The initial perturbation of polar mesospheric summer echoes PMSEs during radio wave heating provides significant diagnostic information about the charged dust layer associated with the irregularity source region. Comparison between the results of computational models and the observation data can be used as a tool to estimate charged dust layer parameters. An analytical model is developed and compared to a more accurate computational model as a reference to investigate the possibilities for diagnostic information as well as insight into the physical processes after heater turn-on. During radio wave heating of the mesosphere, which modifies the background electron temperature, various temporal evolution characteristics of irregularity amplitude may be observed which depend on the background plasma parameters and the characteristics of the dust layer. Turn-on overshoot due to the dominant electron charging process and turn-on undershoot resulting from the dominant ambipolar diffusion process, that can occur simultaneously at different radar frequencies, have been studied. The maximum and minimum of the electron density irregularity amplitude and the time at which this amplitude has been achieved as well as the decay time of irregularity amplitude after the maximum amplitude are unique observables that can shed light on the physical processes after the turn-on of the pump heating and to diagnose the charged dust layer. The agreement between the computational and analytical results are good and indicate the simplified analytical model may be used to provide considerable insight into the heating process and serve as the basis for a diagnostic model after heater turn-on. Moreover, the work proposes that conducting PMSE active experiments in the HF and VHF band simultaneously may allow estimation of the dust density altitude profile, dust charge state variation during pump heating, and ratio of electron temperature enhancement in the irregularity source region.
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 Noctilucent clouds from ground observations, or polar mesospheric clouds in the case of space observations, have first been recorded in 1885 [Backhouse, 1885; Balsley et al., 1983]. Particles responsible for NLCs usually grow sufficiently that they can be observed by lidars [Baumgarten et al., 2008], space born cameras [Russell et al., 2009], and rocket born photometers [Gumbel and Witt, 2001]. Analysis has shown that visible NLC particles typically have the characteristic sizes in the range 20–100 nm [Baumgarten et al., 2008]. Polar mesospheric summer echoes (PMSEs) are strong radar echoes produced by particle clouds which are formed at mesopause altitudes 83–88 km [Rapp and Lübken, 2004]. The electron irregularities are produced as a result of charging onto the irregularity structures in the subvisible particle density [Inhester et al., 1994]. These subvisible particles are located a few kilometers above the NLCs and smaller in size in comparison with those creating NLCs [Robertson et al., 2009]. The Polar Mesosphere Summer Echoes (PMSE), were first observed using the 50 MHz VHF radar at Poker Flat, Alaska [Ecklund and Balsley, 1981] but have subsequently been detected using a number of different radio sounding frequencies including 1.29 GHz, 933, 224, 53.5, 8, 7.6, 4.9, 3.3, 2.43 MHz [Cho and Kelley, 1993; Rottger, 1994; Cho and Rottger, 1997; Ramos et al., 2009; Rapp and Lübken, 2004]. PMSE particles are dusty ice particles and are a combination of dust and ice [Brattli et al., 2009], and special issue devoted to the measurements of ice particles) and for remainder of this paper will be referred to as simply dust particles.
 There has been an extensive effort to measure the dust particles in the mesosphere through space-based and ground-based measurements [Robertson et al., 2009; Chandran et al., 2010; Hervig et al., 2008; Baumgarten et al., 2008]. In recent years, ground based observation techniques have drawn considerable interest in the space science community although it may be lacking in spatial and time resolution. More direct investigations which resolve these issues are possible by sounding rocket. However, with these in-situ methods, the short time frame of the experiments becomes a problem as many of the most interesting mesospheric phenomenon occur over a longer time interval. PMSE heating experiments have possibilities as an alternative approach that can be implemented to diagnose the PMSE source region at much lower cost. The first modulation of PMSE with radio wave heating was reported by Chilson et al. , where it was shown that backscatter could be reduced in intensity as the heater is switched on, and thereafter return to its original strength when the heater was switched off. This behavior was also observed in other PMSE heating experiments and lead to a powerful tool to investigate the physical processes at mesopause altitudes [Belova et al., 2001, 2003; Havnes, 2004; Havnes et al., 2003, 2004].
 The suppression of the radar echoes during the heating with radio waves and enhancement of radar echoes after the pump turn-off was observed and reported by Havnes . It was shown by Rapp and Lübken  that electron temperature enhancements lead to a rapid change of the electron density structure within the dust particle layer. An extensive effort has been made in the past few years to develop theoretical and computational models to study the active modification of PMSE and parameters that may affect the characteristics of these echoes. A theoretical model was first developed by Havnes to predict observations of turn-off overshoot at 224 MHz and predict the basic shape of the so-called Overshoot Characteristic Curve OCC [Havnes, 2004; Havnes et al., 2004]. It was later shown in a more detailed study that diffusion process of ion irregularities may cause an enhancement of radar echoes after the turn-off of radio wave heating[Scales and Chen, 2008]. It has also been predicted [Chen and Scales, 2005] that the radar echoes are expected to be enhanced during the pump heating in lower frequency bands (e.g., 7.9 MHz) due to the enhanced electron charging onto the dust particles.
