2. Background

In this study we consider a rotating magnetized plasma column with constant angular velocity $\boldsymbol {\varOmega }=\varOmega \boldsymbol {e}_{z}$ and static uniform magnetic field $\boldsymbol {B}_{0}=B_{0}\boldsymbol {e_{z}}$. We write $(r,\theta,z)$ cylindrical coordinates on a cylindrical basis $(\boldsymbol {e}_{r},\boldsymbol {e}_{\theta },\boldsymbol {e}_{z})$. The plasma dynamics is described assuming an inviscid and incompressible fluid model. We classically define the Alfvén velocity

(2.1)\begin{equation} \boldsymbol{V}\doteq\boldsymbol{B}_{0}/\sqrt{\mu_{0}\rho }, \end{equation}

where $\mu _{0}$ is the permittivity of vacuum and $\rho$ the mass density of the fluid.

In the simple case where $B_{0}=0$ and $\varOmega \neq 0$ the rotating plasma behaves as an ordinary rotating fluid and inertial waves can propagate. Taking a phase factor $\exp j( \omega t-k_{\Vert }z-k_{\perp }y)$, the dispersion relation for this inertial mode is (Lighthill Reference Lighthill1980)

(2.2)\begin{equation} \omega ={\pm} 2\varOmega k_{\Vert }/\sqrt{k_{\Vert }^{2}+k_{{\perp} }^{2}}. \end{equation}

Conversely, in the case where $\varOmega =0$ but $B_{0}\neq 0$, Alfvén waves can propagate in the magnetized plasma at rest. The dispersion of this torsional mode is (Stix Reference Stix1992)

(2.3)\begin{equation} \omega ={\pm} B_{0}k_{\Vert }/\sqrt{\mu_{0}\rho }={\pm} k_{\Vert }V. \end{equation}

Note that compressional Alfvén waves are not considered here as we are considering an incompressible plasma. The dispersion of uncoupled torsional Alfvén waves and inertial waves is plotted in figure 1 in the $( k_{\Vert }V/\omega,k_{\perp }V/\omega )$ plane for a given frequency $\omega$. Note that we have normalized for convenience the wavevector to $\omega /V$, and that even for the unmagnetized inertial wave branch.

Figure 1.

Uncoupled dispersion of torsional Alfvén waves (TAW) obtained for $B_{0}\neq 0$ and $\varOmega =0$ and of inertial waves (IM) obtained for $\varOmega \neq 0$ and $B_{0}=0$.

In the more general case where both $B_{0}\neq 0$ and $\varOmega \neq 0$, then a strong coupling between inertial waves and torsional Alfvén waves modes rearranges the spectrum and gives rise to two new branches (Lehnert Reference Lehnert1954; Acheson & Hide Reference Acheson and Hide1973). This coupling is expected to be strongest in the regions where the the dispersion curves of the modes absent rotation (torsional Alfvén waves) and absent magnetic field (inertial waves) intersect, which is highlighted in grey in figure 1. Since as already pointed out by Acheson & Hide (Reference Acheson and Hide1973) the group velocity of these new modes for waves such that $\boldsymbol {\varOmega }\boldsymbol {\cdot }\boldsymbol {k}\neq 0$ 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 Reference Rax and Gueroult2021).

3. Rotating Alfvén waves in a rotating plasma

In this section we examine the properties of low-frequency waves carrying OAM in a rotating magnetized plasma.

3.1. 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 laboratory frame to a rotating frame. The other is to perform the study in the laboratory frame starting from first principles. Here we use the first method, similarly to original contributions on MHD waves in rotating conductive fluids (Lehnert Reference Lehnert1954; Hide Reference Hide1969), 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 $2\boldsymbol {\varOmega }\times \boldsymbol {v}$ and the centrifugal forces $\boldsymbol {-\nabla }\psi$ with $\psi =-\varOmega ^{2}r^{2}/2$ must be taken into account.

