Onset and saturation of ion heating by odd-parity rotating-magnetic-fields in a field-reversed configuration Alexandra S. Landsman* Department of Physics, ETH Zurich Samuel Cohen Princeton Plasma Physics Lab Alan Glasser Los Alamos National Lab Heating of figure-8 ions by odd-parity rotating magnetic fields (RMFo) applied to an elongated field-reversed configuration (FRC) is investigated. The largest energy gain occurs at resonances (s ≡ ωR/ω) of the RMFo frequency, ωR, with the figure-8 orbital frequency, ω, and is proportional to s^2 for s - even resonances and to s for s - odd resonances. The threshold for the transition from regular to stochastic orbits explains both the onset and saturation of heating. The FRC magnetic geometry lowers the threshold for heating below that in the tokamak by an order of magnitude. Heating, i.e., stochastic energy gain, of charged particles by time-varying fields is a complex and fundamental phenomenon critically important to as diverse areas of plasma physics as fusion research[1] and plasma processing. Well known are the effects of simple resonances and particle collisionality on the heating of magnetized plasmas. Far less well explored is the role of an inhomogeneous static magnetic-field geometry. Because of its relevance to space plasmas[3], plasma processing[4], and magnetic-confinement controlled-fusion research[5], the field-reversed configuration (FRC, see Fig. 1) – with its poloidal field nulls, lack of toroidal field, and strong field gradients – is an important system in which to explore the effects of magnetic field geometry on particle dynamics under the influence of time-varying fields. Even with axial symmetry, a static FRC allows charged-particle orbits that are regular or ergodic[11]. First studies of single-particle orbits in FRCs assumed time invariance and spatial symmetries that reduced the problem to one or two dimensions, allowing Kolmogorov-Arnold-Mosher (KAM) surfaces to exist[2] and limiting excursions in phase space. The addition of a rotating magnetic field (RMF)[11] breaks the angular invariance of the FRC, creating a three-dimensional system without bounding KAM surfaces and opening the possibility for large excursions in phase space and energy. These excursions can have beneficial results, such as ion heating[7], or detrimental ones, such as loss of confinement. In this paper we present studies of ion orbits in FRCs with RMF applied: the goal is to understand the threshold for chaos and the role of resonances in the non-linear growth and subsequent saturation of ion energy. We restrict attention to the novel odd-parity RMFs (RMFo) because of field closure and encouraging recent experimental results. We show that the same mechanism is responsible for the initial ion heating and its ultimate saturation. Studies of stochastic ion heating by perpendicularly propagating electrostatic waves in tokamaks were performed with similar Hamiltonian techniques and research goals. The results we report are markedly different because of fundamental differences in the magnetic field geometry of the two devices. Earlier papers [7], which used the RMF numerical code to investigate RMFos applied to FRCs, showed that the relevant frequency range for ion heating was broad, |Ω| ~ 0.2 - 2, where Ω ≡ ωR/ωci, ωR is the RMFo frequency, ωci = qBa/mc is the ion-cyclotron frequency in the axial field at the FRC's center, Ba, m is the ion mass, and q is the ion charge. These papers reported significant ion heating even for low relative RMF amplitude, BR: BR/Ba ~ 5 × 10^-4. Phase de-coherence of ion orbits, with respect to the periodic electric fields created by the RMFo, is a necessary condition for ion heating. Strong gradients and regions of field reversal in the FRC provide locations for possible phase de-coherence. For a 10-cm FRC having an ion density of 10^14 cm^-3 and an ion energy of 100 eV, Coulomb collisions will be 10× less frequent than the stochastic effects