 Very recently, investigations of frequency effects on PMSE heating have become important. This is possible at heating facilities with the availability of multiple radar frequencies such as EISCAT in Tromso, Norway, which operate at 224 and 930 MHz as well as the new MORRO radar [La Hoz and Havnes, 2008] which operates at 56 MHz and the 7.9 MHz radar [Mahmoudian et al., 2011]. HAARP in Gakona, Alaska, also has a radar that operates at 139 MHz [Ramos et al., 2009]. La Hoz also recently reported the first joint measurements of PMSE, during heating using the EISCAT facility, with the EISCAT VHF radar at 224 MHz (Bragg scale of 0.67 m) and the MST MORRO radar at 56 MHz (Bragg scale of 2.7 m) [La Hoz et al., 2010]. Weakening of the PMSE after heater turn-on was observed for both radar frequencies. Although an intensification of PMSE was measured by MORRO radar over some intervals of heater turn-off, the enhancement of radar echoes measured by the MORRO radar was modest [La Hoz et al., 2010]. For instance, it has been shown by Mahmoudian et al.  that based on parameters obtained during a recent in-situ measurement, that the intensity of the radar echoes are predicted to be enhanced less after turn-off at 56 MHz compared to 224 MHz which is in agreement with measurements using the MORRO radar [La Hoz et al., 2010].
 Since temporal evolution of radar echoes after the turn-on and turn-off of pump heating varies depending upon what the radar frequency has been used, the irregularity amplitude measured during multifrequency experiments can be exploited to obtain diagnostic information regarding the charged dust layer parameters. Estimating the electron irregularity amplitude under the influence of the pump heating accurately during various times of the heating cycle can shed light on the dust parameters at mesopause altitudes such as dust radius, charge state and density as well as plasma parameters. For instance, the work of Scales and Chen  developed an analytical model for relating the electron irregularity amplitude after turn-off of the radio wave heating to dust layer and plasma parameters. An overall goal of PMSE heating experiments is to unambiguously diagnose the dust layer in the heated volume. This of course requires sufficient observables from all phases of the heating cycle at multiple frequencies and this was not possible in the previous work of Scales and Chen  which just allowed incorporation of information at the turn-off of radio wave heating.
 The object of this study is to develop and validate an analytical model to describe evolution after the turn-on of radio wave heating. This simplified model may then be used to provide more direct diagnostic information about the dust layer. The first attempt to develop an analytical model for temporal evolution of radar echoes associated with heater turn-on was done by Scales and Chen . This work was very limited and will be greatly extended here in order to directly relate electron irregularity amplitude during turn-on of radio wave heating to dust layer and plasma parameters. The second objective is to bring to fruition a methodology for diagnosing the dust layer using a multifrequency measurement using information from all phases during the heating cycle (i.e., both turn-on and turn-off). The organization of the paper is as follows. First, a full computational model is provided as a reference. The next part of the paper is dedicated to the development of an analytical model which is applied to predict the irregularity amplitude after the pump turn-on of radio wave heating as well as providing possibilities for diagnostic information available during the turn-on period of the radio wave. This model is validated with the computational model of the previous section. Application of the simplified analytical models for HF and VHF bands is discussed. Afterward, from the temporal behavior of the electron irregularities during the turn-on of the radio wave, possibilities for using the analytical model to obtain diagnostic information for various charged dust and background plasma quantities from simultaneous PMSE modification experiments in the HF and VHF band is discussed. Finally a summary and conclusions are provided.
2. Benchmark Computational Model
 The first attempt to model the full PMSE heating experiment cycle was done by Havnes  to investigate the radar echo behavior at 224 MHz. The model implemented the time-dependent dust cloud charge model and used Boltzmann electron and ion physics [Havnes et al., 2004; Havnes, 2004; Biebricher et al., 2006; Kassa et al., 2005]. Also, quasi-neutrality and standard orbit-limited motion (OLM) equations are applied to describe the dust particle charging. Although this model successfully reproduced much of the basic behavior observed during experiments at 224 MHz at EISCAT, such as turn-off overshoot effect, the Havnes model was limited to higher radar frequencies as well as smaller dust sizes and lower dust densities because of incorporating the Boltzmann approximation and neglecting finite diffusion time effects. Therefore the original model utilized by Havnes is not able to study the temporal behavior of irregularity amplitude at lower frequencies such as 56 MHz and 7.9 MHz.
 A one-dimensional model was first developed by Scales  and later utilized by Chen and Scales  to study the effects of ionospheric heating by radio waves on plasma irregularities associated with charged mesospheric dust clouds. The model includes electrons, ions and dust. The dust is taken as immobile. In this model, electrons and ions are treated as fluid instead of with the Boltzmann assumption used by Havnes , therefore it enables the study of the evolution of irregularity amplitude at a range of relevant frequency, plasma and dust parameters. Dust particles are also treated as Particle In Cell (PIC) [Birdsall and Langdon, 1991]. The computational model developed by Scales  is solved here by using different numerical algorithms. The governing fluid equations are written in the formulation described by Bernhardt et al. . The continuity equation is solved using an implicit method. The new algorithms not only reduce the computational time substantially but also provide greater accuracy.