We model the evolution of the wave velocity field $\boldsymbol {v}(\boldsymbol {r},t)$ using Euler's equation under the assumptions of homogeneity and zero viscosity:

(3.1)\begin{equation} \frac{\partial \boldsymbol{v}}{\partial t}+( \boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{v}+2\boldsymbol{\varOmega }\times \boldsymbol{v} ={-}\boldsymbol{\nabla}\left( \frac{P}{\rho }+\psi \right) +\frac{1}{\mu_{0}\rho }( \boldsymbol{\nabla}\times \boldsymbol{B}) \times ( \boldsymbol{B+B}_{0}) \end{equation}

and the evolution of the wave magnetic field $\boldsymbol {B}(\boldsymbol {r},t)$ using the Maxwell–Faraday equation under the assumption of perfect conductivity:

(3.2)\begin{equation} \frac{\partial \boldsymbol{B}}{\partial t} =\boldsymbol{\nabla}\times [\boldsymbol{v}\times ( \boldsymbol{B+B}_{0})],\end{equation}

where $\rho$ is the mass density of the fluid and $P$ is the pressure. These dynamical relations are completed by the flux conservation law

(3.3)\begin{equation} \boldsymbol{\nabla }\boldsymbol{\cdot} \boldsymbol{B}=0 \end{equation}

for the magnetic field and the incompressibility relation

(3.4)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{v}=0 \end{equation}

for the velocity field. As already mentioned, this last relation will restrict the plasma behaviour to the Alfvénic dynamics associated with torsional waves. We should also stress that the assumptions made here and in this study only capture specific kinds of pressure equilibrium. Indeed a rotating equilibrium demands a radial gradient in the total pressure (that is either in the plasma pressure or in the magnetic pressure) to balance the centrifugal forces (Hameiri Reference Hameiri1983). In bringing the density under the gradient in (3.1), which will facilitate the analysis of perturbations around this equilibrium, we implicitly assumed a homogeneous density. In this model the pressure gradient must thus come entirely from a temperature gradient. Although restrictive, we work here within this homogeneous assumption to expose new coupling phenomena, and keep in mind that this is only an approximation whose justification remains to be demonstrated.

We now consider a small-amplitude MHD perturbation, propagating along and around the $z$ axis, described by a magnetic perturbation:

(3.5)\begin{equation} \boldsymbol{B}( r,\theta ,z,t) = \mathfrak{B}( r,\theta ,z)\exp (j\omega t) \end{equation}

with respect to the uniform static magnetic field $\boldsymbol {B}_{0} = B_{0}\boldsymbol {e}_{z}$. The wave frequency $\omega$ is assumed smaller than the ion cyclotron frequency and larger than the collision frequency to validate the use of the perfect MHD model (3.1), (3.2). The oscillating magnetic wave $\boldsymbol {B}$ is associated with an oscillating hydrodynamic velocity perturbation $\boldsymbol {v}$:

(3.6)\begin{equation} \boldsymbol{v}( r,\theta ,z,t) =\boldsymbol{u}( r,\theta ,z)\exp (j\omega t) \end{equation}

with respect to rotating frame velocity equilibrium $\boldsymbol {v}_{0}=\boldsymbol {0}$. The pressure $P$ balances the centrifugal force at equilibrium $\boldsymbol {\nabla }( P+\rho \psi ) =\boldsymbol {0}$ and the pressure perturbation is $p( r,\theta,z)\exp j\omega t$. To first order in these perturbations, the linearization of (3.1) and (3.2) gives

(3.7)\begin{gather} j\omega \boldsymbol{u}+2\boldsymbol{\varOmega }\times \boldsymbol{u} ={-}\boldsymbol{\nabla}( p/\rho ) +\frac{1}{\mu_{0}\rho }( \boldsymbol{\nabla}\times \mathfrak{B}) \times \boldsymbol{B}_{0}, \end{gather}

(3.8)\begin{gather}j\omega \mathfrak{B} =( \boldsymbol{B}_{0}\boldsymbol{\cdot} \boldsymbol{\nabla})\boldsymbol{u}. \end{gather}

Flux conservation and incompressibility provide the two additional conditions

(3.9)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}=0, \end{gather}

(3.10)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \mathfrak{B}=0. \end{gather}

Taking the curl of both (3.7) and (3.8) and eliminating $\mathfrak {B}$ gives a linear relation for the velocity perturbation:

(3.11)\begin{equation} \omega^{2}\boldsymbol{\nabla}\times \boldsymbol{u}+2j\omega ( \boldsymbol{\varOmega} \boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u}+( \boldsymbol{V}\boldsymbol{\cdot} \boldsymbol{\nabla})^{2} ( \boldsymbol{\nabla}\times \boldsymbol{u}) =\boldsymbol{0} ,\end{equation}

with $\boldsymbol {V}$ the Alfvén velocity already defined in (2.1). Note that the elimination of the pressure term obtained by taking the curl of (3.7) would not be possible if, as hinted at earlier, accounting for a radial density gradient.

