By Vadim G. Korneev, Ulrich Langer

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**Extra resources for Approximate solution of plastic flow theory problems**

**Example text**

We also note that the definition of the average ( ) has not yet been clearly stated. For a system not in thennal equilibrium, the ensemble average cannot be precisely defined. We shall not dwell on this point, but restrict ourselves to mentioning that the average smears out the discreteness effects to an extent which depends on the given situation of interest. In the discussion below, we shall consider only plasmas very close to an ideal plasma and neglect two-body correlation tenns. The resulting system of equations can then be written as follows: 8F.

VB and d/dt = dx/dt . Vif>. V B - qE) = 0, we obtain q) . [VII b+-(V'xb)] , n dVIl= - ( -V'B+-V' jJ. 17) for E = - V if>. 14) contains a term for the parallel drift which is of higher order in eLf L, and that VII agrees with the usual guiding center velocity along the magnetic field line only in the leading order in eLf L. 13) is identical to the curvature drift in Sect. 2 since b· Vb = -b x (V x b) and V x B = 0: V2 Vc v2 = ~b x (b· Vb) = ~(V x b). 18) The Vlasov description follows as an exact consequence of the preceding guiding center equations of motion.

4) is given, where (Te = -ene + Ze1lj, J = -eneUe + Ze1ljUi and P = Pe + F\. It is noted that the mass density {! and the average velocity U can be approximated by {! 5) U = (ujnjmj + uenem e)/ {! = Uj . 4), the electron inertia term is usually much smaller than the ion inertia term and is therefore negligible. Tben we have the equation {! (c:;; + (u· V)U) = - V P + J x B + aeE . 6) as the equation of motion for one fluid; the plasma is approximated by a single magneto-fluid. Similarly, by taking the sum of the electron continuity equation and the ion continuity equation, we get O{!

### Approximate solution of plastic flow theory problems by Vadim G. Korneev, Ulrich Langer

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