Quasilinear theory of Brillouin resonances in rotating magnetized plasmas

Abstract

Both spin and orbital angular momentum can be exchanged between a rotating wave and a rotating magnetized plasma. Through resonances the spin and orbital angular momentum of the wave can be coupled to both the cyclotron rotation and the drift rotation of the particles. It is, however, shown that the Landau and cyclotron resonance conditions which classically describe resonant energy–momentum exchange between waves and particles are no longer valid in a rotating magnetized plasma column. In this case a new resonance condition which involves a resonant matching between the wave frequency, the cyclotron frequency modified by inertial effects and the harmonics of the guiding centre rotation is identified. A new quasilinear equation describing orbital and spin angular momentum exchanges through these new Brillouin resonances is then derived, and used to expose the wave-driven radial current responsible for angular momentum absorption.

1. Introduction

Understanding how to sustain and control the angular momentum of a rotating magnetized plasma column is a central issue, both for applied and basic plasma physics. On the former, the first successful application of rotating non-neutral plasmas was the magnetron microwave source theorized by Brillouin (1945). Since then, an important motivation for rotating plasma configurations has been and continues to be thermonuclear fusion (Lehnert 1971), both with rotating tokamaks (Ochs & Fisch 2017; Rax, Gueroult & Fisch 2017) and rotating mirrors (Bekhtenev et al. 1980; Hassam 1997; Fetterman & Fisch 2008, 2010; Teodorescu et al. 2010). Besides fusion, rotating plasmas have also attracted attention for ion acceleration (Janes 1965; Janes, Levy & Petschek 1965; Janes et al. 1966) and mass separation (Bonnevier 1966; Krishnan, Geva & Hirshfield 1981; Prasad & Krishnan 1987) as envisioned for instance for nuclear waste cleanup (Gueroult, Hobbs & Fisch 2015), spent nuclear fuel reprocessing (Gueroult & Fisch 2014; Timofeev 2014; Vorona et al. 2015; Dolgolenko & Muromkin 2017) or rare earth element recycling (Gueroult, Rax & Fisch 2018). On the latter, magnetized rotating plasma theory has been shown to be important in understanding pulsar dynamics and radiative transfer (Gueroult et al. 2019b ), rotation augmented gyrotropy (Gueroult, Rax & Fisch 2020) or image rotation (aka Faraday–Fresnel effect) in plasmas (Rax & Gueroult 2021). The adiabatic theory of angular momentum perturbation in rotating magnetized plasmas also provides an interesting realization of a geometrical Berry type phase (Rax & Gueroult 2019).

Two fields configurations can sustain the steady state rotation of a magnetized plasma. One is the Hall configuration with a radial magnetic field and an axial electric field. This is the configuration used notably in stationary plasmas thrusters. The other, illustrated in figure 1, is the Brillouin configuration with an axial magnetic field and a radial electric field. This is the configuration used notably in mass separators (Gueroult et al. 2019c ) and homopolar devices (Barber, Swift & Tozer 1972). In this study we will restrict our analysis to this last configuration and study the quasilinear theory of angular momentum exchange between waves and particles in a rotating Brillouin configuration.

Figure 1. Electric and magnetic fields configurations in a plasma rotating around the $z$ axis, $\alpha$ is the polar angle and $r$ the polar radius.

Steady state angular momentum injection to compensate dissipation in a rotating cylindrical magnetized plasma column is usually envisioned through the use of concentric annular direct current (DC) biased end-electrodes, as illustrated in grey in figure 1. This is the scheme originally proposed by Lehnert (Lehnert 1970, 1973), where electrodes are assumed to drive a DC radial electric field perpendicular to the DC axial magnetic field. It has, however, been recently shown that field penetration through the sheath and along field lines imposes a number of constraints on the achievable electric field (Gueroult, Rax & Fisch 2019a ; Poulos 2019; Liziakin, Gavrikov & Smirnov 2020; Liziakin et al. 2021; Trotabas & Gueroult 2022). Meanwhile, studies in the last decade on rotating mirrors (Fetterman & Fisch 2008, 2010) and rotating tokamaks (Ochs & Fisch 2017; Rax et al. 2017) have underlined the potential to use electromagnetic waves to drive plasma rotation through resonant wave–particle angular momentum absorption, and to compensate for dissipative relaxation. Although very promising, these studies relied on a simple photon picture.

The standard tool to study resonant wave–particle interaction is the quasilinear theory. For an infinite homogeneous magnetized plasma at rest the quasilinear equation is well known (Rax 2011), and has proven particularly useful to evaluate energy absorption and current generation in thermonuclear plasmas (Fisch 1978, 1987). The quasilinear equation for a cylindrical unmagnetized plasma at rest has also been derived (Kaufman 1971), whereas a generalized quasilinear theory for inhomogeneous plasma has recently been laid out (Dodin 2022). However, and while as mentioned above it appears to be key to important applications, the quasilinear theory for a rotating magnetized plasma has to our knowledge not been derived yet. While quasilinear radial transport has been studied within the framework of non-neutral plasmas confinement deploying a so-called ‘rotating wall technique’ (Eggleston & O'Neil 1999; Kiwamoto, Soga & Aoki 2005), these studies were restricted to electrostatic modes. Finite Larmor radius effects were also neglected as an infinite magnetic field was assumed. Lastly, although inertial effects are central to equilibria in Brillouin configurations, these studies neglected inertial effects so that the resonance condition is limited to the axial Doppler-shifted resonance between the plasma rotation and the wave frequency. These restrictions are removed in the present paper. Here we derive the quasilinear equation for a rotating wave and a rotating magnetized plasma, which describes angular momentum absorption/emission within a kinetic framework, and use it to uncover the interplay between orbital angular momentum (OAM), spin angular momentum (SAM) and finite Larmor radius (FLR) effects.

This paper is organized as follows. We begin by recalling in § 2 basic elements on wave and particle angular momentum. We then proceed to derive in § 3 the Hamiltonian description of a magnetized rotating plasma sustained by a radial electric field, and to expose the physical and geometrical meaning of the angle–action variables used to describe the particle dynamics. These canonical angle–action variables are after that used to identify the relation between the canonical and the kinetic angular momentum in § 4, and to model as a perturbed Hamiltonian the effect of a rotating wave perturbation on the rotating particle dynamics in § 5. This formalism is then used to identify a new resonance condition in § 6, and to derive the quasilinear kinetic equation describing the time evolution of the distribution function in action space in § 7. This new kinetic equation is finally used to study OAM and SAM absorption in § 8, and to derive the expression of the wave-driven radial current in § 9. Lastly, § 10 concludes this study and summarizes the main findings.

2. Wave and particle angular momentum

In the following we use $( r,\alpha,z)$, a set of cylindrical coordinates on a cylindrical basis $( \boldsymbol {e}_{r},\boldsymbol {e}_{\alpha },\boldsymbol {e}_{z})$. The associated set of Cartesian coordinates is $( x,y,z)$ on the Cartesian basis $( \boldsymbol {e}_{x},\boldsymbol {e}_{y},\boldsymbol {e}_{z})$. The vertical axis along $z$ is the direction of the background static magnetic field $\boldsymbol {B=}B\boldsymbol {e}_{z}$ and the DC electric field $\boldsymbol {E}=E\boldsymbol {e}_{r}$ is along the radial direction. This is the configuration illustrated in figure 1.

Electromagnetic waves can carry both SAM – associated with right-hand (R) and left-hand (L) circular polarizations – and OAM (Gough 1986; Barnett & Allen 1994; van Enk & Nienhuis 1994; Götte, Barnett & Padgett 2007; Barnett, Babiker & Padgett 2017). The vector field of such a wave has a helical phase front structure and can in general be written as

(2.1)$$\begin{gather} \mathrm{Re}\left[\frac{\boldsymbol{e}_{x}\pm j\boldsymbol{e}_{y}}{\sqrt{2}}\mathcal{E}_{{\pm} }(r)\exp j\left( n\alpha +\beta z-\omega t\right)\right]\!. \end{gather}$$

Here $( \boldsymbol {e}_{x}+j\boldsymbol {e}_{y}) \mathcal {E}_{+}$ is a R circularly polarized field while $( \boldsymbol {e}_{x}-j\boldsymbol {e}_{y}) \mathcal {E}_{-}$ is a L circularly polarized field, the electric field $\mathcal {E}(r)$ is the solution of the radial part of Maxwell's equations, $\omega$ is the wave frequency, $n\in \mathbb {Z}$ is the azimuthal mode number and $\beta \in \mathbb {R}$ is the axial wavevector. An observer located at a fixed point $( r_{0},\alpha _{0},z_{0})$ and probing azimuthally the electric field amplitude $| \mathcal {E}_{\pm }(r_{0})|$ of the wave described by (2.1) will measure a field pattern $| \mathcal {E}_{\pm }(r_{0})|$ rotating at the angular velocity ${\rm d}\alpha /{\rm d}t$ = $\omega /n$. The formal identification of SAM and OAM contents for the wavevector field given in (2.1), as well as the definition of the associated SAM and OAM operators $\hat {\boldsymbol {S}}$ and $\hat {\boldsymbol {L}}$, are discussed in Appendix A. Plasma waves carrying OAM has been an active research topic in the last decade, both in unmagnetized (Mendonça 2012; Chen, Qin & Liu 2017; Bliokh & Bliokh 2022) and magnetized (Shukla 2012; Stenzel & Urrutia 2015; Stenzel 2016) plasmas.

Meanwhile, magnetized charged particles in axisymmetric fields can also carry both SAM and OAM. The former is associated with the cyclotron motion while the latter is associated with the guiding centre motion around the $z$ axis. Note that for classical particles, the separation of angular momentum into cyclotron SAM and drift OAM does not arise from a quantum analysis. It is a simple application of Koenig's theorem which states that the angular momentum of a system can be decomposed into an external orbital part and an internal part. This internal part is nowadays called the spin part for waves and magnetized charges, even within a classical framework. Note also that while the quantum SAM of the charged particles $\pm \hbar /2$ should in principle be considered along with the cyclotron SAM and the drift OAM, it will be neglected in this study as the plasma temperature is assumed to be far larger than $\hbar \omega _{c}$ which is of the order of $10^{-7}$ eV for protons and a magnetic field of 1 T. Consider now more specifically the plasma column shown in figure 1 with a background axial magnetic field $\boldsymbol {B=}B\boldsymbol {e}_{z}$ and a radial electric field $\boldsymbol {E}=E\boldsymbol {e}_{r}$, which leads to a guiding centre $E\times B$ rotation around the $z$ axis. An ion with charge $q$ and mass $M$ is described by (i) its instantaneous position $\boldsymbol {r}$ = $\boldsymbol {R}_{G}$ + $\boldsymbol {\rho }_{L}$, where $\boldsymbol {R}_{G}$ is the guiding centre position and $\boldsymbol {\rho }_{L}( t)$ the Larmor radius, and (ii) its velocity $\boldsymbol {v}$ = $\boldsymbol {V}_{G}$ + $\boldsymbol {v}_{c}$, where $\boldsymbol {V}_{G}$ $\sim$ $( E/B) \boldsymbol {e}_{\alpha }$ is the guiding centre drift velocity and $\boldsymbol {v}_{c}( t)$ = $\omega _{c}\boldsymbol {e}_{z}\times \boldsymbol {\rho }_{L}$ is the cyclotron velocity with $\omega _{c}=qB/M$ the ion cyclotron frequency. The instantaneous angular momentum is defined as $M\boldsymbol {r}\times \boldsymbol {v}$ and its average $\langle \ \rangle$ over the fast cyclotron motion is

(2.2)$$\begin{gather} M\langle (\boldsymbol{R}_{G}+\boldsymbol{\rho }_{L}) \times (\boldsymbol{V}_{G}+\boldsymbol{v}_{c}) \rangle \boldsymbol{\cdot} \boldsymbol{e} _{z}=MR_{G}^{2}\varpi +M\rho _{L}^{2}\omega _{c}=L_{z}+S_{z}, \end{gather}$$

where $\varpi$ = $( E/BR_{G})$ is the angular $E\times B$ drift velocity. The OAM part of (2.2) is $L_{z}$ $=M_{G}\varpi$ with $M_{G}$ = $MR_{G}^{2}$ the guiding centre moment of inertia of the ion with respect to the $z$ axis. The cyclotron spin part $S_{z}$ is defined as $( 2M/q) \mu$ with $\mu =m\omega _{c}^{2}\rho _{L}^{2}/2B$ the Larmor magnetic moment. One thus recovers the classical gyromagnetic factor $q/2M$.

