Hall effects on the Walén relation in rotational discontinuities and Alfvén waves

For Alfvénic fluctuations in magnetohydrodynamics (MHD) the perturbed transverse velocity V t and magnetic field B t can be related by the Walén relation, V t = ± B t /(μ 0 ρ) 1/2 ≡;± V At , where ρ is the plasma density, V At is the transverse Alfvén velocity, and the plus (minus) sign is for antiparallel (parallel) propagation. However, observations of V t and B t for Alfvén waves and rotational discontinuities in the solar wind and at the magnetopause showed an obvious deviation from the relation. In this paper, modifications of the Walén relation for linear and nonlinear Alfvén waves and rotational discontinuities (RDs) are examined in the Hall‐MHD formulation. Let V it (≈ V t ) be the transverse ion velocity and V et be the transverse electron velocity. It is found that V it = ± B t ( z )/(μ 0 ρ 1 ) 1/2 = ±(ρ( z )/ρ 1 ) 1/2 V At ( z ) and V et = ±(ρ 1 /μ 0 ) 1/2 B t ( z )/ρ( z ) = ±(ρ 1 /ρ( z )) 1/2 V At ( z )for RDs in Hall‐MHD, where ρ 1 is the upstream plasma density. The ion and electron Walén ratios are defined as A i = V it /V At and Ae = V et /V At , respectively. It is found in Hall‐MHD that , A i A e = 1 and A i < 1 ( A i > 1) for Alfvén waves and RDs with right‐hand (left‐hand) polarization. The Hall dispersive effect may modify the ion Walén ratio by Δ A i ≈±0.14 for the magnetopause RDs and by Δ A i ≈±0.07 for the interplanetary RDs.