 Two types of dust irregularity shapes are considered to determine the sensitivity on the electron irregularity amplitude. Such a comparison is important since the electron irregularity amplitude temporal behavior within this full computational model will be used to validate the analytical model in the next section. The sinusoidal irregularity model in the uncharged dust is assumed to be [Scales, 2004]:
where nd0 is the undisturbed dust density, δnd0 is the dust density irregularity amplitude which is set to δnd0/nd0 = 0.2 in the simulation, m is the irregularity mode number and l is the system length in the model. The dust density irregularity amplitude observed by rocket probes which, at the length scales considered in the present manuscript, is mostly in the range 0.1–0.2 [Rapp et al., 2003]. Some cases of large variations have been observed but they appear to be comparatively rare. The other irregularity shape considered in this paper is the Gaussian which is incorporated in the uncharged dust density as follows [Havnes, 2004]:
where nd0 is the maximum density of uncharged dust particles, x0 is the location of the center of the dust irregularity and σ is the parameter which determines the width of dust irregularity according to the Bragg scattering condition. The relationship between the irregularities and the radar reflectivity is given by the expression [Røyrvik and Smith, 1984]:
where k = 2π/λ and f are the wave number and frequency of the radio wave, λ is the wavelength, σ(k) denotes the radar scattering cross-section per unit volume, fpe is the electron plasma frequency, n is the exponent of the electron power spectral density, and Sne(k) is the 1-D power spectrum of the electron irregularities. It has been shown that backscatter intensity S can be estimated as S ∝ |ΔNe(k)|2s0 where s0 is the intensity of the incoming radio wave [Nygren, 1996]. It turns out that for the sinusoidal perturbations, Eq.(3) can be simplified to σ(k) ∼ δne2, where δne is the electron density irregularity amplitude [Chen and Scales, 2005]. In this paper both exact and simplified expressions are considered for σ(k) in development and validation of simplified analytical models. A more general expression for radar reflectivity including dust density and charge is provided by Varney et al. . For the work here this would be considered as a second order effect for diagnostic calculations. This should be incorporated into the current model in the future.
3. Sensitivity Study of Dust Irregularity Shape
 The effect of estimated and exact expressions for radar reflectivity is studied in Figure 1. Radar reflectivity calculated using equation(3) are shown with the black line and the blue line represents the irregularity amplitude calculated using σ(k) ≈ δne2. Radar frequencies 7.9, 56, and 224 MHz are studied. The reason for choosing these frequencies is that they show different behaviors during the pump heating cycle and because they are available at EISCAT. In fact, two physical processes control the electron irregularity amplitude: (1) charging of the electrons onto the dust and (2) the ambipolar diffusion process. Since both processes are dependent on the ratio of electron to ion temperature and radar frequency, a distinct evolution of irregularity amplitude is expected to be seen at these three radar frequencies. The diffusion process tends to smooth out irregularities while electron charging process tends to enhance the irregularities. These are assumptions used in the calculations. Electron temperature enhancement during radio wave heating Te/Te0 = 3, dust radii rd = 2 nm, dust density nd = 2 × 109 m−3, electron density ne0 = 109 m−3, ion-neutral collision frequency νin = 105 Hz. Dust radius is assumed to be 2 nm which is consistent with the recent data obtained during in-situ experiments [Robertson et al., 2009]. The curve calculated using the approximate expression shows good agreement with the calculated curve using the exact expression. Therefore, the radar reflectivity calculated in the computational model is not highly sensitive to the approximate and exact expressions of radar reflectivity.
 The effect of the two shapes of dust irregularities on the temporal evolution of radar reflectivity approximated with δne2 during radio wave heating is studied in Figure 2. The Gaussian shape has been used in past modeling studies of the PMSE source region [Lie-Svendsen et al., 2003] while the sinusoidal shape has been used in modeling of ionospheric plasma irregularities (turbulence) where a spectrum of spatial scales are involved [Zargham and Seyler, 1987]. It may therefore be argued the later shape has some advantages for a multifrequency PMSE heating experiment. Figure 2 (left) shows the radar reflectivity after the heater turn-on and the right panel represents the radar reflectivity evolution after the heater turn-off. Dust and plasma densities are assumed to be ne = ni = nd = 109 m−3 and the dust particles are charged up negatively. The ratio of electron temperature enhancement Te/Te0, where Te0 is the electron temperature before the heater turn-on, during heating is 3 [Routledge et al., 2011]. The irregularity scale sizes considered here correspond to the radar frequencies of 7.9, 56, and 224 MHz. In Figure 2, the blue and black lines show the radar reflectivity associated with the Gaussian and sinusoidal irregularities, respectively. As can be seen in Figure 2 (left), irregularity amplitude squared corresponding to the radar frequency of 7.9 MHz associated with the sinusoidal irregularities reaches a larger value in comparison with those calculated using the Gaussian irregularities. It turns out that while both irregularity models show similar estimation of irregularity amplitude in the first 5s after heater turn-on at 7.9 and 56 MHz, the sinusoidal irregularities predict a larger turn-on overshoot amplitude before heater turn-off by about 50% at 7.9 MHz. This difference is less than 10% for radar frequencies 56 and 224 MHz after the pump turn-on. The behavior of radar reflectivity evolution after heater turn-on with both irregularities is reasonably close at 224 MHz. Figure 2 (right) shows the irregularity amplitude after heater turn-off normalized to its amplitude before the pump turn-off. The comparison of radar reflectivity curves calculated for two cases after heater turn-off, which is shown in Figure 2 (right), also shows that sinusoidal irregularities predict a larger turn-off overshoot with respect to the Gaussian irregularities at 224 and 56 MHz which is less than 20%. The irregularity amplitude for both of these two irregularity shapes peaks at the same time. The irregularity amplitude at 7.9 MHz decays with the same rate for both irregularity models after the pump turn-off.
 As will be discussed in the next section measurable parameters that can be used to get diagnostic information about the dust layer are the maximum and minimum amplitude of radar reflectivity, and the time this amplitude is reached. It has been demonstrated that the model for dust irregularities whether sinusoidal or Gaussian is relatively insensitive to providing a sound benchmark for an analytical model. The sinusoidal model will be adopted for use in the next section as will δne2 for radar reflectivity. It can be shown it also provides better agreement with the analytical model. Throughout this paper the term electron density irregularity amplitude is used for δne which refers to square root of radar reflectivity.
