The EISCAT Heating experiments were performed during daytime in November 2014 and October 2017. The Heating facility transmitted a left-handed circularly polarized wave (LHCP, often referred to as O mode) in a beam directed towards magnetic zenith ($\sim \mathrm{78}{}^{\circ }$ elevation south) and cycling 150 s on–85 s off. The Heating beam width at −3 dB was $\sim \mathrm{6}{}^{\circ }$. The term “left-handed” is defined with reference to the geomagnetic field direction: the electric field rotates in the opposite sense to the gyromotion of electrons.

Plasma parameter values in the F region were obtained with the EISCAT UHF incoherent scatter radar. The radar measurements utilized the Beata modulation scheme which includes a 32 bit binary alternating code with a baud length of 20 µs. The UHF radar beam was scanned in steps of 1.0^∘ in the experiment in November 2014 and in steps of 1.5^∘ in October 2017, between eight elevations around magnetic zenith in the plane containing the vertical and with a duration of 5 s in each position. The radar beam width was $\sim \mathrm{0.5}{}^{\circ }$. The pump cycle of 150 s on–85 s off enabled appropriate coverage of the radar measurements throughout the pump-on time, so that after several pump cycles under stable ionospheric conditions the temporal evolution during the pumping could be obtained at all elevations. The radar data analysis provided 5 s temporal resolution and 15–20 km range resolution, depending on the range.

Figure 1 displays measured height profiles for the electron concentration ($\stackrel{\mathrm{̃}}{N_{\mathrm{e}}}$), electron temperature ($\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$) and ion temperature ($\stackrel{\mathrm{̃}}{T_{\mathrm{i}}}$) for the experiment on 25 November 2014 (the tilde denotes measured parameters as opposed to modelled ones). For this case the pump frequency is f₀=6.30 MHz, which is approximately half way between the fourth and fifth electron gyroharmonics in the F region. The transmitted power was 818 kW. For some unknown technical reason, Heating did not transmit a circularly polarized wave during this experiment: the effective radiated power (ERP) was 242 MW in LHCP and 157 MW in right-handed circular polarization (RHCP), assuming a perfectly conducting ground. However, electron heating effects from pumping with LHCP dominate over those with RHCP. Bryers et al. (2013) estimated the height-integrated heating source for O-mode pumping to be approximately a factor of 3 larger than for X-mode pumping, for a pump duty cycle of 50 % and the O-mode pump frequency not near an electron gyroharmonic, with the X-mode frequency near a gyroharmonic (their Fig. 5), however. In addition, the ERP in our experiments for LHCP was larger than for RHCP. We therefore consider the measured heating effects to be representative of pure LHCP pumping.

The $\stackrel{\mathrm{̃}}{N_{\mathrm{e}}}$ profile in Fig. 1a is stable throughout the displayed time interval and does not show modulations due to the pumping. However, the $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ profile in Fig. 1b exhibits clear pump-induced modulations. The HF pumping is marked by white boxes and the red zigzag line indicates the radar elevation scan. The $\stackrel{\mathrm{̃}}{T_{\mathrm{i}}}$ shown in Fig. 1c only exhibits weak pump-induced modulations. The ionospheric conditions and response to the HF pumping in the experiments on 24 October 2017 were similar to those shown in Fig. 1. However, whereas for 2014 the ionospheric critical frequency foF2 was near 8 MHz, well above f₀, foF2 was near f₀ in 2017.

Figure 1Height profiles as a function of time of $\stackrel{\mathrm{̃}}{N_{\mathrm{e}}}$ (a), $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ (b) and $\stackrel{\mathrm{̃}}{T_{\mathrm{i}}}$ (c) during pump cycling on 25 November 2014. The white boxes in (b) show pump-on and the red zigzag line indicates the elevation of the UHF radar which was scanned between 75.2 and 82.2^∘ in 1.0^∘ steps.

