Abstract

Non Resistive Analysis of Rotational Instabilities in FRC and the Unicamp TC-1 m=4 Results

M. A. M. Santiago

Instituto de Física, Universidade Federal Fluminense

24210-340, Niterói, Rio de Janeiro, Brazil

A. S. Assis

Instituto de Matematica, Universidade Federal Fluminense

24020-100, Niteroi, Rio de Janeiro, Brazil

C. A. de Azevedo

Instituto de Física, Universidade do Estado do Rio de Janeiro

20550-013, Rio de Janeiro, RJ, Brazil

P. H. Sakanaka, M. Machida, E. Aramaki, L. A. Berni,

D. O. Campos and R. T. Farias

Instituto de Física 'Gleb Wataghin'

Universidade Estadual de Campinas, UNICAMP

13083-970, Campinas, São Paulo, Brazil

Received September 30, 1997

I. Introduction

The prospect of field reversed configuration (FRC) devices to be competitive with other fusion devices relies on the understanding of the stability properties[1]-[3]. Frieman and Rotenberg [4] developed the original perturbation theory for a flowing equilibrium including the plasma compressibility. We use their basic operator equation to study the rotational mode in field reversed configuration. Other previous work are also concentrated on rotational modes [5]-[8]. According to boundary layer analysis [9], [10], the plasma is governed by ideal MHD with real frequency w on both sides of the singular layer r₀ where the equilibrium magnetic field vanishes. To overcome the accumulation point of the frequency spectrum of the eigenmode, resistivity was introduced in the boundary layer which leads to a growth rate strictly related to resistivity. But previous works [5], [6], indicated that resistivity is not relevant to analyse rotational modes.

Here we use the non resistive model in cylindrical geometry with realistic equilibrium profiles for the density, pressure and magnetic field configuration. We solve the eigenmode equation developed by Frieman [4], using a numerical code constructed with the software "Mathematica" and compare our results with the Compact Torus (TC-1) experiment of UNICAMP, where a m=4 rotational mode is observed.

II. Equilibrium and perturbed equations

We consider a reversed field q-pinch configuration with axial magnetic field and azimuthal rotation W. All variables depend on r only. The equilibrium profiles are given by (see Figures (1) - (3))

where B_z is the magnetic field and is modeled, to be realistic with the TC-1 machine, as

and

where r is the mass density,

is the velocity field, and the rotational frequency W is assumed to be constant.

We use the non resistive MHD and Maxwell equations, which are given by

and

with , where is the current density, p/r^g = const, g is the ratio of specific heats, and .

In equilibrium , we obtain

We use the Lagrangian representation, linearize the equations and introduce the displacement vector , which is considered a small quantity given by

where describes the equilibrium trajectory and describes the displacement from equilibrium, and consider the equilibrium quantities as time independent.

Following the linearization of the basic equations described in [4] we obtain the non hermitian linearized equation of motion in the form

where

and

Assuming normal mode solutions of the form

the equation of motion becomes

This non Hermitian eigenvalue equation will be solved with appropriate boundary conditions, , for r = 0 and r = a through a numerical code developed using the software "Mathematica". The pressure p is obtained solving equation (7) numerically.

III. The UNICAMP TC-1 results

The Compact Torus (TC-1) machine at UNICAMP, [11], is a field reversed q-pinch designed to study FRC formation and stability. The main solenoid is 45 cm long and 16 cm in diameter, and the two mirror field solenoids are 10 cm long and 15 cm in diameter. The base pressure of 1.0 x 10 ^-6 Torr is filled with Hydrogen at (1 to 10) x 10 ^-3 Torr. The 7 kV, 10.8 kJ polarization capacitor bank produces a reverse bias of 1.0 kG. The 25 kV, 0.5 kJ preionization capacitor bank with crowbar ionizes partially the working gas and is interrupted after few oscillations, when the 22 kV, 8.8 kJ main capacitor bank, with rise time of 5 ms, is switched on. The main field decays after the main crowbar is about 30 ms, and the peak magnetic field is 3.6 kG.

The typical plasma parameters of the TC-1 are the following: separatrix radius of 3.5 cm, measured by excluded-flux loops, ion temperature of 180 eV, measured by CIV and SiIV impurity lines, electron density of 1.0 x 10¹⁵ cm^-3 measured by CO₂ laser interferometry and electron temperature of 100 eV estimated by pressure balance. The TC-1 machine and its diagnostic set up are shown in Fig. 4.

One feature of the TC-1 machine is the capability of crowbar on preionization bank, which controls the ionization state before the main phase starts, and avoids the normal oscillations from preionization bank to be present after the start of the main discharge bank.

In Fig. 5, we present end-on framing pictures of implosion and equilibrium phases for filling pressure of 3.3 x 10^-3 Torr and preionization time of 8 ms (one period), where usual m = 0 mode plasma is obtained.

In Fig. 6, it starts the appearance of the m = 4 mode. The filling pressure in this case is 7.5×10^-3 Torr and the preionization time is 16 ms (two periods).