described herein[7]. The question arose whether, in spite of the existence of strong field gradients, ion-cyclotron resonances (ICRs) were important to ion heating. We show that ICRs are important, but with significant differences from the standard ICR picture. More rapid heating occurs at low BR/Ba for figure-8 orbits (see Fig. 1b)) than for cyclotron orbits, though the latter have a more clearly resonant interaction with RMFo. Figure-8 orbits cross the field-reversal and strong-gradient regions (twice) every orbit cycle, possibly losing phase coherence at each traversal. In contrast, cyclotron orbits may only incur phase de-coherence at the less frequent excursions to the axial extremes of their orbits. Betatron orbits have a less non-linear nature and hence are also less well heated than figure-8 orbits. Because figure-8 orbits are representative of a large fraction of ions in hot fusion FRC plasmas and because they represent the physically interesting situa- tion of motion in a double potential well[10], we focus on them. The studies presented herein also clarify why high-energy orbits tend to interact regularly with RMFo, leading, importantly, to a saturation of ion heating by RMFo and a method for tuning ion energy. We follow Ref. [7] by using the same equations for the RMFo and a Solov’ev equilibrium for the FRC, with the notation, see Fig. 1a): R = FRC separatrix radius in midplane; Z = FRC axial half length; κ ≡ Z/R, FRC elongation; r = radial coordinate; z = axial coordinate; φ = azimuthal coordinate; pi = canonical momenta; P ≡ 2pφ/qBaR^2, normalized pφ; A = vector potential of RMF and FRC; ψ = φ − ωRt; k = lπ/κR = axial wave number of the RMFo; l = axial mode number; and time, τ, in units of 2π/ωci. The shape of the effective potential-energy surface on which an ion moves depends on P and z: figure-8 orbits may be confined to the z = 0 subspace, or to a potential-well minimum above or below z = 0, or may oscillate across z = 0. Orbits confined to the z = 0 subspace are amenable to an analytic analysis and are the appropriate choice to analyze because of the RMFo's electric fields, Er and Eφ, there. Each cross-section in z is either a double potential well, allowing both cyclotron and figure-8 orbits, or a raised potential, corresponding to betatron orbits. Cyclotron orbits feel a force towards larger |z|, thus eventually enter a region where the barrier between the double wells is low enough for them to traverse, thereby becoming figure-8 orbits. Since cyclotron orbits interact regularly with RMF, except at these axial extremes, their random fluctuations in energy appear less frequently than for orbits which are always figure-8. Moreover, figure-8 orbits have greater radial excursions, hence gain more energy from the radial electric field of the RMFo. It follows that the heating of figure-8 orbits is an upper limit for the heating of all ions in the FRC and that the threshold for heating is highest in the z = 0 subspace. Extensive numerical simulations, performed with the RMF code, confirmed this. We first examined whether the broad Ω range for heating is due to resonances at the fundamental ICR frequency. As described below, the answer is no. Instead high-harmonic resonances occur because the frequency of the figure-8 orbit is highly nonlinear. (High-harmonic resonances have recently been observed in an RMF experiment.) As the energy of a figure-8 orbit decreases, the ratio s ≡ ωR/ω increases because the ion's frequency, ω, slows down as it gets closer to the phase-space separatrix created by the hump in the double potential well. A set of resonances with the RMFo occurs at integral values of s. Figure 2a) shows the RMF-code-calculated time dependence of ion energy for two values of BR for a 1-keV ion initiated in a figure-8 orbit with zero axial velocity in the z = 0 subspace of an FRC having R = 10 cm, κ = 5, Ω = 0.9, and Ba = 20 kG. Regular motion, with a clear s = 5 component, is seen for BR = 2 