Momentum Conservation in Current Drive and Alpha-Channeling-Mediated Rotation Drive
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Momentum Conservation in Current Drive and Alpha-Channeling-Mediated Rotation Drive Ian E. Ochs1, a) and Nathaniel J. Fisch1 Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08540, USA (Dated: 21 January 2022) Alpha channeling uses waves to extract hot ash from a fusion plasma, while transferring energy from the ash to the wave. Intriguingly, it has been proposed that the extraction of this charged ash could create a radial electric field, efficiently driving E × B rotation. However, existing theories ignore the response of the nonresonant particles, which play a critical role in enforcing momentum conservation in quasilinear arXiv:2201.07853v1 [physics.plasm-ph] 19 Jan 2022 theory. Because cross-field charge transport and momentum conservation are fundamentally linked, this non-consistency throws the whole effect into question. Here, we review recent developments that have largely resolved this question of rotation drive by alpha channeling. We build a simple, general, self-consistent quasilinear theory for electrostatic waves, applicable to classic examples such as the bump-on-tail instability. As an immediate consequence, we show how waves can drive currents in the absence of momentum injection even in a collisionless plasma. To apply this theory to the problem of ash extraction and rotation drive, we develop the first linear theory able to capture the alpha channeling process. The resulting momentum-conserving linear-quasilinear theory reveals a fundamental difference between the reaction of nonresonant particles to plane waves that grow in time, versus steady-state waves that have nonuniform spatial structure, allowing rotation drive in the latter case while precluding it in the former. This difference can be understood through two conservation laws, which demonstrate the local and global momentum conservation of the theory. Finally, we show how the oscillation-center theories often obscure the time-dependent nonresonant recoil, but ultimately lead to similar results. I. BACKGROUND AND INTRODUCTION Such a flow of energy from the alpha particle ash into the waves, as in the case of instabilities, is made possi- A deuterium-tritium fusion reaction produces fast neu- ble by the free energy associated with the radial gradient trons and hot helium ash. While the neutrons quickly of the alpha particles, which are born at the hot reactor leave the device, the charged ash remains confined in the core. A suitably chosen wave can tap this free energy reactor for a time. Initially, this confinement is somewhat by pulling ash outward along this gradient, both captur- beneficial, as the ash is extremely hot (3.5 MeV) com- ing the ash energy and removing the cold ash from the pared to the bulk plasma (∼ 20 keV), and thus transfers reactor, accomplishing two goals with a single process. heat to the bulk via collisions. Because this generic process involves channeling energy However, this situation is not ideal for a few reasons. from alpha particles into waves, while also channeling First, the radial gradient of hot alpha particles represents the alpha particles out of the reactor, it has been termed a large source of free energy, which can drive plasma in- “alpha channeling.” stabilities. Second, the hot ash primarily heats electrons. A blueprint for alpha channeling was first proposed Although some of the heat eventually makes its way to by Fisch and Rax2,3 , using the electrostatic lower hy- ions via electron-ion collisions, this energy flow pattern brid (LH) wave. Sufficiently strong LH waves are useful sets up a situation where the electrons are hotter than the for both current drive and fast ion heating, and also in- ions, which increases pressure and radiative energy loss teract strongly with fusion-born alpha particles. While without increasing the fusion power. Third, once the ash we will focus here on alpha channeling via lower hybrid thermalizes, it no longer provides any heat, but quasineu- waves2–6 , it should be noted that one can also make use trality requires that each alpha particle be accompanied of waves in the ion-cyclotron range of frequencies7–18 , by two electrons, further increasing the pressure without and can further optimize the effect by combining multi- additional fusion power. ple waves19–22 . For all of these reasons, it is desirable to quickly ex- Alpha particles interact with LH waves via a Landau tract the alpha particles from the plasma, while redirect- resonance, wherein twice per orbit in a magnetic field, ing their energy into useful work, such as ion heating and the velocity of the alpha particle matches the phase ve- current drive. Since both of these tasks can be accom- locity of the wave (Fig 1a). Karney23,24 showed that for plished by suitably chosen plasma waves, it makes sense sufficiently strong LH waves, this process leads to a diffu- to try to transfer the ash energy into a these waves. sion in energy, with particles randomly gaining or losing energy with each interaction with the wave. Taken alone, i.e. in a homogeneous plasma, this diffusion leads to net transfer of energy from the wave into the particles, which a) Electronic mail: firstname.lastname@example.org is undesirable for alpha channeling.
2 a) b) Bz ΔX vy ky v⟂ FIG. 1. Schematic view of alpha channeling by a lower hybrid wave. (a) As a hot alpha particle orbits in a magnetic field Bz , it becomes Landau resonant with the LH wave twice per orbit. At this resonance point, it receives a “kick,” either gaining or losing energy. Correlated with this change in energy, the center of its orbit moves as well, resulting in a coupled random walk (diffusion) along a 1D path in energy (K) and space (X). (b) By choosing a wave where this diffusion path connects high-energy particles at the tokamak center (an alpha particle source) to low-energy particles at the edge (a sink), alpha ash can be made to diffuse out while transferring its energy into the wave. Figure (b) is adapted from Ref.1 . What Fisch and Rax showed was that this diffusion traction proposal of Fetterman and Fisch must be looked in energy is coupled to diffusion in the position of the at as a hypothesis, since the theory that they built only particle, so that particles which gain energy move in one looked at the response of the Landau-resonant alpha par- direction, while particles that lose energy move in the op- ticles to the wave, without considering the response of posite direction (Fig. 1a). Formally, the overall diffusion the bulk plasma. Thus, embedded in the theory is an thus takes place on a one-dimensional path in energy- assumption: the bulk particles will not move across mag- position space. Thus, by creating a path that connects netic field lines in response to the wave, leaving only the high-energy particles at the fusing core of the plasma (a alpha particle charge transport. source) with low-energy particles at the plasma edge (a There is a puzzle here, however. Consider the case of sink), the diffusion can be made to extract alpha parti- an electrostatic plane wave. Such a wave has no elec- cles while cooling them (Fig. 1b), thus accomplishing the tromagnetic momentum, which is given by the Poynting goal of alpha channeling. flux, p = S/c2 = E×B/4πc. Thus, as the wave amplifies During this process of wave-mediated ash extraction, due to alpha channeling, momentum must be conserved it is natural to hypothesize that the charge of the ash is in the plasma. pulled out as well. Such charge extraction would create a This momentum conservation has implications for the radial electric field in the magnetized plasma, resulting in transport of charge, which is most familiar from the the- E×B rotation. Driving E×B rotation, and in particular ory of classical collisional transport in magnetized plas- sheared rotation, is useful in a variety of ways for plasma mas. When two particle collide in a uniform magnetic control; it can suppress both large-scale instabilities25–32 field, their gyrocenters move in just such a way that no and small-scale turbulence33–37 , increase the confinement net charge moves across field lines (Fig. 2). As we discuss of mirror plasmas via centrifugal forces38–40 , and even in Section II, this cancellation of net charge transport is be exploited in a variety of plasma mass separation intrinsically linked to the fact that the collision conserves schemes41–50 . Thus, rotation drive via alpha channeling momentum. This link suggests that we must tread with would be an extremely useful tool to have in the plasma care in arguing that alpha channeling pulls charge across control toolbox. field lines, and thus drives rotation. Such rotation due to alpha channeling was first pro- What is ultimately needed is a self-consistent theory posed by Fetterman and Fisch51–53 . In particular, they of alpha channeling, treating both resonant and non- proposed that for a rotating plasma, a stationary electro- resonant particles. Such a theory has been recently static or magnetostatic wave at the plasma edge would developed54–57 for the simple case of alpha channeling appear as a propagating wave in the rotating plasma in a slab-geometry plasma, and the main purpose of this reference frame, pulling out alpha particles and putting paper is to concisely review these developments. How- their energy into the plasma rotation52 . This would re- ever, this paper will also discuss the comparison of this sult in a very efficient direct conversion of energy from new theory to other theories of ponderomotive forces, fusion ash heat into plasma rotation. discussing the advantages and disadvantages of each ap- While this picture is very appealing, the charge ex- proach.
