z - P I N C H W I T H N O N Z E R O H E L I C I T Y

MARTIN URBAN, PETR KULH.&NEK

Department of Physics, Czech Technical University, Technickl 2, 16200 Praha 6, Czech Republic

Received 12 February 1996

The z-pinch equilibrium with non zero azimuthal current was calculated for power dependences in this paper. The pressure equilibrium curves have no maximum in the z- pinch centre and look more like a gass-puff pressure curves. The total current density is formed by grad B drift current density, curvature drift current density and magnetization current density in this case.

1 I n t r o d u c t i o n

z-pinch is a p lasma column with current density along z-axis. The corresponding azimuthal magnetic field causes a Lorentz force opposite to the pressure gradient. In the equilibrium (Bennette equilibrium) the z-pinch is unstable, especially due to the m = 0 and m = 1 instability modes. From the experiments it is obvious that the z-pinch lifetime is much longer than predicted from the MHD theory. An azimuthal component of the current density (axial component of the magnet ic field) is generally supposed to be the stabilizing factor [1-3].

In the second section we discuss the equilibrium of the z-pinch in the presence of both the axial and the azimuthal current densities and the pressure-radius de- pendence will be derived for power dependences of current densities.

In the third section we derive the V B drift current density, the curvature drift current density and the magnetization current density relations f rom the magnetic field of z-pinch and we will show that these three components form the total current density.

2 Equi l ibr ium calculat ion

Let us consider a p lasma column in cylindrical coordinates (r, ~, z) - - see Fig. 1.

Jz

Fig. 1. z-pinch configuration.

Czechoslovak Journal of Physics, Vol. 46 (1996), No. 11 1093

Martin Urban and Petr Kulh~nek

Let us denote:

r0 - - z-pinch radius, l - - z-pinch length, jr (r) - - axial current density, j ~ ( r ) - - azimuthal current density, j o z ( r ) - - axial current density on the surface of the z-pinch, j o ~ ( r ) - - azimuthal current density on the surface of the z-pinch.

Let us assume power dependences of the current densities

(:0) j z ( r ) = jo~ r ,

j~o(r) = jo~o ~o " (1)

The power dependences of the current densities represent both the possibility of the current flowing on the z-pinch surface (a, b > 0; skin effect) and the currents with major part in the z-pinch centre (a, b < 0; hot z-pinch centre) - - see Fig. 2.

2

2_ /o

Oh 0

a=0

a=2

r-L- 1 r0

Fig. 2. Current density power dependence.

The components of the magnetic field can be derived from the Ampere law j~ Bdl = t to I . Using subtitution x = r / r0 we obtain

1 - - x b+l B z ( x ) = p o r o j o ~ b + l A b # - 1 ,

S~(x) = ,0r0j0~, ln(1) A b = - 1 ,

xa+l B ~ ( x ) = # o r o j o z a +-----2 A a # - 2 . (2)

The equilibrium equation

j x B = - g r a d p (3)

1094 Czech. J. Phys. 46 (1996)

z-pinch with non zero helicity

can be rewritten with the help of Maxwell equation rot B = #0j in cylindrical coordinates as

B~,2+ d (B~ ~ Bz 2 ) #---~ ~rr \ 2/J0 + ~ + P = 0. (4)

This equation represents an ordinary differential equation for the hydrostatic pressure with the boundary condition p(r0) = 0. The solution is

( ) 1 Bz(x)2 + a + 2 2 ' p(x)-- 2p0 -~--~B~(x) + C A a # - - 1 . (5)

where C is a constant determined by the boundary condition p(1) = 0. Results for several values of a, b and various ratios of jo~/joz are presented in

Fig. 3.

0.5

2 P ~

Izo ro Joz

Jo~-O 0.1

2 p- - - - - -~ Po ro Joz

Jo~-O

JOz 1000 /

0 0 0 __r 1 0 __r 1

a) ro b) ro

0.11 fio~'O 0.5

2 P ----YT.2 P - - - T ~

po ro Joz Po ro Joz

0 0 0 r 1

c) ro d)

jo -O 1

Oz 2

~ - 1 \

0 r 1 r0

Fig. 3. Equilibrium pressure versus radius dependences: a) a = 0, b = 0, b) a = 2, b = -0.8, c) a = 2 , b = - 0 . 2 , d) a = - 3 , b=2 .

Czech. J. Phys. 46 (1996) 1095

Martin Urban and Petr Kulhs

3 Cur ren t componen t s

Let us consider both the azimuthal and the axial components of the z-pinch magnetic field

B=[0; B~(r); B~(r)]; B = ~ + B ~ (6)

in a cylindrical geometry. The VB drift current density, the curvature drift current density and the magnetization current density relations will be derived in this section. The sum of these three current densities leads to the current equilibrium equation. Therefore it can be concluded that these three components form the total current.