Coupling between wave and particle angular momentum components introduced above can be either adiabatic or resonant. At the fluid level, linear adiabatic coupling is described by the Hermitian part of the dielectric tensor, while linear resonant coupling is described by the anti-Hermitian part of the dielectric tensor and the quasilinear equation. Starting with adiabatic coupling, coupling between wave SAM and particles SAM leads to the classical Faraday rotation (Chen 1984; Rax 2005). Adiabatic coupling between wave SAM and particles OAM leads to the mechanical Faraday – or polarization drag (Jones 1976) – effect whose properties in plasmas have recently been examined (Gueroult et al. 2019b , 2020). Lastly, adiabatic coupling between wave OAM and particles OAM leads to the Faraday–Fresnel rotation and splitting recently uncovered for Trievelpiece–Gould and helicon modes (Rax & Gueroult 2021). Moving on to resonant coupling, coupling between wave SAM and particles SAM is routinely used for electron and ion cyclotron resonance heating (ECRH/ICRH) in tokamaks (Rax 2011), and has also been proposed for mass separation or particle acceleration (Loeb & Friedland 1986; Pendergast et al. 1988; Rax, Robiche & Fisch 2007; Rax & Robiche 2010). Meanwhile, as already mentioned in the introduction, resonant coupling between wave OAM and particles OAM has been proposed to control rotation in magnetic mirrors (Fetterman & Fisch 2008, 2010) and tokamaks (Ochs & Fisch 2017; Rax et al. 2017).

In this study we will build on and extend these results by deriving the quasilinear kinetic equation which will allow us to identify the exact resonance condition, and from there to uncover couplings between waves and particles SAM and OAM, in a cylindrical rotating magnetized plasma. This new resonance condition completes the already identified set of resonant coupling in plasmas: (i) Landau in unmagnetized plasmas; (ii) cyclotron in magnetized plasmas; and (iii) Compton in laser-driven plasmas (Rax 1992). Because our motivation is primarily in rotating mirrors, straight tokamaks and mass filters where the resonant population is the ion population, we will consider a non-relativistic framework. Under this assumption, we will show that FLR effects are responsible for a mixing of OAM and SAM couplings, underlining that rotating magnetized plasmas feature a more complex angular momentum dynamics than unmagnetized plasmas or ordinary neutral matter.

3. Hamiltonian description of a rotating plasma

In this section we lay out the Hamiltonian description of an unperturbed rotation driven by a DC radial electric field in an axially magnetized plasma column. The axial magnetic field is assumed to be produced by a set of coils carrying azimuthal DC currents at the edge of the plasma column. The radial electric field may be generated through DC polarized concentric electrodes at ends of the plasma, provided that the criterion for electric field penetration is fulfilled (Gueroult et al. 2019a ; Poulos 2019; Liziakin et al. 2020, 2021; Trotabas & Gueroult 2022). Alternatively, in a non-neutral plasma (Davidson 2001), the electric field is simply the space charge field and there is no need for concentric electrodes. We focus on the ion population in a quasineutral plasma but results can be easily extended to the electron population in quasineutral and non-neutral plasmas.

3.1. Brillouin modes

An ion of mass $M$ and charge $q>0$, interacts with a static radial linear electric field $\boldsymbol {E}$ and an axial uniform magnetic field $\boldsymbol {B}$ as shown in figure 1 and defined by

(3.1)$$\begin{gather} \frac{q}{M}\boldsymbol{E} =\left(\frac{qE}{Mr}\right) r \boldsymbol{e}_{r}, \end{gather}$$

(3.2)$$\begin{gather}\frac{q}{M}\boldsymbol{B} =\omega_{c}\boldsymbol{e}_{z}. \end{gather}$$

Ion orbits are then a combination of the slow and fast Brillouin rotations (Davidson & Krall 1969; Davidson 2001). The fast $\varOmega _{+}$ and slow $\varOmega _{-}$ angular velocities associated with these fast and slow rotations are given by

(3.3)$$\begin{gather} \varOmega _{{\pm} } ={-}\frac{\omega _{c}}{2}\mp \sqrt{\frac{\omega _{c}^{2}}{4}-\frac{qE}{rM}} \underset{\left|\frac{E}{rB}\right| \ll \omega _{c}}{\approx} -\frac{\omega _{c}~}{2}\mp \frac{\omega _{c}}{2}\pm \frac{E}{rB}, \end{gather}$$

where $E$ is the DC radial electric field at a given radius $r$ and $4qE< Mr\omega _{c}^{2}$ is the classical Brillouin condition (Davidson 2001). These two solutions are plotted as a function of the normalized electric field in figure 2. In the weak electric field regime $| E/(rB)| \ll \omega _{c}$ highlighted in grey in figure 2, the angular velocity of the guiding centre around the $z$ axis reduces to the classical $E\times B$ drift while the angular velocity of the cyclotron motion around the guiding centre reduces to the usual cyclotron motion. This corresponds to the asymptotic limit on the right-hand side of (3.3).

Figure 2. The slow and fast angular velocity as a function of the electric field. A clear separation between guiding centre and Larmor radius is relevant for weak electric field in the grey zone.

From (3.1) the Brillouin limit leads to the simple requirement $\varOmega ^{2}>0$ with

(3.4)$$\begin{gather} \varOmega =\sqrt{\frac{q^{2}B^{2}}{M^{2}}-4\frac{qE}{Mr}}. \end{gather}$$

Note that $\varOmega$, the gyrofrequency $\omega _{c}$ and the wave frequency $\omega$ are all assumed positive throughout this study. Another way to see the condition $\varOmega >0$ is to realize that for a given field configuration ((3.1) and (3.2)) the cutoff mass $M^{*}$ between radially unconfined and radially confined ions is the solution of $\varOmega ^{2}( M^{*}) =0$. Ions such that $M< M^{*}$ remain confined around the axis of the configuration. On the other hand ions such that $M>M^{*}$ are expelled radially at an exponential rate. The assumption in this study of $M< M^{*}$ or $\varOmega >0$ is thus a requirement to study radially bounded trochoidal orbits.

With the definition of $\varOmega$ in (3.4) the usual slow and fast Brillouin modes given by (3.3) rewrite as

(3.5)$$\begin{gather} \varOmega _{{\pm} }={-}\frac{\omega _{c}\pm \varOmega }{2}. \end{gather}$$

One verifies that $\varOmega _{+}+\varOmega _{-}$ = $-\omega _{c}$ and $\varOmega _{+}-\varOmega _{-}$ = $-\varOmega$. Note also that $\varOmega _{+}<0$ and $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {E}=2\varOmega _{+}\varOmega _{-}$. The uniform charge density $2\varepsilon _{0}M\varOmega _{+}\varOmega _{-}/q$ is the small deviation from quasineutrality responsible for the radial electric field.

3.2. Hamiltonian description

Consider now a system of units such that $q=1$ and $M=1$. In this simple system of units, the electric field and the magnetic field given in (3.1) and (3.2) derive, respectively, from the scalar potential

(3.6)$$\begin{gather} \varPhi =\frac{\varOmega ^{2}-\omega _{c}^{2}}{8}(x^{2}+y^{2}) \end{gather}$$

and the vector potential

(3.7)$$\begin{gather} \boldsymbol{A} =\frac{\omega _{c}}{2}(x\boldsymbol{e}_{y}-y\boldsymbol{e}_{x}). \end{gather}$$

The unperturbed Hamiltonian $H_{0}$ is classically the sum of the kinetic energy $\boldsymbol {v}^{2}/2$ plus the potential energy $\varPhi ( \boldsymbol {r})$, that is

(3.8)$$\begin{gather} H_{0}\left( \boldsymbol{p},\boldsymbol{r}\right) =\tfrac{1}{2}\boldsymbol{v}^{2}+\varPhi =\tfrac{1}{2}[\boldsymbol{p-A}(\boldsymbol{r})] ^{2}+\varPhi\left( \boldsymbol{r}\right)\!, \end{gather}$$

where $\boldsymbol {v}$ is the velocity and $\boldsymbol {p}=p_{x}\boldsymbol {e}_{x}+p_{y} \boldsymbol {e}_{y}+p_{z}\boldsymbol {e}_{z}$ is the canonical momentum conjugated to the position $\boldsymbol {r}=x\boldsymbol {e}_{x}+y\boldsymbol {e}_{y}+z\boldsymbol {e}_{z}$ of the ion. In Cartesian coordinates (3.8) rewrites as

(3.9)$$\begin{gather} H_{0}=\frac{1}{2}( p_{x}^{2}+p_{y}^{2}) +\frac{\omega _{c}}{2}( yp_{x}-xp_{y}) +\frac{\varOmega ^{2}}{8}( x^{2}+y^{2}) +\frac{p_{z}^{2}}{2}. \end{gather}$$

This is a quadratic form of the Cartesian momentum and positions variables, so that $H_{0}$ is integrable (Rax 2021).

Let us now introduce the canonical change of variables from the old Cartesian momentum $( p_{x},p_{y},p_{z}=P)$ and positions $(x,y,z)$ to the new actions $( J,D,P)$ and angles $( \varphi,\theta,z)$ variables defined by

(3.10a,b)$$\begin{align} x =\sqrt{\frac{2D}{\varOmega }}\cos \theta -\sqrt{\frac{2J}{\varOmega }}\cos \varphi, \quad y=\sqrt{\frac{2D}{\varOmega }}\sin \theta +\sqrt{\frac{2J}{\varOmega }}\sin \varphi, \end{align}$$

(3.11a,b)$$\begin{align} p_{x} ={-}\sqrt{\frac{\varOmega D}{2}}\sin \theta +\sqrt{\frac{\varOmega J}{2}}\sin \varphi, \quad p_{y}=\sqrt{\frac{\varOmega D}{2}}\cos \theta +\sqrt{\frac{\varOmega J}{2}}\cos \varphi, \end{align}$$

with $J\geq 0$, $D\geq 0$, $\varphi \in [ 0,2{\rm \pi} ]$ and $\theta \in [ 0,2{\rm \pi} ]$. By plugging (3.10a,b ) and (3.11a,b ) into (3.9) one simply gets

(3.12)$$\begin{gather} H_{0}={-}\varOmega _{+}J+\varOmega _{-}D+\tfrac{1}{2}P^{2}. \end{gather}$$

This result is independent of the angles $( \varphi, \theta, z)$ as expected. Note also from (3.5) that the cyclotron (kinetic) part of the energy $-\varOmega _{+}J$ is always positive, but that the drift (potential) part $\varOmega _{-}D$ can be either positive or negative. The particle velocity perpendicular to the magnetic field $\boldsymbol {v}$ defined as $p_{x}\boldsymbol {e}_{x}+p_{y}\boldsymbol {e}_{y}-\boldsymbol {A}$ and the polar radius $r$ defined as $\sqrt {x^{2}+y^{2}}$ are then obtained from a simple substitution of (3.10a,b ) and (3.11a,b ) in the Cartesian definitions, leading to

