A nonlocal wave-wave interaction among Alfven waves in an intermediate-beta plasma

A nonlocal coupling mechanism to directly transfer the energy from large-scale Magnetohydrodynamic(MHD) Alfven waves to small-scale kinetic Alfven waves is presented. It is shown that the interaction between a MHD Alfven wave and a reversely propagating kinetic Alfven wave can generate another kinetic Alfven wave, and this interaction exists in the plasmas where the thermal to magnetic pressure ratio is larger than the electron to ion mass ratio. The proposed nonlocal interaction may have a potential application to account for the observed electron scale kinetic Alfven waves in the solar wind and solar corona plasmas.

💡 Research Summary

The paper investigates a nonlocal three‑wave coupling process in an intermediate‑beta plasma ( mₑ/mᵢ ≪ β ≪ 1 ) that transfers energy directly from a large‑scale magnetohydrodynamic (MHD) Alfvén wave to a small‑scale kinetic Alfvén wave (KAW). The authors consider the interaction MHD Alfvén + KAW₁ → KAW₂, where KAW₁ propagates opposite to the MHD wave. By imposing the standard resonant conditions ωₛ + ω₁ = ω₂ and kₛ + k₁ = k₂, and using the linear dispersion relations for the MHD mode (ωₛ = V_A kₛz) and for KAWs (ω = V_A k_z K, with K = q + ρ²k_⊥² in the intermediate‑beta regime), they derive a constraint (K₁ − s₁)(K₂ − s₂) > 0. The physically relevant case is s₁ = − s₂ = −1, i.e., a counter‑propagating KAW₁. This requires K₂ > 1, which translates to β > mₑ/mᵢ; thus the mechanism only operates when the plasma thermal pressure exceeds the electron‑to‑ion mass ratio.

A two‑fluid model (isothermal electrons and ions) is employed to calculate the nonlinear terms. The momentum equations yield expressions for the perpendicular ion and electron velocities, including the E × B drift, diamagnetic drifts, and the nonlinear Lorentz forces. By combining the continuity and quasi‑neutrality conditions, the authors obtain the nonlinear current density and, through Ampère’s law, a nonlinear wave equation for the scalar potential φ₂ of KAW₂ (Eq. 13). Three nonlinear contributions are identified: (1) ion convective nonlinearity, (2) perpendicular electron nonlinear Lorentz force, and (3) parallel electron nonlinear Lorentz force. The latter dominates the growth of KAW₂.

From the nonlinear dispersion relation they derive the growth rate γ of KAW₂. In the symmetric limit (γ₁ ≈ γ₂ = γ ≪ ω_r) the growth rate is

γ² ∝ V_A⁴ K₂ k₂⊥² K₁ (s₁s₂K₁K₂ + k₁⊥k₂⊥)² (K₁ − s₁)(K₂ − s₂) sin²θ Bₛ⊥² / B₀²,

where θ is the angle between the perpendicular wavevectors of the MHD mode and KAW₁. For the case s₁ = − s₂ = −1 this simplifies to Eq. (18), showing that γ increases with larger perpendicular wavenumber of the generated KAW₂, with more oblique MHD propagation (larger θ), and with higher β (through K₂ > 1). The condition (K₁ − s₁)(K₂ − s₂) > 0 re‑emerges, confirming that only plasmas satisfying β > mₑ/mᵢ permit a positive growth rate.

The authors apply the theory to typical quiet‑Sun coronal parameters: n ≈ 10⁹ cm⁻³, T_i = T_e ≈ 10⁶ K, B₀ ≈ 10 G, giving β ≈ 3.5 × 10⁻² (intermediate‑beta). Observed MHD Alfvén waves in the corona have frequencies around 0.1 Hz, while KAWs are expected in the 10⁻⁵–2.5 Hz range. Using the derived frequency relations they estimate the perpendicular wavenumbers required for resonance. Numerical evaluation shows that the growth rate γ rises sharply with increasing k₂⊥ (i.e., decreasing spatial scale) and with larger obliquity of the MHD wave. Comparing γ with the electron Landau damping rate γ_L, they find that for reasonable amplitudes of the MHD wave (δB/B₀ ≈ 0.2) the growth can overcome damping, allowing the newly generated KAW₂ to reach electron‑scale (λ_e or ρ_e) where it can efficiently heat electrons.

A threshold MHD amplitude is calculated as a function of k₂⊥; the threshold decreases for more oblique MHD propagation and for larger k₂⊥. Since observed coronal Alfvén wave amplitudes exceed these thresholds, the nonlocal coupling is likely active in the solar corona, providing a direct channel for transferring energy from large‑scale motions to electron‑scale kinetic Alfvén fluctuations.

In the discussion the authors emphasize that the mechanism requires coexistence of a large‑scale MHD Alfvén wave and a counter‑propagating KAW, both of which are commonly present in solar wind and coronal environments due to shear flows, reconnection, phase mixing, or beam instabilities. Unlike local three‑wave interactions (which involve comparable scales) or nonlocal decay (large‑scale wave directly decaying into two small‑scale KAWs), the present process bridges disparate scales: a large‑scale MHD wave couples with a small‑scale KAW to generate another KAW at a different small scale. This offers a plausible explanation for the observed presence of electron‑scale KAWs in space plasmas, which cannot be produced efficiently by local cascades because of strong electron Landau damping at those scales.

The paper concludes that nonlocal MHD–KAW coupling is a robust energy‑transfer channel in intermediate‑beta plasmas, potentially important for electron heating in the solar corona and solar wind. Future work is suggested to incorporate kinetic effects beyond the two‑fluid approximation, explore the role of plasma anisotropy, and compare the theoretical predictions with in‑situ spacecraft measurements.