Modelling of the influence of meteoric smoke particles on artificial heating in the D-region

We investigate if the presence of meteoric smoke particles (MSP) influences the electron temperature during artfical heating in the D-region. The presence of MSP can result in height regions with reduced electron density, so-called electron bite-outs, due to charging of MSP by electrons. Artificial heating depends on the height variation of electron density. By transferring the energy of powerful high frequency radio waves into thermal energy of electrons, artificial heating increases the electron temperature. We simulate the influence of the artificial heating by calculating the intensity of the upward propagating 5 radio wave. The electron temperature at each height is derived from the balance of radio wave absorption and cooling through elastic and inelastic collisions with neutral species. The influence of MSP is investigated by including results from a one-dimensional height-dependent ionospheric model that includes electrons, positively and negatively charged ions, neutral MSP, singly positively and singly negatively charged MSP and photo chemistry such as photo ionization and photo detachment. We apply typical ionospheric conditions and find 10 that MSP can influence both the magnitude and the height profile of the heated electron temperature above 80 km, however this depends on ionospheric conditions. During night, the presence of MSP leads to more efficient heating, and thus a higher electron temperature, above altitudes of 80 km. We found differences up to 1000 K in temperature for calculations with and without MSP. When MSP are present, the heated electron temperature decreases more slowly. The presence of MSP does not much affect the heating below 80 km for night conditions. For day conditions, the difference between the heated electron 15 temperature with MSP and without MSP is less than 25 K.


Introduction
MSP are small nanometer-sized dust particles (Hunten et al., 1980;Plane, 2012) that can change the D-region charge balance because they influence the chemical processes through charging of MSP by electrons and ions [cf. Baumann et al. (2015)]. By The presence of MSP in the D-region facilitates the formation of ice particles. These ice particles contribute to the formation of strong radar echoes, called Polar Mesospheric Summer Echoes (PMSE) (Rapp and Lübken, 2004;La Hoz et al., 2006). A heater transmit powerful high-frequency radio waves into the ionosphere during artificial heating experiments. Artificial heating can induces difference phenomena in the ionosphere. Ion upwelling  and artificial optical emission (Kosch et al., 2000;Kosh et al., 2014) are observed in the ionospheric F-region. Observations show that the back-scattered power 35 of PMSE can be modified during artificial heating experiments [cf. Chilson et al. (2000); Havnes et al. (2004) ;Biebricher and Havnes (2012)] where the electron temperature in the ionosphere is increased artificially. In the collisional plasma of the ionospheric D-region electrons absorb the radio wave energy transmitted from the heater and heat up, increasing the electron temperature. As a consequence the intensity of the radio wave reduces (Rietveld et al., 1986;Belova et al., 1995;Kero et al., 2000Kero et al., , 2008. 40 The aim of our study is to numerically model the electron temperature during artificial heating and include the height variation of electron bite-outs by using an ionospheric model (Baumann et al., 2013) with MSP. As a comparison, we also model without MSP. The one-dimensional height-dependent ionospheric model is for quiet ionospheric conditions and includes MSP and photochemistry such as photoionization and photodetachment. We calculate the artificial heating with different radio wave frequencies and higher or lower radio wave power to investigate if this influence the electron temperature and to check 45 the robustness of our results. We will compare night and day conditions to see if a higher electron density during daytime influence the modelled electron temperature. This paper is organized as follows. In part 2 we present a detailed theoretical background and numerically modelling of artificial heating in the D-region. Part 3 gives a brief description of the ionospheric model. In part 4 we introduce the results.
Part 5 present the discussion. Powerful high frequency radio wave can heat up electrons in the ionospheric D-region by artificial heating experiments. The higher temperature of the electrons can lead to various phenomena in the whole ionosphere (e.g. Robinson (1989) and references therein). Artificial heating increases the electron temperature by transferring the radio wave energy into thermal energy of electrons (Rietveld et al., 1986;Kero et al., 2007Kero et al., , 2008. Modelling of artificial heating in D-region altitudes show an increase in electron temperature of a factor of 10 (Belova et al., 1995;Kero et al., 2000). The EISCAT high power high frequency heating facility located in Tromsø, Norway transmit powerful high-frequency radio waves into the ionosphere during artificial heating experiments. The ESICAT radar, also located in Tromsø, Norway, can observe the ionosphere during these heating experiments. The EISCAT heating facility in Tromsø has three different antenna arrays consisting of crossed full-wave dipoles with a frequency range of 3.85-8 MHz. There are 12 transmitters that can adjust the power output from 200 MW to MW 1200, 60 depending on the used radio frequency. The dipoles can transmit ordinary (O) circular polarization mode or extraordinary (X) circular polarization mode (Rietveld et al., 2016). The following model of the heated ionosphere, described in the next section, uses these experimental parameters of the EISCAT heating facility (Rietveld et al., 1993).

