Rotating Alfvén waves in rotating plasmas

Angular momentum coupling between a rotating magnetized plasma and torsional Alfvén waves carrying orbital angular momentum (OAM) is examined. It is demonstrated not only that rotation is the source of Fresnel–Faraday rotation – or orbital Faraday rotation effects – for OAM-carrying Alfvén waves, but also that angular momentum from an OAM-carrying Alfvén wave can be transferred to a rotating plasma through the inverse process. For the direct process, the transverse structure angular rotation frequency is derived by considering the dispersion relation for modes with opposite OAM content. For the inverse process, the torque exerted on the plasma is derived as a function of wave and plasma parameters.


Introduction
Understanding the effect of rotation on plasma dynamics is essential to a wide range of applications.Besides original efforts motivated by microwave generation in magnetrons (Brillouin 1945), it has indeed been shown that rotation could enable new approches to thermonuclear confinement (Rax et al. 2017;Wilcox 1959;Bekhtenev et al. 1980;Ochs & Fisch 2017;Hassam 1997;Fetterman & Fisch 2010, 2008).Rotation has also be found to hold promise for developing plasma mass separation applications (Gueroult et al. 2017(Gueroult et al. , 2019b)), either in pulsed plasma centrifuges (Bonnevier 1966;Krishnan et al. 1981) or in steady-state cross-field rotating plasmas (Ohkawa & Miller 2002;Shinohara & Horii 2007;Gueroult et al. 2014;Fetterman & Fisch 2011;Gueroult et al. 2014), advanced accelerators (Janes 1965;Janes et al. 1965Janes et al. , 1966;;Thaury et al. 2013;Rax & Robiche 2010) and thrusters (Gueroult et al. 2013).But understanding the effect of rotation on plasma dynamics is also essential in a number of environments.Rotation is for instance key to the structure and stability of a number of astrophysical objects (Kulsrud 1999;Miesch & Toomre 2009).In light of this ubiquitousness, and because plasma waves are widely used both for control and diagnostics in plasmas, it seems desirable to understand what the effect of rotation on wave propagation in plasmas may be (Gueroult et al. 2023).In fact the importance of this task was long recognised in geophysics and astrophysics, leading to extensive studies of low frequency MHD waves in rotating plasmas (Lehnert 1954;Hide 1969;Acheson 1972;Acheson & Hide 1973;Campos 2010), and notably of Alfvén waves (Stix 1992).
Meanwhile, following the discovery that electromagnetic waves carry both spin and orbital angular momentum (Allen et al. 1992(Allen et al. , 2016;;Andrews & Babiker 2012), there have been numerous theoretical developments on spin-orbit interactions (Bliokh et al. 2015) in modern optics, which we note are now being applied to plasmas (Bliokh & Bliokh 2022).For spin angular momentum (SAM) carrying waves, that is circularly polarised waves, propagation through a rotating medium is known to lead to a phase-shift between eigenmodes with opposite SAM content (Player 1976;Gueroult et al. 2019aGueroult et al. , 2020)).This phaseshift is then the source of a rotation of polarization or polarization drag (Jones 1976), as originally postulated by Thomson (Thomson 1885) and Fermi (Fermi 1923).For orbital angular momentum (OAM) carrying waves, propagation through a rotating medium is the source of a phase-shift between eigenmodes with opposite OAM content (Götte et al. 2007), leading to image rotation or Faraday-Fresnel Rotation (FFR) (Padgett et al. 2006).
This azimuthal Fresnel drag of OAM carrying waves, which can be viewed as an orbital Faraday rotation of the amplitude, was first derived (Wisniewski-Barker et al. 2014) and observed (Franke-Arnold et al. 2011) in isotropic, nongyrotropic media.In contrast, propagation of OAM carrying wave in a rotating anisotropic (gyrotropic) medium poses greater difficulty since the polarization state and the wave vector direction -which are independent parameters for a given wave frequency in an isotropic medium -become coupled.Yet, it was recently shown that Faraday-Fresnel Rotation (FFR) is also found for the high frequency magnetized plasma modes that are Whistler-Helicon and Trivelpiece-Gould modes (Rax & Gueroult 2021).For such high frequency modes it was found that the main modifications induced by the plasma rotation are associated with Doppler shift and Coriolis effect in the dispersion relation.Interestingly, we note that the result that rotation is the source of an azimuthal component for the group velocity of low frequency waves in magnetized plasmas when Ω • k = 0 was already pointed out in geophysics and astrophysics (Acheson & Hide 1973), but the connection to a Faraday-Fresnel rotation of the transverse structure of the wave did not seem to have been made.An added complexity for these low frequency modes is that one must, in addition to anisotropy and gyrotropy, consider the strong coupling to the inertial mode (Lighthill 1980) that then comes into play.Revisiting this problem, we derive here in this study the expression for FFR for low frequency rotating Alfvén waves in a rotating magnetized plasma.
This paper is organised as follows.After briefly recalling the configuration of interest and previous results in the next section, we construct in section 3 the spectrum of low frequency, small amplitude, fluid waves in a magnetized rotating plasma.The set of linearised Euler and Maxwell equations describes an oscillating Beltrami flow-force free field (Chandrasekhar & Prendergast 1956) whose components are expressed with a cylindrical Chandrasekhar-Kendall (CK) potential (Chandrasekhar & Kendall 1957;Yoshida 1991).Then, in section 4, these orbital angular momentum carrying waves are shown to display a FFR under the influence of the plasma rotation.Section 5 focuses on the inverse problem when the orbital angular momentum of the wave is absorbed by the plasma.We derive in this case the torque exerted by this wave on the fluid driven as a function of the wave and plasma parameters.Finally section 6 summarises the main findings of this study.

