3, ˜ r = 0 . , C s = 0 . ,
1, and with the magnetisation parameter δ i = 1. As a graph-ical confirmation of the above discussion, we can onlysee multiple maxima in the ˜ v θ direction for ˜ r > / | ˜ ω s | ,and in the ˜ v z direction for | ˜ ω s | >
2, with the appropriatebounds marked by the horizontal/vertical white lines.Aside from multiple maxima in the orthogonal direc-tions, the DF can also be ‘two-peaked’. That is, the DFcan have two isolated peaks in (˜ v z , ˜ v θ ) space. This isseen to occur for figures (5d, 5g, 5h). Hence, f i is seento be ‘two-peaked’ when k = 1 for both ˜ r > / ˜ ω i and˜ r < / ˜ ω i . However, we do not see a two-peaked DFfor k = 0 .
1. This seems to suggest that the strongerguide field ( k = 1) correlates with multiple peaks. Physi-cally, this may correspond to the fact that a homogeneousguide field is consistent with a Maxwellian DF centredon the origin in (˜ v z , ˜ v θ ) space, given that a Maxwelliancontributes zero current. Hence, if the ‘main’ part/peakof the DF is centred away from the origin, then theMaxwellian contribution from the guide field could con-tribute a secondary peak. These secondary peaks areseen to be more pronounced when ˜ C i is larger, i.e. thecontribution from the second term from the DF is greater.Figures (6) and (7) show the electron DFs for k = 0 . k = 1 respectively, for all combinations of ˜ ω e = 1 , r = 0 . ,
2, and C e = 0 . ,
1, and with the magnetistaionparameter δ e = δ i (cid:112) m e /m i ≈ / √ T i = T e . In general wesee DFs with fewer multiple maxima in velocity spacethan the ion plots, which is physically consistent withthe electrons being more magnetised, i.e. more ‘fluid-like’. In particular we see no multiple maxima in figure7, the case with the stronger background field.Note that when the electrons to have the same mag-netisation as the ions, i.e. δ e = δ i = 1, then thesemarked differences in the velocity-space plots disappear,and we observe a qualitative symmetry f i (˜ v θ , ˜ v z , r ) ∝ f e ( − ˜ v θ , − ˜ v z , r ). V. SUMMARY
In this paper we have calculated one-dimensional colli-sionless equilibria for a continuum of magnetic field mod-els based on the Gold-Hoyle flux tube, with an additionalconstant background field in the axial direction. Thisstudy was motivated by a desire to extend the existingmethods for solutions of the ‘inverse problem in Vlasovequilibria’ in Cartesian geometry, to cylindrical geome-try. Initial efforts focussed on solving for the exact force-free Gold-Hoyle field, but this seems impossible due tothe positive definite centrifugal forces. The Gold-Hoylefield in particular was chosen as it represents the ‘natural’analogue of the Force-Free Harris Sheet in cylindrical ge-ometry, a magnetic field whose VM equilibria have beenthe subject of recent study, .A background field was introduced, and an equilibriumdistribution function was found that reproduces the re-quired magnetic field, i.e. solves Amp`ere’s Law. It is0the presence of the background field that allows us tosolve Vlasov’s equation and Amp`ere’s Law, and it ap-pears physically necessary as it introduces an ‘asymme-try’; namely an extra term into the equation of motionwhose sign depends explicitly on species. In contrast tothe ‘demands’ of insisting on a particular magnetic field,no condition was made on the electric field. The distri-bution function allows both electrically neutral and non-neutral configurations, and in the case of non-neutralitywe find an exact and explicit solution to Poisson’s equa-tion for