Numerical study of anomalous absorption of O mode waves on magnetic field‐aligned striations

Simple expressions that describe mode conversion and anomalous absorption of ordinary (O) mode waves injected at angles between the vertical and magnetic zenith to upper hybrid (UH) oscillations in the presence of field‐aligned density striations are presented. The absorption takes place in a region above the UH resonance layer where the striations allow trapped eigenmodes, leading to excitation of large‐amplitude UH waves. The derivation of the expressions is guided by dimensional analysis and numerical simulations. The results are relevant in interpreting high‐latitude heating experiments where anomalous absorption due to striations plays a crucial role.


Introduction
The interaction of HF radio waves with the ionospheric plasma in the vicinity of the upper hybrid (UH) resonance layer plays a critical role in the physics of ionospheric modifications and artificial ionospheric turbulence. It is believed that the UH interaction region gives rise to features such as the downshifted maximum and continuum in the stimulated electromagnetic emissions [Mjølhus, 1998;Leyser, 2001]. Early experiments at Platteville [Cohen and Whitehead, 1970], Tromsø [Stubbe et al., 1982;Jones et al., 1984;Stocker et al., 1993], and in Russia [Getmantsev et al., 1973] showed a drastic increase of absorption of the ordinary (O) mode pump and test waves after a few seconds if the wave frequencies were below the maximum UH frequency of the ionosphere. It has been observed that when the electromagnetic beam is directed along the magnetic zenith (MZ), parallel to the magnetic field lines [Gurevich et al., 2002], or between the MZ and the Spitze angle [Rietveld et al., 2003;Honary et al., 2011], there is an increase of the electron temperature in the heated region. The observations are consistent with the formation of smallscale magnetic field-aligned striations, followed by mode conversion of the O mode pump and test waves to UH waves trapped in the striations [Graham and Fejer, 1976;Vas'kov and Gurevich, 1976;Dysthe et al., 1982;Jones et al., 1984;Mjølhus, 1985;Gurevich et al., 1995Gurevich et al., , 1996. The UH oscillations are then responsible for heating the electrons and increasing further the amplitude of the striations [Lee and Fejer, 1978;Dysthe et al., 1982;Inhester, 1982;Gurevich et al., 1995]. Striations measured in situ by a sounding rocket flown through the heater beam at Arecibo [Kelley et al., 1995] had a typical width of 7 m, a mean depletion depth of about 6%, and a separation of about 15 m.
The aim of this paper is to numerically study the efficiency of mode conversion of an O mode to UH waves on field-aligned striations and its dependence on ionospheric plasma parameters. The study concentrates on cases where the resonant UH interaction and absorption processes take place in a separate region below the turning point of the O mode wave. The results have implications on the electron heating at the UH layer and on the fraction of the electromagnetic wave energy that reaches the critical layer of the O mode where it can excite Langmuir turbulence.
ionosphere, its frequency ω and horizontal wave vector component k x are conserved. Since the wave is injected from a ground-based transmitter in free space, we have At the altitude where the wave propagates parallel to the magnetic field, we also have is the local magnitude of the wave vector. At this altitude, the O mode propagates as a left-hand polarized (L) mode wave, obeying the dispersion relation is the electron plasma frequency and ω ce = eB 0 /m e the electron cyclotron frequency. Here n 0 is the local electron number density, e is the magnitude of the electron charge, m e is the electron mass, and ϵ 0 is the electric permittivity in vacuum. Eliminating k and k x from equations (1)-(3) gives For example, using ω = ω pe and eliminating ω pe in equation (4), we find the Spitze angle (5) where the O mode is efficiently converted to a Z mode wave as it reaches the critical layer [e.g., Mjølhus, 1990]. In equation (5) and eliminating ω pe in equation (4) gives the optimal incidence angle that results in O mode propagation parallel to the magnetic field when it reaches the UH layer. For typical parameters at High Frequency Active Auroral Research Program (HAARP), θ = 14.5°, ω ce = 9.09 × 10 6 s À 1 , and ω = 20.1 × 10 6 s À 1 we find χ S = 8.0°and χ UH = 9.6°. Hence, χ UH is between the Spitze and MZ. This is consistent with the maximum electron heating observed at an incidence angle between the Spitze and MZ [e.g., Honary et al., 2011].