G. The energy fluctuations are small (~ 15%) for BR = 2 G and large, > 100%, for BR = 20 G. For BR = 2 G, the fast Fourier transform (FFT) of ion energy, Figure 2 b), shows sharp peaks in frequency space, indicative of regular motion. The separation between peaks, Δf, is 0.207 ± 0.002, in units of ωci. For BR = 20 G, the FFT shows broadband noise at a 30× higher absolute level. Under these conditions, betatron (also shown in Fig. 2b)) and cyclotron orbits (not shown) display regular motion — sharp peaks in their FFTs — with energy fluctuations less than 7%, even for BR = 20 G. Figure 2c) shows Δf vs Ẽ, initial energy normalized to En ≡ q^2 Ba^2 R^2 / 2m, for three values of initial P and low BR. Below P = 0.25 orbits may be cyclotron or figure-8; above P = 0.25 orbits are betatron. A logarithmic drop in Δf is seen for P = 0.1 and 0.2 at an energy corresponding to the phase-space separatrix energy, at the transition of cyclotron into figure-8 orbits. Little change in Δf occurs for betatron orbits. In heating, the variance of energy, and therefore the maximum energy, Emax, will increase with time. Figure 3a) shows Emax attained by an initially figure-8 orbit as a function of time for four BR values and the same FRC parameters as in Fig. 2. Emax displays saturation behavior quickly, implying phase coherence growing with increasing energy. The threshold for heating is at BR ~ 3 G, above which Emax grows ∝ BR^1.5. Figure 3b) compares Emax vs BR for figure-8 (P = 0.22), betatron (P = 0.26), and cyclotron (P = 0.19) orbits initiated in the z = 0 subspace at the same radial position, r/R = 0.8131, and the same energy, 1000 eV, for a simulation time of τ = 10^4. Heating of figure-8 orbits occurs at lower BR than for cyclotron or betatron orbits. At BR ~ 20 G, FFTs of energy for figure-8 ions initiated at higher energy, E > 0.03En ~ 30 keV, in the z = 0 subspace show sharp peaks and little further gain in energy. The regular interaction with RMF0 of these higher energy figure-8 orbits may be understood by the greater separation between resonances in phase-space[6, 9] with increasing energy (or ω). We have calculated the RMFo-induced energy gain of a figure-8 orbit in a single half period of its motion as a first-order correction to the one-dimensional motion along r. For P < 0.25, the shape of the effective potential, V(r), is a double well[11], corresponding to cyclotron orbits (in either well) inside the phase-space separatrix and to figure-8 orbits (moving across both wells) outside the phase-space separatrix. The figure-8 orbit is approximated by motion in a symmetric double well: ρ ≡ r/R = ρ0 + a1 cos[ω(t − t0)] + a2 cos[3ω(t − t0)], with a1/a2 ~ 10. The amplitudes of oscillation, a1 and a2, are determined by the total energy. The energy change from the interaction with the field is given by dH/dt = qE⃗ · v⃗ = q(Ervr + Eφvφ), with cE⃗ = −∂A⃗/∂t. After some algebra and integrating over a single half oscillation we get the energy gain from the interaction with the Er and Eφ components of the RMF. The biggest en- ergy change occurs for a resonance between the RMFo and the Fourier components of the orbital motion, i.e., ωR/ω = s, where s is an integer. The Er- and Eφ-induced radial and azimuthal portions of the energy change (ΔE) in a single half oscillation for an s resonance are: ΔEr = H0 [ ∑_{n=1, n≠s}^9 Fn(ω) cos(Ψ0) + (F̂0 + Fs) sin(Ψ0) ] (1) ΔEφ = H0 [ ∑_{n=0, n≠s}^8 Qn(ω) cos(Ψ0) + Qs(ω) sin(Ψ0) ] (2) where Fn(ω) = Cn [(-1)^{s+n} - 1] n / (s^2 - n^2), F̂0 = - (C0/s) ((-1)^s - 1), Fs = (π/2) Cs, Qs(ω) = (π/2) (ωci/ω) Ks, Qn(ω) = Kn [(-1)^{s+n} - 1] (s / (s^2 - n^2)) (ωci/ω), Ψ0 = φ - ωRt0, t0 is the initial time, H0 = mkR^3 ωci ωR BR / 2Ba, and the Cn depend on ρ0, a1 and a2. Since the total energy is H ~ (1/2)m(Rωa1)^2 and |Ω| ~ 1, the relative fluctuations in energy during an oscillation are of order, max ΔEodd/H ~ O(10^2 s BR/Ba), and max ΔEeven/H ~ O(10 s^2 BR/Ba). These predict significant energy gain for