4. Analytical Model for Irregularity Temporal Evolution
 As was discussed in the previous section, the ambipolar diffusion process tends to suppress the radar echoes and the electron charging process acts to enhance the backscattered signal during the radio wave heating process. The diffusion timescale can be approximated by [Chen and Scales, 2005]:
where νin, λ, and vthi are the ion-neutral collision frequency, irregularity wavelength and ion thermal velocity, respectively. Equation (4) shows that the diffusion timescale depends on Te/Te0; therefore, after the radio wave heating turn-on, the time decreases. The diffusion timescale is of order of 1s for irregularities observed in the 50 MHz range. The timescale for electron attachment onto the dust immediately after heater turn-on is approximated [Chen and Scales, 2005]:
where k can be approximated as , rh = Te/Te0 is the ratio of electron temperature enhancement, and x = eϕd/kTe denotes the equilibrium normalized floating potential of the dust prior to the radio wave heating. Therefore, while at lower radar frequencies (e.g., 7.9 MHz) the electron charging onto the dust dominates the ambipolar diffusion process and enhances the irregularity amplitude after the pump turn-on, at higher radar frequencies (e.g., 224 MHz) the diffusion timescale is less than the electron charging timescale, therefore, suppression of irregularities is predicted. It should be mentioned that during the heating cycle, the dust reaches a new charge state which can be obtained using the equilibrium condition. The reduction rate of electron density due to the electron charging is denoted by τr = 1/(knd0). It is noted that this rate is related to the initial dust charging time by τr = τcne/nd.
4.1. Layer Characteristics Modified by Radio Wave Heating
 Radio wave heating of the PMSE modifies at least four important parameters associated with the dusty plasma heated region. These parameters are electron temperature Te, electron density ne, electron density irregularity amplitude δne and dust charge number Zd. Important parameters that are assumed unmodified by the radio wave heating are the dust density nd, dust radius rd and dust irregularity amplitude . This subsection provides relationships between the modified and unmodified parameters that are used along with the analytical model for the electron irregularity amplitude (the primary measurable parameter) of the following section to diagnose the heated region. In other words, to develop the analytical model, unmodified parameters (such as nd, rd and ni) are assumed to be known in order to calculate modified parameters. In sections 4.5 and 4.6, the analytical model along with the measurable parameters (electron density irregularity amplitude δne variation) will be implemented to predict unmodified parameters such as nd, rd, and as well as the modified parameter Δne0 to diagnose the layer.
 There are models that can estimate the ratio of electron temperature increase Te/Te0 with good precision [Senior et al., 2010]. Therefore, to get an estimation of electron density irregularity amplitude during the pump heating, we start off finding the value of ne, ϕd (dust floating potential) and Zd before heater turn-off. After the turn-on of the radio wave heating, the electron temperature will be enhanced and electron current on to the dust increases rapidly and dominates the ion current during the pump turn-on period. Then, the ion current increases and another equilibrium is reached again with |Ii| ≈ |Ie| as result of decreased floating potential during continued heating. The electron Ie and ion currents Ii can be estimated with the following expression for the negatively charged dust particles [e.g., Havnes, 2004]:
Here, rd is the dust radius, vte(i) electron (ion) thermal velocity and ϕd dust floating potential. me and mi are electron and ion mass, respectively. Therefore, the following expression can be written for the steady state after the heater turn-on using equations (6) and (7) and |Ii| = |Ie| [Chen and Scales, 2005]:
where and T = Te ≈ Ti represents the electron or ion temperature before heating that they are assumed to be equal. rh is the ratio of electron temperature increase after the heater turn-on and ϕd is the new level of dust floating potential. It should be noted that after the pump turn-on, the reduction in the ion density is of the order of a few percent and negligible. Considering the steady state after the heater turn-on and negligible variation of ion density during heater turn-on cycle, the floating potential and electron density are the two unknown (modified) parameters in equation (8). According to the quasi-neutrality condition, the plasma densities and dust density can be written in the following form:
Considering that dust is assumed to be unmodified during the heating period and variation of ion density is very small in comparison with the electron density variation, the dust charge and electron density are the only two unknown parameters in this equation. The relation of the dust floating potential and the dust radius to the number of charges on the dust particle can be written in the form [Scales, 2004]:
This expression can be used to relate the floating potential and the charge number on the dust at the steady state before the heater turn-off. Therefore, using equations (8), (9) and (10) and solving them, the three unknown modified parameters, dust charge number Zd, electron density ne0 and dust floating potential ϕd can be estimated in the plasma steady state after the pump turn-on. The electron density ne0 is used in the analytical model to be described shortly.
 For the shorter irregularity wavelengths, the diffusion time is sufficiently small and electrons and ions can be modeled with the Boltzmann approximation. This implies that the normalized electron and ion irregularity amplitude before the pump turn-off can be written in terms of the electrostatic potential irregularity amplitude δϕ as [Scales and Chen, 2008]
where and are the normalized electron and ion irregularity amplitudes. Considering the Poisson equation for the irregularities in the Fourier spectral domain, the dust charge variation during the radio wave heating can be written in this form ΔZd ≈ − (1 + λDe2k2)Δδne/δnd for irregularities in the VHF band where Δδne and δnd are electron irregularity amplitude variation during heating and dust irregularity amplitude, respectively. It will be discussed in section 4.6 that Δδne is observable and δnd may be calculated from the analytical model.
