We present the results of numerical simulations of artificial ducts during high-power HF heating performed by a novel model accounting for the effect of self-action. This effect interferes with the HF-plasma matching in the heated region and hence with electron heating. The model satisfactorily explains recent experimental observations. It helps for choosing the heating parameters optimal for duct formation, such as proper duration of the heating pulse and its frequency. It also suggests that distortion of the ducts caused by the self-action effect can be avoided by down-chirping the heating frequency. The down-chirping rates needed to suppress such distortions are evaluated.
Milikh et al.'s [2010b] numerical model, a modification of the SAMI2 code [Huba et al., 2000], reproduces the ducts' magnitude and propagation velocity consistent with the existing observations. However, the current modification of the model does not account for the effect of self-action. Indeed, the efficiency and location of the heating source are defined by theF2-peak plasma densityne. The latter changes significantly in the course of high-power heating, thereby affecting HF pump wave - ionosphere interaction and hence the electron heating rate. Therefore, the current model lacks self-consistency and may be used only for semi-qualitative estimates.
 This paper presents the results of our efforts to expand on the Milikh et al. [2010b]model by including the self-action effect. A new model explains some experimental observations so far remained poorly understood. It also helps for choosing optimal heating parameters for duct formation such as proper duration of the heating pulse and its frequency.
2. Model Description
 SAMI2 is an inter-hemispheric, Eulerian grid-based 2D model, which simulates the plasma along the entire dipole magnetic line. The original code did not include an electron heating source by the HF-pump radio wave.Milikh et al. [2010b] applied a simple “hot brick” model of a Gaussian F2-region source with vertical and horizontal extent. Within the irradiated spot centered atz0 and r0 = (x0, y0), the heating rate per electron is
Here P is the transmitter radiated power, μ is the pump wave absorption coefficient (0 < μ < 1), Δr = (Δx, Δy), Aeff ≈ ΔxΔy is the horizontal cross section of the HF beam at half maximum, Δx, y ≈ z0 tan(θx, y) and θx(θy) is the beam angular half-width in the magnetic N-S (E-W) direction. This approximation uses the fact that ionospheric electrons are heated mainly via ohmic heating by the pump-excited plasma waves, rather than purely by the pump electric field. This occurs near altitudesz0 = zp or zuhwhere the O-mode pump frequencyfO matches either the plasma frequency fp or upper hybrid resonance (UH) , respectively [e.g., Fejer, 1979]. Here fc is the electron gyrofrequency and kHz.
 For fO far from multiples of fc, i.e., ∣fO/fc − s∣ ≫ 10−2(s = 1, 2, 3 …), Rietveld et al.  found that the electron temperature Temaximizes when the radio beam points toward the magnetic zenith (MZ). For MZ-injections at HAARP, pump waves offO > 4 fc ≈ 5.5 MHz reflect below the UHR altitude, thereby preventing the UH matching [e.g., Mishin et al., 2004]. Otherwise, the heating is associated with so-called anomalous absorption near the UH altitudezuh [e.g., Gurevich et al., 1996; Gustavsson et al., 2005]. As of yet the duct formation experiments have used O-mode pumping withfO≤ 5 MHz, slightly below the F2-peak critical frequencyfOF2. Therefore, hereafter we focus solely on the anomalous absorption mechanism caused by the HF-exciteduhwaves coupled with field-aligned density irregularities [e.g.,Gurevich et al., 1996]. This comes with a caveat that the effects of accelerated suprathermal electrons [e.g., Gustavsson et al., 2005] are neglected.
 A new model uses two postulates. First, the source is centered at z0 = min (zuh(t)) defined at each step from the matching condition fuh(z, t) = fO with the density profile changing with time. Second, the saturated plasma wave spectrum is taken unaffected by a slow, the timescale of ∼1 min modification of the density profile. This allows us to hold the spatial rate of absorption of the incident pump power (ka) during the heating. As a result, for each time step the HF energy flux Sin the absorption region can be described by a quasi-stationary equation
where is the incident HF flux at the lower boundary zmin. The HF power deposition and HF heating rate per electron at z > zmin are defined by the loss of the HF flux per unit length
Initially the anomalous absorption is considered to occur between zuh and the reflection height zp. When the critical frequency fOF2 reduces, making the upper hybrid resonance impossible, the anomalous absorption is assumed to vanish entirely.
 The absorption rate kais defined by the conversion of the pump wave to uh waves coupled with field-aligned density irregularities. As yet, the conventional models [e.g.,Gurevich et al., 1996] give only semi-qualitative estimates for the rateka(z) and its characteristic gradient scale Δz defined by the vertical extent of the absorption region about zuh(t). However, our simulations show that so long as Δz−1 ≫ d ln(Te)/dz, the actual value of Δz is quite insignificant. Nor does a particular shape of QHF(z) play any role, as the fast heat transport will smooth out any sharp Te gradients well before the density profile starts changing. Only the total deposited energy matters. The corollary to above is that we can safely use a Gaussian profile for QHF(z) centered at z0(t), which is adjusted at each time step. Furthermore, as d ln(Te)/dz ∼ 0.01–0.02 km−1, we assume Δz ∼ 10 km, thereby avoiding spurious numerical effects due to the model grid step of 1–2 km.