Simulation Setup
We next consider a two-dimensional simulation geometry in the x-z plane with the magnetic field directed along the negative z axis. For the auroral zone geometry, where the geomagnetic field is directed almost vertically downward, O mode propagation along the z axis is justified up to a few wavelengths away from the critical layer of the O mode. The ambient ionospheric (ion and electron) number density profile is where n 0,max is the number density at the F 2 peak located at the altitude z = z max , and L n0 is the ionospheric length scale. The total plasma density, including a striation superimposed on the ambient density at x = 0, is taken as n x; z ð Þ ¼ n 0 z ð Þ 1 À δñ str exp Àx 2 =D 2 str À Á Â Ã , where δñ str is the relative density depletion and D str is the transverse size of the striation. Figure 1 shows a density striation with D str = 2 m, δñ str = 0.1, n 0,max = 1.436 × 10 11 m À 3 (corresponding to the peak plasma frequency f oF2 ¼ 3:40 MHz), z max = 242 km, and L n0 = 31.62 km. For periodic boundary conditions along the x direction, Figure 1 represents a grid of striations separated by the transverse simulation box size L x = 12 m. The used pump frequency ω = 20.1 × 10 6 s À 1 (f 0 = 3.20 MHz) corresponds to a vacuum wavelength λ 0 = c/f 0 = 93.75 m. For the chosen parameters, the critical altitude where the pump frequency equals the ambient plasma frequency is at z O = 230.96 km (the dotted line in Figure 1). The ambient magnetic field B 0 = 5.17 × 10 À 5 T corresponds to electron cyclotron frequency ω ce = 9.09 × 10 6 s À 1 (1.45 MHz). The UH resonance where the pump frequency equals the UH frequency in the unperturbed ionosphere is at z UH = 223.27 km (the dashed line in Figure 1). The critical surface ω = ω UH is Geophysical Research Letters 10.1002/2015GL063751 indicated with a solid line in Figure 1 is the UH resonance frequency and is the local electron plasma frequency. In Figure 1 the upper boundary of the UH interaction layer is indicated with a dash-dotted line. Notice that the O mode will propagate up to or near the critical altitude for angles smaller than or close to the Spitze angle.
In order to have well-defined, transmitted, and absorbed waves through the UH region this paper focuses on cases where the UH interaction region is below the critical altitude of the O mode, i.e., the dotted line should be at a higher altitude than the dash-dotted line in Figure 1. The separation between the altitudes of the critical and UH layers is approximately z O À z UH ≈ Y 2 L UH , where the local length scale is L UH = 1/|d ln (n 0 )/dz| at z = z UH , and the separation between the bottom and top of the UH interaction region is Δz UH ≈ δñ str L UH . Hence, the condition for the separation of the UH interaction region from the O mode critical layer is For example, for a relative striation density depletion of δñ str = 0.1, the condition is fulfilled for Y > 0.32. For the typical electron cyclotron frequency of 1.45 MHz at HAARP and relative striation depth δñ str = 0.1, the UH interaction region is separated from the O mode critical layer only for transmission frequencies below 4.5 MHz If Y 2 < δñ str , the critical altitude of the O mode wave will be located within the UH interaction region, making the interpretation of the absorption process more complicated.
On the fast UH and O mode timescale, we assume that the ions are immobile and contribute only to a neutralizing background of the plasma. For the high-frequency electron dynamics and the electromagnetic field, we solve the dynamical equations for the slowly varying complex envelopes, with the substitution ∂/∂t → ∂/∂t À iω. There is a clear separation of length scales in the problem, where the electromagnetic waves have wavelengths of the order 100 m, while the UH waves have typical wavelengths of less than a meter. This poses a computational problem since the numerical scheme must resolve both length scales, while the time step is limited by the Courant condition to the smallest grid size, which in our case is in the x direction. To resolve this problem, we follow the techniques presented in

Geophysical Research Letters
10.1002/2015GL063751 Eliasson [2013] and separate the electric field into an electrostatic, curl-free part E ES = À ∇ϕ, where the scalar potential is obtained from Poisson's equation and an electromagnetic, divergence-free part E EM defined by where A is the vector potential defined by its relation to the magnetic field B = ∇ × A and the Coulomb gauge ∇ Á A = 0. The electromagnetic electric field is obtained from The overbar denotes spatial averaging in the x direction, so that the electric current and the electromagnetic fields E EM and A depend only on the vertical coordinate z and on time, but not on the transverse coordinate x. This allows us to take about 50-100 times longer time steps, since the Courant condition now limits the time step by Δt ≲ Δz/c instead of the much more restrictive Δt ≲ Δx/c.