(3.13)$$\begin{align} \boldsymbol{v} & =\left( -\varOmega _{+}\sqrt{\frac{2J}{\varOmega }}\sin \varphi -\varOmega _{-}\sqrt{\frac{2D}{\varOmega }}\sin \theta \right) \boldsymbol{e}_{x}\nonumber\\ & \quad +\left( -\varOmega _{+}\sqrt{\frac{2J}{\varOmega }}\cos \varphi +\varOmega _{-}\sqrt{\frac{2D}{\varOmega }}\cos \theta \right) \boldsymbol{e}_{y} \end{align}$$

and

(3.14)$$\begin{gather} r^{2}=x^{2}+y^{2}=2\frac{J+D}{\varOmega }-4\frac{\sqrt{JD}}{\varOmega }\cos \left( \theta +\varphi \right). \end{gather}$$

Having identified a set of canonical angles $( \varphi,\theta, z)$ and actions $( J,D,P)$ variables describing the ion interaction with the electric and magnetic field given in (3.6) and (3.7), we can now try to shed light onto the physical meaning of the new variables. Starting with $z$ and $P$, they are, respectively, the usual Cartesian coordinate and momentum $P=Mv_{z}$, and their physical interpretation is thus straightforward. The meaning of $( J,\varphi )$ and $(D,\theta )$ is on the other hand less obvious. To help our interpretation, figure 3 shows the ion motion in the $(x,y)$ plane when $D>J$. The instantaneous position of the ion is $\boldsymbol {r}=r\boldsymbol {e}_{r}=\boldsymbol {OC}$ and it can be viewed as the sum of a rotating Larmor radius $\boldsymbol {GC}$ plus a rotating guiding centre $\boldsymbol {OG}$. From figure 3, $\theta$ is the anticlockwise angle between $\boldsymbol {e}_{x}$ and $\boldsymbol {OG}$, and $\varphi$ is the clockwise angle between $-\boldsymbol {e}_{x}$ and $\boldsymbol {GC}$. We then find from (3.10a,b ) that the guiding centre $| \boldsymbol {OG}|$ = $\sqrt {2D/\varOmega }$, and that the Larmor radius $| \boldsymbol {GC}|$ = $\sqrt {2J/\varOmega }$. Equation (3.14) is just the law of cosines applied to the $\boldsymbol {OGC}$ triangle with respect to the grey angle in figure 3.

Figure 3. Physical meaning of the angle $(\varphi,\theta )$ and actions $(J < D)$ variables in real $(x, y)$ space.

The geometrical interpretation proposed above for $( J,\varphi )$ and $(D,\theta )$ based on figure 3 assumed $D>J$. If one now considers $J>D$, the canonical description ((3.10a,b )–(3.12)) is still valid, but the picture of the orbit is to be replaced by the one shown in figure 4. As we will show in the next section these two regimes $J\lessgtr D$ can be discriminated based on the sign of the particle canonical angular momentum. In effect most of the physical interpretations made in this study will be argued with the ordering $D>J$ in mind as it is the most intuitive, but one should keep in mind that all the relations are valid in both cases $J\lessgtr D$.

Figure 4. Physical meaning of the angle $(\varphi,\theta )$ and actions $(J > D)$ variables in real $(x, y)$ space.

Finally, one verifies that Hamilton's equations

(3.15a,b)$$\begin{gather} \frac{{\rm d}\theta }{{\rm d}t}=\frac{\partial H_{0}}{\partial D}=\varOmega _{-}, \quad \frac{{\rm d}\varphi }{{\rm d}t}=\frac{\partial H_{0}}{\partial J}={-}\varOmega _{+}, \end{gather}$$

lead to the expected classical Brillouin results (Davidson 2001). The minus sign for the fast (cyclotron) rotation is simply due to the choice of a clockwise angle for $\varphi$ (the anticlockwise choice for $\theta$). It must be stressed here though that the Larmor radius angle $\varphi$ does not rotate at the cyclotron frequency $-\varOmega _{+}\neq \omega _{c}$. Similarly the $\theta$ angle of the guiding centre does not rotate with the $E\times B$ velocity $\varOmega _{-}\neq -E_{r}/rB$. This is the consequence of inertial effects. The interpretation of the motion as a slow $E\times B$ drift $\varOmega _{-}\approx$ $-E_{r}/rB$ plus a fast cyclotron rotation $\varOmega _{+}\approx -\omega _{c}$ is thus only meaningful in the weak electric field limit $| E_{r}/B| \ll r\omega _{c}$ highlighted in grey in figure 2.

3.3. Weak field limit

In order to develop a deeper physical understanding of the weak field regime, which is the one of experimental interest for quasineutral plasmas applications, let us write the ion orbit as a combination of an $E\times B$ slow rotation plus a fast cyclotron rotation. We introduce the guiding centre radius $R_{G}$ and the Larmor radius $\rho _{L}$:

(3.16)$$\begin{gather} x =R_{G}\cos \left( -\frac{E}{rB}t\right) -\rho _{L}\cos (\omega _{c}t), \end{gather}$$

(3.17)$$\begin{gather}y =R_{G}\sin \left( -\frac{E}{rB}t\right) +\rho _{L}\sin (\omega _{c}t). \end{gather}$$

Equations (3.16) and (3.17) are simply a rewriting of (3.10a,b ) in the weak field limits $| E_{r}/rB| \ll \omega _{c}$. In this weak field limit $\varOmega \approx \omega _{c}$, and the actions $J$ and $D$ can be, respectively, related to the cyclotron orbit magnetic flux $\varPsi _{L}$ = ${\rm \pi} \rho _{L}^{2}B$ and the guiding centre orbit magnetic flux $\varPsi _{G}$ = ${\rm \pi} R_{G}^{2}B$, with

(3.18)$$\begin{gather} D \approx \frac{\omega _{c}}{2}R_{G}^{2}=\frac{\varPsi _{G}}{2{\rm \pi} }, \end{gather}$$

(3.19)$$\begin{gather}J \approx \frac{\omega _{c}}{2}\rho _{L}^{2}=\frac{\varPsi _{L}}{2{\rm \pi} }. \end{gather}$$

Note that these two magnetic fluxes are the first and third adiabatic invariant of Alfven's theory. The usual link between action variables and adiabatic invariants is thus recovered.

Assuming further $J\ll D$, that is a small Larmor radius, and reintroducing momentarily the ion mass and charge $M$ and $q$ for clarity, one gets in the weak field regime

(3.20)$$\begin{gather} -\varOmega _{+}J \approx \frac{M}{2}v_{c}^{2}, \end{gather}$$

(3.21)$$\begin{gather}\varOmega _{-}D \approx q\varPhi \left( R_{G}\right) +\frac{M}{2}\frac{E^{2}}{ B^{2}}. \end{gather}$$

Plugging these results into (3.12) yields

(3.22)$$\begin{gather} H_{0}\underset{|\frac{E}{rB}| \ll \omega _{c}}{\approx}\frac{M}{2}v_{c}^{2}+\frac{1}{2M}P^{2}+\frac{M}{2}\frac{E^{2}}{B^{2}}+q\varPhi. \end{gather}$$

The Hamiltonian thus reduces in this limit to the sum of four terms: the cyclotron kinetic energy $Mv_{c}^{2}/2$; the electrostatic potential energy $q\varPhi (R_{G})$ of the guiding centre; the drift energy $ME^{2}/2B^{2}$; and the parallel kinetic energy .

If one considers now a wave perturbation $\delta H_{0}$ of the Hamiltonian $H_{0}$, (3.22) shows that in the weak field limit this perturbation is associated with an increase or a decrease of (i) the kinetic cyclotron energy $Mv_{c}\delta v_{c}$, (ii) the potential energy $q\delta \varPhi$ and (iii) the axial kinetic energy $Mv_{z}\delta v_{z}$. The structure of the unperturbed Hamiltonian (3.12) indeed offers in this case the possibility to transfer axial linear momentum, kinetic energy $-\varOmega _{+}\delta J\approx Mv_{c}\delta v_{c}$ or/and potential energy $\varOmega _{-}\delta D\approx q\delta \varPhi$ from rotating waves to rotating particles. When the weak field approximation is no longer valid, inertia effects make this picture more intricate, and the angle–action variables $( \varphi,\theta,z)$ and $( J,D,P)$ then provide the right framework to understand the dynamics. In order to simplify the algebra, we define $\boldsymbol {J}$, $\boldsymbol {\theta }$ and $\boldsymbol {\varOmega }$ such that

(3.23)$$\begin{gather} \boldsymbol{J} =\left( J,D,P\right)\!, \end{gather}$$

(3.24)$$\begin{gather}\boldsymbol{\theta }=\left( \varphi ,\theta ,z\right)\!, \end{gather}$$

(3.25)$$\begin{gather}\boldsymbol{\varOmega } =\frac{{\rm d}\boldsymbol{\theta }}{{\rm d}t}=\frac{\partial H_{0}}{ \partial \boldsymbol{J}}=\left( -\varOmega _{+},\varOmega _{-},P\right)\!. \end{gather}$$

To conclude this section we note that as $H_{0}$ is integrable there exists an infinite set of canonical angle and action variables. The particular choice of $\boldsymbol {\theta }$ and $\boldsymbol {J}$ is simply motivated by their straightforward geometrical interpretation, as illustrated in figures 3 and 4, and their clear physical meaning in the weak field regime.

4. Canonical and kinetic angular momentum

From (3.10a,b ), (3.11a,b ) and (3.13)), the $z$ components of the canonical angular momentum and of the kinetic angular momentum, respectively, write

(4.1)$$\begin{gather} L_{C}=xp_{y}-yp_{x}=D-J \end{gather}$$

and

(4.2)$$\begin{gather} L_{K}=xv_{y}-yv_{x}=\frac{2\varOmega _{-}}{\varOmega }D+\frac{2\varOmega _{+}}{\varOmega }J+\frac{2\omega _{c}}{\varOmega }\sqrt{JD}\cos \left( \theta +\varphi \right). \end{gather}$$

As noted in the previous section, the ordering of $J$ and $D$ depends on the sign of $L_{C}$. Specifically, $L_{C}>0$ leads to the orbit topology illustrated in figure 3 while $L_{C}<0$ leads to the orbit topology illustrated in figure 4. Note also that $L_{C}$ is independent of time as a consequence of the cylindrical symmetry, but that $L_{K}$ is a function of time since $\theta +\varphi$ = $( \varOmega _{-}-\varOmega _{+}) t$.

A physical interpretation of (4.2) can be brought up by considering the moment of inertia of a rotating ion with mass $M=1$. Seeing again the rotating ion as the sum of a rotating guiding centre and a rotating Larmor radius, the guiding centre moment of inertia is $M_{G}$ = $2D/\varOmega$, while the Larmor radius moment of inertia is $M_{L}$ = $2J/\varOmega$. The associated angular momenta are $M_{G}\,{\rm d}\theta /{\rm d}t$ = $M_{G}\varOmega _{-}$ ($\theta$ is anticlockwise) and $-M_{L}\,{\rm d}\varphi /{\rm d}t$ = $M_{L}\varOmega _{+}$ ($\varphi$ is clockwise). One verifies that the sum of these two angular momenta $M_{G}\varOmega _{-}$ + $M_{L}\varOmega _{+}$ indeed matches $\langle L_{K}\rangle$ computed from (4.2), where the average $\langle {}\rangle$ is over the angle $\theta +\varphi$.

Defining the magnetic flux

(4.3)$$\begin{gather} \varPsi = \oint_{\mathcal{C}}\boldsymbol{A}\boldsymbol{\cdot}\,{\rm d}\boldsymbol{l=}\frac{\omega _{c}}{2}\oint \left( x\,{\rm d}y-y\,{\rm d}x\right) = {\rm \pi}\omega _{c}r^{2} \end{gather}$$

with $\mathcal {C}$ a contour along a $\Delta \alpha =2{\rm \pi}$ full turn of the orbit and $r^{2}$ given by (3.14), then (3.14), (4.1) and (4.2) can be further used to write

(4.4)$$\begin{gather} L_{C}=L_{K}+\frac{\varPsi }{2{\rm \pi} }. \end{gather}$$

We thus see that the criteria $L_{C}\lessgtr 0$, which was identified as determining the type of orbit topology (either that shown in figure 3 or that shown in figure 4), can be recast as $2{\rm \pi} L_{K}\lessgtr -q\varPsi$. This last condition can be interpreted as an ordering between the kinetic energy and the magnetic coupling.

