Powerful electromagnetic high-frequency (HF) waves transmitted into the ionosphere excite geomagnetic field-aligned plasma density striations. The structuring in the plasma into striations absorbs substantial power from the injected pump wave when it has ordinary (O)-mode polarization and its frequency is below the maximum upper hybrid (UH) frequency of the ionosphere . Typical transverse (to the magnetic field) sizes of small-scale striations are a few meters up to tens of meters, while their parallel sizes are tens of kilometers due to the strongly anisotropic mobility of the electrons in the magnetic field . The striations are typically associated with local density depletions of the order of 5–10 %, and are observed to be separated by a few tens of meters.

In in situ measurements at the Arecibo Observatory it was found that the spatial structure of the plasma density at large scales across the magnetic field is due to organization of the small-scale striations into bunches of a few hundred meters across . At high latitudes, optical emissions show self-organization into filaments of a few kilometers across during pumping in at magnetic zenith .

The small-scale striations result from pump-driven UH waves, which as the thermal instability develops, become self-localized in the density depletions of the striations . This occurs at altitudes where the pump frequency is below the UH frequency outside the plasma depletion but above the local UH frequency in a region inside the depletion. Large-amplitude UH waves are excited at altitudes where the resonance frequency of the trapped UH waves equals the transmitted frequency . However, it is theoretically predicted that the localized UH oscillations are not perfectly trapped in the depletions but radiate electromagnetic waves in the Z-mode branch that escape from the striations .

The electromagnetic interaction of several striations was first studied in the Wentzel–Kramers–Brillouin (WKB) approximation for the localized UH field, which showed that the coupling depends crucially on the phasing of the mean Z-mode wave between the striations . One-dimensional theory predicts that the Z-mode leakage from a single striation is so strong that a striation cannot be excited with the power of presently available pump transmitters . However, it is proposed that in a system of parallel striations there would be partial influx of Z-mode waves to a given depletion from neighboring striations, to partly compensate for the Z-mode leakage and thus enable excitation of striations with the available pump power. It has been proposed that for a system of about 40 striations, the total Z-mode leakage and the damping by collisions are of the same order.

Using a scale separation technique, the amplitude of the Z-mode wave has been obtained analytically in terms of eigenfunctions of the localized UH field in a single density depletion in one dimension . Several scattering processes are included. The electromagnetic ordinary-mode pump wave scatters off the density depletion into localized UH oscillations. For a symmetric striation in one dimension, the pump wave only excites UH modes with the wave electric field being an even function in space, while odd modes are not excited . The UH oscillations in turn scatter off the cavity into Z-mode waves, which also can scatter off the depletion into localized UH oscillations. The theory also includes the scattering of the pump wave directly into Z-mode waves on the density depletion. For a system of density cavities , the scattering of the pump wave directly into Z-mode waves leads to the excitation of odd UH modes for an asymmetric distribution of one-dimensional cavities, although the original pump field and the depletions are symmetric. The system of striations is thus electromagnetically coupled by the radiation of Z-mode waves out from the striations and the influx from neighboring striations and the strongly inhomogeneous UH turbulence is embedded in a sea of Z-mode waves.

However, in ionospheric radio wave experiments the transverse profiles of the striations are believed to be two-dimensional and axisymmetric rather than one-dimensional. As this case corresponds to odd UH resonances in one dimension, the analytic one-dimensional results have been used for a first two-dimensional treatment. It is predicted that the Z-mode leakage is significantly weaker for a two-dimensional distribution of parallel striations compared to the one-dimensional case .

This is because the odd UH modes in two-dimensional cavities correspond to dipole distributions of the localized fields while the even modes in the one-dimensional case correspond to monopole field distributions. The weaker dipole radiation of the UH field compared to the monopole radiation of even modes results in weaker Z-mode leakage and therefore larger UH amplitude in the two-dimensional case.