3 to get around these issues. By applying our simple elec- B trostatic theory of the quasilinear force, we show how in the plane-wave initial value problem of alpha channeling, the resonant charge extraction identified by Fetterman51 is canceled out by a nonresonant charge transport in the opposite direction, eliminating the rotation drive effect. In Section VII, we turn our attention to the problem in multiple dimensions. We extend our quasilinear the- ory to this case, discussing how the addition of spatial fluxes to the conservation laws changes the fundamen- tal constraints of the problem. In particular, we show how the Reynolds and Maxwell stress combine to can- cel the nonresonant charge transport, allowing resonant charge extraction and rotation drive consistent with local momentum conservation. Furthermore, using a Fresnel analysis and our conservation laws, we show how the force on the plasma needed to extract these particles is ulti- FIG. 2. When two particles collide in a uniform mag- mately provided by the antenna that launches the wave. netic field, their gyrocenters move such that no net charge Finally, in Section VIII, we compare our analysis to the moves. This cancellation ultimately comes from the fact that oscillation center theories of ponderomotive forces64,65 . momentum is conserved in the collision. Because a planar These theories can make the recoil force difficult to cal- electrostatic wave has no electromagnetic momentum, as the culate, but also offer some general perspectives on the wave amplifies or damps, momentum should be conserved in steady-state problem. the plasma. Thus, the wave amplification or damping can be seen as mediating a momentum-conserving “collision” be- tween particles in the plasma, making charge transport ques- tionable. II. MOMENTUM CONSERVATION IN A MAGNETIZED PLASMA SLAB We begin in Section II by reviewing the deep link To examine perpendicular charge transport, we con- between momentum conservation and charge transport sider the simplest possible plasma setup. Namely, we in a magnetized plasma58,59 . To eliminate confound- consider a slab plasma, uniform along y and z, with a ing processes that might muddy the subsequent analy- magnetic field B along the axis ẑ. Any quantities which sis, we work in the simplest possible magnetic geometry: vary in space (such as the electric potential) vary only a slab with a uniform magnetic field. In such a geome- along the x direction. Thus, x can be thought of as the try, we show that large radial charge transport requires “radial” coordinate, y as the “poloidal” coordinate, and an azimuthal force to be applied to the plasma. In Sec- z as the “axial” or “toroidal” coordinate. We are inter- tion III, we review the momentum conservation principles ested in charge transport along x, which will lead to a in plasma wave problems, showing how planar electro- change in the y-directed E × B drift, i.e. the plasma static waves cannot produce a net force on the plasma. rotation. In Section IV, we see how these conservation laws play Consider the motion of a charged particle in such a out in the quasilinear theory of the bump-on-tail insta- uniform magnetic field. The magnetic field is given by the bility. This familiar textbook60–62 example shows the vector potential A = Bxŷ. Thus, ignoring the ẑ direction importance of the nonresonant response in enforcing mo- and working in the x-y plane, the particle motion exhibits mentum conservation63 . Analysis of this case allows us two invariants, from the energy and y-directed canonical to derive a simple form of the quasilinear force, consis- momentum: tent with momentum conservation and the kinetic theory 1 H = mv 2 + φ(x) (1) of the bump-on-tail instability, but valid for any planar 2 electrostatic wave in a homogeneous plasma. qB py = mvy + x. (2) As an immediate application, this force reveals how c currents can be driven by ion acoustic waves in plasmas The upper and lower bounds of the orbit can be used to despite the nonresonant recoil, even in the absence of find the gyrocenter. When φ(x) is a linear function of collisions. We explore this current drive in Section V, x, i.e. when the radial electric field is constant, then the which can be skipped by readers interested only in alpha gyrocenter position is simply given by: channeling and rotation drive. c With our groundwork laid, we return in Section VI xgc = py . (3) to the problem of alpha channeling. We discuss what qB has historically made a self-consistent linear and quasi- Now consider a collision between any number of par- linear theory of alpha channeling challenging, and how ticles, but which conserved momentum. The total cross
4 field charge transport will be given by: X c X System qi ∆xgc,i = ∆py,i = 0. (4) i B i Thus, there is no net movement of charge due to the WEM, pEM EM Field collision: movement of net charge requires momentum input, i.e. a net force on the plasma. WM , p M Oscillating Particle A. A Note on Viscosity Energy Interestingly, this conclusion relaxes if the electric field Resonant varies in x. As we show in Appendix A, the relation WP , p P Particle WRP, pRP between gyrocenter and energy becomes nonlinear as Energy soon at the electric field has a nonzero second deriva- tive, allowing for net charge transport in a momentum- Thermal Unimportant conserving collision. However, this charge transport is Energy ordered down by 2 relative to the motion of each indi- for 1D problem vidual charge, where ∼ ρi /L, with ρi the gyroradius and L the characteristic scale length of the electric field variation. This scaling leads to cross field-currents con- FIG. 3. Schematic depiction of the difference between elec- tromagnetic and Minkowski energy and momentum. The sistent with the Braginskii perpendicular viscosity59,66 Minkowski energy and momentum incorporate the oscillat- and collisional gyrokinetics67,68 , and can be seen as a ing motion of the particles, and can sometimes form a closed Larmor-radius-scale random walk of momentum. system with the nonresonant particles. A broader discussion of the multiple ways to view these small cross-field currents, including in the two-fluid59 , MHD, and particle pictures, is given in Ref.57 . How- The second form of energy and momentum often ever, for the purposes of this paper, we will be focused on discussed is known as the generalized Minkowski69 or whether alpha channeling leads to uncompensated charge plasmon62 energy and momentum, which incorporates transport, i.e. to 0th order net x-directed currents inde- the oscillating motion of the plasma particles: pendent of the flow shear variation length L. The present analysis suggests that this requires a net force to be ap- WM = ωr I; piM = kri I. (8) plied to the plasma along the y direction: a