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Coulomb Logarithm

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For test particles of mass ma and charge ea = Zae scattering off field particles of mass mp and charge = Zpe, the Coulomb logarithm is defined

as A = In A = ln(rmax/Tmin)- Here rmin is the larger of eaep/mapu2 and h/2mol(3U, averaged over both particle velocity distributions, where mayg = mam/3/(ma + mp) and u = vQ: - v^; rmax = (4tt ^ n7e72//cT7)_1/2, where the summation extends over all species 7 for which u2 < VT-y2 = kT^/ra^. If this inequality cannot be satisfied, or if either itcc;CQ,< rmax or ULJcp-1 < rmax, the theory breaks down. Typically A ~ 10-20. Corrections to the trans­port coefficients are 0(A-1); hence the theory is good only to 10% and fails when A 1.

The following cases are of particular interest:

(a)   Thermal electron—electron collisions

Aee = 23 - In(ne1/2Te"3/2), Te < lOeV; = 24 - In(ne1/2Te_1),   Te > lOeV.

(b)   Electron—ion collisions

Aei=Aie = 23 - In (ne1/2ZT;3/2),                                               T^e/m; < Te < 10Z2 eV;

= 24 - In (ne1/2Te_1) ,                                                              Time/mi < 10Z2 eV < Te

= 30 - In (ni1/2Ti~3/2Z2/i_1), Te < TiZme/mi.

(c) Mixed ion—ion collisions

\i> = \'i = 23 - In

ZZ\ijl + p') fmZ2 nvz/2x 1/2

pTif + p'Ti                          + T,/ J

(d) Counterstreaming ions (relative velocity vd = (3dc) in the presence of warm electrons, kTi/m^/cT^/ /m^/ < dd2 < kTe/me

Xu> = \'i = 35 - In

ZZ\p + p') Zne V/2

 

Fokker-Planck Equation

Dfa df

Dt dt

+ v V/"+F. Vv/'

ar

at

coll

where F is an external force field. The general form of the collision integral is

(a/°ydt)con = - V vv • JaX/3, with

2 2

Ta\/3 0 x eP / ,3 / / 2 ,                                                 x -

J x = ZTTAap------------------- I a v (u I — uu)u

TUn,

— r(v)Vvz/Vz) - -/^(v)Vv/a(v)

WI/3

(Landau form) where u = vz — v and I is the unit dyad, or alternatively,

JqX/3 = 47TAa/3//«(v)VvH(v) - ivv • /"(v)VvVvG(v)

mzv^                                                                                  2

where the Rosenbluth potentials are

G(v) = I fP(v')ud\'

H(y) = 1 +

m.

rf3 / /x -1 ,3 / j (y )u a v

If species a is a weak beam (number and energy density small compared with background) streaming through a Maxwellian plasma, then

■cx\(3 _

m,

v

ac\(3

/iOL

— — V

2

cx\(3

ma + rap

1 at\(3 ( 2 ,      X vy r-a

[V I - VVj • Vv/ .

VV • Vv/'

B-G-K Collision Operator

For distribution functions with no large gradients in velocity space, the Fokker-Planck collision terms can be approximated according to

Dfe

-J^- = Vee(Fe - fe) + Vei(Fe - /e);

Dfi

= Vie(Fi - fi) + va(Fi - fi).

The respective slowing-down rates v^^ given in the Relaxation Rate section above can be used for v^p, assuming slow ions and fast electrons, with e re­placed by Ta. (For uee and va, one can equally well use u±, and the result is insensitive to whether the slow- or fast-test-particle limit is employed.) The Maxwellians Fa and Fa are given by

*a-na\2*kTa) eXP\ [ 2 kTa J/' F -n ( ma V/2c::pf

*a~na\2KkTa) 6XPl L 2kTa \ /'

where na, va and Ta are the number density, mean drift velocity, and effective temperature obtained by taking moments of fa. Some latitude in the definition of Ta and va is possible;20 one choice is Te = Ti, Ti = Te, ve = v^, v^ = vc.

Transport Coefficients

Transport equations for a multispecies plasma:

dana

—--------------------------------------------------- h na V • va = 0;

dt

ma floe

dav( dt

= -Vpa - V • Pa + Zc

ena E + -va x B + Ra; c

\UoC clt™ + PaV ' Va = _V " CI a: - Pa ■ Vva + Qa.

Here da / dt = d/dt + va -V; pa = kTa, where k is Boltzmann's constant; Rq = ^^ Ra/3 and Qa = ^^ Qa/3, where Ra/3 and Qa/3 are respectively

the momentum and energy gained by the ath species through collisions with the /3th; Pa is the stress tensor; and qa is the heat flow.

The transport coefficients in a simple two-component plasma (electrons and singly charged ions) are tabulated below. Here || and _L refer to the di­rection of the magnetic field B = bB; u = ve — v^ is the relative streaming velocity; ne = rti = n; j = —new is the current; ujce = 1.76 X 107B sec-1 and iuCi = (me/mi)iuce are the electron and ion gyrofrequencies, respectively; and the basic collisional times are taken to be

Te = v ll ;----------------------------------------------- = 3.44 x 105—---------- sec.