4.2. Analytical Model for Electron Irregularities During Heating
 The purpose of the present investigation is the use of temporal evolution of the electron density irregularity amplitude after the turn-on of radio wave heating to diagnose the charged dust layer. As was discussed before, the amplitude of electron irregularities after heating depends on diffusion and charging processes. Considering that the charging process tends to increase and diffusion process tends to suppress the amplitude of irregularities, depending upon which process has the shorter timescale after heater turn-on, the amplitude of irregularities may be suppressed or enhanced. Figure 3a shows the schematic of these characteristics after the pump turn-on. Therefore, as can be seen in Figure 3a, there is diagnostic information that can be inferred regarding the charged dust layer. Depending on what the radar frequency or electron irregularity wavelength is, the irregularity may be enhanced or suppressed following the heater turn-on and the maximum or minimum of irregularity amplitude, which is shown by δnemax and δnemin, respectively, is the unique characteristic that can be used as a diagnostic tool. The time at which the maximum or minimum amplitude has been reached τmax, τmin, are parameters that can place a diagnostic bound on the dust parameters and shows the transition from the domination of electron charging process to the domination of ambipolar diffusion process. The timescale for the decay of the electron irregularity amplitude after heater turn-on, when the maximum is reached, is another characteristic associated with the active modification of PMSEs.
 A detailed study of radio wave modulation with local plasma at mesopause altitudes was done by Havnes et al.  and Kassa et al.  where it is shown that the ion density is not much affected during the heating but the electron density is reduced by a large factor within the bite-out. This is because the heated electrons attach themselves rapidly to the dust and become depleted as the dust is charged more negatively while the heavy and colder ions collide at a much slower rate.
 Assuming the electron density neeq before the heater turn-on, due to increasing and rapidly dominating electron current during the pump turn-on, the electron density reduces. The electron density is assumed to reach ne0 which is the electron density after steady state during heating. The electron density variation during radio wave heating is shown in Figure 3b. This happens when the floating potential decreases during continued heating and another equilibrium is reached again with Ie ≈ Ii. As was mentioned before, the electron density before heater turn-off ne0 can be obtained using the equilibrium condition equation (8), the quasi-neutrality condition equation (9) and the expression of dust charge number equation (10).
 The electron density variation with time can be approximately modeled with the diffusion and the dust charging, that corresponds to reduction of the electron density, during the initial turn-on of the radio wave. The temporal behavior of electron density at a fixed spatial point can be written as follows:
where turn-on of the radio wave is referenced to time t = 0. At t = 0, the electron density has its equilibrium value neeq and for a longer time period after the heater turn-on, the second term in equation (12) shows the electron density at the steady state dominates the first term. ne0 is described by the solution of equations (8), (9) and (10) as discussed in the section 4.1. Another physical process that may have a secondary effect on electron irregularity amplitude before radio wave heating turn-off is the recombination of the electrons and ions. The recombination of electrons and ion is shown in the form of since it causes the reduction of electron density. The recombination time where α is the recombination rate and rh is the ratio of electron temperature increase after the turn-on of radio wave heating. The recombination process is assumed to be important at the later time after heater turn-on. Considering that the effect of the recombination process on the initial evolution of irregularities is negligible and the subject of this work is to study the temporal evolution of the irregularities at early time after heater turn-on, the effect of the recombination process is neglected. Therefore the temporal evolution of the electron density can be written as:
 The background and perturbation components of electron and dust densities can be written in the form n0 + δn0, respectively. The fluctuation amplitude of electron density right before heater turn-on is assumed to be normalized to its value before heater turn-on and is one. δne0 is the electron irregularity amplitude before heater turn-off. It should be noted that the irregularities on the electrons and dust are 1800 out of phase while ions and dust have in phase irregularities [Lie-Svendsen et al., 2003; Scales and Chen, 2008]. As a result, equation (13) can be simplified to
By simplifying equation (14) we can write the following expression:
where B = (neeq − ne0) ≡ Δne0 is the variation of the electron density after heater turn-on and A = (δneeq − δne0) which represents the variation of electron density irregularity amplitude during the pump turn-on period. is the dust irregularity amplitude normalized to the background dust density. The analytical model can be used as a diagnostic tool to estimate some parameters associated with the charged dust layer such as dust density, dust radius and ratio of electron temperature enhancement during heating using the temporal behavior of electron irregularity amplitude or the amplitude of the backscattered signal during the PMSE heating experiment. The analytical expression mentioned in equation (15) is incorporated for all figures which will be presented after this section. This model also will be simplified for HF and VHF bands in the following sections.
 The mesopause temperature for both ions and electrons is taken to be Te = Ti = 130K. Proton hydrates with mass between 59 and 109 proton masses are the dominant ion compositions at this height range. O2+ ions are more numerous than NO+. NO+ and O2+ together can be slightly more dense than the proton hydrates at 88 km and above [Kopp et al., 1985]. It should be noted that the variation of ion mass from 50 to 100 proton masses does not have a significant impact on the irregularity amplitude evolution during heating. The ion-neutral collision frequency is of order 105s−1 [Lie-Svendsen et al., 2003]. The variation of ion-neutral collision frequency is predicted to be between 3 × 104s−1 and 3 × 105s−1 in the altitude range 80–90 km [Turunen et al., 1988]. The electron-neutral collision frequency temperature dependence is assumed to be νen ∼ Te and recombination rate dependence on temperature is taken to be α ∼ Te−1/2. The size and density of dust particles are assumed to vary from 1–3 nm and 1–3 × 109m−3, respectively. It should be noted that these parameter ranges are based on recent experimental observations [Robertson et al., 2009]. The dust irregularity amplitude normalized to the background dust density is taken to be 0.2.
