Second Harmonic Electromagnetic Wave Emissions from a Turbulent Plasma with Random Density Fluctuations

In the solar wind, electromagnetic waves at the harmonic plasma frequency 2ω p can be generated as a result of coalescence between forward- and backward-propagating Langmuir waves. A new approach to calculate their radiation efficiency in plasmas with external background density fluctuations is developed. The evolution of Langmuir wave turbulence is studied by solving numerically the Zakharov equations in a two-dimensional randomly inhomogeneous plasma. Then, the dynamics of the nonlinear electric currents modulated at frequencies close to 2ω p are calculated, as well as their radiation into harmonic electromagnetic waves. In the frame of this non-self-consistent approach where all transformations of Langmuir waves on density inhomogeneities are taken into account, the electromagnetic wave radiation rate (emissivity) is determined numerically as well as analytically, providing in both cases similar results. Moreover, scaling laws of the harmonic wave emissivity as a function of the ratio of the light velocity to the electron plasma thermal velocity are found. It is also shown how the emissivity depends on the average level of density fluctuations and on the isotropic/anisotropic character of the Langmuir waves’ and density fluctuations’ spectra.


Introduction
Observations in the solar wind have reported electromagnetic waves emitted during type III solar radio bursts at the second harmonic of the electron plasma frequency ω p (e.g., Wild et al. 1953;Fainberg et al. 1972;Stewart et al. 1974;Kaiser 1975;Dulk 1985;Suzuki & Dulk 1985;Robinson & Cairns 1998).They are sometimes accompanied by radiation at the fundamental frequency (Dulk et al. 1980) and commonly emitted in the kilometric and hectometric wavelengths' ranges (e.g., Alvarez & Haddock 1973).
To date, several processes have been proposed to explain harmonic electromagnetic radiation by Langmuir wave turbulence generated by electron beams released by flares.First, in the frame of the weak turbulence theory, it was argued that forward  and backward ¢  propagating Langmuir waves, which can for example be involved in the electrostatic decay  ¢ + ¢    , where ¢  is an ion-acoustic wave, can merge to produce electromagnetic waves  at frequency 2ω p according to the coalescence process + ¢     (e.g., Ginzburg & Zheleznyakov 1958;Cairns & Melrose 1985;Robinson & Cairns 1994).In the framework of another approach, linear mode conversion was invoked to explain such electromagnetic emissions.Indeed, a Langmuir/Z-mode wave incident on a density gradient of size larger than the skin depth c/ω p can be transformed into an electromagnetic wave at the same frequency.In such a process, the reflected Langmuir waves can couple with the incident ones to provide electromagnetic waves emitted at 2ω p (e.g., Kim et al. 2008).A third process, the so-called antenna radiation mechanism of localized Langmuir waves, was also proposed (Malaspina et al. 2013).In this case, the nonlinear second-order current associated with the velocity and the density perturbations of the electron beam population is shown to produce an electromagnetic radiation at frequency 2ω p .Finally, processes based on the strong turbulence theory were also investigated (Galeev & Krasnoselʼskikh 1976;Kruchina et al. 1980), as electromagnetic radiation from intense localized Langmuir waves (Malaspina et al. 2010) or solitons (Papadopoulos & Freund 1978;Hafizi & Goldman 1981), for example.More recently, a theoretical model of generation of harmonic emissions of type III solar radio bursts in strongly inhomogeneous plasmas was proposed (Tkachenko et al. 2021).
Two-dimensional particle-in-cell (PIC) simulations were also performed, evidencing electromagnetic emissions at 2ω p resulting from Langmuir waves radiated by an electron beam (Kasaba et al. 2001;Umeda 2010;Henri et al. 2019;Lee et al. 2019;Krafft & Savoini 2021).Recently, Henri et al. (2019) studied the efficiency of the process leading to harmonic emissions and the directivity of this radiation, whereas Lee et al. (2019) solved the electromagnetic weak turbulence equations in the presence of a beam and compared these results with PIC simulations.Moreover, Krafft & Savoini (2021) studied for the first time the radiation of harmonic electromagnetic waves in a plasma with random density fluctuations, compared it to the homogeneous plasma case, and demonstrated that a three-wave coalescence process actually takes place in both cases.On another hand, using fluid simulations and strong turbulence conditions, Akimoto et al. (1988) evidenced electromagnetic harmonic emissions due to the transverse instability of Langmuir solitons, in agreement with the theoretical predictions.
Solar wind and coronal plasmas contain density fluctuations δn of various average levels n n 0 2 1 2 and scales, as revealed by several observations (Celnikier et al. 1987;Kellogg et al. 1999;Krupar et al. 2018Krupar et al. , 2020)).As the frequency of harmonic electromagnetic waves is well above the plasma frequency, they are expected to be very weakly scattered by the fluctuations during their radiation.However, this is not the case for the beam-driven  and backscattered ¢  Langmuir waves (resulting from electrostatic decay or wave transformations on the fluctuations), which are merging through the coalescence process + ¢    .Indeed, the modifications of the Langmuir spectra due to the presence of density fluctuations can be large and the subsequent amount of energy carried by the harmonic electromagnetic waves at saturation can be significantly modified, compared to the case of a homogeneous plasma (Krafft & Savoini 2021).Moreover, the random modifications of the wave-wave resonance conditions induced by Langmuir wave scattering on density fluctuations can influence the occurrence of the wave coalescence process.
This paper studies the impact of background plasma's random density fluctuations and electron temperature on the emissivity of the electromagnetic waves radiated at 2ω p , by developing a semianalytical and semi-numerical theory to calculate the radiation efficiency of electromagnetic waves in a turbulent inhomogeneous plasma.This model describes the generation of harmonic electromagnetic waves through the action of two mechanisms: The first is the linear conversion of Langmuir waves during their interactions with background density fluctuations; the transformations they encounter influence significantly on their spectra (broadening, diffusion, ...), producing in particular backscattered Langmuir waves (Krafft et al. 2015;Krafft & Volokitin 2016, 2019;Volokitin & Krafft, 2016;Krafft & Savoini 2021;Krafft & Volokitin 2021).The second mechanism is the coupling of two Langmuir waves at the origin of the nonlinear current generating the magnetic field of harmonic electromagnetic waves.We note that such effects were also considered in works using the quasilinear theory equations (e.g., Ratcliffe & Kontar 2014).However, our study presents an essential difference with such works where averaging on random wave phases is performed.Indeed, we take into account the significant modifications of Langmuir spectra due to the waves' transformations on the density fluctuations, which strongly influence the harmonic electromagnetic radiation.Moreover, the model provides a first-order differential equation that can be integrated numerically with a method developed by the authors (Volokitin & Krafft 2020), enabling for the first time to obtain the radiation rate (emissivity) of the harmonic electromagnetic waves in a randomly inhomogeneous plasma, as well as its dependence as a function of the average level of density fluctuations, the electron plasma temperature, and the Langmuir waves' and density fluctuations' spectra.Finally, this work allows for generalization of the scaling laws provided by the weak and strong turbulence theories, developed for homogeneous plasmas, and it extends their applicability to new situations.

