Spectrum of Langmuir fluctuations trapped in a cavity of a collisional plasma

The Langmuir fluctuations trapped and enhanced inside a cavity of the electron concentration in a plasma are considered. Stationary fluctuation spectrum is calculated with regard to finite collisional damping of the Langmuir waves and nonequilibrium fluctuation source. It is shown that the fluctuation spectrum obtained exhibits a frequency modulation due to the reflection and superposition of the elementary Langmuir waves in the irregularity. The modulation depth and period as a function of the irregularity size and the electron–ion collision frequency is calculated. These results may help to find from experimental observations such important plasma parameters as the irregularly size and lifetime, as well as the electron collision frequency.

Spatial localization can be pronounced particularly in case of the Langmuir fluctuations trapped inside plasma density cavities. The Langmuir fluctuations constitute a set of electrostatic plasma waves with a frequency close to the Langmuir frequency. If the damping of these waves is small enough they can propagate in a cavity. But there exists a frequency interval when the waves reflect at the turning points and return back to the cavity. In this case, if a steady source drives the fluctuations, their intensity may be enhanced significantly. The behavior of fluctuations depends on the collision frequency of electrons with heavy particles. Small collisional damping of the Langmuir waves creates conditions for the modulational instability of the Langmuir waves [27]. This kind of instability results in development of the strong Langmuir turbulence in which the fluctuation level is determined by nonlinear processes.
At the same time, collisions can play an important role in achieving steady state fluctuations. As is shown in [28], due to the collisions, an instability threshold for the plasma wave arises both in laser and space plasmas. For example, in the ionosphere, the collisions become important under a wide set of conditions when the fluctuation electric field is of the order or less than 1 mV m −1 [28] and the cavity size is more than 10 m [22] (i.e. for mid-and large scale irregularities). Note that only such irregularities can be resolved by incoherent scatter radars in the experiments on plasma diagnostics with the help of the Thomson scattering.
In this paper we calculate stationary amplitudes and spectrum of the Langmuir fluctuations captured in a parabolic cavity of the plasma density in the presence of the electron-ion collisions. Our results are applicable for arbitrary distribution functions of the electrons, provided the parametric instabilities are not excited. The fluctuation spectrum as a function of the collision frequency, the irregularity parameters, and the fluctuation source is presented. It is shown that collisions restrict fluctuations intensity and smear the discrete spectrum which is obtained in the collisionless approximation [22,24,25,29].
Earlier the Langmuir wave trapping in the presence of the collisional damping was treated for the Langmuir waves generated by the resonant mixing of 2 transverseelectromagnetic waves [30]. In the paper [30] only regular Langmuir wave with a fixed frequency and wavenumber was analyzed. Such an analysis is insufficient for the fluctuation problem, as it does not take into account a set of the multiple Langmuir waves and their correlation mechanism.
In the present work we consider another generation mechanism which is appropriate to the real fluctuation problem. The mechanism uses a fluctuation source calculated in [31,32] which depends on nonequilibrium plasma parameters and has a broad set of wavenumbers and frequencies exciting the fluctuations. This source does not uses some additional assumptions about specific forms of the collision integrals (e.g., so-called model collision integrals [5,33,34]) which restrict the generality of the results and do not ensure explicit formulation of the application conditions for the results obtained.
The fluctuation source found in [31,32] is applicable for arbitrary (not only maxwellian) distribution functions of particles and takes into account collective effects in a plasma. This source was used in [31,32] to calculate fluctuation spectra in the case of a homogeneous plasma. In the present paper, we show that the same fluctuation source can be used to solve an inhomogeneous problem.

Excitation of the Langmuir waves by the fluctuation source
Consider a plasma layer where the electron concentration n changes in the direction x. Let the ambient electron density n x ( ) depend on the coordinate x in the following way where n 0 is the density at the minimum point. The parameter b determines the irregularity size. For example, if the electron density in the ionosphere changes 10% at a distance of 100 m then b 4.47 10 5 =´cm −1 . If this distance is 1 km, b 4.47 10 6 =´cm −1 . The parabolic model (1) is a good approximation for real irregularities, as the most intensive fluctuations are formed in the vicinity of the density minimum.
