Three-wave interactions of dispersive plasma waves propagating parallel to the magnetic field

Three-wave interactions (M ↔ P + Q) of plasma waves propagating parallel to the mean magnetic field at frequencies below the electron cyclotron frequency are considered. We consider Alfvén–ion-cyclotron waves (L), fast-magnetosonic–whistler waves (R), and ion-sound waves (I). Especially the weakly turbulent low-beta plasmas like the solar corona are studied, using the cold-plasma dispersion relation for the transverse waves (L, R) and the fluid-description of the warm plasma for the longitudinal waves (I). We analyse the resonance conditions for the wave frequencies ω and wavenumbers k (i.e., ωM = ωP + ωQ and kM = kP ± kQ, and the interaction rates u of the waves for all possible combinations of the three wave modes, and list those reactions that are not forbidden. One of the waves has to be longitudinal and two transverse. This demonstrates an attractive feature of the theory: the conservation of angular momentum is implicitly built in. In a low-beta plasma, non-zero reaction rates are obtained for (i) L ↔ I + L− and R ↔ I + R− in a wide frequency range extending from the MHD frequency range to the resonances of the waves; (ii) for L ↔ L + I in more narrow frequency range, where at least one of the L waves is in the dispersive frequency range; and (iii) I ↔ L + R± close to the resonance of the L mode. The reaction types (ii) and (iii) have, to our knowledge, not been discussed before in low-beta plasmas. In high-beta plasmas, reactions I ↔ R + L− and I ↔ L + R− are the main reactions, extending down to the MHD frequencies and discussed earlier for the non-dispersive case, but new reactions involving dispersive waves are found in high-beta plasmas as well: I ↔ L + R and R ↔ I + R± are now possible in limited frequency ranges involving at least one dispersive transverse wave. We discuss the implications of the discovered new reactions to turbulent cascading in space plasmas.