 As a result, the vertical profile of the volume heating rate is taken as
3. Simulation Results and Discussion
 To model HF-induced ducts, we specified the effective radiated power and beam pattern of the HAARP heater. Ionograms taken just before the HF-heating by the HAARP digisonde provide the ambientF2 region density profile. The input parameters of the model are chosen to reproduce the ambient F2 region prior to the heating. Similar to the HF- heating experiments we chose the heating frequency to be slightly below the foF2. This is done to avoid the underdense heating if foF2 drops below the heating frequency due to variations with time.Figure 1 shows the calculations made by using the modified heating model. Figure 1adescribes the evolution of the heating source, which gradually moves upward to the peak of the F2 layer due to the plasma transport caused by the HF-heating. Trace 1 corresponds to preheating, while traces 2. 3. 4, 5, and 6 correspond to 1.5, 6, 10.5, 10.75, and 11 minutes into the heating, respectively.Figure 1b shows that the electron density peak reduces and moves slowly upward. Its reduction rate is significantly lower than that in the unmodified case not shown here. Unlike the “hot brick” model, the density peak never drops below the ambient UHR level. Here trace 1 again corresponds to preheating, but traces 2. 3. 4, 5, correspond to 1.5, 6, 10.5, and 15 minutes into the heating, respectively. Note that the last few traces in Figure 1a have the time resolution better than those in Figure 1b. This is required since in the former the heating source starts abruptly dropping about 10 minutes into heating as the density approaches the UHR. Hereafter, all plots in Figure 1b follow the same trace times as above. Figure 2 shows latitudinal changes of the electron temperature as could be detected by a polar orbit satellite flying at 660 km. Figure 2ashows the first 15 minutes of HF-heating when the electron temperature increases as the constant “hot brick” source continuously pumps energy into the system. In the modified model the electron temperature also grows although not as fast as in the “hot brick” case (seeFigure 2b). After about 10 minutes into heating the plasma density reduces to the extent that the UH resonance is barely reachable, so the heating in the immediate vicinity of the MZ reduces significantly. However it is still strong on the periphery of the heated region near latitudes of about 60.7° and 62.1°. This leads to a temperature drop in the center of heated spot, thus forming a two-peak latitudinal profile. The electron density also experiences drop in the center of heated spot (Figure 2c).
 This pattern could be detected by a satellite overflying the heated region. Unfortunately a satellite flyover lasts less than a few tens of seconds, which is too short to detect changes in the electron density generated by the self-action of the heating source. Nevertheless this effect could explain earlier observed differences in the ducts' shape observed at HAARP by DMSP satellites (seeFigure 2d). A duct with a single peak was detected when the satellite passed the HAARP MZ at 6 min into heating, while the other duct with two peaks was detected at 12 min into heating.
 Note that double-peaked horizontal profiles ofTe(x) and ne(x) are obtained in our simulations if the initial position of the anomalous-absorption region is close to theF2 peak, so that the plasma-resonance and UHR layers can disappear as the electron density drops due to the HF-heating. Decreasing the heating wave frequency by about 5% removes this effect, and single-peak horizontal profiles are formed since the resonance regions persist during the heating interval (not shown).
 The self-action effect can be suppressed by slowly reducing the heating frequency (down-chirping), and thus keeping the source of anomalous heating intact (Figure 3). Figure 3ashows profiles of the self-consistent heating sourceqHF(z); Figure 3b shows latitudinal profiles of the temperature variation. Since the frequency of the heating wave fO is proportional to the square root of the electron density neat the reflection height one can find that the down-chirping frequency rate is
From the data presented in Figure 2we find that the average value of the down-chirping rate dfO/dt ≃ −30 kHz/min. Figure 3illustrates that by using down-chirping, with a slightly lower than the above rate (−16 kHz/min), the anomalous heating is kept intact. The lower chirping rate is sufficient since we only desire that the UHR region does not disappear due to the HF-heating.
 We have shown that the modification of the plasma density profile due to high-power HF heating change the absorption efficiency of the pump wave, thereby changing the HF heating source. This self-action effect distorts the duct's shape and eventually reduces the plasma density in the heated region below the upper hybrid height, thereby eliminating anomalous absorption as a source of electron heating. The corollary to that, confirmed by satellite observations, is that the strongest ducts are not caused by long duration of heating pulses.
 In fact, plasma transport from the heated region caused by long heating will reduce its thickness and subsequently cause a drop in the absorption efficiency of the heating radio waves. Thus there exists the optimal heating time for the duct formation. Namely, a few minutes heating pulses are more efficient than continuous heating. The optimal heating frequency for duct formation is slightly below fOF2. Finally, by down-chirping the heating frequency one can suppress distortion of the ducts caused by the self-action effect.
 The self-action effect could be detected by a satellite overflying the heated region. The latter lasts less than a few tens of seconds, which is too short to detect changes in the electron density inside ducts generated by the above effect. An interesting opportunity for in-situ studies of the temporal changes of ducts due to self-action could be opened by the Russian space mission called “Resonance” which is planned to be launched in 2014. The satellites will move along the magneto-synchronous orbit which allows them to stay in HAARP's magnetic flux tube for about an hour [Demekhov et al., 2003].
 GM and AV were supported by DARPA via a subcontract N684228 with BAE Systems and also by the ONR grants NAVY.N0017302C60 and MURI N000140710789. EM was supported by the Air Force Office of Scientific Research. The authors are grateful to Dennis Papadopoulos for helpful discussions.
 The Editor thanks Hannah Vickers and an anonymous reviewer for assisting with the evaluation of this paper.