Heating of the ISM by Alfvén-wave damping - Theoretische Physik IV ...

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 )