Background
In this study we consider a rotating magnetized plasma column with angular velocity Ω = Ωe z and static uniform magnetic field B 0 = B 0 e z .We write (r, θ, z) and (x, y, z) cylindrical and Cartesian coordinates on cylindrical (e r , e θ , e z ) and Cartesian (e r , e θ , e z ) basis, respectively.The plasma dynamics is described assuming an inviscid and incompressible fluid model.We classically define the Alfvén velocity V .= B 0 / √ µ 0 ρ where µ 0 is the permittivity of vacuum and ρ the mass density of the fluid.
In the simple case where B 0 = 0 and Ω = 0 the rotating plasma behaves as an ordinary rotating fluid and inertial waves can propagate.Taking a phase factor Uncoupled dispersion of torsional Alfvén waves (TAW) obtained for B0 = 0 and Ω = 0, and of inertial waves (IM) obtained for Ω = 0 and B0 = 0.
exp j ωt − k z − k ⊥ y , the dispersion relation for this inertial mode (IM) is (Lighthill 1980) Conversely, in the case where Ω = 0 but B 0 = 0, Alfvén waves can propagate in the magnetized plasma at rest.The dispersion of this torsional mode (TAW) is (Stix 1992) Note that compressional Alfvén wave (CAW) are not considered here as we are considering an incompressible plasma.The dispersion of uncoupled TAW and IM is plotted in Fig. 1 in the k V /ω, k ⊥ V /ω plane for a given frequency ω.In this figure the grey zones indicate regions of strong coupling between TAW and IM.Note that we have normalised for convenience the wave-vector to ω/V , and that even for the unmagnetized IM branch.
In the more general case where both B 0 = 0 and Ω = 0, then a strong coupling between IM and TAW modes rearranges the spectrum and gives rise to two new branches (Lehnert 1954;Acheson & Hide 1973).Since as already pointed out by Acheson & Hide (1973) the group velocity of these modes for waves such that Ω • k = 0 has an azimuthal component, then we expect Fresnel-Faraday Rotation as recently identified for Trivelpiece-Gould and Whistler-Helicon high frequency electronic modes (Rax & Gueroult 2021).

Rotating Alfvén waves in a rotating plasma
In this section we examine the properties of low frequency waves carrying orbital angular momentum in a rotating magnetized plasma.

