T HE UNIV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Aeronautical and Astronautical Engineering Final Report THE GENERAL CONSERVATION LAWS OF A DILUTE PLASMA R oS.B. Ong ORA Project 02929 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION OFFICE OF SPACE SCIENCES GRANT NO. NSG-22-59 WASHINGTON, D. C administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR July 1963

FOREWORD This report represents the final report on the NASA Grant NSG-22-59. It is essentially a continuation of the work done on the general study of the physics of a low density ionized gas assuming no collisions. The aim of this study is to familiarize ourselves with the properties of such a dilute plasma in the hope to gain insight into phenomena such as instabilities, discontinuities and the all-important derivation of the "magnetohydrodynamic" equations from a kinetic theoretical point of view. The project is directed by Professor R.S.B. Ong of The University of Michigan. The contents of this report will be condensed and submitted for publication shortly. iii

LIST OF SYMBOLS r = (rx9ryyrz) position vector pertaining to electrons R = (Rx,Ry,Rz) position vector pertaining to ions = (vxVy, vz) velocity vector pertaining to electrons = (Vx,VyVz) velocity vector pertaining to ions 4( Iri-r21) Coulomb potential between two electrons (.Itl1-t2lI) Coulomb potential between two ions d( Ir-RI) Coulomb potential between an electron at position r and an ion at position R e electron charge m electron mass M ion mass t time E(r) induced electric field It(r) induced magnetic field

1. INTRODUCTION In the case of Coulomb forces in a dilute plasma the effect of weak interactions is more important than the effect of single collisions. in order to give a good evaluation of these multiple interactions the two-particle distribution function is used. The usual expression for the effect of binary collisions in the kinetic equation is then replaced by an integral containing the various two-particle velocity distribution functions. In the case of a dilute, fully ionized hydrogen plasma the basic equations are of the following forms (for the meaning of the symbols see page v). af C(r Tv;t) + vJ -. E() + (vl x.i f. ~ t f+rce m v 3cm = e dr2dv2 a ( Ir-r2 I) a f2(rl,r2,vl,2;t) (1.1) m 'r,1 J tion defined in such a way that frlvl;t)drldvl (where d = drlxdrlydrlz ~-~z~ and dvl = dvlxdvlydvlz) gives the probability of finding an electron at a given instant t located within the element drl of the coordinate space and having velocity vectors with their end points in the element d-v> of the velocity space. The subscript 1 under the space coordinate r and velocity v indicates that a typical electron is being considered. The function f2(rl,rvl,,v2;t) is defined in such a way that f2 dld~ 2dld- 2 gives the 1

probability of finding electron 1 at the instant t located within drid'v together with electron 2 located within dr2dv2 at the same instant. The integration with respect to dr2 and d$2 in the multiple integral must be extended over the entire coordinate space and velocity space available to electrons. The function f2(r,R,v,V;t) is defined in such a way that f2 drdRdvdV gives the probability of finding electron 1 at the instant t located within drldvl together with an ion located within dtdt at the same instant. Note that capital letters refer to ion positions and velocities while lower case letters indicate electron positions and velocities. The integration with respect to dR and dV must be extended over the entire coordinate space and velocity space available to ions. An equation similar to (1.1) applies for the distribution function Fl(R,V;t) for single ions. Furthermore Eq. (1.1) is one equation of a hierarchy of equations for the multi-particle distribution functions of increasing order; the analogous Bogoliubov-Born-Green-Kirkwood-Yvon equations for a dilute plasma.l Along with Eq. (1.1) we shall be concerned with the equation for f2( rl, r2, l, v2;t) which has the form:

f rr v72t) + Vl f f d__ fa(!:lt,~1; V2;t) r+ v la f2 + 2 a f2 + mF X(l)=,,e a 4( |rl-r2 a 2+ m 6r 1o v m ar2a av2e + te2 c~dv3dr3 1 6rc( jr -r3 6) a (1.2) + - 4( Ir-r3) --- f3( rl,2, r3,, V 2l v3;t 4ite2 J d 4te2 a. m [~ +? ar, r e 6vlc + a 4( Ir2-RD) f3 rl,, r2,AfVl2t) 6r2Cf m r vl m Jr2~ av2a 4r~aa d( Irl-r1 ") a -~-!~ The three-particle distribution functions f3 are defined similarly as the definition of f2 and the integrations with respect to dr3 and dv3 must be extended over the entire coordinate space and velocity space available to electrons and those with respect to dR and dV again over the entire coordinate space and velocity space available to ions. A similar equation also holds for the corresponding two-particle distribution function F2(R1,R2,V1,V2;t) for the ions. Since this equation is exactly analogous to Eq. (1.2), hence there is no need to write it down. HOwever, we will be concerned with the equation for the electron-ion distribution function f2(rl,Rl,vl, Vl;t) which is of the form: 3