Description of model
This section describes the physical background of the electron heating process. For the implementation of these processes 65 we rely on earlier work done by Rietveld et al. (1986); Belova et al. (1995); Kero et al. (2000); Kassa et al. (2005);Kero et al. (2007). Note that the model described in this section only cover the lower ionosphere. The heater transmits a powerful high frequency radio wave that propagates through the cold, magnetized, collisional plasma of the ionospheric D-region. The intensity I, or energy of the radio wave varies with height h according to: where k is the absorption coefficient, given as: In Eq. 2, ω is the angular frequency of the heating radio wave, Im(n) is the imaginary part of the refractive index n and c is the speed of light. When integrated, Eq. 1 in combination with Eq. 2, yields the following expression for the intensity: where ERP is the effective radiated power. For solving Eq. 3 we need an expression for the refractive index n. We can derive the refractive index by using the Appleton-Hartree dispersion relation, which describes the radio waves propagation in a cold magnetized plasma and which can be applied to the ionospheric D-region. It describes the refractive index as: where θ is the angle between the wave vector and the direction of the magnetic field. Here, (+) and (−) represents the 80 ordinary and extraordinary polarization modes, respectively. Note that the refractive index is complex n = n 1 + in 2 . If the imaginary part is less than zero, the wave is damped. The wave damping is caused by wave energy loss through absorption by the plasma while the wave propagates through the ionosphere. Due to its lower mass, electrons absorb the energy and are thus heated, while ions and neutrals remain unheated in comparison. The dimensionless X, Y and Z are normalized frequencies, defined as: where N e is electron density,e is unit charge, ε 0 is the permittivity of vacuum, m e is electron mass, B is the Earth's magnetic field and ν en is the electron-neutral collision frequency. How the electron-neutral collision frequency from Eq. 7 depends on 90 electron temperature is taken from Dalgarno et al. (1967): Through ν en the refractive index depends on the electron temperature. The electron-neutral collision frequency is high in the 95 D-region due to the relatively low electron density in comparison to the neutral density. Therefore, electron Omic heating is the dominant D-region ionospheric response to heating. In Omic heating, electrons oscillating parallel to the radio wave electric field collides with neutrals. This causes a phase shift between the direction of the radio wave electric field and the direction of electron oscillation. Overall, electrons are scattered in a random direction. This random motion of electrons leads to absorption of wave energy, where the wave energy is transferred into thermal energy of electrons, increasing the electron temperature.

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To find the increased electron temperature we use the electron energy balance equation, which describes local electron energy conservation. Solving the electron energy equation gives us the electron temperature time variation due to energy input from the heater and cooling through collisions with neutrals. We have neglected thermal conductivity due to high neutral density in the D-region and neglected plasma transport. The electron energy equation is then given as: The electrons loose energy, and are thus cooled, through elastic and inelastic collisions with neutral species, where the inelastic collisions can excite vibrational and rotational states. The sum of all energy losses is given by the energy loss function 110 L(T e ); these are the electron cooling rates. Our cooling rates include vibrational and rotational excitation of molecular oxygen (Pavlov, 1998b) and of molecular nitrogen (Pavlov, 1998a), excitation of fine structure levels of atomic oxygen (Pavlov and Berringston, 1999) and elastic collisions between electrons and neutral species (Schunk and Nagy, 1978). Due to the low electron density in the D-region, we neglect electron-ion collision. More detailed descriptions of the electron cooling rates are in the appendix. When the heater is switched on, the electron temperature increases from its initial temperature, which is 115 equal to the neutral temperature in the D-region, to a higher heated electron temperature. The heating time for this temperature increase is less than 100 ms due to the high collisions frequency ν en in the D-region. After less than 100 ms the electron temperature has reached thermal equilibrium where dT e /dt = 0. In cases where the heating modulation time is much longer than the heating time for the electron temperature, we can simplify Eq. 9 to: Q(T e ) − L(T e ) = 0 (11) 120