problem for a planar electrostatic wave, which has no momentum. Here, ωr and kr are the real components of the wave frequency and wavenumber, and I is the wave action, given for electrostatic waves by: III. CONSERVATION LAWS ∂Dr I = WEM , (9) Before launching into a discussion of momentum con- ∂ωr servation in waves problems, it is good to clarify what is meant by momentum (and, for that matter, energy). In where Dr is the real part of the dispersion function, de- plasma waves, two types of energy and momentum are fined in detail in Section IV. Although the Minkowski common to discuss. The first are the energy and mo- energy and momentum do not generally form a part of mentum associated with the electromagnetic fields in the a closed system, in a collisionless, homogeneous plasma, plasma: they form a closed system with the resonant particles, in the sense that: E2 + B2 1 ijk WEM = ; piEM = Ej Bk . (5) 8π 4πc d d i pM + piRP = 0. (WM + WRP ) = 0; (10) These form a closed system with all the particles con- dt dt stituting the plasma, which (in the sub-relativistic limit) The resonant particle energy and momentum are given in have energy and momentum: Z Z the same way as the nonresonant energy and momentum 1 from Eqs. (6), but with the integration only over the WP = mv 2 f dv; piP = mv i f dv, (6) 2 resonant region. These two conservation relations, summarized in where f (t, x, v) is the distribution function. In a homo- Fig. 3, will allow for a very clear interpretation of the geneous plasma, this conservation manifests as: evolution of the bump-on-tail instability, which will guide d d i our subsequent analysis of the alpha channeling problem. pEM + piP = 0. (7) (WEM + WP ) = 0; dt dt For a more detailed discussion, see Ref.56 .
5 Usually, we get the real and imaginary frequencies Nonresonant Shift by Taylor expanding this dispersion relation in small |ωi |/|ωr |, yielding real and imaginary components of the dispersion relation: X Diffusive Flattening 0=1+ Drs (13) s X ∂Drs 0= iωi + iDis , (14) s ∂ωr Here, Dis and Drs are the real and imaginary parts of Ds when evaluated at real ω, k. FIG. 4. Quasilinear evolution of the bump-on-tail instabil- Now, we can get the average force on each species over ity. Resonant particles lose momentum, which shows up as a wave cycle by taking the average of the field-density Minkowski momentum in the wave. However, the wave has no electromagnetic momentum, and so the correlation: qs h i Fsi = qs ns E i = Re Ẽ i∗ ñs (15) IV. BUMP-ON-TAIL INSTABILITY AND 2 SELF-CONSISTENT QUASILINEAR FORCES ∂Drs = 2WEM k i Dis + ωi . (16) ∂ωr The detailed kinetic mathematics of the bump-on-tail instability are worked out in several good textbooks60–62 ; Here, the second line used the definition of Ds and Ẽ i = here, we will simply give a brief overview of the relevant −ik i φ̃. features of the problem. Note that Eq. (16) automatically satisfies momentum The bump-on-tail instability is an electrostatic kinetic conservation, since if we sum over all species, the term instability in a homogeneous, unmagnetized plasma. It in brackets is simply the imaginary part of the dis- occurs for electron Langmuir waves, with the ions taken persion function (Eq. (14)), which must by definition as a neutralizing background. The instability is triggered be 0. As explained in more detail in Refs.54–56 , the when, near the phase velocity vp = ωr /kr , the particle first term represents the resonant diffusion (and can be distribution satisfies v∂f /∂v > 0; i.e. when there are shown to be consistent with the absorption of Minkowski more particles at high energy than low energy (black line, momentum56 ), and the second term is the nonresonant Fig. 4). Then, the wave-induced diffusion flattens out recoil response. Thus, the nonresonant recoil naturally the bump, transferring energy and momentum out of the arises in this momentum-conserving quasilinear frame- resonant particles. Keeping in mind our conservation re- work, which forms the basis for our subsequent analysis. lation from Eq. (10), this energy and momentum appear as Minkowski momentum in the wave. If one just looked at the resonant electrons, it would V. CURRENT DRIVE BY WAVE-MEDIATED MOMENTUM EXCHANGE thus look as if the plasma was losing energy and mo- mentum. However, from Eq. (7), we know that the to- tal energy and momentum in the plasma must remain The quasilinear theory of the Section IV leads to an constant, since the wave has no electostatic momentum. important and immediate result in current drive, which Thus, the nonresonant particle distribution must shift, we explore in this section. This section can thus be canceling the momentum loss from the resonant particles, skipped by readers interested only in questions of cross- to satisfy both momentum conservation equations—and field charge transport and rotation drive. this is exactly what happens (Fig. 4). Consider again the Langmuir wave from Section IV, It turns out that we can actually derive a very sim- but now without the bump-on-tail, so that the wave ple expression54,55 for the quasilinear force on the plasma damps rather than amplifies. Because of the lack of mo- that demonstrates this result, and generalizes to any elec- mentum in a purely electrostatic planar field, equal and trostatic wave. We start with the linear theory, which for opposite momenta, and thus equal and opposite currents, an electrostatic wave begins with the Poisson equation: will be driven in the resonant and nonresonant electron X populations. To get net current, therefore, some of this −∇2 φ = 4π q s ns , (11) momentum has to be transferred into the ions. Such mo- s mentum exchange can be provided by collisions. For in- Fourier transforming gives the general form of dispersion stance, since Langmuir waves have a high phase velocity relation for an electrostatic wave: relative to the electron thermal velocity, and because the X 4πqs ñs Coulomb collision frequency with the background ions 0=D ≡1+ Ds ; Ds ≡ − 2 . (12) scales as v −3 , the resonant current driven in the tail k φ̃ s electrons will be much longer-lived than the nonresonant