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The used pump cycle in combination with the radar scan cycle enabled measurement of the temporal evolution of the ionospheric parameters at all radar elevation angles throughout the pumping. Figure 2 shows the temporal evolution of height profiles of $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ for the different elevation angles of the UHF radar, starting at t=5 s after pump-on for the experiments on 25 November 2014. Such measurements require reasonably stable ionospheric conditions during several pump cycles (see Fig. 1), as the radar, scanning eight elevations between 75.2 and 82.2^∘, samples the interaction region at a given elevation at different times after pump-on in different pump pulses. With measurements at sufficiently many pump pulses the temporal evolution can then be traced throughout the duration of pump-on (t=0–150 s) at all elevations.

As seen in Fig. 2, $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ enhancements occurred already within the first few seconds of pump-on. The high $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ around 300 km in the first 5 s data dump is likely not real but due to HF enhanced ion acoustic lines on the topside ionosphere. During the following few tens of seconds $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ was further enhanced at all elevations and in a wider altitude range. Notice also the slow conduction of electron heat toward increasing altitudes with time, up to 300–400 km altitude, as can be seen in Fig. 1 too. The strongest $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ enhancements occurred at the elevations 77.2 to 79.2^∘, around magnetic zenith ($\sim \mathrm{78}{}^{\circ }$). This is also where the $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ enhancements extended toward the highest altitudes. Differences in the enhanced $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ profiles can be discerned even though the radar elevation changes by only 1.0^∘.

Figure 2Height profiles of $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ (colour coded) versus time for the different elevation angles of the UHF radar between 75.2 and 82.2^∘ on 25 November 2014 (11:03:45–13:00:00 UT). Pump-on was from t=0 to t=150 s.

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Figure 3 displays $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ height profiles versus time for the experiment on 24 October 2017. As for Fig. 2, the high $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ around 300 km in the first 5 s data dump is likely not real. In this experiment f₀=6.2 MHz, which again is approximately half way between the fourth and fifth electron gyroharmonics in the F region. The transmitted power was 734 kW and the ERP was 471 MW (LHCP). The radar was scanned in steps of 1.5^∘ from 74.56 to 85.06^∘, which is a larger range of elevations than that covered by the 1.0^∘ steps in Fig. 2. $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ enhancements due to the HF pumping again occurred already in the first 5 s radar data integration after pump-on and $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ was the highest at elevations 77.56 and 79.06^∘, closest to magnetic zenith ($\sim \mathrm{78}{}^{\circ }$). The gaps in the plots are because the ionospheric conditions were not stable long enough to give sufficient data to obtain the full temporal evolution at all elevations. However, the results for $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ are similar to those in Fig. 2 for the experiments on 25 November 2014.

Figure 3Height profiles of $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ (colour coded) versus time for the different elevation angles of the UHF radar between 74.56 and 85.06^∘ on 24 October 2017 (12:00:00–12:43:00 UT). Pump-on was from t=0 to t=150 s.

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To get information on the source that underlies the observed electron temperature enhancements, we model the electron heating rate through the fluid equations (Shoucri et al., 1984). As the measurements of the UHF incoherent scatter indicate no major pump-induced effects in $\stackrel{\mathrm{̃}}{N_{\mathrm{e}}}$ and $\stackrel{\mathrm{̃}}{T_{\mathrm{i}}}$ (Fig. 1), the fluid equations can be reduced to the electron energy equation:

where T_e(z,t) is the modelled electron temperature, $\widehat{\mathit{z}}$ is the unit vector in the direction of the geomagnetic field, k_B is Boltzmann's constant, ${\mathit{\kappa }}_{\mathrm{e}}(T_{\mathrm{e}},z,t)$ is the electron heat conductivity, Q_HF is the HF pump wave energy deposition to the electrons, Q_e(z,t) is the background electron heating rate (mainly from photoelectrons), and $L_{\mathrm{e}}(T_{\mathrm{e}},z,t)$ is the electron cooling rate due to elastic and inelastic collisions with ions and neutrals.

