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PHYSICS OF PLASMAS 24, 012502 (2017) Equilibrium properties of hybrid field reversed configurations
M. Tuszewski,a) D. Gupta, S. Gupta, M. Onofri, D. Osin, B. H. Deng, S. A. Dettrick, K. Hubbard, H. Gota, and TAE Team
Tri Alpha Energy, Inc., P.O. Box 7010, Rancho Santa Margarita, California 92688, USA
(Received 4 October 2016; accepted 2 December 2016; published online 5 January 2017)
Field Reversed Configurations (FRCs) heated by neutral beam injection may include a large fast ion pressure that significantly modifies the equilibrium. A new analysis is required to characterize such hybrid FRCs, as the simple relations used up to now prove inaccurate. The substantial contributions of fast ions to FRC radial pressure balance and diamagnetism are described. A simple model is offered to reconstruct more accurately the equilibrium parameters of elongated hybrid FRCs. Further modeling requires new measurements of either the magnetic field or the plasma pressure. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4972537]
I. INTRODUCTION
A Field Reversed Configuration (FRC) is a very high beta compact torus formed without toroidal magnetic field.1,2 The equilibrium properties of elongated thermal FRCs con- fined inside long cylindrical flux conservers can be estimated to good accuracy with simple analytical formulae. The main FRC parameters are derived as functions of time, and are included in experimental databases soon after each dis- charge. Some FRC confinement properties are also included in the databases, as derived from global analyses3–5 based on the above formulae.
Powerful neutral beams have been used recently in the C- 2 and C-2U devices to heat and sustain FRCs.6,7 These plas- mas are hybrid FRCs that have comparable thermal and fast ion pressures. The fast ions modify significantly the FRC equilibrium, and the simple relations used up to now are inac- curate. A new analysis is required to infer the equilibrium properties of hybrid FRCs from available measurements.
Numerical simulations coupled to Monte Carlo codes have been developed to model hybrid FRCs.8,9 These calcu- lations are sophisticated tools that can be used to model a posteriori selected FRC data. A relatively fast interpretative FRC analysis has been recently proposed,10 but this model does not include fast ions. A simpler analysis of hybrid FRC equilibrium would be useful to compile more accurate data- bases of all FRC discharges. The purpose of the present paper is to motivate and start development of such a model.
Standard FRC analysis, suitable for experiments with- out neutral beam injection, is briefly reviewed in Section II. The main effects of fast ions on the equilibrium of hybrid FRCs are presented in Section III. A simple model of elon- gated hybrid FRCs is described in Section IV. The main results are discussed in Section V, and they are summarized in Section VI.
II. STANDARD FRC ANALYSIS
After formation, most FRC parameters decay on relatively long transport timescales. Meanwhile, radial and axial FRC equilibria require only a few Alfven transit times. For example,
a)Author to whom correspondence should be addressed. Electronic mail: mgtu@trialphaenergy.com
decay times are a few milliseconds in the C-2 device, while Alfven transit times are a few microsecond. Hence, the FRC remains in equilibrium during its decay phase.
Three simple relations are approximately valid at the midplane (z 1⁄4 0) of elongated, purely thermal FRCs confined inside long cylindrical flux conservers.1 These relations per- mit to evaluate many FRC equilibrium parameters as func- tions of time with just a few measurements.
First, radial pressure balance is approximately pþB2/ (2l )1⁄4B 2/(2l ), where p is the plasma pressure and B is
1070-664X/2017/24(1)/012502/7/$30.00 24, 012502-1
Published by AIP Publishing.
0w0
the axial (z) magnetic field. This relation holds at any radial location. The constant right hand side is evaluated at the flux conserver (r 1⁄4 rw), where p is zero. At the FRC field null (r 1⁄4 R), one obtains
kTR1⁄4 Bw2=ð2l0neRÞ;
(1)
where TR is the total (T 1⁄4 Te þ Ti) temperature and where ne 1⁄4 ni is assumed. TR can be evaluated from Eq. (1) in most FRCs experiments, since n and B are usually available
eR w
from multi-chord side-on interferometry and from a wall
magnetic probe, respectively.
Second, axial balance between field-line tension and
FRC plasma pressure yields the midplane “average beta condition”11
hbi 1⁄4 1–ðrs=rwÞ2=2; where b 1⁄4 2l0p/Bw2 is the external plasma
(2)
that Eq. (2) yields values of hbi close to unity.
Third, the separatrix radius rs can be estimated from
excluded flux measurements. The excluded magnetic flux is Ð2
beta, rs
FRC separatrix radius, and hbi is the volume-averaged exter- nal beta value. Typical values of rs/rw are about one half, so
DU 1⁄4 prw Bw Uw, where Bw is measured by a probe and Uw 1⁄4 B2prdr is the magnetic flux measured by a loop, both located at r 1⁄4 rw. The excluded flux radius is defined1,11 as rDU 1⁄4 (DU/p/Bw)1/2. The integral in Uw is from r 1⁄4 rs to r 1⁄4 rw for an FRC. One can approximate Uw 1⁄4 p(rw2 rs2)Bw, for an elongated FRC with negligible plasma pressure outside of the separatrix, to obtain
is the
r1⁄4r : s DU
(3)