Nonlinear wave interactions as a model for naturally enhanced ion-acoustic lines in the ionosphere

TL;DR

This paper proposes a model where nonlinear wave interactions, specifically Langmuir wave decay, explain the naturally enhanced, asymmetric ion-acoustic lines observed in the ionosphere, supported by weak-turbulence analysis.

Contribution

It introduces a new explanation for NEIALs based on Langmuir wave decay, using weak-turbulence theory to identify key parameters for the process.

Findings

Langmuir decay can produce asymmetric ion lines

Weak-turbulence approximation identifies critical parameters

Model explains observed NEIAL asymmetry

Abstract

Incoherent radar scatter from the ionosphere will, for equilibrium conditions, show two symmetric ion-acoustic lines, one for each direction of wave propagation. Many observation, from the EISCAT Svalbard Radar (ESR) for instance, demonstrate that the symmetry of this ion line can be broken, accompanied by an enhanced, nonthermal, level of fluctuations, i.e., Naturally Enhanced Ion-Acoustic Lines (NEIALs). Several models have been proposed for explaining these naturally enhanced lines. Here, we consider one of these, suggesting that decay of electron beam excited Langmuir waves gives rise to enhanced asymmetric ion lines in the ionosphere. We use a weak-turbulence approximation, and identify crucial parameters for Langmuir decay processes to be effective in generating the observed signals.

Tables

Table 1. Table 1: Varying parameters

Altitude, km	$T_e$, K	$T_i$, K	$f_{pe}$, MHz	$n_{gas}$,cm-3	$ℒ$, km
400	3100	1400	4.0	4$\times {10}^8$	$\infty$
400	3100	1400	4.0	4$\times {10}^8$	$+30$
250	2200	1100	5.0	4$\times {10}^9$	$-30$

Equations

$$\frac{\partial f}{\partial t}=\frac{4{\pi }^2{e}^2}{m^2}\frac{\partial }{\partial v}\frac{W_k}{v}\frac{\partial f}{\partial v},$$

$$\frac{\partial W_k}{\partial t}-\frac{{\omega }_{pe}}{\mathcal{L}}\frac{\partial W_k}{\partial k}=\frac{\pi {\omega }_{pe}^3}{n{k}^2}{W}_k\frac{\partial f}{\partial v}-2{\nu }_{ce}{W}_k$$

$$+S{t}_{decay}(W_k,W_k^s)+S{t}_{ion}(W_k),$$

$$\frac{\partial W_k^s}{\partial t}=-2{\gamma }_k^s{W}_k^s$$

$$-\alpha {{\omega }_k^s}^2\int \left(\frac{W_{k^{\prime }-k}}{{\omega }_{k^{\prime }-k}}\frac{W_k^s}{{\omega }_k^s}-\frac{W_{k^{\prime }}}{{\omega }_{k^{\prime }}}\left(\frac{W_{k^{\prime }-k}}{{\omega }_{k^{\prime }-k}}+\frac{W_k^s}{{\omega }_k^s}\right)\right)$$

$$\times \delta ({\omega }_{k^{\prime }}-{\omega }_{k^{\prime }-k}-{\omega }_k^s)d{k}^{\prime },$$

$$S{t}_{decay}(W_k,W_k^s)=$$

$$\alpha {\omega }_k\int {\omega }_{k^{\prime }}^s[\left(\frac{W_{k-k^{\prime }}}{{\omega }_{k-k^{\prime }}}\frac{W_{k^{\prime }}^s}{{\omega }_{k^{\prime }}^s}-\frac{W_k}{{\omega }_k}\left(\frac{W_{k-k^{\prime }}}{{\omega }_{k-k^{\prime }}}+\frac{W_{k^{\prime }}^s}{{\omega }_{k^{\prime }}^s}\right)\right)$$

$$\times \delta ({\omega }_k-{\omega }_{k-k^{\prime }}-{\omega }_{k^{\prime }}^s)-$$

$$\left(\frac{W_{k+k^{\prime }}}{{\omega }_{k+k^{\prime }}}\frac{W_{k^{\prime }}^s}{{\omega }_{k^{\prime }}^s}-\frac{W_k}{{\omega }_k}\left(\frac{W_{k+k^{\prime }}}{{\omega }_{k+k^{\prime }}}-\frac{W_{k^{\prime }}^s}{{\omega }_{k^{\prime }}^s}\right)\right)$$

