Dispersion Relation for Alfvén Waves in a Viscous, Diffusive Plasma

Propagation of Alfvén waves in the solar plasma has been a topic of scientific interest for a long time. We have derived a dispersion relation ω4 + ω2[(ν2 + η2)k4 − v2 A k2] + iω(ν + η)v2 A k4 + (νηv2 A k6 + ν2η2k8) = 0 for shear Alfvén waves in a viscous and diffusive plasma. The MHD equations for viscous and diffusive plasma are ρ ∂ → v ∂t + ρ( → v .∇) →v= 1 μ (∇× → B)× → B +ρν∇ → v Momentum equation (1) ∂ → B ∂t = ∇× (→v × → B) + η∇ → B Induction equation (2) ∇. → B= 0 Magnetic flux conservation (3) where → v is the velocity, → B the magnetic field and ρ, μ, η, ν are, respectively, the mass density, magnetic permeability, magnetic diffusivity and the coefficient of viscosity. Let us consider small perturbations from the equilibrium [1]: ρ = ρ0 + ρ1 → v= → v 1 → B= → B0 + → B1 and linearize the equations (1) through (3) by neglecting squares and products of the small quantities (denoted by subscript 1). Here, the quantities with subscript 0 represent their values in equilibrium. Notice that the equilibrium velocity v0 is zero [1]. After linearization, we have the corresponding relations as the following: ρ0 ∂ → v 1 ∂t = 1 μ (∇× → B1)× → B0 +ρ0ν∇ → v 1 (4) ∂ → B1 ∂t = ∇× (→v 1 × → B0) + η∇ → B1 (5) ∇. → B1= 0 (6) Here, the magnetic field → B0 is taken uniform as well as time independent. On differentiating equation (4) with respect to time and then substituting the expressions for ∂ → v 1/∂t and ∂ → B1/∂t from equations (4) and (5), we get ∂2 → v 1 ∂t2 = v A { ∇× [∇× (→v 1 ×B̂0)] } × B̂0 + η μρ0 ∇× (∇ → B1 × → B0) + ν μρ0 ∇[(∇× → B1)× → B0] + ν ∇→v 1 (7) Visiting Associate of IUCAA, Pune, India