Paper: Third and Fourth Harmonics of Electromagnetic Emissions by a Weak Beam in a Solar Wind Plasma with Random Density Fluctuations
Authors: C. Krafft, P. Savoini
Figure caption: Examples of cross sections of the auto- and cross-bicoherences b  c  of the magnetic and electric fields in the parallel wavenumber and frequency planes. The left (right) column refers to homogeneous (inhomogeneous) plasma. For each panel, the bottom shows the maxima of bicoherence (b  c ), their location in frequency (ω 1, ω 2) and wavenumber (k  x1, k  x2) spaces, and the corresponding coalescence processes. Note that, as mentioned in the text, theory provides for our parameters the following values for the wavevectors’ moduli of ﹩{{ \mathcal H }}_{2}﹩, ﹩{{ \mathcal H }}_{3}﹩, and ﹩{{ \mathcal H }}_{4}﹩: k 2 λ  D  ≃ 0.035, k 3 λ  D  ≃ 0.056, k 4 λ  D  ≃ 0.077. Note that for ΔN = 0.05, spectral broadening due to density fluctuations can shift wavenumbers from their theoretical values. (a) ΔN = 0, ω  p  t = 3292, b  c  ≃ 0.75, (k  x1 ≃ 0.217, k  x2 ≃ − 0.187), ﹩{ \mathcal L }\left({k}_{x1}\right)\,+{{ \mathcal L }}^{{\prime} }({k}_{x2})\to {{ \mathcal H }}_{2}\left({k}_{x1}+{k}_{x2}\simeq 0.03\right);﹩                    b  c  ≃ 0.6, (k  x1 ≃ 0.0564, k  x2 ≃ 0.0347), ﹩{{ \mathcal H }}_{2}\left({k}_{x2}\right)+{{ \mathcal L }}^{{\rm{{\prime} }}}(-{k}_{x1}-{k}_{x2}\simeq -0.091)\,\to {{ \mathcal H }}_{3}\left(-{k}_{x1}\right)﹩. (b) ΔN = 0, ω  p  t = 4867, b  c  = 0.9, (k  x1 ≃ 0.074, k  x2 ≃ − 0.11), ﹩{ \mathcal L }\left({k}_{x1}\right)+{{ \mathcal L }}^{{\prime} }({k}_{x2})\to {{ \mathcal H }}_{2}\left({k}_{x1}+{k}_{x2}\simeq -0.036\right);﹩                    b  c  = 0.76, (k  x1 ≃ 0.134, k  x2 ≃ − 0.099), ﹩{ \mathcal L }\left({k}_{x1}\right)+{{ \mathcal L }}^{{\prime} }({k}_{x2})\to {{ \mathcal H }}_{2}\left({k}_{x1}+{k}_{x2}\simeq 0.035\right);﹩                    b  c  ≃ 0.62, (k  x1 ≃ 0.0911, k  x2 ≃ − 0.0564), ﹩{{ \mathcal H }}_{2}\left({k}_{x2}+{k}_{x1}\simeq 0.035\right)+{{ \mathcal L }}^{{\prime} }(-{k}_{x1})\to {{ \mathcal H }}_{3}\left({k}_{x2}\right);﹩                    b  c  = 0.7 (k  x1 ≃ 0.052, k  x2 ≃ − 0.078), ﹩{{ \mathcal H }}_{3}\left({k}_{x1}\right)+{{ \mathcal L }}^{{\prime} }({k}_{x2}-{k}_{x1}\simeq -0.13)\to {{ \mathcal H }}_{4}\left({k}_{x2}\right).