5. Hamiltonian description of a rotating wave

We now consider a wave perturbation associated with a rotating and propagating potential

(5.1)$$\begin{gather} \phi \left( \boldsymbol{r},t\right) =\mathrm{Re}[\phi ( r) \exp j( n\alpha +\beta z-\omega t) ], \end{gather}$$

and a $R$ or $L$ vector potential

(5.2)$$\begin{gather} \boldsymbol{a}_{{\pm} }\left( \boldsymbol{r},t\right) =\mathrm{Re} \left[a\left( r\right) \exp j\left( n\alpha +\beta z-\omega t\right) \frac{\boldsymbol{e}_{x}\pm j \boldsymbol{e}_{y}}{\sqrt{2}}\right], \end{gather}$$

where $n\in \mathbb {Z}$ and $\beta \in \mathbb {R}$. The function $a( r)$ is the real amplitude solution of the radial part of Maxwell–Ampère equation. The Maxwell–Faraday equation is fulfilled through $\mathcal {E} =-\partial \boldsymbol {a}_{\pm }/\partial t$ and $\mathcal {B}=\boldsymbol {\nabla } \times \boldsymbol {a}$ where $\mathcal {E}$ and $\mathcal {B}$ are the wave electric and magnetic fields. The function $\phi ( r)$ is the solution of Poisson equation. Such solutions of Maxwell–Ampère and Poisson equations were recently identified for the whistler or helicon branch and the Trievelpiece–Gould modes in a rotating plasma (Rax & Gueroult 2021). As shown in Appendix A, the wave $\boldsymbol {a}_{\pm }( \boldsymbol {r},t)$ is both an SAM and OAM eigenfunction since

(5.3)$$\begin{gather} (\hat{L}_{z}+\hat{S}_{z}) \boldsymbol{\cdot} \boldsymbol{a}_{{\pm} }=\left( n\mp 1\right) \boldsymbol{a}_{{\pm} }, \end{gather}$$

and the scalar potential wave $\phi ( \boldsymbol {r},t)$ is an OAM eigenfunction.

The perturbed Hamiltonian $H$ describing the interaction of an ion with the DC confining fields ((3.6) and (3.7)) and the rotating RF waves ((5.1) and (5.2)) then writes as

(5.4)$$\begin{gather} H=\frac{1}{2}\left( \boldsymbol{p-A}\right) ^{2}+\varPhi -\left( \boldsymbol{p-A} \right) \boldsymbol{\cdot} \boldsymbol{a}_{{\pm} }+\frac{\boldsymbol{a}_{{\pm} }^{2}}{2}+\phi . \end{gather}$$

In the following we neglect the second-order ponderomotive part of the interaction

(5.5)$$\begin{gather} \boldsymbol{a}_{{\pm} }^{2}/2\ll \boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{a}_{{\pm} } \end{gather}$$

but keep the first-order dipolar coupling $\boldsymbol {v}\boldsymbol {\cdot } \boldsymbol {a}_{\pm }+\phi$ which is responsible for the quasilinear resonant exchange of energy and momentum between waves and particles. This separation between dipolar and ponderomotive perturbations is usual and we will not explore here the interplay between these two couplings (Ochs & Fisch 2021a , b , 2022, 2023) which is associated with the transient build up of the wave. Defining

(5.6)$$\begin{gather} \boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{a}_{{\pm} }+\phi = v_{x}a_{{\pm} x}+v_{y}a_{{\pm} y}+\phi, \end{gather}$$

and assuming that $V_{n} \ll H_{0}$, we then write

(5.7)$$\begin{gather} H=H_{0}\left( \boldsymbol{J}\right) +V_{n}\left( \boldsymbol{J},\boldsymbol{\theta } ,t\right)\!, \end{gather}$$

that is that $H$ is decomposed into an unperturbed part $H_{0}$ given in (3.12) and the wave perturbation. Using (3.13), (5.1) and (5.2), one gets for the wave perturbation

(5.8)$$\begin{align} V_{n} & =\mathrm{Re}\left[{\mp} ja\left( r\right) \left[ \varOmega _{+}\sqrt{\frac{J}{ \varOmega }}\exp \left({\mp} j\varphi \right) -\varOmega _{-}\sqrt{\frac{D}{\varOmega } }\exp \left({\pm} j\theta \right) \right] \exp j\left( n\alpha +\beta z-\omega t\right)\right]\nonumber\\ & \quad +\mathrm{Re}\left[\phi \left( r\right) \exp j\left( n\alpha +\beta z-\omega t\right) \right]. \end{align}$$

The last step is to write both $a( r)$ and $\phi (r)$ in terms of $( J,D)$ and $( \varphi,\theta )$ to write $V_{n}$ in a form suitable to carry out the quasilinear analysis. This requires finding a convenient basis to express $a( r)$ and $\phi ( r)$. For quasilinear theory in an homogeneous plasma at rest, this basis is a Fourier set of plane waves associated with translation invariance. Within the framework of random phase approximation each Fourier component then acts separately in the quasilinear diffusion operator which is a sum over the square of the amplitude of these Fourier components.

In the case of interest the Fourier–Bessel expansion seems more natural given the cylindrical symmetry of the problem. An added motivation for this choice is that recent studies on the OAM Faraday–Fresnel effect (Rax & Gueroult 2021) have shown that the eigenmodes of the whistler branch in a rotating plasma are of the type described by (5.2) with in this case $a( r)$ the ordinary Bessel function ${\rm J}_{n}( kr)$, and that the eigenmodes of the Trievelpiece–Gould branch in a rotating plasma are of the type described by (5.1) with in this case $\phi ( r)$ proportional to ${\rm J}_{n}(kr)$, with $k$ fulfilling in each case an appropriate dispersion relation. Lastly, the Fourier–Bessel expansion theorem states that all the other branches of the plasma waves spectrum in a rotating plasma can similarly be written with waves of the type (5.1) and (5.2), and that $a( r)$ can be represented by $\tilde {a}( k)$ as the sum $a(r)$ = $\int k\tilde {a}( k) {\rm J}_{n}( kr) \,{\rm d}k$ with $\tilde {a}( k)=\int ra( r){\rm J}_{n}( kr) \,{\rm d}r$. Thus, without loss of generality, we consider transverse and longitudinal rotating and propagating cylindrical waves of the type

(5.9)$$\begin{gather} \boldsymbol{a}_{{\pm} }\left( \boldsymbol{r},t\right) =\mathrm{Re}\left[\frac{\mathcal{E} \left( \omega \right) }{j\omega }{\rm J}_{n}\left( kr\right) \exp j\left( n\alpha +\beta z-\omega t\right) \frac{\boldsymbol{e}_{x}\pm j\boldsymbol{e}_{y}}{\sqrt{2}}\right] , \end{gather}$$

(5.10)$$\begin{gather}\phi \left( \boldsymbol{r},t\right) =\mathrm{Re}\left[\phi \left( \omega \right) {\rm J}_{n}\left( kr\right) \exp j\left( n\alpha +\beta z-\omega t\right)\right] . \end{gather}$$

Here $\mathcal {E}( \omega )$ and $\phi ( \omega )$ are the spectral component of the transverse electric field and potential of a given cylindrical wave packet. The final quasilinear operator, quadratic in $\mathcal {E}$ and $\phi$, will ultimately be summed over the full $\omega$ and $k( \omega )$ spectra. In this study though we do not specify the $\omega$ spectra and the $k( \omega,\beta,n)$ dispersion. To provide a general result we instead analyse the quasilinear dynamics under the influence of a single cylindrical component ((5.9) and (5.10)). The quasilinear effect of a wave packet is simply the sum over the effects of each Fourier–Bessel component as in the plane wave case.

In a homogeneous magnetized plasma at rest the Jacobi–Anger expansion is used to identify the harmonic cyclotron resonances of a plane wave. For the cylindrical waves described by (5.9) and (5.10) we instead consider the triangle $\boldsymbol {OGC}$ in figures 3 and 4 and apply Graf's addition theorem

(5.11)$$\begin{gather} {\rm J}_{n}\left( kr\right) \exp \left( jn\alpha \right) =\sum_{l={-}\infty}^{l={+}\infty }{\rm J}_{l+n}\left( k\sqrt{\frac{2D}{\varOmega }}\right) {\rm J}_{l}\left( k\sqrt{\frac{2J}{\varOmega }}\right) \exp j\left[ \left( l+n\right) \theta +l\varphi \right] \!. \end{gather}$$

In the limit of zero Larmor radius the sum on the right-hand side reduces to the $l=0$ term only, whereas $l\neq 0$ terms capture FLR effects. One verifies that the larger $k$ and the larger $J/D$, the more terms are needed to approach the left-hand side in (5.11). Plugging this result into (5.7) and (5.8), the Hamiltonian writes as

(5.12)$$\begin{gather} H=H_{0}\left( \boldsymbol{J}\right) +\mathrm{Re}\left[\sum_{l={-}\infty }^{l={+}\infty }V_{nl\sigma }\left( J,D\right) \exp j\left[ \left( l+\sigma \right) \varphi +\left( l+n\right) \theta +\beta z-\omega t\right] \right]\!, \end{gather}$$

where we introduced $\sigma \in [ -1,0,+1]$ the SAM index such that the L, R and scalar wave perturbations $V_{nl\sigma }$ are given by

(5.13)$$\begin{align} V_{n,l,\sigma ={+}1} & =\mathcal{E}\frac{\varOmega _{+}}{\omega }\sqrt{\frac{J}{ \varOmega }}{\rm J}_{l+n}\left( k\sqrt{\frac{2D}{\varOmega }}\right) {\rm J}_{l}\left( k\sqrt{ \frac{2J}{\varOmega }}\right) \nonumber\\ & \quad-\mathcal{E}\frac{\varOmega _{-}}{\omega }\sqrt{\frac{D}{\varOmega }} {\rm J}_{l+1+n}\left( k\sqrt{\frac{2D}{\varOmega }}\right) {\rm J}_{l+1}\left( k\sqrt{\frac{ 2J}{\varOmega }}\right), \end{align}$$

(5.14)$$\begin{align}V_{n,l,\sigma ={-}1} & =\mathcal{E}\frac{\varOmega _{-}}{\omega }\sqrt{\frac{D}{ \varOmega }}{\rm J}_{l-1+n}\left( k\sqrt{\frac{2D}{\varOmega }}\right) {\rm J}_{l-1}\left( k \sqrt{\frac{2J}{\varOmega }}\right) \nonumber\\ & \quad-\mathcal{E}\frac{\varOmega _{+}}{\omega }\sqrt{\frac{J}{\varOmega }} {\rm J}_{l+n}\left( k\sqrt{\frac{2D}{\varOmega }}\right) {\rm J}_{l}\left( k\sqrt{\frac{2J}{ \varOmega }}\right) , \end{align}$$

(5.15)$$\begin{align} V_{n,l,\sigma =0} & =\phi {\rm J}_{l+n}\left( k\sqrt{\frac{2D}{\varOmega }}\right) {\rm J}_{l}\left( k\sqrt{\frac{2J}{\varOmega }}\right) . \end{align}$$

In order to simplify the notation we define the vector

(5.16)$$\begin{gather} \boldsymbol{N}=\left[ \left( l+\sigma \right) ,\left( l+n\right) ,\beta \right] \end{gather}$$

and use it as an index to specify $n$, $l$ and $\sigma$. The Hamiltonian $H$ in (5.7) then writes in compact form as

(5.17)$$\begin{gather} H\left( \boldsymbol{J},\boldsymbol{\theta },t\right) =H_{0}\left( \boldsymbol{J}\right) +\sum_{\boldsymbol{N}}V_{\boldsymbol{N}}\left( \boldsymbol{J}\right) \exp j\left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\theta}-\omega t\right) , \end{gather}$$

where we dropped the $\mathrm {Re}$ mention for readability and $\sum _{\boldsymbol {N}}= \sum _{l=-\infty }^{l=+\infty }$. This implies that both the OAM azimuthal number $n$ of the cylindrical waves ((5.9) and (5.10)) and the SAM number $\sigma$ remain fixed as we study separately L, R and potential waves.