Classical modes
Two methods can be used to identify and describe the coupling between the angular momentum of a rotating plasma column and the angular momentum of a wave propagating in this rotating magnetized plasma.One is to consider the transformation laws of the various parameters from the lab frame to a rotating frame.The other is to perform the study in the lab frame starting from first principles.Here we will use the first method, similarly to original contributions on MHD waves in rotating conductive fluids (Lehnert 1954;Hide 1969), and solve the perfect MHD dynamics to calculate the rotating plasma linear response for the low frequency branches where the coupling between the fields and the particles is large.By working in the co-rotating frame (R) rather than in the laboratory frame (L), both the Coriolis force 2Ω × v and the centrifugal forces −∇ψ with ψ = −Ω 2 r 2 /2 must be taken into account.
We model the evolution of the wave velocity field v (r, t) using Euler's equation under the assumption of zero viscosity and the evolution of the wave magnetic field B (r, t) using Maxwell-Faraday's equation under the assumption of perfect conductivity where ρ is the mass density of the fluid and P is the pressure.These dynamical relations are completed by the flux conservation law for the magnetic field and the incompressibility relation for the velocity field.As already mentioned this last relation will restrict the plasma behaviour to the Alfvénic dynamics associated with torsional waves.
We then consider a small amplitude magnetohydrodynamic perturbation, propagating along and around the z axis, described by a magnetic perturbation with respect to the uniform static magnetic field B 0 = B 0 e z .The wave frequency ω is assumed smaller than the ion cyclotron frequency and larger than the collision frequency to validate the use of the perfect MHD model Eqs.(3.1, 3.2).The oscillating magnetic wave B is associated with an oscillating hydrodynamic velocity perturbation v v (r, θ, z, t) = u (r, θ, z) exp(jωt), (3.6) with respect to rotating frame velocity equilibrium v 0 = 0.The pressure P balances the centrifugal force at equilibrium ∇ (P + ρψ) = 0 and the pressure perturbation is p (r, θ, z) exp jωt.To first order in these perturbation the linearisation of Eqs.(3.1) and (3.2) gives Flux conservation and incompressibility provide the two additional conditions Taking the curl of both Eqs.(3.7, 3.8) and eliminating B gives a linear relation for the velocity perturbation Now if ones Fourier analyses this velocity perturbation as a superposition of plane waves (3.12) Coupled dispersion of magnetoinertial waves (MI) and inertial waves (IM).
that is to say put the emphasis on the linear momentum dynamics rather than on the angular momentum one, one recovers the two branches of Alfvenic/Inertial perturbations in a rotating plasma (Lehnert 1954;Acheson & Hide 1973).Specifically, plugging Eq. (3.12) into Eq.(3.11) and then taking the cross product jk× of this algebraic relation one obtain the dispersion relation These two branches, which are illustrated in Fig. 2, have been widely investigated within the context of geophysical and astrophysical magnetohydrodynamics models.For short wavelengths the Ω = 0 torsional Alfvén wave (TAW) splits into inertial (IM) and a magneto-inertial (MI) waves.For long wavelengths, that is in the grey zone in Fig. 2, inertial terms dominate the dispersion and the IM mode is found to reduce to its zero rotation behaviour already shown in Fig. 1.Note finally that the torsional Alfvén wave is recovered for large k ⊥ where a local dispersion becomes valid as opposed to small k ⊥ where the large wavelength allows the wave to probe the large scale behaviour of the rotation.

Beltrami flow
Instead of this usual procedure using a full Fourier decomposition as given by Eq. (3.12), we start here by considering a travelling perturbations along z of the form u (r, θ, z) = w(r, θ) exp(−jk z). (3.14) Note that this is analog to what was already done by Shukla (2012) to study OAM carrying dispersive shear Alfvén waves though in this earlier study the paraxial approximation and a two fluid model were used, and the plasma was considered at rest (i.e. non-rotating).Plugging Eq. (3.14) in the dispersion relation for a rotating plasma Eq. (3.11) gives where we have defined From Eq. (3.8) the oscillating magnetic field then writes (3.17) The two modes identified in Fig. 2 can be recovered from Eq. (3.16).More specifically, for k V > ω Eq. (3.15) describes an Alfvén wave modified by inertial effect.Conversely for k V < ω Eq. (3.15) describes an inertial wave modified by MHD coupling.In the following we will focus on the Alfvén wave dynamics and thus assume K > 0. Equation (3.15) is characteristic of a Beltrami flow (Chandrasekhar & Prendergast 1956).As such u can be written in terms of the so called Chandrasekhar-Kendall (CK) potential Φ (Chandrasekhar & Kendall 1957) as where the CK potential is solution of the scalar Helmholtz equation One verifies that the three components of Eq. (3.18) are independent.
Before examining the structure of OAM carrying modes through the CK potential, two additional results can be obtained from Eq. (3.16).First, for the Fourier decomposition used above, plugging Eq. (3.13) in Eq. (3.16) gives (3.20) Second, we can derive the dimensionless group-velocity dispersion coefficient which we will use later to explicit the axial wave vector difference for two eigenmodes with opposite OAM content.

