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
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,
                                                              (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:                       is undesirable for alpha channeling.
Momentum Conservation in Current Drive and Alpha-Channeling-Mediated Rotation Drive

         a)                                                    b)

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.

                                                               to get around these issues. By applying our simple elec-

                                                               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.
   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

field charge transport will be given by:
             X               c X                                                           System
                 qi ∆xgc,i =       ∆py,i = 0.              (4)
                             B i
Thus, there is no net movement of charge due to the                  WEM, pEM                  EM
collision: movement of net charge requires momentum
input, i.e. a net force on the plasma.                                                                        WM , p M
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:
                                                                                       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 .

                                                               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:
                                      Diffusive Flattening                       0=1+        Drs                     (13)
                                                                                  X         ∂Drs
                                                                             0=        iωi        + iDis ,          (14)

                                                               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)
   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

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

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
                                           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
                         ωpi                                       become force-free, which is enforced by the generation
              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

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
                                                                                      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

                       ! "#! $"# (&, #! , #$ )



                                                                  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
                              ∂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
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

                                                                      y                                     Uniform
     0.010                                                                               Dilute   Ramping
                                                                                                              with                Conversion

                                                                                Vacuum   Plasma   Density               Density
     0.005                                                                                                  Resonant                Layer

                                                                            x                               Particles
                                                                 z, B
                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
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
                                                                        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

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
  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

          3                                                          0.16        quiver Lagrangian sense), or the change in the mean ki-
              a)                    b)                               0.14        netic momentum hmnui of the plasma particles. In con-
          1                                                          0.10        trast, P represents a canonical momentum with complex

          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
          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

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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