The total electron density variation N x t , ( )consists of the unperturbed density n(x) and small amplitude fluctuations n x t , d ( ) The Langmuir fluctuations electric field E d can be described by the equation derived in [21] L )is a random function (fluctuation source) calculated in [31,32] on the basis of the Klimontovich equations [4] . The operator L x , w  ( ) describes propagation of the Langmuir wave in the presence of the collisional damping which is represented by the conductivity σ. This term is related to the efficient collision frequency of electrons ef n where e e and m e are the electron charge and mass, respectively, and p0 w is the Langmuir frequency at the minimum density point. v xe in (3) is the root-mean-square velocity of electrons in the x-direction.
x , 0 e w ( )is the part of the longitudinal dielectric function of the plasma without the spatial dispersion and collisional terms (both are taken into account by the operator L  ). The electric field fluctuation E d in (3)is a function of ω and x. In the stationary case it is convenient to apply the Laplace transform to the time-dependent In this equation 0 D > , and the limit 0 D  should be taken in the final results. The inverse transform for (6) is as follows For simplicity, we will present results only for the frequencies 0 w > . The case of the negative frequencies is quite similar.
As mentioned above, the most intensive fluctuations are formed near the density minimum, i.e., in the vicinity of the point x=0. This means that the frequency ω of the fluctuations is close to the electron Langmuir frequency p0 w and varies within narrow limits: . 8 In addition to that, the Landau damping in our case is negligible (see below). Under these conditions the ωand x-dependence of all the terms except x , 0 e w ( )can be neglected. Note that the ω and x dependence of S  remains weak and it will have no effect on the amplitude calculations. We retain this dependence here just for the obtaining correct relationship between the scattered spectrum an the statistical moment of the fluctuation source S  (see below).
The right-hand side of (3) contains integration with respect to the wavenumber k x . Since equation (3) is linear, we can solve it for a fixed k x and then make the integration in the final results: This is a one-dimensional equation for the elementary wave driven by the source in the right-hand side with a fixed k x and ω. Equation (9) will be solved with the help of the WKB approximation. Represent the fluctuation field E d in the form . Along with the presentation of the electric field (7) the minus sign in this expression corresponds to the wave propagation in the positive direction when 0 w > . The exponential containing the phase changes with x much faster than Q and ψ. Substitute the relationship (10) into (9). As a result we find the zero-order approximation for the solution of equation Note that x , y w ( ) is the wavenumber of the Langmuir wave. At the points x d 1,2 =  the value x , y w ( ) is equal to 0. Thus, the wave going outside the irregularity reflects here and returns back, executing a cyclic propagation. It is seen from (11) that the solution (10) is valid when p0 w w > . Moreover, the backscattering of the probe wave by the Langmuir fluctuations is possible only when , 0 y w ( )is equal or greater than the twice wavenumber of the probe wave (see below). Hence, we will consider the fluctuations with a finite frequency shift p0 w w satisfying this condition. In such a case the distance d 2 between the turning points x 1,2 is much more than the Langmuir wavelength x 2 , p y w ( ) anywhere inside the irregularity except a relatively small vicinity of the turning points. This makes the WKB solution accurate enough in the most part of the irregularity. As for the turning points, their influence will be discussed later.
The equation for the first order solution of (9) is as follows As the plasma wavelength is significantly less than the distance between the turning points d 2 , the exponential term in (12) is a fast-oscillation function except the points where the derivative of the phase is equal It is seen from this equation that a parabolic layer has 2 such points x 3 and x 4 when We call these points stationary or plasma wave generation points. It is seen from equation (14) that the coordinate of a stationary point x 3,4 satisfies the condition k x , ). In fact, this is the synchronism condition meaning that the plasma wave is generated in a small region where the wavenumber of this wave ψ is equal to that of the fluctuation source k x . Note that ω is the frequency of the fluctuation source component playing a role of external driver for the plasma wave. It is interesting to note that in our case the system 'Langmuir wave + fluctuation source' behaves like a mechanical vibratory system driven by an external harmonic force [30].