The aim of this paper is to investigate numerically the linear-mode conversion of an O-mode wave to UH waves on clusters of two-dimensional striations, and how the leakage of Z-mode waves affects the amplitude of the trapped UH waves. Both large- and small-amplitude striations are considered, and the influence of the size of striation clusters on the amplitude of the trapped UH waves is investigated. Section 2 describes the mathematical model used in the numerical work. The properties of trapped electrostatic UH oscillations in a single striation are discussed in Sect. 3, while the coupling to Z-mode waves in clusters of striations are numerically investigated in Sect. 4. Finally, the results are discussed and conclusions are drawn in Sect. 5.

An O-mode polarized continuous wave injected into the overhead ionosphere along the magnetic field lines will excite UH resonances at quantized heights , where the transmitted frequency matches one of the local resonances of the UH waves trapped in the striations. This leads to the excitation of large-amplitude UH waves and to anomalous absorption of the O-mode wave. When an O-mode wave propagates parallel to the ambient magnetic field, it is in the form of a left-hand circularly polarized (L)-mode wave with its electric field directed perpendicularly to the ambient magnetic field. For simplicity we do not take into account the effects of the vertical stratification of the plasma on the propagation of the L-mode or on the interaction with the striations. The L-mode wave is instead represented by an externally imposed left-hand polarized dipole electric field ẼL=ẼL(x^-iy^), where x^ and y^ are unit vectors in the x and y directions. We use a two-dimensional simulation geometry in the x–y plane, transverse to the ambient magnetic field B0=z^B0, where z^ is the unit vector along the z axis. The simulation box size is 200 m in both the x and y directions for simulations using 1 striation, while somewhat larger box sizes of 250 m and 300 m, respectively, are used for clusters of 7 and 19 striations (see Fig.  below). A pseudo-spectral method is used to calculate derivatives in space accurately, and a fourth-order Runge–Kutta scheme is used to advance the solution in time.

Superimposed on the background electron and ion density are magnetic field-aligned small-scale striations that are associated with localized density depletions at a fraction of the background density. The total ion density is of the form n(r)=n0+ns(r), where ns(r)=-αn0∑jexp⁡[-(x-xj)2Dstr2-(y-yj)2Dstr2] describes the ion density profiles of the striations, (xj,yj) is the central position of each striation, α is the relative amplitude, n0 is the background ion number density, and Dstr is the transverse size of the striations.

The physics involves very disparate length scales. The typical transverse size Dstr is a few meters, while the electromagnetic wave with a frequency of a few MHz has a local wavelength of a few tens of meters, and UH waves trapped in the striations can have wavelengths of about a meter or less. For the electromagnetic model, the different length scales pose a challenge on the numerical scheme, which has to resolve both the large and small scales, while using a sufficiently short time step Δt to maintain stability due to the Courant condition determined by the smallest scale, Δt≲Δx/c=Δy/c, where Δx and Δy are the grid sizes in the x and y directions, and c is the speed of light in vacuum. The wave number, and hence the wavelength can be estimated by using the UH dispersion relation. At the center of the large-amplitude striations used in the numerical work, the electron density is 90 % of the ambient density, and therefore at the bottom of the striation; where the upper hybrid waves have the shortest wavelength, we can use the dispersion relation ω2=0.9ωpe2+ωce2+3vTe2κk2 for the wave frequency ω, the electron plasma frequency ωpe, electron cyclotron frequency ωce, electron thermal speed vTe, wave number k and a kinetic correction coefficient κ (defined below). Using, for example, the frequency equal to the ambient UH frequency outside the striation, ω2=ωpe2+ωce2, and eliminating ω, we obtain 0.1ωpe2=3vTe2κk2. For the simulation parameters ωpe=20.12×106, vTe=2.46×105 m s-1, and κ≈2.5, we solve for the wave number to obtain k≈9.4m-1, corresponding to an upper hybrid wavelength of 0.67 m. In order to resolve the UH waves with more than two grid points per wavelength, the grid sizes are set to Δx=Δy= 0.2 m in all simulations except in Figs. 4 and 7, where Δx=Δy= 0.4 m were used since the wave frequency was below the UH frequency and the wavelengths were longer. The small grid-size puts a limit on the time step. To relax the Courant condition, we here follow the strategy outlined by and use a coarser resolution for the electromagnetic wave. In doing so, the electric field E=EES+EEM is divided into one curl-free part EES=-∇ϕ primarily associated with electrostatic waves, and one divergence free part EEM=-∂A/∂t associated with electromagnetic waves, where ϕ and A are the scalar and vector potentials, respectively, and using the Coulomb gauge ∇⋅A=0. High Fourier components of the electromagnetic field E⟂ and A, corresponding to the wave vector components kx and ky having magnitudes larger than a maximum wave number kmax, are set to zero. This corresponds effectively to representing the solution on a coarser grid Δx‾=Δy‾=π/kmax. In order to resolve the electromagnetic wave, the value of kmax has to be larger than the typical wave number of the electromagnetic wave. We choose kmax≈0.8 m-1, while the shortest wavelength of the Z-mode in the simulations is 15 m, corresponding to a wave number of 0.4m-1, and hence the electromagnetic wave is well resolved. The effective coarser grid size is Δx‾=Δy‾=π/kmax≈4 m, and we can use about time steps that are 10 times longer, making the simulations feasible on a standard single processor workstation. The time step used in the simulations using the electromagnetic models is Δt=5×10-9s, while for simulations using an electrostatic model, the time step is essentially limited by the inverse of the upper hybrid frequency and is taken to be Δt=5×10-8s.