For a purely rotating wave, characterized by $n$, $\sigma$ and $\beta =0$, the structure of the relation (5.17) reveals a number of selection rules between the wave-induced small increments of actions and energy. These selection rules provide also the branching ratio between the exchange of the actions. To see this consider Hamilton's equations:

(5.18)$$\begin{gather} \frac{{\rm d}H}{{\rm d}t} =\frac{\partial H}{\partial t}={-}j\omega \sum_{\boldsymbol{N}}V_{ \boldsymbol{N}}\left( \boldsymbol{J}\right) \exp j\left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{ \theta }-\omega t\right)\!, \end{gather}$$

(5.19)$$\begin{gather}\frac{{\rm d}L_{C}}{{\rm d}t} =\frac{\partial H}{\partial \varphi }-\frac{\partial H}{ \partial \theta }=j\left( \sigma -n\right) \sum_{\boldsymbol{N}}V_{\boldsymbol{N} }\left( \boldsymbol{J}\right) \exp j\left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\theta} -\omega t\right)\!. \end{gather}$$

A wave-induced small variation of the ion energy $\delta H$ thus implies a small variation of the angular momentum $\delta L_{C}$ through

(5.20)$$\begin{gather} \frac{\delta L_{C}}{\delta H}=\frac{n-\sigma }{\omega }. \end{gather}$$

Using $\delta H=\varOmega _{-}\delta D-\varOmega _{+}\delta J$ we can express the branching ration between the cyclotron kinetic energy channel and the potential energy channel as

(5.21)$$\begin{gather} \frac{\delta J}{\delta H} =\frac{l+\sigma }{\omega }, \end{gather}$$

(5.22)$$\begin{gather}\frac{\delta D}{\delta H} =\frac{l+n}{\omega }. \end{gather}$$

These relations provide a first basic tool to optimize phase space engineering. For example one may want to set up a radial current ($\delta D\neq 0$) but avoid ICRH heating ($\delta J=0$). In this case, (5.21) and (5.22) point to waves such that $l+\sigma =0$ but $l+n\neq 0$. However, the best strategy to optimize power deposition among the various energy channels is to consider the kinetic equation.

6. Brillouin resonances

6.1. Resonance condition

The Hamiltonian (5.17) makes it possible to identify the conditions for resonant coupling. For this we simply substitute the unperturbed motion $\boldsymbol {\theta }= \boldsymbol {\varOmega }t + \boldsymbol {\theta }_{0}$ into the oscillating phase $j( \boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\theta }-\omega t)$ of each $V_{\boldsymbol {N}}\exp j( \boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\theta }-\omega t)$ perturbation. We then obtain the phase factor $j( \boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\varOmega }-\omega ) t$ + $j\boldsymbol {\theta }_{0}$ which stops to rotate and becomes stationary when the resonance condition

(6.1)$$\begin{gather} \omega = \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega} ={-}\left( l+\sigma \right) \varOmega _{+}+\left( l+n\right) \varOmega _{-}+\beta P, \end{gather}$$

is fulfilled. This resonance condition replaces the classical Landau-cyclotron condition

(6.2)$$\begin{gather} \omega -\beta v_{z}=m\omega _{c} \end{gather}$$

with $m \in \mathbb {Z}$ and where $v_{z}$ is the velocity along the magnetic field. Because this is the slow and fast Brillouin modes that are involved in (6.1) rather than the cyclotron frequency we call (6.1) the Brillouin resonance condition. One verifies as expected that (6.1) reduces to (6.2) when the static electric field (3.1) cancels. Rewriting (6.1) as

(6.3)$$\begin{gather} \omega -\beta P-(n+\sigma)\varOmega_{-} = l(\omega_{c}+2\varOmega_{-}) +\sigma \omega_{c}, \end{gather}$$

the left-hand side is simply the Doppler-shifted wave frequency considering both the axial translation and the azimuthal rotation (Garetz 1981; Courtial et al. 1998). The first term on the right-hand side $l(\omega _{c}+2\varOmega _{-})$ can then be interpreted as normal ($l>0$) or anomalous ($l<0$) Doppler effect modified by inertial effects. Indeed $\omega _{c}+2\varOmega _{-}$ is the gyrofrequency corrected by the Coriolis force for an ion rotating at the angular frequency $\varOmega _{-}$. The classical yet subtle picture of normal/anomalous Doppler effect can be extended to the cases where the helical motion of the guiding centre ($P,\varOmega _{-}$) is slower or faster than the axial ($\omega /\beta$) and azimuthal ($\omega /n$) phase velocities (Nezlin 1976).

At resonance $\exp j( \boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\theta }-\omega )t= \exp j( \boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\theta }_{0})$. Some particles gain energy/momentum while others loose energy/momentum, with the sign of the variation determined by the phase factor $\mathrm {Re}(\exp j( \boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\theta }_{0}))$. This diffusive behaviour of the actions is described within the framework of random phase approximation where we average over $\exp j( \boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\theta }_{0})$ the square of the action variations to construct the quasilinear diffusion equation. The construction of this kinetic description leading to the quasilinear equation can be done either through a Lagrangian or a Eulerian point of view in phase space (Rax 2021). Here we will use the latter, as reviewed in Appendix B.

Note finally that the resonance condition (6.1) can be recovered from a simple photon picture. For this we recall that a photon associated with a wave described by (2.1) carries an energy $\hbar \omega$ and a linear momentum along the $z$ axis $\hbar \beta$. When this photon is absorbed by a rotating ion the variation of the particle energy $H_{0}$ and linear momentum $P$ are given by

(6.4)$$\begin{gather} \delta H_{0}=\hbar \omega ,\quad\delta P=\hbar \beta . \end{gather}$$

Besides energy and axial linear momentum, the photon associated with the wave (2.1) also carries an OAM plus SAM angular momentum $(n\mp 1)\hbar$ along the $z$ axis (see Appendix A). When this photon is absorbed by a rotating ion the change of the ion canonical angular momentum $L_{C}$ is

(6.5)$$\begin{gather} \delta L_{C}=\left( n\mp 1\right) \hbar . \end{gather}$$

Equation (3.9) reveals the harmonic oscillators structure of the Hamiltonian $H_{0}$. We can thus draw an analogy with the Hamiltonian of the Landau levels of a magnetized quantum particle to conclude that, at the quantum level, the changes of the action $J$ and $D$ can only be an integer multiple of $\hbar$, that is

(6.6a,b)$$\begin{gather} \delta J=n_{J}\hbar ,\quad\delta D=n_{D}\hbar \end{gather}$$

with $( n_{J},n_{D}) \in \mathbb {Z}^{2}$ a pair of integers. In fact we are considering the quasiclassical limit with large quantum numbers: $n_{J}+1/2\sim n_{J}$ and $n_{D}+1/2\sim n_{D}$. Then, from (3.12) and (4.1), one finds

(6.7)$$\begin{gather} \delta L_{C} = \delta D - \delta J, \end{gather}$$

(6.8)$$\begin{gather}\delta H_{0} = \varOmega _{-}\delta D-\varOmega _{+}\delta J +\delta P^{2}/2. \end{gather}$$

These two relations together with the semiclassical expansion $\delta P^{2}$ = $2\hbar \beta P+O( \hbar ^{2})$ finally lead to

(6.9)$$\begin{gather} \omega -\beta P={-}n_{J}\varOmega _{+}+\left( n_{J}+n\mp 1\right) \varOmega _{-}+O\left( \hbar \right)\!. \end{gather}$$

If $\hbar = 0$, we recognize a relation similar to our resonance condition identified in (6.1) with $n_{J}$ = $l\pm 1$, supporting the simple photon picture.

6.2. Diffusion paths

Besides the resonance lines (6.1), we can also identify from the Hamiltonian the diffusion paths along which resonant energy–momentum exchanges take place in actions space. Restricting the study to a single $V_{\boldsymbol {N}}(\boldsymbol {J})$ resonant coupling in (5.17), Hamilton's equations near the resonance (6.1) write as

(6.10)$$\begin{gather} \left. \frac{{\rm d}\boldsymbol{J}}{{\rm d}t}\right| _{\boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega} =\omega } ={-}j \boldsymbol{N}V_{\boldsymbol{N}}\exp j\boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\theta }_{0}, \end{gather}$$

(6.11)$$\begin{gather}\left. \frac{{\rm d}H}{{\rm d}t}\right| _{\boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega} =\omega } ={-}j\omega V_{\boldsymbol{N}}\exp j\boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\theta }_{0}. \end{gather}$$

Resonant actions variation (wave kicks) $\delta \boldsymbol {J}$ associated with a resonant energy variation $\delta H$ are thus related by

(6.12)$$\begin{gather} \omega \delta \boldsymbol{J}=\boldsymbol{N}\delta H. \end{gather}$$

Practically $\delta H$ can be taken as $\delta H_{0}$ within the two time scales quasilinear framework reviewed in Appendix B. In fact, without invoking the ordering between the secular quasilinear evolution and the fast $\omega$ oscillation, we can simply evaluate a variation of $H_{0}$ in (3.12) and take into account (6.12) to obtain

(6.13)$$\begin{gather} \delta H_{0}=\boldsymbol{\varOmega }\boldsymbol{\cdot} \delta \boldsymbol{J}=\boldsymbol{\varOmega }\boldsymbol{\cdot} \boldsymbol{N}\frac{\delta H}{\omega }=\delta H \end{gather}$$

since $\omega =\boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\varOmega }$. Thus, the resonant wave kicks $\delta \boldsymbol {J}$ associated with a resonant energy exchange $\delta H_{0}$ between the rotating wave and the rotating particle are given by

(6.14)$$\begin{gather} \left[ \delta J,\delta D,\delta P\right] =\left[ \left( l+\sigma \right) ,\left( l+n\right) ,\beta \right] \frac{\delta H_{0}}{\omega } \end{gather}$$

for a given $V_{\boldsymbol {N}}$ coupling. In the weak field limit this reduces to

(6.15)$$\begin{gather} [\omega_{c}\rho_{L}\delta\rho_{L},\omega_{c}R_{g}\delta R_{g},\delta v_{z}] =\left[ \left( l+\sigma \right) ,\left( l+n\right) ,\beta \right] \frac{\delta H_{0}}{M\omega }, \end{gather}$$

where we have temporarily reintroduced the ion mass $M$ and the cyclotron frequency $\omega _{c}$.