The key feature of the fluctuation problem considered here is that we have a set of Langmuir waves with different frequencies ω and wavenumbers ψ. The stationary point position depends on ω and the fluctuation source wavenumber k x . As the fluctuation source contains a broad set of k x and ω (see below, section 4) we need to summarize the effect of all the stationary points distributed over the interval determined by the second line of equation (14) and the condition equation (15). Thus, the stationary points giving contribution to the total fluctuation spectrum turn to be distributed in the interval ( d -, d + ), i.e. between the turning points x 1 and x 2 .
Equation (12) can be solved using the stationary phase approximation. Expand T x ( ) in the vicinity of the stationary points as a Taylor series Thus, near the stationary points the solution of equation (12) can be found with the help of the expansion (16). Far from the stationary points the phase T x ( ) varies fast enough and causes short-period oscillations of the rhs in (12). Such oscillations suppress any substantial effect of the rhs on Q x , w ( ). In view of that, outside the vicinity of the stationary points the equation (12) reduces to the homogeneous form This equation describes a plasma wave propagating freely after it leaves the generation region localized near the point x 3 or x 4 . The wave exhibits collisional damping represented by the exponential argument.
Here J x x , 3,4 ( ) determines the damping magnitude on the way from a generation point to the point x. The damping of the wave passing a full cycle back and forth inside the irregularity is given by the exponent exp -G ( ). In order to find the factor Q 3,4 in (18) we go back to equation (12). In a small vicinity of the turning points, where the plasma wave is generated, the damping effect is negligible. Hence in calculating Q 3,4 we can omit the term with x , g w ( )in the lhs of (12). In this case the solution of (12) reduces to a simple integration of the rhs of this equation Here the integration limits are extended to ¥. In fact, the convergence of the integral to it is limit takes place when the integration interval x D meets the condition T x x 1 . This relationship gives an estimation for the generation interval of the plasma wave at the stationary points x T x 20 The integral in (19) reduces to the and the factor Q 3,4 takes the form ). This means that the the stationary phase approximation brakes down when the stationary points are close to the density minimum point. Nevertheless, due to the subsequent integration with respect to k x the divergence becomes integrable and has no effect on the total result. Numerical integration shows that the total plasma wave amplitude gets the main contribution almost uniformly from all the permissible wave numbers k , 0 x y w < ( )(see (15)). This enables us to estimate approximately the generation interval x g D (20) by taking the difference k , 0 Evaluation of this expression for the ionospheric parameters demonstrates that x g D is much less than the distance between the turning points d 2 . This justifies the stationary phase approximation employed here.
3. Spatial evolution of the waves in the parabolic irregularity in the presence of collisional damping To obtain the last relationship in (24) we used the expression (10) for E d and took the derivative of the exponential only, as it is the fastest varying term.
For simplicity of the presentation, we will give analytical expressions only for the amplitude n d coming out of the generation point x 3 in the positive direction. Other cases are quite similar. Combining the equations (10), (18), (22) and (24) we find The amplitude (25) is valid until the wave reaches the turning point and reflects back. The full first-cycle amplitude is a sum of four amplitudes where the subscripts in the first line denote generation points (x 3 or x 4 ) and the direction in which the plasma wave leaves the vicinity of the stationary point (+ or −). Figure 1 demonstrates that, as the waves propagate in the irregularity, their normalized amplitude decreases because of the collisional dumping. Due to the turning points x 1,2 the waves cannot leave the irregularity, and they pass a number of cycles until the full damping occurs. The solid lines in figure 1 denote the propagation of all the waves in the positive direction. The propagation in the negative direction can be treated exactly in the same way due to the symmetry of the problem.
After the reflection at the point x 1 the wave with the amplitude A x , figure 1) travels in the negative direction (dashed curve) reflects at x 2 , and then goes in the positive direction (solid curve). The amplitude of the wave for the last stage at the interval x x x 2 3 < < is given by  The total amplitude of each wave is the sum of the amplitudes from all the cycles [30]. In order to obtain the total amplitude we should find the first-cycle amplitude and then multiply this by the factor calculated in [30] where Γ determines the one-cycle damping (see (18)), and w F( )is the full phase shift of the wave passed one cycle.  Then the total amplitude A taking into account the plasma wave circulation inside the irregularity can be easily found by (28). Finally we arrive at the expression for the electron density fluctuations trapped in the irregularity The amplitude A is proportional to the fluctuation source via the factors Q 3 and Q 4 , i.e., we can write where B is the regular (non-random) factor of the amplitude A. The amplitude A (or B) is fully determined by the first-cycle amplitudes of the type (25), (27) and the equations (26) and (28). The amplitudes B 1 ( ) related to the full amplitudes A 1 ( ) and eventually to A are presented in appendix A for all the intervals x and both directions of the source wavenumber k x .