$$\times \delta ({\omega }_k-{\omega }_{k+k^{\prime }}+{\omega }_{k^{\prime }}^s)]d{k}^{\prime },$$

$$S{t}_{ion}(W_k)=$$

$$\beta {\omega }_k\int \frac{({\omega }_{k^{\prime }}-{\omega }_k)}{v_{Ti}|k-k^{\prime }|}\frac{W_{k^{\prime }}}{{\omega }_{k^{\prime }}}\frac{W_k}{{\omega }_k}\exp \left[-\frac{({\omega }_{k^{\prime }}-{\omega }_k{)}^2}{2v_{Ti}^2|k-k^{\prime }{|}^2}\right]d{k}^{\prime },$$

$$\alpha =\frac{\pi {\omega }_{pe}^2(1+3T_i/T_e)}{4n\kappa T_e},\beta =\frac{\sqrt{2\pi }{\omega }_{pe}^2}{4n\kappa T_i(1+T_e/T_i{)}^2},$$

$${\gamma }_k^s=\sqrt{\frac{\pi }{8}}{\omega }_k^s\left[\frac{v_s}{v_{Te}}+{\left(\frac{{\omega }_k^s}{k{v}_{Ti}}\right)}^3\exp \left[-{\left(\frac{{\omega }_k^s}{k{v}_{Ti}}\right)}^2\right]\right].$$

Full text

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KONTAR and PÉCSELI \titlerunningheadEnhanced ion acoustic lines

\paperid \authoraddrPermanent addresses: E. P. Kontar, Kelvin Building, University of Glasgow, Glasgow G12 8QQ, UK (eduard@astro.gla.ac.uk); H. L. Pécseli, University of Oslo, Physics Department, Blindern, N-0316 Oslo, Norway (hans.pecseli@fys.uio.no)

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1 Introduction

Incoherent scatter radars are a standard tool for investigating the properties of the Earth’s ionosphere, and in the case of thermal equilibrium, the backscatter can be analyzed in terms of the fluctuations-dissipation theorem from basic thermodynamics and statistical mechanics (Sedgemore-Schulthess and St.-Maurice, 2001). Very often it is found, however, that the ionospheric plasma is out of equilibrium, and that the ion-line signal is distorted (Buchert et al., 1999; Grydeland et al., 2003). Several models can account for this feature: the symmetry of the natural ion-line is broken, for instance, if a current is flowing through the plasma (Sedgemore-Schulthess and St.-Maurice, 2001), or if an electron beam enhances electron plasma waves (Langmuir waves) significantly above the thermal level, so that ion acoustic waves are excited by parametric decay (Tsytovich, 1995). Such models were invoked by Forme (1999) and Forme et al. (2001).

Although the observations as such are unambiguous (Buchert et al., 1999), the interpretation is made difficult by several problems: the observed features are often sporadic in nature and can vary with time as well as altitude, on second and km scales, respectively (Sedgemore-Schulthess and St.-Maurice, 2001). Also, we bear in mind that the radar is usually obtaining backscatter at one selected wave-vector ${\bf k}_{R}$, which is related to the scattering wave-vector ${\bf k}_{B}$ by selection rules, which for a mono-static radar gives ${\bf k}_{B}=2{\bf k}_{R}$. For the ESR-radar (Buchert et al., 1999), we have a transmitter frequency of 500 MHz, giving $\lambda_{B}\equiv 2\pi/k_{B}=0.299$ m. For an altitude $\sim 400$ km, with electron temperatures of $T_{e}\approx 3000$ K, and plasma densities of $n_{e}\approx 2\times 10^{11}$ m*-3*, we find Debye lengths $\lambda_{De}\approx 8.5\times 10^{-3}$ m, i.e. $\lambda_{B}/\lambda_{De}\sim 35$, or $k_{B}\lambda_{De}\approx 0.18$. In order to observe the ion sound wave we should have the sound wavelength $\lambda_{s}$ equal to $\lambda_{B}$, implying that the primary Langmuir wavelength $\lambda_{L}$ is approximately $\frac{1}{2}\lambda_{B}$. This means that in a decay process, we cannot usually, at a given altitude, expect to observe the first generation Langmuir waves simultaneously with the sound waves forming the low frequency part of the decay products. In the case where we have a “cascade” of decaying waves (Tsytovich, 1995), we might observe one or the other of the decay products, and it might very well be the second or third generation that is observed, instead of the first one. For the parameters mentioned before we have $k_{L}\lambda_{De}\approx 0.09$. The plasma conditions are strongly variable (Fontaine, 2002) and it is not always obvious under which conditions the enhanced ion-lines are observed.

