Towards a rational dynamics of plasmas
Citation for published version (APA):
Benach, R. (1974). Towards a rational dynamics of plasmas. Technische Hogeschool Eindhoven.
https://doi.org/10.6100/IR157038
DOI:
10.6100/IR157038
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Published: 01/01/1974
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TOWARDS A RATIONAL DYNAMICS OF PLASMAS
PROEFSCHRI FT
ter verkrijging van de graad van doctor in de technische wetenschappen aan de Teehnische Hogeschool Eindhoven, op gezag van de rector magnificus, prof.cir. ir. G. Vossers, voor een commissie aangewezen door het college van dekanen in het operbaar te verdedigen op vrijdag 24 mei 1974 te 16.00 uur
door
Robert Benach
geboren te Cleveland
Dit proefschrift is goedgekeurd door de prornotoren
* Prof. d r. I . Mu 11 er
en
Prof.dr. L.H.Th. Rietjens
* Department of Mechanics and Materials Science The Johns Hopkins University
That urg'd by thee, I turn'd the tuneful art From sounds to things, from fancy to the heart; For Wit's false mirror held up Nature's light; Shew'd erring Pride, whatever is, is right; That Reason, Passion, answer one great aim; That true Self-love and Social are the same; That Virtue only makes our Bliss below; and all our Knowledge is, ourselves to know."
Alexander Pope
To the memory of my mother, -Estel le G. Benach, and to Josje and Dave, my dearest companions, I dedicate with deep respect and love this contribution to the knowledge and hopefully the betterment of mankind.
vii
PREFACE
The two general problems which stand in the background of this study are of an interdependent nature. On the vie hand, of the "states of matter" from which a macrocosm may be composed, there must be a specification of the particular physical system to be considered here. On the other, there is the task of securing a suitable means for treating the behavior of at least that system.
Clearly, the first point is concerned with the problem of what the system in question~· Here I direct attention towards the "vague" concept of plasma
which I assume is not a state of matter. but rather, a condition that may on occasion pervade those states of matter that are commonly recognized: viz., gases, I iquids and sol ids.
The second point in turn is related to the complementary problem of describing what a (class of) system(s) does. Here too, as in the case of the first point, there are numerous basic issues of concern that require further investigation. Indeed, not only the subject matter of plasma physics but also the present means for its treatment appear to be fluid-I ike.
A theoretical treatment of problems relating to the gross (i.e. bulk) behavior of dynamic systems requires a complete set of macroscopic equations. For gaseous plasmas, a special class of electromagnetic fluids, there appears at this time to be essentially H~o general approaches by means of ~~hich the
desired relationships are secured. The one is of a particle nature and is based upon a (non-)equi I ibrium statistical continuum development; and, the other is a classical (i.e. I inear) continuum mechanical approach.
It is a foremost purpose of this study to consider the possibility of deriving the basic governing system equations from a "new" (for plasma physics) viewpoint: that being rational mechanics and rational therm odynam-ics 1. In order to ascertain the possible potential of a modern continuum mechanical treatment, the methodology of rational mechanics wil I be used to develop a particular non-equilibrium mixture theory for the case of a one-temperature, chemically reactive, non-relativistically moving, magnetizable and dielectric fluid medium.
Herein consideration is given specifically to the matter of securing for a plasma bulk a set of general constituent, mixture and electromagnetic field balance and jump balance relationships. And this, in addition to the proposal of a set of non-1 inear mechanical, electromagnetic and energetic constitutive equations which have been taken as being appropriate to the model of the system employed. On the basis of said model a set of equations emerges from the considerations here which is suitable for further theoretical and empirical investigation of a broad class of natural and laboratory plasma systems.
This study, in reflecting the tradition of rational mechanics, attempts to set down a pre! iminary conceptual framework suitable for the systematic investigation of plasmas. Although the theory developed here is of modest mathematical sophistication and treats mixtures of fluids with relatively simple physical properties, it is nonetheless directed towards'the ga·ining of an eventual "breakthrough in understanding" .of plasma systems.
It is henceforth to be understood that these two aspects of modern continuum theory are meant with the mention of only the former. Note further that footnotes are subsequently placed at the end of the respec-tive chapters to which they relate.
ix
Further, while recognizing the value of well-motivated and executed special case studies, the purpose of this endeavor is taken to be served with the presentation of such results as ~sually form the starting point
for those considerations. A treatment of these matters in greater depth or an investigation of their applicability must await the results of future developments.
Acknowledgements
It is not possible for me to express completely and honestly with words alone my deep sense of indebtedness to those who have contributed most meaning-fully to the realization of this dissertation.
How to say thank you to Professor Muller for his encouragement, his untiring guidance and his innumerable kindnesses; to Dr. Alts for valuable discussions; to Professor Erlcksen for helpful conversations; to Professor Truesdel I for his interest and the example of scholarship he sets; and to my wife, Jos, for her love, forbearance and optimism how indeed do I thank you all? To each of you I say simply, yet sincerely, thank you.
It is also a pleasure to express my appreciation to the "Technische Hogeschool Eindhoven" for the facilities placed at my disposal during the period in which I prepared the greater part of this work. To Mrs. H.K. van der Putten-Bosscher, Miss Th.J.M. van den Hurk and Mrs. E.E.F.M. Baselmans-Weijers I wish to state my gratefulness for their painstaking efforts taken
in the typing of the final manuscript; and to Mr. H.F. Koolmees for the preparation of the figures.
Robert Benach
NOMEtKLATURE
A. INTRODUCTION
1. On representative plasma systems 2. Position of the problem
a. Purpose b. Problem Footnotes to Chapter A B. BALANCE EQUATIONS a. Co-ordinate systems b. Material systems
c. Material integral kinematics i. Volume integrals ii. Surface integrals 1. I ntegra I ba 1 ances
a. Volume balances b. Surface balances 2. Differential balances
a. Continuous system balances b. Discontinuous system b.alances 3. Mass balance equations
a. Constituents b. Mixture
4. The charge-current equations
a. Coulomb's law: the charge flux equations b. Charge balance equations
i. Constituents ii. Mixture
c. Ampere-Maxwel 1 's law: the current equations
xv 10 10 14 22
32
35 39 42 42 46 49 49 5152
53 53 54 54 57 58 60 63 63 64 66xii
5, Electromagnetic field equations a. Magnetic flux equations b. Faraday's law of induction 6. Linear momentum balance equations
a. Constituents b. Mixture
c. Electromagnetic field
d. Mixture-electromagnetic field 7. Angular momentum balance equations
a. Constituents b. Mixture
8. Energy balance equations a. Constituents
b. Mixture
c. Electromagnetic field
d. Mixture-electromagnetic field 9. Entropy balance equat Ions
a. Constituents b. Mixture
10. Selected equations: a recapitulation Footnotes to Chapter B
C. CONSTITUTIVE EQUATIONS
1. Constitutive equation representations a. Non-I inear representations b. Linearized representations 2. Appendices Footnotes to Chapter C
68
69
69
70 71 72 7374
75 75 78 8182
84
8586
87 8789
91 93 101 106 109 119 124 142D. A MATTER OF PRINCIPLE: ENTROPY
1. Entropy principles
2. Admissible thermodynamic processes ·r V
3.