Now if one Fourier-analyses this velocity perturbation as a superposition of plane waves:

(3.12)\begin{equation} \boldsymbol{u}( \boldsymbol{r}) \exp j\omega t=\exp [j( \omega t- \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{r})] , \end{equation}

that is to say put the emphasis on the linear momentum dynamics rather than on the angular momentum dynamics, one recovers the two branches of Alfvénic/inertial perturbations in a rotating plasma (Lehnert Reference Lehnert1954; Acheson & Hide Reference Acheson and Hide1973). Specifically, plugging (3.12) into (3.11) and then taking the cross-product $j\boldsymbol {k}\times$ of this algebraic relation, one obtains the dispersion relation

(3.13)\begin{equation} \omega^{2}-( \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{V})^{2}={\pm} 2\omega ( \boldsymbol{\varOmega} \boldsymbol{\cdot} \boldsymbol{k}) / | \boldsymbol{k}| .\end{equation}

The two solutions, which are illustrated in figure 2, have been widely investigated within the context of geophysical and astrophysical MHD models. They are also in agreement with the solutions obtained from the local dispersion relation for magneto-rotational instabilities (Goedbloed, Keppens & Poedts Reference Goedbloed, Keppens and Poedts2019, § 13.5.1) in the appropriate limit. For short wavelengths, the $\varOmega =0$ torsional Alfvén wave splits into an inertial and a magneto-inertial wave. For long wavelengths, that is, in the grey zone in figure 2, inertial terms dominate the dispersion and the inertial wave is found to reduce to its zero rotation behaviour already shown in figure 1. Note finally that the torsional Alfvén wave is recovered for large $k_{\perp }$ where a local dispersion becomes valid as opposed to small $k_{\perp }$ where the large wavelength allows the wave to probe the large-scale behaviour of the rotation.

Figure 2.

Coupled dispersion of magneto-inertial waves (MI) and inertial waves (IM).

3.2. Beltrami flow

Instead of this usual procedure using a full Fourier decomposition as given by (3.12), we start here by considering a travelling perturbation along $z$ of the form

(3.14)\begin{equation} \boldsymbol{u}( r,\theta,z) =\boldsymbol{w}(r,\theta) \exp ({-}jk_{\Vert }z).\end{equation}

Note that this is analogous to what was already done by Shukla (Reference Shukla2012) 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 (3.14) in the dispersion relation for a rotating plasma (3.11) gives

(3.15)\begin{equation} \boldsymbol{\nabla}\times \boldsymbol{u}=\mathcal{K}\boldsymbol{u} ,\end{equation}

where we have defined

(3.16)\begin{equation} \mathcal{K}( k_{\Vert },\omega )\doteq 2\frac{\varOmega }{\omega }k_{\Vert }\left( \frac{k_{\Vert }^{2}V^{2}}{\omega^{2}}-1\right)^{{-}1}. \end{equation}

From (3.8) the oscillating magnetic field is then written as

(3.17)\begin{equation} \omega \mathfrak{B}={-}\sqrt{\mu_{0}\rho }k_{\Vert }V\boldsymbol{u}. \end{equation}

The two modes identified in figure 2 can be recovered from (3.16). More specifically, for $k_{\Vert }V>\omega$ (3.15) describes an Alfvén wave modified by inertial effect. Conversely for $k_{\Vert }V<\omega$ (3.15) describes an inertial wave modified by MHD coupling. In the following we focus on the Alfvén wave dynamics and thus assume $\mathcal {K}>0$.

Equation (3.15) is characteristic of a Beltrami flow (Chandrasekhar & Prendergast Reference Chandrasekhar and Prendergast1956). As such $\boldsymbol {u}$ can be written in terms of the so-called CK potential $\varPhi$ (Chandrasekhar & Kendall Reference Chandrasekhar and Kendall1957) as

(3.18)\begin{align} \boldsymbol{u} & =\frac{1}{\mathcal{K}}\boldsymbol{\nabla}\times (\boldsymbol{\nabla}\times\varPhi\boldsymbol{e}_{z}) +\boldsymbol{\nabla}\times\varPhi\boldsymbol{e}_{z}\nonumber\\ & ={-}\left[\frac{1}{\mathcal{K}}\boldsymbol{\nabla}\times\boldsymbol{e}_{z} \times\boldsymbol{\nabla}+\boldsymbol{e}_{z}\times\boldsymbol{\nabla}\right]\varPhi , \end{align}

where the CK potential is a solution of the scalar Helmholtz equation

(3.19)\begin{equation} \Delta \varPhi +\mathcal{K}^{2}\varPhi =0. \end{equation}

One verifies that the three components of (3.18) are independent.