In Fig. 7, we show the case with same preionization sequence as in Figure 6, but with a Hydrogen filling pressure of 10 x 10^-3 Torr, where the start of framing pictures were delayed by 2 ms.

One can notice from the last two framing pictures that during the implosion phase the plasma is circular and it becames more square shaped and rotates around the axis at near maximum compression. After this phase, the plasma rotation is stopped by the leak of particles from the square corners touching the wall, but still mantaining the mode structure. The total time of the sequence of events leading to an square plasma is about 2 ms.

The plasma azimuthal rotation inferred from Fig. 7 before the wall touching is 1.4 x 10⁶ rad/s, as obtained from the square plasma rotation during two sequences of frame pictures.

IV. The rotational modes

We use our non resistive MHD model to calculate the rotational modes using TC-1 parameters. We solved equation (13) using typical field reversed equilibrium magnetic profile, to analyze the m = 2 and m = 4 modes. Taking the magnetic field at the edge B(r_w) to be 3.5 kG, the separatrix radius r₀ be 3.5 cm, the peak plasma density n(r₀) to be 10¹⁵ cm^-3, the Alfvén transit time t_A is 0.1 ms. Since the measured plasma rotation is W¢ = 1.4 x 10⁶ rad/s, the normalized rotation W₀ is of the order of 0.2. Estimating the plasma length L to be 20 cm, and considering l = 2L, the normalized wavevector k r₀ is 0.5. The real and imaginary parts of the eigenfrequency w = w¢/W¢ = w¢t_A/W₀ are plotted in Fig. 8, for the m=2 mode, as a function of the wavevector k r₀ showing a decreasing Im(w¢) t_A /W₀ and an increasing Re(w¢) t_A /W₀, and we define g = Im(w). At kr₀ ® 0, Fig. 8 shows the Re(w)/g = 1 limit , for the first radial mode, which agrees with the earlier work of Freidberg and Wesson [1]. Firstly, we note that Re(w) is different from mW in the long wavelength and small kr₀ limit. This shows a fundamental error in boundary layer analysis which assumes the marginal stability expansion w = mW+ ig in solving the kr₀ limit. The expansion would be adequate in the short wavelength limit. Second, the result of [1] is based on a q-pinch with parallel bias where B(r) is nonzero everywhere. They used an incompressible perturbed fluid model and assumed a more restricted expansion of the perturbed variables . In our case we assume compressibility and the general expansion of the perturbed quantities obtaining the same general eigenmode equation as in [4].

The eigenfrequency w¢t_A /W₀ is plotted as a function of rotation W₀ in Fig. 9 (W₀ = 0.20), for the m=2 mode, showing that g goes to zero as rotation goes to zero. The displacement x_r , real and imaginary parts, are plotted in Figs. 10 and 11 respectively as a function of radius r/r₀. For the m = 4 mode, Fig. 12 shows the real and imaginary parts of the frequency w¢t_A /W₀ as a function of the wavevector kr₀. The corresponding displacement x_r, real and imaginary parts, is shown in Figs. 13 and 14 as a function of radius r/r₀. In Figure 15, the eigenfrequency w¢t_A /W₀ is plotted against the rotation W₀ which shows again that g® 0 in the zero rotation limit.

Taking W₀ = 0.20 for the TC-1 machine, Fig.15 gives Im(w¢) t_A /W₀ = 0.30 for the n=0 radial mode (fundamental mode). This corresponds to g^-1 of the order of 1.3 ms and the measured growth time is of the order of 2.0 ms. The appearance of the m = 4 mode rather than the m=2 mode in the TC-1 can be justified by the larger growth rates in Fig. 15 than those in Fig.9.

V . Conclusions

We used a one dimensional MHD model to study the rotational instability in FRC plasmas taking into account the plasma compressibility and comparing our results with the TC-1 machine of Unicamp.

Although the m = 2 mode is commonly observed in many machines, [12], we have only seen the m = 4 mode in TC-1. However, the presence of this mode is compatible with the hybrid code of Harned [6], which has predicted high m modes. We emphasize that the appearance of the m=4 mode in TC-1 depends decisively on the timing of the capacitor banks, whereas filling pressure plays a less critical role.

Acknowledgements

This work was supported by the Conselho Nacional de Desenvolvimento Científico Tecnológico (CNPq), the Financiadora de Estudos e Projetos (FINEP), and the Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP).

References

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[9] B. Coppi, R. M. O. Galvão, R. Pellat, M. Rosenbluth and P. Rutherford, Sov. J. of Plasma Phys. 2, 533 (1976).

[10] R. M. O. Galvão and M. A. M. Santiago, Phys. of Fluids 24, 661 (1981).

[11] D. O. Campos, M. Machida, S. V. Lebedev, M. Kantor, S. A. Moshkalyov and L. A. Berni, Braz. J. Phys. 26, 747 (1996).

[12] A. L. Hoffman, J. T. Slough and D. G. Harding, Phys. of Fluids 26, 1226 (1983).

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