The HF component of the electric field is assumed to take the form E=(1/2)(Ẽ(z,t)exp⁡(-iω0t)+Ẽ*(z,t)exp⁡(iω0t)), where Ẽ represents the slowly varying complex envelope of the HF field, and ω0 is the transmitted frequency of the L-mode wave, and the asterisk denotes complex conjugation. Similar assumptions are made for the HF magnetic field, scalar potential, and the electron density and velocity fluctuations, which are linearly coupled to the HF electric field. Hence, the time derivatives on the fast timescale are transformed as ∂/∂t→∂/∂t-iω0 in the governing equations for the envelopes of the HF fields. We assume that the HF current is carried by the electrons, while the ions are stationary and contribute only to the neutralizing background and to the density profiles of the striations. The complex-valued envelopes of the electromagnetic fields are then obtained from the linearized evolution equations ∂Ã∂t=iω0Ã-ẼEM and ∂ẼEM∂t=iω0ẼEM-c2∇2Ã-eε0∇-2∇×[∇×(nṽe)], where e is the magnitude of the electron charge and ε0 is the electric vacuum permittivity. Here, ∇-2 denotes the inverse of the Laplacian operator, which is efficiently calculated in Fourier space using a pseudospectral method. The envelope of the electrostatic field is ẼES=-∇ϕ̃, where the scalar potential ϕ̃ is obtained from Poisson's equation, ∇2ϕ̃=eε0ñe. The HF electron dynamics is governed by the electron continuity and momentum equation ∂ñe∂t=iω0ñe-∇⋅(nṽe) and ∂ṽe∂t=iω0ṽe-emeẼ+ẼL+ṽe×B0-3vTe2nκ∇ñe-νeṽe, respectively, where Ẽ=ẼES+ẼEM is the self-consistent electric field, me is the electron mass, νe is the effective electron collision frequency due to collisions with neutrals and ions, vTe=(kBTe/me)1/2 is the electron thermal speed, Te is the electron temperature, and kB is Boltzmann's constant. The coefficient κ=ω02/(ω02-4ωce2), where ωce=eB0/me is the electron cyclotron frequency, is a dispersive effect derived from kinetic theory , in which the UH wave is one of many electron Bernstein modes. For ω0<2ωce, the UH wave changes topology and becomes a backward wave as part of the first electron Bernstein mode, and in this case the UH waves are not trapped in density depletions. We use ω0/ωce≈2.5 in the numerical treatment below. To absorb the escaping Z-mode radiation, an absorbing layer is introduced near the boundaries; see Appendix A for details.