The relation (6.12) implies that there exists a linear combination of the actions which is invariant under the time evolution prescribed by the wave coupling $V_{\boldsymbol {N}}\exp j( \boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\theta }-\omega t)$, namely

(6.16)$$\begin{gather} \delta \left( \boldsymbol{N}\times \boldsymbol{J}\right) =0. \end{gather}$$

Focusing on Brillouin resonances rather than on Landau resonances, we restrict the analysis to the case $\beta P=0$. For a single $V_{\boldsymbol {N}}$ ((6.12) and (6.16)) then identify a diffusion path in action space $(J,D)$. Specifically, the path passing through a resonant point $({\rm J}_{0},D_{0})$ writes as

(6.17)$$\begin{gather} \left( l+n\right) \left( J-{\rm J}_{0}\right) =\left( l+\sigma \right) \left( D-D_{0}\right). \end{gather}$$

This is illustrated in figure 5. The diffusion paths (6.17) are invariant under the dynamics driven by a single $V_{\boldsymbol {N}}$ coupling. Quasilinear diffusion in action space takes place along these diffusion paths provided that the resonant condition (6.1) is fulfilled and that $|V_{nl\sigma }( {\rm J}_{0},D_{0}) |$ is not too small. Figure 5 also represents the isoenergy lines, i.e. points in $(J,D)$ space where $H_{0}$ is constant. For $\varOmega >\omega _{c}>0$ the electric field is confining and we can consider a canonical equilibrium distribution function

(6.18)$$\begin{gather} F_{0}\left( \boldsymbol{J}\right) =\frac{\varOmega ^{2}-\omega _{c}^{2}}{4\sqrt{ 2{\rm \pi} }\left( k_{B}T\right)^{{5}/{2}}}\exp \left(-\frac{H_{0}\left( \boldsymbol{J} \right) }{k_{B}T} \right), \end{gather}$$

where $T$ is the temperature and we have taken the normalization $\int \,{\rm d}\boldsymbol {J}F_{0}=1$. The corresponding density levels in $(J,D)$ space are colour coded in grey in figure 5. For $0<\varOmega \leq \omega _{c}$ the canonical distribution function (6.18) cannot be normalized as the potential $\phi (r)$ is not confining but instead either flat ($\varOmega =\omega _{c}$) or hill shaped ($\varOmega <\omega _{c}$).

Figure 5. Isoenergy levels $H_0$ and diffusion paths in $(J, D)$ action space.

6.3. Small Larmor radius limit

When the Larmor radius $\sqrt {2J/\varOmega }$ is small compared with the radial wavelength $2{\rm \pi} /k$, that is

(6.19)$$\begin{gather} k\sqrt{\frac{J}{2\varOmega} }<1, \end{gather}$$

the coupling coefficients $V_{n,l,\sigma }$ in (5.13), (5.14) and (5.15) can be simplified by using the small parameter expansion

(6.20)$$\begin{gather} {\rm J}_{l}\left( k\sqrt{\frac{J}{2\varOmega} }\right) \sim \frac{\left( k\sqrt{\dfrac{J}{2\varOmega} }\right) ^{l}}{l!}. \end{gather}$$

We see that in this limit the $l=0$ term is the most effective since it is associated with the largest $V_{n,l,\sigma }$. The resonance condition (6.1) for $l=0$ then gives $\omega -\beta P$ = $-\sigma \varOmega _{+}+n\varOmega _{-}$ and the fundamental $l=0$ Doppler-shifted cyclotron resonance term $\sigma \varOmega _{+}$ is only due to the SAM content of the wave.

If one assumes further $\sigma =0$, as in the scalar case (5.10), and $\beta P=0$, cyclotron and Landau terms are avoided and one gets get a pure OAM harmonic Brillouin–Landau resonance

(6.21)$$\begin{gather} \omega =n\varOmega _{-} \end{gather}$$

between the drift rotation $\varOmega _{-}$ and the wave OAM. This is the optimal choice for the sustainment of a plasma column rotation. The physical interpretation of (6.21) is simple: the wave angular velocity ${\rm d}\alpha /{\rm d}t$ = $\omega /n$ is equal to the particle guiding centre angular velocity $-{\rm d}\theta /{\rm d}t$ = $\varOmega _{-}$.

6.4. Special examples

Let us illustrate the physics behind Brillouin coupling through three selected examples. Consider first a potential wave $\sigma =0$, that is (5.10), with $\beta =0$ and the small Larmor radius approximation such that $l=0$. In these conditions $\boldsymbol {N}$ = $[ 0,n,0]$ and (6.14) writes as

(6.22a,b)$$\begin{gather} \frac{\delta J}{\delta H_{0}}=0,\quad\frac{\delta D}{\delta H_{0}}=\frac{1 }{\varOmega _{-}} \end{gather}$$

since the Brillouin resonance (6.1) implies $\omega$ = $n\varOmega _{-}$. This result can be interpreted as follows. For absorption $\delta H_{0}>0$, the wave energy is transferred to potential energy through the wave-induced radial dynamics of the guiding centre in the electrostatic potential $\varPhi (r)$. For emission $\delta H_{0}<0$, the potential energy of the particles is passed on to the wave energy. This type of instability is used for microwave generation in magnetrons. We note also that the simultaneous cooling ($\delta H_{0}<0$) and ash removal ($\delta D>0$) of alpha particles in a rotating tokamak, through free energy extraction (Fisch & Rax 1992, 1993; Fisch & Herrmann 1994, 1995; Herrmann & Fisch 1997), is optimal in these wave conditions. The ratio of energy extraction to radial expulsion is then adjusted through $\varOmega _{-}$.

Consider now a vectorial wave $\sigma =\pm 1$, that is (5.9), with $\beta =0$, no OAM ($n=0$) and the small Larmor radius approximation such that $l=0$. In these conditions $\boldsymbol {N} = [ \sigma,0,0]$ and (6.14 ) writes as

(6.23a,b)$$\begin{gather} \frac{\delta D}{\delta H_{0}}=0,\quad\frac{\delta J}{\delta H_{0}}={-}\frac{ 1}{\varOmega _{+}} \end{gather}$$

since the Brillouin resonance (6.1) implies $\omega = -\sigma \varOmega _{+}$ (remember that $\varOmega _{+}<0$). In this case the wave energy is simply passed into cyclotron energy $-\varOmega _{+}\delta J$. This is the case of pure ICRH modified by inertial effects.

Consider finally a potential wave $\sigma =0$, that is (5.10), with $\beta =0$ and no OAM ($n=0$), but with FLR effects $l\neq 0$. In these conditions $\boldsymbol {N}$ = $[ l,l,0]$ and (6.12) writes as

(6.24a,b)$$\begin{gather} \frac{\delta D}{\delta H_{0}}=\frac{1}{\varOmega },\quad\frac{\delta J}{ \delta H_{0}}=\frac{1}{\varOmega } \end{gather}$$

since the Brillouin resonance (6.1) implies $\omega = -l\varOmega _{+}$ $+$ $l\varOmega _{-}$ = $l\varOmega$. This result can be interpreted as follows. For absorption $\delta H_{0}>0$ the wave energy is transferred to both kinetic energy $-\varOmega _{+}\delta J$ and potential energy $\varOmega _{-}\delta D$ as shown by (3.12). The partitioning between these two energy channels must, however, fulfil conservation of canonical angular momentum $\delta L_{C}=0$, which from (4.1) implies $\delta D= \delta J$. Conservation of $L_{C}$ is indeed the consequence of the invariance through rotation of the Hamiltonian $H_{0}+V$, since here the wave carries neither OAM ($n=0$) nor SAM ($\sigma =0$).

7. Quasilinear theory in a rotating magnetized plasma

When a wave propagates in a plasma, two time scales are associated with the quasilinear ordering. One is the fast linear response that is described at the kinetic level by Vlasov's equation, whose solution gives the refractive/dispersive part of the dielectric tensor. The other is the slow, angle averaged, quasilinear evolution of the action distribution function $F( \boldsymbol {J},t)$ which is described at the kinetic level by the quasilinear equation. The standard quasilinear equation (B10) is derived in Appendix B.

With the Hamiltonian (5.17), the evolution of the action distribution function $F( \boldsymbol {J},t)$ in a magnetized rotating plasma is given by

(7.1)$$\begin{gather} \frac{\partial F}{\partial t}=\frac{{\rm \pi} }{2}\sum_{\boldsymbol{N}}\left( \boldsymbol{ N}\boldsymbol{\cdot }\frac{\partial }{\partial \boldsymbol{J}}\right) \left[ \left| V_{ \boldsymbol{N}}\right| ^{2}\delta \left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega}-\omega \right) \right] \left( \boldsymbol{N}\boldsymbol{\cdot} \frac{\partial }{\partial \boldsymbol{J}}\right) F\left( \boldsymbol{J},t\right)\!, \end{gather}$$

where the $V_{\boldsymbol {N}}$ are those derived in (5.13), (5.14) and (5.15). The operator $\boldsymbol {N}\boldsymbol {\cdot } \partial _{\boldsymbol {J}}$ involved in this slow (with respect to $1/\omega$) diffusion in action space writes as

(7.2)$$\begin{gather} \boldsymbol{N}\boldsymbol{\cdot }\frac{\partial }{\partial \boldsymbol{J}}=\left( l+\sigma\right) \frac{\partial }{\partial J}+\left( l+n\right) \frac{\partial }{\partial D}+\beta \frac{\partial }{\partial P}. \end{gather}$$

We normalize $F$ by taking $\int \,{\rm d}\boldsymbol {J}F( \boldsymbol {J},t) =1$ and consider that $F( \boldsymbol {J}=+\infty,t) =0$ and $F(J<0,D<0,P=-\infty,t) =0$.

Introducing the resonant particles density $\rho$, the power per unit volume $W$ lost or gained by the wave (and gained or lost by the plasma) is

(7.3)$$\begin{gather} \frac{W}{\rho} =\frac{{\rm d}}{{\rm d}t}\int \,{\rm d}\boldsymbol{J}H_{0}\left( \boldsymbol{J}\right) F\left( \boldsymbol{J},t\right) =\int \,{\rm d}\boldsymbol{J}H_{0}\frac{\partial F}{\partial t} \end{gather}$$

where the integral is to be taken over $-\infty < J,D,P<+\infty$. Using (7.1) and integrating by parts the operator (7.2) gives

(7.4)$$\begin{gather} \frac{W}{\rho} ={-}\frac{{\rm \pi} }{2}\int \,{\rm d}\boldsymbol{J}\omega \sum_{\boldsymbol{N}}[ \left| V_{ \boldsymbol{N}}\right| ^{2}\delta \left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega}-\omega \right) ] \boldsymbol{N}\boldsymbol{\cdot} \frac{\partial F}{\partial \boldsymbol{J}}. \end{gather}$$

We recognize in (7.4) the power balance given by the dissipative part of the collisionless dielectric tensor. A second integration by parts then gives the density of power

(7.5)$$\begin{gather} W= \frac{{\rm \pi} }{2}\rho\int \,{\rm d}\boldsymbol{J}F\left( \boldsymbol{J},t\right) \omega \sum_{\boldsymbol{N}}\boldsymbol{N}\boldsymbol{\cdot }\frac{\partial }{\partial \boldsymbol{J}}[\left| V_{\boldsymbol{N}}\right| ^{2}\delta \left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega} -\omega \right)]. \end{gather}$$

Defining $w_{\boldsymbol {N}}( \boldsymbol {J})$ through

(7.6)$$\begin{gather} W = \rho \sum_{\boldsymbol{N}}\int \,{\rm d}\boldsymbol{J}F\left( \boldsymbol{J}\right) w_{\boldsymbol{N}}\left( \boldsymbol{J}\right)\!, \end{gather}$$

equation (7.5) then gives

(7.7)$$\begin{gather} w_{\boldsymbol{N}}\left( \boldsymbol{J}\right) =\frac{{\rm \pi} }{2}\omega \sum_{\boldsymbol{N}}\boldsymbol{N}\boldsymbol{\cdot} \frac{\partial }{\partial \boldsymbol{J}}[ \left| V_{\boldsymbol{N}}\right| ^{2}\delta \left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega}-\omega \right) ]. \end{gather}$$

This quantity is thus interpreted, for a single component $V_{\boldsymbol {N}}$, as the power exchanged by a particle at $\boldsymbol {J}$ with this $V_{\boldsymbol {N}}$ component of the wave interaction. Equation (6.12) can hence be rewritten as

(7.8)$$\begin{gather} \left. \frac{\left\langle \delta \boldsymbol{J}\right\rangle }{\delta t}\right| _{\boldsymbol{N}}=\frac{\boldsymbol{N}}{\omega }\frac{\delta H_{0}\left( \boldsymbol{J}\right) }{\delta t}=\frac{\boldsymbol{N}}{\omega }w_{\boldsymbol{N}}\left( \boldsymbol{J}\right)\!, \end{gather}$$

where $\langle \ \rangle$ is an average over the fast phase of the wave and the $\boldsymbol {N}$ index restricts the wave kick $\delta \boldsymbol {J}$ to a single component $V_{\boldsymbol {N}}$. Plugging (7.7) into (7.8) and using the tensorial product notation $\boldsymbol {\otimes }$ yields