-f2(r1,R1,v1,Vlt) + v -- f2 + Vl f2 at rl aRlC e +e 1 [ f2 -m,( 1 +c [vlxHx](j 2 +M,e() +c- [VlxH] l aF( m c f 'J M c v1 = ~~~ - ma 4( Irl-rl) avfl 4i4e2 + "-* m arvloR r1 _ 4xe2 dR22dV 2 a 4( I rl-Rl a f3(R1R2;vVV2;t) m ar1 r avl 4Tre2 + + 6v d + + + +.+ M 6aRl avl, 2 dR2dV2 -— 4( JIrl-R1D) -~ ---f3(rRl,R2;v1,V1,V2; t) 11 ja 6vlc' rade2Va ~ ( I rl-R I) The problem is how to deal with these functions without running into an endless chain of higher order distribution functions. Obviously, expressions must be found giving the f3s in terms of the f1s and f s for both electrons and ions. This will then yield a bona fide kinetic equation for a dilute plasma which is analogous to the Boltzmann equation for neutral gases. For the general case of a spatially non-homogeneous plasma this leads to a rather complicated kinetic equation. For the special case of a spatially homogeneous plasma we have the less complicated Balescu-Guernsey equation.2 At any rate, in order to derive the macroscopic magneto-hydrodynamic equations from kinetic theoretical principles two preliminary studies are required. 1. Study of the approach to equilibrium. 4~. D~eriv~~atow4

The study of the approach to equilibrium should reveal the character of the relaxation times, which in turn would be very useful in the attempt to seek for approximate solutions to the complicated kinetic equation. The conservation laws are usually obtained by taking moments of the kinetic equation. They are equations giving relations between the various macroscopic variables, e.g., density, temperature, mass average velocity, pressure, etc. Unfortunately they form an open set of equations and in order to close them we need to overcome our ignorance concerning the characteristic relaxation times of a dilute plasma. However, it is important to note that only the complete set of macroscopic (in this case magneto-hydrodynamical) equations are of physical interest. Hence, although we are still far from obtaining the full set of magneto-hydrodynamic equations derived from first principles of physics, yet it is of interest to study the open set of general macroscopic conservation equations. This is usually obtained by taking moments of the kinetic equation. However, it is also possible to obtain these conservation equations directly from the B-B-G-K-Y hierarchy, i.e., from Eqs. (1.1), (1.2), (1.3), and the corresponding ones for the ions. They will clearly exhibit the role which the two-particle distribution functions play in the various macroscopic quantities. 5

2, DERIVATION OF THE GENERAL CONSERVATION LAWS We start with Eq. (1. 1) of the preceding section. Integrating with respect to vl from -o to +oo and assuming that the distribution vanishes as v1 goes to oo we obtain the equation: i-' n(r,t) + i (n(r,t)u(r,t)]} = 0 (2.1) where 00 n(rt) f(,;t) dv = the number density. — 00 00 urt) - _ v1fldvl = the macroscopic density. -00 The terms in (1.1) involving the two-particle distribution function f2 do not contribute anything, for e.g., 00oo arl ava 00 00 dr2dv2 a ( rr2 |,J(,) a -00 Vjar1c 00 00 00 dr~dV~2 4( r1_r2 fOd'c = J d r2dV2 f2 dv1ldv12dvl3 + a f2 dv12dvl3dvll +-00 aLl a1 0 v 00 + L f2 dvl3dvlldVl2 = 0 r13 "-0 v3ls since fdvi = 0 = -00 The subscript 1 in n and u is not needed anymore and to conform with the usual notation for the macroscopic quantities it is therefore omitted. Equation (2.1) yields the equation for the conservation of mass: 7

p(r,t) + -~- [p( r, t) u( r, t) ] = O (2.2) where p(r,t) - mn(r,t) = electron mass density. Equation (1.1) is now Imultiplied by Vli and integrated with respect to v1 from -oo to +0o. We then obtain: m L (nui) + m K m (at(nui) + m ~ (nuiu) + a- Pi (r,t) + en(Ei -+ [uxH]i) = 4te2 dvld2drVli a 4( 1I1-21) a f2(rl,r2,V1, v2;t) -00 ar61 avl2X (2.3) 00 - 4e2 d1dVdRvi ( R) --— a 2( 1,, l,V,;t) with 00 PijK(+,t) --- | dv UiUjfl(, v;t) (2.4) —.O In (2.3) we have put vi = ui+Ui; U = thermal velocity. Pij(r,t) is the familiar expression of the pressure tensor due to the kinetic motion of the molecules. Using (2.1) Eq. (203) may be written in the form: Dui l r P pD P= - ia en Ei + c [uxHi 00 + 4ire2 drdV2dr2vli - ( ri-r21) a f2(-rlr2,vl, v2,t) -00 arla av (2.5) - 4e2 dvldtdRvli a r( Irl ) f2(1 1,t 00 ar10 av4a with D -a Dt at ar, 8