Before examining the structure of OAM-carrying modes through the CK potential, two additional results can be obtained from (3.16). First, for the Fourier decomposition used above, plugging (3.13) in (3.16) gives

(3.20)\begin{equation} \frac{\mathcal{K}^{2}}{k_{\Vert }^{2}} =1+\frac{k_{{\perp} }^{2}}{k_{\Vert }^{2}}>1. \end{equation}

Second, we can derive the dimensionless group-velocity dispersion coefficient

(3.21)\begin{equation} \frac{\omega }{\mathcal{K}}\frac{\partial \mathcal{K}}{\partial \omega } ={-}\frac{k_{\Vert }}{\mathcal{K}}\frac{\partial \mathcal{K}}{\partial k_{\Vert }}=\frac{k_{\Vert }^{2}V^{2}+\omega^{2}}{k_{\Vert }^{2}V^{2}-\omega^{2}} ,\end{equation}

which we use later to make explicit the axial wavevector difference for two eigenmodes with opposite OAM content.

3.3. Structure of OAM-carrying modes

Because we are interested in waves carrying OAM around $z$ and linear momentum along $z$, we search for solutions of the form

(3.22)\begin{equation} \varPhi ( r,\theta ,z) =\phi ( r) \exp [{-}j( m\theta+k_{\Vert }z)],\end{equation}

where $m\in \Bbb {Z}$ is the azimuthal mode number associated with the OAM of the wave. From (3.19) the radial amplitude of this rotating CK potential $\phi (r)$ must be a solution of the Bessel equation

(3.23)\begin{equation} \frac{1}{r}\frac{{\rm d}}{{\rm d} r}\left( r\frac{{\rm d} \phi }{{\rm d} r}\right) -\frac{m^{2}}{r^{2}}\phi +( \mathcal{K}^{2}-k_{\Vert }^{2}) \phi =0. \end{equation}

Since as shown in (3.20) $\mathcal {K}^{2}>k_{\Vert }^{2}$, $\phi (r)$ is in general the combination of Bessel functions of the first and the second kind and order $m\in \Bbb {Z}$, $J_{m}$ and $Y_{m}$. Yet, the finite value of $\phi$ at $r=0$ demands restricting the physical solution to Bessel functions of the first kind $J_{m}$ so that we find

(3.24)\begin{equation} \phi ( r) =J_{m}( \alpha r) ,\end{equation}

with the cylindrical dispersion relation

(3.25)\begin{equation} \alpha^{2}+k_{\Vert }^{2}=\mathcal{K}^{2}( k_{\Vert },\omega ) .\end{equation}

Note that, like the ordinary plane wave (3.12) used in the standard analysis, the cylindrical Bessel waves (3.24) cannot be normalized.

Putting these pieces together one finally gets

(3.26)\begin{align} \boldsymbol{v}& =\left[\frac{1}{\mathcal{K}}\boldsymbol{\nabla}\times \boldsymbol{e}_{z}\times\boldsymbol{\nabla}+\boldsymbol{e}_{z}\times\boldsymbol{\nabla}\right] J_{m}( \sqrt{\mathcal{K}^{2}-k_{\Vert }^{2}}r) \exp[ j( \omega t-m\theta -k_{\Vert }z)]\nonumber\\ & ={-}\frac{\omega \boldsymbol{B}}{\sqrt{\mu_{0}\rho }k_{\Vert }V} . \end{align}

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 $m\pm 1$. All these Bessel functions have the same radial dependence, namely $\sqrt {\mathcal {K}^{2}( k_{\Vert },\omega )-k_{\Vert }^{2}}r$, where $\mathcal {K}( k_{\Vert },\omega )$ is given by (3.16).

4. 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 the azimuthal coordinate $\theta$ changes, with

(4.1)\begin{gather} r |_{L}=r|_{R}, \end{gather}

(4.2)\begin{gather}z |_{L}=z|_{R}, \end{gather}

(4.3)\begin{gather}\theta |_{L}=\theta |_{R}+\varOmega t. \end{gather}

Since the axial wavevector is unchanged $k_{\Vert } |_{R}=k_{\Vert } |_{L}$, the phase of the wave in the plasma rest frame

(4.4)\begin{equation} \omega t-k_{\Vert }z\pm m \theta |_{R} \end{equation}

becomes

(4.5)\begin{equation} ( \omega \mp m\varOmega ) t-k_{\Vert }z\pm m\theta |_{L} \end{equation}

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 $R$:

