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References 1. D. V. 93 (ed. by M. A. Leontovich) Consultant Bureau, New York 1966 K. 4 The MIT Press, Cambridge, Mass. 1 Magnetohydrodynamic Equations for Two Fluids Plasmas can be described as magnetohydrodynamic two °uids of ions and electrons with mass densities ½mi , ½me , charge density ½, current density j, °ow velocities V i , V e , and pressures pi , pe . 4. The number density of ion ni , the ion mass density ½m;i , and the ion °ow velocity V i (r; t) are expressed as follows: ni (r; t) = Z fi (r; v; t)dv; (5:1) ½mi (r; t) = mi ni (r; t); R vfi (r; v; t)dv V (r; t) = R = fi (r; v; t)dv (5:2) 1 ni (r; t) Z vfi (r; v; t)dv: (5:3) We have the same expressions for electrons as those of ions.

6 Equilibrium In order to maintain a hot plasma, we must con¯ne and keep it away from the vacuumcontainer wall. The most promising method for such con¯nement of a hot plasma is the use of appropriate strong magnetic ¯elds. An equilibrium condition must be satis¯ed for such magnetic con¯nement systems. 5) indicates that B and rp are orthogonal, and the surfaces of constant pressure coincide with the magnetic surfaces. 6) shows that the current-density vector j is everywhere parallel to the constant-pressure surfaces.

Since vk2 ¿ v? for the trapped particle, the r component of the toroidal drift vdr of trapped particle is given by r_ = vdr sin µ = 2 m v? 4) dvk ¹m @B =¡ ; dt m @l v_ k = ¡ The solution is d dt à ¹m r v 2 Bp sin µ: ∙B0 sin ∙l = ¡ ? m R 2R B0 m r+ v qBp k r ¡ r0 = ¡ ! 9 Banana orbit of ion Here r = r0 indicates the radial coordinate of turning point by mirror e®ect. 9). 4 as follows: 2 mvk2 =2 1 mv? (b £ rB) + (b £ (b ¢ r)B) v G = vk b + (E £ b) + B qB 2 qB 2 (3:46) and 2 ¹m = mv? =(2B) = const: When the electric ¯eld E is static and is expressed by E = ¡rÁ, the conservation of energy m 2 2 ) + qÁ = W (v + v?

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