Coulomb Logarithm

For test particles of mass m_a and charge e_a = Z_ae scattering off field particles of mass mp and charge = Zpe, the Coulomb logarithm is defined

as A = In A = ln(r_max/Tmin)- Here r_mi_n is the larger of e_aep/m_apu² and h/2m_ol(3U, averaged over both particle velocity distributions, where m_ayg = m_am_/3/(m_a + mp) and u = v_Q: - v^; r_max = (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,< r_max or ULJcp^-1 < r_max, 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(n_e^1/2T_e"^3/2), T_e < lOeV; = 24 - In(n_e^1/2Te^_1),   T_e > lOeV.

(b)   Electron—ion collisions

Aei=A_ie = 23 - In (n_e^1/2ZT;^3/2),                                               T^e/m; < T_e < 10Z² eV;

= 24 - In (n_e^1/2T_e^_1) ,                                                              Tim_e/mi < 10Z² eV < T_e

= 30 - In (ni^1/2Ti~^3/2Z²/i^_1), T_e < TiZm_e/mi.

(c) Mixed ion—ion collisions

\i> = \'i = 23 - In

ZZ\ijl + p') fmZ² n_vz^/2x ^1/2

pT_if + p'Ti                          ⁺ T,/ J

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

^Xu> = \'i = 35 - In

ZZ\p + p') Zn_e V^/2

 

Fokker-Planck Equation

Df^a df

Dt dt

+ v V/"+F. V_v/'

ar

at

coll

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

(a/°ydt)_con = - V v_v • J^aX/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)V_vz/V^z) - -/^(v)Vv/^a(v)

WI/3

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

J^qX/3 = 47TA_a/3//«(v)V_vH(v) - iv_v • /"(v)V_vV_vG(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

m_a + rap

1 at\(3 ( 2 ,      X vy r-a

[V I - VVj • V_v/ .

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

Df_e

-J^- = Vee(F_e - f_e) + Vei(F_e - /_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 T_a. (For u_ee 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 F_a and F_a are given by

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

*^a~^na\2KkT_a) ^6XPl L 2kT_a \ /'

where n_a, v_a and T_a are the number density, mean drift velocity, and effective temperature obtained by taking moments of f_a. Some latitude in the definition of T_a and v_a is possible;²⁰ one choice is T_e = Ti, Ti = T_e, v_e = v^, v^ = v_c.

Transport Coefficients

Transport equations for a multispecies plasma:

d^an_a

—--------------------------------------------------- h n_a V • v_a = 0;

dt

m_a floe

d^av₍ dt

= -Vp_a - V • P_a + Z_c

en_a E + -v_a x B + R_a; c

\^UoC clt™ ^{+ PaV} ' ^{Va = _V} " CI a: - Pa ■ Vv_a + Q_a.

Here d^a / dt = d/dt + v_a -V; p_a = kT_a, where k is Boltzmann's constant; Rq = ^^ R_a/3 and Q_a = ^^ Qa/3, where R_a/3 and Q_a/3 are respectively

the momentum and energy gained by the ath species through collisions with the /3th; P_a is the stress tensor; and q_a 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 = v_e — v^ is the relative streaming velocity; n_e = rti = n; j = —new is the current; uj_ce = 1.76 X 10⁷B sec^-1 and iu_Ci = (m_e/mi)iu_ce 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—-------- p^1/2 sec.

nX

In the limit of large fields (co_car_a 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 ;

R_u = ne(j||/cr|| + jx/crx); cr|| = 1.96ct_l; &l = ne²r_e/m_e;

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

2uJ_cet _e

Q,

3m_P nk

— (t_e - to;

mi r_e Qe — ~Qi R u:

q; = Vy (kTi) - Kⁱ_LV±(kT_i) + K\h x V_±(kTi)-

k , = 3.9

nkTin

2nkTi

rrii

e . e

= 0.71n/cT_eu,, +

K,_± =

rriiuj .Ti

^ka —

5nkTi

2miUJni

3nkT_e

2uJr.pT f

b X ui;

thermal gradient heat flux

electron thermal conductivities

species)

c\_T = V|| (kT_e) - K^e_±V_±(kT_e) - K^b x V±(kT_e)-,

5nkT_e

Km = 3.2

nkT_er_{e e} nkT_e ----------------------- ; k_± = 4.7               

m,

hi

m_eLU_c*T_e

A

2m_P(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 = 0.73nkT_er_e] r)₁ = 0.51

nkT_P

^3 =

e

=

nkT_e

nkT_e

 

r?2 = 2.0

n/cT_e

^ce^Te

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

<9I>₇- d^fc 2

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

dxk dxj 3

When B = 0 the following simplifications occur:

R_u = nej/crn; R_T = -0.71 nV(/cT_e); = V(feTi);

Qu = 0.71n/cT_eu; q^ = -n\V(kT_e)- P_jk = -r)₀W_jk.

For cu_ceT_e 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, uj_cer 1, condition (2) is replaced by Lu I and L \_ \/lr_e (L± > r_e 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 r_e Ad, or 8tv n_em_ec² B²; (5) relative drifts u = v_a — v/5 between two species are small compared with the

thermal velocities, i.e., u² <C kT_a / m_a, 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 = n₀a_s (kT_a /m_a) ,

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. — kT_a /TTlcy ZVq, ,

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

PiD_e - PeDi (Ti+T_e)DiD_e jja = -------------  = -------------------------- ,

Pi — Pe                                                                          TiD_e + T_eDi

where p_a = e_a /is the mobility. The conductivity cr_a satisfies cr_a =

Tla 6_q Pa .

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

where b = B/ /i and

cr* = n^ea²/m_au_a\

ex                                                    ex 2// 2,                   2 \

cr_x = cry zy_a /(i/_a +cu_ca);

ex                                                    ex             / / 2 .             2 x

cr_A = cry                                                                         +

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