Appendix D-1. The Lagrange multipliersFootnotes to Chapter D
E. ON EQUILIBRIUM PROPERTIES OF FLUID MIXTURES INTERACTING WITH AN ELECTROMAGNETIC FIELD
1. On equilibrium
2. Equilibrium and the entropy principle in V
3.
Stress tensor4. Chemical potential 5. Gibbs-Duhem equations
6. Chemically (non-)reacting fluid mixtures a. Law of mass action
7. Gibbs' equation Footnotes to Chapter E
F. ON SPECIFIC PLASMA SYSTEM FIELD EQUATIONS
1. System interior equations
a. Balance, conservation and constitutive equations b. On mass and charge transport
2. System boundary and interface equations
3.
On solutions and applications4. On empirical considerations Footnotes to Chapter F 151 151 155
172
176
178
178
179
183
184
191191
1971
98
201 204 205 205211
216 219 224 226G. CONCLUSIONS AND CLOSURE REFERENCES VITA xiv 237 239
255
NOMENCLATURE
The following 1 ist of symbols is me.•.nt to be complete with the excep -tion of almost al 1 of the general material response coefficients and the invariants in terms of which they are derived. These quantities appear for the most part in Chapter C and can easily be recognized there from the con-text of the discussion on constitutive equations.
Symbol Latin a a. I A b. J b. I a b. E1 b. 01 b. Oa I 3!J 8 a 8. I alphabet Description
A constant with the dimensions of time Arbitrary polar vector
Arbitrary field
Arbitrary polar vector
Specific total external body force of a constituent
Specific body force of the electromagnetic
field in a mixture
Specific non-electromagnetic body force of a mixture
Specific non-electromagnetic body force of a constituent
mean normal "magnetic stress"
Constituent body
Magnetic flux density
Place where defined or introduced Sect. C. 1.a. Sect. D.1. Sect. C. 1 . a . (882) 1 (891) ( 888) (883) ( E27) l Sect. B.a. Sect. B. 5.
IB .• I J c( t) c a c. I
c
a -S c a -SC c ac
ac
..
a'J di ( t) dk. a J d l. I D. I 'Dl e e I e e a e E (eSM) -SE e a a xviMagnetic stress deviator
Singular line
Mass fraction of a constituent Arbitrary axial vector
Production rate density of mass for a constituent
Mass density supply by s(t) for a constituent
Charge density supply by s(t) for a constituent
Material curve for a constituent
Time-dependent vector with the dimensions of time
Deformation rate tensor of a constituent Differential element of c(t)
Charge potential
Partial charge potential
Fundamental unit of electric charge; Equilibrium relationship of e1 Basis vector
Production rate of total energy density of a constituent
Power expended by an electromagnetic field
(Non-)electromagnetic supply by s(t) of the total energy of a constituent
(E27)2 Sect. B.c.ii. Sect. B. b. Sect. C.1.a. Sect. B .. ~. Sect. B.3. (B58) Sect. B.b. (B153) (C3)1 (B34) 3 Sect. B. c. ii. (844) (B45) Sect. B.b. Sect. E.2. Sect. B.a. (D18) (8128) (B 141) 3 (8130)
-SE e E, EM
E.
I&.
·I a f g G G. I I Cl i. j ' k •• I ( t) J a Supply of£ by s(t) ERegions of "low"-density gas plasmas El ec t r i c fie l d
Electromotive intensity of a constituent
Mixture electromotive intensity
Arbitrary scalar field on s(t) tensor function(al)
Deformation gradient of a constituent
Arbitrary scalar field on s(t) Region of "very low"-density gases "Abraham vector"
Arbitrary vector field on s(t) Current potential
Partial current potential
Spatial vector component indices Reduced set of fundamental invariants Fundamental invariants
Phase interface
Jacobian of a constituent
Magnetization current density of a constituent
Polarization current density of a constituent
Sect. A.1. Sect. B.4.a. (BBO) 1 Chpt. C.
( B3)
( B72) 1 Sect. A.1. (B92) (B72) 2 (B67) 1 (B67) 2 Sect. B.a. (C27) (C16) (C12) Sect. B. c. i. (B4)(B53)
(B54)
J~ I k. K b I 1 k a lk L Lk a Lk Ll • m m a
m
.
