THE LILEY FORM FOR THE PRESSURE TENSORIN MAGNETISED PLASMA

Roger J. Hosking

School of Mathematical Sciences

University of Adelaide

Email: marjhg@gmail.com

Contents

1. Quirks of History

2. Plasma Pressure Tensor

3. Applications

4. Epilogue

Bibliography

1. Quirks of History

“Transport Processes in a Plasma” (Braginskii, 1965) isone of the most cited articles in plasma physics. Claritythroughout its English translation provided the Westernresearch community with an early insight into the basisof macroscopic plasma models beyond the ideal or evenresistive formulations previously widely used.

The microscopic foundation is a Boltzmann-type kineticequation that alternatively may be written

∂fs∂t

+ c·∇fs + as(r, c, t)·∇cfs =δfsδt

, (1)

defining the time evolution of a velocity distributionfunction fs(r, c, t) in phase space (r, c) for each particlespecies s, where as is the particle acceleration.

For a particle mass ms and charge es the acceleration is

as(r, c, t) = g +esms

[E(r, t) + c×B(r, t)] ,

if both gravitational and electromagnetic componentsare included (where g, E and B denote gravitational,electric and magnetic fields) – cf. Chapman & Cowling(1970).

The symbol ∇c denotes the gradient operator relativeto the independent velocity vector c, analogous to ∇relative to the position vector r in phase space, and theterm on the right-hand side δfs/δt represents the timerate of change of the velocity distribution function dueto the microscopic particle collisions.

Any macroscopic field equation of interest correspondsto taking some moment of equation (1). If w = c− c0

denotes the peculiar velocity for the species relative tosome reference velocity c0, the moment correspondingto any related function Ψ(w) is defined by

ns〈Ψ〉 =

∫fs(r, c, t) Ψ(w) dc , (2)

including the particle number density (number ofparticles in a unit volume)

ns(r, t) =

∫fs(r, c, t) dc .

One may identify c0 with mass-weighted mean velocityv defined via the total density and total mass flux

ρ = Σsρs , ρv = Σsρsvs ,where the summation is over all species with density ρsand mean flow velocity vs = 〈c〉. Then we may invoke

∂(ns〈Ψ〉)∂t

+∇·(ns〈(v + w) Ψ〉)

− ns〈(fs +esms

w×B−w·∇v)·∇wΨ〉 = Cs(Ψ) ,

with fs = g +esms

[E + v×B]− dv

dt, Cs(Ψ) =

∫δfsδt

Ψ dc ,

(3)

which is a generalised equation of change from (1).

Familiar fundamental field quantities are related to thelower moments defined by (2) for each species s ofparticle mass ms — viz.

ρs ≡ nsms (density)

us ≡ 〈w〉 (mean relative velocity)

ps ≡ ns〈1

3msw

2〉 (pressure)

ps ≡ ns〈msww〉 (pressure tensor)

qs ≡ ns〈1

2msw

2w〉 (thermal flux) .

Other higher moments are much less familiar!

Thus if Ψ is identified with the quantitiesms ,msw,

12msw

2,ms{ww}, 12msw2w etc. successively,

then for each species the general equation of change(3) produces the basic equations of continuity (mass),momentum, thermal energy, the traceless part of thepressure tensor ts , heat conduction vector qs , etc. etc.

Now let us proceed to the far less cited results due toLiley, Hosking & Marinoff and Callen et al., as has nowbeen presented in considerable detail in Chapter 2 ofHosking & Dewar (“Fundamental Fluid Mechanics andMagnetohydrodynamics”, Springer, 2016).

2. Plasma Pressure Tensor

Various representations for the collisional terms on theright-hand sides of the moment equations have beenconsidered, but on tensor rank alone two simple linearexpressions for each species s are

Cs(ms{ww}) = −∑j

ϑsjτsj

tj ,

Cs(

1

2ms(w

2 − 5kTs

ms)w

)= −

∑j

(1

τsjRj + ζsjρjuj

),

with the coefficients {τsj , ϑsj , ζsj} functions of numberdensity and temperature.

An asymptotic closure like Chapman-Enskog for neutralgas transport (Chapman & Cowling, 1970) or one ofvarious parameter ordering procedures for plasmas maybe considered, but there is another “careful andelegant” approach (Hazeltine & Meiss, 2003) – notablythe Grad “thirteen moment” approximation, whichcorresponds to a truncation of the hierarchy of momentequations up to the equation for the heat flux qs whereterms involving two higher level quantities are omitted.

[Similar higher “n moment” (n > 13) closures have alsobeen considered, including even more of the hierarchy ofmoment equations and then neglecting terms involvingeven higher moments beyond the corresponding set.]