Several models have been proposed (Rietveld et al., 1991; Wahlund et al., 1992), but the decay model may be the most promising, at least for explaining some observed features (Sedgemore-Schulthess and St.-Maurice, 2001; Grydeland et al., 2003). One problem with the Langmuir decay model seems to be that very short wavelength primary Langmuir waves are sometimes needed to account for the observations, below a few tens of Debye lengths, $\lambda_{De}$. It might be possible to find a low velocity electron beam which generates unstable waves for $v<4v_{Te}$, but the decay Langmuir wave (“daughter wave”) obtained from these will be strongly Landau damped, implying that the growth rate of the decay instability becomes negligible. Also, the effects connected with plasma inhomogeneity have been largely ignored, although a consistent treatment of the ionospheric plasma density gradient can bring new understanding of the observational results.

We discuss here the decay model in detail, including the most important additional effects, such as the nonlinear Landau damping and the effect of a large scale density gradient, to demonstrate a set of criteria for the decay mechanism to be a viable explanation for the enhanced ion-acoustic waves. These processes are here modeled by a self-consistent set of dynamic equations, which are solved numerically. A self-consistent description of beam-plasma interactions, Langmuir wave scattering off ions (nonlinear Landau damping), and plasma inhomogeneity can bring qualitatively new results.

2 Beam-driven Langmuir turbulence

The basic equations are here solved for an initial value problem, which is standard for these types of problems. This gives a simplified alternative to the full problem, while at the same time retaining the important physics. Qualitatively, we might transform temporal evolutions to spatial evolutions, using the beam velocity for the transformation, i.e., $x\sim tv_{b}$. The analysis is restricted to one spatial dimension for practical computational reasons. We do not expect this to pose any serious limitation, in part also because the growth rate for the decay instability is known to have a maximum for aligned wave-vectors (Thornhill and ter Haar, 1978). Also, for the observations relevant to the present study (Grydeland et al., 2003) the radar beam is basically directed along the magnetic field lines. Using the assumptions mentioned above, we can write the system of kinetic equations of weak turbulence theory (Vedenov et al., 1962; Kontar, 2001b; Kontar and Pécseli, 2002)

[TABLE]

where $f(v,t)$ is the spatially averaged electron distribution function. $W(k,t)$ is the spectral energy density of Langmuir waves and plays the same role for waves as the electron distribution function for particles. This equation accounts for the quasi-linear relaxation of the electron beam.

For an inhomogeneous plasma, with a density gradient along the direction of beam propagation ${\cal L}=\omega_{pe}/(\partial\omega_{pe}/\partial x)^{-1}$, there are two distinct processes affecting Langmuir waves (Kontar, 2001a). First, the local plasma frequency $\omega_{pe}$ is spatially dependent leading the instability growth-rate to change with distance. Second, and more important, a small spatial movement of Langmuir waves with group velocity $\partial\omega_{L}/\partial k<<v_{Te}$ leads to a substantial change in the wavenumber of the plasma waves, since $\omega_{L}(k,x)\approx\omega_{pe}(1+\Delta n/n+\frac{3}{2}(k_{L}\lambda_{De})^{2})$. By taking the wave frequency constant along the ray path for a stationary density profile, we find, as an estimate, the variation of the Langmuir wavenumber to be $(k_{L1}\lambda_{D1})^{2}\approx(k_{L2}\lambda_{D2})^{2}+\frac{2}{3}\Delta n/n$, where we follow the wave from positions labeled 1 to 2, having a density difference of $\Delta n$. This turns out to be an effective mechanism for transferring wave energy from large to small wavenumbers or vice-versa (Kontar and Pécseli, 2002), depending on the sign of $\Delta n$. This process has seemingly been ignored in previous studies of the enhanced ion-lines.