Now it can be shown that 42ea d 6( ~1~2 )- f( 4re jdvldv2dr2vli Irlrf2(l1r,2 2;t) = a (2.6) where Pi(r, t) is defined by hr ij( r00 id 00 P.(,t) = 2re2 dp j d 4( r-rzJ) d. vdv f(+;t).... r 2 ri, re, V1z, t) p2 dp o O -oo ( 27) Also, -,4tea dvdVdRvli - ( Irl- f,) a — _V;t) -00 arla avla ara (2.8) where Pi2( rt) is defined by _.'ijt,'t) = -2Pe2 p ' Pi= -2ice dp? P2 dp 4( jra-RI), d2x' dv1dVf2( r1,R,v,V;t) (2.9) 1'2 ----- with p -r~1 and. is defined such that r1 = r + (k-D) p r = r + 2-p) Furthermore p' R-P1 and k' is defined such that r = r + ('-P) P R = r + k P4l(r,t) is the stress tensor due to the potential between pairs of elecij trons and Pij (r,t) is the stress tensor due to the potential between pairs of electrons and- ions. Thus Eqo (2.5) becomes: 9

Dui 1 +-> + PDt - Pic- - en(Ei +c uxH i Dt a a c where Pij(r,t) = Pij(r,t) + Pij(r,t) + Pj(r,t) Hence Pi( rt) is the total stress tensor and it is symmetric. Proof of Equation (2.6): (See S. T. Choh and G. E. Uhlenbeck, "The Kinetic Theory of Phenomena in Dense Gases," U.M.R.I. Report, Feb. 1958). We define the pair density distribution: 00 n2( r;t) ) n2( rlr2;t) = | dvld~2f2( rl r2V~1 V2;t) -00 n2(1l,1:2;t) is a symmetric function of the two points r, and r2. — 00 00 dv1dv2d 2vfi ia 4( jrj rl) i 2 = dp Pi d n2(r1,r2;t) where p r2-rl. Now: 6.r 00 pp i = 2rce2 dp C dN -d. a. n2(P + (o-p)., r + P. t) p2 dp o a P P But an2 P an2 ari Pi a6 Therefore, -a- = P2e d dp t {n2(,r+p; - r-p,r;t) ] = 42te2 do i dp n2( l, +2;t) p dp because of symmetry of n2. Q.E.D. 10

Equation (2.8) can be derived in a similar manner. Equation (2.10) is the equation for the conservation of momentum. To get the equation for the transport of the kinetic energy density we multiply (1.1) by 1 mv2l and integrate over vl. After rearranging terms and making use of the equation (2.10) scalarly multiplied by u we obtain: p( ~ +(D qe + Po Do a (u- P( ) 00 = 4e2 dRdVdlvl ( r-R ( rR, t) (2.11) '~mJ l oo t - 00 ~ vtce2 dr2dv2dvlV0e I r )f2( where qi( r, t) t U2Uifl(rv;t) dv = the kinetic part of the heat current density. 00 E ( r, t)- 2- dvlUlfl(rl,vl;t) = thermal energy density. -00 l/ "~~~~ui ~~~~u-N (2.12) Dij(?:t 2 (;r — +Y arj) = deformation tensor. Pij(2,t) = (rt) + P(r P ij ij Next we derive the transport equation for the potential energy density. For this we need to use Eqs. (1.2) and (103) of the B-B-G-K-Y hierarchy as described in Section 1. Multiplying Eq. (1.2) by 4)e24( 1r1-2 r) and integrating over r2, l, and v2 we obtain: 11

a + Ua (4) + a 4 r00 -4Tce2 j dr2dvldv2v1 (flar Ir- r2I)f2(rlAr2vl'v2;t) -00 (2.13) 00 + 4ce2 - dr2dldd2v2' a ( (rl -2v)f2(1, r2 vl v2;t) *00 ar2C1 where 00 E(rt) = 4ie2 dr2dvdv2( Irl- r2)r2(rlr2 V2; -00 = the potential energy density resulting from the potential between pairs of electrons. (r, t) - 4ne2 dP22d2dtl( I|:L- 2 |) Ulif2(' 1, ' + +t) the electron-electron potential part of the heat current density. Equation (2.13) is the transport equation for the potential energy density resulting from the potential between pairs of electrons. In a similarly way we obtain the transport equation for the potential energy density resulting from the potential between single electrons and single ions. For this we multiply Eq. (1.3) from Section 1 by -41ce21( Ir-RI) and integrate over v, R, V. The result is: a + (uD) + a D -4ce2 va a ( )f2(R, t (2.14) -00 00 - 4rce2 dRdVdVc - 4( Il-RI)f2(,R,,V;t) 12

