Tomographic imaging system for measuring impurity line emission in a field-reversed configuration
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 PHYSICS OF PLASMAS 21, 022506 (2014) Rotational stability of a long field-reversed configuration
D. C. Barnesa) and L. C. Steinhauer
Tri Alpha Energy, Rancho Santa Margarita, California 92688, USA
(Received 4 December 2013; accepted 30 January 2014; published online 18 February 2014)
Rotationally driven modes of long systems with dominantly axial magnetic field are considered. We apply the incompressible model and order axial wavenumber small. A recently developed gyro-viscous model is incorporated. A one-dimensional equilibrium is assumed, but radial profiles are arbitrary. The dominant toroidal (azimuthal) mode numbers ‘ 1⁄4 1 and ‘ 1⁄4 2 modes are examined for a variety of non-reversed (B) and reversed profiles. Previous results for both systems with rigid rotor equilibria are reproduced. New results are obtained by incorporation of finite axial wavenumber and by relaxing the assumption of rigid electron and ion rotation. It is shown that the frequently troublesome ‘ 1⁄4 2 field reversed configuration (FRC) mode is not strongly affected by ion kinetic effects (in contrast to non-reversed cases) and is likely stabilized experimentally only by finite length effects. It is also shown that the ‘ 1⁄4 1 wobble mode has a complicated behavior and is affected by a variety of configuration and profile effects. The rotationally driven ‘ 1⁄4 1 wobble is completely stabilized by strong rotational shear, which is anticipated to be active in high performance FRC experiments. Thus, observed wobble modes in these systems are likely not driven by rotation alone. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4865409]
I. INTRODUCTION
Long systems with dominantly axial magnetic field have small magnetic shear and are susceptible to instabilities driven by rotation, because there is little field line bending associated with radial plasma displacements. This issue received much attention earlier.1–6 The most definitive study of such modes in the “theta pinch” case was carried out by Freidberg and Pearlstein.7 The equilibria studied in Ref. 7 are those with a main (axial) magnetic field which varies strongly with cylindrical radius because of high plasma b, but does not reverse direction. Seyler8 subsequently consid- ered resonant ion kinetic effects, and showed instability extends beyond thresholds computed with the earlier finite Larmor radius (FLR) models.
Modes associated with rotation were observed in the earliest field reversed theta pinch experiments,9 and were soon found to be the limit of lifetime of field reversed config- urations (FRC’s). Thus, a number of theoretical studies were carried out. The magnetohydrodynamic (MHD) model with rotation was applied by Ishimura,10 and by Spencer and Tuszewski.11 Harned carried out hybrid particle in cell (PIC) simulations12 of this instability in the FRC case in which there is no axial variation of either the equilibrium or unsta- ble plasma motion. As noted by Tuszewski in his review,13 Harned’s results differed qualitatively from the earlier non- reversed predictions of Ref. 7, and the discrepancy was not completely explained at that time.
While the bulk of the FRC work focused on the ‘ 1⁄4 2 (‘ is azimuthal or toroidal mode number in this work) elliptical distortion of the column which is most troublesome, the ‘ 1⁄4 1 wobble is also of interest in other situations in which the ‘ 1⁄4 2 is well controlled, and was also of primary interest in
a)Email: coronadocon@msn.com
the earlier non-reversed studies. Consequently, both classes of these modes are considered in the present work.
Since the time of the earlier studies, there has been no study of FRC rotational modes using a fluid model with ki- netic ion closure. The difficulty was to obtain a suitable FLR closure for the case of magnetic field passing through zero, leading to a region of unmagnetized ions. This difficulty has recently been resolved14 by dividing the ions into those which are magnetized and those which are unmagnetized and applying an appropriate kinetic calculation to both classes to obtain a FLR type closure appropriate for the low- collisionality limit realized in modern FRC experiments.
In the present work, a comprehensive fluid kinetic ion closure model is developed and applied to FRC rotational instabilities. The calculations are simplified considerably by restricting at the outset attention to incompressible plasma motion. While such a model might be suspect at the extremely high b considered here, Freidberg15 has shown recently that corrections associated with compressibility are ignorable under a broad range of circumstances, including those of the present considerations.
Appropriate to the long equilibria of interest, we con- sider perturbations of an infinitely long FRC equilibrium, and include finite length effects through an axial wavenum- ber k, which is ordered small (as subsequently quantified). After deriving the governing differential equation and boundary conditions to determine unstable modes, we apply this theory to previous and new situations. Application to previous situations validate the model. The non-reversed results of Ref. 7 are reproduced semi-quantitatively. New results for the FRC reproduce the results of Ref. 12.
Extensive results for the FRC are obtained. First, the effects of finite k on the results of Ref. 12 are obtained. The rigid-rotor profiles of all previous studies are extended by consideration of other profiles. First, retaining rigid ion rota- tion, the electron rotation profiles are modified to incorporate
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