With negligible plasma drift along the geomagnetic field as measured with the UHF radar, the convective terms in Eq. (1) can be neglected, giving (Löfås et al., 2009; Gustavsson et al., 2010)

The heating rate of the electrons due to the electromagnetic pump wave consists of two parts:

where Q_Ω is the ohmic heating due to collisional damping of the pump wave and Q_AA is the heating due to the anomalous absorption of the wave associated with the excitation of, for example, upper hybrid turbulence and associated small-scale density striations. The ohmic heating rate is the time-averaged product of the pump electric field E₀ and induced electric current σ_ijE₀, where σ_ij is the conductivity tensor: $Q_{\mathrm{\Omega }}=(\mathrm{1}/\mathrm{2})Re[{\mathit{E}}_{\mathrm{0}}^{*}\cdot ({\mathit{\sigma }}_{ij}{\mathit{E}}_{\mathrm{0}}\left)\right]$ (Gustavsson et al., 2010). At the relatively high ERP levels used in the experiments, Q_AA gives the dominating contribution to Q_HF and may be several times larger than Q_Ω (Bryers et al., 2013).

In the present treatment we obtain a model T_e(z,t) of the observed $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}(z,t)$ by integrating the electron energy Eq. (2). The electron heating rate Q_HF(z,t) due to the HF pumping is modelled by a one-dimensional and asymmetric Gaussian along the geomagnetic field. Q_HF(z,t) has its maximum Q_m at range z₀ and has independent upper (σ_u) and lower (σ_l) half-widths (Senior et al., 2012; Bryers et al., 2013):

where $t_{\mathrm{on}}\le t\le t_{\mathrm{off}}$ is the time during which HF pumping occurs. This leads to a parameter estimation problem in the model parameters Q_m, z₀, σ_l, σ_u, and τ that we solved by weighted nonlinear least squares:

where $T_{\mathrm{e}}(z,t,{\mathrm{par}}_{\mathrm{HF}})$ is obtained by integrating Eq. (2) with $Q_{\mathrm{HF}}(z,t,{\mathrm{par}}_{\mathrm{HF}})$ and ${\mathit{\sigma }}_{\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}}$ is the standard deviation of the observed electron temperature.

When integrating Eq. (2) we used the observed range profiles for $\stackrel{\mathrm{̃}}{T_{\mathrm{i}}}$ and $\stackrel{\mathrm{̃}}{N_{\mathrm{e}}}$ as they evolve in time at each elevation. For example, L_e depends on both T_i and N_e and both the left-hand side of Eq. (2) and κ_e depend on N_e. As the initial condition we took a smoothed $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ range profile measured just before pump-on. Further, we used mixed boundary conditions, taking at the lower boundary $T_{\mathrm{e}}=\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}(z=\mathrm{150}\mathrm{km},t)$ as given by the UHF radar measurements at z=150 km slightly before pump-on at t=t_on and at the upper boundary $\partial T_{\mathrm{e}}/\partial z(z=\mathrm{500}\mathrm{km},t)=\mathrm{0}$. The fixed temperature at the lower boundary follows from the observations with the additional theoretical justification that at such low altitudes T_e and T_i are both approximately equal to the neutral temperature due to the high collision frequencies. The upper boundary condition too is based on the observations and corresponds to a balance between upward heat flux out from the ionosphere and downward heat flux from the magnetosphere into the ionosphere.

The temporal evolution of the modelled T_e altitude profile for the elevation angles scanned by the radar is obtained by integrating Eq. (2) with the optimal parameters for Q_HF(z,t). The results are shown in Figs. 4 and 5, which correspond to the measurements in Figs. 2 and 3, respectively. T_e(z,t) is enhanced for all elevations already within the first seconds after pump-on at t=0 s. Slow conduction of the electron heat is seen both upward and downward in altitude and T_e(z,t) reaches the highest values near magnetic zenith ($\sim \mathrm{78}{}^{\circ }$). The modelling results in Figs. 4 and 5 agree qualitatively with the measurements in Figs. 2 and 3, respectively.

Figure 4Modelled temporal evolution of the altitude profile of the electron temperature T_e(z,t) for the different elevation angles in the experiments on 25 November 2014 (Fig. 2).

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Figure 5Modelled temporal evolution of the altitude profile of the electron temperature T_e(z,t) for the different elevation angles in the experiments on 24 October 2017 (Fig. 3).