I a L2 • L3 xviiiTotal current density of a constituent
Non-convective current density of a canst i tuent
Conduction current density of a mixture Magnetization current density of a mixture
Polarization current density of a mixture
Total current density of a mixture
Non-convective current density of a mixture
Generalized conduction current of a mixture
Arbitrary vector field on s(t)
Chemical potential in non-equi 1 ibrium of a constituent
Specific body couple of a constituent Specific body couple of a mixture Characteristic geometric length Specific total angular momentum of a canst i tuent
Specific total angular momentum of a mixture Pederson, scalar, Hal 1 electrical
conductivities
Unit atomic mass
Unit binormal vector to c{t) Mass of a constituent
Interaction force density of a constituent
(657)
(655)
(
662
)
1
(660)(661)
(662)3,4 (662)2 (670)(680)2
(E42) Sect. 6. 7. a. Sect. A. 1 . ( B99) 1 (Bl 14) l Footnote F21 Sect. 6.3.a. Sect. B.c. ii. Sect. B.b._s
m. I a -S m. I -SE m. I M, M - Me L M a Mk a Ml mk. a J mkj n n a n e iii\
a -S nk a_s
nk ND 0 0 .. ( t) I J pSupply by s(t) of 1 inear momentum for a constituent
Supply by s ( t) of mixture 1, 'ear momentum
Supply of Gi by s(t)
Regions of "medium"-density gas plasmas
Magnetic susceptibility of a constituent
Magnetization density of a constituent
Magnetization density of a mixture Couple stress tensor of a constituent
Couple stress tensor of a mixture
Total number of independent chemical reactions Number density of a constituent
Electron number density
Unit normal vector
Product ion rate density of Lk a Supply of Lk by s ( t)
a
Supply of Lk by s ( t)
Number density of charged particles in a Debye sphere
Origin of a co-ordinate system Time-dependent orthogonal tensor
Gas kinetic pressure;
Pressure of a mixture and an electromagnetic field
(684)
(689)
Sect. B.6.c. Sect. A.1. (C32) Sect. B. 4 • b. i .(659)
(699)2 (6114) 2 Sect. B.3.a. Sect. B.b. Sect. A.1. Sect. B. c. (6100)2 (Bl 01) ( 6117) Sect. A.1. Sect. B.a. (C3) 1 Sect. A.1. (E29)* p a
*
p P. I a P. q a a r R a s I xxPressure of a constituent in the absence of an electromagnetic field
Pressure of a mixture in the absence of an electromagnetic field
Volume polarization density of a constituent
Volume polarization density of a mixture
(Hass) specific charge
Internal energy flux of a constituent
Internal energy flux of a mixture
Inner part of qj
Charge density of a constituent
Volume density of free charge of a constituent
Free charge density on s(t) of a constituent
Charge density of a mixture
Volume density of free charge of a mixture Surface density of charge of a mixture Free charge density on s(t) of a mixture
Specific body heating of a constituent
Specific body heating of a mixture Inner part of r
Net production rate density of n a Three-dimensional Euclidean space Euc 1 i dean space-time
Total number of constituents
(E26) l Sect. B.4. ( B43) Sect. B. b. (B126) (B136) (B136)2 (B4 7) 2
(B8)
Sect. B.4. (B47) (B42) l (B52) (B42)2 (B 127) (Bl 37) (Bl 52) 2 (B35) Sect. B.a. Sect. B.a. Sect. B. a.*
s s ( t)s
s*
s
as
a t t* tij a t .. E'J t.. IJ T T, T .. I J T~. IJ u. I a u n T - MeTotal number of electrically charged constituents
Singular surface
Region of "high"-density gas plasmas; A scalar field;
An absolute (polynomial) scalar invariant Poynting vector
A generic scalar
A scalar (field) under a Euclidean trans formation
A surface point
Invariant (polynomial) functions Material surfa~e for a constituent
Time
Unit tangent vector to c(t) t under a Euclidean transformation Stress tensor of a constituent
Maxwell -stress tensor
Stress tensor of a mixture (Absolute) temperature
Regions of "high-to-medium" density gas plasmas
A tensor field
T ij under a Euclidean transformation
Diffusion velocity of a constituent
Speed of displacement of s(t) Sect. 8.b. Sect. 8. c. i. Sect. A. 1. (C 1) 1 (C 11) (B 141) 2 (C9) (C4) Sect. 8. c. ii. (C12) Sect. 8.b. Sect. 8.a. Sect. 8. c. ii. (C3)2 (881)2 (893) (887) Sect. 8.9.a. Sect. A. 1. (C 1) 3 (C4)4 (812) (818)
..
u v. I a xxi i Velocity of propagation of s(t) Velocity of a constituent v.I Mass average velocity of a mixture a
v
.
I a v~ I w •• a'Jw
a x...
x I * x. Ix
a az
a Greek alphabet a a,a,y, ...Material volume for a constituent
Velocity difference of a constituent
A vector field
A generic vector
A vector (field) under a Euclidean transformation
Spin tensor of a constituent
Molecular weight of a constituent
Spatial co-ordinates
Position vector of a point in R 3 x. under a Euclidean transformation
I
Material co-ordinates of a constituent
Charge multiplicity of a constituent
An independent homogeneous chem i ca. I reaction Material vector component indices
Sect. 8. c. i. (89) ( 811) Sect. 8.b. (810) (C 1) 2 (C9) (C4) Sect.
c.
1. Sect. D. 1. Sect. 8.3.a. Sect. 8.a. Sect. 8.a. (C3) 1 Sect. 8.a. Sect. 8.b. Sect. 8.3.a. Sect. 8.a.8
s.
8 e y a Cl y as
y a ys
y6
.
k a nS'\
a nS cSk cSkj £ a £ I £ * ~ £ Ehx)
a a Plasma betaElectron, ion Hal I parameters
Specific entropy "production" rate of a constituent
Stoichiometric coefficient of a constituent
Entropy supply density by s(t) for a constituent
Specific entropy production of a mixture Entropy density supply by s(t) for a mixture
Production rate of nk a Supply of nk by s(t)
a
Supply of nk by s(t) Kronecker de 1 ta
Specific total energy of a constituent Internal energy density of a mixture Inner part of £
Specific (equl l ibrium) internal energy of a mixture in the absense of an electromagnetic field
Energy density of an electromagnetic field Vacuum permittivity Alternator Configuration of a constituent Sect. A. 1 . Sect. A.1. (8147)2 Sect. B. 3. a. Sect. B.9.a. (Bl 50) ( B 155) (8109)2 (B 112) (B 118) (834)3 (8125) (Bl 35) (8135)2 (E2 l) ( B 141) l Sect. B.4.a. (84) Sect. B.a.
T\ a T\
*
T\"·
e"L.
e"
o
{'.(}. ck s t. I. Bk t. • µ b µo vc' e ~L I t. \) .c I!Cs,
tl a -a 71 p a p '¥ a v aYJ
B4 xx ivSpecific entropy on a constituent
Specific entropy of a mixture
Specific (equilibrium) entropy of a mixture in
the absence of an electromagnetic field
Electron, ion mean free path Electron, ion Larmor radius Debye length
Ct.th reaction rate density
Lagrange multipliers
Chemic:il potential in equilibrium of a
constituent
Vacuum permeability
Electron, ion coll is ion frequency
Surface density of ~
a
Arbitrary surface-related field property of a constituent
Mean pressure of a mixture
Mass density of a constituent
Mass density of a mixture
Volume supply density of'¥
a (B146) (Bl 52) l (E18) Sect. A. 1. Sect. A. 1. Sect. A.1. Sect. B.3.a. (Dl 7) 1 _4 (E43) Sect. B.4.e. Sect. A.1. ( B21) (B21) (045) (B5) (B6) (B26)
aw
v a I; o. I a 'I; o. I aI
T' T e u c' u .c e I T'l' a T-a ~~ I a~~
I a ~M;(x.
t) a a 'iiv (x, t) a '!' a E '!' * '!'Volume production rate of '!'
a
Surface supply density of_
Surface production density of~ a Summation
Electron, ion col Ii s ion time
Electron, ion col I is ion frequency
Total surface supply densi.ty by s(t) of '!'