In contradistinction to the standard equilibrium calculations [5], the considera- tions presented here are based on the drift model (current caused by the movement of the gyration centres, Larmor rotation averaged). This approach is suitable even for nonequilibrium pinches with time dependent electric field E(t , r). It was proved [6] (for the pinch with zero helicity only) that fourth component of the total current density has to be included: polarization drift current density. This model enables us to perform electric field calculations.

3.1 VB drift current density

VB drift velocity [4]

mv 2 B x V B v v B - 2---B- B 3 (7)

results in the current density

<z > ( , o,miv . ) ne vex B~oB~, j = q,~n~v~, = + ~ B4 [0 ; B~ ; -B~] .

(s) Using

( 1 } ~_ ( 1 mv~) kT = = I-- (9) mv2_t. 2 , -~ 2

(one longitudinal and two perpendicular degrees of freedom), the current density equation can be rewritten as

(10) I

jVB = (nekTe + nikTi) B~~176 + BzB'z B4 [0; Bz ; -B~o].

After a simple manipulation we get

(11) [ 28 4 ~r(B )[0; S~; -B~] = 0; 2B 4 (B2); p B ~ o (B2) 2B 4

1096 Czech. J. Phys. 46 (1996)

z-pinch with non zero helicity

3.2 C u r v a t u r e drif t c u r r e n t d e n s i t y

The curvature drift velocity is given by the relation [4]

mv~ R x B vR = (12)

qB 2 R 2 '

where R is the magnetic field curvature radius. For the field shape (6) it can be calculated as

= . (13)

Substituting (13) into (12) the curvature drift velocity becomes

mv~ B~ ( 0 ; - B , ; B~) (14) v a - qr B 4

The corresponding current density can be calculated in a similar way as above. The result is [ pB ]

pB~ [ 0 ; - B z " B~] = 0; �9 . (15) j a = ~ , r B 4 , "~'~1

3.3 Magnet iza t ion current densi ty

Magnetization current density can be derived from the expression

jM = V x M , (16)

where magnetization M is given by relation

m v ~ B (17) M=Ena<,~); p- 2B B '

# being magnetic moment of the particle. After a simple manipulation we derive

[ d l d J M : - 0 ; - ~ r \ B 2 ] , r d r \ B 2 ] J " (18)

Let us calculate the Lorentz force density fL corresponding to the current den- sities (11), (15), (18):

fL = (jVB + j R +jM) x B . (19)

Substituting from (11), (15), (18) and (6), after differentiation and other straight- forward manipulations we obtain

0; 0] which is standard equilibrium condition for the cylindrical plasma column. From this we can conclude that in the equilibrium the total current density has just three components: VB drift current density, curvature current density and magnetization current density.

Czech, J. Phys. 46 (1996) 1097

M. Urban and P. Kulhs z-pinch with non zero helicity

4 C o n c l u s i o n

The z-pinch equilibrium with non zero azimuthal currents was calculated in this paper. In addition to the Bennette equilibrium the azimuthal current forms a plasma stabilizing factor. Both axial and azimuthal currents were treated as a power dependent. Magnetic field was derived from the Ampere law and the pressure-radius dependence .from standard equilibrium equation. The pressure equi- librium curves in the presence of azimuthal current are different from the Bennette ones. The pressure maximum does not occur in the z-pinch centre - - see Fig. 3. In some cases the p(0) value is only a small fraction of the maximum pressure. The z-pinch behaves more like a gas-puff. In our opinion, the real z-pinch equilibrium corresponds to the configuration with azimuthal current on the surface (b > 0) and axial current in the centre (a < 0). It may follow from the "force free" condition [4].

The current densities corresponding to grad B drift, curvature drift and magne- tization for non zero axial magnetic field were calculated in this paper. From the equilibrium equation it was concluded that the three current density components mentioned above form the total z-pinch current.

This research has been conducted as part of the research project "Magnetic Pinches Stable Structure Study" and have been supported by GACR grant No. 202-95-0178.

References

[1] Kube~ P. et al.: IEEE Trans. on Plasma Science PS-21 (1993) 605.

[2] Kube~ P. et al.: IEEE Trans. on Plasma Science PS-22 (1994) 986.

[3] KulhAnek P.: in Proc. 17th Symp. on Plasma Phys. and Technol., Prague 1995. Czech. Tech. University and Inst. Plasma Phys., Acad. Sci. CzR, Prague, 1995, p. 144.

[4] Perrat A.L.: Physics of the Plasma Universe, Springer-Verlag, Berlin, 1992.

[5] Braginskij S.I.: in Plasma Physics (Ed. Kadomtsev B.), Mir, Moscow, 1981, p. 91 (in English).

[6] Hakr J. et al.: in Proc. XXI Int. Conf. Ion. Gases, Bochum 1994. Bochum Techn. University, Bochum, 1994, p. 381.

1098 Czech. J. Phys. 46 (1996)