Three-wave interactions in magnetized warm-fluid plasmas: General theory with evaluable coupling coefficient

Resonant three-wave coupling is an important mechanism via which waves interact in a nonlinear medium. When the medium is a magnetized warm-fluid plasma, a previously unknown formula for the coupling coefficients is derived by solving the fluid-Maxwell's equations to second order using multiscale perturbative expansions. The formula is not only general but also evaluable, whereby numerical values of the coupling coefficient can be determined for any three resonantly interacting waves propagating at arbitrary angles. To illustrate how the general formula can be applied, coupling coefficient governing laser scattering is evaluated as one example. In conditions relevant to magnetized inertial confinement fusion, Raman and Brillouin instabilities are replaced by scattering from magnetized plasma waves when lasers propagate at oblique angles. As another example, coupling coefficient between two Alfvén waves via a sound wave is evaluated. In conditions relevant to solar corona, the decay of a parallel Alfvén wave only slightly prefers exact backward geometry.

Figure 1

Wave dispersion relations (a) and polarization angles (b) in magnetized warm-fluid electron-ion plasma when $\langle \mathbf{k},{\mathbf{B}}_0\rangle ={30}^{\circ }$. The two electromagnetic (EM) waves are elliptically polarized R wave (blue) and L wave (red), which become transverse and approach the light cone $\omega =ck$ when $ck\to \infty$. The other four branches are plasma waves, which become longitudinal when $ck\to \infty$. In this limit and when $\langle \mathbf{k},{\mathbf{B}}_0\rangle \to {90}^{\circ }$, the yellow branch (P) is the upper-hybrid (UH) wave and the purple branch is the lower-hybrid (LH) wave. In the opposite limit $ck\to 0$, the purple branch is the fast (F) wave, the green branch is the Alfvén (A) wave, and the cyan branch is the slow (S) wave. For all dispersion branches to be visible on the same scale (${10}^{12}$ rad/s), the mass ratio $m_i/m_e=5$ is artificial. The plasma density is $n_e=n_i={10}^{18}{\text{cm}}^{-3}$; the plasma temperature is $T_e=T_i=3.2$ keV; the polytropic index is adiabatic ${\xi }_e={\xi }_i=3$; the magnetic field is $B_0=2.5$ MG such that $|{\mathrm{\Omega }}_e|/{\omega }_{pe}\approx 0.8$ and $v_A/c_s\approx 4$, where $v_A$ is the Alfvén speed and $c_s$ is the sound speed.

Figure 2

Maps of frequency downshift (a) and normalized growth rate (b) when an L wave decays to P and F daughter waves (insets). Plasma parameters are the same as in Fig. 1. The pump wave (green dot) has frequency ${\omega }_1=75$ Trad/s, and propagates at ${\theta }_1={30}^{\circ }$ with respect to ${\mathbf{B}}_0$. The P-daughter wave propagates at polar angle ${\theta }_2$ and azimuthal angle ${\varphi }_2$. Due to the presence of ${\mathbf{B}}_0$, backscattering is not the strongest. Moreover, special angles exist where the coupling is suppressed.

Figure 3

Decay rates of a pump laser in a deuterium plasma via $\mathrm{R}\to \mathrm{R}$ (a), (c) and $\mathrm{R}\to \mathrm{L}$ (b), (d) scattering. The rates are in units of Raman backscattering, and the curves are color-coded by frequency downshifts. The 351-nm pump laser propagates along a 30-T magnetic field (a), (b) or a 300-T field (c), (d), and the scattered light propagates at angle ${\theta }_2=\langle {\mathbf{k}}_2,{\mathbf{B}}_0\rangle$. The plasma density is $n_e=n_i=4.5\times {10}^{20}{\text{cm}}^{-3}$, the temperature is $T_e=400$ eV and $T_i=150$ eV, and the polytropic index ${\xi }_e={\xi }_i=3$. Since $|{\mathrm{\Omega }}_e|\ll {\omega }_p$, scattering mediated by the P branch is close to Raman; since $v_A<c_s$, scattering mediated by the S branch is close to Brillouin. Additionally, the laser can scatter from the F branch, which is energy suppressed in weak magnetic field.

Figure 4

Resonant coupling between two Alfvén waves via the sound wave in solar corona type plasma with $c_s/v_A\approx 0.35$. The pump Alfvén wave propagates along ${\mathbf{B}}_0$ with frequency $f_1$, while the daughter Alfvén wave propagates obliquely at angle ${\theta }_2$. The frequency of the daughter wave $f_2/f_1$ (a) and the coupling coefficient $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\parallel }$ (b) have weak dependence on both ${\theta }_2$ and $f_1$. Consequently, the parametric decay rate is only slightly larger for exact backward scattering.