figure-8 orbits over a single oscillation, even for a relatively low amplitude RMF, BR/Ba ~ 10^-3. The energy gain for s - even resonances has an s^2 dependence while s - odd energy gain has a linear dependence on s. Resonances with an odd value of s show better heating than s - even resonances, especially at lower values of s, where the ion energy is higher. Thus, the heating observed for figure-8 orbits at higher energies results primarily from an overlap of odd-s resonances. Using the condition for exponential separation of trajectories[12], we now determine the threshold for the ergodicity of ion trajectories, essential to convert energy gain to stochastic heating. The change in energy over an oscillation is used to map the dynamics: Ej+1 = Ej + ΔE(tj); tj+1 = tj + π / ω(Ej+1) where tj is the time of the start of successive ion oscillations at ρ = ρmax ≡ ρ0 + a1 and ΔE(tj) is ΔEr(tj) + ΔEφ(tj), with the substitutions Ψ → Ψj, Ψj = φ - ωRtj, and ΔE → ΔE(tj). The dynamics will be chaotic if exponential separation of trajectories, i.e., K > 1, occurs, where K = max |dtj+1/dtj - 1|. In dimensionless variables, Ẽ = (m/b^2 R^2)E and ω̃ = mω/b = 2ω/ωci, where b = qBa/2c. K for odd and even resonances are: Kodd ≈ 8πs (1 / kR) (BR / Ba) (dω̃(Ẽ) / dẼ) (3) Keven ≈ (π/2) s^2 (kR) (BR / Ba) (dω̃(Ẽ) / dẼ) (4) Based on these, increasing the axial wavenumber, k, of the RMFo should lower the chaos threshold for s - odd resonances while raising the threshold for s - even resonances. This is borne out by numerical simulation. Fig. 3c) shows dω̃(Ẽ)/dẼ vs. Ẽ for figure-8 orbits having P = 0.15. At energies very close to the separatrix, dω̃(Ẽ)/dẼ grows as (Ẽ - Ẽh)^(-5/6), where Ẽh is the energy at the phase-space separatrix. The large growth of dω̃(Ẽ)/dẼ near the separatrix corresponds to a large increase in non-linearity of the figure-8 orbital frequency. The greatest rate of stochastic heating is expected to occur for lower energy figure-8 orbits where the values of s and dω̃(Ẽ)/dẼ are higher. As BR is increased, the stochastic region above the phase-space separatrix will broaden. Close to the separatrix, even very low amplitudes of BR should produce chaotic orbits. Eqns. 3) and 4), combined with Fig. 3c), can be used to estimate the relative amplitude of BR needed to produce stochasticity and heating. For example, s = 3 resonance occurs at Ẽ ≈ .0185, corresponding to dω̃(Ẽ)/dẼ ≈ 25, see Fig. 3c). Using Eqn. 3) and kR ~ 1, chaotic trajectories are to be expected for all s - odd resonances with s ≥ 3 and BR/Ba ≥ 5 · 10^-4, for Ω ~ 1, the assumption used in the derivation. These findings approximately agree with the numerical findings. Changing the value of a P changes the scale of Ẽ, but does not have a substantial effect on the value of dω̃(Ẽ)/dẼ at different resonances. Thus P determines the energy range over which figure-8 orbits get heated, with greater energy range for lower values of P, while not affecting the approximate structure of phase space. In Fig. 3c), all s > 4 resonances are located to the left of Ẽ ≈ 0.013, hence occur over the interval δẼ ~ 0.003. This leads to much greater chaos closer to the phase-space separatrix where the closely spaced resonances overlap. Thus, lower-energy figure-8 orbits are more chaotic and much better heated by the RMFo than the higher energy ones. Fig. 3d) shows this effect for two values of P: 0.17 and 0.2. Figure-8 orbits are not further heated once their energy reaches (or initially exceeds) the curved line appropriate for each P value. The simplifications on which Eqns. 3) and 4) are based become less accurate at BR/Ba > 0.001. Among the clearest differences between these results for the FRC and those reported for the tokamak are: 1) The non-linearities for the FRC arise from the double potential well and field gradient and their direct effects on the particle orbit. Those in the tokamak arise from trapping in the wave field – hence require a stronger wave field – and resonance between the cyclotron motion and the wave field, resulting in a single large resonance and large first-order islands. In contrast, close spacing in phase space between resonances of a figure-8 orbit leads to an overlap between resonances and the observed stochastic heating for figure-8 orbits in FRCs. The importance of the time-varying field, E, in the tokamak analysis is to create a small nonlinearity in this 2-D system (of the order of E/B) which leads to resonances between the two degrees-of-freedom, not between the E field and the ion trajectory. 