6 current. Thus, a net current is produced on collisional IAW wake timescales70–73 . However, such a situation spells trouble for current drive by other waves, which don’t exhibit collisional Fe timescale separation between resonant and nonresonant particles. For instance, ion-acoustic waves (IAWs) have E s vshock Shock p a phase velocity vp ∼ Cs ≈ ZTe /mi vthe , so that resonant electrons are right in the middle of the thermal distribution, and thus lack a collisional timescale sepa- L !B/!t ration from the nonresonant particles. One might then think that it is impossible to drive currents with IAWs. However, there is a major difference between the IAW E and the electron Langmuir wave: both ions and electrons participate in the IAW. Thus, momentum conservation Fe alone does not say where the nonresonant recoil will go; it could go either to electrons or ions. If the recoil goes to the ions, it will drive negligible current, leaving uncom- FIG. 5. Mechanism of magnetogenesis by ion-acoustic waves. A shock propagates through the ISM, producing ion acoustic pensated resonant electron current, and thus net current waves in its wake. These waves produce a force Fe electrons, drive. inhomogeneous on a scale L in the ISM. A compensating in- Our quasilinear force from Eq. (16) can quickly tell us homogeneous electric field E arises to cancel this force. This where this recoil goes. Combining this equation with the field has curl (consider the integral of the field over the loop definition of the Minkowski momentum from Eqs. (8-9), s), and thus induces a magnetic field B. Figure from Ref.87 . we find the very compact expression54 : dj i X qs dpi s dpi X qs tend to produces ion-acoustic waves in their wake76–78 . = =− M (γ̄s − η̄s ) (17) dt s ms dt dt s ms Thus, the same shocked systems that are predicted to produce magnetic fields in astrophysics via the Bier- where we have now defined the species resonant (γ̄s ) and mann battery79–86 , such as those around super-nova nonresonant (η̄s ) response coefficients: remnants82,83 , could produce magnetic fields via IAW- driven currents, providing an alternate possible mecha- Dis ∂Drs /∂ωr nism for the origin of cosmological seed magnetic fields. γ̄s ≡ P ; η̄s ≡ P . (18) s0 Dis 0 s0 ∂Drs /∂ωr 0 This intriguing possibility is explored in Ref.87 . To cal- culate the magnetic field growth due to the IAW current, For a wave that damps on multiple species, these coef- one must account for the self-inductance of the plasma ficients take a value between 0 and 1, representing the that fights the creation of a magnetic field. This can fraction of the resonant and nonresonant response that be accomplished, as in magnetohydrodynamics, by ne- goes to each species. For a typical, long-wavelength IAW, glecting the electron inertia in the electron momentum we have: equation. Thus, the electron momentum equation must 2 ωpi become force-free, which is enforced by the generation 1 Dri = − ; Dre = , (19) of an electric field. If this electric field has curl, a mag- ω2 k 2 λ2De netic field is generated. The change in the magnetic field can be calculated by combining Faraday’s Law and the so that η̄i = 1 and η̄e = 0. Thus, the entire nonresonant electron momentum equation to obtain87 : response is in the ions, and the driven resonant electron current is uncompensated. This quick result is strength- ∂B c Fe ened in Ref.54 , where we show that for any IAW, the = ∇ × (v × B) − ∇ × . (20) ∂t e ne electron resonant current γ̄e is always significantly larger than the electron nonresonant response η̄e , even when Here, Fe is the force on the electrons from everything taking into account next-order corrections. other than the large-scale electric and magnetic fields. If Because this current drive mechanism relies on the mo- this force is the pressure force, one obtain the Biermann mentumless wave field to rearrange momentum between battery; if it is the quasilinear force from the IAW, one the different plasma species, it has been termed ”wave- obtains a corresponding “IAW Battery.” mediated momentum exchange”. This method of current A possible scenario for magnetogenesis by this IAW drive was first discovered using a much more complex ki- battery is shown in Fig. 5. A shock from a super nova netic calculation by Manheimer74 for IAWs, and Kato75 expands into the surrounding intersteller medium (ISM), for IAWs and LH waves; here, we see that it follows very encountering inhomogeneities in the medium. These in- straightforwardly from the form of the force in Eq. (16). homogeneities lead in turn to inhomogeneities in the wave The case of the ion-acoustic wave is intriguing, because strength, and thus in the force experienced by the elec- shocks, which are ubiquitous in astrophysical systems, trons. A curl in this force results in a compensating curl
7 in the electric field to keep the electrons force-free, and This last case was examined in full nonlinear detail by thus to the generation of a magnetic field. Depending on Karney, who derived a diffusion coefficient for the res- the wave parameters, this process can sometimes produce onant particles in perpendicular velocity space. To de- fields faster than the Biermann battery mechanism87 . rive this diffusion coefficient, Karney solved for the dy- namics of a particle undergoing Larmor rotation in the presence of an electrostatic wave of arbitrary strength. VI. LINEAR AND QUASILINEAR THEORY OF ALPHA Because the nonlinear mathematics involved were very CHANNELING complicated, the theory was limited to planar waves with constant amplitude in time—precluding a self-consistent quasilinear theory. However, the theory did give a diffu- Now we return to the subject of rotation drive by alpha sion coefficient for the resonant particles, which can be channeling. The case of the bump-on-tail instability in written in perpendicular energy space as54 : Section IV suggests that the nonresonant particle will gain any momentum lost by the resonant alpha particles. 2 vp2 m2α qα kφa The analysis of Section II, meanwhile, suggests that in DKK ≡ p H (v⊥ − vp ) , the absence of net momentum input, no charge transport 2 mα 2 − ω2 k 2 v⊥ r is possible. (21) To see how this plays out in the case of alpha channel- p ing by a plane wave, we would like to make use of our where φa is the wave amplitude, v⊥ = 2K/mα , and simple quasilinear theory from Eq. (16). However, mak- H(x) is the Heaviside function. ing use of this theory requires a linear theory, which until Using the existence of this diffusion, Fisch and Rax2,3 recently did not exist for alpha channeling. used the relation between momentum and gyrocenter to show that every time a particle receives a kick ∆K in perpendicular energy, it receives a correlated kick A. Historical Results and Challenges ∆X/∆K = k × b̂/mα ωΩα in gyrocenter position. Wave amplification occurs when there are more particles at high than low energy along this path, i.e. when: The primary reason that this linear theory has proved elusive is that the pole associated with the perpendicular ! ∂ k × b̂ ∂ Landau resonance disappears from the dispersion rela- + · Fα0 (K, X) > 0, (22) tion as soon as there is any magnetic field in the plasma. ∂K mα ωΩα ∂X This puzzling disappearance of Landau damping as soon as a plasma is even infinitesimally magnetized has been where Fα0 (K, X) is the lowest-order distribution function termed the “Bernstein-Landau paradox,” since it involves for the resonant alpha