an electric field that decays like 1 /r far from theaxis. We note here that the type of solutions derived inthis paper could - after a Galilean transformation - beinterpreted as 1D BGK modes with finite magnetic field(see Refs. 60–63 for example, to provide some context).An analysis of the physical properties of the DF wasgiven in Section IV, with some detailed calculations inAppendix B. The dependence of the sign of the chargedensity (and hence the electric field) on the bulk ion andelectron rotational flows was analysed, with a physical in-terpretation given. Essentially the argument states thatthe electric field exists in order to balance the differencein the centrifugal forces between the two species. TheDF was found to be able to give sub-unity values of theplasma beta, should this be required/desirable given therelevant physical system that it is intended to model.The final part of the analysis focussed on plotting theDF in velocity space, for certain parameter values, andat different radii. Mathematical conditions were foundthat determine whether or not the DF could have multi-ple maxima in the orthogonal directions in velocity space,and these are corroborated by the plots of the distribu-tion functions. For certain parameter values, the DFwas also seen to have two separate, isolated peaks. Thisnon-thermalisation suggests the existence of microinsta-bilities, for a certain choice of parameters.Further work could involve a deeper anlysis of the prop-erties of the distribution functions and their stability.This work has also raised a fundamental question: ‘isit possible to describe a one-dimensional force-free col-lisionless equilibrium in cylindrical geometry?’ Prelimi-nary investigations seem to suggest that it is not possible. ACKNOWLEDGMENTS
O.A. would like to thank both Professor A.W. Hood ofthe University of St Andrews and Professor P.K. Brown-ing of the University of Manchester for encouraging dis-cussions. The authors gratefully acknowledge the sup-port of the Science and Technology Facilities CouncilConsolidated Grants ST/K000950/1 and ST/N000609/1,as well as Doctoral Training Grant ST/K502327/1. Wealso gratefully acknowledge funding from LeverhulmeTrust Research Project Grant F/00268/BB.
Appendix A: Moments of the DF
In this appendix we calculate the zeroth and first or-der velocity space moments of the DF, necessary for thecharge density and the current density respectively. SeeTable 1 for a clarification of all dimensionless quantitiesdenoted by a tilde, ˜ . The number density of species s isgiven by the zeroth moment of the DF; n s = (cid:90) f s d v s = n s e − ˜ φ s × (cid:104) e ( ˜ U zs +˜ r ˜ ω s ) / e ˜ U zs ˜ A zs e ˜ ω s ˜ r ˜ A θs + C s e ˜ V zs / e ˜ V zs ˜ A zs (cid:105) (A1)The following sum gives the charge density, σ = (cid:88) s q s n s = (cid:88) s n s q s e − ˜ φ s × (cid:104) e ( ˜ U zs +˜ r ˜ ω s ) / e ˜ U zs ˜ A zs e ˜ ω s ˜ r ˜ A θs + C s e ˜ V zs / e ˜ V zs ˜ A zs (cid:105) (A2)We take the v z moment of the DF to calculate the z − component of the bulk velocity, u zs = v th,s n s (cid:90) ˜ v zs f s d ˜ v s , = n s v th,s n s e − ˜ φ s (cid:20) ˜ U zs e ˜ U zs ˜ A zs e ( ˜ U zs +˜ r ˜ ω s ) / e ˜ ω s ˜ r ˜ A θs + ˜ V zs C s e ˜ V zs / e ˜ V zs ˜ A zs (cid:21) , (A3)for n s the number density. The following sum gives the z − component of the current density, j z = (cid:88) s q s n s u zs = (cid:88) s n s q s v th,s e − ˜ φ s × (cid:18) ˜ U zs e ˜ U zs ˜ A zs e ( ˜ U zs +˜ r ˜ ω s ) / e ˜ ω s ˜ r ˜ A θs + ˜ V zs C s e ˜ V zs / e ˜ V zs ˜ A zs (cid:19) . (A4)By taking the v θ moment of the DF we can calculatethe θ − component of the bulk velocity, u θs = v th,s n s (cid:90) ˜ v θs f s d ˜ v s , = ˜ r ˜ ω s n s v th,s e − ˜ φ s n s e ( ˜ U zs +˜ r ˜ ω s ) / e ˜ U zs ˜ A zs e ˜ ω s ˜ r ˜ A θs , (A5)for n s the number density. The following sum gives the θ − component of the current density, j θ = (cid:88) s q s n s u θs = (cid:88) s n s q s v th,s ˜ r ˜ ω s e − ˜ φ s × e ˜ U zs ˜ A zs e ( ˜ U zs +˜ r ˜ ω s ) / e ˜ ω s ˜ r ˜ A θs . (A6)1 Appendix B: Looking for multiple maxima1. Maxima of the DF in v θ space The ˜ p rs dependence of the DF is irrelevant to our dis-cussion, and as such can be integrated out. We can alsoneglect the scalar potential φ . The reduced DF, ˜ F s , indimensionless form is˜ F s = (( √ πv th,s ) /n s ) e ˜ φ s (cid:90) ∞−∞ f s dv r , which then reads˜ F s = exp (cid:40) − (cid:34)(cid:18) ˜ p θs ˜ r − ˜ A θs (cid:19) + (cid:16) ˜ p zs − ˜ A zs (cid:17) (cid:35)(cid:41) × (cid:104) exp (cid:16) ˜ ω s ˜ p θs + ˜ U zs ˜ P zs (cid:17) + C s exp (cid:16) ˜ V zs ˜ P zs (cid:17)(cid:105) . (B1)We have written ˜ F s in terms of the canonical mo-menta, and so we search for stationary points given by ∂ ˜ F s /∂ ˜ p θs = 0, equivalent to ∂ ˜ F s /∂ ˜ v θs = 0. Setting ∂ ˜ F s /∂ ˜ p θs = 0 gives˜ p θs − ˜ r ˜ A θs = ˜ ω s ˜ r C s e − ˜ ω s ˜ p zs e − ˜ ω s ˜ p θs = A Be − ˜ ω s ˜ p θs := R (˜ p θs ) . (B2)To derive a necessary condition for multiple maxima, weanalyse the RHS of equation (B2), R (˜ p θs ). This func-tion is bounded between 0 and A, and is monotonicallyincreasing. Hence, using techniques similar to those in ,a necessary condition for multiple maxima in the DF isthat max ˜ p θs R (cid:48) (˜ p θs ) > . (B3)This condition can be shown to be equivalent to A ˜ ω s / > ω s > r − ⇐⇒ ˜ r > / | ˜ ω s | (B4)This demonstrates that for sufficiently small ˜ r , there can-not exist multiple maxima. Equivalently, this conditionwill always be satisfied for some ˜ r , and as such is just acondition on the domain, in ˜ r , for which multiple maximacan occur. This condition is not sufficient however, as itcould still be the case that there exists only one pointof intersection (and hence one maximum), depending onthe value of B . It is seen that R has unit slope at˜ p ± θs = 1˜ ω s × (cid:104) ln (2 B ) − ln (cid:16) A ˜ ω s − ± (cid:112) A ˜ ω s ( A ˜ ω s − (cid:17)(cid:105) . (B5)Clearly R has unit slope for two values of ˜ p θs . Aftersome graphical consideration of the problem, it becomesapparent that B should be bounded above and below formultiple maxima. After elementary consideration of the functional form of (B2), for example with graph plottingsoftware, we see that multiple maxima in the ˜ v θ directioncan only occur, for a given ˜ r , when B (and hence ˜ v z )satisfies these inequalities for ions˜ p + θi − R (˜ p + θi ) − ˜ r ˜ A θi > , ˜ p − θi − R (˜ p − θi ) − ˜ r ˜ A θi < , (B6)and these for electrons˜ p + θe − R (˜ p + θe ) − ˜ r ˜ A θe < , ˜ p − θe − R (˜ p − θe ) − ˜ r ˜ A θe > . (B7)