4V27rnAe4                                                                                      n X

where A is the Coulomb logarithm, and

.>>3/2

ti

3 y^-jkTj) 4^/nn Xe4

rji 3/2

= 2.09 X 107—-------- p1/2 sec.

nX

In the limit of large fields (cocara 1, a = i, e) the transport processes may be summarized as follows:21

momentum transfer frictional force

electrical conductivities

thermal force

ion heating

electron heating ion heat flux

ion thermal conductivities electron heat flux

frictional heat flux

Rei— Rie = R — Ru + Rt ;

Ru = ne(j||/cr|| + jx/crx); cr|| = 1.96ct_l; &l = ne2re/me;

Rt = —0.71nV|| (kTe)------------- b x V±(kTe)-

2uJcet e

Q,

3mP nk

— (te - to;

mi re Qe — ~Qi R u:

q; = Vy (kTi) - KiLV±(kTi) + K\h x V±(kTi)-

k , = 3.9

nkTin

2nkTi

rrii

e . e

= 0.71n/cTeu,, +

K,± =

rriiuj .Ti

ka —

5nkTi

2miUJni

3nkTe

2uJr.pT f

b X ui;

thermal gradient heat flux

electron thermal conductivities

species)

c\T = V|| (kTe) - Ke±V±(kTe) - K^b x V±(kTe)-,

5nkTe

Km = 3.2

nkTere e nkTe ----------------------- ; k± = 4.7               

m,

hi

meLUc*Te

A

2mP(jj,

p —

1 XX--

~^{WXX + Wyy) - ^{WXX - Wyy)

- V3 WXy-

Pyy —

~ — (WXX + Wyy) + ~{WXX ~ Wyy)

+ mWXy\

Pxy

773

Pyx = -TjlWxy + ~{WXX - Wyy)-,

 

i > —

1 XZ -

Pzx = ~mWxz - rj^Wyz;

 

Pyz

Pzy = ~mWyZ + r]4Wxz-

 

1 > —

1 ZZ -

~r)oWzz

 

(here the z axis is defined parallel to B);

ion viscosity

r?o = 0-9 QnkTiTi] r]\

3 nkTi

10CJ 2 Ti '

 

QnkTi 5cc; 2r,- '

nkT;

 

2cu,

;

electron viscosity

nkTi

^ci

e

?70 = 0.73nkTere] r)1 = 0.51

nkTP

^3 =

e

=

nkTe

nkTe

 

r?2 = 2.0

n/cTe

^ceTe

For both species the rate-of-strain tensor is defined as

<9I>7- d^fc 2

W7^ = —- H--------------------------------------------------------- SjkV • v.

dxk dxj 3

When B = 0 the following simplifications occur:

Ru = nej/crn; RT = -0.71 nV(/cTe); = V(feTi);

Qu = 0.71n/cTeu; q^ = -n\V(kTe)- Pjk = -r)0Wjk.

For cuceTe 1 coci^i-, the electrons obey the high-field expressions and the ions obey the zero-field expressions.

Collisional transport theory is applicable when (1) macroscopic time rates of change satisfy d/dt <C 1/r, where r is the longest collisional time scale, and (in the absence of a magnetic field) (2) macroscopic length scales L satisfy L I, where I — vr is the mean free path. In a strong field, ujcer 1, condition (2) is replaced by Lu I and L \_ \/lre (L± > re in a uniform field),

where Ly is a macroscopic scale parallel to the field B and L±_ is the smaller of B/\\7±B\ and the transverse plasma dimension. In addition, the standard transport coefficients are valid only when (3) the Coulomb logarithm satisfies A 1; (4) the electron gyroradius satisfies re Ad, or 8tv nemec2 B2; (5) relative drifts u = va — v/5 between two species are small compared with the

thermal velocities, i.e., u2 <C kTa / ma, kTp/mp] and (6) anomalous transport processes owing to microinstabilities are negligible.

Weakly Ionized Plasmas

Collision frequency for scattering of charged particles of species a by neutrals is

a\0 /, rri / X 1/2 = n0as (kTa /ma) ,

where no is the neutral density and                           is the cross section, typically

5 X 10-15 cm2 and weakly dependent on temperature.

When the system is small compared with a Debye length, L < Ad, the charged particle diffusion coefficients are

D<y. — kTa /TTlcy ZVq, ,

In the opposite limit, both species diffuse at the ambipolar rate

PiDe - PeDi (Ti+Te)DiDe jja = -------------  = -------------------------- ,

Pi — Pe                                                                          TiDe + TeDi

where pa = ea /is the mobility. The conductivity cra satisfies cra =

Tla 6q Pa .

In the presence of a magnetic field B the scalars p and a become tensors, Ja = <Ta E = crffEy + al Ex + cr^E X b,

where b = B/ /i and

cr* = n^ea2/maua\

ex                                                    ex 2// 2,                   2 \

crx = cry zya /(i/a +cuca);

ex                                                    ex             / / 2 .             2 x

crA = cry                                                                         +

Here cr± and crA are the Pedersen and Hall conductivities, respectively.

Авторы: 1379 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 1908 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я