particles. a discrepancy between the Bernstein (magnetized) and As we move to develop a self-consistent linear and Landau (unmagnetized) kinetic dispersion relations88,89 . quasilinear theory of alpha channeling, our main targets The Bernstein-Landau paradox is a mathematical arti- are the diffusion coefficient in Eq. (21) and the amplifi- fact more than anything else. It arises from the fact that, cation condition in Eq. (22). in calculating the forces in the linear magnetized theory, the force is always evaluated at the unperturbed parti- cle position. Because of this, a particle never forgets its B. Linear Theory phase with respect to the wave as it goes around repeat- edly on its gyro-orbit, and so unless there is a resonance The requisite self-consistent linear theory of alpha between the gyroperiod and the wave period, the particle channeling, for a poloidal plane wave (k k ŷ), is devel- will gain and lose equal amounts of energy in a periodic oped in Ref.55 . The basic idea is to construct a theory oscillation. In the language of dynamical systems, the for the alpha particles in a magnetized plasma that con- particle is trapped on a small cycle, or “island,” in phase tains the Landau resonances perpendicular to the mag- space24 , and thus only oscillatory motion is possible. netic field (Fig. 1) that underlie the diffusion process in Diffusion, i.e. stochastic motion of the particle through alpha channeling. Since these resonances exist in the un- phase space, can only occur if these islands are destroyed. magnetized kinetic theory, our approach is to transform While it is true that, in sufficiently magnetized plas- the unmagnetized kinetic theory, which lives naturally mas, some particles will remain trapped in these islands, in local phase space coordinates xi ≡ fα (x, y, vx , vy ), to there are a multitude of mechanisms which can allow gyrocenter-energy coordinates X i ≡ Fα (X, Y, K, θ). To loss of phase memory and stochastic diffusion in a phys- model the nonlinear loss of gyrophase structure, we force ical plasma. This can occur when there are multiple the zeroth-order alpha particle distribution function to overlapping waves whose islands overlap (the “Chirikov be independent of gyrophase θ, and average all equations criterion”)90,91 , when random collisions introduce suffi- over θ. Indeed, a similar approach has been used before cient stochasticity92 , or when a single wave has a large in homogeneous plasmas to calculate damping on alpha enough amplitude to trigger nonlinear physics23,24 . particles92–94 ; in contrast, here we allow the distribution
8 ! "#! $"# (&, #! , #$ ) #% #$ FIG. 7. Simulated single-particle trajectories in the x-y FIG. 6. Equivalence of alpha channeling and the bump-on-tail plane of (a) hot particles (v0 = 3.5vp ) and (b) cold par- instability. If a distribution Fα0 (X, K) for which alpha chan- ticles (v0 = 0.03vp ) in a growing electrostatic wave. Lines neling occurs, i.e. for which ωi > 0 in Eq. (23), is examined are particle positions, and triangles are orbit-averaged gyro- in local coordinates, then the local distribution fα0 exhibits R center positions. Axes are normalized to ρps ≡ ω/kΩs , i.e. a bump-on-tail instability, where sgn(vp ) dvx ∂f /∂vy > 0. x̃ = x/ρps and ỹ = y/ρps . The color indicates time, light to dark. The hot particles diffuse stochastically. The cold particles exhibit a clear shift in gyrocenter toward positive x̃, function to be inhomogeneous along X, i.e. to have a which ultimately cancels the charge transport from the hot radial alpha particle gradient. alpha particles. Figure from Ref.55 . Treating the ions and electrons as cold fluids in the LH frequency range Ωi ωLH Ωe , we find that the alpha particles introduce an imaginary component to the dis- neling amplification condition, transforming the unmag- persion relation, resulting in an imaginary frequency55 : netized quasilinear kinetic theory yields the diffusion 2 4 π ωpα ωLH 2 Z ∂Fα0 ky ∂Fα0 equation for the resonant alpha particle distribution55 : ωi = 2 |k |3 mα dvx + , 2 ωpi y ∂K mα ωr Ωα ∂X X ∗ ,K ∗ " # ∂Fα0 d KK d (23) = D Fα0 (24) ∂t t dK path dK path where X ∗ ≡ x + vp /Ωa , K ∗ ≡ m vx2 + vp2 /2. The term d ∂ ky ∂ in parenthesis can be recognized as precisely the alpha ≡ + . (25) channeling amplification condition from Eq. (22), but dK path ∂K ms ωr Ωα ∂X now weighted properly over a distribution of alpha par- ticles, rather than simply a single particle. Eq. (23) rep- Thus, the theory recovers diffusion along the channeling resents the first calculation of a linear amplification rate diffusion path, definitively establishing that the theory from the alpha channeling process. captures the alpha channeling process. The amplification condition for LH wave alpha chan- Since the new linear theory captures the alpha parti- neling was originally derived by demanding that the gyro- cle response as an imaginary component of the disper- center distribution of alpha particles have more particles sion relation Diα , we can apply the simple quasilinear at high energy than low energy along the gyrocenter dif- theory force theory from Eq. (16). Since for the bulk 2 fusion path. In contrast, Eq. (23) is just a coordinate components we have Dre = ωpe /Ω2e , Dri = −ωpi 2 /ω 2 , transform of the imaginary frequency result from Lan- the recoil force (as in the case of ion acoustic waves in dau damping or the bump-on-instability. Nevertheless, Section V) will be entirely in the ions, and lead to an these two very different approaches lead to the same am- F × B drift (and corresponding charge transport) that plification condition, and thus, evidently, represent the entirely cancels the current from the resonant alpha par- same physical process. In other words, if we take a dis- ticles. The presence of this drift in a time-growing plane tribution function Fα0 (X, K) in gyrocenter coordinates wave is confirmed55 by single-particle simulations (Fig. 7) for which alpha channeling occurs, and we look at it lo- of the full Lorentz force in a growing high-frequency elec- cally in physical coordinates fα0 (x, vx , vy ), we will find trostatic wave, which agree well with the predictions of a bump-on-tail instability along vy (Fig. 6). Establish- quasilinear theory (Fig. 8). ing the equivalence of these two instabilities, which were As a result of this analysis, we see is that charge cannot thought to be entirely distinct, is a rewarding result of be extracted, and thus E × B rotation cannot be driven, developing the linear theory. by a purely poloidal plane wave that grows in time. C. Quasilinear Theory VII. SPATIALLY STRUCTURED WAVES Just as transforming the unmagnetized linear kinetic However, this conclusion is not the whole story. Often, theory to gyrocenter coordinates yielded the alpha chan- in problems of plasma control, we drive a wave from the
