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They form an accelerated electron beam detached from themain body of the distribution.5.6 Collisions in Fully Ionized Plasmas5.6.4171Numerical Values of ηExact computations of η which take into account the ion recoil in each collision andare properly averaged over the electron distribution were first given by Spitzer. Thefollowing result for hydrogen is sometimes called the Spitzer resistivity:ηk ¼ 5:2 105Z ln Λohm-mT ðeVÞ3=2ð5:76ÞHere Z is the ion charge number, which we have taken to be 1 elsewhere in thisbook. Since the dependence on M is weak, these values can also be used for othergases.

The subscript jj means that this value of η is to be used for motions parallel toB. For motions perpendicular to B, one should use η⊥ given byη⊥ ¼ 2:0ηkð5:77ÞThis does not mean that conductivity along B is only two times better thanconductivity across B. A factor like ω2c τ2 still has to be taken into account.

Thefactor 2.0 comes from a difference in weighting of the various velocities in theelectron distribution. In perpendicular motions, the slow electrons, which havesmall Larmor radii, contribute more to the resistivity than in parallel motions.For KTe ¼ 100 eV, Eq. (5.76) yieldsη ¼ 5 107 ohm-mThis is to be compared with various metallic conductors:copper . . . . .

. . . . . . . . . . η ¼ 2 108 ohm-mstainless steel . . . . . . ::η ¼ 7 107 ohm-mmercury . . . . . . . . . . . . ::η ¼ 106 ohm-mA 100-eV plasma, therefore, has a conductivity like that of stainless steel.5.6.5Pulsed CurrentsWhen a steady-state current is drawn between two electrodes aligned along themagnetic field, electrons are the dominant current carrier, and sheaths are set up atthe cathode to limit the current to the set value. When the current is pulsed,however, it takes time to set up the current distribution. It was shown by Stenzeland Urrutia that this time is controlled by whistler waves (R-waves), which musttravel the length of the device to communicate the voltage information.1725.75 Diffusion and ResistivityThe Single-Fluid MHD EquationsWe now come to the problem of diffusion in a fully ionized plasma.

Since thedissipative term Pei contains the difference in velocities vi ve, it is simpler to workwith a linear combination of the ion and electron equations such that vi ve is theunknown rather than vi or ve separately. Up to now, we have regarded a plasma ascomposed of two interpenetrating fluids. The linear combination we are going tochoose will describe the plasma as a single fluid, like liquid mercury, with a massdensity ρ and an electrical conductivity 1/η.

These are the equations of magnetohydrodynamics (MHD).For a quasineutral plasma with singly charged ions, we can define the massdensity ρ, mass velocity v, and current density j as follows:ρ ni M þ ne m nð M þ m Þð5:78Þ1Mvi þ mvev ðni Mvi þ ne mve Þ ρMþmð5:79Þj eðni vi ne ve Þ neðvi ve Þð5:80ÞIn the equation of motion, we shall add a term Mng for a gravitational force.

Thisterm can be used to represent any nonelectromagnetic force applied to the plasma.The ion and electron equations can be written∂vi¼ enðE þ vi BÞ ∇ pi þ Mng þ Pie∂tð5:81Þ∂ve¼ enðE þ ve BÞ ∇ pe þ mng þ Pei∂tð5:82ÞMnmnFor simplicity, we have neglected the viscosity tensor π, as we did earlier. Thisneglect does not incur much error if the Larmor radius is much smaller than thescale length over which the various quantities change. We have also neglectedthe (v · ∇)v terms because the derivation would be unnecessarily complicatedotherwise.

This simplification is more difficult to justify. To avoid a lengthydiscussion, we shall simply say that v is assumed to be so small that this quadraticterm is negligible.We now add Eqs. (5.81) and (5.82), obtainingn∂ðMvi þ mve Þ ¼ enðvi ve Þ B ∇ p þ nðM þ mÞg∂tð5:83ÞThe electric field has cancelled out, as have the collision terms Pei ¼ Pie. We haveintroduced the notationp ¼ pi þ peð5:84Þ5.7 The Single-Fluid MHD Equations173for the total pressure. With the help of Eqs.

(5.78)–(5.80), Eq. (5.83) can be writtensimplyρ∂v¼ j B ∇ p þ ρg∂tð5:85ÞThis is the single-fluid equation of motion describing the mass flow. The electricfield does not appear explicitly because the fluid is neutral. The three body forces onthe right-hand side are exactly what one would have expected.A less obvious equation is obtained by taking a different linear combination ofthe two-fluid equations.

