Theory of ion dynamics and heating by magnetic pumping in FRC plasma
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 PHYSICS OF PLASMAS 25, 072510 (2018)
Theory of ion dynamics and heating by magnetic pumping in FRC plasma
J. Egedal,1 H. J. Monkhorst,2 E. Lichko,1 and P. Montag3
1Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA 2TAE Technologies, Inc., Foothill Ranch, California 92688, USA
3Department of Physics, MIT, Cambridge, Massachusetts 02139, USA
(Received 25 May 2018; accepted 6 July 2018; published online 26 July 2018)
A new mathematical framework is developed for efficient evaluation of ion heating during magnetic pumping of field-reversed configuration (FRC) plasmas. The topology of ion trajectories is examined through the use of action variables applied to a typical FRC equilibrium. Orbit bifurcations are iden- tified and characterized analytically. These bifurcations are associated with the breakdown of the action variables as adiabatic invariants. The associated pitch-angle-mixing is important to the effi- ciency of magnetic pumping and is included in a new model for the kinetic behavior of the ions dur- ing the heating process. Published by AIP Publishing.
Heating of magnetically confined plasmas to the high ion temperatures required for thermonuclear fusion is a major technical challenge. While only the temperature of the ions needs to be high to achieve the desired level of fusion reactiv- ity, the required confinement time is often so long that their interactions with electrons inevitably, and concomitantly, raise the latter temperature to comparable values. This ion- electron interaction enables techniques that primarily heat either electrons or ions, followed by thermal interchange. Many traditional techniques such as high-energy neutral beam injection as well as current drive mostly heat the electrons.1 Common heating methods also include the application of radio frequency waves that interact with plasma ions or elec- trons, somewhat selectively depending on their frequencies. Here, the penetration of the waves into the core of the plasma is a major issue,2 as well as the occurrence of resonances.3–5
The present paper is concerned with the heating of field- reversed configuration (FRC) plasmas through the application of periodic changes of the confining magnetic field. This pro- cess is known as magnetic pumping (MP) and is potentially applicable to the operation of a steady-state plasma fusion reactor. It was first proposed by Schl€uter,6 then analyzed and extended by Berger7 for adiabatic plasmas. Depending on the relative magnitudes of Larmor, collision, transit, and magnetic driving period, the latter work identifies four heating mecha- nisms.2 Given the radical departure of FRC plasma from the main-stream tokamak, mirror or even spheromak plasma con- figurations, it is not clear a priori that the above MP heating results have any bearing on FRC plasmas.
FRC plasmas, although finite in length, contain highly inhomogeneous magnetic structures. Internal field lines are closed, and there are special points where the field vanishes. They often have axial segments with nearly parallel field lines, with a very steep radial behavior, and an r – z magnetic surface on which the axial component reverses sign— whence, the name field-reversed configuration. Thanks to its good stability, confinement, and technical properties, it is the configuration of choice in developing a fusion reactor by TAE Technologies, Inc.8
The effectiveness of magnetic pumping requires that the ions are subject to a mechanism which isotropizes their distri- bution function. For example, estimates for the effectiveness of MP through collisional isotropization are readily obtained.7,9 Meanwhile, the FRC configuration includes a sig- nificant number of confined ion orbits for which the magnetic moments are not adiabatic invariants. These orbits rapidly iso- tropize the ion distribution function rendering MP potentially attractive for heating of an FRC reactor. The intrinsic pitch angle mixing of the non-adiabatic FRC ions also appears well suited to explain the results of the FRC Injection Experiment (FIX) experiments of MP heating of FRC plasmas in Yamanaka et al.10 A striking finding in these experiments is the negligible heating of electrons during their long-pulse shots, interpreted as being representative of MP.
The starting point for our analysis is the recent study by Lichko et al.,11 which applied MP as a means of heating the solar wind. In generalizing this framework to the FRC geom- etry, we will only consider “slow” MP, xzs=vth 􏰅 1, with x being the MP frequency, zs being the FRC length, and vth being the ion’s thermal speed, indicating that typical ions complete many bounces during each pump cycle time 2p/x. Using the methods of multiple time scales,12 the ion distribu- tion function, f, will be approximately constant along bounce orbits. Given that these orbits can be characterized by their constants of motion (COM) variables, f can then be expressed as a function of these COM variables. Note that this approach eliminates all transit-time effects considered by Berger,7 but pitch angle mixing intrinsic to FRC orbits permits MP to remain efficient in this low frequencies limit. This lower pump frequency is important as it renders MP well suited for practical implementations.
The paper is organized as follows: In Sec. II, we discuss the properties of ion orbits in the FRC equilibrium. Distinct FRC orbit topologies are characterized in a parameter space of adiabatic invariants, and methods are obtained for evaluat- ing changes in ion speed for general magnetic perturbations. A kinetic formulation of MP is provided in Sec. III including a derivation and evaluation of a pitch-angle-mixing operator applicable to ions undergoing bifurcations in the topology of
1070-664X/2018/25(7)/072510/18/$30.00 25, 072510-1 Published by AIP Publishing.

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