9 y Uniform 0.010 Dilute Ramping Region Ramping Mode with Conversion pi Vacuum Plasma Density Density 0.005 Resonant Layer X/ x Particles z, B 0.000 10 20 30 40 50 60 t i1 FIG. 8. Change in gyrocenter position X for the particle in Fig. 7b due to the slow ramp-up of the electrostatic wave. The gyro-period-averaged position in the simulations (filled black diamonds) agrees well with the predicted shift55 (gray dashed FIG. 9. Model for the structured wave analysis from Ref.56 . line) due to the nonresonant recoil. Figure from Ref.55 . Darker color represents higher density. A vacuum evanes- cent wave (black), consisting of a forward-decaying “incident” wave (dotted gray) and backward-decaying “reflected” wave outer radial plasma boundary, and are interested in what (dashed gray), converts to a “transmitted” slow wave at the this wave looks like in steady state. Understanding what plasma / vacuum interface. This slow wave then continues charge transport looks like for this state requires analyz- to propagate inward as the plasma density increases until it ing waves with spatial structure. For simplicity, we will hits the lower hybrid resonance, where it mode converts into consider absorption of a damped wave; however, the core an electrostatic lower hybrid wave (blue). Wave action is results apply also to amplification of a wave due to alpha conserved during the in-plasma propagation and mode con- channeling. version. This wave then propagates back to a uniform region To look at this new situation, we will consider the containing resonant particles, where it damps. Figure from model from Ref.56 , shown in Fig. 9. This simple model Ref.56 . is motivated by the coupling of waveguides for lower hy- brid current drive95,96 . For such waves, an evanescent Looking at momentum conservation, we are primarily wave in a vacuum region converts into a plasma slow interested in the appearance of the stress terms Π: wave as the plasma density ramps up past the point where |ωpe | > |ω|. It then propagates up to the mode- ij 1 i j 1 ij 2 i j 1 ij 2 conversion layer, where it reflects as an electrostatic lower ΠEM = − E E − δ E +B B − δ B 4π 2 2 hybrid wave97 . We make two main simplifications to (28) dramatically reduce the problem complexity. First, we Z assume all damping occurs in a uniform region with res- ΠijP = mv i v j f (t, x, v)dv (29) onant particles, so that we do not have to account for changing wave parameters in the resonant region. Second Πij i j M = pM vg , (30) we take the plasma to have a sharp boundary, where the evanescent wave converts to a slow wave. This reduces where we have introduced the group velocity of the wave: the coupling calculation to a Fresnel-style98,99 boundary- matching condition, dramatically simplifying the mathe- ∂Dr /∂kri vgi = − . (31) matics. ∂Dr /∂ωr For the purposes of charge transport, we are interested A. Conservation Relations Revisited in the force along the symmetry direction y. As discussed in Ref.56 , for the purposes of this calculation, the tensors (TEM + TP ) and (TM + TRP ) both represent closed The conservation relations from Section III relax con- systems—a fact we make extensive use of in the ensuing siderably in a plasma with spatial structure. The general analysis. conservation relations are best expressed in terms of the energy-momentum tensor (EMT) of the system: W S/c B. Local Momentum Conservation T= . (26) cp Π In the presence of a wave, the kinetic stress tensor gives In a closed system, conservation laws follow from the rise to a stress component due to the oscillating motion vanishing of the 4-divergence of this tensor, expressed in of the particles, known as the Reynolds stress: Cartesian coordinates in flat spacetime as: E 1 1 ∂ µ0 ∂ µi D Πij i j i j T + T = 0. (27) Rey = mn 0 u 1 1 = mn0 Re ũa ũa , u (32) c ∂t ∂xi 2 Energy conservation corresponds to the µ = 0 portion of where ũ1 is the local complex velocity amplitude of the this equation, and momentum conservation to the µ = wave. This Reynolds stress plays an important role in 1 − 3 components. calculating turbulent forces on a plasma100–106 . The
10 presence of the Maxwell stress tensor and the Reynolds stress dramatically change the conservation calculations. As shown in Ref.56 , for the lower hybrid wave, the two combine in just such a way as to cancel the nonreso- nant (recoil) force on the plasma, leaving only the res- onant force, in a way consistent with momentum con- servation. Thus, in the spatially-structured, steady-state problem of alpha channeling, resonant particles can be extracted, and E × B rotation can be driven, thanks to the torque exerted on the plasma by the Maxwell and FIG. 10. Oscillation center theories track the slow dynamics Reynolds stresses. of the central position around which a particle quickly oscil- lates in a wave field. C. Global Momentum Conservation center. It is important to see whether the theories are One might still wonder where this torque is coming compatible and lead to the same conclusions. Thus, in from; is there an externally-applied net torque on the Chapter 7 of Ref.57 , we performed a detailed comparison plasma consistent with this force on the resonant alpha of the two types of theory. particles? If not, then this local force on the plasma There are actually (at least) two distinct types of os- must be compensated somewhere else within the plasma, cillation center theory in the literature: one64 which leading to momentum rearrangement within the plasma comes from separating out the quiver motion of a par- (and thus the generation of shear flow), but no net spin- ticle in the wave Lagrangian, and one, exemplified by up of the plasma. Dewar65 , which comes from applying a non-secular, near- To look at this question, we turn in Ref.56 to a Fres- identity canonical transform to the particle coordinates nel model of the wave transition between the vacuum and in the wave-particle interaction. Each of these theories the plasma. Now, the evanescent wave in the vacuum has has somewhat different features, as we explore in Ref.57 a well defined electromagnetic momentum flux (Maxwell for the simple case of an electrostatic wave in an unmag- stress), but no well-defined Minkowski flux. However, it netized plasma, which we summarize briefly here. is possible to show that, for a lower hybrid wave, the x-directed flux of y-directed electromagnetic momentum A. Quiver Lagrangian Πyx EM through the vacuum, is equal to the x directed flux of y-directed Minkowski momentum Πyx M within the plasma. Because the Minkowski EMT forms a closed We will start by examining the quiver-Lagrangian the- system with the resonant particle EMT, this momentum ory of the oscillation center64 . Since it