A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review and f and g can be found, for example, in Kanellopoulos et al [146]. Equation (3.20) can be rewritten as a pair of coupled first order ordinary differential equations [147, 148] for rξ r and p ∗ , the total perturbed pressure, p ∗ = p 1 + B o · B 1 , (3.22) µ o which is in a suitable form for employing a shooting code [149]. The boundary conditions that apply are (rξ r ) r=o = 0 and, for a plasma cylinder of radius a centred inside a perfectly conducting wall of radius r w , B 1 and p ∗ are continuous at r = a and B 1r is zero at r = r w . Unlike a diffuse pinch or a tokamak the pure Z-pinch, i.e. with no equilibrium z component of magnetic field, has no mode rational surfaces, which would lead to regular singular points for equation (3.20) at zeros of f . When f is zero it is zero for all r, and this corresponds to the m = 0 mode which is a purely interchange mode. As a result we can employ the relatively simple theory for the m = 0 mode to find the equilibrium that gives marginal stability by postulating that there is no net change in energy when plasma on one hoop-like flux tube of volume V 1 is exchanged with that on a neighbouring tube of volume V 2 and carrying the same magnetic flux, B 1 A 1 = B 2 A 2 = through their respective cross-sectional areas A 1 and A 2 . There will be no change in magnetic energy with such an interchange, while the plasma interchange will obey an adiabatic law pV Ɣ = constant. On interchange the increase in internal energy is 1 Ɣ − 1 ⌊p 1 V Ɣ 1 V Ɣ 2 dp dr = Ɣp B θ r d dr V 2 − p 1 V 1 + p V 2 Ɣ 2 V Ɣ 1 V 1 − p 2 V 2 ⌋ . For V 2 = V 1 + δV and p 2 = p 1 + δp, δV ≪ V 1 , δp ≪ p this becomes to second order δpδV + Ɣ p 1 (δV ) 2 V 1 On equating this to zero and writing δp = δr dp/dr and V = 2πrδφ/B θ we obtain the local condition for zero internal energy change ( ) r , (3.23) B θ which together with the pressure balance, equations (2.1) and (2.2), gives dp dr =−B θ ∂ µ o r ∂r (rB θ), (3.24) which leads to the Kadomtsev’s marginal stability condition [83] − r p dp dr = 2Ɣ 1+ 1 , (3.25) Ɣβ 2 where β(r) is 2µ o p/B 2 θ . To find the pressure profile we integrate equation (3.23) togive p/p 0 = Z Ɣ where Z(r) = B θ(r) r ( r B θ ) . (3.26) r=0 Defining a scale length r o = [µ o Ɣp o /(Ɣ − 1)] 1/2 (r/B θ ) r=0 and x = r/r o , equation (3.24) can be integrated to give 1 − Z Ɣ−1 = x 2 Z. (3.27) This gives the Kadomtsev profile for p(r), which is characterized by a centrally peaked current density as well as pressure, and which must extend as far as the conducting wall, i.e. no vacuum region. It is plotted in figure 17. 35
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 17. The Kadomtsev pressure profile which is marginally stable to the m = 0 sausage instability [149, figure 1]. Figure 18. Growth rate γ versus ka for the m = 0 and m = 1 modes for a flat current profile [149, figure 2]. It follows that a pinch with uniform current density is unstable to both m = 0 and m = 1. Tayler [38] showed that it is however stable for m 2. The early theory of Kruskal and Schwarzchild [37]ofm = 1 instability for a skin current equilibrium was generalized by Tayler for all m showing that it was unstable to m = 0 and m = 1 for all values of ka but for m 2 there is a value of ka below which it is stable. For the case where the current density varied as r n , Tayler showed that a sufficient condition for stability is m 2 2n +4. (3.28) With the advent of computers a more detailed study of stability could be undertaken, and Coppins [149] has computed the growth rates and eigenfunctions for the flat current profile and for the Kadomtsev profile. Figure 18 shows the growth rate versus ka for the m = 0 and 36
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