The Field-Reversed Configuration (FRC) is a high-beta compact toroidal in which the external field is reversed on axis by azimuthal plasma The FRC is

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1 and Stability of Field-Reversed Equilibrium with Toroidal Field Configurations Atomics General Box 85608, San Diego, California P.O. APS Annual APS Meeting of the Division of Plasma Physics 42st City, October 23-28, Québec Yu. A. Omelchenko, M. Schaffer, P. Parks

2 The Field-Reversed Configuration (FRC) is a high-beta compact toroidal in which the external field is reversed on axis by azimuthal plasma The FRC is primarily confined by poloidal fields. The possibility of the of self-generated toroidal field inside the separatrix has been presence ignored in the previous stability analysis of such configurations. largely studies have not identified a physical mechanism that would Those Recent FRC formation and stability simulations [1] performed with a hybrid, PIC code FLAME [2] for a limited set of toroidal modes 3-D, Here we shift focus to demonstrating the enhanced stability of the FRCs toroidal fluxes to a number of kink modes, including the most with Abstract plasma current. explain the robust tilt stability of experimentally produced FRCs. (m = 0; 1) revealed the important role of toroidal field effects in mitigating FRC tilting. disruptive tilt mode. GENERAL ATOMICS

3 toroidal self-field in FRC-like compact toroids (CTs) as generating both in theoretical [5],[1] and experimental [6],[7] studies. shown Introduction The FRC kink instability (including the m=1 tilt) has been studied in MHD and kinetic regimes under the assumption of no toroidal both self-field inside the separatrix. MHD theory predicts FRC confinement degradation and destruction on Alfven timescale. Stabilization by profile shaping has been shown to the be eventually ineffective [3]. Kinetic simulations of FRC stability employ hybrid schemes but use a single-fluid Grad-Shafranov equilibrium (assuming traditionally = 0) as a starting point [4]. These studies focus only on the kinetic B resulting predominantly from large orbit phase mixing. effects This work examines the importance of the Hall term normally ignored in MHD description. This is the most significant mechanism in the GENERAL ATOMICS

4 is a hybrid, PIC code created to simulate plasma phenomena in the FLAME cylindrical (r; ;z) geometry. It follows low-frequency, quasi-neutral 3-D, motions described by the Maxwell equations in the Darwin plasma approximation: ion species are represented by full-orbit macro-particles and the Multiple electrons are modeled as a massless collisional fluid (Ohm's law): plasma Simulation = cr E; r B = 4ß c (j e + j i ) = j e + j e B E e c en rp e ; = 4ß(ν ei + ν en )! 2 : (2) pe en e 1 1 fl = fl 1 r(p ev e ) p e rv e + Q e ; Q e = j 2 e: (3) this work we ignore the electron thermal effects (p e = 0). In GENERAL ATOMICS

5 @B ^ r 1 = e en h c ( 4ß r B j i) B crp e the axisymmetric case ( = 0, B = rψ r + I r ; I = rb ) there are In physical drivers of B, i.e. the Hall effect (nonuniform poloidal B), several h r Λψ ψ; i r 2 h ri ; + predominant mechanism in generating B is through the Hall term. It The as scales v Hall v A B -Generation = 0) i nonlinear terms and ion/electron thermal = r2 n e r 2 I e r 2 n + S i + S e (5) where [X; Y ] = (rx r ) ry; r Λ = r 2 (6) cb ο c=! pi A 4ßenRv r ci R ' ' (7) R electron pressure effect is small: S e ο ^ (rn e rt e ) ο 0. The GENERAL ATOMICS

6 Axisymmetric FRCs with different values of parameter s Λ (the number ion gyroradii between the O-point and the separatrix) are formed by of Initialize a distribution of ion macro-particles with uniform 1. and density profiles. temperature Apply an external electric field in the -direction. Run the code until 2. field reversal on axis takes place. the Turn off the applied electric field, decrease the plasma resistivity and 3. the code until an approximate equilibrium is reached. run FRCs obtained in this work are similar to ones produced by simulating imploding -pinch discharge [1]. The FRC parameters used in this an FRCs with the same s Λ are generated twice: (1) by imposing B = 0; (2) solving the model equations self-consistently. by FRC Formation Setup applying a new numerical technique : study are summarized in Table 1. GENERAL ATOMICS

7 The simulation is run long enough to demonstrate nonlinear 1. of initially unstable kink modes. saturation Each self-consistent stability run is accompanied by a "reduced" 2. with B = 0. The comparative analysis of these runs is simulation in understanding the role of B in preventing rapid FRC helpful termination. 3-D FRC Equilibrium Setup Axisymmetric FRCs are projected into 3-D space by distributing the and particle inventory uniformly in the -dimension. The m» 4 field toroidal modes are currently enabled in the system. An initial m = 1 perturbation is applied to ion axial velocities (of order A ). Its amplitude is chosen to be proportional to ψ 2 (it vanishes at 0:02v the separatrix, ψ = 0). The purpose of this study is twofold: GENERAL ATOMICS

8 s Λ = 4 s Λ = 10 Run Radius 15 cm 15 cm Wall B z 1:8 kg 5 kg Background Density cm 3 0: cm 3 Peak Temperature 160 ev 200 ev Ion Length 40 cm 40 cm Separatrix Radius 10:5 cm 10 cm Separatrix Radius 7:5 cm 7 cm O-Point 2 2 Elongation Gyroradius 0:7 cm 0:3 cm Ion Velocity 1: cm=s 1: cm=s Alfven Growth Time 1:6 μs 1:3 μs MHD Size (z; r; m) Grid PIC (3-D) 700; ; 000 N Table 1 (Summary of FRC Runs)