Let us multiply Eq. (5.81) by m and Eq. (5.82) by M andsubtract the latter from the former. The result isMmn∂ðvi ve Þ ¼ enðM þ mÞE þ enðmvi þ Mve Þ B m∇ pi∂tþ M∇ pe ðM þ mÞPeið5:86ÞWith the help of Eqs. (5.78), (5.80), and (5.61), this becomes Mmn ∂ j¼ eρE ðM þ mÞneη j m∇ pi þ M∇ pe þ enðmvi þ Mve Þe ∂t nBð5:87ÞThe last term can be simplified as follows:mvi þ Mve ¼ Mvi þ mve þ Mðve vi Þ þ mðvi ve Þρj¼ v ðM mÞnneð5:88ÞDividing Eq. (5.87) by eρ, we now haveE þ v B ηj ¼ 1 Mmn ∂ jþ ðM mÞ j Bþm∇ pi M∇ pe ð5:89Þeρ e ∂t nThe ∂/∂t term can be neglected in slow motions, where inertial (i.e., cyclotronfrequency) effects are unimportant. In the limit m/M ! 0, Eq.

(5.89) then becomesE þ v B ¼ ηj þ1ð j B ∇ pe Þenð5:90ÞThis is our second equation, called the generalized Ohm’s law. It describes theelectrical properties of the conducting fluid. The j B term is called the Hallcurrent term. It often happens that this and the last term are small enough to beneglected; Ohm’s law is then simply1745 Diffusion and ResistivityE þ v B ¼ ηjð5:91ÞEquations of continuity for mass ρ and charge σ are easily obtained from the sumand difference of the ion and electron equations of continuity. The set of MHDequations is then as follows:ρ∂v¼ j B ∇ p þ ρg∂tð5:85ÞE þ v B ¼ ηjð5:91Þ∂ρþ ∇ ðρvÞ ¼ 0∂tð5:92Þ∂σþ∇ j¼0∂tð5:93ÞTogether with Maxwell’s equations, this set is often used to describe the equilibrium state of the plasma.

It can also be used to derive plasma waves, but it isconsiderably less accurate than the two-fluid equations we have been using. Forproblems involving resistivity, the simplicity of the MHD equations outweighs theirdisadvantages. The MHD equations have been used extensively by astrophysicistsworking in cosmic electrodynamics, by hydrodynamicists working on MHD energyconversion, and by fusion theorists working with complicated magnetic geometries.5.8Diffusion of Fully Ionized PlasmasIn the absence of gravity, Eqs.

(5.85) and (5.91) for a steady state plasma becomejB ¼ ∇pð5:94ÞE þ v B ¼ ηjð5:95ÞThe parallel component of the latter equation is simplyEjj ¼ ηjj jjjwhich is the ordinary Ohm’s law. The perpendicular component is found by takingthe cross-product with B:E B þ ðv⊥ BÞ B ¼ η⊥ j B ¼ η⊥ ∇ pE B v ⊥ B2 ¼ η ⊥ ∇ p5.8 Diffusion of Fully Ionized Plasmas175v⊥ ¼E B η⊥ 2∇ pB2Bð5:96ÞThe first term is just the E B drift of both species together. The second term is thediffusion velocity in the direction of ∇p.

For instance, in an axisymmetriccylindrical plasma in which E and ∇p are in the radial direction, we would havevθ ¼ ErBvr ¼ η⊥ ∂ pB2 ∂rð5:97ÞThe flux associated with diffusion isΓ⊥ ¼ nv⊥ ¼ η⊥ nðKT i þ KT e Þ∇nB2ð5:98ÞThis has the form of Fick’s law, Eq. (5.11), with the diffusion coefficientD⊥ ¼η⊥ nΣKTB2ð5:99ÞThis is the so-called “classical” diffusion coefficient for a fully ionized gas.Note that D⊥ is proportional to 1/B2, just as in the case of weakly ionized gases.This dependence is characteristic of classical diffusion and can ultimately be tracedback to the random-walk process with a step length rL.

Equation (5.99), however,differs from Eq. (5.54) for a partially ionized gas in three essential ways. First, D⊥ isnot a constant in a fully ionized gas; it is proportional to n. This is because thedensity of scattering centers is not fixed by the neutral atom density but is theplasma density itself. Second, since η is proportional to (KT)3/2, D⊥ decreases withincreasing temperature in a fully ionized gas. The opposite is true in a partiallyionized gas.

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