Page 2 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-2 Egedal et al.
Phys. Plasmas 25, 072510 (2018)
their orbit motion. In Sec. IV, the kinetic model is applied to a particular MP scenario demonstrating the potential of this heating technique. The paper is concluded in Sec. V.
II. PROPERTIES OF FAST ION ORBITS IN FRC PLASMA
A. The FRC equilibrium
50
0 -1000
To study the properties of ion orbits in a reactor relevant FRC, we consider the Solov􏰀ev equilibrium13
Wðq; zÞ 1⁄4 qA/ 1⁄4 W0 with the magnetic field
q2! q2 z2!
q2 1 􏰄 q2 􏰄 z2 ; (1)
sss
0 z 1000
given by B 1⁄4 r 􏰅 A, and W0 1⁄4 B0q2s=2. Here, B0 is the magnetic field at q 1⁄4 z1⁄40. Our initial orbit analysis will be limited to a highly elongated
FIG. 1. Solov􏰀ev equilibrium applied in the analysis. The red ion trajectory is calculated for v1⁄43, representative of a 9keV proton. The flux surfaces highlighted in green (W 1⁄4 1173) and blue (W 1⁄4 400) will be considered later when calculating the rates of pitch-angle-mixing for non-adiabatic orbits.
in z. Consider the average magnetic field components hjBzjiq and hjBq jiq , where the averaging is over one fast q-oscilla- tion. Given the constraint of high elongation in Eq. (2), it is clear that hjBz jiq 􏰇 hjBq jiq , and in turn, it then follows that the kinetic energy in the z-direction, / v2z , and the perpendic- ular energy, / v2? 􏰆 v2/ þ v2q, are decoupled at the time scale of a single q oscillation. However, v2z ðzÞ and v2?ðzÞ are cou- pled at the longer timescale characterizing z oscillations, nat- urally obeying v2z þ v2? 1⁄4 v2.
The fast oscillations in the q-direction are well described by the effective potential formalism of St€ormer15 and Landsman et al.16 For this we notice that from p/ 1⁄4 qW þ mqv/, we can solve for v/ as a function of ðq; z; p/ Þ
v/ðq;z;p/Þ1⁄4p/ 􏰄qWðq;zÞ: (3) mq
The effective potential is then given by Veff 1⁄4 mv2/=2.
Because v2? is constant at the time scale of the fast q oscilla-
tions, it follows that the ions are confined to regions where
v2 􏰈v2 , where we now consider v2 1⁄4v2ðq;z;p Þ to be a /? ///
function of q for fixed values of ðz; p/ Þ.
Figures 2(a)–2(d) present the structure of the effective
potential for a fixed value of p/ representative of ions in the bulk of the equilibrium. Panel (d) shows color contours of constant log10 ðv2/ Þ as a function of ðq; zÞ, while the profiles of v2/ ðq; z; p/ Þ in panels (a)–(c) correspond to the vertical cuts marked in panel (d). The profile for the midplane in panel (a) includes a local maximum of v2/ characterized by
@v2/1⁄40; @2v2/<0: (4) @q @q2
Ions with sufficiently small values of v2? can be trapped in the potential wells on either sides of the maximum and fol- low standard cyclotron orbits described by regular guiding center theory. Meanwhile, ions with larger values of v2? are not confined by the local potential wells. Instead, these tra- verse the local wells with bounce points in the regions of q marked c in panel (a) and are denoted by figure-eight orbits (referring to the shapes of their trajectories projected on the xy-plane). The figure-eight orbits can be understood as a
reactor relevant device where
qs
􏰋􏰆z􏰃1: (2)
s
Then, in the later sections, we will generalize the study to include a characterization of pitch-angle scattering that occurs at a finite elongation.
For numerical examples, we will apply a configuration with B0 1⁄4 1 T. The ion mass m is normalized by the proton mass, the ion speed is normalized by that of a 1 keV proton (4:4 􏰅 105 m/s), and q is normalized to the proton charge. Furthermore, t is normalized by 1=xci 1⁄4 m=ðqB0 Þ ’ 1 􏰅10􏰄8 s, and spatial dimensions are normalized by the Larmor radius of a 1keV proton in the B0 magnetic field (qi0 ’ 4:6 􏰅 10􏰄3 m). The equilibrium is consistent with the confinement of ions with energies in the range of 100 keV (corresponding to v 1⁄4 10). The analysis is kept general in terms of the temperature of the ions. Only in the final sec- tion, we introduce particular Maxwellian distributions char- acterized by their thermal speed vth. In turn, this permits the results on MP to be cast in terms of the FRC s-parameter.14
For calculating ions orbits, we integrate the resulting
Newton’s law in dimensionless variables, which simply read
x 1⁄4 x_ 􏰅 B. Furthermore, we will consider a device with a
radius of 0.40 m, such that q 1⁄4 0:40=0:0046 ’ 87, and we s
will use zs 1⁄4 10qs, resulting in the configuration shown in Fig. 1, with the projection of a typical 9keV proton orbit superposed.
B. Effective potential for fast q-oscillations
For a static equilibrium, magnetic fields do no work and the particle speed v is constant at the time scale of a single bounce or collision time. However, dynamic changes of the equilibrium are associated with inductive electric fields which allow the particle energies to evolve at the longer MHD time- scales governing the equilibrium. Meanwhile, for cylindri- cally symmetric systems, the canonical angular momentum, p/ 1⁄4 qW þ mqv/, is an exact constant of motion even for a dynamically evolving equilibrium.
Again, the orbit in Fig. 1 is typical of the FRC, com- prised of rapid oscillations in q and much slower oscillations