In the electrostatic limit, the system behaves as a Sturm–Liouville problem which allows a set of un-driven standing UH waves trapped in the striations with resonance frequencies below the ambient UH frequency. To investigate these resonances, we employ Eqs. ()–() in the electrostatic (ẼEM=0 and Ã=0) and collisionless (νe=0) limits with no driving field (ẼL=0). We assume the solution to be proportional to exp⁡(iδωt), so that ∂/∂t→-iδω where δω is a frequency shift, and denote the total frequency ω=ω0+δω. Eliminating ñe and ṽe from Eqs. ()–() gives 3vTe2κ∇2ϕ̃+ω2[(1-Y2-Xnn0)ϕ̃-ψ̃]=0, where X=ωpe2/ω2, Y=ωce/ω, and ψ̃ is defined via ∇2ψ̃+X[∇⋅(∇nn0ϕ̃)+iYz^⋅(∇nn0×∇ϕ̃)]=0.

In the ambient plasma where n=n0, we have in the long wavelength limit ∇=0 so that 1-Y2-X=0, or ω2=ωUH2, where ωUH=(ωpe2+ωce2)1/2 is the UH resonance frequency and ωpe=(n0e2/(ϵ0me))1/2 is the electron plasma frequency. UH waves with ω<ωUH can be trapped in striations where n<n0 locally in space (provided ω>2ωce). In this case ω works as an eigenvalue for the set of eigenfunctions ϕ̃ and ψ̃.

We next restrict the investigation to one cylindrically symmetric striation centered at x=y=0. Introducing cylindrical coordinates x=rcos⁡θ and y=rsin⁡θ, we assume that the potential is of the form ϕ̃(r,θ)=Φ̃(r)exp⁡(iNθ) and ψ̃(r,θ)=Ψ̃(r)exp⁡(iNθ), where N=0,±1,±2,… are azimuthal-mode numbers, and the background plasma density n(r) depends only on the radial coordinate r. Inserted into Eqs. () and (), this leads to the system 3vTe2κr2[r∂∂r(r∂Φ̃∂r)-N2Φ̃]+ω2[(1-Y2-Xnn0)Φ̃-Ψ̃]=0 and r∂∂r(r∂Ψ̃∂r)-N2Ψ̃+X(r∂∂r-YN)(rn0∂n∂rΦ̃)=0. The term proportional to YN in Eq. () shows that the symmetry is broken between positive and negative azimuthal-mode numbers in the presence of an ambient magnetic field. Appropriate boundary conditions for N≠0 are Φ̃=0 and Ψ̃=0 at r=∞ and at r=0. For N=0, the boundary condition for Φ at r=0 can be taken ∂Φ̃/∂r=0, while Eq. () reduces to ∂Ψ̃∂r+Xn0∂n∂rΦ̃=0, and it is only possible to impose the boundary condition Ψ̃=0 at r=∞, giving Ψ̃=∫r∞Xn0∂n∂rΦ̃dr, which is used to eliminate Ψ̃ in Eq. ().

To the lowest order, neglecting terms containing derivatives of n (hence setting Ψ̃=0), we obtain from Eq. () the time-independent, cylindrical Schrödinger equation 3vTe2κr2[r∂∂r(r∂Φ̃∂r)-N2Φ̃]+ω2(1-Y2-Xnn0)Φ̃=0, which is a simplified model for trapped UH waves in a cylindrically symmetric striation.