will already re- all ends up in the resonant particles. Thus, the force quire a fair amount of complexity, we will consider only on the resonant particles that allows charge extraction is electrostatic waves in an unmagnetized plasma. To get ultimately provided by the electromagnetic fields prop- the Lagrangian that describes this theory, we separate agating through the vacuum at the plasma edge. As a out the oscillating motion of a particle from its smooth result, the entire plasma spins up due to the interaction motion that changes on a longer timescale, obtaining: with the wave, in a way consistent with local and global 1 1 momentum conservation. L̄ ≈ mV 2 + m ṽ 2 + q hx̃ · E(X, t)i . (33) 2 2 In the case of alpha channeling, this analysis suggests that electromagnetic energy will flow from the plasma Here, X and V represent the oscillation center position, into the antenna as a result of alpha channeling. This and x̃ and ṽ describe the quiver motion, which can be raises the intriguing possibility of performing direct con- solved simply at constant X, V. Thus, the latter two version of alpha particle energy into electrical energy at terms combine to form a ponderomotive potential: the antenna, which could be a much more efficient way 1 q 2 k 2 |φa (X)|2 to usefully harvest the energy than any thermal process. Φ(X, V) = , (34) 4 m (k · V − ω)2 which gives the ponderomotive force via the Euler- VIII. OSCILLATION CENTER PONDEROMOTIVE Lagrange equations. THEORIES We show in Appendix B that a naive application of this oscillation-center theory can yield erroneous conclu- Although the above theory holds together well, and ex- sions about the ponderomotive force, but a more careful plains how charge extraction is consistent with momen- calculation agrees with the fluid theory. As we show, tum conservation, there are other, completely different forces in the two theories differ by three terms. The first approaches to calculating ponderomotive forces in the two—the Reynolds stress and the polarization stress— literature, which focus on the motion of the oscillation were identified by Gao107 in a cold fluid plasma. We show
11 3 0.16 quiver Lagrangian sense), or the change in the mean ki- 2 a) b) 0.14 netic momentum hmnui of the plasma particles. In con- 0.12 1 0.10 trast, P represents a canonical momentum with complex Vy/vth 0 0.08 F(V) dependence on the ponderomotive potential, and with 1 0.06 hPi =6 hpi, as evidenced by Eq. (36). Thus, while the 0.04 2 0.02 Hamiltonian theories hold together theoretically, and el- 3 0.00 egantly conserve energy and momentum65 , backing out 2 0 2 2 0 2 the physical forces and currents from the solution in Vx/vth Vx/vth Hamiltonian phase space is a very complex task. FIG. 11. Oscillation center velocity distribution F (V), as given by Eq. (B34), (a) far from the wave region, and (b) in the wave region. As explained in the appendix, the distri- C. Steady State bution in (b) is√calculated for WEM /pth0 = 0.35, kx vth /ω = ky vth /ω = 1/4 2. In the wave region, the distribution re- Nevertheless, Dewar’s formulation is very useful for us, duces in magnitude, and also stretches out along k, resulting because it suggests something fundamental about pon- in anisotropy. This anisotropy provides the stress that allows deromotive forces from steady-state waves, and their ef- a net force to be exerted on the plasma by the wave. fects on particle orbits. In the absence of a wave, in a plasma confinement device, the orbit is determined by several constants of motion, including the particle en- that in a warm plasma, which is necessary wave packet ergy and its momentum along the symmetry directions. propagation into an unmagnetized plasma or for purely- For instance, in a periodic cylinder, the orbit would be perpendicular modes in a magnetized plasma, there is determined by the energy, and the momenta along the also a third term of discrepancy, namely the kinetic stress θ and z directions. Thus, the change in the orbit is of the oscillation centers themselves. Only when account- determined by hdH/dti, hdpθ /dti, and hdpz /dti. How- ing for these three terms do the theories agree, and sup- ever, in Dewar’s theory, hHi = hH − K2 (S)i, where K2 port the conclusion that the nonresonant force along y is a “renormalization energy” that depends only on the vanishes in steady state. function S. For a steady-state wave, H is constant in time. Furthermore, because S is defined to not grow secularly, i.e. to be a fixed function of the wave am- B. Near-Identity Transform plitude, it is also constant in time. Thus, hHi will be contant in time. Similarly, looking at Eq. (36), we see The basic idea behind Dewar’s theory is to use a that hdp/dti is equal to hdP/dti plus terms that depend canonical transformation, familiar from Hamilotonian on hdS/dti, so that by the same argument, in steady mechanics108,109 , to transform from the physical particle state, hpi is a constant function of hPi. Furthermore, coordinates (x, p) to an oscillation center coordinate sys- hdP/dti = h−dH/dX i = 0 on a closed orbit. Thus, De- tem (X , P). This coordinate system is chosen so as to war’s theory suggests that in steady state, nonresonant only include the gradient ponderomotive potential and particles on closed orbits remain on those closed orbits, the resonant force, without any of the reactive time- leaving only the resonant cross-field current. dependent terms that can prove confusing. To second In summary, Dewar’s theory is not the best for un- order in wave amplitude, these coordinate systems are derstanding and calculating the reactive forces that are linked by the transformation65 : applied to nonresonant particles in time-growing waves. However, once the consistency between the models has ∂S(x, p, t) ∂S(x, p, t) ∂ 2 S(x, p, t) been established, it offers a deep explanation for the X =x+ − · (35) ∂p ∂x ∂p∂p seemingly fortuitous disappearance of the nonresonant ∂S(x, p, t) ∂S(x, p, t) ∂ 2 S(x, p, t) recoil force for steady-state, boundary-driven waves. P =p− + · . (36) ∂x ∂x ∂p∂x Here, S is a quantity that defines the near-identity trans- IX. CONCLUSIONS AND FUTURE DIRECTIONS formation, so we demand that hSi = 0 at first order. In these coordinates, the “force” dP/dt from the waves This paper reviewed recent work that addressed the on the particles consists of only two terms: the gradi- question of whether alpha channeling, in transport- ent of the ponderomotive potential, and the resonant ing resonant ash across field lines, also transports net diffusion65 . However, we have to be careful how we in- charge across field lines. Such charge transport would terpret this, because the ponderomotive force as we intu- drive E × B rotation51–53 , which can be advantageous itively think of it cannot be simply identified with dP/dt. for plasma stabilization, turbulence reduction, enhanced This is because, in general, what we think of intuitively confinement in magnetic mirrors, and plasma centrifuga- as the ponderomotive force is either the change in the tion. In the end, we found that, at least for electrostatic kinetic momentum mV of the oscillation center (in the waves, the answer depends on how the waves enter the