Implementation of model
To compute the electron temperatures during heating we numerically solve Eq. 11. At the first height the intensity is I 0 = ERP/4πh 2 , the undamped radio wave. We then compute Q(T e ) from Eq. 10 by using the intensity I 0 , where Q(T e ) is a function of T e . We use the intensity I 0 to solve Q(T e ) − L(T e ) = 0 for the electron temperature by using an algorithm that combines the inverse quadratic interpolation method, bisection method and secant method (Brent, 1973;Forsythe et al., 125 1977). By solving Q(T e ) − L(T e ) = 0 we find the zero-point of Eq. 11, which gives us a new electron temperature. This new, modified electron temperature changes the refractive index since the electron-neutral collision frequency depends on electron temperature. With the changed refractive index, we recalculate the intensity, taking into account the loss due to absorption. We compute the intensity numerically by approximating the integral in Eq. 3 as a sum: where the first part ERP/4πh 2 represents the undamped radio wave and the part in the exponential function represents the damping effect due to absorption. The distance between each height is ∆h = (h ) − (h − 1). For our case ∆h = 1 km and ∆h is constant for all altitudes. In the next iteration, the intensity has changed, so there is a new zero-point for Q(T e ) − L(T e ) = 0, which we compute. In Fig. 1 we show Q -L as a function of T e with I 0 , where the zero-point is illustrated as a blue-coloured star. Also in Fig. 1 we show the changed intensity, illustrated as I 1 and the zero-point for Q -L with I 1 is marked as a magenta-135 coloured star. We see in Fig. 1 that the zero-point of Q -L is different for I 0 and I 1 . This process with a new, modified electron temperature, which changes the intensity, is repeated in an iteration scheme. The neutral temperature is the starting point in the iteration scheme. The iteration scheme is repeated until the change in the electron temperature is very small, i.e. when T e converges. This equation visualizes the iteration for the intensity:

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where I 0 = ERP . Here dI(h) represents absorption at heights below us and dI(h+1) represents absorption at the current height. Before we move to the next height, we sum all the absorption, so that for the next height we take into account all absorption below. In the next height, we repeat the procedure described for the first height and calculate the heated electron temperature and the new intensity. This is done for all heights, moving upward from the initial height to the final height. Our altitude range is 60-120 km. The ionospheric D-region varies in altitude range from about 50 km to 100 km, however, we 145 model up to 120 km to see if the electron temperature at altitudes above 100 km is influenced by the presence of MSP at lower altitudes below.
We model the height-dependent heated electron temperature with initial height profiles for the following parameters: Earth's magnetic field, electron density, neutral temperature, neutral densities of molecular nitrogen, molecular oxygen and atomic oxygen. For Earth's magnetic field, we use a dipole approximation (Brekke, 2013) . The magnetic field goes into Eq. 6, which 150 we use to compute the refractive index in Eq. 4. We compare day and night conditions to see if a higher ionization level, as during day condition, have an influence on the heated electron temperature. The used electron density height profiles during day and night conditions comes from an ionospheric model by Baumann et al. (2013). The neutral temperature and neutral densities are from MSISE-90 model (Hedin, 1991;Picone et al., 2002)

Background ionospheric model
Here in this part we give a brief description of a one-dimensional height-dependent ionospheric model for the D-region, which includes MSP, developed by Baumann et al. (2013). For the full description, see Baumann et al. (2013) and references therein.
The one-dimensional height-dependent ionospheric model is run for quiet ionospheric conditions between heights of 60-120 km and includes electron, positively and negatively charged ions, neutral MSP, singly positively and singly negatively charged

Night conditions 180
This section present results for the electron temperature modelled during artificial heating with and without the presence of MSP for night condition. The main results are that from 80 km and above the heated electron temperature is higher when MSP are present, and this applies to all cases with different frequencies and ERP. In Fig. 4 we show results for electron density influenced by MSP. As a comparison, we ran the model without the influence of MSP. We see in Fig. 4 that there is a reduction in electron density, an electron bite-out, due to the presence of MSP, predominantly between heights 80-100 km. There is an 185 electron bite-out between 70-80 km, but it is significantly smaller than between 80-100 km. Between 100-120 km, the electron bite-outs are not present. We see that electron bite-outs changes the height profile of the electron density. are reaction rates coefficients. For details on the individual reactions see Baumann et al. (2013) are present. The shape of the height profile varies as well, where the heated electron temperature decreases more slowly when 190 MSP are present so that the shape of the height profile is more flat. Without MSP, the electron temperature decreases faster and it has a different overall shape. Below 80 km, the heated electron temperature is the same with and without MSP. A comparison of the four different cases show similar results for the heated electron temperature in Fig. 5. The electron temperature is higher when MSP are present for all five different cases and the shape is also similar. In addition, for all five cases the heated electron temperature with and without MSP are the same up to 80 km. We also see that a higher ERP results in a higher electron 195 temperature, where T e reaches almost 3000 K for case 4 with ERP 1200 MW.  km is 700 K. For the lower ERP of 200 MW with frequency 4 MHZ (case 1) or ERP of 300 MW with frequencies 5.5 MHZ (case 2), the difference in T e is between 200-500 K at 95 km.
In Fig. 5 a small feature appears in some of the plots when the electron temperature is around 1500 K. The location of the feature appears at different altitudes, varying between 80-110 km. Out of the four different cases that we considered for 205 comparison (4 cases with MSP and 4 cases without MSP, so 8 all together), the feature appears in 4 out of 8 plots, 1 with MSP and 3 without MSP.