(4.6)\begin{equation} \left. \begin{array}{c@{}} \varPhi_{+}|_{R} =J_{m}( \alpha r) \exp (j[( \omega -m\varOmega ) t-( k_{\Vert }-\delta k_{\Vert })z-m\theta |_{R}]),\\ \varPhi_{-}|_{R} =J_{{-}m}( \alpha r) \exp (j[( \omega +m\varOmega ) t-( k_{\Vert }+\delta k_{\Vert })z+m\theta |_{R}]). \end{array}\right\} \end{equation}

These transform in the CK potentials in the laboratory frame $L$:

(4.7)\begin{equation} \left. \begin{array}{c@{}} \varPhi_{+}|_{L} =J_{m}( \alpha r) \exp (j[\omega t-( k_{\Vert }-\delta k_{\Vert }) z-m\theta |_{L}]), \\ \varPhi_{-}|_{L} =J_{{-}m}( \alpha r) \exp (j[\omega t-( k_{\Vert }+\delta k_{\Vert }) z+m\theta |_{L}]), \end{array}\right\} \end{equation}

as a result of the rotational Doppler shift $\theta |_{L}=\theta |_{R}+\varOmega t$. These Alfvén CK potentials $\varPhi _{\pm }|_{L}$ can be driven by a multicoil antenna similar to that used to study Whistler–Helicon modes (Stenzel & Urrutia Reference Stenzel and Urrutia2014, Reference Stenzel and Urrutia2015a,Reference Stenzel and Urrutiab,Reference Stenzel and Urrutiac; Urrutia & Stenzel Reference Urrutia and Stenzel2015, Reference Urrutia and Stenzel2016; Stenzel & Urrutia Reference Stenzel and Urrutia2018; Stenzel Reference Stenzel2019). The radial field pattern is then a superposition of $+m$ and $-m$ Bessel amplitudes $J_{\pm m}( \alpha r)$, where $\alpha$ is associated with the radial modulation of the antenna currents (Rax & Gueroult Reference Rax and Gueroult2021). The antenna then sets both the radial wavevector $\alpha$ and the frequency $\omega$, whereas the axial wavevectors $k_{\Vert }\pm \delta k_{\Vert }$ are solutions of the rotating frame dispersion relation. From (3.25),

(4.8)\begin{gather} \alpha =\sqrt{\mathcal{K}^{2}( k_{\Vert }-\delta k_{\Vert },\omega -m\varOmega ) -( k_{\Vert }-\delta k_{\Vert })^{2}}, \end{gather}

(4.9)\begin{gather}\alpha =\sqrt{\mathcal{K}^{2}( k_{\Vert }+\delta k_{\Vert },\omega +m\varOmega ) -( k_{\Vert }+\delta k_{\Vert })^{2}}. \end{gather}

Since we assume $\omega \gg \varOmega$ and $k_{\Vert }\gg \delta k_{\Vert }$, we can Taylor-expand (4.8) and (4.9) to get $\delta k_{\Vert }$, leading to

(4.10)\begin{equation} \frac{k_{\Vert }}{\mathcal{K}}\delta k_{\Vert }=\frac{\delta k_{\Vert }}{2}\frac{\partial \mathcal{K}( \omega ,k_{\Vert }) }{\partial k_{\Vert }}+\frac{m\varOmega}{2} \frac{ \partial \mathcal{K}( \omega ,k_{\Vert }) }{\partial \omega }. \end{equation}

Equation (3.21) can then be used to finally write the axial wavevector difference $\delta k_{\Vert }$ for two modes with the same frequency $\omega$, the same radial amplitude $| J_{m}( \alpha r) |$ and equal but opposite azimuthal number $| m|$ as

(4.11)\begin{equation} \frac{\delta k_{\Vert }}{k_{\Vert }}=\frac{1}{2}m\frac{\varOmega }{\omega }\frac{1+\dfrac{k_{\Vert }^{2}V^{2}}{\omega^{2}}}{1-\dfrac{k_{\Vert }^{2}}{\mathcal{K}^{2}} +\left( 1+\dfrac{k_{\Vert }^{2}}{\mathcal{K}^{2}}\right) \dfrac{k_{\Vert }^{2}V^{2}}{\omega^{2}}} ,\end{equation}

where $\mathcal {K}( k_{\Vert },\omega,\varOmega )$ is given by (3.16). This implies that there will be a difference in the axial phase velocity $\omega /( k_{\Vert }\pm \delta k_{\Vert })$ of these two modes, and because these two modes rotate in opposite directions due to their opposite azimuthal mode number, the transverse structure of the sum of these modes will rotate. This is the Fresnel drag–Faraday orbital rotation effect. Specifically, if one launches a wave which is a superposition of $+m$ and $-m$ modes such that at the antenna location $z=0$