Equation (30) is the Langmuir wave excited by the fluctuation source with a definite wavenumber k x . As noted above, in order to find the total fluctuations n x , d w ( )produced by all the permissible k x , we must integrate n results from the condition of the existence of stationary points (15). Thus, the procedure presented here makes it possible to find the stationary amplitude of the Langmuir fluctuations trapped in the irregularity in the presence of a finite collisional damping.

Spectrum of Langmuir fluctuations in the parabolic irregularity
In order to calculate the frequency and wavenumber spectrum let us consider the second moment of the electron density fluctuations n t x n t x n n , , , t t x x 1 1 2 2 , , where the angle brackets denote the average. The moment n n d d ( ) depends only on the difference of the time variables t t 1 2 due the stationary character of the problem considered. According to the Laplace-Fourier transform (6) and (7), the frequency spectrum is defined as n n n n It is shown in [4] that the spectrum (34) The second moment of the fluctuation source here can be found in a way similar to that used in equation (35) (see [4]) The equation (38) is applicable for arbitrary distribution functions f e i , which can differ from the Maxwell distributions due to various nonequilibrium processes in a plasma. In this paper, we will not consider specific mechanisms of generation of the Langmuir fluctuations. Instead, we concentrate on the effect of a parabolic irregularity on the fluctuation spectrum. It is shown in [31] that the spectral function of the fluctuation source changes only slightly with a small variation of the electron density, as it takes place in the vicinity of the density minimum point (1). That's why we use the expression for the fluctuation source (37) valid in the homogeneous case.
Substitute equation (37) into (36) and integrate the δ-function with respect to k x,2 Numerical calculations presented below show that the integration with respect to k x,1 in (40) gives a result slowly varying at a distance of the Langmuir wavelength. At the same time, the exponential in the equation (40) is a rapidly oscillating function of x x 1 2 with the characteristic scale of the order of the Langmuir wavelength (remind that ψ is the Langmuir wavenumber). Therefore, the exponential factor will give the main contribution to the spectrum of the Langmuir fluctuations in the wavenumber space.
Introduce new variables x x ) which represent the small and large scales in space, respectively. Then the full frequency and wavenumber spectrum of the Langmuir fluctuations can be found in the following way where ξ is the wavenumber variable in the spectrum, and the slow dependence on x is retained. The exponential in equation (40) is the only factor depending on the 'fast' variable χ: the integral in this exponential can be reduced to the form )changes negligibly at a distance χ being of the order of the Langmuir wavelength. Then the integral in equation (41) with respect to χ can be expressed in terms of the δ-function )is the function presented in appendix B.

Discussion of the results
The formula (44) is the frequency and wavenumber spectrum of the Langmuir fluctuations. According to onedimensional approximation considered here, the y-and z-components of the wavenumber are omitted in the expression for n n e e x , , ) . The spectrum (44) is proportional to the spectral function of the fluctuation source (38).
As we pointed out above, the fluctuation source used here is applicable for non-maxwellian distribution functions. In this paper, we do not concentrate on the role of various non-equilibrium processes in the generation of fluctuations. Here we only note that non-maxwellian distributions can increase the fluctuation source intensity, and make it is dependence on ω and k more complicated. An example of the ionospheric plasma in which the electron distribution function is anisotropic (in contrast to the Maxwell distribution) is given in [31]. It is shown in this paper that such an anisotropy results actually in an enhancement of the spectrum intensity and makes the spectrum anisotropic, as well.