Nonlinear Electric Current
Let us consider a plasma source where low-intensity Langmuir wave turbulence is developed, and which includes density fluctuations δn at an average level few percent of the average background plasma density n 0 , with characteristic wavelengths much larger than those of Langmuir waves.Let us assume that all Langmuir wave energy converted into electromagnetic waves leaves the inhomogeneous plasma volume and propagates conservatively through an external homogeneous plasma.Electric currents with modulation frequency close to 2ω p , which are generated in the plasma source due to the nonlinearity of the electrons' oscillations in the Langmuir waves' fields, radiate electromagnetic wave energy at the same frequency.
The nonlinear (second-order) current j (2) can be calculated using the hydrodynamic description; in a background plasma with slow density fluctuations δn/n 0 , its envelope where j  is the potential's envelope ( of the Langmuir waves in the linear approximation; −e < 0 and m e are the electron charge and mass.In (1), the current ( )  j 2 calculated in a homogeneous plasma has been replaced by ( ) Galeev & Krasnoselʼskikh (1976).However, the correction provided by δn/n 0 turns out to be small and can be omitted, as one can verify by a simple estimation and as proven by the simulations presented below.Further, similarly to calculations performed for electromagnetic wave radiation at the fundamental plasma frequency ω p (Volokitin & Krafft 2020), we calculate the growth rate of the energy of electromagnetic waves generated by electric currents with a modulation frequency of 2ω p .
In a nonmagnetized plasma, the evolution of the magnetic field B with the electric current density j is described by the equation (e.g., Volokitin & Krafft 2020) Extracting from (2) the slowly varying envelope Then, using (1), one obtains the following equation in the wavevector space: B 2 and Equation (4) can be solved analytically if its right-hand side is known.If, as assumed, electromagnetic radiation freely leaves its region of generation and if its intensity is too low to be able to trigger a counter-effect on the Langmuir wave turbulence, then the determination of the right-hand side of Equation (4)i.e., of the nonlinear current responsible for the generation of electromagnetic waves-can be considered independently.This is achieved in the next section, where Langmuir wave turbulence is calculated numerically in an inhomogeneous plasma by solving the Zakharov equations.The integration of Equation (4) provides that where the second right-hand side term is a particular solution of (4).Equation (7) can be solved analytically by using the method provided by the authors in a previous work (Volokitin & Krafft 2020), as shown below.