a Total I ine supply density by c{t) of _
a (826) (828) (828) (86) Sect. A.I. Sect. A.1. (826) (828) Flux of '!' through (826) a a Flux of _ through a Magnetic f I ux a
Continuous motion of a constituent
Volume density of'!' a
Arbitrary volume-related field property of a constituent
Specific equilibrium free energy of a mixture Specific (equilibrium) free energy of a mixture in the absence of an electromagnetic
fie Id (828) (873) Sect. B.a. Sect. 8.c. i. (816) (E8) ( E 15)
w c' w .c e I w p Ilk a Ilk Subscripts a, b, c, ... e, A, p Superscripts * E L xxvi
Electron, ion cyclotron frequency
Plasma frequency
Specific internal angular momentum of a constituent
Specific internal angular momentum of a mixture
Arbitrary constituent E 1 ec t ran, ion cons t i t ue n t Axial, polar vector
Material derivative of a constituent Material derivative of a mixture
Volume production rate density of a given field quantity for a constituent
Surface supply density of a particular constituent field quantity
Average value of a field quantity
Field quantity considered in the absence of an electromagnetic field
Field quantity in equilibrium Linearized field quantity
Sect. A. 1 . Sect. A.1. (Bl 09) l (B122) Sect. B.a. Sect. A.1. (C4)2-3 (B14) (B15) Sect. B. 3. a. Sect. B.3.a. (E15) Sect. E.2. Footnote C35
Mi see 11 aneous
Determinant ( C4)
3
Jump brackets Sect. 8. c.
a
Derivative with
at respect to ti me (89)
< Less than (C34)
<< Much less than Sect. A. I.
Equals Sect. A. 1.
> Greater than Footnote 872
>> Much greater than Sect. A. 1.
~ Greater than or equal to (8151)
c.< Proportional to Sect. A. I. CP Infinity Sect. B.a. n Intersection (B 1 7) 2 ( )· ( ) Scalar product (C26) ( ) x ( ) Vector product (854)
A. INTRODUCTION
A~
1may be understood 2· to be any collect ion whatsoever of
charged particles (e.g. electrons and io ..• 1 for which there exists co-operative behavior. This coherent material response is due essentially to the collective nature of the mutual space-charge (i.e. Coulomb) interactions between said paricles.
A. 1. On representative plasma systems
Both natural as wel I as laboratory plasmas find, at least in a
statistical continuum description, a partial characterization which is
based upon the assumed existence of certain "microscopic" 3 parameters. The quantities which are commonly taken in particle models of plasmas
include, among others, the respective particle number densities, the system related Debye and mean free path lengths, the Larmor radii and
the charged particle cyclotron frequencies, and soforth.
To the extent that the theoretical concepts from which these various quantities derive are physically meaningful, one can uti I ize these
para-meters in terms of their relative magnitudes and numerical values as
criteria for identifying the different plasma regimes. Indeed, it is
possible in a general sense .to note some of the different types of
4
plasma systems by setting down, as in Table 1 , some of their character-istic properfy values; and, pictorially as is done in Fig. 1 5
The general topography of gaseous plasma physics having been given,
it is now worthwhile to note further the possibility of classifying the
DIMENSION ElECUON NEUTRAL TEMPUATURE MAGNETIC PLASMA COLLISION ElfCTAON OE BYE CONDUCTIVITY
DENSITY DENSITY FIELD FREQUENCY FREQUENCY CYCLOTRON LENGTH FREQUENCY L "
.
" T•
w p u WH>-o
u < -3 m-3 OK T -1 -1 -1 -1 -1 m m..
,
..
,
..
,
m ohm m LABORATORY 10-1 1020 1022 4 10-1 1012 108 10 10-6 4 Ga1 diuhorge 10 10 10 N MFD direct eneJgy 10- 1 1021 1024 103 1012 1012 1011 10-1con version plas mo 1 1-10
Thermonuclear plasma 10-1 1022 0 108 10 1013 106 1012 10 -5 107 10-1 1029 2 1016 1014 1010 10-12 6 Liquid l't1etol 0 10 1 10 5 PACE 1011 107 105 -8 105 -5 103 105 lntuplonetory plasmo 0 10 10 10 Sunspot 107 1017 1020 tx104 1 1010 106 1011 Hi5 104 6 29 104 -3 1016 1016 1010 1013 6 Eort h interior 10 10 0 10 10
18 16 7~ 14
.
' 2!
t2 10 -1 0 log 10 I k '•I. eVFig. 1. Approximate regions of diverse
important gaseous plasmas. (From [1966, 8, p. 8]. Used with permission of McGraw -Hi 11 Book Company.) 11 10 MC
t
-" E,
ct
!' ·~ c 3~"
"
c 2 ~..
~
0 Center of sun 25 10 '•"
' 20 10""""
v "'""a
ce
;:;"
10 15 o; .~ "' :~ ~ o;"'
10 10 105L_~,~-'-O:::::L-~~~.L-~~~.LL"-"""-...;;."":'.;.;.;;.~~.L-,~~~"'-:~~_,__, G 10 1. 10 102 103 10 10 Temperature electron vollsFig. 2. A characteristic length classification
of regions of deuterium gas magneto-fluid-dynamics.
(By permission of A.R. Kantrowitz [1960, 4, p. 960].)
mater i a 1 response poss i bi 1 it i es of said p 1 asmas. Motivated by the needs of astrophysical and control led thermonuclear fusion plasma research,
6
Kantrowitz & Petschek prepared, on the ba.sis of particle arguments, the particular "equilibrium" classification given in Fig. 2.
Because of their relative importance in influencing plasma behavior, the thermodynamic variables of electron number density (n) and
tempera-e
.!.!!!:!. (T) were taken, together with the strength of the magnetic flux
density
(B),
as the parameters in terms of which this classification is expressed. The specific choice of deuterium here fol lows from practical advantages it offers: viz., it has only one valence electron and can thus form only one ion sort; and, it is suitable for investigation of high-temperature plasma behavior as occurs, e.g., in fusion research.An additional important condition placed upon this classification is that of the ratio (B) 7 of the ideal gas kinetic
to that of the magnetic flux density field (82/2µ0)
pressure (p = nkT) e being equal to unity.
Indeed, special emphasis has been placed upon this minimum limit case for plasmas of magneto-fluid-dynamics (MFD) wherein the coupling between the plasma and the electromagnetic field is strong and below which
8 "useful" MFD is not to be expected
9
Let AD denote the Debye length for electrons This important plasma quantity defines for an equilibrium plasma a Debye sphere of radius AD about a given charged particle; whkh, given a "sufficient" number (ND=
n
A
~)
of charged particles therein, shields that particle from the Coulomb force field due to other charged particles lying out~side said sphere. The parameter AD is a measure for the relative impor-tance of space-charge effects in a given system here and serves thus
5
as a criterion for identifying those systems which are (not) to be considered as plasmas. Conventionally, if L denotes a characteristic geometric length of a particular syste1" 'cf. Table 1), then only those
1 0
systems for which AD cc L are cal led gaseous plasmas. For the pur-pose of Fig. 2, L has been taken as being 1 cm
For a statistical continuum approach to the study of plasmas to be physically sensible, it is at least necessary that theaveragesof the various quantities involved (e.g. charged particle number densities) exist over a volume
A~.