It is convenient to introduce a characteristic frequencyω to represent the time derivative term in the resultingmoment equation for the traceless component of theplasma pressure tensor t, which may then be written inthe notationally convenient form

t− 2{t×a} = −2µs (4)

where the symmetric and traceless generalised rate ofdeformation (or strain) tensor s is given by

2p s = −2ρ{f u}+ 2p{∇v}+ t∇·v

+ 2{t·∇v}+∇·h +∑j 6=s

ϑsjτsj

tj , (5)

with coefficient µ =p τ

ωτ + ϑand vector a =

eB

m

τ

ωτ + ϑ.

Note that a proportional to the gyrofrequency firstlyreflects the time derivative and secondly the collisionalterms, with one or the other predominant when theparameter ωτ is respectively sufficiently large or small.Note also we refer here to collisions between particlesof the particular species under consideration (with itsappropriate factor ϑ typically of order one), and anycollisional coupling between the different species isrepresented by the summation for j 6= s incorporated ins. The form (4) isolates the term {t×a} proportionalto the magnetic field B, which enables us to obtain t asan explicit function of s below — even though there aretwo terms involving t in the generalised deformationtensor (5), a point examined further in the case of asimple ion-electron plasma in Hosking & Dewar (2016).

Using tensor identities, we may re-express (4) as

t =− 2µ

(1+|a|2)(1+4|a|2)[ (s+2{s× a})(1+|a|2)

+ 6 ({s·aa}+2{{s·aa}×a})+6 s:aa{aa} ] . (6)

The Cartesian representation of this Liley form (6)appears in Chapman & Cowling (1970), who assumed auniform magnetic field B — but we now recognise that(6) is an invariant result (valid in any system ofcoordinates) without any restriction on themagnetic field (even null points). Eq. (4) and theresult (6) obviously always reduce to t = −2µs for aneutral species and for charged species if there is nomagnetic field (i.e. for a = 0), when we recover thefamiliar shear viscosity form on identifying s = {∇v}.

However, the terms involving the vector a usuallydominate for charged particles in the presence of amagnetic field, producing characteristically anisotropicplasma viscosity contributions. Thus on noting

− 2µ

(1 + | a|2)(1 + 4| a|2)=µ

2

[−| a|−4 +

5

4| a|−6 + · · ·

],

in magnetised plasma wherever |a| � 1 the generalexplicit form (6) may be expanded to obtain

t = t‖ + tg + t⊥ + · · · , (7)

successively specifying the leading parallel, cross ortransverse (gyroviscous or “FLR”) and perpendicularviscosity components.

Consequently, we obtain the quite succinct forms

t‖ = −3µ s:b̂b̂{b̂b̂} ,

tg = − µ

|a|

{s×b̂ + 6{s·b̂b̂}×b̂

},

t⊥ = − µ

2|a|2

[s + 6{s·b̂b̂} − 15

2s:b̂b̂{b̂b̂}

].

Note: the operator { } is defined by

{F} ≡ 1

2(F + FT )− 1

3F:I I

for any dyadic F, where FT denotes its transpose.As Tr F ≡ F:I and Tr I = 3, the result is a tracelesssymmetric dyadic.

The result (6) was originally obtained by Bruce Lileywhen I first worked with him at ANU in 1968, and theabove successive components in a sufficiently largemagnetic field (|a| � 1) were afterwards identified atCulham (Hosking & Marinoff, 1973). Alternative formslater obtained by Callen et al. (1987) immediatelyfollow by expanding the above forms, and exhibit all ofthe terms in the traceless component of the tensor inBraginskii (1965) – i.e. where (apart from his Delphiccoefficients) the W0 corresponds to t‖, the sum of W1

and W2 to t⊥, and the sum of W3 and W4 to tg .Incidentally, in the absence of collisions, an anisotropicplasma pressure p = p‖b̂b̂ + p⊥I⊥ was earlier discussedby Chew, Goldberger & Low (1956).

As previously indicated, the Grad “thirteen moment”approximation renders another moment equation thatreadily produces an explicit result for the heat fluxvector q (Herdan & Liley, 1960). Extensions of thehierarchy to higher moments include contributions byRamos (2005, 2007), who followed Braginskii and someothers in defining the moments with reference to thespecies flow velocity vs . However, in a simple plasmaconsisting of electrons and only one ion species, themean velocity we used is effectively the ion velocitythat corresponds to the predominant plasma viscositycontribution invoked (i.e. v ' vi since me � mi).

3. Applications

A simple variational form demonstrates that the growthrates of ideal and resistive instabilities in magnetisedplasma may be reduced by viscosity (Hosking & Dewar,2016). An early calculation showed “resistive-g modes”with azimuthal mode numbers m ≥ 1 may be stabilisedin the reverse field pinch by parallel viscosity (Hosking& Robinson, 1979), and further nonlinear calculationsaddressed the m = 0 mode (Hender & Robinson, 1981).