 The accuracy of the analytical model is examined for the variation of radar frequency, the degree of temperature enhancement during heating and dust density. Figure 4 represents the irregularity amplitude after heater turn-on associated with the computational and analytical model for varying radar frequency which corresponds to the variation of electron irregularity wavelength according to the Bragg scattering condition. Figure 4 shows the electron irregularity amplitude 5s before the pump turn-on when the equilibrium of irregularities has been achieved. The radio wave heating is turned on at t = 25s for heating cycle 30s. As can be seen, the simple analytical model has very reasonable agreement for 30s after the turn-on in comparison with the full computational model. It is much superior to previous analytical models for irregularity behavior during turn-on [e.g., Scales and Chen, 2007]. The computational and analytical models predict an enhancement of electron irregularities for larger irregularity scale sizes 20 and 10m which correspond to radar frequencies 7.9 and 56 MHz. The τmax estimated by the analytical model are 12 and 2.9s for radar frequencies 8 and 56 MHz, respectively. The τmax according to the computational results are 13.3 and 3.1s which shows good agreement with the analytical model. While the computational model predicts a small enhancement of irregularity amplitude after the turn-on at 134 MHz, the analytical model shows a suppression. According to the analytical model results, the decay of irregularity amplitude after the peak at 7.9 and 56 MHz or after heater turn-on at 224 and 930 MHz is faster in comparison with the computational result and shows that the analytical model accentuates the diffusion process slightly. In summary, it turns out that the diffusion timescale in the analytical model is somewhat shorter for 134, 224 and 930 MHz in comparison with the computational model. The significance will be discussed in more detail shortly.
4.3. Application to Temporal Evolution of Electron Density Irregularity Amplitude in the VHF Band
 According to the equations of electron charging and ambipolar diffusion timescales, only the diffusion time depends on the dust density and it reduces as the dust density increases. It turns out that at higher radar frequencies (e.g., 224 MHz), the electron density irregularity amplitude which suppresses rapidly after the pump turn-on due to the short diffusion timescale, may increase in amplitude after continued heating as the result of the electron charging process for certain dust density and radii ranges. This has been observed both in experimental data and computational models [Naesheim et al., 2008]. This enhancement of electron irregularity amplitude requires relatively small dust densities and radii. In the case of continued heating and at the time the ambipolar diffusion has ceased, the charging of electrons on to the dust takes the control of the electron irregularity amplitude, therefore temporal evolution can then be approximately described by:
The solution for the temporal evolution of electron density irregularity amplitude can be expressed as:
where corresponds to the dust irregularity amplitude associated with the VHF radar. Considering the assumption τd ≪ τr for the smaller irregularity wavelengths (corresponding to the higher radar frequencies), the time at which the minimum amplitude has been reached after the turn-on of the radio wave can be obtained from the condition ∂ δne(t)/∂ t = 0 and equation (15):
where Δne0 and Δδne1 are the electron density variation and electron density irregularity amplitude variation during the pump heating (Figures 3a and 3b) and Δδne1 = δne0 − δneeq. It should be noted that Δδne1 can be measured during the active experiment just by comparing the strength of the radar echo before the heater turn-off to its value at the steady state before the heater turn-on. This expression can be used to approximate the minimum amplitude of the electron irregularities after the turn-on of the radio wave heating as a result of the diffusion process. Substituting equation (18) into (15) implies the following expression for the minimum amplitude of electron irregularities in the VHF band during heating:
where Δδne1 = δne01 − δneeq. Since at higher radar frequencies or shorter electron irregularity wavelengths, the ambipolar diffusion timescale is much less than the electron charging timescale, the approximate analytical expression can be used to investigate the behavior of the evolution of the dust associated electron irregularities at the initial turn-on of the radio waves. The condition for no enhancement of the electron irregularities at the initial turn-on of the radio wave, typically observed in VHF experimental observations, ∂δne/∂t < 0 for all t yields the condition:
By assuming , this expression can be simplified to:
This condition imposes a lower diagnostic bound on the ratio of the electron density reduction rate to ambipolar diffusion rate which can be used as a quantitative condition to estimate the electron density reduction during heating by measuring the electron density irregularity amplitude variation according to the radar echo strength.
 The effect of the varying dust density on the temporal evolution of the electron density irregularity amplitude is shown in Figure 5 for dust densities 0.8 × 109, 1.6 × 109, and 2.4 × 109 m−3 with dust radius 3 nm. Electron density and the ratio of electron temperature increase during heating are assumed to be 109 m−3 and 3, respectively. Figure 5 (left) shows the computational results and Figure 5 (right) represents the analytical results. It is evident that the analytical model performs well in predicting the general behavior of irregularity amplitude after heater turn-on. The primary difference is that the rate of the irregularity's decay in analytical results is faster than computational results. This shows that the diffusion timescale is longer in the computational model. This can be explained based on equation (4) which shows the electron density dependency of diffusion timescale. Therefore, the electron density changes gradually with time in the simulation after the pump turn-on till it reaches its steady state value before pump turn-off, while the diffusion timescale used in the analytical model is calculated by making the assumption that electron density drops to its steady state value instantly. As a result, there is a tendency for the analytical model to slightly overestimate τmin in this regard.
Figure 6 compares the variation of the electron irregularity amplitude with the ratio of electron temperature increase during the pump heating which is obtained using the both computational and analytical models. According to Figure 6, the electron irregularity has been suppressed at 224 MHz by increasing the ratio of electron temperature enhancement after the heater turn-on which is also predicted by equation (11) where depends on 1/Te. The irregularity amplitude decreases in the first 2s after the pump turn-on as a result of the diffusion process and then starts to increase which is due to the electron charging process as is described by equation (17). As can be seen from both computational and analytical models, irregularity amplitude increases more for continued heating at smaller electron temperature enhancement ratios. This is also consistent with the approximate expressions for ambipolar diffusion and electron charging timescales, Equations (4) and (5), where diffusion depends on 1/rh and electron charging . This shows that for smaller values of rh, the charging timescale reduces more in comparison with diffusion timescale and leads to a larger electron irregularity amplitude with the continued heating.
