Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence - I. Turbulent damping rates and feedback on background fluctuations

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Table 2.

Turbulent damping of quasi-parallel Alfvén waves with parallel wavelength $λ_{‖}^{w}$ $\lambda_{\|}^{w}$ .

Background	Scale range of interaction		Scaling of the turbulent damping rate $(Γ_{turb}^{w})$ $\mathbf(\Gamma_{\mathrm{turb}}^{\,w})$ :
cascade

(acronym)	$λ_{‖, \max}^{w}$ $\lambda_{\mathrm{\\|,max}}^{w}$	$λ_{‖, \min}^{w}$ $\lambda_{\mathrm{\\|,min}}^{w}$	“FG04”	“L16”	This work
Without scale-dependent alignment

sub-Alfvénic injection (M_A, 0 < 1):
(W0)	M_A, 0 ℓ₀	$M_{A, 0}^{4} ℓ_{0}$ $M_{\rm A,0}^4\,\ell_0$	–	$ω_{A, 0} M_{A, 0}^{8 / 3} {(λ_{‖}^{w} / ℓ_{0})}^{- 2 / 3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{8/3}\,(\lambda_{\\|}^{w}/\ell_0)^{-2/3}$	$ω_{A, 0} M_{A, 0}^{8 / 3} {(λ_{‖}^{w} / ℓ_{0})}^{1 / 3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{8/3}\,(\lambda_{\\|}^{w}/\ell_0)^{1/3}$
(GS95)	$M_{A, 0}^{4} ℓ_{0}$ $M_{\rm A,0}^4\,\ell_0$	$S_{0}^{- 1} ℓ_{0}$ $S_{0}^{-1}\,\ell_0$	–	$ω_{A, 0} M_{A, 0}^{2} {(λ_{‖}^{w} / ℓ_{0})}^{- 1 / 2}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^2\,(\lambda_{\\|}^{w}/\ell_0)^{-1/2}$	$ω_{A, 0} M_{A, 0}^{4}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{4}$

trans-Alfvénic injection (M_A, 0 ≈ 1):
(GS95)	ℓ₀	$S_{0}^{- 1} ℓ_{0}$ $S_{0}^{-1}\,\ell_0$	$ω_{A, 0} {(λ_{‖}^{w} / ℓ_{0})}^{- 1 / 2}$ $\omega_{\mathrm{A,0}}\,(\lambda_{\\|}^{w}/\ell_0)^{-1/2}$	–	ω_A, 0

super-Alfvénic injection (M_A, 0 > 1):
(K41)	ℓ₀	$M_{A, 0}^{- 3} ℓ_{0}$ $M_{\mathrm{A,0}}^{-3}\,\ell_0$	–	$ω_{A, 0} M_{A, 0} {(λ_{‖}^{w} / ℓ_{0})}^{- 2 / 3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}\,(\lambda_{\\|}^{w}/\ell_0)^{-2/3}$	$ω_{A, 0} M_{A, 0} {(λ_{‖}^{w} / ℓ_{0})}^{- 2 / 3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}\,(\lambda_{\\|}^{w}/\ell_0)^{-2/3}$
(GS95)	$M_{A, 0}^{- 3} ℓ_{0}$ $M_{\mathrm{A,0}}^{-3}\,\ell_0$	$S_{0}^{- 1} ℓ_{0}$ $S_{0}^{-1}\,\ell_0$	–	$ω_{A, 0} M_{A, 0}^{3 / 2} {(λ_{‖}^{w} / ℓ_{0})}^{- 1 / 2}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{3/2}\,(\lambda_{\\|}^{w}/\ell_0)^{-1/2}$	$ω_{A, 0} M_{A, 0}^{3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{3}$


With scale-dependent alignment


sub-Alfvénic injection (M_A, 0 < 1):
(W0)	M_A, 0 ℓ₀	$M_{A, 0}^{4} ℓ_{0}$ $M_{\rm A,0}^4\,\ell_0$	–	$ω_{A, 0} M_{A, 0}^{8 / 3} {(λ_{‖}^{w} / ℓ_{0})}^{- 2 / 3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{8/3}\,(\lambda_{\\|}^{w}/\ell_0)^{-2/3}$	$ω_{A, 0} M_{A, 0}^{8 / 3} {(λ_{‖}^{w} / ℓ_{0})}^{1 / 3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{8/3}\,(\lambda_{\\|}^{w}/\ell_0)^{1/3}$
(B06)	$M_{A, 0}^{4} ℓ_{0}$ $M_{\rm A,0}^4\,\ell_0$	$M_{A, 0}^{8 / 7} S_{0}^{- 5 / 7} ℓ_{0}$ $M_{\mathrm{A,0}}^{8/7}\, S_{0}^{-5/7}\,\ell_0$	–	–	$ω_{A, 0} M_{A, 0}^{24 / 5} {(λ_{‖}^{w} / ℓ_{0})}^{- 1 / 5}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{24/5}\,(\lambda_{\\|}^{w}/\ell_0)^{-1/5}$
(TMT)	$M_{A, 0}^{8 / 7} S_{0}^{- 5 / 7} ℓ_{0}$ $M_{\mathrm{A,0}}^{8/7}\, S_{0}^{-5/7}\,\ell_0$	$S_{0}^{- 1} ℓ_{0}$ $S_{0}^{-1}\,\ell_0$	–	–	$ω_{A, 0} M_{A, 0}^{4} S_{0}^{1 / 2} {(λ_{‖}^{w} / ℓ_{0})}^{1 / 2}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{4}\,S_{0}^{1/2}\,(\lambda_{\\|}^{w}/\ell_0)^{1/2}$