9 We focus on the enhanced stability properties of FRC-like with a certain amount of toroidal field generated during configurations The most significant effect in generating toroidal field is the Hall effect. terms of the MHD energy principle this corresponds to adding In Since the Hall term tends to bend field lines predominantly in the direction (in contrast to Alfven-driven plasma motion), this toroidal not result in increasing poloidal field curvature. Therefore, this does is always stabilizing. effect The magnitude of the Hall effect scales as r ci =R (R is the characteristic gradient length). Thus, the self-generated toroidal field should magnetic FRC Global Stability I (Theory) the FRC formation. another stabilizing field-bending term to the energy equation. be an important stabilizing factor in low-s (kinetic) plasmas. We also explore stability properties of "fusion-scale" FRCs (s Λ ο 10). GENERAL ATOMICS

10 run. Time histories of magnetic field modes self-consistent 5(a,b,c)) corraborate this finding. (Fig. FRC Global Stability II (Results) All simulations have been run long enough to demonstrate the nonlinear of the kink/tilt instability through kinetic phase mixing and saturation FRC shape relaxation. As shown in Fig. 1(a,b) the low-s self-consistent FRC does not exhibit confinement degradation during the instability. significant Fig. 2(a,b) illustrates the dynamics of the "zero-b " kinetic FRC. Its poloidal structure undergoes more deformation compared to the internal The "fluid-like" FRCs are shown in Fig. 3(a,b,c) and Fig. 4(a,b). The FRC is still more robust in conserving its poloidal flux. self-consistent In all cases the kinks are stabilized by virtue of ion phase mixing in the dimension and FRC plasma axial expansion (Fig. 7). toroidal GENERAL ATOMICS

11 The kink modes are the most disruptive toroidal modes that can cause FRC confinement degradation. The ideal MHD theory predicts rapid FRC termination. On the contrary, this study demonstrates the fast robustness with respect to generating such modes, including the FRC The FRC tilt stabilization is achieved through nonlinear kinetic ion (toroidal phase mixing) resulting in an FRC plasma relaxation motion The tilt-driven confinement degradation is effectively prevented in the FRC and reduced in the high-s case by virtue of toroidal self-field low-s (the Hall term effect), which increases the CT local helicity generation destabilizing bad field curvature regions. without The results of this study are in good agreement with available evidence of robust kinetic FRC stability and existence of experimental Conclusion most feared tilt mode. in the poloidal plane and expansion in the axial direction. toroidal fluxes in the FRC-like configurations. GENERAL ATOMICS

12 Figure 1a: Poloidal Flux (s Λ = 4, B 6= 0)

13 Figure 1b: Plasma Density (s Λ = 4, B 6= 0)

14 Figure 1c: Toroidal Field (s Λ = 4, B 6= 0)

15 Figure 2a: Poloidal Flux (s Λ = 4, B = 0)

16 Figure 2b: Plasma Density (s Λ = 4, B = 0)

17 Figure 3a: Poloidal Flux (s Λ = 10, B 6= 0)

18 Figure 3b: Plasma Density (s Λ = 10, B 6= 0)

19 Figure 3c: Toroidal Field (s Λ = 10, B 6= 0)

20 Figure 4a: Poloidal Flux (s Λ = 10, B = 0)

21 Figure 4b: Plasma Density (s Λ = 10, B = 0)

22 Figure 5a: Mode Evolution (s Λ = 4, B 6= 0)

23 Figure 5b: Mode Evolution (s Λ = 4, B 6= 0)

24 Figure 5c: Mode Evolution (s Λ = 4, B = 0)

25 Figure 6a: Mode Evolution (s Λ = 10, B 6= 0)

26 Figure 6b: Mode Evolution (s Λ = 10, B 6= 0)

27 Figure 6c: Mode Evolution (s Λ = 10, B = 0)

28 Figure 7: FRC Axial Expansion

29 Yu.A. Omelchenko and R.N. Sudan, J. Comp. Phys. 133, [2] (1997). 146 E.J. Horowitz, D.E. Shumaker, and D.V. Anderson, J. [4] Phys. 84, 279 (1989). Comp. References [1] Y.A. Omelchenko, Phys. Plasmas 7, 1443 (2000). [3] L.C. Steinhauer, Phys. Plasmas 6, 2734 (1999). [5] D.W. Hewett, Nucl. Fusion 24, 349 (1984). [6] K. Wira and Z.A. Pietrzyk, Phys. Fluids B 2, 561 (1990). [7] G. Durance and I.R. Jones, Phys. Fluids 29, 1196 (1986).

30 FIGURE CAPTIONS Figure 1: Time evolution of the self-consistent kinetic FRC: (a) poloidal flux function ψ; (b) plasma density; (c) toroidal field. Figure 2: Time evolution of the "zero-b " kinetic FRC: (a) poloidal flux function ψ; (b) plasma density. Figure 3: Time evolution of the self-consistent fluid-like FRC: (a) poloidal flux function ψ; (b) plasma density; (c) toroidal field.

31 Figure 4: Time evolution of the "zero-b " fluid-like FRC: (a) poloidal flux function ψ; (b) plasma density. Figure 5: Time histories of magnetic energy modes of the kinetic FRC: (a) poloidal field; (b) toroidal field; (c) B = 0. Figure 6: Time histories of magnetic energy modes of the fluid-like FRC: (a) poloidal field; (b) toroidal field; (c) B = 0. Figure 7: Time history of FRC plasma axial length evolution. The kink instability saturates through FRC axial expansion.

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