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Figure 6 displays the corresponding Q_HF versus elevation angle for the experiment on 25 November 2014 (Fig. 2) in panels (a) and (b) and for 24 October 2017 (Fig. 3) in panels (c) and (d). Figure 6a and c show the column-integrated Q_HF (blue) as well as the profile of the transmitted Heating beam (red) and Fig. 6b and d depict the altitude profiles of Q_HF. These modelling results are for the case after that steady state was reached in the 150 s pump-on period. The white and black lines in Fig. 6b and d show the altitudes of the plasma and upper hybrid resonances, respectively, as obtained from the ion and plasma lines. The altitude separation between the two resonances is larger in Fig. 6d for which f₀ was near foF2 than in Fig. 6b for which f₀ was well below foF2.

Figure 6Modelled electron heating rate Q_HF (eV m⁻³ s⁻¹) during steady state versus radar elevation angle for 25 November 2014 (a, b) and 24 October 2017 (c, d). Panels a and c display the column-integrated Q_HF (blue dots, with connecting lines to guide the eye) and the relative intensity of the transmitted Heating beam (red) assumed to propagate in vacuum. The elevation corresponding to magnetic zenith is indicated by the dashed line and labelled MZ. Panels b and d show the altitude profiles of Q_HF. The white line indicates the altitude of the plasma resonance where f_p=f₀ and the black line shows the upper hybrid resonance height at which the upper hybrid frequency equals f₀. Note that the elevation scale is different in panels a and b and c and d.

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The column-integrated Q_HF in Fig. 6a is maximum at 78.2^∘ and in Fig. 6c at 77.5^∘, which are the same elevations at which the observed electron temperature reached the highest values (Figs. 2 and 3, respectively). Also, the column-integrated Q_HF is maximum at the elevation closest to magnetic zenith ($\sim \mathrm{78}{}^{\circ }$, labelled by MZ in the plots).

Further, as seen in Fig. 6a and c, the angular extent of the Q_HF profile is smaller than that of the Heating beam. Q_HF follows the profile of the Heating beam at elevations lower than magnetic zenith ($\sim \mathrm{78}{}^{\circ }$), while at higher elevations it is more confined to magnetic zenith than the Heating beam. For reference, the Spitze angle is about 6^∘ from the vertical or at about 84^∘ elevation.

As seen in Fig. 6b and d, the altitude profile of Q_HF is generally asymmetric, with Q_HF decreasing steeply with increasing altitude above the maximum and declining more gradually with decreasing altitude below the maximum. Only in Fig. 6d for 77.5^∘ (at magnetic zenith) does Q_HF decrease more slowly toward high altitudes than toward lower altitudes. Also, Q_HF decreases steeply for increasing elevations beyond about 80^∘, towards the vertical. This decrease in Q_HF with increasing elevation is steeper than what would be expected from the point view of the width of the Heating beam in vacuum (see Fig. 6a and c).

It is notable that Q_HF reaches larger values at the two elevations next to magnetic zenith compared to at the elevation nearest to magnetic zenith. In Fig. 6b, Q_HF is slightly higher at 77.2 and 79.2^∘ than at 78.2^∘, while the Q_HF profile is more extended in altitude at 78.2^∘. Figure 6d shows larger differences, with Q_HF higher at 76.0 and 79.0^∘ than at 77.5^∘, while the Q_HF profile is more extended in altitude at 77.5^∘. Thus, despite the maximum of the Q_HF profile being slightly lower at magnetic zenith compared to at the two nearest neighbouring elevations, the column-integrated Q_HF is maximum at magnetic zenith (Fig. 6a and c) for both experiments.

We have presented experimental and modelling results concerning electron heating and the ionospheric plasma response to HF pumping near magnetic zenith. The experiments were performed with the EISCAT Heating facility and measurements of the plasma response were done with the EISCAT UHF incoherent scatter radar. The Heating beam was tilted in the magnetic zenith direction and the UHF radar was scanned between eight positions around this direction to study the electron-heating efficiency. The electron heating rate Q_HF(z,t) and associated electron temperature T_e(z,t) due to the HF pumping were modelled by integrating the energy Eq. (2) and fitting the model parameters with respect to the measurements of $\stackrel{\mathrm{̃}}{N_{\mathrm{e}}}$, $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}$ and $\stackrel{\mathrm{̃}}{T_{\mathrm{i}}}$.

