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Electromagnetic emissions on the third harmonic of the plasma frequency ω p have been reported through the prevalence of sort II and sort III photo voltaic radio bursts (e.g., Zlotnik 1978, Zlotnik et al. 1998, Cairns; 1986), even when hardly ever and generally controversially. just lately the *Wind* The spacecraft detected a number of occasions exhibiting third harmonic emissions in domestically noticed sort III interplanetary bursts close to 1 AU (Reiner and MacDowald; 2019). For the primary time, the impression of plasma density fluctuations on the wave coalescence mechanisms answerable for the era of $mathcalH_3$ and $mathcalH_4$ harmonics electromagnetic waves at $3omega_p$ and $4omega_p$, proven.

In a two-dimensional simulation field, a beam of electrons with velocity $v_b=0.25c$ and density $n_b/n_0=5×10^-4$ ($n_0 $ is the background ion density) propagating alongside the $x$ axis generates a Langmuir wave turbulence in (i) an inhomogeneous plasma typical of sort III outburst areas within the photo voltaic wind, with random fluctuations of background density $delta n$ of imply degree $Delta N=langle (delta n/n_0)^2rangle ^1/2simeq 0.05$ and lengths waves a lot bigger than Langmuir waves, and (ii) the identical plasma however with out initially utilized density fluctuations. Particulars in regards to the bodily and numerical parameters got in latest articles (Krafft and Savoini, 2021, Krafft and Savoini, 2022).

**Determine 1:** Wave spectra within the airplane $(k_x,k_y)$, in logarithmic scales. (Left) Homogeneous plasma ($Delta N=0$) at asymptotic time $omega_pt=8760$: (a) $|B_zk|^2$ at $omega_k ∼3omega p : wave mathcalH 3 . (b) $|B_zk|^2$ in $omega ksimeq 4omega p$: wave $mathcalH_4$. (Proper) Inhomogeneous plasma ($Delta N=0.05$) at asymptotic time $omega pt=7150$: (c) $|B_zk|^2$ in $omega_ ksimeq 3omega p : wave $mathcalH_3$. (d) $|B_zk|^2$ in $omega_ksimeq 4omega_p$: wave $mathcalH_4$. All variables are normalized.

Fig. 1 reveals, in asymptotic instances, the spectral electromagnetic power densities $|B_zk|^2$ within the airplane $(k_x,k_y)$, in $omega _ okay }simeq nomega p$ ($n=3.4$), for $Delta N=0$ (left column) and $Delta N=0.05$ (proper column). You’ll be able to clearly see the round rings that signify the electromagnetic waves $mathcalH_3$ and $mathcalH_4$. Whereas for the case of homogeneous plasma these distributions present some angular dependence, they present dispersion, broadening and isotropization for the non-homogeneous case, as a result of transformations of Langmuir waves in density fluctuations, in addition to a noticeable discount in density. of power of the waves in comparison with $ Delta N=0$.

**Determine 2.** Temporal variations of wave energies (black curves) and merchandise between them (coloured curves), on a logarithmic scale. (a) $Delta N=0$: time variations of $W_mathcalH_2, $ $W_mathcalH_3$ and $W_mathcal H_4$ (black). (b) $Delta N=0$ : Time variations of $ W_mathcalL$ and $W_mathcalL^prime $ (black), $W_mathcal H 3W_mathcalH_4,$ and $W_mathcalH_2W_mathcalH_3$, labeled by $H_3ast H_4$ (inexperienced), $H_2ast H_3$ (pink). (c) $Delta N=0$ : Time variations of $W_mathcalH_2$ (black), in addition to $W_mathcalH_3W_ mathcalL,$ $W_mathcalH_3W_mathcalL^prime$ and $W_mathcalH_3W_ mathcalL^prime prime $, labeled by $H_3ast L$ (inexperienced), $H_3ast L^prime $ (blue), and $H_ 3ast L^prime prime $ (pink); the blue curve is doubled (that’s, multiplied by one other coupling issue) to point out the connection $W_ mathcalH_2propto W_mathcalH_3W_ mathcalL ^prime $ that happen at totally different instances. (d) $Delta N=0$ : Time variations of $W_mathcalH_3$ (black), in addition to $W_mathcalH_2W_ mathcalL,$ $W_mathcalH_2W_mathcalL^prime $ and $W_mathcalH_2W_ mathcalL^prime prime $, labeled by $H_2ast L$ (inexperienced), $H_2ast L^prime $ (blue), and $H_ 2ast L^prime prime $ (pink). (e) $Delta N=0.05$ : Time variations of $W_mathcalH_2,$ $W_mathcalH_3$ and $W_mathcal H_4$ (black). (f) $Delta N=0.05$ : Time variations of $W_mathcalH_2$ (black), $W_mathcalH_3W_mathcal L$, $W_mathcalH_3W_ mathcalL^prime $, and $W_mathcalH_3W_mathcal L^prime prime $, labeled by $H_3ast L$ (inexperienced), $H_3ast L^prime $ (blue), and $H_ 3 ast L^prime prime $ (pink). The energies are normalized by the kinetic power of the preliminary beam.