4.4. Application to Temporal Evolution of Electron Density Irregularity Amplitude in the HF Band
 At higher frequency regimes (above 224 MHz), which would imply smaller irregularity wavelengths, the diffusion timescale is much less than the charging timescale and electron irregularities are suppressed quickly after the heater turn-on before they have chance to be enhanced by the charging process. But at lower frequencies such as 7.9 MHz where the charging timescale is less than the diffusion timescale initially, electron irregularity amplitudes starts to grow until the point at which the diffusion timescale becomes comparable to the charging timescale. After this point, the electron irregularity amplitude approaches to the steady state value before turn-off of the radio wave where the dust charge reaches a new charge state by satisfying the equilibrium condition Ie + Ii = 0. The approximate temporal behavior after the electron irregularity amplitude has reached its maximum is given by:
 Therefore the timescale for the decay of the irregularities after the maximum value is reached is the ambipolar diffusion time τd. In this case the time at which the maximum amplitude is reached after the turn-on of the radio wave can be estimated by making the assumption τr ≪ τd. Hence τmax obtained from ∂ δne(t)/∂ t = 0 is:
where corresponds to the dust irregularity amplitude associated with the HF radar and Δδne2 = |δne02 − δneeq|. This expression can be used to approximate the maximum amplitude of the electron irregularities after the turn-off of the radio wave heating. Substituting equation (23) into (15) implies the following expression for the maximum amplitude of electron irregularities in the HF band:
 These three characteristics of the irregularity temporal evolution after the turn-on may be used for diagnostic information. They are the time at which the maximum amplitude is reached (equation (23)), the maximum amplitude achieved (equation (24)) and the timescale of decay after the maximum amplitude has been reached. The condition determines the condition for enhancement of the electron irregularity after the turn-on of the radio wave. For the time period of about a few tenths of second after the turn-on of radio wave heating, this leads to the following condition:
Using the parameters presented in Figure 4, radar frequencies 7.9 MHz and 56 MHz with imply the right hand side of this equation to be approximately 8 and 80, respectively. The calculated ratio τr/τd ≈ 2 and 34 compared to the right hand side of equation (25) which validates the condition.
Figure 7 compares the effects of dust densities on the evolution of irregularity amplitude during heating at 7.9 MHz in more detail. Figure 7 indicates how the dust density can alter the ambipolar diffusion and electron charging timescales such that the electron irregularity amplitude may be suppressed after the pump turn-on at 7.9 MHz. While the models represent the enhancement of radar echoes for nd = 109 and 2 × 109 m−3, a suppression of irregularity amplitude has been observed after the pump turn-on for 3 × 109 m−3 which shows a similar behavior to temporal evolution of radar echoes at 224 MHz.
 The comparison of the computational and analytical models for various electron to ion temperatures are shown in Figure 8. The plasma parameters are ne = 109 m−3, rd = 2 nm, nd = 2 × 109m−3 and rh has been changed from 2 to 8. The results show good agreement between the two models and only the analytical model predicts a slower decay of irregularity amplitudes for rh = 4, 6 and 8 with respect to the computational results. According to the result of the computational model, the τmax is 30, 13.2, 6.8 and 4.5s for the temperature enhancement ratio 2, 4, 6 and 8, respectively, at the radar frequency 7.9 MHz. The time at which maximum amplitude has been achieved calculated using equation (19) is 30, 15, 5.5 and 4.9s which shows that the analytical model provides very reasonable accuracy regarding the estimation of τmax.
4.5. Incorporation of Information at Turn-Off
 Utilizing the measurable parameters after heater turn-off with the parameters associated with the pump turn-on during a multifrequency experiment may increase degrees of freedom to get more diagnostic information. Considering [Scales and Chen, 2008], which represents the timescale for the decay of the irregularities after the turn-off overshoot maximum is reached, the reduction rate of the ion density due to the dust charging from ion flux τri can be estimated using the data collected during an active VHF PMSE experiment. Therefore, the electron density reduction time after the pump turn-on is given by:
Using the time at which the maximum value is reached after the pump turn-on τmax during HF PMSE experiment to estimate the electron reduction time is another way of estimating of this parameter (i.e. τr ≈ τmax). As shown in Figure 12 of Chen and Scales  the electron charging time is very short and at the time the maximum amplitude is reached, the plasma is expected to have reached the steady state. Another parameter that can be measured in the active PMSE experiment is the time at which the maximum electron irregularity amplitude is reached after the turn-off of the radio wave heating and can be written as [Scales and Chen, 2008]:
ΔZd0 denotes the net gain in electron charges during radio wave heating and τdi is the diffusion time given by equation (4) for Te ≈ Te0. The maximum amplitude of the electron irregularities is estimated to be [Scales and Chen, 2008]:
where . Since all parameters of equation (28) are measurable, this equation can be used to approximate the degree of the electron temperature enhancement rh = Te/Te0. An alternative simple analytical expression to estimate the upper and lower limit of heated electron temperature by measuring the reflection coefficient, which is proportional to the electron gradient squared, was derived by Kassa et al. . It has been shown that the condition for turn-on overshoot τd/τc ≫ 1 and turn-off overshoot τd/τc ≪ 1 may provide a possible range of dust radius. Therefore, conducting PMSE heating experiments in the HF and VHF band simultaneously, where turn-on overshoot in the HF band and turn-off overshoot in the VHF band are observed, and using [Chen and Scales, 2005], where , , may possibly give fairly accurate bounds for rd in comparison with recent rocket data [Robertson et al., 2009].