Langmuir Wave Turbulence Modeling
In our previous works (Volokitin & Krafft 2018;Krafft & Volokitin 2019;Volokitin & Krafft 2020), we presented a twodimensional (2D) modeling of Langmuir wave turbulence in a plasma with external density fluctuations that was used, in particular, to determine the efficiency of electromagnetic radiation at the fundamental plasma frequency ω p .This model also makes it possible to calculate the 2D nonlinear current ( ) (1) as a function of time, by using the density fluctuations δn(r, t) and the scalar potentials' envelopes ( ) j  r t , obtained by solving the high-frequency Zakharov equation: where λ D is the electron Debye length.We use this equation with applied density fluctuations δn, so that it becomes linear.
For the solar wind plasmas considered here, where the weakness of the turbulence parameter permits the ponderomotive effects to be neglected, the second low-frequency Zakharov equation can be also used in its linearized form: i 1 2 is the ion-acoustic velocity; m i is the proton mass; T e and T i are the background electron and ion temperatures; and the electric field of Langmuir waves ( ( ) i t p can be obtained by means of the slowly varying envelope ( ) j  r t , .Then, the combination of Equations (8) and (9) enables us to calculate non-selfconsistently the evolution of the Langmuir waves' potential's envelope j  , providing the nonlinear current ( )

 j
2 .In the model, Langmuir wave turbulence is maintained at a stationary level, and the total wave energy is fixed.No mechanisms responsible for wave generation and damping are included.The turbulence parameter ranges as 10 −5  W L /n 0 T e  10 −4 , where W L is the energy density carried by the Langmuir wave turbulence, corresponding to electron beams with very weak relative densities satisfying n b /n 0  10 −5 .We note that three-wave nonlinear interactions involving low-frequency waves cannot manifest, due to the absence of ponderomotive force in Equation (9).Thus, the main mechanism described by Equations ( 8) and (9) consists of Langmuir waves' transformations on the density fluctuations, whereas the interaction processes between the scattered Langmuir waves responsible for the generation of the nonlinear current and the magnetic field are described by Equations (4) and (5).
For simplicity, and because the nature of low-frequency oscillations does not matter here, those are represented by ionacoustic waves whose dynamics is calculated.Let us recall that the applied and induced density fluctuations are contained in the term δn/n 0 .The fluctuations applied initially to the background plasma density form a wide spectrum of ion-acoustic waves described by the linear Equation (9), and their corresponding spectra 2 are Gaussian distributions with random phases.The characteristic scales λ δn of these fluctuations largely exceed the Langmuir wavelengths λ L (λ δn ∼ 10-50λ L ), and their average levels n n 0 2 1 2 vary within the range 0.01-0.05.In the absence of high-frequency pressure, the main effects induced by density fluctuations consist in spreading and broadening of wave spectra, i.e., diffusion of wave energy over wavevectors' scales and propagation angles.
Our simulations are performed for Langmuir waves with wavevectors lying in the range where Landau damping is not essential.However, due to the scattering of these waves on the density fluctuations, their wavenumbers can be modified so that a small part of them can fall in the damping region.Our model does not take this fact into account, as those waves have large wavevectors and are not able to participate in electromagnetic radiation.
Large boxes of sizes of several thousands of Debye lengths are used in simulations (L x = 16, 000λ D , L y = 8000λ D ), with N x , N y = 1024-4096 grid points along each direction.Initially, the Langmuir waves are distributed mostly within the parallel wavevectors' range 0.01 k x λ D 0.15, with perpendicular wavevectors lying in the interval − 0.1  k y λ D  0.1, in agreement with typical Langmuir spectra obtained by PIC simulations with electron beams (Krafft & Savoini 2021).

Radiated Magnetic Field
As a result of the numerical modeling, two-dimensional distributions of the potential ( ) j  r t , i and the density δn(r, t i ) are obtained for sequences of times t (1), and using fast Fourier transforms, to determine their spectra ( ) In the 2D geometry considered here, the magnetic field radiated at 2ω p has only one component directed along the z-axis, i.e., ( ) ( ) , where z is the corresponding unitary vector.Then, using (7), we can write that x y , 2 at time t.It should be noted that the right-hand side of (11) is calculated directly using the Fourier transforms of the spatial distributions of nonlinear currents provided by the solutions of Equations ( 8) and (9), which significantly reduces the computing time.Then the magnetic field can be calculated at any time t as follows (Volokitin & Krafft 2020): and the integral in the second right-hand side term can be expressed as These analytical expressions are used to compute numerically the time evolution of the radiated magnetic field ( )


B zk 2 and thus the electromagnetic energy density as well as the emissivity, as shown below.

