By Kenro Miyamoto
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Extra resources for Fundamentals of Plasma Physics and Controlled Fusion
9) is hpi ¡ pa = 1 2 ((B'v )a + (Br2 + Bz2 )a ¡ hB'2 i): 2¹0 Here h i indicates the volume average and pa is the plasma pressure at the plasma boundary. 2 0 . 2 0 i=2¹0 is called the poloidal beta ratio ¯p . When pa = 0, ¯p is ¯p = 1 + 2 ¡ hB 2 i B'v 2B'v ' ¼ 1 + 2 hB'v ¡ B' i: 2 B! 0 B! 5 Displacement of the plasma column. : B' and B'v are the toroidal magnetic ¯elds in the plasma and the vacuum toroidal ¯elds respectively. When B' is smaller than B'v , the plasma is diamagnetic, ¯p > 1. When B' is larger than B'v , the plasma is paramagnetic, ¯p < 1.
1). The r component of j £ B = rp leads to the equation on Ã : L(Ã) + ¹0 r 2 where L(Ã) ´ Ã @p(Ã) ¹2 @I 2 (Ã) + 02 =0 @Ã 8¼ @Ã @ 1 @ @2 r + 2 @r r @r @z ! (6:15) Ã: This equation is called Grad-Shafranov equation. The current density is expressed in term of the function of the magnetic surface as jr = ¡1 @I(Ã) ; 2¼r @z ¡1 j' = ¹0 1 = ¹0 r Ã jz = @ 1 @Ã 1 @ 2 Ã + @r r @r r @z 2 Ã 1 @I(Ã) ; 2¼r @r ! ¹2 ¹0 r 2 p0 + 02 (I 2 )0 8¼ =¡ L(Ã) ¹0 r ! or j= I0 B + p0 re' ; 2¼ (6:16) L(Ã) + ¹0 rj' = 0: The functions p(Ã) and I 2 (Ã) are arbitrary functions of Ã.
5) becomes Ã 3 3 3 X X @Á @F X @F @Ak ¡ vk +q vk + @t @xk @xi @xi i=1 k=1 i=k ! 8), independent variables are transformed from (qi ; pi ; t) to (xj ; vj ; t) and 1 @vj (xk ; pk ; t) = ±ij ; @pi m @vj (xk ; pk ; t) q @Aj =¡ ; @xi m @xi @vj (xk ; pk ; t) q @Aj =¡ : @t m @t 4 Velocity Space Distribution Function ¢ ¢ ¢ 36 We denote F (xi ; pi ; t) = F (xi ; pi (xj ; vj ; t); t) ´ f (xj ; vj ; t)=m3 . 11) is reduced to @f X @f + @t @vi i + µ Ã X X i k ¡q m ¶ @Ai X vk + @t k @Á @Ak ¡ vk @xi @xi ! Ã X @f @f + @xk @vi i q @f = m @vi µ ±f ±t ¶ µ ¡q m ¶ @Ai @xk !