12 plasma, with different results for plane waves that grow In Section VIII, we compared our quasilinear model to in time, as opposed to waves which enter from the plasma the variational Lagrangian and Hamiltonian theories of boundary. the oscillation center, for the simple case of an electro- To arrive at these results, in Section II, we considered static wave in an unmagnetized plasma. We found that charge transport along x due to collisions in a magne- the resulting ponderomotive forces in the two models dif- tized plasma slab, finding that it was very hard to move fered by the terms arising from the Reynolds, polariza- charge across field lines, since it required a net force to tion, and oscillation center stresses, but ultimately agreed be applied along y. In Section III, we saw how this pre- on the form of ponderomotive force. Furthermore, an sented a problem for charge transport by a planar elec- analysis of Dewar’s near-identity transform theory sug- trostatic wave, which contains no momentum. In Sec- gested that the vanishing of the nonresonant recoil re- tion IV, we saw how this momentum conservation played action for steady-state waves was likely to be a general out in the bump-on-tail instability, motivating the devel- result across many systems, at least for nonresonant par- opment of a self-consistent quasilinear theory for alpha ticles on closed orbits in phase space. channeling. After a foray into current drive theory in This paper thus provides a cohesive and simple theory Section V, in Section VI we developed the first linear of rotation driven by alpha channeling, which demystifies theory of alpha channeling as a prerequisite to evaluat- how momentum conservation works out while allowing ing the quasilinear forces on the plasma. In so doing, charge extraction. Since it was intended to resolve a core we showed that alpha channeling is physically equivalent mystery, our analysis throughout was deliberately sim- to the bump-on-tail instability, exposing a deep connec- ple, taking every simplifying assumption possible while tion between two thought-to-be-distinct processes. The retaining the core features of the problem. Thus, there resulting quasilinear theory allowed us to confirm the ex- are many generalizations and extensions possible, which istence of the nonresonant recoil reaction in the case of will be required in order to quantitatively connect to fu- a time-growing plane wave, which canceled the charge ture experiments. transport from the resonant ash, preventing the E × B First, and likely most straightforward, is to extend rotation drive that was predicted by earlier theories. As the analysis from electrostatic waves to electromagnetic a silver lining, this return current was in the ions; thus, waves. This is necessary to handle one of the systems even though no rotation was driven, there was a different originally envisioned by Fetterman, in which stationary beneficial effect, in that for every alpha particle brought magnetic perturbations at the plasma edge provide the out by alpha channeling, two ions were brought into the moving wave in the rotating plasma frame52,53 . Although core plasma, fueling the fusion reaction. it requires us to move from a scalar to a vector theory, the In Section VII, we generalized the model to allow for essential features of the theory will be similar, but with spatially damped waves. In this multi-dimensional prob- the susceptibility in many cases taking over role of the lem, the Maxwell and Reynolds stresses played a promi- dispersion function. Of course, for an electromagnetic nent role, leading to the disappearance of the poloidal wave, the electromagnetic field does have momentum, nonresonant force and its associated charge transport in somewhat altering the momentum balance and magni- steady state. Thus, extraction of charge by a steady- tude of the recoil in the plane wave initial value prob- state, boundary-driven wave was shown to be consis- lem. However, the general principles from the oscillation tent with momentum conservation. A Fresnel model of center theories suggest that the core conclusions in the the wave propagation from vacuum to plasma, combined electromagnetic case will remain fairly similar: namely, with a wave action conservation principle, confirmed that there will be some canceling charge transport for a time- the momentum and energy required for this charge ex- dependent plane wave due to the nonresonant recoil, but traction were ultimately provided by electromagnetic en- no recoil for the steady-state boundary-value problem, ergy and momentum flux in the evanescent wave trav- with momentum conservation in the latter case enforced eling through the vacuum gap between the antenna and by a combination of Reynolds and Maxwell stress. Thus, plasma. Applied to alpha channeling, this analysis sug- rotation drive will likely be possible for Fetterman’s mag- gested the possibility of direct conversion of alpha parti- netostatic edge coil scheme52,53 as well. cle energy into electric energy, as the alpha particle am- As we further generalize to consider the generation of plification results in an outwardly-directed Poynting flux, shear flow in a plasma, the plasma must by definition be- driving electromagnetic energy into the antenna. come inhomogeneous, since the rotation profile is a func- Taken together, the analysis in Sections VI-VII re- tion of radius. This inhomogeneity will impact both the vealed a core difference between the plane-wave initial dissipation and the evolution of k. Handling this situa- value problem of alpha channeling, where net charge can- tion correctly will require integration of the momentum- not be extracted, and thus rotation cannot be driven, conserving channeling theory with geometrical optics. and the steady-state boundary-value problem, where it To fully solve for the rotation drive, it will be neces- can be. In contrast to earlier models that focused only sary to combine this improved theory with a theory of on the resonant particles, the analysis here respects the rotation dissipation. This dissipation can be due to a va- momentum conservation principles between the various riety of collisional processes59,110 , or “anomolous” trans- plasma subsystems. port such as that due to the stochasticity of the magnetic
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