Numerical Simulations
Simulations are performed for various initial spectral distributions of Langmuir waves' energy and plasma density fluctuations, as well as different values of the average level ΔN and the velocity ratio c L = c/v T , where v T is the electron plasma thermal velocity.We note that some of these simulations were used previously by the authors (Volokitin & Krafft 2018) to study the emissivity of electromagnetic emissions at the fundamental plasma frequency.
Because the problem considered here is not self-consistent, the amplitudes of the electric field's envelopes j = - ~ E can be normalized arbitrarily.The simulation results are presented below in dimensionless quantities according to the following normalization: The normalized Fourier components E k (E xk , E yk ) of the dimensionless electric field envelope E(E x , E y ) are given by Normalizing the potential and the magnetic field according to , respectively, one can write Equation (4) in a dimensionless form In what follows, the normalized current spectrum J k is used in the numerical integration of Equation (3) to determine the spectrum of electromagnetic waves at the second harmonic . 17 Then, using Equations ( 12)-( 13), we get The effect of temperature on the rate of magnetic field generation manifests itself through the parameter c L = c/v T , which is included in the rapidly oscillating factor under the integral in (17) and (18).Then one can calculate ( ) b t k at any time, as well as the normalized electromagnetic energy: where we define, for ω k = 2ω p , the normalized electromagnetic and magnetic energies by the dimensionless parameters η and μ, respectively: Below, amplitudes and energies of electromagnetic waves are presented in figures using the dimensionless quantities | | b k , η, and μ, which depend only on the shape of the Langmuir wave spectrum and not on the level of wave turbulence W L /n 0 T e .If the average level of density fluctuations is not very small (0.05  ΔN  0.01), the scattering and the subsequent diffusion of Langmuir waves in the wavevector space becomes the main process, which is faster than other nonlinear processes.This allows us to use a non-self-consistent approach, by building a linearized theoretical model that neglects the effects of Langmuir waves on the density fluctuations and the counter-effects of electromagnetic waves on the Langmuir wave turbulence, which is valid if the level of wave turbulence is small enough, as supposed here.The linearization performed allows us to normalize all terms of equations using the Langmuir wave intensity W L , in order to study the radiation rate dη/dt for a wide set of parameters.Our model does not include the beam itself, but one has to note that W L is proportional to the beam density n b .Then, the analytical formulas obtained in (17)-( 20) can also be considered as normalized by the beam energy, at a given velocity.As our model is not self-consistent, W L is a free parameter, and the radiation rate of harmonic electromagnetic waves can be calculated if W L is known (i.e., if the beam characteristics are known), as well as the density fluctuations' spectrum.
The values used below for c L , i.e., 40 c L 70 are significantly smaller than the realistic ones in the solar wind.This is due to numerical reasons, as larger values of c L would require the use of smaller time steps of integration Δt in Equations ( 16)-( 18); doing so would preclude the detailed numerical studies, presented below, that determine the dependence of the radiation rates on ΔN, c L and the initial waves' and density fluctuations' spectra.Nevertheless, one can use such values of c L to find the power-law index α allowing for the proportionality of dμ/dt and dη/dt to a c L , as shown in the figures in this section, especially given that the value of α obtained numerically matches with that derived using analytical calculations (see Section 4), which are valid in the case of the solar wind.
Figure 1 2 , as harmonic electromagnetic emissions are produced by the coalescence of two Langmuir waves.
Let us first consider the case of both isotropic initial spectra of Langmuir waves and plasma density fluctuations (see also the left column of Figure 6).Figure 2 shows, for four values of c L , the growth with time of the magnetic and electromagnetic wave energies μ(t) and η(t) (20), which occurs unevenly, due to the evolution of the Langmuir waves' spectra.During a first stage, a fast increase of μ(t) and η(t) can be observed (right column), corresponding to the fast growth of Langmuir waves scattered on the density fluctuations (see also Krafft & Volokitin 2019;Volokitin & Krafft 2020;Krafft & Volokitin 2021).Later on, μ(t) and η (t) grow almost linearly (however, small deviations remain) and more slowly while Langmuir spectra stabilize, exhibiting at this stage only small perturbations around a mean level.Using the same approach as in our previous work dealing with fundamental