2) In the tokamak, heating occurs at 10× higher values of Ω (over 20 vs 1 in the FRC) and lower values of ωR/kvi,thermal (~ 1 vs 10). 3) The threshold for heating in the FRC is lower by the factor s dω̃(Ẽ)/dẼ, through which the effect of the FRC's double effective-potential well is clear. In summary, the energy gain in an orbital period due to RMFo was calculated for a figure-8 orbit in an FRC. Resonances of ωR with ω produce significant energy gain. Odd-s resonances more effectively heat for high energy (lower s) figure-8 orbits. The energy gain in a oscillation was used to map the dynamics and a criterion for the exponential separation of trajectories was used to find the threshold for chaotic orbits. K, the measure of the rate of trajectory separation, increases with BR. At higher energies, the orbits are less chaotic due to both a lower value of s and, more importantly, to a decreased nonlinearity reducing dω̃(Ẽ)/dẼ. * [email protected] [1] T.H. Stix. Waves in Plasmas. American Institute of Physics, 1992. [2] A.H. Lichtenberg and M.A. Lieberman. Regular and Chaotic Dynamics. Springer-Verlag, 1992. [3] J. Chen. Nonlinear dynamics of charged particles in the magnetotail. J. Geophys. Res., 97:15011, 1992. [4] I.Yu, Kostukov, and J.M. Rax. Stochastic heating in field-reversed low pressure discharges. Phys. Plasmas, 7:185, 2000. [5] M. Tuszewski. Review Paper: Field Reversed Configurations. Nuclear Fusion, 28:2033, 1988. [6] A.S. Landsman, S.A. Cohen, M. Edelman, and G.M. Zaslavsky. Nonlinear resonance and chaotic trajectories in magnetic field-reversed configuration. Communications in Nonlinear Science and Numerical Simulations, 10:617, 2005. [7] S.A. Cohen and A.H. Glasser. Ion Heating in the Field-Reversed Configuration by Rotating Magnetic Fields near the Ion-Cyclotron Resonance. Phys. Rev. Letters, 85:5114, 2000. [8] S.A. Cohen, A.S. Landsman and A.H. Glasser. Stochastic ion heating in field-reversed configuration geometry by rotating magnetic fields. Physics of Plasmas 14, 072508 (2007). [9] A.S. Landsman. Single ion dynamics inside the magnetic field-reversed configuration. Princeton University Doctoral Dissertation, 2005. [10] A.I. Neishtadt, V.V. Sidorenko, and D.V. Treschev Stable periodic motions in the problem of passage through separatrix Chaos, 7:2-11, 1997. [11] A.S. Landsman, S.A. Cohen and A.H. Glasser. Regular and stochastic orbits in a highly prolate field-reversed configuration. Physics of Plasmas 11, 947 (2004). [12] G.M. Zaslavsky. Physics of Chaos in Hamiltonian Systems. Imperial College Press (1998). FIG. 1. a) Shape of FRC magnetic field in z-r plane. κ = 4.1. b) Shapes of typical cyclotron, betatron and figure-8 orbits in z = 0 plane. FIG. 2. Results of numerical simulation of ion trajectories in an FRC with RMFo. a) Figure-8 ion energy versus time for two values of BR. b) FFTs of ion energy for the two cases shown in a) and also for a betatron orbit. c) Separation between resonances, Δf, versus normalized initial energy for P = 0.1, 0.2 and 0.3, at low BR. FIG. 3. a) Maximum energy attained versus time for four values of BR. b) Maximum energy attained by τ = 10^4 versus BR by three types of orbits. c) dω̃(Ẽ)/dẼ versus Ẽ for figure-8 orbits. P = 0.15. d) Curves: Threshold Ẽ for saturation of stochastic heating versus BR/Ba, based on Eqns 3) and 4): P = 0.17 (upper curve) and P = 0.2 (lower curve); Data points, dots and stars, from RMF-code.