The solution of the eigenvalue problem provides a set of eigenfrequencies ω=ω1,ω2,…, and corresponding trapped waves for ωj<ωUH. By choosing the pump frequency equal to one of the resonances, ω0=ωj, the respective UH mode is pumped resonantly. First, simulations of Eqs. ()–() are carried out in the electrostatic limit (EEM=0 and A=0) and are compared with solutions of the time-independent systems Eqs. ()–() and (), and the results are presented in Figs.  and . Details of the numerical methods used to solve the time-independent equations are given in Appendix B. We consider one case of a deep striation with a relatively deep striation with α=0.1 (Fig. 1) and one case of a shallow striation with α=0.01 (Fig. 2). In both cases, the striation has the transverse size Dstr= 2 m. We use the ambient plasma parameters Te=4000K, B0=4.8×10-5T and n0=1.272×1011m-3, giving vTe=2.46×105 m s-1, ωce=8.44×106s-1, ωpe=20.12×106s-1, and ωUH=21.82×106s-1. We use κ=2.67 in all cases. To excite UH oscillations in the simulations, a short driving pulse of the form ẼL(t)=EL0sin⁡(πt/2×10-5) for 0≤t≤2×10-5s and ẼL(t)=0 for t>2×10-5s with frequency ω0=21.35×106s-1 and the reference amplitude EL0=1V/m is used to excite UH oscillations at the beginning of the simulations.

Figure a shows the spectrum of trapped electrostatic oscillations in a single deep striation with α=0.1. Here, the plasma density at the center of the striation is 10 % lower than the ambient density, leading to the local plasma and UH frequencies ωpe=19.09×106s-1 and ωUH=20.87×106s-1, respectively. Hence, the frequencies of the trapped UH waves are clamped between 20.87×106s-1 and 21.82×106s-1, indicated by vertical dash-dotted and dashed lines in Fig. a–c. The total simulation time is 5ms, which gives reasonable frequency resolution of the wave spectrum. The wave energy is concentrated at frequencies correlated with the eigenfrequencies (resonances) of the system (Eqs. –) shown in Fig. b. For comparison, the spectrum for the Schrödinger equation (Eq. ) is shown in Fig. c. The spectral peaks in Fig. a are closely aligned with the resonances corresponding to N=-1 of the system (Eqs. )–() in Fig. b, as indicated by vertical lines. The spatial profiles of the radial eigenmodes for the azimuthal-mode number N=-1 are shown in Fig. d–i. Eigenmodes with higher radial-mode numbers R correspond to higher eigenfrequencies (but below the ambient UH frequency), they have larger number of extrema, their largest amplitude is at the outer edge of the striation, and they are less localized in space since their frequencies are closer to ωUH. In a similar manner as for one-dimensional striations , the number of resonances in two-dimensional striations can be roughly estimated (using the Schrödinger equation) as M=112πκλDe2n0∫∫(-ns)dxdy=αDstr212κλDe2. For the used plasma parameters α=0.1, Dstr=2m, κ=2.67, and λDe=1.22×10-2m, we have M=84, to be compared with the 82 and 81 resonances found in Fig. b and c, respectively. Hence, the number of trapped eigenmodes is relatively large and forms more or less a continuum of waves in the deep striation.

In contrast, the shallow striation has only a few resonances, as seen in Fig. b and c, and only two peaks with ω<ωUH are visible in the frequency spectrum in Fig. a. For the shallow striation, the local plasma and UH frequencies at the center of the striation are ωpe=20.02×106s-1 and ωUH=21.73×106s-1, respectively. Hence, for this case the frequencies of the trapped UH waves are between 21.73×106s-1 and 21.82×106s-1, indicated by vertical dash-dotted and dashed lines in Fig. a–c. Visible in Fig. a are two discrete peaks for ω<ωUH, one of which is clearly correlated with the resonances for N=-1 in Fig. b, while a continuum of frequency components corresponding to un-trapped waves is visible in Fig. a for ω>ωUH. The radial profiles of the trapped modes in Fig. d–h are relatively extended in space since the resonance frequencies are close to ωUH.