2. Maxima of the DF in v z space We shall once again use the reduced DF defined inequation (B1) in our analysis. Thus, we shall consider ∂ ˜ F s /∂ ˜ p zs = 0, which is equivalent to ∂ ˜ F s /∂ ˜ v zs = 0. Set-ting ∂ ˜ F s /∂ ˜ p zs = 0 gives˜ p zs − ˜ A zs = ˜ U zs + C s ˜ V zs e − ˜ ω s (˜ p zs +˜ p θs ) C s e − ˜ ω s (˜ p zs +˜ p θs ) = A B e − D ˜ p zs + A B e − D ˜ p zs := R (˜ p zs ) + R (˜ p zs ) = R (˜ p zs ) , such that A = ˜ U zs , A = ˜ V zs ,B = C s e − ˜ ω s ˜ p θs = B − , D = ˜ ω s = − D . To derive a necessary condition for multiple maxima, weanalyse the RHS of equation (B8). Each R function isbounded and monotonic. Once again using techniquessimilar to those in , a necessary condition for multiplemaxima in the DF is thatmax ˜ p zs ( R (cid:48) (˜ p zs ) + R (cid:48) (˜ p zs )) > . (B8)After some algebra this condition can be shown to beequivalent to ˜ ω s / > | ˜ ω s | > . (B9)This condition is not sufficient however, as it could stillbe the case that there exists only one point of intersec-tion, depending on the value of B (= 1 /B ). The tran-sition between 3 points of intersection and one occurs atthe value of B for which the straight line of slope unitythrough ˜ p zs = 0 just touches R (˜ p zs ) + R (˜ p zs ) at thepoint where it also has unit slope. It is readily seen that R + R has unit slope at˜ p ± zs = 1˜ ω s × (cid:104) ln (2 B ) − ln (cid:16) ˜ ω s − ± (cid:112) ˜ ω s (˜ ω s − (cid:17)(cid:105) . (B10)We see again that R has unit slope for two values of ˜ p zs .Once again, after some graphical consideration of the2problem, it becomes apparent that B should be boundedabove and below for multiple maxima. After elementaryconsideration of the functional form of (B8), for examplewith graph plotting software we see that multiple max-ima in the ˜ v z direction can only occur, for a given ˜ r , when B (and hence ˜ v θ ) satisfies these inequalities for ions˜ p + zi − R (˜ p + zi ) − ˜ A zi > , ˜ p − zi − R (˜ p − zi ) − ˜ A zi < , (B11)and these for electrons˜ p + ze − R (˜ p + ze ) − ˜ A ze < , ˜ p − ze − R (˜ p − ze ) − ˜ A ze > . (B12) TABLES & FIGURES FOLLOW s subscript refers to particles of species s . Variable Dimensionless formParticle Hamiltonian ˜ H s = β s H s Particle angular momentum τ p θs = m s v th,s ˜ p θs Particle z -Momentum p zs = m s v th,s ˜ p zs Vector potential q s A = m s v th,s ˜ A s Scalar Potential ˜ φ s = q s β s φ Bulk rectilinear flows v th,s ˜ U zs = U zs , v th,s ˜ V zs = V zs Bulk angular frequency τ v th,s ˜ ω s = ω s Particle velocity v = v th,s ˜ v s TABLE II: The fundamental parameters of the equilibrium.The s subscript refers to particles of species s . Macroscopic Microscopicparameter Meaning parameter Meaning B Characteristic magnetic field strength m s Mass of particle τ Measure of the twist of flux tube q s , q Charge, magnitude of charge k Strength of the background field β s = 1 / ( k B T s ) Thermal beta γ (cid:54) = 0 ,
1, 0 < γ < v th,s Thermal velocity U zs , V zs Bulk rectilinear flows δ s ( r ) , δ s Magnetisation parameters ω s Bulk angular frequency n s Normalistaion of particle number(a) k = 0 . k = 0 . k = 1 FIG. 1: The twist (normalised by τ / (2 π )) of the GH+B field for three values of k . 1a shows the twist for k < / (a) B for k = 0 . B for k = 0 . B z reversal, given0 < k < / FIG. 2: 2a and 2b show the GH+B magnetic field in the xy plane, for two values of k . The curved arrows indicatethe direction of the ˜ B θ components, whilst the blue-black-red shading denotes the magnitude and direction of the˜ B z component. The k = 0 . B z field direction and as such is a Reversed Field Pinchwhilst k = 0 . B z at ˜ r = 0. 2c shows the radius at which ˜ B z changes its direction, for 0 < k < /
2. ˜ B z does not reverse for k ≥ / (a) | B | for k = 0 . | B | for k = 0 . | B | for k = 1 FIG. 3: 3a-3c show the magnitude of the GH+B magnetic field for k = 0 . , . k = 1 respectively, normalised by B . For k < . | ˜ B | → k from above, whereas for k ≥ / | ˜ B | → k from below. (a) (˜ ω i , ˜ r, C i ) = (1 , . , .