Day conditions
This section presents results for electron temperature modelled during artificial heating with and without the presence of MSP for day condition. Figure 7 shows electron density with and without MSP. As for night conditions the electron density 210 come from the ionospheric model. We see that the electron bite-outs are not present in Fig. 7. Figure 8 shows the heated electron temperature for case 1-4. We see that the heated electron temperature is the same with and without MSP. We find that the absolute difference between the electron temperature modelled with and without MSP is less than 25 K for all cases 1-4.

Discussion
Both Kero et al. (2007) and Senior et al. (2010) found that current theoretical models most likely overestimate artificial heating in the D-region compared to observations of the heating effect in the D-region. Why the theoretical models overestimates artificial heating in the D-region remains an open question. Kero et al. (2007) studied how artificial heating influences cosmic 220 radio noise absorption and found that the observed enhancement of cosmic radio noise absorption during heating is lower than predicted theoretically. Senior et al. (2010) used a cross-modulation technique with the EISCAT radar and found that the model overestimates the diagnostic wave absorption. In the study by Senior et al. (2010), the authors note that electron biteouts located at PMSE layer altitudes might influence the model, but that the influence is probably small, because the bite-outs are located too high in altitude. They investigate the influence of the electron bite-outs by scaling the whole electron density 225 profile with a factor of 2 or 0.5. However, they do not include the height variation of the electron density profile when electron bite-out are present. We find that electron bite-outs are only present at certain altitudes. The magnitude of the electron bite-outs varies within these altitude, for instance, the electron bite-out is significantly larger between 80-100 km compared to between 70-80 km. In our study, we have modelled the electron temperature during heating and included the height variation of the electron bite-outs. We have included the height variation of electron bite-outs by using the ionospheric model with MSP, which electrons, ions and neutral and charged MSP. This enables us to have a more realistic representation of the height variation of the electron bite-outs. In a future study we will make a detailed comparison of our results to observations of the electron temperature during heating. This detailed comparison can investigate if the presence of MSP can explain the discrepancy between model and observations. 235 Figure 5 for night conditions show that the electron temperature is higher and decreases more slowly when MSP are present.
An explanation for why the heated electron temperature decreases more slowly is that with electron bite-outs at certain altitudes, the heating above these heights will be increased since less of the wave energy is absorbed within the electron bite-outs. The absorption of wave energy depend on electron density and the absorption decreases with decreasing electron density. We see this effect in Fig. 9, which shows absorbed radio wave energy as a function of height. Here, less wave energy is absorbed when 240 MSP are present. More wave energy is absorbed at higher altitudes, slightly above where the electron bite-outs are largest in magnitude. The cooling rates also depend on electron density and decreases at higher altitudes due to a lower electron-neutral collision frequency since the neutral density is lower. The electron cooling -heating equality is reached at higher electron temperatures as more wave energy remains in the MSP case compared to the case without MSP. An electron bite-out at lower altitudes can lead to an increased electron temperature at higher altitudes above.

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The results of our study shows that the frequency of the transmitted radio wave only plays a minor role, lower frequency only slightly shifts the start of the heated ionosphere to a higher altitude. We also see that the increased electron temperature due to the presence of MSP extends up to 120 km in the E-region. Our model for the heated electron temperature might not be applicable for the E-region, however, this is beyond the scope of this paper. The results from this study agrees with Kassa et al. (2005), where an electron bite-out inserted as a linearly decreasing 'toy model' between 84-86 km during PMSE conditions 250 resulted in an increased modelled electron temperature within and above the electron bite-out. Figure 8 shows that day condition electron temperature is the same with and without MSP. This indicate that for day condition, MSP are less important for the heated electron temperature. A higher ionization level, and thus a much higher electrons density, means that loss of electrons, like electron attachment to MSP, is less important. Generally, the electron temperature is lower for day conditions. This is because the electron density is higher during the day, also at lower heights. The electron 255 density in Fig. 7 is 2.5·10 6 m −3 at 60 km for day conditions, while for night conditions the electron density in Fig. 4 is 2.6·10 4 m −3 at 60 km. With a higher electron density as during day conditions, the radio wave energy is absorbed already at lower heights.
In Fig. 5 there is a feature in some of the plots of the heated night condition electron temperature. This feature can resembles a small second maximum, or it might just be an artefact. It appeared when we included the temperature dependence of the 260 cooling rates for vibrational excitation of molecular nitrogen; the values from Pavlov (1998a) that we use are different for the temperatures 300 ≤ T e ≤ 1500 K and for those T e > 1500 K. The feature that we note is at electron temperature around 1500 K. The feature disappears if we apply the same values for vibrational excitation of molecular nitrogen over the entire range of temperatures and disregard the difference for the T e ≤ 1500 K case. Kero et al. (2008) found a second maximimum in the EISCAT incoherent scatter observations for the heated electron temperature in the D-region, which they could not explain.