We next carry out a set of simulations using a continuous wave (CW) driving L-mode wave in clusters of striations to investigate the mode conversion of the L-mode wave to UH waves, and the coupling to Z-mode waves. Figure  shows the background ion number density for cases with 1, 7 and 19 striations with transverse size Dstr= 2 m, where groups of striations are organized in hexagonal patterns. The chosen central distance ≈25 m is consistent with the rocket experiment at Arecibo 1992, where a rocket was flown through the heated region . mention a mean spacing between the filaments across the magnetic field along the path of the rocket to be s=45 m. This is roughly supported by the mean distance between filaments seen in their Fig. 1 and the one-dimensional spectrum in their Fig. 2 which has a spectral break at k≈0.15m-1 corresponding to a perturbation wavelength of about 40 m. The mean width of the striations at half maximum was measured to be w=15 m. Then the number of striations per unit area can be estimated to be ns=1/(ws), and the mean distance between striations in the plane perpendicular to the magnetic field to be d=1/ns=ws≈26 m.

Figures  and  show the results of simulations using 1 deep striation with α=0.1. The reference amplitude of the L mode is ẼL=EL0=1V/m, and the driving frequency ω0=21.127×106s-1, corresponding to the resonance frequency for N=-1 and R=1 in Fig. b. Simulations are carried out using the fully electromagnetic model and an electrostatic model (setting à and Ẽ⟂ to zero), and for cases without collisions (νe=0) and with collisions (νe=103s-1). For the collisionless case (top panels in Fig. ), the electrostatic field shown in Fig. b increases linearly with time to almost 103EL0 at t=5ms. The simulations including collisions (bottom panels) show a saturation amplitude of the electrostatic field at about 300EL0 (Fig. d). Figure  shows Z-mode waves escaping the striation and propagating to the simulation boundaries where they are absorbed. By using the cold plasma dispersion relation for X-mode waves, c2k2=(ω02-ωpe2)2-ωce2ω02ω02-ωUH2, the wavelength λ=2π/k of the escaping Z-mode wave is estimated to be λ≈59.23 m. However, as seen in Fig. b and d, there is only a slight difference in the electrostatic wave amplitude between the electrostatic and fully electromagnetic simulations. Hence, for the deep striation, the Z-mode leakage plays only a minor role for the UH amplitude, and collisions are more important. If the driving L-mode amplitude EL0 is of the order of 1V/m, the UH wave amplitudes would rapidly exceed the threshold for nonlinearity, which is only a few V/m for ionospheric conditions. In a nonlinear model, the large-amplitude UH waves would excite parametric wave couplings to lower hybrid waves (see, e.g.,  for laboratory conditions and for ionospheric conditions), and the profile of the striation would be modified by the heating of the plasma.

Simulation results using a short driving pulse are shown in Fig.  for clusters using 1, 7 and 19 deep striations with α=0.1. In the simulations we use the reference amplitude ẼL=1V/m of the external L-mode wave, with a rise time of about 10-5s, indicated with dotted lines in Fig. . We choose the pump frequency ω0=21.35×106s-1, which is equal to the local UH frequency at the edge of the striation where the plasma density is 95 % of the ambient density. A number of different scattering processes can be identified in Fig. . The L mode is converted to UH waves trapped in the striations, which gives a rapid growth of the electrostatic field in Fig. b. As seen in Fig. a, the electromagnetic wave amplitude rises initially on the same fast timescale as the pump wave, which indicates that the pump is also scattered directly to Z-mode waves. Visible in Fig. b are oscillations in the amplitude of the UH wave with a typical periodicity of 0.5-1×10-4s. These oscillations are consistent with groups of trapped UH waves being reflected off the edges of the striations. This can be understood by using a one-dimensional ray-tracing picture of the UH wave, whose wave frequency ω and wave number k are related through ω2=ωpe2(1+ns(x)n0)+ωce2+3κvTe2k2. A wave packet at position x(t) with wave number k(t) obeys approximately the equations of motion: dxdt=∂ω∂k=3κvTe2kω, dkdt=-∂ω∂x=-ωpe22ωn0∂ns∂x=-ωpe22ωn0αxDstr2exp⁡(-x2Dstr2). This coupled system for the wave packet describes a nonlinear classical oscillator. For small oscillations x2/Dstr2<1, the equations can be combined to the harmonic oscillator equation d2xdt2=-Ω2x, where the oscillation frequency is Ω=3καωpeωvTeDstr. Using the plasma parameters vTe=2.46×105 m s-1, ω=21.82×106s-1, κ=2.5, α=0.1 and Dstr=2 m gives Ω=105s-1 with a periodicity of 2π/Ω≈0.6×10-4s. This periodicity is consistent with the typical modulation periods of the UH oscillations in Fig. . While there are some small but visible differences in the time development for different numbers of striations using the fully electromagnetic model in Fig. 6a and b, there is almost no difference between the different cases using the purely electrostatic model in Fig. 6c. Hence, as expected, the coupling between striations is through Z-mode radiation.