1) (b) (˜ ω i , ˜ r, C i ) = (1 , , .
1) (c) (˜ ω i , ˜ r, C i ) = (3 , . , .
1) (d) (˜ ω i , ˜ r, C i ) = (3 , , . ω i , ˜ r, C i ) = (1 , . ,
1) (f) (˜ ω i , ˜ r, C i ) = (1 , ,
1) (g) (˜ ω i , ˜ r, C i ) = (3 , . ,
1) (h) (˜ ω i , ˜ r, C i ) = (3 , , FIG. 4: Contour plots of the f i in (˜ v z , ˜ v θ ) space for an equilibrium with field reversal ( k = 0 . < . ω i , ˜ r, C i ) and δ i = 1. The white horizontal/vertical lines indicate the regions in which multiple maximain either the ˜ v z or ˜ v z directions can occur, if at all. A single line indicates that the ‘region’ is a line.5 (a) (˜ ω i , ˜ r, C i ) = (1 , . , .
1) (b) (˜ ω i , ˜ r, C i ) = (1 , , .
1) (c) (˜ ω i , ˜ r, C i ) = (3 , . , .
1) (d) (˜ ω i , ˜ r, C i ) = (3 , , . ω i , ˜ r, C i ) = (1 , . ,
1) (f) (˜ ω i , ˜ r, C i ) = (1 , ,
1) (g) (˜ ω i , ˜ r, C i ) = (3 , . ,
1) (h) (˜ ω i , ˜ r, C i ) = (3 , , FIG. 5: Contour plots of f i in (˜ v z , ˜ v θ ) space for an equilibrium without field reversal ( k = 1 > . ω i , ˜ r, C i ) and δ i = 1. The white horizontal/vertical lines indicate the regions in which multiple maximain either the ˜ v z or ˜ v z directions can occur, if at all. A single line indicates that the ‘region’ is a line. (a) (˜ ω e , ˜ r, C e ) = ( − , . , .
1) (b) (˜ ω e , ˜ r, C e ) = ( − , , .
1) (c) (˜ ω e , ˜ r, C e ) = ( − , . , .
1) (d) (˜ ω e , ˜ r, C e ) = ( − , , . ω e , ˜ r, C e ) = ( − , . ,
1) (f) (˜ ω e , ˜ r, C e ) = ( − , ,
1) (g) (˜ ω e , ˜ r, C e ) = ( − , . ,
1) (h) (˜ ω e , ˜ r, C e ) = ( − , , FIG. 6: Contour plots of f e in (˜ v z , ˜ v θ ) space for an equilibrium with field reversal ( k = 0 . < . ω e , ˜ r, C e ) and δ e ≈ / √ v z or ˜ v z directions can occur, if at all. A single line indicates that the ‘region’ is a line.6 (a) (˜ ω e , ˜ r, C e ) = ( − , . , .
1) (b) (˜ ω e , ˜ r, C e ) = ( − , , .
1) (c) (˜ ω e , ˜ r, C e ) = ( − , . , .
1) (d) (˜ ω e , ˜ r, C e ) = ( − , , . ω e , ˜ r, C e ) = ( − , . ,
1) (f) (˜ ω e , ˜ r, C e ) = ( − , ,
1) (g) (˜ ω e , ˜ r, C e ) = ( − , . ,
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