emission (Volokitin & Krafft 2018, 2020), we define the radiation rate of the electromagnetic (magnetic) harmonic energy as the slope ) of the linear stage of the time evolution of η(t) (μ(t)).We note that such a definition is reasonable only if, at late times, the plasma turbulence parameter W L /n 0 T e can be considered stationary, which occurs at ω p t  5000 in our simulations.At earlier times, the evolution of the Langmuir wave spectra as a result of wave scattering on density fluctuations is not achieved.Green (cyan) lines superimposed on μ(t) (η(t)) show the linear approximation of its asymptotic time dependence, obtained after removing fast fluctuations by averaging.Small deviations from the linear dependence indicate that the simulated wave turbulence has not reached a perfect stationary state.Indeed, Langmuir waves are initiated in the form of a narrow Gaussian wave packet in k-space, but later on, as discussed in detail in previous works (Krafft & Volokitin 2019;Volokitin & Krafft 2020), a significant spectral broadening occurs during their scattering on density inhomogeneities, until a quasi-stationary state is established and further variations in the spectrum are small.
Moreover, the variation of dη/dt and dμ/dt with 1/c L is presented in the left column of Figure 2, showing that the following scaling law is satisfied: Such dependence of the radiation rates h  and m  with c L = c/v T was also found, with good accuracy, for simulations with other values of characteristic parameters.For comparison, Figures 3  and 4 show similar results obtained for a larger average level of density fluctuations ΔN = 0.04, when (i) the initial Langmuir wave spectrum is isotropic but the density fluctuations' spectrum is strongly anisotropic (Figure 3), as well as when (ii) both initial spectra are anisotropic (Figure 4).Similar conclusions can be stated, whatever ΔN, c L , and the initial density fluctuations' spectra are.Indeed, Figure 5 shows the dependence m µ -  c L 4 for five simulations with the values ΔN = 0.01, 0.02, 0.03, 0.04, and 0.05, for the case when the initial waves' and density fluctuations' spectra are both isotropic.Here, α is the calculated power-law index m h µ a -  c , L , which is close to the expected one, i.e., α ; 4 ; s + 2, where s is the dimension of the system (here, s = 2).The scaling index is derived analytically below in Section 4, for 2D and 3D geometry, and was obtained in earlier works for homogeneous plasmas only, in the framework of weak turbulence theory (Aamodt & Drummond 1964;Sagdeev & Galeev 1969).The variations of α with ΔN as well as its scattering around the theoretically derived power-law index are discussed below in detail.
We performed a series of calculations with different levels of density fluctuations and used three types of initial conditions (see Figure 6): (i) isotropic Langmuir waves' and density fluctuations' spectra (left column), (ii) isotropic Langmuir wave spectrum, and strongly anisotropic spectrum of density fluctuations of characteristic spatial scales many times larger along x-the external magnetic field direction-than along y(middle column), and (iii) both anisotropic Langmuir wave and density spectra, with the spectrum of waves extended along the positive x direction (right column).In the latter case, the anisotropic wave spectrum mimics the situation when the Langmuir wave turbulence is generated by a type III electron beam.
Figure 7 shows the dependence of m  on the average level of density fluctuations ΔN, for different values of c L , in the case when both initial wave and density spectra are isotropic; due to the scaling m µ -  c L 4 highlighted above, it is suitable to present the variation of ( ) m  c 70 weakly depends on ΔN within the range 0.02  ΔN  0.04, whereas its value is approximately doubled at ΔN = 0.01; at larger ΔN = 0.05, the radiation efficiency tends to decrease.When simulations are performed with, initially, a strongly anisotropic density fluctuations' distribution and an isotropic spectrum of Langmuir waves (Figure 6, middle column), the results obtained in Figure 8 show small differences from the previous ones (Figure 7), as ( ) m  c 70 L 4 also decreases with increasing ΔN.Finally, Figure 9 shows the variation of ( ) m  c 70 L 4 with ΔN when both initial wave and density spectra are strongly anisotropic (Figure 6, right column).The dependence of the emission efficiency on ΔN exhibits a trend opposite to that of the previous cases (Figures 7 and 8); indeed, ( ) m  c 70 L 4 increases with increasing ΔN, from ΔN = 0.01 up to ΔN = 0.05.The differences observed between the three cases with different initial wave and density distributions can be explained by the different nature of the time evolution of the Langmuir wave spectra.Before discussing this in greater detail (see below), it is necessary to present the analytical calculations providing the electromagnetic radiation rate and the power-law index α.