The electron dynamics is governed by the continuity and momentum equations, respectively, and where is the electron thermal speed, T e is the electron temperature, k B is Boltzmann's constant, and κ = 1/(1 À 4Y 2 ) accounts for a kinetic dispersive effect for UH branch of the electron Bernstein waves that becomes important for low pump frequencies within a few electron cyclotron harmonics [e.g., Lominadze, 1981]. A pseudo-spectral method is used to calculate the spatial derivatives accurately, with typical grid sizes Δx = 4 cm in the transverse direction and Δz = 17 m in the longitudinal direction, and a standard fourth-order Runge-Kutta scheme is used to advance the solution in time, with a time step of Δt = 10 À 8 s. The size of the simulation box is 13 km in the z direction, and different widths are used in the x direction to match the widths of the striations. To resonantly drive an upward propagating (opposite to the direction of B 0 ) L mode wave below the UH interaction region, a right-hand circularly polarized external field E ext ¼x þ iŷ ð Þ exp ik ð Þexp À z À z 0 ð Þ 2 =D 2 ext h i is used, centered at z 0 near the bottom of the simulation box and with a vertical width D ext = 250 m. The wave number k, given by equation (3), matches that of the propagating L mode wave. A damping layer for the electromagnetic wave is introduced near the top of the simulation box to absorb O mode waves that have propagated through the UH interaction region. The external field is switched on at the start of the simulation t = 0.

Numerical Results
Key physics characteristics of the O to UH mode conversion are illustrated in Figures 2-4 at time t = 0.1 ms into the simulations. The O mode wave injected from the bottom side is continuously converted to UH waves, and the UH wave amplitude gradually increases with time. The fraction of O mode wave energy converted to UH waves quickly reaches a steady state, and for times larger than 0.1 ms, the amplitude profiles of the electromagnetic wave undergo only very slight changes. To study the absorption of the O mode wave, it is therefore sufficient to run the simulations up to t = 0.1 ms and then record the results. heights where the wave frequency is equal to quantized eigenfrequencies of trapped UH waves, as discussed by Mjølhus [1998]. (ii) The absorption layer Δz UH starts at the altitude where ω = ω UH outside the striation, but its extent is bracketed by the altitude where locally ω = ω UH at the center of the striation. (iii) As the striation width increases, the absorption width increases occupying a larger fraction of Δz UH , the number of layers where the UH waves are resonantly absorbed increases, and the resonant layers move closer to each other. The total number of UH resonances (both odd and even) in a striation is roughly given by [Mjølhus, 1998] but due to symmetry, only even (M/2) resonances are excited. Here λ De = v Te /ω pe is the electron Debye length. In our case we have δn=n 0 ¼ δñ str exp Àx 2 =D 2 str À Á , and the integral can be evaluated as For the parameters used in Figure 2, we have λ De ≈ 0.014 m, δñ str = 0.1, and κ = 5.5. For the cases D str = 1 m, 2 m, and 4 m, the number of even resonances are M/2 ≈ 2.3, 4.5, and 9, respectively, which is consistent with Figure 2, where respectively 2, 4, and 8 resonances are visible. (iv) Most importantly, in all three cases, the relative absorption, as seen by the value of E y at Figure 2 (bottom) remains the same. The y component of the electric field decreases about 50% in amplitude above the UH layer, implying that about 75% of the wave energy is absorbed and converted to UH waves.
When instead changing the electron temperature (cf. Figure 3), the wavelength of the UH wave (Figure 3, top) increases with increasing temperature; for lower temperatures the number of resonant layers increases and the resonant layers move closer to each other. Similar to Figure 2, the absorption layer is constrained to Δz UH starting where ω = ω UH outside the striation up to an altitude below where, locally, ω = ω UH at the center of the striation. Most importantly, the relative absorption remains the same for different temperatures, as seen by the value of E y at Figure 3 (bottom). In contrast, we see in Figure 4 that an increase of δñ str from 0.05 to 0.15 leads to a reduction of the O mode amplitude above the UH interaction region by almost an order of magnitude. Hence, the striation depth is a crucial parameter for the anomalous absorption of the O mode wave. For deeper striations, the interaction region Δz UH increases, leading to a larger number of resonant layers, while the distance between the layers depend only weakly on δñ str .