This requirement is customarily satisfied 11 by the introduction into the considerations of such an approach, of the condition that ND» 1. Further, using AD it is possible to introduce the (electron) plasma frequency w (oc nt). This important parameter isp e
characteristic of the rate of longitudinal osci I lat ions of electrons in
~ plasma due to electrostaiic fields; and, as such, is a measure of the number density of that particle.
In terms of models for the collisional interactions between particles it becomes possible to secure expressions for the approximate average distance traversed between particle encounters. These lengths are called
mean free paths and they are important characteristic microscopic prop-erties of a given system. Here, they are indicated respectively for the electrons and the one type of ion involved by A and A. The otherwise
e i
possible role played by neutral particles has, it may be observed, been taken in Fig. 2 as being negligible since the gas in question is "fully" ionized 12. Related to these col I is ions of, but not necessarily between, electrons and ions are respectively the times 1 (= 1/v ) and 1 (= l/v ) ;
c . .c
e e 1 1
where v and v represent the indicated particle col I is ion frequencies. c .c
....
As is wel I-known, the trajectories of charged particles in a B field are in general helices. As such, the motion of a charged particle can be decomposed into two parts. The one part is a motion para I lei or anti -paral lei to
B;
while, the other is a circular motion with a charge -dependent direction in a plane normal toii.
Characteristic of thislatter motion for the case of electrons (ions) is the cyclotron freguency
wc(~c) and its related _L.;;;a_r __ m-'o_r_r..;;a...;d_i...;u~s >-L(~L). Lastly, I introduce here
e 1 e 1
for the electrons and ions the not unimportant Hall parameters
B
(=13 e
wcT = wc/vc = A/AL) and B. These last quantities are (where defined)
ee e e ee i
characteristic for a given plasma of, for example, the degree of an
iso-tropy induced respectively in the electron and ion gases by the
B
field related to that system.Although Fig. 2 relates most directly to a deuterium plasma and not to any of the other gaseous plasmas found in the systems of Fig. 1, it is
worthwhile nonetheless to consider both figures together. It is necessary
in this regard to appreciate the fact that, at least in principle, it is
possible to prepare for any of the other (e.g. laboratory) plasma regions of Fig. I a classification such as that of Fig. 2. At most it is to be
expected that a certain shifting would occur of the lines of Fig. 2 which define the regions thereof to be discussed now. The qualitative features of that figure, which I wish to exploit for i I lustrative purposes here would, however, remain 14
"High"-density gas plasmas: region S
15
For such plasmas , electron and ion col I is ion processes dominate over those of a space-charge electric or magnetic field origin. The velocity distribution functions of these particles are max1•ellian at the
7
"same" temperature; and, the isotropy of the system is evidenced in parti-cular by the fact that with B << 1 and B << 1, its transport properties
e
(e.g. electrical conductivity} are sca'ars.
"High-to-medium" density gas plasmas: regions T and T-Me
-+
With decreasing n and increasing T, the B field assumes control over e
the electron motion while the ion behavior, as in region S, is influenced predominantly by collisions involving those particles. Here too, the respective particle types possess maxwel I ian velocity distribution func-tions, but not necessarily at the same
te~perature
16• Noting in Fig. 2the I ine dividing regions of unequal temperature, it is seen that T-Me is the portion of region T where electrons and ions enjoy the same tempera-tu re.
In this case
B
> 1 whileB
< 1; the consequence of which is that thee · i
plasma has mixed transport properties. That is, the system now has ani-sotropic (i.e. tensor) material response with regard to those properties (e.g. electrical conductivity, diffusion and thermal conductivity} which are strongly dependent upon the electrons. On the other hand, however, the isotropy of the plasma is preserved here in terms of the still scalar ion-dependent properties (e.g. vJscosity}.
"Medium"-density gas plasmas: regions M and M-Me
In the M region, for which n and T can respectively be yet lower and
e -+
higher, the ions too succumb to the now irresistable influence of the B field upon their motion. Thus, with
B
>> 1 and B >> 1, the anisotropy ofe i
the plasma induced by the
B
field is total, for all of the transport properties of the system are now of a tensor character. Not unimportantis the fact that the tendency which arose in region T for the electrons and ions to uncouple themselves energetically continues here. Aside from the region M-Me where the rate of energy transfer between these gases is presumed sufficiently high to keep them at the same tempera-ture, it is possible (at least in priciple) that in region Meach gas
17 assumes a well-defined temperature of its own
The extent to which this actually occurs, if indeed it does occur, is now dependent upon additional factors related to the plasma system. With the decline in importance of the collisional processes, there
IB h · ·I· f 1. . 1 d
appears t e poss1b1 1ty o , e.g., non- 1near part1c e an wave-wave interactions which constitute a fundamentally different means of dissipating energy; and consequently of equilibrating the electron and ion temperatures.
"Low"-density gas plasmas: regions EM and E
->
For the EM region the B field is still the major factor influencing the behavior of the electrons; and thus, with AL < L, the condition S >> 1
e e
holds. Here, however, ~L > L and the space-charge electric field has thus I
essentially taken over the role of control I ing the ion motions. In this case the transport properties of a given s~stem, if defined, remain anisotropic.
With a yet further diminishing of n the motion of the electrons, e
I ike that of the ions, becomes primarily dependent upon said electric field; it having replaced the hitherto important
B
field. This is the region E for which AL > L and ~L > L.9
"Very I ow"-dens i ty gases: region G
When n and Tare such that AD > L, the system of interest has passed
e
from a plasma condition to that of an ionized, but col I isionless 19 , gas of very low density. Such systems lie in region G; but, due to the fact that a (statistical) continuum description of them is not possible, any consideration of these systems 1 ies beyond the scope of this study.
Boundaries
The low-density (i.e. lower) boundary of this classification is, as discussed above, that line for which AD> L. Next, the low-temperature (i.e. left) boundary here occurs at the line denoting a 50 per cent
degree of ionization 20. With increasing n the high-density (i.e. Upper} e
boundary is defined essentially at al 1 points where the plasma must be treated with quantum mechanical methods; this being due to the electron
spin-spin interaction degeneracy which appears.
A further increase of n, although not shown here would 21 indicate
e
the passage of the system through the 1 iquid and sol id phases of matter, through a region of relativistic degeneracy, and eventually into the
region of matter existing •t nuclear densities; such as white giant and
white dwarf stars, novae, etc. While it does not seem to be known if these latter circumstances of matter satisfy the definition of a plasma
used here, plasma phenomena are known 22 to exist under certain con-di tions in the former region of matter (i.e. 1 iquids and sol ids) that
is "somewhat" more condensed than the gaseous systems considered here.