Analytical Determination of the Radiation Efficiency
In this section, let us return back to dimensional quantities and 3D geometry.In agreement with numerical simulations' results presented above, the term δn/n 0 in Equation (1) brings a minor correction; therefore, in what follows, we can use the approximated expression (5).After symmetrization, one gets where k 1 and k 2 (j  k 1 and j  k 2 ) are the wavevectors (potential's envelopes) of the Langmuir waves that coalesce to produce harmonic electromagnetic waves with the wavevector k.
Integrating Equation (4), we find, similarly to Equation (7), the following expression (in which, for convenience, we write

Bk instead of
where Ω k is given by (6) in its normalized form.Then, expressing the electric field envelope as , we can single out the phases of the waves linearly changing with time according to the dispersion law as , so that If the plasmon amplitudes are constant and the phases are correlated as, for example, in the absence of density fluctuations (δn = 0), then 7 The Astrophysical Journal, 964:65 (13pp), 2024 March 20 where the superscript " * " indicates the complex conjugates.Let us average this expression over random phase fluctuations, assuming that the correlation between the phases of plasmons propagating in opposite directions decreases with time according to where the symbol áñ designs ensemble averaging.This assumption is quite often true for random processes, with ) vary more slowly than oscillating terms of the integrand, we can take them out of the integrals and get, after averaging: where   At the asymptotic stage when t is large, one can calculate that if ν is not too large.According to (28)-( 30), the amplitudes of electromagnetic oscillations grow linearly with time, as shown by the numerical simulations.The most rapid growth occurs in eigen electromagnetic waves, for which the condition | | n DW  k is fulfilled with high accuracy.Non-eigen electromagnetic oscillations can also grow under the influence of Langmuir waves' beats, but these forced oscillations do not exhibit linear growth with time and their energy remains small.
We note that the above calculations closely replicate those usually done in the framework of weak turbulence theory, and accordingly, they lead to a similar result.However, the assumptions of our model, which considers plasmas with external density fluctuations, are different from those that underlie the weak turbulence theory.Moreover, we note that an expression similar to (28) is also obtained in the theory of strong Langmuir turbulence (Kruchina et al. 1980); in addition, the latter work quite reasonably assumes that ν weakly depends on k, i.e., ( ) n w ~Dn n p , where . In any case, the smallness of ν allows us to use the asymptotic approximation (30).Then, summing on k, we obtain the time derivative of the magnetic energy in the following form: As k = k 1 + k 2 , and due to the smallness of the electromagnetic wavevector k, i.e., , so that the rate of growth of the magnetic energy is Then, using , where V s is the volume of the system of dimension s, we get the radiation rate of the magnetic energy in the   where  s is the integral over angles in cylindrical (2D case, s = 2) or spherical (3D case, s = 3) coordinates, ( ) ò ò ò q q q q q q j = = where θ is the angle between k 1 and k ( ).In Finally, the radiation efficiency of the magnetic energy can be obtained in normalized variables as a function of the spectra of the forward-and backward-propagating Langmuir waves, showing that, for s = 2, the power-law index is α = 4, as obtained above by numerical simulations.One gets the same scaling for the electromagnetic energy, i.e., h µ -  c L 4 .For s = 3 (3D geometry), one recovers that h m µ -   c , L 5 , which was also obtained in the framework of the weak and the strong turbulence theories for plasmas without external density fluctuations, as mentioned above (Aamodt & Drummond 1964;Kruchina et al. 1980).

Comparison between Theory and Simulation Results
The scaling laws h m µ -   c , L 4 derived from the numerical simulations (Figures 2 to 5) diverge from the theoretical calculations presented above only by relatively small values, which typically do not exceed 10%-15%.One reason for such a discrepancy is due to the unaccuracies inherent to the numerical integration of Equations ( 12)-( 13) with a finite time step Δt.Reducing Δt below the values used in this work (2 ω p Δt 5.4) was hardly acceptable in terms of storage and computing time, for the numerous cases considered here with various values of ΔN, c L , and initial wave and density spectra.However, as shown by the numerical analysis we performed (not presented here), the values of Δt used ensure an accuracy of a few percent, which is quite satisfactory.The most significant numerical discrepancies with respect to the analytical results appear when both the conditions ΔN  0.01 and c L  70 are fulfilled.Another source of computational inaccuracies is the discreteness of the Langmuir wave spectra (due to the finite box size), which manifests mostly at large c L when the number of waves contributing to the radiation  decreases and the fluctuations of the radiated magnetic field grow.The third and most significant reason for the small deviation of the simulation results from the theoretical scaling laws lies in the temporal evolution of the Langmuir wave spectra.Even if the shapes of two Langmuir wave spectra are initially the same, they become different when the wave turbulence steady state is reached, and in fact, their broadening depends on ΔN (Volokitin & Krafft 2020).Fourth, in all the considered cases, a stationary state for electromagnetic waves is not established during the simulation time (ω p t  12,000), such that their radiation efficiency cannot be determined with an excellent accuracy.
At a given Langmuir wave turbulence level, the type of initial wave spectrum has a noticeable effect on the radiation efficiencies m  and h  .Let us discuss their dependences on the average level of density fluctuations ΔN (see Figures 7-9).The normalized Equation (36) can be written as follows: where s = 2 in 2D geometry.This formula allows us to understand how ΔN affects m  in the case of an isotropic Langmuir wave spectrum | | E k 2 1 .As shown in the upper row of Figure 10, such spectrum broadens with time while its shape remains unchanged.Then, taking into account that Langmuir wave energy is conserved, i.e., , one can write in the 2D case that where Δk is the spectral broadening.According to Krafft & Volokitin (2021), one can write at the wave turbulence steady state that ( ) , so that one gets from (37) and (38) the scaling law m ~D  N 1 . The results shown in Figures 7 and 8, obtained with isotropic Langmuir spectra, qualitatively agree with this dependence; small discrepancies are most likely due to the impact of spectral broadening on the scaling index.
When the Langmuir wave spectrum is initially anisotropic (see the bottom row of Figure 10), it also broadens with time but conserves its anisotropy, which is responsible for the small radiation rate at small ΔN (see Figure 9).Indeed, as a result of anisotropy overlap only partially, which leads to the decrease of the radiation efficiency m  , proportional to the Langmuir wave energy product , with decreasing ΔN, and to its growth with increasing ΔN (see Figure 9); correspondingly, a higher degree of isotropy leads to an increase in the overlap between the spectra 1 .This reasoning is in good agreement with the results shown in Figure 10, where the time evolutions of three different Langmuir wave spectra are illustrated.