For the small density variation $\Delta n/n<(k\lambda_{De})^{2}$, we can ignore the frequency variations, while retaining the wavenumber shift due to inhomogeneity in the equation for the evolution of Langmuir waves in the WKB (geometrical optics) approximation

[TABLE]

where $-({\omega_{pe}}/{\cal L})\,{\partial W_{k}}/{\partial k}$ accounts for the effective acceleration $-\partial\omega_{pe}/\partial x$ of a Langmuir wave packet by a density gradient. This term is well known, for instance, from geometrical optics, and is caused by the spatial variation of the index of refraction. This acceleration can imply that waves excited by the beam can reach phase velocities above the beam velocity, after which they will be re-absorbed by Landau damping.

The system (1) and (2) self-consistently describes the resonant interaction $\omega_{pe}=kv$ of electrons and Langmuir waves. The first term on the right hand side of (2) accounts for the waves generated by the electron beam. A simple damping term with collision frequency $\nu_{ce}$ accounts for the collisional damping of the waves. We included the collisional damping of Langmuir waves with $\nu_{ce}=\nu_{ei}+\nu_{en}$, where $\nu_{ei}=(34+4.18\ln(T_{e}^{3}/n_{e}))n_{e}T_{e}^{-3/2}$ is for electron-ion collisions and $\nu_{en}=5.4\times 10^{-10}n_{gas}T_{e}^{1/2}$ accounts for electron neutral collisions (Itikawa, 1973). The former dominates the latter at higher altitudes while electron-neutral collisions are a faster process below the F region maximum.

The third term $St_{decay}$ accounts for the change of wave energy due to decay processes $L\rightarrow L^{\prime}+S$, with $L$ and $S$ denoting Langmuir and ion sound waves, respectively. We introduced also $St_{ion}$ to account for the nonlinear Landau damping, which is also known as scattering off ions, $L+i\rightarrow L^{\prime}+i^{\prime}$ (Tsytovich, 1995). These processes are efficient in redistributing energy within the spectra. The nonlinear processes accounted for by $St_{decay}$ and $St_{ion}$ are effective in the scattering of primary, beam generated waves with wavenumber ${\bf k}$, into secondary Langmuir waves with wavenumber ${\bf k}^{\prime}\approx{\bf-k}$. The decay of Langmuir waves leads to generation of the ion-sound waves with ${\bf k}_{s}\approx 2{\bf k}$.

The time evolution of ion acoustic fluctuations, with a spectral energy $W^{s}_{k}$, is described by

[TABLE]

with

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

Here $\gamma^{s}_{k}$ is the Landau damping rate of ion-sound waves, containing both electron and ion components, $v_{s}=\sqrt{\kappa T_{e}(1+3T_{i}/T_{e})/m_{i}}$ is the sound speed, while $W(k,t)$ and $W^{s}(k,t)$ are the spectral energy densities of Langmuir waves and ion-sound waves, respectively. It turns out that for parameters relevant here, Landau damping of ion-sound waves dominates collisional damping. The relative importance of the various terms in (3) can be studied by including them in the numerical calculations one at a time. Since the equations are non-linear, this procedure can give an estimate only.

3 Numerical Results

The set of equations (1)–(7) were solved numerically using the method described by Kontar and Pécseli (2002), here for downward propagating electron beams.

Common parameters for the numerical analysis are $n_{b}=20$ cm*-3*, $v_{b}=3\times 10^{8}$ cm/s ($\sim 26$ eV), with the subscript $b$ indicating “beam”. As a reference parameter set we take those listed in Table 1. Relevant electron-neutral collision frequencies can be deduced and extrapolated from known results, as summarized by, for instance, Itikawa (1973). Since the dominant ion in the ionosphere at 250-400km is O+, the effective ion mass $m_{i}$ was taken to be 16 proton masses. We attempted to use representative plasma parameters, so that the numerical results can be used for qualitative as well as quantitative comparisons with observations.