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The feature in Fig. 5 might be a second maximum or it might be an artefact caused by problems in the numerical modelling when switching between values for T e ≤ 1500 K and T e > 1500 K. Whether the feature is an artefact or not is unknown at the present and can be investigated further. The feature is not seen in the day condition electron temperature in Fig. 8.

Conclusions
The  Code availability. A function that computes the electron temperature and radio wave intensity during artificial heating, which includes the electron cooling rates, will be made available.  Rietveld et al. (1986); Gustavsson et al. (2010)]. In addition, atomic oxygen [O] plays an important role through the impact excitation of fine structure levels of its ground state (see Pavlov and Berringston (1999) and references given there). We here repeat the cooling rates that are used. The sum of the electron cooling rates are the energy loss function, given as: The unit of L(T e ) are in Jm −3 s −1 .
To describe the excitation of fine structure levels of atomic oxygen, we use Eq. 15 from Pavlov and Berringston (1999): The units of equation A2 The S ij denote the transitions between the three fine structure levels of the atomic oxygen ground state.
For vibrational excitation of molecular nitrogen, we use Eq. 11 from Pavlov (1998a) for a Boltzmann distribution: where E 1 = 3353 K is the energy of first vibrational level of [N 2 ] and we assume that the vibrational temperature is equal to the neutral temperature. The units of L vib (N 2 ) is eVcm −3 s −1 . Here, Q 0v describes excitation transitions from ground states and Q 1v describes excitation transitions from the first vibrational state. For Q 0v and Q 1v , we implement Eq. 19 and Eq. 20 from Pavlov (1998a), respectively: log Q 0v = A 0v + B 0v T e + C 0v T e 2 + D 0v T e 3 + F 0v T e 4 − 16 (A8) where the coefficients A 0v , B 0v , C 0v , D 0v , F 0v to compute Q 0v and A 1v , B 1v , C 1v , D 1v , F 1v to compute Q 1v come from tables in Pavlov (1998a). For Q 0v from Table 1 for 300 ≤ T e ≤ 1500 K and from Table 2 for T e > 1500 K. For Q 1v from Table   3 for 1500 ≤ T e ≤ 6000 K. However, there is no table for Q 1v for T e < 1500 K. Both Q 0v and Q 1v have units eVcm 3 s −1 .

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For vibrational excitation of molecular oxygen we use Eq. 8 from Pavlov (1998b), which assume a Boltzmann distribution: in units eVcm −3 s −1 and where E 1 = 2239 K is the energy of the first vibrational level of [O 2 ] and we set T vib = T n . Here Q 0v describes excitation transitions from ground states. Q 0v come from Eq. 11 in Pavlov (1998b): where the coefficients A v , B v , C v , D v , F v , G v as a function of vibrational level come from Table 1 of Pavlov (1998b).  Figure A1. Night condition electron cooling rates with MSP as a function of height for a heated electron temperature. The frequency is 5.5 MHz and ERP is 600 MW. The legend show the different cooling rates as described in this section.

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L el (O 2 ) = N e [O 2 ]1.21 · 10 −18 √ T e (T e − T n )(1 + 3.6 · 10 −2 √ T e ) (A17) L el (O) = N e [O]7.9 · 10 −19 √ T e (T e − T n )(1 + 5.7 · 10 −4 T e ) In Fig. A1 and Fig. A2 we present height profiles for electron cooling rates for night conditions. We show electron cooling rates for a heated electron temperature. Figure A1 show cooling rates where MSP are present, while Fig. A2 shows cooling 335 rates where MSP are not present. The frequency is 5.5 MHz and ERP is 600 MW. Acknowledgements. We would like to thank Ove Havnes for providing us with a program from Meseret Kassa, which computes the electron temperature during heating. We have compared our results to the program from Meseret Kassa. In addition, we would like to thank Antti Kero for suggesting to look at how the presence of MSP can influence artificial heating. Lastly, we would like to thank Björn Gustavsson for helping us with the theory of artificial heating.