I am thus lead to remark that in general, plasma, rather than being a 23
only as a condition thereof. This contention is strengt:1ened somewhat by the fact that even with the high-temperature (i.e. right) boundary of Fig. 2 given as ·that line for which the speed of electrons closely
24
approximates that of I ight, it is possible to further treat the properties of that gas using the concepts of relativity theory. Although relativistic plasmas do exist 25, they too, "like" the quantum plasmas mentioned above, are not considered here.
A.2. Position of the problem
A.2.a. Purpose
The formal description of the mechanics of a gaseous MFD plasma has developed essentially along I ines analogous to those found in the class
i-i h . f f I . d Th . f I 1 . d d. . 1 . 2 6 ff ca mec an1cs o u1 s. 1s more u y mature 1sc1p 1ne o ers ·two methods for attacking a particular problem: the particle approach and the classical continuum approach. An investigator of plasma behavior may find upon examining the physical conditions characteristic of a certain problem (e.g. the particle densities) that one or the other of these treatments is to be preferred.
Statistical continuum approach
27
A statistical continuum approach aims in part to determine, via a treatment of the presumed microscopic nature of a given system's material components, the quantities which enter into the macroscopic system
. 28
equations (e.g. the so-cal led ensemble averaged parameters ). For the case of, say, gaseous plasmas, this approach concerns itself also with the physical composition of a given system, the "structure" of its
ma-11
terial components, and the microscopic nature of the interactions which involve these components.
When the plasma density, for exampl,., is such that an assumption of "continuum" properties of the system is physically reason<.ble (such as in regions S, T, T-Me, M and M-Me here 29 ) one can augment the individual
partiile approach (needed in regions E, EM and G 30) by certain
statisti-cal considerations in order to secure the macroscopic equations of that system. Of importance--in general, but in particular for this study, is the fact that in the area of gaseous plasma physics deemed suitable for laboratory MFD energy conversion purposes (cf. Fig. 1), research based
31
upon such an approach is carried out even though the condition N
0 >> 1
is clearly not satisfied.
With regard to plasma systems in general, partially ionized gases or otherwise, it is a no less significant point here that the present body of microphysical knowledge of, e.g. interparticle potential energy
f unctions . 32 , non-a 1a at1c part1c d. b . . 1 e 1nteract1ons . . 33 an d process cross-sections 34, appears to be insufficient to permit the establishment of an adequate statistical continuum model for various plasmas of interest
35 . The consequences of such a situation are reflected, e.g., in the results of any classical mean free path theory 36; which type of theory,
it may be further noted, is itself open to doubt 37. It seems, in
addi-38
t ion, that the Chapman-Cowling formal ism which is an important factor in contemporary plasma physics research, has also been questioned 39 with regard to both its degree of relative generality as wel I as its appl
Generally speaking, for this approach seen in regard to plasma physics, the concepts of (non-)equi librium classical kinetic theory and
statistical mechanics are thus seen to serve as one of the bases from
which the conventional MFD system equations have been obtained.
Classical continuum mechanical approach
The statistical continuum approach was,most characteristically, predicated upon a presumed knowledge of the particulate nature of a
material system. This is not, however, the only manner in which the
macroscopic behavior of plasma systems may be studied.
The plasma investigator also has the option, under appropriate
con-ditions, of attempting to describe a given system by means of a
classi-cal continuum mechaniclassi-cal approach 40. This alternative technique, which
for MFD is neither more nor less general than the statistical continuum 41
method , provides for the examination of a given system by assuming
from the beginning a macroscopic point of view. In "neglecting" the
particle nature of matter, the so-called physical insight into the
be-havior of the system in question is taken to be relatableto a hopefully
sufficient degree of carefully obtained macro-phenomenology.
Of special interest here is the fact that, regardless of the necessity
42
of a multi-fluid description of MFD plasmas , the classical continuum
mechanical approach has es:abl is~ed in this problem area essentinlly
43
only a single-fluid treatment of plasmas . Further yet, when the
devel-opment of the system mechanics is complemented by classical irreversible
h d . 44 . d . 45
t ermo ynam1cs cons• erat1ons, serious objection has been raised
13
For an MFD problem thus, a statistical continuum approach treats the plasma in terms of its various material components; attempting thereby to derive the macroscopic equa• ·~ns of a given system from considerations relating to the individual particle natures of said components. On the other hand, the method of classical continuum mech-anics seeks the basic macroscopic relations by initially neglecting the discrete material nature of a system, then postulating the necessary kinematical and dynamical expressions, and finally supplementing the (semi-)empirical information relevant to a particular problem.
Both of these approaches, although differ(ng in attitude, have been
46
shown for MFD circumstances , to arrive at essentially the same macro-scopic equations. Unfortunately, both approaches also appear 47 to be unable to provide a (adequate) general macroscopic description of the (non-) linear dissipative, i.e. irreversible thermodynamic, processes which are related to non-equilibrium plasma systems.
On the basis of these considerations, together with the earlier mentioned shortcomings of these two approaches to the study of dynamic plasma behavior, I conclude the following: viz., that it is desirable to examine the feasibility of securing a formal ism that could offer for the same problem area a description with fewer limitations than that of the statistical continuum method, and a more detailed insight into the macro-physical character of a given plasma system than that given by the classical continuum mechanical method. This constitutes essentially the purpose of this study.
A.2.b. Problem
The general problem which this study attempts to provide a cont
ri-bution towards solving is that of theestabl ishment of a unified,
self-consistent and "exact" theory of the non-linear macroscopic mechanics,
electrodynamics and energetics of finite and bounded real multi-continua
plasma dynamic systems. In terms of the actual systems concerned (cf.
Fig. 1), this probl_em poses in general the necessity of describing the
material behavior of real, chemically reactive, radiative, anisotropic,
compressible, inhomogeneous, magnetizable, polarizable, non-equi I ibrium
and non-I inear electro-magneto-mechano-thermo-dynamic mixture systems.
I hasten to point out that this is not just an arbitrary
hypotheti-cal problem. It is concerned with real material systems and its
solu-tion in some physically more adequate sense than has been presented to
48
date is not only highly desirable, but perhaps even urgent
This problem unquestionably reflects a framework within which a
generalization of the contemporary approaches to laboratory plasma
physics may be sought. However, and not surprisingly, for numerous
phys-ical and mathematical reasons it does not appear to be possible to bring
the problem as stated above to a "satisfactory" closure at this time.