(7.9)$$\begin{gather} \frac{\left\langle \delta \boldsymbol{J}\right\rangle }{\delta t}=\frac{{\rm \pi} }{2}\frac{\partial }{\partial \boldsymbol{J}}\boldsymbol{\cdot} \sum_{\boldsymbol{N}}\boldsymbol{N\otimes N}\left| V_{\boldsymbol{N}}\right| ^{2}\delta \left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega}-\omega \right)\!. \end{gather}$$

This relation (7.9) is just the usual Einstein's relation (Fraiman & Kostyukov 1995) between the wave-induced drift coefficient $\langle \delta \boldsymbol {J}\rangle /\delta t$ and the wave-induced diffusion coefficient

(7.10)$$\begin{gather} \frac{\left\langle \delta \boldsymbol{J\otimes }\delta \boldsymbol{J}\right\rangle }{2\delta t}=\frac{{\rm \pi} }{2}\sum_{\boldsymbol{N}}\boldsymbol{N\otimes N}[ \left| V_{\boldsymbol{N}}\right| ^{2}\delta \left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega}-\omega \right) ] \end{gather}$$

used to write the kinetic equation (7.1) in the classical Fokker–Planck form with a drift and a diffusion coefficient

(7.11)$$\begin{gather} \frac{\partial F}{\partial t}={-}\frac{\partial }{\partial \boldsymbol{J}}\boldsymbol{\cdot} \left[ \frac{\left\langle \delta \boldsymbol{J}\right\rangle }{\delta t}F-\frac{\partial }{\partial \boldsymbol{J}}\boldsymbol{\cdot} \frac{\left\langle \delta \boldsymbol{J\otimes }\delta \boldsymbol{J}\right\rangle }{2\delta t}F\right] \end{gather}$$

rather than in the completely equivalent and more usual quasilinear form

(7.12)$$\begin{gather} \frac{\partial F}{\partial t}=\frac{\partial }{\partial \boldsymbol{J}}\boldsymbol{\cdot} \left[ \frac{\left\langle \delta \boldsymbol{J\otimes }\delta \boldsymbol{J}\right\rangle }{2\delta t}\boldsymbol{\cdot} \frac{\partial F}{\partial \boldsymbol{J}}\right] \end{gather}$$

used here in (7.1). Einstein's relation

(7.13)$$\begin{gather} \frac{\left\langle \delta \boldsymbol{J}\right\rangle}{\delta t} = \partial_{\boldsymbol{J}}\boldsymbol{\cdot}\frac{\left\langle \delta \boldsymbol{J\otimes }\delta \boldsymbol{J}\right\rangle}{2\delta t} \end{gather}$$

is a consequence of microreversibility (Rax 2021).

8. Angular momentum absorption

Short of solving the quasilinear equation (7.1), a clear understanding of the mechanism of angular momentum absorption (or emission) can be gained through the analysis of the quasilinear guiding centre radial velocity $\langle \delta D\rangle /{\rm d}t$ and quasilinear ICRH $\langle \delta J\rangle /{\rm d}t$ identified in (7.8) and (7.9). In general the effect of the wave is indeed two-fold: it both drives a radial drift current $\delta D$ and provides ICRH $\delta J$. Yet, this last channel should be avoided or at least minimized for fluid rotation sustainment. This can in principle be done through the choice of a suitable wave.

8.1. Absorption from global angular momentum conservation

For a single particle, the SAM ($S_{z}$) and OAM ($L_{z}$) lost by the wave during the resonant wave–particle interaction are gained by the particle in the form of canonical angular momentum $L_{C}$. From (4.1)

(8.1)$$\begin{gather} {-}\frac{\left\langle \delta L_{C}\right\rangle }{\delta t} =\frac{ \left\langle \delta J\right\rangle }{\delta t}-\frac{\left\langle \delta D\right\rangle }{\delta t} \end{gather}$$

which using (7.9) rewrites as

(8.2)$$\begin{gather} {-}\frac{\left\langle \delta L_{C}\right\rangle }{\delta t} =\left( \sigma -n\right) \frac{{\rm \pi} }{2}\sum_{\boldsymbol{N}}\boldsymbol{N}\boldsymbol{\cdot} \partial _{\boldsymbol{J}}[ \left| V_{\boldsymbol{N}}\right| ^{2}\delta \left( \boldsymbol{N}\boldsymbol{\cdot} \boldsymbol{\varOmega}-\omega \right) ]. \end{gather}$$

Under the simple photon picture developed at the end of § 6.1, global angular momentum conservation for the full system wave plus particle and a single $| V_{\boldsymbol {N}}|$ coupling coefficient thus writes as

(8.3a,b)$$\begin{gather} \delta \left. L_{z}\right| _{{\rm wave}}={-}n\frac{\delta H_{0}}{\omega },\quad\delta \left. S_{z}\right| _{{\rm wave}}=\sigma \frac{\delta H_{0}}{\omega }. \end{gather}$$

Quasilinear theory brings additional insights in that it allows to relate the wave's change in angular momentum to the angular momentum absorption by a distribution function $F( \boldsymbol {J},t)$. Specifically, averaging (8.2) over the distribution of actions in the plasma and integrating by parts gives

(8.4)$$\begin{gather} \left. \frac{{\rm d}\left( L_{z}+S_{z}\right) }{{\rm d}t}\right| _{{\rm wave}}=\left( n-\sigma \right) \frac{{\rm \pi} }{2}\iint \,{\rm d}J\,{\rm d}D\sum_{l}\left| V_{\boldsymbol{N}}\right| ^{2}\boldsymbol{N}\boldsymbol{\cdot} \partial _{\boldsymbol{J}}\left. F\right | _{P=P_{l}}. \end{gather}$$

Here $P_{l}$ is the resonant axial momentum fulfilling the relation $\boldsymbol {N}\boldsymbol {\cdot } \boldsymbol {\varOmega }=\omega$ for given wave field and DC field configurations, that is

(8.5)$$\begin{gather} \beta P_{l}=\omega +\left( l+\sigma \right) \varOmega _{+}-\left( l+n\right) \varOmega _{-}. \end{gather}$$

Note that since $-\infty < P<\infty$ there is always a solution $P_{l}$ to (8.5) for a given $( l,n) \in \mathbb {Z}^{2}$. Note also that (8.4) can be equivalently derived from (8.3a,b ) using (7.4).

The angular momentum absorption coefficient derived in (8.4) can be evaluated by considering the kinetic evolution of $F( \boldsymbol {J},t)$ given in (7.1) together with a relaxation term associated with collisions. Assuming that the plasma equilibrium is only weakly perturbed by the wave we can consider that $F\sim F_{0}$ as given in (6.18).

8.2. Physical picture

To develop a deeper physical understanding of quasilinear angular momentum exchange we define the average kinetic angular momentum

(8.6)$$\begin{gather} \left\langle L_{K}\right\rangle =\langle xv_{y}-yv_{x}\rangle _{\theta +\varphi }=\frac{2\varOmega _{-}}{\varOmega }D+\frac{2\varOmega _{+}}{\varOmega }J \end{gather}$$

and the average magnetic flux through the orbit

(8.7)$$\begin{gather} \left\langle \varPsi \right\rangle =\omega _{c}{\rm \pi} \langle x^{2}+y^{2}\rangle _{\theta +\varphi }=2{\rm \pi} \frac{\omega _{c}}{\varOmega } D+2{\rm \pi} \frac{\omega _{c}}{\varOmega }J. \end{gather}$$

According to the quasilinear prescription, the bracket $\langle {}\rangle$ indicates an angle average of (3.14), (4.2) and (4.3). Meanwhile, (4.4) gives a relation for the canonical angular momentum variation

(8.8)$$\begin{gather} \delta L_{C}=\delta \left\langle L_{K}\right\rangle +\delta \left\langle \varPsi \right\rangle \!/2{\rm \pi} . \end{gather}$$

From (8.6) and (8.7) the kinetic and magnetic components $\delta \langle L_{K}\rangle$ and $\delta \langle \varPsi \rangle$ then write as

(8.9)$$\begin{gather} \varOmega \delta \left\langle L_{K}\right\rangle =2\varOmega _{-}\delta D+2\varOmega _{+}\delta J, \end{gather}$$

(8.10)$$\begin{gather}\varOmega \delta \left\langle \varPsi \right\rangle =2{\rm \pi} \omega _{c}\delta D+2{\rm \pi} \omega _{c}\delta J. \end{gather}$$

A physical interpretation of these results can be obtained as follows, where we focus again on the more intuitive ordering $D>J$.

Starting with the kinetic component (8.9), recall from § 4 that an ion with mass $M=1$ displays a guiding centre moment of inertia $M_{G}$ = $2D/\varOmega$ with respect to the $z$ axis, and a moment of inertia of the cyclotron motion $M_{C}$ = $2J/\varOmega$ with respect to the guiding centre. Recall also that the guiding centre of this ion rotates at ${\rm d}\theta /{\rm d}t = \varOmega _{-}$ whereas the cyclotron rotation takes place at the angular frequency $\varOmega _{+} = -{\rm d}\varphi /{\rm d}t$. Now, because these two angular velocities are set by the fields ((3.1) and (3.2)), the variation of the kinetic angular momentum of the ion $\langle L_{K}\rangle = M_{G}\varOmega _{-} + M_{C}\varOmega _{+}$ ($\theta$ is anticlockwise and $\varphi$ is clockwise) must come from a variation of the moments of inertia and not of the angular velocities, so that

(8.11)$$\begin{gather} \frac{\delta \left\langle L_{K}\right\rangle }{\delta t}=\frac{ \delta M_{G}}{\delta t}\varOmega _{-}+\frac{\delta M_{C}}{\delta t}\varOmega _{+}= \frac{2\varOmega _{-}}{\varOmega }\frac{\delta D}{\delta t}+\frac{2\varOmega _{+}}{\varOmega }\frac{\delta J}{\delta t}, \end{gather}$$

which is precisely (8.9).

