## Advanced Plasma Theory, Volume 25 |

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Page 57

We now wish to consider the limit of infinite conductivity a -> oo, so that (8) is

replaced by (8) E+-VxB=0. c However, in a certain sense a -> oo corresponds to

very large t which seems to contradict our previous assumption. However, in this

case, a large means t » TiLWjq2)-1 so (l)-(7), and (8a) should be valid if ^-.r«T«T'

o>\ = inne2jm where n, e and m are electron density, charge and mass, and L is a

macroscopic length. The first half of this inequality is usually well

fluid ...

We now wish to consider the limit of infinite conductivity a -> oo, so that (8) is

replaced by (8) E+-VxB=0. c However, in a certain sense a -> oo corresponds to

very large t which seems to contradict our previous assumption. However, in this

case, a large means t » TiLWjq2)-1 so (l)-(7), and (8a) should be valid if ^-.r«T«T'

o>\ = inne2jm where n, e and m are electron density, charge and mass, and L is a

macroscopic length. The first half of this inequality is usually well

**satisfied**so ourfluid ...

Page 77

(23) ^eiFoqdwdq^O. It is easily shown from (11) that the time derivative of (22) is

zero if (23) is

of (23) gives where P° = m (t)-o-»',n)(v-a- », n)/°d2p, is the zero order pressure.

This may be obtained more simply by taking a moment of the unexpanded

Boltzmann equation for ions and electrons dotting with n and subtracting (see ref.

[13]). It is easily shown that (25) V° = p±(I — nn) + p,nn, where (26) p±=

mlwF02ndu-dq ...

(23) ^eiFoqdwdq^O. It is easily shown from (11) that the time derivative of (22) is

zero if (23) is

**satisfied**. (This is just (do-lfit)+V-J-l = 0.) Similarly the time derivativeof (23) gives where P° = m (t)-o-»',n)(v-a- », n)/°d2p, is the zero order pressure.

This may be obtained more simply by taking a moment of the unexpanded

Boltzmann equation for ions and electrons dotting with n and subtracting (see ref.

[13]). It is easily shown that (25) V° = p±(I — nn) + p,nn, where (26) p±=

mlwF02ndu-dq ...

Page 207

We now replace QrSPr— P,8Qr on the right-hand side of (A-4.6) by the variation

of the expression (A-4.3) expressing this variation in terms of 8Pr, 8Qr. If we now

separate terms of first order in w, we find that (A-4.6) is

that we may

function V1(Pr,Q,,t) which is to

by means of the equations r) TT1 ?, TP We now wish to find second-order terms

of ...

We now replace QrSPr— P,8Qr on the right-hand side of (A-4.6) by the variation

of the expression (A-4.3) expressing this variation in terms of 8Pr, 8Qr. If we now

separate terms of first order in w, we find that (A-4.6) is

**satisfied**to this order andthat we may

**satisfy**the requirement (A-4.7) El= 0, by introducing a generatingfunction V1(Pr,Q,,t) which is to

**satisfy**(A-4.8) and from which Pi, Ql are obtainedby means of the equations r) TT1 ?, TP We now wish to find second-order terms

of ...

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adiabatic invariant amplitude approximation assumed Boltzmann equation boundary conditions boundary layer calculated cathode charge coefficient collision column components consider const constant contraction corresponds courbe current density Debye length derived differential equations diffusion discharge dispersion relation distribution function double adiabatic theory effect eigenvalue electric field electromagnetic waves electrostatic energy principle equations of motion equilibrium expand experimental finite fluid theory frequency given Hence hydromagnetic inertia-limited instability integral interaction ionized Kruskal Kulsrud l'axe magnétique lignes limit linear theory lowest order magnetic field Maxwell's equations mode negative ions nonlinear obtain Ohm's law parameter particle perturbation Phys plasma oscillations plasma physics Poisson's equation potential pressure problem produced quantities radial region satisfied saturation current self-adjointness solution solving stability surface temperature thermal tion transverse wave values vanish variables vector velocity Vlasov equation waves in plasmas zero zero-order