Conclusion
The radiation efficiency of electromagnetic waves emitted at frequency 2ω p is calculated owing to a 2D numerical model based on the Zakharov equations as well as by performing analytical calculations.It is supposed that Langmuir wave turbulence is produced in a plasma with random density fluctuations and radiates harmonic electromagnetic waves as a result of the coalescence between forward-and backwardpropagating Langmuir waves, which are then escaping out of the source into a homogeneous plasma.The model, which allows the description of all linear processes of Langmuir waves' transformations on density fluctuations, is not selfconsistent, as the electromagnetic waves are supposed not to exert any back-effect on the Langmuir wave spectra.
The magnetic energy carried by the harmonic waves radiated by the nonlinear current of the randomly inhomogeneous plasma source can be calculated as a function of time.Electromagnetic waves grow without reaching saturation, being continuously produced by the Langmuir wave turbulence through linear conversion mechanisms that transfer electrostatic energy into electromagnetic energy.
The analysis of the numerical simulations has been accompanied by analytical calculations based on a method developed previously by the authors.Both studies lead to the same results, i.e., (i) the emissivity of harmonic waves is found to scale as a c L (α = 4 and α = 5 in 2D and 3D geometry, respectively), where c L = c/v T is the ratio of the light velocity to the electron plasma thermal velocity, (ii) the initial Langmuir wave spectrum has a noticeable influence on the power index α, and (iii) the emissivity of harmonic waves depends as 1/ΔN on the average level of density fluctuations ΔN for initially isotropic Langmuir spectra (for anisotropic ones, a rather weak increase of emissivity is observed with increasing ΔN).
Then, it is possible to determine, for given measured or analytically modeled Langmuir spectra, the emissivity of harmonic electromagnetic waves radiated in the solar wind, for various electron temperatures, average levels of density fluctuations, and volumes of the emitting source.The scattering of Langmuir waves on plasma density fluctuations, which produces backscattered Langmuir waves, is much more fast than nonlinear wave interaction processes-and in particular, than the so-called electrostatic decay that also generates backward-propagating Langmuir waves.In such an optically thin plasma source as considered here, this fact should have a strong importance, because the escaping harmonic waves transport out of the source a large amount of energy carried by the Langmuir wave turbulence.Then, less energy remains available for other processes, including the electrostatic decay, which could therefore not play a decisive role in actual solar wind plasmas where random density fluctuations are present.
Finally, even if the scaling laws derived for plasmas with external random density fluctuations are in agreement with those provided by the weak and strong turbulence theories, developed for homogeneous plasmas, this work allows them to be generalized and their applicability to be extended to new situations.where ( ) dn e 1 and ( ) dv e 1 are the oscillations of the electron density and velocity at plasma frequency ω p , in the linear approximation; ( )  dn e 2 are ( ) dv e 2 are the second-order fast electron density and velocity perturbations; n 0 is the density of the background plasma; and −e < 0 is the electron charge.
The Poisson equation, as well as the electron hydrodynamic and conservation equations, provide at first order of perturbations that where j (1) is the first-order potential.The perturbation ( ) dv e 2 can be calculated using the electron motion equation at second order:

L
presents the spectral distributions of the normalized magnetic wave amplitude | ( )| b t k at four different times, for c L = 50 and a weakly inhomogeneous plasma with ΔN = 0.01.According to the rapidly oscillating exponential in the integrand of (18), one can expect that electromagnetic waves grow most strongly at normalized wavevectors satisfyis observed in Figure1via the formation of a circular emission pattern of normalized radius k ; 0.035, very close to the dimensionless wavenumber  for c L = 50.In particular, there must be enough pairs (k 1 , k 2 ) of Langmuir waves satisfying the condition

Figure 1 .
Figure 1.Spectral distributions, in the k x − k y space, of the normalized magnetic wave field amplitude | ( )| b t k at times ω p t = 11, 81, 405, and 759, for c L = 50 and ΔN = 0.01.All variables are normalized.

Figure 2 .
Figure 2. (Top right panel) Time variation of the normalized magnetic wave energy μ(t) radiated at frequency 2ω p (solid, dashed, dashed-dotted, and dotted lines), for four values of c L = 40, 50, 60, and 70, respectively.Linear interpolations (green lines) are superimposed to μ(t). (Top left panel) Variation of m m =  d dt with 1/c L , in logarithmic scales; numerical results are represented by black stars; the scaling laws a c L are shown in blue (green) for α = 3 (α = 4).(Bottom right panel) Time variation of the normalized electromagnetic wave energy η(t) radiated at frequency 2ω p (solid, dashed, dashed-dotted, and dotted lines), for four values of c L = 40, 50, 60, and 70, respectively.Linear interpolations (cyan lines) are superimposed to η(t).(Bottom left panel) Variation of h h =  d dt with 1/c L , in logarithmic scales.Numerical results are represented by diamonds, and the scaling laws a c L are shown in blue (green) for α = 3 (α = 4).Initial plasma density fluctuations' and Langmuir wave energy's spectra are isotropic, with an average level of random inhomogeneities ΔN = 0.02.