The Z-mode leakage is more significant for shallow striations. Figure  shows the amplitudes of the electromagnetic and electrostatic fields for simulations using clusters of 1, 7 and 19 striations having the relative depth α=0.01. The driving frequency ω0=21.7858×106s-1 for the electromagnetic simulations in Figs. 7a and b is near that of the N=-1 and R=1 mode in Fig. b, which drives the mode resonantly. A purely electrostatic simulation shown as the dotted line in Fig. b, d uses a slightly higher frequency of ω0=21.799×106s-1, which drives the purely electrostatic mode resonantly. The main result of the simulations is that the larger number of striations organized in a cluster leads to a larger amplitude of the UH wave. As seen in Fig. b, the electrostatic wave amplitude reaches 7EL0 for 1 striation, 23EL0 for 7 striations, and 75EL0 for 19 striations. Hence, the amplitude increases about a factor of 3 for each layer of striations in the cluster. The larger amplitudes of the UH oscillations for larger clusters is due to influx of Z-mode radiation from neighboring striations within the cluster, which partially compensates for the Z-mode leakage and leads to a longer confinement time of the wave energy for a larger number of striations in the cluster. For the slightly higher frequency used in Fig. 7c and d, the electrostatic amplitude in Fig. 7d is almost the same as in Fig. 7b for 1 striation, while it is significantly lower in Fig. 7d for the less damped case of 19 striations.

The amplitudes of the electromagnetic wave fields at t=5ms (Fig. ) show the influence of groups of shallow striations with α=0.01 on the radiation field. Here the distance between nearest neighbor striations is 24.53 m and the wavelength λ=2π/k of the Z mode is estimated using the dispersion relation (Eq. ) to be λ≈13.46 m. The most important observation is that for the larger cluster of 19 striations, shown in Fig. c, the amplitude of the radiated field is relatively small compared to the field within the cluster of striations.

The mode conversion of an L-mode wave to UH waves on small-scale striations has been investigated with numerical simulations. In particular we have addressed how the amplitude of the trapped UH wave depends on the leakage to Z-mode waves escaping the striation. The leakage to Z-mode waves is important for small-amplitude striations and may arrest the growth of the striations. The Z-mode leakage is inhibited in groups of striations by multiple scattering of the Z-mode wave. In clusters of striations, the Z-mode leakage is inhibited by multiple scattering of the Z mode and UH waves on striations. In this case, the UH wave may reach significant amplitude, beyond the threshold for parametric instabilities leading to UH and lower hybrid turbulence, and to thermal instabilities further enforcing the striations. For large-amplitude striations, the mode conversion to UH waves is more efficient and the Z-mode leakage is less important. In this case the UH amplitude quickly reaches the threshold for nonlinearity and processes other than Z-mode leakage become more important. The present study is relevant for ionospheric high-frequency pump experiments, where the anomalous absorption of O and L-mode waves on field-aligned striations is important and the striations are observed to be clustered in bunches a few hundred meters to kilometers across . The self-consistent formation of clusters of striations and the nonlinear evolution of the system in the presence of Z-mode leakage are interesting questions which we hope to address in future works. In particular, large-amplitude UH waves exceeding the threshold for nonlinearity lead to upper hybrid and lower hybrid turbulence , which would dynamically change the spatial profiles of the striations and the resonance conditions for the partially trapped UH waves.