#### 4.1. Gaussian Radar Beams

[24] Plasma motion transverse to the radar wave vector does not give rise to a Doppler shift, as seen from equation (4). However, typical radar beams have significant divergences and thus can be viewed as a superposition of plane waves with a range of wave vectors centered around the beam axis. Therefore, a wave vector in the beam generally has a nonzero component transverse to the beam axis, which, however, for a narrow beam is much smaller than that along the axis. This implies that the scatter will be affected by plasma motion transverse to the beam. For a Gaussian beam there will be wave vector components both parallel and antiparallel to a given flow direction so that the Doppler shift will broaden the fluctuation spectrum detected by the radar.

[25] Figure 1 illustrates schematically a unidirectional plasma flow transverse to a Gaussian radar beam. The dashed circle indicates the full width at half maximum (FWHM) of the beam. Figure 2 shows two spectra of ion acoustic fluctuations, or ion lines, for a plasma drift *v*_{x} = 1.0 km/s in the *x*-direction transverse to the radar line-of-sight and a beam with a Gaussian intensity profile (Figure 1). For typical F-region conditions this velocity is of the order of the ion-acoustic velocity, which determines the overall width of the incoherent scatter spectrum. The dashed spectrum is for the central single transmitted wave vector that is determined by the radar frequency and radar line-of-sight. As the plasma motion occurs transverse to the radar wave vector the ion line spectrum is unaffected by the flow. The solid-line spectrum inFigure 2 is for a transmitted Gaussian beam with a FWHM of 1.2° and is slightly weaker than the dashed spectrum. In this case the radar wave vector generally has components both parallel and antiparallel to the plasma motion, depending on where in the beam, which causes a Doppler broadening of the ion line integrated over the beam cross section. For the narrow beam considered here, the slight spectral broadening implies that the ion line peaks appear about 14 Hz closer to the radar frequency whereas the base of the spectrum is slightly widened.

[26] A broader beam and/or a higher flow velocity would increase the Doppler broadening. The EISCAT UHF and EISCAT Svalbard antenna dishes produce beams with divergences of 0.5° and 0.6°, respectively, so that the spectral broadening would be roughly only half of that at PFISR. On the other hand, flows with higher Mach numbers *M* have been observed on occasions, higher than *M* ≈ 10 (electric fields up to 600 mV/m) [*Lanchester et al.*, 1998]. If the effect is not taken into account in the data analysis (as has been the case so far, to our knowledge), the ion temperature gets overestimated accordingly.

[27] A twisted beam with a FWHM approximately that of the Gaussian beam used for Figure 2 would give approximately the same broadening of the ion line spectrum as the Gaussian beam, in the case of unidirectional transverse flow (Figure 1).

#### 4.2. Twisted Radar Beams

[28] As seen from equation (4), plasma flow strictly transverse to the radar wave vector does not cause a Doppler shift of a backscattered signal. However, according to equation (6) a rotational frequency shift (Δ*ω*_{R}) may occur if the plasma has an angular velocity (Ω ≠ 0) relative to the radar and the radar beam carries angular momentum (*j* ≠ 0).

[29] Figure 3illustrates schematically the special case of a rotating plasma flow with the rotational axis coincident with the radar beam axis. The dashed circles indicate the FWHM of a ring-shaped Laguerre–Gaussian beam.Figure 4 shows two ion line spectra computed for the rotating plasma flow in Figure 3. As in Figure 2, the dashed spectrum is for a single transmitted radar wave vector. Since in this case the electromagnetic field does not carry angular momentum the frequency spectrum is not shifted by the rotating plasma. Further, since the flow velocity is everywhere perpendicular to the scattering wave vector the Doppler shift is zero. The solid-line spectrum is for a transmitted twisted Laguerre–Gaussian beam and exhibits a rotational frequency shift of Δ*f*_{R} = Δ*ω*_{R}/2*π* = 0.1 kHz. As seen from equation (6), Δ*ω*_{R} is determined by both Ω and *j* = *s* + *l*. For example, such a shift can be obtained for a twisted beam with *l* = 10, circular polarization with *s* = 1 and a plasma rotation of Ω = 57 s^{−1}. This vortical plasma motion corresponds to an azimuthal speed of *v*_{ϕ} = 5.7 km/s at *ρ* = 0.1 km and *v*_{ϕ} = 57 km/s at *ρ* = 1.0 km. Increasing the angular momentum of the beam (larger |*l*|) increases the rotational frequency shift; alternatively, smaller angular velocities in the plasma can be detected for a given frequency resolution. We emphasize that the rotational frequency shift in Figure 4 is a result of the interaction of the transmitted beam carrying OAM and the plasma flow. OAM need not be detected in the received scattered radiation.

[30] It should be noted that strictly speaking even a linearly polarized Gaussian radar beam will result in a slightly broadened frequency spectrum since the beam consists of equal numbers of photons with opposite spin *s* = ±1, that is, the beam can be considered a superposition of two beams with opposite circular polarization. And according to equation (6) the spectrum will be upshifted for *s* = 1 and downshifted for *s* = −1, even though *l* = 0. For Ω = 57 s^{−1}, we obtain Δ*f*_{R} = ±9 Hz, which was neglected for the dashed spectrum in Figure 4.