Figures 1 and 2 show the evolution of beam-plasma system in a homogeneous plasma. The distributions of electrons, Langmuir and ion-sound waves are shown for various times. The most rapid process in the system is the quasi-linear relaxation, followed by wave decay, while here the scattering off ions plays only a secondary role. The turbulence exhibits fine structure both in the Langmuir and ion-sound wave spectra (Figure 1). This structure appears due to the decrease of the absolute value of $k$ at every decay by $\Delta k_{d}\lambda_{De}=2\sqrt{m_{e}/m_{i}}\sqrt{1+3T_{i}/T_{e}}/3$ and scattering off ions by $\Delta k_{s}\lambda_{De}=2\sqrt{m_{e}/m_{i}}\sqrt{T_{i}/T_{e}}/3$.

The overall picture changes dramatically when plasma inhomogeneity is accounted for (Figure 3). For the parameters in Figure 3 we have $\Delta k_{d}\lambda_{De}\approx 0.005$ and $\Delta k_{s}\lambda_{De}\approx 0.002$, and for the density gradient ${\cal L}=30$ km the wavenumber shift rate is $dk/dt\sim\omega_{pe}/{\cal L}\sim 10$ cm*-1* s*-1*. As an electron beam propagates downwards above the F maximum, the generated Langmuir waves experience a decrease in $k$, facilitating further decay of Langmuir waves with the same $k$ (Figure 3). Below the F-region maximum, the wavenumber decrease due to decay and scattering can be compensated by the density inhomogeneity for approximately $0.5$ ms. Similarly, an increasing plasma density can compensate an increase in $k$ along the beam direction in Figure 4.

4 Discussions and Conclusions

Density gradient: The steady state plasma distribution has a “natural” large scale density gradient, which changes sign when going from below to above the F-region maximum. The scale length of this gradient is some tens of kilometres. In addition, we might expect that under disturbed conditions, we might have gradients with shorter scale-lengths, where the direction of these can have both signs. The shift of Langmuir waves in $k$-space is strong enough to produce substantial changes in local Langmuir turbulence spectra as illustrated in Figures 3 and 4. The results in these figures demonstrate that an interpretation of NEIALs in terms of Langmuir decay (Forme, 1999; Forme et al., 2001) is most convincing in regions with vanishing or negative density gradients, corresponding to altitudes around or below the F maximum.

Scattering off ions: The decay of Langmuir waves into ion-acoustic and back-scattered Langmuir waves is normally most efficient for plasmas with $T_{e}>>T_{i}$. The growth rate of the decay or parametric instability should be higher than the ion-acoustic damping rate, producing a threshold for the instability. The damping rate is strongly dependent on $T_{e}/T_{i}$, which can be small for the lower parts of the F-region, while scattering off ions has no such threshold and dominates for $T_{e}/T_{i}\sim 1$. The enhanced level of Langmuir waves with $k^{\prime}_{L}\approx-k_{L}$ makes the term $W_{k}W_{k-k^{\prime}}$ in (3) efficient and thus stimulates the decay. In contrast to a pure parametric instability with a single shoulder, it gives a close to simultaneous growth of two ion-acoustic shoulders, as in Figure 3. In this case the radar backscatter has the best possibility of indicating an enhancement of both ion lines.

We found that for altitudes above $\sim 150$ km, the quasi-linear beam relaxation due to the collective effects is faster than that caused by collisions. The decay instability, the scattering off ions, and the shift due to a plasma inhomogeneity act on a longer time-scale. It can be seen in Figures 1 and 2 that the decay leads to antisymmetric structures associated with multiple decay in the spectra of Langmuir turbulence.

We have also considered the case where the background plasma is nonthermal, here simulated by using a superposition of two Maxwellians with different temperatures, consistent with observations (Guio and Lilensten, 1999). We have found the most significant effect of a small density (1% or less) warm electron component (temperature $\sim 20\times$ the background temperature $T_{e}$) to be a change in the Landau damping of the backscattered Langmuir wave. When this wave is damped, the ion signal becomes weaker. The present results were derived for downwards propagating electron beams, but qualitatively, the results apply for upgoing beams as well, if the sign of ${\cal L}$ is changed.

Acknowledgements.

The present work was carried out as a part of the project “Turbulence in Fluids and Plasmas”, conducted at the Centre for Advanced Study (CAS) in Oslo in 2004/05. We also thank Anja Strømme for valuable discussion in the early stages of this work. Discussions with Jan Trulsen are appreciated.