(4.12)\begin{equation} \varPhi |_{z=0}=J_{m}( \alpha r) (( \exp [j( \omega t-m\theta |_{L})] +({-}1)^{m}\exp [j( \omega t+m\theta |_{L})] ), \end{equation}

the wave transverse amplitude rotates as it propagates along $z>0$ with an angular velocity along the propagation axis

(4.13)\begin{equation} \frac{{\rm d} \theta }{{\rm d} z}|_{L}=\frac{\delta k}{m}=\frac{1}{2} \frac{\varOmega }{\omega }k_{\Vert }\mathcal{K}^{2} \frac{k_{\Vert }^{2}V^{2}+\omega^{2}}{k_{\Vert }^{2}V^{2} ( \mathcal{K}^{2}+k_{\Vert }^{2}) +\omega^{2}( \mathcal{K}^{2}-k_{\Vert }^{2})}.\end{equation}

This CK potential rotation is illustrated in figure 3 for the case $m=4$. Equations (4.11) and (4.13) quantify 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 Reference Rax and Gueroult2021). The $1/m$ factor in (4.13) comes from the fact that the image constructed from the superposition of $\pm m$ modes has a $2m$-fold symmetry.

Figure 3.

Fresnel drag–Faraday rotation of the CK potential describing an Alfvén–Beltrami wave with $m=\pm 4$ after a propagation along a path $z={\rm \pi} /4( \textrm {d}\theta / \textrm {d} z)$.

To conclude this section it was shown that besides the Fresnel–Faraday rotation associated with a phase-velocity difference for $m$ and $-m$ modes, there can also be a splitting of the envelope of an $( m,-m)$ wave packet if the group velocity for co-rotating ($m$) and counter-rotating ($-m$) modes were different (Rax & Gueroult Reference Rax and Gueroult2021). We note that this second effect is also present here for Alfvén waves in rotating plasmas. Indeed, given a radial wavevector $\alpha$ the dispersion relation is $\mathcal {D}=\mathcal {K}^{2}-k_{\Vert }^{2}-\alpha ^{2}=0$, so that from (3.21) the axial group velocity is given by

(4.14)\begin{equation} -\frac{\partial \mathcal{D}}{\partial k_{\Vert }}/\frac{\partial \mathcal{D}}{\partial \omega }=\frac{k_{\Vert }}{\mathcal{K}\partial \mathcal{K}/\partial \omega }-\frac{\omega }{k_{\Vert }},\end{equation}

and one verifies from (3.16) that the group velocity for a mode $(k_{\Vert }+ \delta k_{\Vert },m)$ and that for a mode $(k_{\Vert }- \delta k_{\Vert },-m)$ are different. Rather than deriving here an explicit formula for the Fresnel–Faraday splitting, we consider in the next section the inverse Fresnel–Faraday effect associated with wave absorption.

5. 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 OAM absorption mechanisms can be considered. One is resonant collisionless absorption; the other is collisional absorption. The former was recently studied in Rax, Gueroult & Fisch (Reference Rax, Gueroult and Fisch2023) 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 applies. In both cases, collisional or collisionless, each time an energy $\delta \mathcal {U}$ is absorbed by the plasma, an axial angular momentum $\delta L=m\delta \mathcal {U}/\omega$ is also absorbed by the plasma (Rax et al. Reference Rax, Gueroult and Fisch2017, Reference Rax, Gueroult and Fisch2023). The rate of decay of the wave angular momentum is hence equal to the wave-induced density of torque on the plasma $\varGamma =\textrm {d} L/\textrm {d} t$. In steady state, this angular momentum transfer $\textrm {d} \varGamma /\textrm {d} t$ is balanced by viscous damping of the velocity shear and Ohmic dissipation of the radial charge polarization sustaining the rotation. This dissipation is larger in the collisional case considered here than in the collisionless regime considered in Rax et al. (Reference Rax, Gueroult and Fisch2023).