Structure of OAM carrying modes
Because we are interested in waves carrying orbital angular momentum around z and linear momentum along z, we search for solutions of the form where m ∈ Z is the azimuthal mode number associated with the orbital angular momentum of the wave.From Eq. (3.19) the radial amplitude of this rotating CK potential φ(r) must be solution of the Bessel equation Since as shown in Eq. (3.20)K 2 > k 2 , φ(r) is in general the combination of Bessel functions of the first and the second kind and order m ∈ Z, J m and Y m .Yet, the finite value of φ at r = 0 demands to restrict the physical solution to Bessel functions of the first kind J m so that we find with the cylindrical dispersion relation Note that, like the ordinary plane wave Eq. (3.12) used in the standard analysis, the cylindrical Bessel waves Eq. (3.24) can not be normalised.
Putting these pieces together one finally gets The components in the plasma frame of a rotating Alfvén wave with azimuthal mode number m thus have an amplitude proportional to combination of Bessel functions of the first kind and of orders m and m ± 1.All these Bessel functions have the same radial dependence, namely K 2 k , ω − k 2 r, where K k , ω is given by Eq. (3.16).

Direct rotational Fresnel drag-orbital Faraday rotation
Let us now rewrite these perturbations as seen from the laboratory frame.We use the index R for the rotating plasma rest frame and L for the laboratory frame.The radial Eulerian coordinates r and z are unchanged through this change of frame or reference, but azimuthal coordinates θ changes, with Since the axial wave-vector is unchanged k R = k L , the phase of the wave in the plasma rest-frame in the laboratory frame.
Equipped with these transformations we can now describe the conditions to observe Fresnel-Faraday Rotation.For this we consider two CK potentials describing two Alfvén modes with opposite OAM content in the rotating frame R These transform in the CK potentials in the laboratory frame L as a result of the rotational Doppler shift θ| L = θ| R + Ωt.These Alfvén CK potentials Φ ± | L can be driven by a multicoil antenna similar to that used to study Whistler-Helicon modes (Stenzel & Urrutia 2014, 2015b,a,c;Urrutia & Stenzel 2015, 2016;Stenzel & Urrutia 2018;Stenzel 2019).The radial field pattern is then a superposition of +m and −m Bessel amplitudes J ±m (αr) where α is associated with the radial modulation of the antenna currents (Rax & Gueroult 2021).The antenna then sets both the radial wave-vector α and the frequency ω, whereas the axial wave-vectors k ± δk are solutions of the rotating frame dispersion relation.From Eq. (3.25) (4.9) Since we assume ω ≫ Ω and k ≫ δk we can Taylor expand Eq. (4.8) and Eq.(4.9) to get δk , leading to Eq. (3.21) can then be used to finally write the axial wave-vector difference δk for two modes with the same frequency ω, the same radial amplitude |J m (αr)| and equal but opposite azimuthal number |m| as where K k , ω, Ω is given by Eq. (3.16).This implies that there will be a difference in the axial phase velocity ω/ k ± δk of these two modes, and because these two modes rotates in opposite direction due to their opposite azimuthal mode number, the transverse structure of the sum of these modes will rotate.This is Fresnel drag-Faraday orbital rotation effects.Specifically, if one launches a wave which is a superposition of +m and −m modes such that at the antenna location z = 0 the wave transverse amplitude rotates as it propagates along z > 0 with an angular velocity along the propagation axis This CK potential rotation is illustrated in Fig. 3 for the case m = 4. Eqs.(4.11, 4.12) quantifies the direct Faraday-Fresnel effect of Alfvén waves in rotating plasmas, completing the similar results previously obtained for Trivelpiece-Gould and Whistler-Helicon modes (Rax & Gueroult 2021).The 1/m factor in Eq. (4.12) comes from the fact that the image constructed from the superposition of ±m modes has a 2m-fold symmetry.
To conclude this section it was shown that besides the Fresnel-Faraday Rotation associated to a phase velocity difference for m and −m modes, there can also be a spliting of the envelope of a (m, −m) wave packet if the group velocity for co-rotating (m) and counter-rotating (−m) modes were different (Rax & Gueroult 2021).We note that this second effect is also present here for Alfvén waves in rotating plasmas.Indeed, given a radial wave-vector α the dispersion relation is D = K 2 − k 2 − α 2 = 0, so that from Eq. (3.21) the axial group velocity is given by the Fresnel-Faraday splitting, we consider in the next section the inverse Fresnel-Faraday effect associated with wave absorption.

