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

For Alfvenic fluctuations in magnetohydrodynamics (MHD) the perturbed trans verse velocity V-t and magnetic field B-t can be related by the Walen relat ion, V-t = +/-B-t/(mu(0)rho)(1/2) = +/-V-At, where rho is the plasma densit y, 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 th e 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 formulat ion. 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 ra tios 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) polarizati on. 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) approxim ate to +/-0.07 for the interplanetary RDs.