Specifically, we introduce two dissipative collisional couplings to our dissipation-less system (3.7), (3.8), namely finite viscosity $\rho \mu$ and finite resistivity $\mu _{0}\eta$. We follow the notation of Taylor (Reference Taylor1989) (devoted to Alfvén wave helicity absorption) and introduce the magnetic diffusion coefficient $\eta$ and the kinematic viscosity $\mu$. Ohm's law is then written as $\boldsymbol {E}+\boldsymbol {v\times B}=\mu _{0}\eta \boldsymbol {j}$ and the system (3.7), (3.8) becomes

(5.1)\begin{gather} j\omega \boldsymbol{u}+2\boldsymbol{\varOmega }\times \boldsymbol{u} ={-}\boldsymbol{\nabla}( p/\rho ) +\frac{1}{\mu_{0}\rho }( \boldsymbol{\nabla}\times\mathfrak{B}) \times \boldsymbol{B}_{0}+\mu \Delta \boldsymbol{u}, \end{gather}

(5.2)\begin{gather}j\omega \mathfrak{B}=( \boldsymbol{B}_{0}\boldsymbol{\cdot} \boldsymbol{\nabla})\boldsymbol{u}+\eta \Delta\mathfrak{B}. \end{gather}

Since we assume weak dissipation, the resistive term $\eta \Delta \boldsymbol {B}$ in the Maxwell–Faraday equation and the viscous term $\mu \Delta \boldsymbol {u}$ in the Navier–Stokes equation can be evaluated with the dispersive properties of the non-dissipative dispersion relation. Within the bounds of this perturbative expansion scheme ($\mathcal {K}^{2}\eta \ll \omega$ and $\mathcal {K}^{2}\mu \ll \omega$), and for the perturbation $\boldsymbol {u}( r,\theta,z) =\boldsymbol {w}(r,\theta ) \exp (-jk_{\Vert }z)$ already given in (3.14), we get from (3.15), (3.17) the non-dissipative Laplacians

(5.3)\begin{gather} \Delta \boldsymbol{u} ={-}\mathcal{K}^{2}\boldsymbol{u}, \end{gather}

(5.4)\begin{gather}\Delta \mathfrak{B}={-}\mathcal{K}^{2}\mathfrak{B}. \end{gather}

Plugging these results into (5.1), (5.2) yields the system

(5.5)\begin{gather} j\omega \boldsymbol{u}+2\boldsymbol{\varOmega }\times \boldsymbol{u} ={-}\boldsymbol{\nabla} ( p/\rho ) +\frac{1}{\mu_{0}\rho } ( \boldsymbol{\nabla}\times \mathfrak{B} ) \times \boldsymbol{B}_{0}-\mathcal{K}^{2}\mu \boldsymbol{u}, \end{gather}

(5.6)\begin{gather}j\omega \mathfrak{B}= ( \boldsymbol{B}_{0}\boldsymbol{\cdot} \boldsymbol{\nabla} ) \boldsymbol{u}-\mathcal{K}^{2}\eta \mathfrak{B}, \end{gather}

where now viscous and resistive dissipation introduce a local relaxation.

We then take the rotational of the first equation and eliminate $\mathfrak {B}$ using the second equation to get

(5.7)\begin{equation} [( j\omega +\mathcal{K}^{2}\mu )( j\omega +\mathcal{K}^{2}\eta )] \boldsymbol{\nabla}\times \boldsymbol{u}+2j( j\omega +\mathcal{K}^{2}\eta) k_{\Vert }\varOmega \boldsymbol{u}+k_{\Vert }^{2}V^{2}\boldsymbol{\nabla}\times \boldsymbol{u=0}. \end{equation}

After some algebra we find that the linearized dissipative regime of velocity and field low-frequency oscillations is now described by

(5.8)\begin{gather} \boldsymbol{\nabla}\times \boldsymbol{u} = [ \mathcal{K}_{R} ( k_{\Vert },\omega )-j\mathcal{K}_{I} ( k_{\Vert },\omega ) ] \boldsymbol{u}, \end{gather}

(5.9)\begin{gather}( \omega -j\mathcal{K}^{2}\eta ) \boldsymbol{B} ={-}\sqrt{\mu_{0}\rho }k_{\Vert }V\boldsymbol{u}, \end{gather}

rather than by the collisionless equations (3.15), (3.17), where we have defined the two real wavevectors $\mathcal {K}_{R}\approx \mathcal {K}\gg \mathcal {K}_{I}$ through

(5.10)\begin{equation} \mathcal{K}_{R} ( k_{\Vert },\omega ) -j\mathcal{K}_{I} ( k_{\Vert },\omega ) =2\varOmega \frac{ ( \omega -j\mathcal{K}^{2}\eta ) k_{\Vert }}{k_{\Vert }^{2}V^{2}- (\omega -j\mathcal{K}^{2}\mu ) ( \omega -j\mathcal{K}^{2}\eta ) }.\end{equation}

We then consider an initial-value problem with a weakly decaying wave of the form