As a rule, this is a nonresonant function in the frequency range of the Langmuir waves This condition is equivalent to that for the position of discrete spectrum lines discussed in [22,24,25,29] for the case of a collisionless approximation. Unlike [22,24,25,29] in our case of a collisional plasma the spectrum lines have finite width and maximum values. It is seen that these lines become less distinct with an increase in Γ due to a decrease in the maximum value P 1 exp The factor P x , 0 w ( ) determines ωand x -dependence of the spectrum part proportional to the δ-function. It is seen from (45) that the spectrum intensity increases when the parameter b is reduced which means the irregularity scale increase. Figure 2 shows the factor P x , 0 w ( ) as a function of the coordinate x for 3 fixed frequencies ω. These frequencies are taken so that they match the condition (47) for the local maxima in the spectrum. It is seen from figure 2 that the Langmuir fluctuations occupy larger region in the irregularity when the relative frequency shift becomes greater. The zero points for P 0 in this picture coincide with the turning points for a chosen frequency ω.
It is necessary to emphasize that the relative frequency shift Another characteristic which can be measured experimentally, is the interval between neighboring maximum positions. From (48) we can find for this interval . The curves are normalized to the value P , 0 0 w ( ) with 0.03 dw » . The parameter v xe p0 w is taken to be equal to 1 cm.
. The Debye length for the ionospheric plasma can be found using the International Reference Ionosphere model [36]. In particular, for the ionospheric F-layer the Debye length is usually of the order of 1cm. Knowing the Debye length we can directly obtain the irregularity parameter b from the measurements of the interval It is important that the Thomson scattering process selects only the fluctuations with the wave vectors parallel to the scattering vector [34]. Thus, if the scattering vector is parallel to the density gradient of the irregularity, the observable scattering spectrum will depend only on the fluctuations propagating along the density gradient, which is the case considered in this paper. Such a geometry is appropriate for observing the most pronounced effects of plasma irregularities. If the scattering vector makes a nonzero angle with the density gradient, we will observe the fluctuations obliquely propagating in the irregularity. In this case the Langmuir wave amplification due to the superposition of incident and reflected waves inside the irregularity decreases, and the frequency spectrum of the scattered wave will be not modulated.
The one-dimensional model considered in this paper can be valid for real 3D plasma if the distance d 2 between the turning points (11) is sufficiently less than the radius of curvature for the surface where the electron density n r ( ) is constant. For a 100 m irregularity (b 4.47 10 5 =´cm −1 , see the beginning of section 2) and the relative frequency shift 10 2 dw =we obtain that d 2 is of the order of the irregularity size=100 m. Then the radius of curvature should be more than 100 m. Such flat-layered irregularities are present both in the ionosphere [40,41] and in the laboratory [23].
If the flat-layer approximation is not valid, we should consider excitation of Langmuir fluctuations in a 3D plasma cavity. The solution of the problem in this case is much more complicated but in a qualitative sense the main conclusions presented above remain valid. Similarly to 1D case, the plasma waves trapped in the cavity will be represented by a set of eigenfunctions satisfying the quantization conditions. This will result in the frequency modulation of the electron density spectrum. The modulation depth will depend on the collision frequency in a way similar to that shown in figure 3, i.e. the modulation depth will decrease with an increase in the collision frequency. In order for the frequency modulation to be observed in Thomson scattering experiments, the scattering volume should normally include only one plasma cavity. In the opposite case the frequency maxima in the observable spectrum formed by many cavities will be blurred. At the same time, the total intensity of the spectrum will be enhanced when the collision frequency decreases.

Conclusions
We have calculated the spectrum of Langmuir fluctuations trapped in a parabolic irregularity of plasma density in the presence of collisional damping. Stationary fluctuation amplitudes are expressed in terms of the fluctuation source which is applicable to non-maxwellian distribution functions of particles. The source contains a broad range of wavenumbers driving a set of elementary waves which form the full fluctuation pattern.
The explicit expression for the fluctuation spectrum is modulated with respect to the frequency variable. This modulation results from reflection of the elementary Langmuir waves at the turning points and superposition of these waves. The depth and the period of the modulation depend on the electron collision frequency and the irregularity size. Observing these modulation characteristics in the experiments on the electromagnetic wave scattering by the fluctuations may provide a possibility to find such important plasma parameters as the electron collision frequency, the size and the lifetime of the irregularity.