Parallel viscosity has been referred to as “Braginskiiviscosity” by various authors working in the field of solarphysics, following Hollweg (1985, 1986) who exploredthe Braginskii form W0 with the collisional coefficientη0 = 0.96niTiτi for application to the solar corona.

Craig and Litvinenko at the University of Waikato haveinter alia shown that energy dissipation due to parallelviscosity during magnetic merging can substantiallyexceed that due to resistivity, formerly considered toexplain the observed energy release in solar flares butwas “found severely wanting”. For example, this wasshown in transient reconnecting plasmas (Craig, 2010),with a formal exact solution for steady state merging(Craig & Litvinenko, 2009), and dynamic coalescence(Craig & Litvinenko, 2010).

However, adoption of the Liley form to deal with theinherent magnetic nulls could prove to be a significantnew feature in such calculations.

4. Epilogue

Bruce Liley’s PhD thesis was the source of the articleHerdan & Liley (1960), which represents the beginningof the theoretical work that eventually led him to theLiley form (6). When I joined him in 1968 under specialleave for a few months from the Flinders University ofSouth Australia, Bruce simply gave me his result on ascrap of paper and mentioned several tensor identitiesused in its derivation – and asked me to check it outindependently. The first two Exercises in Hosking &Dewar (2016) illustrate my subsequent derivation; andwhen he later transferred to the University of Waikato,I had become so interested I chose to follow him there!

Our families had quickly become good friends – butwhen Bruce and his wife Margaret stayed at our house(he had came to Flinders to give a seminar), he said hewas surprised at that possibility if an opportunity cameup, as my wife and I were living in our home town inAdelaide. However, a second chair in Mathematics atthe University of Waikato was advertised, to which Iwas eventually appointed.

[When I told Peter Karmel (then the Flinders UniversityVice-Chancellor) I thought I was too young to apply, hesaid that he was much the same age when appointed toa chair – and not to go if one was not offered to me.]

On arrival at Waikato in mid-1971, I worked with Bruceas he wrote up a Physics Report alongside Hosking &Marinoff (1973), research begun a little earlier when Iproceeded on leave from Flinders University to CulhamLaboratory. That Report was referenced there as alsointended for publication in the same journal (PlasmaPhysics), but Bruce did not manage that as he hadassumed various time-consuming responsibilities in thedevelopment of the new School of Science and as aProfessorial representative on the University of WaikatoCouncil. It may be that rendition of the Liley form inChapter 2 of Hosking & Dewar (2016) will remedy thishistorical omission at last, especially to provide a basisfor discussion of magnetic nulls in magnetised plasma.

photograph.pdf

Bruce Sween Liley

Published in: Robert L. Dewar; Ian Hutchinson; R. S. “BAS” Pease; Alan Ware; Physics Today 55, 80-81 (2002)DOI: 10.1063/1.1485600Copyright © 2002 American Institute of Physics

Figure: Bruce Sween Liley

Braginskii. S.I. 1965 In M.A. Leontovich (Ed.)Reviews of Plasma Physics, vol. 1, pp. 205-311,Consultants Bureau, New York.

Chapman, S. & Cowling, T.G. 1970The Mathematical Theory of Non-Uniform Gases,Third Edition, Cambridge University Press.

Hosking , R.J. & Dewar, R.L. 2016 FundamentalFluid Mechanics and Magnetohydrodynamics,Springer.

Hazeltine, R.D. & Meiss, J.D. 2003Plasma Confinement, Dover.

Hosking, R.J., & Marinoff, G.M. 1973Plasma Physics 15, 327-341.

Callen, J.D., Qu, W.X., Siebert, K.D., Carreras,B.A., Shaing, K.C. & Spong, D.A. 1987 PlasmaPhysics and Controlled Thermonuclear Research,Volume 2, IAEA, Vienna.

Chew, G.F., Goldberger, M.I. & Low, F.F. 1956Proc. Roy. Soc. London A 236, 112-118.

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Ramos, J.J.2005 Physics of Plasmas 12, 112301;2007 Physics of Plasmas 14, 052506.

Hosking, R.J. & Robinson, D.C. 1979Controlled Fusion and Plasma Physics (Proc. 9thEurop. Vonf. Oxford, 1979) BP6 61

Hender,T.C. & Robinson, D.C. 1981Nuclear Fusion 21, 755-758.

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Craig, I.J.D. 2010Astron. Astrophys. 515, A96.

Craig, I.J.D. & Litvinenko, Y. E.2009 Astron. Astrophys. 501, 755-760;2010 Astrophys. J. 725, 886-893.

Dewar, R.L., Hutchinson, I., Pease, R.S. & Ware. A.2002 Physics Today 55, doi: 10.1063/1.1485600,American Institute of Physics.