Hall effects on the Walen relation in rotational discontinuities and Alfven waves

For Alfvenic fluctuations in magnetohydrodynamics (MHD) the perturbed transverse velocity V-t and magnetic field B-t can be related by the Walen relation, V-t = +/-B-t/(mu(0)rho)(1/2) = +/-V-At, where rho is the plasma density, V-At is the transverse Alfven velocity, and the plus (minus) sign is for antiparallel (parallel) propagation. However, observations of V-t and B-t for Alfven 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 Walen relation for linear and nonlinear Alfven waves and rotational discontinuities (RDs) are examined in the Hall-MHD formulation. Let V-it (approximate to V-t) be the transverse ion velocity and V-et be the transverse electron velocity. It is found that V-it = +/-B-t(z)/(mu(0)rho(1))(1/2) = +/-(rho(z)/rho(1))(1/2) V-At(z) and V-et = +/-(rho(1)/mu(0))B-1/2(t)(z)/rho(z) = +/-(rho(1)/rho(z))(1/2) V-At(z) for RDs in Hall-MHD, where rho(1) is the upstream plasma density. The ion and electron Walen ratios are defined as A(i) = V-it/V-At and A(e) = V-et/V-At, respectively. It is found in Hall-MHD that V-it . V-et = V-At(2), A(i)A(e) = 1 and A(i) < 1 (A(i) > 1) for Alfven waves and RDs with right-hand (left-hand) polarization. The Hall dispersive effect may modify the ion Walen ratio by Delta A(i) approximate to +/-0.14 for the magnetopause RDs and by Delta A(i) approximate to +/-0.07 for the interplanetary RDs.