﹩ (c) ΔN = 0, ω  p  t = 2743, b  c  = 0.7, (ω 1 = 2.93, ω 2 = 2.01), ﹩{{ \mathcal H }}_{2}\left({\omega }_{2}\right)\,+{{ \mathcal L }}^{{\prime} }\to {{ \mathcal H }}_{3}\left({\omega }_{1}\right)﹩. (d) ΔN = 0, ω  p  t = 4858; (bottom): b  c  = 0.76, (ω 1 = 3.02, ω 2 = 2.00), ﹩{{ \mathcal H }}_{2}\left({\omega }_{2}\right)+{{ \mathcal L }}_{1}\to {{ \mathcal H }}_{3}\left({\omega }_{1}\right);﹩ (top): b  c  = 0.32, (ω 1 = 4.01, ω 2 = 3.01), ﹩{{ \mathcal H }}_{3}\left({\omega }_{2}\right)+{{ \mathcal L }}_{1}\to {{ \mathcal H }}_{4}\left({\omega }_{1}\right)﹩. (e) ΔN = 0.05, ω  p  t = 2369, b  c  = 0.56, (k  x1 ≃ 0.0564, k  x2 ≃ − 0.031), ﹩{{ \mathcal H }}_{2}\left(-{k}_{x2}\right)+{{ \mathcal L }}^{{\prime} }(-{k}_{x1}\,+{k}_{x2}\,\simeq -0.087)\to {{ \mathcal H }}_{3}\left(-{k}_{x1}\right);﹩                    b  c  ≃ 0.51, (k  x1 ≃ 0.06, k  x2 ≃ − 0.035), ﹩{ \mathcal L }\left({k}_{x2}+{k}_{x1}\simeq 0.025\right)+{{ \mathcal L }}^{{\prime} }(-{k}_{x1})\to {{ \mathcal H }}_{2}\left({k}_{x2}\right)﹩. (f) ΔN = 0.05, ω  p  t = 3917, b  c  = 0.85, (k  x1 ≃ 0.161, k  x2 ≃ − 0.126), ﹩{ \mathcal L }\left({k}_{x1}\right)+{{ \mathcal L }}^{{\prime} }({k}_{x2})\,\to {{ \mathcal H }}_{2}\left({k}_{x1}+{k}_{x2}\simeq 0.035\right);﹩                    b  c  = 0.78, (k  x1 ≃ 0.0347, k  x2 ≃ − 0.065), ﹩{{ \mathcal H }}_{2}\left({k}_{x1}\right)+{{ \mathcal L }}^{{\prime} }(-{k}_{x1}+{k}_{x2}\simeq -0.099)\,\to {{ \mathcal H }}_{3}\left({k}_{x2}\right);﹩                    b  c  = 0.52, (k  x1 ≃ 0.047, k  x2 ≃ − 0.078), ﹩{{ \mathcal H }}_{3}\left({k}_{x1}\right)+{{ \mathcal L }}^{{\prime} }(-{k}_{x1}+{k}_{x2}\,\simeq -0.125)\,\to {{ \mathcal H }}_{4}\left({k}_{x2}\right)﹩. (g) ΔN = 0.05, ω  p  t = 2108, b  c  = 0.4, (ω 1 = 3.00, ω 2 = 2.01), ﹩{{ \mathcal H }}_{2}\left({\omega }_{2}\right)+{{ \mathcal L }}_{1}\to {{ \mathcal H }}_{3}\left({\omega }_{1}\right).﹩(h) ΔN = 0.05, ω  p  t = 4421; (bottom): b  c  = 0.6, (ω 1 = 2.97, ω 2 = 1.98), ﹩{{ \mathcal H }}_{2}\left({\omega }_{2}\right)+{{ \mathcal L }}_{1}\to {{ \mathcal H }}_{3}\left({\omega }_{1}\right)﹩; (top): ﹩{b}_{c}=0.4,({\omega }_{1}\,=3.96,{\omega }_{2}=2.95),{{ \mathcal H }}_{3}\left({\omega }_{2}\right)+{{ \mathcal L }}_{1}\to {{ \mathcal H }}_{4}\left({\omega }_{1}\right)﹩. All variables are normalized.