[31] We also consider a more moderate plasma rotation with Ω = 2 s^{−1} for which the azimuthal plasma speed at *ρ* = 0.5 km is *v*_{ϕ} = 1 km/s. According to equation (6), for the lowest nonzero beam angular momentum (*l* = 0 and *s* = 1, for example, a circularly polarized Gaussian beam) we obtain a minute rotational frequency shift of the ion line of Δ*f*_{R} = 0.3 Hz.

[32] It may be mentioned that whereas in the present treatment we only consider large scale plasma flows of the order of the radar beam diameter or larger it may be speculated that small scale vortices exist which many together filling the radar beam cross section could result in larger frequency shifts than what is discussed here. Smaller vortices may rotate faster and therefore give larger rotational frequency shifts when scattering a radar beam.

[33] Twisted beams can be used to detect sheared plasma flows transverse to the beam direction, such as associated with auroral arcs [*Haerendel et al.*, 1996]. Auroral arcs have been seen to be associated with sheared flows sometimes as large as 8 km/s [*Lanchester et al.*, 1998]. Optical measurements [*Maggs and Davis*, 1968; *Borovsky et al.*, 1991] suggest that the spatial scales of such flows could become as small as 0.1 km, the width of narrow arcs. The beam width of a typical incoherent scatter radar is a few kilometers in the F region. Thus, sheared velocity fields could well be present within such radar beams.

[34] We consider an arc in the *xz*-plane that is just passing the radar beam axis which we take to be parallel to the geomagnetic field and in the*z*-direction, the beam axis being in the plane of the shear. The flow speed in the*x*-direction is*v*_{x} for *y* > 0 and −*v*_{x} for *y* < 0, as schematically illustrated in Figure 5. The dashed circles indicate the FWHM of a ring-shaped Laguerre–Gaussian beam. To obtain a rotational frequency shift of Δ*f*_{R} = 0.1 kHz, a OAM beam mode with *l* = 10 (*s* = 1) and beam width of 1.0° requires a sheared drift speed with *v*_{x}≈ 22 km/s. It may be noted that with a Gaussian beam a sheared flow cannot be discriminated from a unidirectional flow transverse the radar line-of-sight (Figure 2).

[35] As seen from equation (3), a more narrow twisted beam is associated with larger effective wave vector components in the azimuthal direction (*k*_{x,y} ∝ 1/*ρ*). Thus, a more narrow beam shows a larger frequency shift for the same drift speed. For a given frequency resolution, more narrow beams can therefore be used to detect smaller speeds. This is in contrast to the case of a rotational plasma flow aligned with the radar beam axis (Figure 4) for which there is no dependence on the radar beam opening angle.

[36] For a rotational flow with the axis of rotation aligned with the radar beam axis the flow velocity is everywhere transverse to the radar wave vector, so that the Doppler shift is zero. However, this is not the case when the plasma flow is sheared (Figure 5). In fact, for typical ionospheric parameter values, the transverse wave vector components due to a nonzero beam opening angle give a broadening of the scattered spectrum that exceeds the rotational frequency shift, in addition to a significant change in the spectral shape. Figure 6 displays the effect of the beam opening angle on the ion line spectrum (solid line) for a transmitted twisted beam scattering off the sheared plasma flow described above. The twisted beam has *θ* = 1.0°, *l* = 10 and *s* = 1. For comparison, when effects of the beam opening angle are neglected the spectrum is only slightly distorted and shifted by the rotational frequency shift of 0.1 kHz (dashed spectrum), as a result of the azimuthal wave vector components in equation (3)associated with the angular momentum content of the transmitted beam interacting with the plasma flow. A Gaussian beam would give rise to a similar ion line as the solid-line spectrum inFigure 6, but without the effects in the dashed spectrum.

[37] Langmuir and ion acoustic wave packets may themselves occur as electrostatic beams that carry OAM [*Mendonça et al.*, 2009a, 2009b]. The waves in such beams are grouped so as to exhibit the azimuthal phase variation characteristic of OAM content discussed above. For electrostatic twisted beams the OAM constitutes the total angular momentum, in contrast to the electromagnetic case for which the angular momentum has contributions from the polarization too. It has been predicted that a Langmuir (plasmon) vortex or ion acoustic (phonon) vortex can be excited and its OAM controlled by two oppositely propagating twisted electromagnetic beams in stimulated Raman or Brillouin scattering [*Mendonça et al.*, 2009b].

[38] Twisted electrostatic beams in the plasma can be detected by twisted radar beams. We expect that naturally excited Langmuir or ion acoustic vortices would be composed of a distribution of OAM modes with different *l*. Plasma beams with a certain mode number *l* = *l*_{p} could be selected by a suitable combination of transmitted and received radar OAM modes. With twisted beams there is in addition to the usual matching conditions on the frequencies and wave vectors, a condition also for the OAM mode number, as discussed by *Mendonça et al.* [2009b] for the case of nonlinear stimulation of twisted electrostatic modes:

where *l*_{t}, *l*_{r}, and *l*_{p} are the OAM mode numbers for the transmitted, received and plasma beam, respectively. Particularly, if *l*_{r} ≠ *l*_{t} the coherent backscatter has occurred from a plasma vortex with *l*_{p} ≠ 0.

[39] Whereas the rotational frequency shifts expected for radar scatter from the considered large scale plasma flows are small, as seen above, the coherent scatter from twisted ion acoustic and Langmuir beams occurs at the usual ion and plasma line frequencies, however, only at the appropriate combination of transmitted and received twisted radar beam modes, *l*_{t} and *l*_{r}.