Heating of the ISM by AlfvÃ©n-wave damping - Theoretische Physik IV ...

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2 Felix Spanier 3 Damping Processes For a given damping rate the energy dissipation rate is given by: ∫ ɛ i = d 3 kPyy2γ A i (6) Four different processes have been included in the calculation: • Collisionless Landau damping • Viscous damping • Joule heating • Ion-Neutral collisions 3.1 Joule Heating Joule heating is related to the resistivity of the plasma and the currents. We take the formula from Braginskii (1965) Fig. 1. Damping rates at θ = 2 ◦ resulting in σ ⊥ = ω2 pe 4πν e , σ ‖ = 1.96σ ⊥ (7) γ J (k) = ν ec 2 k 2 2ω 2 pe ( cos 2 θ + 0.51 sin 2 θ ) (8) Joule heating is neglected in favor of viscosity. 3.2 Viscosity Viscosity was proposed by Hollweg (1985), the parameter η 0 cancels out due to the incompressibility of Alfvén waves γ V (k) = k2 ( (η p 1 2m p n + ηe 1) sin 2 θ + (η p 2 + ηe 2) cos 2 θ ) (9) e The electron contribution is small compared to the proton contribution Introducing γ V (k) ≃ 0.15 k BT p k 2 c 2 τ i ( sin 2 m p c 2 (Ω p τ i ) 2 θ + 4 cos 2 θ ) (10) k c = Ω p V A , κ = k k c we find for Joule and viscous damping together γ V +J = 10 −7 κ 2 ( sin 2 θ + 4 cos 2 θ ) (11) As Joule and viscous damping have the same k 2 dependence and similar angular depence we may sum them up. Integrating with 6 gives ɛ V +J = 4 · 10 7 1 + s 3 − s (δB κ 3−s A) 2 kc 2 max − κ 3−s min κ 1+s max − κ 1+s H V +J (Λ, s) (12) min H V +J (Λ, s) = 3F ( 2+s 2+s 2 , 1; 3; 1 − Λ−1) 8F ( 3F ( 2 2+s 2 , 1 2 ; 5 + , 2; 4; 1 − Λ−1 ) 2 ; 1 − Λ−1) 8F ( 2+s 2 , 1 2 ; 5 2 ; 1 − (13) Λ−1 )
Heating of the ISM by Alfvén-wave damping 3 resulting in ɛ V +J = 10 −39 erg s −1 cm −3 (Λ = 1) (14) This gives only a small contribution, so it has no influence on ISM heating. 3.3 Collisionless Landau We are using the damping rate for obliquely propagating shear Alfvén waves (Ginzburg 1961, p.218, Eq. (14.56)) γ L = ≃ Inserting into Eq. (6) yields ( π ) 1/2 ω 3 v e tan 2 θ 8 Ω 2 p V A sin 2 θ + 3(ω 2 /Ω 2 p) cos 2 θ × ( vi 2 /ve 2 + (sin 2 θ + 4 cos 2 θ) exp[−VA/(2v 2 i 2 cos 2 θ)] ) ( π ) 1 2 m e v e k c κ 3 cos θ + sin 2 θ 8 m p sin 2 θ + 3κ 2 cos 4 θ (15) Approximations ɛ L = 1.1 · 10 −5 1 + s 2 − s v ek c (δB A ) 2 κ2−s max − κ 2−s min κ 1+s max − κ 1+s H L (Λ, s) (16) min H L (Λ, s) = 3F ( 2+s 2 , 1; 3; 1 − Λ−1) 8F ( 2+s 2 , 1 2 ; 5 (17) 2 ; 1 − Λ−1) H L (Λ ≫ 1) ≃ const (18) H L (Λ ≪ 1) ∝ Λ 1/2 (19) For given ISM parameters we find ɛ L ≃ 3.8 · 10 −42 erg s −1 cm −3 (Λ = 1) (20) Collisionless Landau damping can be ignored for any value of Λ. Comparison to Fast Magnetosonic waves Lerche & Schlickeiser (2001): 3.4 Ion-Neutral R(Λ = 1) = ɛA L (Λ = 1) ɛ M L (Λ = 1) ≃ (δB A) 2 10−20 (δB M ) 2 (21) R(Λ ≫ 1) ≃ 10 −20 Λ s/2 (δB A) 2 R(Λ ≪ 1) ≃ 10 −20 Λ −s/2 (δB A) 2 (δB M ) 2 (22) (δB M ) 2 (23) γ N (k) ≃ ν N cos 2 θ , κ ≥ κ N cos θ (24) ν N = 4 · 10 −9 n H Hz (25) κ N = ν N [Hz] (26) B[G]
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