where 00 t =(+ e r-R), R, V;t) -00 = the potential energy density resulting from the potential between pairs of electrons and ions. qi( t) = -4ie2 j dRdVdUi ( r-R) f2( r V;t) -00 the electron-ion potential part of the heat current density. Equation (2.14) is the transport equation for the potential energy density resulting from the potential between pairs of electrons and ions. The transport equation for the total electron potential energy is obtained by adding (2.13) and (2.14). After rearranging terms and making use of the equation for the conservation of mass this may be written as: Dt (p ) rg = 4ce2 j, a +( -00 vjvl + e dr2dv2dv a ( -2) f2( t1,d2,1,t2,t) (2.15) -00 6r2 00 - 4ie2 dd~dtvc a ( |- V _oo 6ara - 00 - 4e dRdVd a-RI) f2R,,V t) Combining (2.11) and (2.15) we obtain the transport equation for the total electron energy density. This may be written as: 13

P D ( -) + qr a = PopDa + ay (uppp) 00 4~e2,. ~ b + d4te2 dr2dv2dvlv2 - ( ll-r2 ) f2(2l,126rl)2;t) 00- ar20Y - 4ire2 d~didvV f2(r,,,V;t) _00 6RU where e(r,t) = E (, t) + EP(rt) = the electron energy density. q ~~i( rt) + qPi(r,t) the electron heat current density. Now, 4lhe2 JdS 2dv2dvlv20 V A-. r r2 l2f2( rl=r2,V-2;t) -00 6r2U arr if we define Ji(l,t) 2e dOldv = - dk O d4 -2te2 p r dpvldv2 0 o -00 Moreover, = _uT f2( rvlv2;t)dVldV 0 0 — 00 -2need jm Piid df() -i(,t - 2,ce2 p2dkJ dp dA Ucf2vldvad -e 6re2~10

Similarly, -4Te2 ddVdVa ~ ( ( I-R)f2(, R,,V;t) if we define (,t) - -2iTe2 d' PiPa d'PT Vf2(rl,R,vl,V;t)dvdV i p,2 dp' o P o -00 Again, letting Vi = vi+?t we have A2' e = -2f+e2v dp + f ~~00 p 00 2 ite v.- d P,2 dp' 'Rv 0 0 -co o -2 dp' - o O -oo pP Dj() +aa = PD a -f - a (2.19) raHence, i we definer where i.(r-,t) -2+e d'Pp c d_%d' f2( R_, Y,,V; t) ddV P,2 dp' ' ' Hence using (2.17) and (2.18) we can write (2.16) in the form: Hence, if we define the heat current density for the electrons. z5

P ' )+ T Q = PaDa + (U.-vp) 2 pir (2.20) Note that the heat current density vector Qi(r,t) depends only on the thermal velocities of the electrons and ions. In a completely similar way we can obtain the conservation laws analogous to Eqs. (2.2), (2.10), and (2.20) for the ions. Hence, in summary, the conservation laws for electrons and ions may be written in the form: + -a — (pk+) = o (2.21) + + + Dui _ - — + P en. Ei + I [u-x"]i) (2.22) Dt E c DtP (p+) ark a& -P[FDp (ru. (2.23) where the superscript plus refers to the ions and minus to the electrons. By summing up over the electrons and ions in (2.21), (2.22), and (2.23) we can write the conservation laws for a dilute plasma considered as a mixture of electrons and ions. We would then obtain equations analogous to those derived by Chandrasekhar for the simpler case where the collisionless Landau-Vlasov equations were used as the starting point.4 16

REFERENCES 1. R. L. Guernsey, "The Kinetic Theory of Fully Ionized Gases," Dissertation, The University of Michigan (1960). 2. S. T. Choh and G. E. Uhlenbeck, "The Kinetic Theory of Phenomena in Dense Gases," U.M.R. I. Report (1958). 3. G. E. Uhlenbeck and G. W. Ford, "Lectures in Statistical Mechanics, " Amer. Math. Soc. Publ. (1963). 4. S. Chandrasekhar and S. K. Trehan, "Plasma Physics." 17