(5.11)\begin{equation} \boldsymbol{v}= \boldsymbol{u}\exp [j( \omega +j\nu ) t], \end{equation}

with $\omega \gg \nu$, and with the structure

(5.12)\begin{equation} \boldsymbol{v}=\left[\frac{1}{\mathcal{K}_{R}-j\mathcal{K}_{I}} \boldsymbol{\nabla}\times\boldsymbol{e}_{z}\times\boldsymbol{\nabla}+ \boldsymbol{e}_{z}\times\boldsymbol{\nabla}\right]J_{m}( \alpha r) \exp (j[( \omega +j\nu ) t-m\theta -k_{\Vert }z]), \end{equation}

where $\alpha$ is a real number, $\omega$ and $k_{\Vert }$ are given and the damping rate $\nu ( \omega,k_{\Vert },\mathcal {K})$ is to be determined from the weak dissipation expansion of the dispersion relation

(5.13)\begin{equation} \alpha^{2}+k_{\Vert }^{2}= [ \mathcal{K}_{R} ( k_{\Vert },\omega +j\nu )-j\mathcal{K}_{I} ( k_{\Vert },\omega +j\nu ) ]^{2}, \end{equation}

obtained by plugging this solution in (5.8). Taylor-expanding this last relation for $\nu \ll \omega$, the lowest-order real part gives the collisionless dispersion

(5.14)\begin{equation} \alpha^{2} ( k_{\Vert },\omega ) =\mathcal{K}_{R}^{2} ( k_{\Vert },\omega )-k_{\Vert}^{2}\approx \mathcal{K}^{2} ( k_{\Vert },\omega )-k_{\Vert }^{2} ,\end{equation}

while the lowest-order imaginary part gives a relation for the decay rate $\nu$:

(5.15)\begin{equation} \nu ( k_{\Vert },\omega ) \frac{\partial \mathcal{K}_{R} ( \omega ) }{\partial \omega } =\mathcal{K}_{I} ( k_{\Vert },\omega ) \approx \frac{\mathcal{K}^{3}}{\omega }\left[\eta+ (\mu +\eta)\left({\frac{k_{\Vert }^{2}V^{2}}{\omega^{2}}-1}\right)^{{-}1}\right]. \end{equation}

Here we took $\partial \mathcal {K}_{R}/\partial \omega \approx \partial \mathcal {K}/\partial \omega$ and used (3.21).

Finally, (5.15) can be used to write an equation for the evolution of the wave energy density $\mathcal {U}$:

(5.16)\begin{equation} \frac{{\rm d} \mathcal{U}}{{\rm d} t}={-}2\nu \mathcal{U}={-}2\mathcal{K}_{I}\left( \frac{\partial \mathcal{K}_{R}}{\partial \omega }\right)^{{-}1}\mathcal{U}. \end{equation}

For a rotating Alfvén wave, this energy density $\mathcal {U}$ has three distinct components:

(5.17)\begin{equation} \mathcal{U}=\frac{ \langle B^{2} \rangle }{2\mu_{0}}+\frac{\varepsilon_{0}}{2} \langle ( \boldsymbol{v}\times \boldsymbol{B}_{0} )^{2} \rangle +\frac{\rho }{2} \langle v^{2} \rangle ,\end{equation}

where $\langle{\;} \rangle$ indicates an average over the fast $\omega$ 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)\begin{equation} \mathcal{U}=\frac{\rho }{2}\left[ \langle \boldsymbol{v}^{2}\rangle \left( 1+ \frac{V^{2}}{c^{2}}\left( 1+\frac{k_{\Vert }^{2}c^{2}}{\omega^{2}}\right) \right) -\left\langle \left( \boldsymbol{v}\boldsymbol{\cdot} \frac{\boldsymbol{V}}{c}\right) ^{2}\right\rangle \right]. \end{equation}

Combining equations (5.16), (5.17) and the relation between energy and angular momentum absorption, one finally gets

(5.19)\begin{equation} \varGamma =2\rho \frac{m}{\omega }\mathcal{K}_{I}\left( \frac{\partial \mathcal{K}_{R}}{\partial \omega }\right)^{{-}1}\left[ \langle \boldsymbol{v}^{2}\rangle \left( 1+\frac{V^{2}}{c^{2}}\left( 1+\frac{k_{\Vert }^{2}c^{2}}{\omega^{2}}\right) \right) -\frac{\langle ( \boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{V})^{2}\rangle }{c^{2}}\right], \end{equation}

where $\mathcal {K}_{R}$ and $\mathcal {K}_{I}$ are given by (5.10) and $\boldsymbol {v}$ is given by (3.26).