trans-Alfvénic injection (M_A, 0 ≈ 1):
(B06)	ℓ₀	$S_{0}^{- 5 / 7} ℓ_{0}$ $S_{0}^{-5/7}\,\ell_0$	–	–	$ω_{A, 0} {(λ_{‖}^{w} / ℓ_{0})}^{- 1 / 5}$ $\omega_{\mathrm{A,0}}\,(\lambda_{\\|}^{w}/\ell_0)^{-1/5}$
(TMT)	$S_{0}^{- 5 / 7} ℓ_{0}$ $S_{0}^{-5/7}\,\ell_0$	$S_{0}^{- 1} ℓ_{0}$ $S_{0}^{-1}\,\ell_0$	–	–	$ω_{A, 0} S_{0}^{1 / 2} {(λ_{‖}^{w} / ℓ_{0})}^{1 / 2}$ $\omega_{\mathrm{A,0}}\,S_{0}^{1/2}\,(\lambda_{\\|}^{w}/\ell_0)^{1/2}$

super-Alfvénic injection (M_A, 0 > 1):
(K41)	ℓ₀	$M_{A, 0}^{- 3} ℓ_{0}$ $M_{\mathrm{A,0}}^{-3}\,\ell_0$	–	$ω_{A, 0} M_{A, 0} {(λ_{‖}^{w} / ℓ_{0})}^{- 2 / 3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}\,(\lambda_{\\|}^{w}/\ell_0)^{-2/3}$	$ω_{A, 0} M_{A, 0} {(λ_{‖}^{w} / ℓ_{0})}^{- 2 / 3}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}\,(\lambda_{\\|}^{w}/\ell_0)^{-2/3}$
(B06)	$M_{A, 0}^{- 3} ℓ_{0}$ $M_{\mathrm{A,0}}^{-3}\,\ell_0$	$M_{A, 0}^{- 6 / 7} S_{0}^{- 5 / 7} ℓ_{0}$ $M_{\mathrm{A,0}}^{-6/7}\,S_{0}^{-5/7}\,\ell_0$	–	–	$ω_{A, 0} M_{A, 0}^{12 / 5} {(λ_{‖}^{w} / ℓ_{0})}^{- 1 / 5}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^{12/5}\,(\lambda_{\\|}^{w}/\ell_0)^{-1/5}$
(TMT)	$M_{A, 0}^{- 6 / 7} S_{0}^{- 5 / 7} ℓ_{0}$ $M_{\mathrm{A,0}}^{-6/7}\,S_{0}^{-5/7}\,\ell_0$	$S_{0}^{- 1} ℓ_{0}$ $S_{0}^{-1}\,\ell_0$	–	–	$ω_{A, 0} M_{A, 0}^{3} S_{0}^{1 / 2} {(λ_{‖}^{w} / ℓ_{0})}^{1 / 2}$ $\omega_{\mathrm{A,0}}\,M_{\mathrm{A,0}}^3\,S_{0}^{1/2}\,(\lambda_{\\|}^{w}/\ell_0)^{1/2}$

Notes. Summary of the scaling relations for the turbulent damping rate $Γ_{turb}^{w}$ $\Gamma_{\mathrm{turb}}^{\,w}$ of a quasi-parallel Alfvén wave in background MHD turbulence for the different regimes mentioned in Section 4. The damping rates obtained in this work are compared with tose available in the existing literature, namely in Farmer & Goldreich (2004) (FG04) and in Lazarian (2016) (L16). The notation ω_A, 0 = v_A, 0/ℓ₀ has been used. In all the regimes with dynamic alignment, a transition to a tearing-mediated range (TMT) has been assumed, i.e., that $S_{0} ≫ M_{A, 0}^{- 4}$ $S_0\gg M_{\mathrm{A,0}}^{-4}$ ( $S_{0} ≫ M_{A, 0}^{3}$ $S_0\gg M_{\rm A,0}^3$ ) holds for sub-Alfvénic (super-Alfvénic) injection.

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