Differences in the plasma response were observed for radar elevations differing by only 1^∘ (Fig. 2). The pump-induced measured $\stackrel{\mathrm{̃}}{T_{\mathrm{e}}}(z,t)$ enhancements (Figs. 2 and 3), the modelled T_e(z,t) (Figs. 4 and 5) and the associated column-integrated Q_HF (Fig. 6) were all found to maximize in the magnetic zenith direction ($\sim \mathrm{78}{}^{\circ }$ elevation). Further, the angular width of the Q_HF profile, with a full width at half maximum (FWHM) of about 4^∘ around magnetic zenith, was less than that of the HF beam, which suggests that some focusing of the Heating beam occurred.

Pedersen et al. (2008) obtained the angular distribution of the optical emission production efficiency by HF beam-swinging experiments at HAARP. The optical emission production efficiency peaked at magnetic zenith with a FWHM of 7^∘, for which the HAARP beam width and many other experiment variables were accounted. This FWHM of 7^∘ is larger than the two cases for Q_HF in Fig. 6. The HAARP experiments used an ERP of 32.1 MW at 2.83 MHz and 42.4 MW at 3.3 MHz, thus, both lower ERP and lower f₀ than in the present EISCAT experiments. It is plausible that self-focusing effects were larger at the higher ERP in the present experiments, which could give a narrower region of pump-induced enhancements.

It has been proposed that filamentary plasma density ducts can guide a transmitted LHCP wave, entering the ionosphere in the O mode, as an L-mode wave along the geomagnetic field (Leyser and Nordblad, 2009; Nordblad and Leyser, 2010). The L mode is an LHCP electromagnetic wave mode with the wave vector parallel or anti-parallel to the ambient magnetic field. For a homogeneous and cold magnetized plasma the refractive index (n_∥) parallel to the ambient magnetic field is given by $n_{\parallel }^{\mathrm{2}}=\mathrm{1}-f_{\mathrm{p}}^{\mathrm{2}}/f(f+f_{\mathrm{e}})$, where f_p is the electron plasma frequency and f_e is the electron gyrofrequency. Whereas the O mode has a cutoff at f_p=f₀, the L mode has the cutoff frequency $f_{\mathrm{L}}=-f_{\mathrm{e}}/\mathrm{2}+(f_{\mathrm{p}}^{\mathrm{2}}+f_{\mathrm{e}}^{\mathrm{2}}/\mathrm{4}{)}^{\mathrm{1}/\mathrm{2}}$, which corresponds to $f_{\mathrm{p}}\approx f_{\mathrm{0}}+f_{\mathrm{e}}/\mathrm{2}$ for $f_{\mathrm{p}}^{\mathrm{2}}\gg f_{\mathrm{e}}^{\mathrm{2}}$ for a pump wave at frequency f=f₀. Thus, an electromagnetic wave in the L mode can propagate at higher plasma densities than in the O mode.

L-mode propagation can occur when the background plasma density gradient near the plasma resonance is parallel to the geomagnetic field, instead of the density gradient for example being vertical as in a horizontally stratified ionosphere. Such a condition with the density gradient being magnetic field-aligned can occur in density ducts, either natural or pump-induced. In the L mode the pump wave can propagate upwards, passing through the plasma resonance on its way to the cutoff at $f_{\mathrm{p}}\approx f_{\mathrm{0}}+f_{\mathrm{e}}/\mathrm{2}$ if the plasma is sufficiently dense. With its perpendicular electric field, strong pumping of upper hybrid phenomena localized in small-scale density striations and related anomalous electron heating can occur at higher altitudes and deeper into the plasma compared to the case of an O-mode wave which therefore could contribute to the strong plasma response observed in magnetic zenith (Leyser and Nordblad, 2009; Nordblad and Leyser, 2010).