4.6. Calculation of Dust Layer Diagnostic Information
 Considering the two types of distinct temporal behavior of radar echoes at HF (e.g., 7.9 MHz) and VHF (e.g., 56, 134 and 224 MHz) during active modification of PMSE, conducting these types of experiments in HF and VHF frequency band may provide enough observables to estimate dust layer parameters as well as background plasma quantities. The observable parameters during the experiment after the pump turn-on are τmax, τmin, δnemax, δnemin, τd, and Δδne1,2 as described in Figure 3a. Observables after turn-off includes the ion density reduction period τri, maximum amplitude of the electron irregularities after the turn-off of the radio wave heating δnemax,OFF, and the time at which the maximum amplitude is reached after the turn-off τmax. These observables may be used in the method described here to provide information on the dust density, charge state, electron density variation during the radio wave heating and the degree of the electron temperature enhancement.
 As discussed in section 4.5, τri can be used to estimate the electron reduction time τr and the maximum irregularity amplitude after the heater turn-off δnemax,OFF may be used to predict the temperature enhancement ratio during heating rh. Then, observing τmax, τmin, δnemax, δnemin, Δδne1,2, and τd during a two frequency experiment and using equations (18), (19), (23) and (24) gives a system of 4 equations that can be solved for the electron density variation during heating Δne0, and the dust irregularity amplitude associated with VHF radar δnd01 and associated with HF radar δnd02. It can be noted that this solution is facilitated by the fact that the mean electron density is the same in both frequency bands. After some mathematical manipulation of equations (9) and (11), it turns out that the dust density nd0 also can be estimated using:
where ne and Te are the electron density and temperature before active perturbation and e denotes the electron charge. Δδne1 corresponds to the electron irregularity amplitude variation and k1 = 2π/λ1 is the irregularity wavelength at the Bragg scatter of VHF radar. δnd01 and Δne0 are calculated from the model. To access the capability of the analytical model in predicting the dust and plasma parameters, the computational results for 7.9 and 224 MHz shown in Figure 4 are used as the possible case that may be observed in the experiment. Then, the predicted parameters by the technique just described is compared with the computational model parameters and shown in Table 1 which show very reasonable agreement.
Table 1. Computational Model Parameters Comparison With Those Obtained by the Analytical Model
Predicted by Analytical Model
1.1 × 109m−3
≈ 0.5 × 109m−3
0.2 × nd
≈0.27 × nd
 The irregularities in electron density are believed to be formed by fluctuation in the dust density. Looking at PMSE radar data represents a band of scattered signal over altitude range about 1 km which requires electron number density fluctuations at the Bragg scale [Kassa et al., 2005; Naesheim et al., 2008; Ramos et al., 2009]. As shown in Figure 9, the overlapped region of HF and VHF PMSEs can be divided out to N subregions [Ramos et al., 2009]. Therefore, the proposed active PMSE experiment in the HF and VHF bands simultaneously may be implemented to estimate the altitude profile of dust density by comparing the average of HF and VHF PMSE signal on the subdivided regions and using equations (18), (19), (23) and (24).
 Although the focus of this paper is on the multifrequency PMSE heating experiment in the VHF and HF band, it may be possible to perform this type of experiment for 2 radar frequencies in the VHF band, where a significant difference in the amount of suppression of radar reflectivity after heater turn-on and turn-off overshoot amplitude is expected. Dust layer parameters may be estimated. In this case equations (18) and (19) can be written for both frequencies where τmin1, τmin2, δne1min, δne2min, and τri are measurable in the experiment.
 Using ground based ionospheric heating facilities to produce an artificial enhancement in electron temperature is shown as a rich source of diagnostic information for charged dust layers in the earth's upper atmosphere. The dependency of the backscatter signal strength after the turn-on and turn-off of the radio wave heating to the radar frequency is an unique phenomenon that can shed light on the unresolved issues associated with the basic physics of the natural dust layer. This work has attempted to provide further physical insight into the physical processes associated with temporal evolution of the electron irregularities during the turn-on of the radio wave heating and can be seen to be complementary to past work that has considered the physical processes after the turn-off of radio wave heating. The new analytical model is able to describe the temporal evolution of electron irregularities during the early phase of the heating cycle. The simplified analytical models here provide quite reasonable agreement with full computational results. It turns out that active PMSE heating experiments involving multiple observing frequencies at 7.9 (HF), 56, and 224 MHz (VHF) may contribute further diagnostic capabilities since the temporal evolution of radar echoes is substantially different for these frequency ranges. Measuring radar echoes at multiple frequencies imposes enough information to estimate important plasma and dust parameters. Analytical expressions for observable parameters associated with the radio wave turn-on, τmax, τmin, δnemax, δnemin, Δδne, and τd as well as τri, δnemax,OFF and τmax after the pump turn-off, during active perturbation of PMSEs are derived here that may provide information on the dust layer such as dust density altitude profile, dust density irregularity amplitude, dust charge state variation, and degree of electron temperature enhancement during radio wave heating. It has been shown that predicted enhancement of irregularity amplitude after heater turn-on in the HF band is the direct manifestation of the dust charging process in the space. Therefore further active experiments of PMSEs should be pursued in the HF band to illuminate the fundamental charging physics in the space environment and get more insight on this unique medium.
 This work was supported by the National Science Foundation.