Hence, I have not attempted to consider said problem in the degree of
generality expressed above; but, l propose instead to examine it here
in the more restricted sense which I shal I now discuss.
Rational mechanical approach
The discipline of rational mechanics, or modern natural philosophy
15
(dating from the mid-1950's 49); but, its tradition lies essentially in the eigthteenth, and to a lesser degree the nineteenth and early twen-tieth, century efforts of a handful of en 50. It is now, as an
exten-51
sion of geometry , fundamentally a mathematical science with the pur-pose of establishing a rigorous theoretical foundation for the study of the physical behavior of material continua.
The basis here for a non-I inear field theory treatment of gross irreversible material behavior rests essentially upon three general sets of assumptions. The first concerns itself with the topological character of the space involved; while the second and third relate, respectively, to the kinematics and dynamics of the class of systems whose behavior is to be described.
Based upon this foundation rational mechanics in particular strives, via use of an axiomatic approach to· its subject matter, to secure phys-ical ly wel I-motivated theoretphys-ical models of material systems which are
l . . If . 52 d I 53 · h . ·
exp 1c1 t, se -consistent , an compete 1n t e1r construction. Th . 1s approach, as exp a1ne by I . d T rues e d I I 54 , consists o . f f our e ements: 1 viz., primitive quantities, definitions, general axioms, and proved theorems. In terms of the physical primitive quantities which are de -fined only to the extent that the mathematical properties are given, the definitions of additional system-related quantities can be made. Relationships satisfied by these two elements are physical axioms ("laws") and here they are of t1~0 types: kinematical and dynamical.
Axioms of kinematics are represented by the set of balance equations proposed for the type of system under consideration; and, they are val id for the class of al I such systems. On the other hand, the axioms of
dynamics are represented by the set of "appropriately invariant" and
thermodynamically restricted constitutive equations postulated; which expressions make the theory system-specific in terms of its material
response possibilities.
55
Experience and not, of course , experiment is the guide which
motivates those assumptions as are made. Rational mechanics draws its physical motivation from the body of common experience (e.g. "phenomen-ology'') relevant to the specific problem area(s) under study; while,
its theoretical motivation derives in part and to varying degrees from
the conceptual backgrounds of those other disciplines which concern
themselves with the same subject matter. Here (cf. Fig. 3) these are,
e.g., statistical and classical continuum mechanics. With the estab -1 ishment of the proved theorems and their related coral laries 56 for a given theory, the formal structure of that particular rational mech ani-cal description is taken to be complete.
In the relatively young di sci pl ine of rational mechanics there are a number of non-trivial open questions which serve to make this· field a controversial one at the present time. For example, the number of basic principles involved and their precise formulation 57 , together
with the (degree of) applicability of an axiomatization procedure to physical theories 58 remain debatable subjects. Regarding both of these points, I shall endeavour to fol low an enl igthened course in the sequel 59
It is on the basis of two points that I propose here a rational mechanical approach to the problem of describing the general material behavior of plasma systems as a, if not the most, worthwhile of the presently available alternatives. The first point to be noted is the
KINETIC THEORY RATIONAL MECHANICS PHENOMENOLOGY
CLASSICAL
CONTINUUM MECHANICS
BALANCE AND JUMP
BALANCE EQUATIONS
CONSTITUTIVE EQUATIONS
STATISTICAL MECHANICS
BALANCE ANO JUMP BALANCE EQUATIONS CONSTITUTIVE EQUATIONS
Fig. 3. I I I
I
FIELD EQUATIONSL ________ ,
r
-FIELD EQUATIONS I I I I I - _.Jfact that a theory formulated thusly possesses in contrast to numerous,
if not the majority, of plasma theories secured otherwise, not only a conceptual simplicity, clarity, rigor and a not undesirable aesthetic qua I ity; but also, with regard to the eventual establishment of results
60
of permanence , the capacity to aid in making itself as self-corre
ct-61
ing as might possibly be expected The second, independent and here also important point is the recognition of the striking theoretical and experimental successes of this discipline in other areas of experience 62 and these with regard to problems the solution of which lies beyond
63 the present capacity of the traditional approaches to provide
Rational mechanics strives towards a unification via mathematics of the diverse physical sciences; and hence, it does not restrict its
interests and efforts to any one of them in particular. It is in the above indicated sense that the word "rational" is employed here. I mean hereby no offence to those persons who by inclination or circumstance
select to treat physical problems in terms of ad hoc model~, approxi -mative methods, "physical intuition", and soforth.
As observed earlier, the establishment of a theoretically "adequate" description of the gross non-1 inear irreversible thermodynamic behavior of fluid mixtures is a problem meriting study for its own sake; but also one deserving attention on the basis of practical considerations. With the appl icabi 1 ity of rational mechanics to the investigation of gaseous plasma systems constituting a main purpose of this study, I now proceed to further delimit the scope of same. The restrictions placed upon this study concern, naturally enough, both its content and its genera Ii ty.
19
Regarding content, as seen in relation to the aforementioned general description, I wish to present in greater or lesser degree the salient points pertinent to a particuli' rational mechanical theory: viz., that for a one-temperature, chemically reactive (but rad iation-less), non-relativistically moving, magnetizable dielectric fluid mix-ture. The point of view adopted here is that the basis for said treat-ment is given in the form of a complete set of system balance and jump balance equations together with the "appropriate" constitutive equa-tions indicated therein. The solution of these relationships for the as yet undetermined field quantities, under the boundary and initial conditions that delimit the problem of interest, is taken to fulfi I the main objective of a thermodynamic constitutive theory for the prob-lem posed 64
The content of the treatment of the constitutive equations wi I 1 be based upon certain selected rational thermodynamic and modern consti tu-tive theoretical principles. Due, however, to the as yet unresolved
.nature of various fundamental problems hereto related, the content of the rational mechanical treatment of constitutive equations wi 11 not have the "absolute" character of generality as found in the set of balance equations secured by employing the same formal ism.
Among these problems are, e.g., the mixed invariance properties (Euclidean-Galilean/Lorentz invariance of the mechanical/electromagnetic field equations) of the system, and the problem of "correctly" identi
fy-65
ing and using entropy concepts in a modern continuum theory . Regard -less of these difficulties, the results that can be derived are, with all due respect for the various points of interpretation, of equal or greater macroscopic generality than those found from conventional approaches.
Liu
c
MUiler 66 have recently examined the problem of describingthe material behavior of single simple 67 heat conducting continua in
an electromagnetic field. In .this study I undertake the specific task of extending, in some respects, their study to a particular case of
non-simple heat conducting Eulerian fluid mixtures which are also in an electromagnetic field.