Inverse rotational Fresnel drag and angular moment absorption
In a perfectly conducting inviscid plasma there is no power absorption.The power exchange between the oscillating electromagnetic field and the plasma is purely reactive.To obtain an irreversible (active) angular momentum absorption, on needs a dissipative mechanism.Two different wave orbital angular momentum absorption mechanisms can be considered.One is resonant collisionless absorption, the other is collisional absorption.The former was recently studied in Rax et al. (2023) through quasilinear theory and will not be considered here.Instead, we consider in this section a weakly dissipative plasma where the ideal MHD hypothesis of perfect conductivity is relaxed and the inviscid assumption of zero viscosity no longer apply.In both case, collisionnal or collisionless, each time an energy δU is absorbed by the plasma, an axial angular momentum δL = mδU/ω is also absorbed by the plasma (Rax et al. 2017(Rax et al. , 2023)).The rate of decay of the wave angular momentum is hence equal to the wave induced density of torque on the plasma Γ = dL/dt.In steady-state, this angular momentum transfer dΓ/dt is balanced by viscous damping of the velocity shear and Ohmic dissipation of the radial charge polarisation sustaining the rotation.This dissipation is larger in the collisionnal case considered here than in the collisionless regime considered in Rax et al. (2023).
Specifically, we introduce two dissipative collisional coupling to our dissipation-less system Eqs.(3.7, 3.8), namely finite viscosity ρµ and finite resistivity µ 0 η.We follow the notation of Taylor (1989) (devoted to Alfvén wave helicity absorption) and introduce the magnetic diffusion coefficient η and the kinematic viscosity µ.Ohm's law then writes E + v × B = µ 0 ηj and the system Eqs.(5.4) Plugging these results into Eqs.(5.1, 5.2) yields the system where now viscous and resistive dissipation introduce a local relaxation.
We then take the rotational of the first equation and eliminate B using the second equation to get (5.7) After some algebra we find that the linearised dissipative regime of velocity and field low frequency oscillations is now described by rather than by the collisionles Eqs.(3.15, 3.17), where we have defined the two real wave . (5.10) We then consider an initial value problem with a weakly decaying wave of the form v = u exp [j (ω + jν) t] (5.11) with ω ≫ ν, and with the structure where α is a real number, ω and k are given, and the damping rate ν ω, k , K is to be determined from the weak dissipation expansion of the dispersion relation obtained by plugging this solution in Eq. (5.8).Taylor expanding this last relation for ν ≪ ω, the lowest order real part gives the collisionless dispersion while the lowest order imaginary part gives a relation for the decay rate ν (5.15) Here we took ∂K R /∂ω ≈ ∂K/∂ω and used Eq.(3.21).Finally, Eq. (5.15) can be used to write an equation for the evolution of the wave energy density U (5.16) For a rotating Alfvén wave, this energy density U has three distinct components where indicate an average over the fast ω oscillations.The first term on the right hand side is the magnetic energy, the second term is the electric energy and the third term is the kinetic energy.This energy density can be rewritten using the Alfvén velocity V and the velocity of light c as (5.18) Combining Eq. (5.16), Eq. ( 5.17) and the relation between energy and angular momentum absorption, one finally gets where K R and K I are given by Eq. (5.10) and v is given by Eq. (3.26).

Conclusion
Building on previous contibutions studying Alfvén waves in rotating plasmas in geophysical and astrophysical settings, we have examined here the dynamics of orbital angular momentum (OAM) carrying torsional Alfvén waves in a rotating plasma.It is found that two new couplings between the orbital angular momentum of the Alfvén waves and the angular momentum of the rotating plasma exist.
One is Fresnel-Faraday rotation (FFR), that is a rotation of the transverse structure of the wave due to the medium's rotation, which had already been predicted for the high frequency electronic modes that are Trivelpiece-Gould and Whistler-Helicon modes (Rax & Gueroult 2021).Extending these earlier contributions, direct Fresnel-Faraday rotation (FFR) for torsional Alfvén waves in a rotating plasma is described by Eqs.(4.11) and (4.12).It is the orbital angular momentum analog of the polarization drag effect for spin angular momentum waves (Jones 1976;Player 1976).An important distinction found here though is that while rotation did not introduce new high frequency modes so that FFR for Trivelpiece-Gould and Whistler-Helicon modes was simply the consequence of the interplay between Coriolis force and rotational Doppler shift (Rax & Gueroult 2021), the strong coupling to the inertial mode that exists for Alfvén waves in rotating plasmas complexifies this picture.
The second coupling is the inverse effect through which the OAM carrying wave exerts a torque on the plasma.Inverse FFR is described by Eqs. (5.10) and (5.19).This inverse effect is akin to the spin angular momentum inverse Faraday effect but for the orbital angular momentum of the wave.It is found that for a plasma with non-zero collisional absorption the damping of an OAM carrying wave is the source of a torque on the plasma.
Looking ahead, these results suggest that direct FFR could in principle be used to diagnose plasma rotation with Alfvén waves.Conversely, it may be possible to utilise inverse FFR to sustain plasma rotation through Alfvén waves angular momentum absorption.The detailed analysis of these promising prospects is left for future studies.