Figs.2a and 2e current the temporal variations of the magnetic energies $W=(1/2)iint B^2(x,y)dxdy$ of the waves $mathcalH_2, $ $mathcalH_3$ and $mathcalH_4$ (see additionally Krafft and Savoini, 2021, Krafft and Savoini, 2022), for $Delta N=0$ and $ Delta N=0.05$, respectively. The power relationships $W_mathcalH_3/W_mathcalH_2$ and $W_mathcalH_4/W_mathcal H_3$ are in line with observations of radio bursts. The presence of density fluctuations reduces the energies transported asymptotically by electromagnetic harmonics by lower than an order of magnitude. The effectivity to extract power from $mathcalH_2$ ($mathcalH_3$) waves to generate $mathcalH_3$ ($mathcalH _4$) is considerably bigger for $Delta N=0.05$ than for $Delta N=0$. In homogeneous plasma, the dominant course of producing the harmonic $mathcalH_3$ is the coalescence of $mathcalH_2$ with a Langmuir wave, $mathcalH _ 2}+% mathcalL^prime longrightarrow mathcalH_3$ , the place the backscattered wave $mathcalL^prime$ comes from the primary cascade from the electrostatic decomposition $mathcalL longrightarrow mathcalL^ prime + mathcalS^prime $, as proven in Fig. 2b the place $W_mathcal L^ prime propto W_mathcalH_2W_mathcalH_3$ and in Figs. 2(cd) the place $W_mathcalH_2propto W_mathcalH_3W_mathcalL^prime $ and $W_ mathcalH_3propto W_mathcalH_2W_mathcalL^prime $ . At later instances, coalescence $mathcalH_2+mathcalL^prime prime longrightarrow mathcalH_3$ happens with the forward-propagating wave $mathcal L^prime prime$ produced by the second cascade $mathcalL^prime longrightarrow mathcalL^prime prime+mathcal S^ prime prime$, similar to $W_mathcalH_2propto W_mathcalH_3W_mathcalL^prime prime $ (Fig. 2c) and $W_mathcalH_3propto W_mathcalH_2W_mathcalL^prime prime $ (Determine 2nd). The identical conclusions may be drawn for the era of the harmonic $mathcalH_4$ (not proven right here). When plasma comprises density fluctuations, the Langmuir wave energies are damped after saturation. Nevertheless, the harmonics $mathcalH_3$ and $mathcalH_4$ may be generated by wave coalescence regardless of heterogeneities that randomly modify the resonance circumstances of the waves by power transport in k-space. The dominant processes are the identical as for homogeneous plasma, that’s, the coalescence of harmonics $mathcalH_2$ ($mathcalH_3$) with a Langmuir wave that generates $mathcal H_3$ ($mathcalH_4$) (see additionally Fig. 2f the place $W_mathcalH_2 propto W_ mathcalL^prime W_mathcalH_3$).

**Based mostly on a latest article** by C. Krafft and P. Savoini,”*Third and Fourth Harmonics of Electromagnetic Emissions by a Weak Beam in a Photo voltaic Wind Plasma with Random Fluctuations in Density*“, ApJL 934 L28 (2022) doi: 10.3847/2041-8213/ac7f28

**References**

cairns **1986**, J. Geophys. Res., 91, 2975, doi: 10.1029/JA091iA03p0297568

Krafft, C. and Savoini, P. **2021**ApJL, 917:L23, doi: 10.3847/2041-8213/ac179570

Reiner, M.J. and MacDowall, R.J. **2019**SoPh, 294, 91, doi: 10.1007/s11207-019-1476-973

Zlotnik, IE **1978**Ast. soviet, 55, 399

Zlotnik, EY, Klassen, A., Klein, KL, Aurass, H., Mann, G. **1998**A&A, 331, 1087.

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