The interpretation of the magnetic component

(8.12)$$\begin{gather} \frac{1}{2{\rm \pi} }\frac{\delta \left\langle \varPsi \right\rangle }{\delta t}=\frac{\omega _{c}}{\varOmega }\frac{\delta D}{\delta t}+\frac{\omega _{c}}{\varOmega }\frac{\delta J}{\delta t}, \end{gather}$$

requires an analysis of both the $\delta D$ and $\delta J$ terms. As we will now show, these two terms can be interpreted in terms of two different torques exerted on an ion in the background magnetic field. Starting with $D$, two pictures can be invoked. The first one is to consider the axial torque due to the magnetic force exerted on a charge $q=1$ moving radially. The radial velocity of this charge is

(8.13)$$\begin{gather} \frac{{\rm d}}{{\rm d}t}\left(\sqrt{\frac{2D}{\varOmega}}\right) \boldsymbol{e}_{r}, \end{gather}$$

so that this torque writes as

(8.14)$$\begin{gather} \sqrt{\frac{2D}{\varOmega }}\boldsymbol{e}_{r}\times \left[\frac{{\rm d}}{{\rm d}t}\left(\sqrt{\frac{2D}{\varOmega}}\right)\boldsymbol{e}_{r}\times \omega _{c}\boldsymbol{e}_{z}\right] ={-}\frac{\omega _{c}}{\varOmega }\frac{{\rm d}D}{{\rm d}t}\boldsymbol{e}_{z}. \end{gather}$$

The second is to consider an azimuthal electromotive force (emf) for a $q=1$ charge distributed along a rotating circle with radius $\sqrt {2D/\varOmega }$, associated with the variation of the loop surface as a result of the radial motion. This electric inductive field $E_{{\rm emf}}$ is also the source of an axial torque

(8.15)$$\begin{gather} \sqrt{\frac{2D}{\varOmega }}\boldsymbol{e}_{r}\times E_{{\rm emf}}\boldsymbol{e}_{\alpha }={-}\frac{1}{2{\rm \pi} }\frac{{\rm d}\varPsi _{G}}{{\rm d}t}\boldsymbol{e}_{z}={-}\frac{\omega _{c}}{\varOmega }\frac{{\rm d}D}{{\rm d}t}\boldsymbol{e}_{z}, \end{gather}$$

where we have introduced the magnetic flux through the guiding centre orbit $\varPsi _{G}$ and applied Faraday's law

(8.16)$$\begin{gather} {-}\frac{{\rm d}\varPsi _{G}}{{\rm d}t}=\oint E_{{\rm emf}}\boldsymbol{e}_{\alpha }\boldsymbol{\cdot }\,{\rm d}s \boldsymbol{e}_{\alpha }=2{\rm \pi} \sqrt{\frac{2D}{\varOmega }}E_{{\rm emf}}. \end{gather}$$

One verifies that both analyses give the same axial torque experienced by an ion as a result of the wave-driven radial motion, which is precisely the first term on the right-hand side in (8.12). Moving on finally to the $\delta J$ term in (8.12), a similar current loop picture can be brought up but by considering a $q=1$ charge distributed this time along the Larmor radius $\sqrt {2J/\varOmega }$. The magnetic flux through this varying Larmor radius in indeed $\varPsi _{L}=2{\rm \pi} (\omega _{c}/\varOmega ) J$, whose time derivative precisely gives back the second term on the right-hand side in (8.12)

In summary, the first term on the right-hand side of (8.8) corresponds to a change of the moment of inertia of the particle as a result of the quasilinear radial drift and Larmor radius evolution. The second term on the right-hand side of (8.8) corresponds to torques which result from a change in magnetic fluxes. Both of these terms, interpreted here in the weak field limit, must be balanced by the transfer of a corresponding angular momentum from the wave. Note, however, that the wave quasilinear primary effect is not a torque, it is a wave-driven radial current.

9. Radial current generation

Two types of model can be constructed from the quasilinear kinetic equation derived in (7.1). One option is to balance the quasilinear evolution of the distribution function with a collisional evolution at the kinetic level and then to average the solution $F( \boldsymbol {J})$ to obtain a steady state fluid picture. The other is to average the quasilinear dynamics to derive the fluid quasilinear flows of mass, charge and momentum, and then to balance these fluid flows with the dissipative terms involved in classical transport theory. The latter option is used in the following, where we further assume for simplicity axial homogeneity along $z$.

Let us write

(9.1)$$\begin{gather} \boldsymbol{\varGamma }\left( r\right) = \varGamma _{r}\boldsymbol{e}_{r}+\varGamma_{\alpha }\boldsymbol{e}_{\alpha }+\varGamma _{z}\boldsymbol{e}_{z} \end{gather}$$

the wave-driven particle flux which results from the absorption of the wave power $w_{\boldsymbol {N}}( r)$ at radius $r$. The radial flow $\varGamma _{r}$ is due to the wave-driven guiding centre radial velocity, which we showed is proportional to $\langle \delta D\rangle /\delta t$. The azimuthal flow $\varGamma _{\alpha }$ is a small diamagnetic effect associated with inhomogeneous wave-driven ICRH proportional to $\langle \delta J\rangle /\delta t$. The axial flow $\varGamma _{z}$ is simply the wave-driven current from classical current generation (Fisch 1978, 1987) due to $\langle \delta P\rangle /\delta t$. The amplitude of these fluxes is governed by the evolution equation (7.8).

An analytical expression for the radial flow $\varGamma _{r}$ can be derived if focusing once again on the familiar limit $J< D$. In this limit the average radial position of a resonant particle is $r^{2}=2D/\varOmega$, and the average Lagrangian radial velocity is hence $r\varOmega \,{\rm d}r/{\rm d}t$ = ${\rm d}D/{\rm d}t$. From (7.8) the phase averaged evolution of $D$ is proportional to the absorbed power $w_{\boldsymbol {N}}$ associated with the $l$ harmonic Brillouin resonance for a rotating wave with azimuthal number $n$, and one has

(9.2)$$\begin{gather} \left. \frac{\left\langle \delta D\right\rangle }{\delta t}\right| _{\boldsymbol{N}}=\frac{l+n}{\omega }\left. w_{\boldsymbol{N}}\right| _{D={\varOmega r^{2}}/{2}}\!. \end{gather}$$

Introducing back the particle mass $M$, and using the power density absorbed by the plasma $W_{\boldsymbol {N}}$ (in W m$^{-3}$) rather than $w_{\boldsymbol {N}}$ (in Watt), the wave-driven resonant particle flux is

(9.3)$$\begin{gather} \varGamma _{r}=\frac{\left( l+n\right) }{rM\varOmega \omega }W_{\boldsymbol{N}} \quad\text{s}^{{-}1}\,\text{m}^{{-}2}. \end{gather}$$

Consider now that the wave power density $W_{\boldsymbol {N}}( t)$ is turned on adiabatically at $t$ = $-\infty$ with $W_{\boldsymbol {N}}( -\infty ) =0$. The wave moves some resonant particles across the magnetic field, which leads to a radial current ${\rm J}_{W}( t)$ such that ${\rm J}_{W}( t=-\infty ) = 0$ and ${\rm J}_{W}( t=0) = q\varGamma _{r}$. The resulting time evolution of the radial electric field can be described as follows. From an electrical point of view, the build-up corresponds to a capacitive electric field build-up in a dielectric media, akin to the charging of a cylindrical capacitor. By considering the plasma as a dielectric with low frequency permittivity

(9.4)$$\begin{gather} \varepsilon = 1 + \left(\frac{\omega _{pi}}{\omega}\right)^{2} \approx \left(\frac{\omega _{pi}}{\omega}\right)^{2} \end{gather}$$

with $\omega _{pi}$ the ion plasma frequency, the electric field $\boldsymbol {E}( t)$ throughout this transient phase is thus determined from Maxwell–Ampère equation

(9.5)$$\begin{gather} \varepsilon _{0}\frac{\omega _{pi}^{2}}{\omega _{ci}^{2}}\frac{\partial \boldsymbol{E}}{\partial t}+{\rm J}_{W}\left( t\right)\boldsymbol{e}_{r} =\boldsymbol{0}. \end{gather}$$

From a mechanical point of view this build-up phase corresponds to an angular momentum input via the ${\rm J}_{W}( t)\boldsymbol {e}_{r} \times \boldsymbol {B}$ force, and this momentum is converted into plasma $E\times B$ drift. Indeed, integrating (9.5) over the transient phase gives

(9.6)$$\begin{gather} \int_{-\infty }^{0}{\rm J}_{W}\left( t\right)\boldsymbol{e}_{r} \times \boldsymbol{B}\,{\rm d}t={-}N_{p}M\frac{\boldsymbol{E}_{0}\times \boldsymbol{B}}{B^{2}} \end{gather}$$

with $\boldsymbol {E}_{0}=\boldsymbol {E}( t=0)$, $M$ the ion mass and $N_{p}$ the ion density, which confirms this momentum balance. Note that ion diamagnetic effects have been neglected in writing (9.5). Note also that because the conductivity along magnetic field lines is generally far larger than the conductivity across the field lines, charges rapidly move away from the wave active regions along the field lines, which in turn become equipotential.

Finally, for $t>0$, the charge separation induced by the wave is short circuited by the plasma perpendicular conductivity (Helander & Sigmar 2005; Kolmes et al. 2019; Rax et al. 2019). In this steady-state dissipative regime the wave-driven current $q\varGamma _{r}\boldsymbol {e}_{r}$ is balanced by a weak discharging Ohmic current $\boldsymbol {J}_{\text {conduction}}$, with

(9.7)$$\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot} [ q\varGamma _{r}\boldsymbol{e}_{r}+\boldsymbol{J}_{\text{conduction}}] =0. \end{gather}$$

Thus, as opposed to axial current generation, the steady-state is determined by the geometry of the plasma. The examination of this problem is left for a future study.

10. Conclusions

Although angular momentum exchange between a wave and a rotating plasma is of importance both to astrophysics (Goldreich & Julian 1969; Julian 1973; Ferrière 2006) and laboratory plasmas (Kostyukov et al. 2002; Shvets, Fisch & Rax 2002; Thaury et al. 2013), a kinetic model of this interaction had to our knowledge never been proposed. In this study we addressed this issue and derived the quasilinear equation describing the interaction between a rotating wave and a rotating magnetized plasma. We further used this kinetic model to analyse angular momentum absorption/emission and to understand the interplay between OAM, SAM and FLR effects.

First, a canonical angle–action Hamiltonian description of the unperturbed Brillouin rotation dynamics in a rotating plasma is derived. The identification of angle–action variables allows to separate the fast part of the unperturbed motion (angles) from the constant (integrable system) or slow (adiabatic system) part of the unperturbed motion (actions). Through this process we identify three canonical actions $D$, $J$ and $P$. The latter is the classical momentum along $\boldsymbol {B}$. The first two $D$ and $J$ are shown in the weak field limit, that is, for a cross-field drift frequency $E/(rB)$ small compared with the ion gyrofrequency, to be related to the magnetic flux through the cyclotron orbit and the guiding centre orbit.

Then, the wave–particle coupling is expressed in terms of these angle–action variables in the form of a perturbed Hamiltonian. This approach made it possible to identify a new resonance condition, which generalizes the classical Landau-cyclotron resonance to the case of a rotating plasma interacting with a rotating wave. This new condition, (6.1), is referred to as Brillouin resonance. It notably expresses FLR effects through (5.13) and (5.14), (5.15), which are found to be responsible for the occurrence of harmonic ($l$) Brillouin resonances. Together with this resonance condition, diffusion paths in action space were identified, and particular examples were exposed in the weak field limit. Finally, the quasilinear equation (7.1) which describes energy–momentum exchange as a time evolution of the actions distribution function was derived by averaging the kinetic response of the plasma to the perturbation over the fast part of the motion (angles).

By analysing the variation in canonical actions $D$ and $J$ predicted by the quasilinear equation, a physical picture for momentum absorption was finally proposed. Specifically, angular momentum from the wave was shown to be transferred to the plasma either as a change of the inertia tensor of the plasma, or as a magnetic flux variation ((8.11) and (8.12)). This analysis also showed that the radial flux can be identified as the source of angular momentum injection in the plasma. An interesting prospect is the generalization of this work to magnetic field inhomogeneities. The use of action-angle coordinates for the motion in a straight magnetic field with constant gradient (Brizard 2022) may for instance enable to capture bounce resonances in a mirror geometry rather than the uniform $z$ translation Doppler shift considered here.

Finally, since the radial current is proportional to the absorbed power, sustaining steady-state rotation with waves will require adjusting the wave power deposition profile in a way that (9.7) is fulfilled. The optimization of power deposition will be the object of future studies, but we note for example that for a resonance $\boldsymbol {N}$ in a rotating magnetized plasma cylinder with large conductivity along the field lines $\eta _{\shortparallel }$ and weak conductivity across the field lines $\eta _{\perp }$, (9.7) gives

(10.1)$$\begin{gather} W_{\boldsymbol{N}}\left( r\right) \approx \eta _{{\perp} }\frac{M^{2}}{q^{2}}\frac{\varOmega \omega }{\left( l+n\right) }\left( \frac{\omega _{c}^{2}-\varOmega ^{2}}{4}\right) r^{2}. \end{gather}$$

How this power density can be deposited in the plasma requires information on what the normal modes of a rotating plasma column are, and in particular the dispersion relation characterizing these modes, beyond the simple case of an aligned rotator (Gueroult et al. 2019b , 2020; Gueroult, Rax & Fisch 2023). This important question will be addressed in forthcoming studies.