Figure 3 .
Figure 3. (Top right panel) Time variation of the normalized magnetic wave energy μ(t) radiated at frequency 2ω p (solid, dashed, dashed-dotted, and dotted lines), for four values of c L = 40, 50, 60, and 70, respectively.Linear interpolations (green lines) are superimposed to μ(t). (Top left panel) Variation of m m =  d dt with 1/c L , in logarithmic scales; numerical results are represented by stars; lines with scaling a c L are shown in blue (green) for α = 3 (α = 4).(Bottom right panel) Time variation of the normalized electromagnetic wave energy η(t) radiated at frequency 2ω p (solid, dashed, dashed-dotted and dotted lines), for four values of c L = 40, 50, 60, and 70, respectively.Linear interpolations (cyan lines) are superimposed to η(t).(Bottom left panel) Variation of m m =  d dt with 1/c L , in logarithmic scales; numerical results are represented by diamonds; lines with scaling a c L are shown in blue for α = 3 and green for α = 4.The initial Langmuir wave energy spectrum is isotropic, whereas the density fluctuations' spectrum is anisotropic, with random inhomogeneities of average level ΔN = 0.04.

Figure 4 .
Figure 4. (Top right panel) Time variation of the normalized magnetic wave energy μ(t) radiated at frequency 2ω p (solid, dashed, dashed-dotted, and dotted lines), for four values of c L = 40, 50, 60, and 70, respectively.Linear interpolations (green lines) are superimposed to μ(t). (Top left panel) Variation of m m =  d dt with 1/c L , in logarithmic scales; numerical results are represented by stars; lines with scaling a c L are shown in blue (green) for α = 3 (α = 4).(Bottom right panel) Time variation of the normalized electromagnetic wave energy η(t) radiated at frequency 2ω p (solid, dashed, dashed-dotted, and dotted lines), for four values of c L = 40, 50, 60, and 70, respectively.Linear interpolations (cyan lines) are superimposed to η(t).(Bottom left panel) Variation of m m =  d dt with 1/c L , in logarithmic scales; numerical results are represented by diamonds; lines with scaling a c L are shown in blue for α = 3 and green for α = 4. Initial Langmuir wave energy and density fluctuations' spectra are both anisotropic, with random inhomogeneities of average level ΔN = 0.03.

Figure 5 .
Figure 5. Variation of the radiation rate m  with 1/c L , in logarithmic scales, for five different values of the average level of density fluctuations ΔN = 0.01, 0.02, 0.03, 0.04, and 0.05 (see the inset); red dashed straight lines indicate the best fits.The initial Langmuir wave and density fluctuations' spectra are both isotropic.The power-law index α found is indicated in the inset for each case.

Figure 7 .
Figure 7. Variation of the radiation rate ( )m  c 70 L 4 with average level of density fluctuations ΔN, four values of c L = 40, 50, 60, and 70.The initial Langmuir wave and density fluctuations' spectra are both isotropic (left column of Figure 6).

Figure 9 .
Figure 9. Variation of the radiation rate ( )m  c 70 L 4 with the average level of density fluctuations ΔN, for four values of c L = 40, 50, 60, and 70.The initial Langmuir wave and density fluctuations' spectra are both anisotropic (right column of Figure 6).

Figure 8 .
Figure 8. Variation of the radiation rate ( )m  c 70 L 4 with the average level of density fluctuations ΔN, for four values of c L = 40, 50, 60, and 70.The initial Langmuir wave spectrum is isotropic, whereas the density fluctuations' spectrum is anisotropic (middle column of Figure 6).

Figure 10 .
Figure 10.Time evolution of the three Langmuir wave spectra shown at t = 0 in Figure 6.(Upper row) Isotropic Langmuir wave spectrum, at ω p t = 10, 1000, and 3000, which keeps its isotropic character during time evolution.(Middle row) Isotropic Langmuir wave spectrum, at ω p t = 27, 2700, and 8100, tending to isotropy with time.(Bottom row) Anisotropic Langmuir wave spectrum, elongated initially along the magnetic field direction, at ω p t = 27, 2700, and 8100, and keeping its anisotropy with time.All variables are normalized.The average level of density fluctuations is ΔN = 0.04.For each row, the density fluctuations' spectrum used is shown at t = 0 in the bottom row of Figure 6.
where j(2)is the second-order potential.Moreover, the Poisson and the electron conservation equations are written at second order as