Scaling Analysis
From Figures 2 and 3, we conclude that the absorption of O mode waves to UH waves on striations does not significantly depend on D str and T e while keeping other parameters constant. By dimensional analysis of the governing equations, one finds that the physics of the system depends only on four local dimensionless parameters: η, δñ str , Δz UH /λ 0 . (Δz UH is the vertical width of the UH interaction region, between the dashdotted and dashed lines in Figure 1), and Y = ω ce /ω 0 . Note that Δz UH also depends on D str , Y, and parameters defining the ionospheric profile. By performing a set of simulations similar to the ones in Figures 2-4 for different combinations of the parameters, we have found a simple expression for the transmission coefficient (transmitted intensity above the striation divided by incident intensity below the striation) estimated as the squared amplitude ratio of the transmitted electric field above the UH interaction region E yT and E yT0 with and without striation, respectively. The formula is valid for typical ionospheric parameters 0.2 < Y < 0.46, η < 0.5, and δñ str ≲ Y 2 (cf. equation (7)).
A comparison between numerically obtained values of the transmission coefficient and the expression (15) is shown in Figure 5, with good agreement. As seen in Figure 5a, the transmission coefficient in general decreases with increasing packing factor η, since there are more striations per given area to absorb wave energy. However, for large values η > 0.25, the absorption becomes less efficient and the curves in Figure 5a have a minimum at η ≈ 0.35. This is reflected in the term proportional to η 2 in the exponential in   (15) and is due to close-packing effects which changes the average dielectric properties of the plasma and reduces the effectiveness of absorption for densely packed striations. On the other hand, we see in Figures 4, 5a, and 5b that an increase of δñ str from 0.05 to 0.1-0.15 can decrease the transmitted electric field by more than a factor 10 and the corresponding transmission coefficient by a few orders of magnitude. An increase of δñ str increases both the depth of the striation locally and the vertical width of the UH interaction region (cf. Figure 4). To first order, Δz UH ≈ δñ str L UH , and hence, the expression in the exponential of (15) is approximately proportional to δñ 2 str . Therefore, most of the injected wave energy will be absorbed when the striations have grown to a significant amplitude. Finally, the results in Figure 5c, indicating a decrease of T with Δz UH , were obtained by keeping δñ str , Y, and η constant, while halving and doubling the ionospheric vertical length scale L n0 , so as to decrease and increase Δz UH proportionally.
The numerical results are in qualitative agreement with the theoretical expression for the transmission coefficient for small amplitude striations, corresponding to δñ str L UH /D str ≪ 1, given in equation (2a) of Mjølhus [1985], which for parallel propagation of the O mode along the striations can be written as where the angle brackets denote spatial averaging and δn is the density perturbation due to the striations. In our case, δn=n 0 ¼ Àδñ str exp Àx 2 =D 2 str À Á þ δñ str η=2 ð Þ ffiffiffi π p , where we have subtracted the mean density and a spatial average over the simulation box gives , which also recovers the close-packing effect governed by the η 2 term in equation (15); the coefficient ffiffiffiffiffiffiffi ffi π=2 p ≈1:25 in front of η 2 is comparable to the coefficient 1.4 obtained numerically in equation (15). The decrease of the transmission coefficient for the relative decrease of the magnetic field Y seen in Figure 5b is also consistent with equation (16). In general, T + A + R = 1, where A and R are the absorption and reflection coefficients, respectively [e.g., Mjølhus, 1985]. Mjølhus [1985] also predicted a significant reflection of O mode waves by the UH layer for wave propagating into denser plasma, which would limit the absorption coefficient to A < 0.5. A reflection by the UH interaction region would give rise to a standing wave pattern below the UH layer, which, however, is not seen in Figures 2-4. Hence, the numerical results indicate that R ≈ 0 and A = 1 À T. The numerical results are consistent with experiment [e.g., Cohen and Whitehead, 1970;Stubbe et al., 1982], where the amplitude of reflected O mode waves on the ground drops about 10-15 dB after the striations have had time to develop. If there is no direct reflection by the UH layer, the O mode must travel through the UH layer twice: first, as an upgoing wave into denser plasma to the turning point of the O mode, and then as a downgoing wave propagating out from the plasma. If the transmission coefficient is the same for upgoing and downgoing waves, the total transmission coefficient is T tot ≈ T 2 where T is given by equation (15). As is evident from Figure 5, an injected wave could then be absorbed and decreased in power several tens of decibels before returning back to ground. data supporting the figures can be requested from Bengt Eliasson, e-mail: bengt.eliasson@strath.ac.uk.
The Editor thanks Thomas Leyser and an anonymous reviewer for their assistance in evaluating this manuscript.