For systems of this type where the mixture may contain a (a= 1,2,
s) constituents, I take the foremost objective of a theory such as that contemplated above to be the following: viz., the calculation, as functions of position and time, of the field quantities
p
Ci::,
tl.
T(i::,
t), a v. (~. t). I a E. (~, t) I and B. I (~, t).Here, the fields are respectively the mass density of constituent a, the absolute temperature of the constituents and the mixture, the
velo-city of constituent a, the total electric field taken with respect to a
stationary observer, and the magnetic flux density.
In a field theory the macroscopic properties of a material system
are considered to be field quantities. Generally speaking, al I of the relationships which relate the different field quantities to one another
are field equations. In this study (cf. Fig. 3) the term "field equ a-tions" shall be taken to mean the set of balance and jump balance
equations for al I the various macroscopic properties of a given system;
together with a set of constitutive equations taken for a specific
material system.
The possible meaningfulness of a transition from the more fami I iar
approaches of plasma analysis to that of rational mechanics is not
21
necessarily made evident with the determination of a set of balance and jump balance equations (cf. Chpt B). Under less general circumstances than those considered here, said results when contrasted with those of more conventional approaches may give the false impression that rational
mechanics .has little new to offer.
All the (dis)similarities to other treatments notwithstanding, the statement above would constitute a prematurely formed conclusion by an
individual who does not yet realize that any treatment whatsoever of only the kinematics of a system represents at best only a partial, albeit important, descriptive element of that system. Thus, Chpt. B shall present the essentials, for this study, of and related to the rational mechanical kinematics of a mixture of material and
electro-magnetic field continua. It becomes then in greater measure.the burden
most particularly of Chapters C and D treating, respectively, consti-tutive theory and the problem (introduced in Chpt. B) of entropy, to-gether with the subsequent chapters, to make clear in some respects the possible favorabi lity above the more traditional methods of a rational mechanical investigation of plasma continua behavior.
Footnotes to Chapter A
The origin of this word seems to I ie with the introduction of the word
"protoplasm" in the nineteenth century by the biologist Purkinje. The meaning of "first plasma" was subsequently given to this initial term by von Mohl, a botanist. In the particular context of plasma physics here, the word "plasma is commonly attributed to Tonks & Langmuir [1929, 1, p.196, footnote SJ; who, perhaps, in studying arc discharges
observed the jelly-I ike behavior of the medium and were motivated to use this word.
2 Kunkel [1966, 8, p.3]. Note, there exist other definitions of plasma in the I iterature (cf. for example Sutton & Shetman [1965, A, p.6]}, but these are essentially special cases of that taken here (cf. Kunkel [ 1966, 8, p. 5 J) .
3 Shkarofsky, Johnston & Bachynski[1966, 12, pp.2-3]_make a finer dis-tinction between "microscopic" and "macroscopic" than that which I employ
here. On the basis of an argument related to the combination- and
division-invariance of charges, the Debye length and the plasma frequen-cy are considered to be macroscopic parameters there.
4 This table is adapted from that given by Bueren [1966, 2]. The values
for magneto-fluid-dynamic plasmas added here were calculated from selected experimental results presented in [1966, 9].
5 Kunkel [ 1966, 8, p. 8]. Here too, the region denoted for magneto-fluid dynamics (MFD) direct energy conversion (DEC) systems has been added.
It is perhaps worthwhile to observe that only gaseous plasmas
are represented in this figure. Other known plasma systems which could supposedly be introduced into, and hence generalize, this overview include, among others, the fol lowing:
23
liquid metals (Kirko [1965, 10)), semi-conductors and semi-metals
(Anker-Johnson [1966, 1 )) and metals ([1965, 6)). 6
Kantrovitz & Petschek [1957, 3, p.5). ·. 's classification, which is slightly adapted, is but one of a number of various types of classifications that exist in the I iterature of plasma physics. 7 The "plasma beta" is discussed, e.g., by Glasstone & Lovberg [1960,
5, pp.52-53) and Kral 1 & Trivelpiece [1973, 5, p.7).
8 Sutton & Sherman [1965, 17, p.10). I note that there exist at
present several names for the subject area taken under consideration. Examples include plasma dynamics, magneto-gas-dynamics, magnetohydro-dynamics, and soforth. 1 select the term "magneto-fluid-dynamics" for two reasons: first, it reflects all fluid and fluid-like media; and, it permits a distinction to be made with, e.g., magneto-sol id-dynamics. This last mentioned subject which relates to, say,
magneto-elastic media does not enter into the considerations of this study.
9 A Debye length is, of course, definable for ions. Further yet, with
some quantum mechanical modifications, the concept of Debye length is adaptable to the case of sol id-state plasmas.
lO Kunkel [1966, 8, p.4), Shkarofsky, Johnston & Bachynski [1966, 12, p.2) and Spitzer [1962, 8, p.22).
11 Kunkel [1966, 8, p.6), Shkarofsky, Johnston & Bachynski [1966, 12, p.3) and Kral 1 & Trivelpiece [1973, 5, p.4). With regard to meaning
of this "plasma approximation" condition, there appears to be some difference of opinion in the 1 iterature. The last two cited references seem to hold to the position that with diminishing N0 one can speak
of a medium with very low N
0, say less than or about equal to unity, is not valid. Gottlieb [1965, 8, p.46], on the other hand, expresses
the view that the number of charged particles per cm- 3 may
be
~
number and the system in question is sti 11 a plasma as long as itsatisfies the condition A
0 << L. I might further add that inasmuch
as the degree of ionization does not enter into the definition of
plasma employed here, it seems reasonable to state that if there are
"enough" neutral particles in that cm- 3 , a continuum description
appears to be possible. Lastly, it may be noted that regardless of the relative importance of this point, not al 1 authors (cf., e.g. Sutton &
Sherman [1965, 17]) are explicit with regard to their introduction of
it into their respective treatments.
12 Alfven
& Falthammer [ 1963, 1, p. 134 and p. 180] state that a
p
lasm~
is slightly (highly) ionized if the degree of ionization is less (greater) than 1 per cent.13 5utton & Sherman [ 1965, 17, p.156].
14
Petchek [1958, 4, p.967]. 15
Cf., for example, Alfven & Falthammer [1963, 1, pp.169-1970] with regard to the density-dependence of plasma properties.
16
Depending upon the relative (in)ability of the bulk electrons, ions and possibly neutral particles to exchange energy between themselves at a sufficiently high rate, the respective kinetic temperatures of
these gases may be (un)equal; in which case the system in question is said to be in a condition of thermal (non-)equilibrium.
An awareness that multiple-temperature plasmas can exist is evi-denced at least as early as 1929 (cf. Tonks & Langmuir [ 1929, 1, p. 201]). Kinetic theory considerations of this possibility were given