The HF pump-induced electron heating rate Q_HF obtained for our experiments exhibited an interesting dependence on the elevation angle near magnetic zenith (Fig. 6). Q_HF is maximum at the elevations next to magnetic zenith. At magnetic zenith, the Q_HF profile is more extended in altitude, such that the column-integrated Q_HF is maximum in this direction. These results are consistent with the pump wave propagating in the L mode in magnetic zenith and in the O mode at angles deviating from the zenith direction. As a wave in the L mode propagates to higher altitudes than in the O mode, electron heating can occur in a more extended altitude range for L-mode propagation, thereby giving maximum column-integrated electron heating in magnetic zenith. The large difference in the Q_HF profile between magnetic zenith and the adjacent elevations is consistent with that the pump wave in the O mode has a much lower reflection height than in the L mode; the O-mode reflection height is well below the plasma resonance for elevations near magnetic zenith. In magnetic zenith the pump wave is guided by magnetic field-aligned density ducts in the L mode, but at the adjacent elevations the pump wave makes too large an angle to the magnetic field for trapping of the HF wave in the duct, and thus guiding, to occur so that instead the pump wave propagates in the O mode.

Evidence of L-mode propagation of the EISCAT Heating beam has previously been obtained as transionospheric propagation for $f_{\mathrm{0}}<foF\mathrm{2}<f_{\mathrm{0}}+f_{\mathrm{e}}/\mathrm{2}$, in which case an L-mode wave would not be reflected but pass through the ionospheric plasma density peak. This was observed by direct measurement on the CASSIOPE spacecraft (Leyser et al., 2018) and indirectly by EISCAT UHF radar observations of ion acoustic lines in the topside ionosphere (Rexer et al., 2018).

Figure 6 also displays the altitude of the plasma resonance (white lines in Fig. 6b and d). The position of the Q_HF profile relative to the plasma resonance is not fully understood. In Fig. 6b, Q_HF is maximum slightly above the plasma resonance at magnetic zenith. This is consistent with a pump wave in the L mode being able to propagate well above the plasma resonance, whereas an O-mode wave cannot. Further, an L-mode wave has its electric field perpendicular to the geomagnetic field all the way up to its reflection height, so that pumping of upper hybrid turbulence can occur in an extended altitude range. In the O mode, on the other hand, the electric field turns to parallel to the geomagnetic field close to the reflection height, which favours excitation of Langmuir turbulence that generally causes less electron heating than upper hybrid turbulence.

However, in Fig. 6d all electron heating appears to occur well below even the upper hybrid resonance height (black line). In this case foF2 was near f₀, whereas for Fig. 6b it was well above f₀, which is consistent with that the altitude separation between the plasma and upper hybrid resonances is larger in Fig. 6d. We do not have any explanation for why electron heating seemed to occur at such low altitudes in this case.

Gurevich et al. (2002) developed a theory for self-focusing of the electromagnetic pump wave propagating in the O mode on geomagnetic field-aligned density striations. An important mechanism in the nonlinear pump beam self-focusing is the trapping of pump rays near the magnetic zenith direction in the large-scale density depletions within the beam, as previously was found in numerical studies (Gurevich et al., 1999). The results were shown to be consistent with observations of pump-induced optical emissions at HAARP (Pedersen et al., 2003). However, the possibility of propagation of the pump wave in the L mode, deeper into the plasma than what is possible in the O mode, was not considered. Whereas the nonlinear self-focusing of the pump beam is an important mechanism, particularly for guiding the pump beam in magnetic zenith, it does as it stands not seem to account for the difference that we have found in the altitude distribution of the electron heating rate in magnetic zenith compared to that just about 1^∘ away from this direction (Fig. 6). We therefore suggest that such theories for self-focusing are developed to include the possibility of L-mode propagation.

Access to the raw data may be provided upon reasonable request to the authors.

BG, TBL and MTR performed the experiments. BG developed the theoretical model and performed the numerical analysis. TR contributed to the data analysis and data presentation. TBL carried out most of the interpretation and prepared the paper. All the co-authors helped in the interpretation of the results, read the paper and commented on it.

The authors declare that they have no conflict of interest.

This paper was edited by Nick Sergis and reviewed by two anonymous referees.