© The Authors 2024

1. Introduction

Turbulent magnetized plasmas permeate a wide range of space and astrophysical environments (e.g., Quataert & Gruzinov 1999; Schekochihin & Cowley 2006; Brandenburg & Lazarian 2013; Bruno & Carbone 2013; Ferrière 2020). Understanding the properties of the turbulent cascade, and how the fluctuation energy is transferred from injection to dissipation scales, thus heating the plasma and also producing nonthermal particles in the process, is a relevant task in itself since it can elucidate the role that turbulence plays in the dynamics and thermodynamics of several astrophysical systems. Inspired by the seminal work of Kolmogorov (1941) in hydrodynamics, turbulence in magnetized plasmas has been the object of several theoretical efforts aimed at obtaining universal scaling for its fluctuations on large (fluid) magnetohydrodynamic (MHD) scales (e.g., Iroshnikov 1963; Kraichnan 1965; Goldreich & Sridhar 1995; Ng & Bhattacharjee 1997; Galtier et al. 2000; Cho & Lazarian 2002; Boldyrev 2006; Lazarian et al. 2012; Chandran et al. 2015; Mallet et al. 2015; Boldyrev & Loureiro 2017; Mallet et al. 2017; Cerri et al. 2022; Schekochihin 2022). At the same time, these astrophysical environments are also populated with cosmic rays (CRs), which are charged particles with supra-thermal (relativistic) energies that pervade the interstellar, intergalactic, and intracluster media (e.g., Brunetti & Jones 2014; Amato & Blasi 2018; Faucher-Giguère & Oh 2023; Ruszkowski & Pfrommer 2023) and get scattered by magnetic-field fluctuations (Ginzburg & Syrovatskii 1964; Berezinsky et al. 1990). While cosmic-ray transport partly depends upon the properties of pre-existing turbulence (e.g., Schlickeiser & Miller 1998; Chandran 2000; Lerche & Schlickeiser 2001; Yan & Lazarian 2002, 2008; Teufel et al. 2003; Shalchi & Schlickeiser 2004; Fornieri et al. 2021; Lazarian & Xu 2021; Lemoine 2023; Kempski et al. 2023), CRs can also generate their own scattering fluctuations through streaming instabilities (e.g., Kulsrud & Pearce 1969; Lee 1972; Skilling 1975; Gary 1993; Bell 2004; Amato 2011; Weidl et al. 2019a, 2019b; Marcowith et al. 2021). The level at which self-generated fluctuations saturate depends on a balance between the instability growth and the damping mechanisms that these waves are subjected to. Depending on the Galactic environment, the damping processes that were originally considered are the ion-neutral (IN) damping (Kulsrud & Pearce 1969) and the nonlinear Landau (NLL) damping (Lee & Völk 1973). These cosmic-ray driven Alfvén-wave (AW) packets, however, also interact with pre-existing fluctuations of the turbulent environment in which they are generated. This interaction has been suggested to represent another source of damping, the process called turbulent damping, for which a CR-generated AW packet is cascaded to dissipation by its nonlinear interaction with background fluctuations. This damping mechanism was originally proposed by Farmer & Goldreich (2004), and subsequently extended by Lazarian (2016) to account for different regimes of background turbulence. However, an important parameter that has not been taken into account in these previous works is the strength of the nonlinear interaction (usually referred to as the nonlinearity parameter χ) between the AW packet and pre-existing turbulent fluctuations. This parameter is indeed different from (and typically much smaller than) the nonlinear parameter describing the regime of background turbulence, which also needs to be taken into account (as done by Lazarian 2016). We show here that taking this difference into account completely changes the estimate of the turbulent damping rate, which is almost always much lower than any rate derived previously. Moreover, the damping rate and its scaling strongly depend upon the properties of background turbulence. In the previous literature, only what we can call the classic theories of MHD turbulence have been taken into account: isotropic “Kolmogorov-like” turbulence, hereafter the K41 regime (Kolmogorov 1941); a weak cascade of Alfvénic fluctuations, hereafter the W0 regime (Ng & Bhattacharjee 1997; Galtier et al. 2000); and a critically balanced Alfvénic cascade, hereafter the GS95 regime (Goldreich & Sridhar 1995). However, advanced theories of MHD turbulence that extend the above classic picture have been formulated in the past years. This is the case, for instance, of a theory that incorporates dynamic (i.e., scale-dependent) alignment of fluctuations in a critically balanced Alfvénic cascade, hereafter the B06 regime (Boldyrev 2006), which can intrinsically lead at even smaller scales to a regime usually referred to as tearing-mediated turbulence, hereafter the TMT regime (i.e., a regime where magnetic reconnection mediates the generation of smaller-scale fluctuations, Boldyrev & Loureiro 2017; Mallet et al. 2017). It is also worth mentioning that the conditions under which critical balance and the associated cascades develop (i.e., GS95, B06, and TMT regimes) may not cover all the possible scenarios in MHD turbulence (e.g., see discussion in Oughton & Matthaeus 2020). However, analytical (phenomenological) scaling of turbulent fluctuations and their anisotropy can be only derived for these cases. Moreover, several numerical simulations and in situ measurements in the solar wind have provided solid evidences for these regimes (e.g., Chen 2016; Sahraoui et al. 2020; Schekochihin 2022, and references therein). Therefore, it is of interest to derive turbulent damping rates for all these theories. The results obtained here have indeed implications for the cosmic-ray self-confinement, since its effectiveness for CR scattering is the result of a competition between different damping mechanisms and a balance between the most-relevant damping rate and the growth rate of the CR streaming instability (e.g., Farmer & Goldreich 2004; Blasi et al. 2012; Lazarian 2016; Kempski & Quataert 2022; Xu & Lazarian 2022). For instance, by adopting the rates obtained in Farmer & Goldreich (2004) and Lazarian (2016), turbulent damping can compete with or even dominate over the IN and NLL damping processes, depending on the properties of the Galactic environment under consideration and on the CR energy (see, e.g., Nava et al. 2019; Kempski & Quataert 2022; Recchia et al. 2022; Xu & Lazarian 2022, and references therein); this picture can be significantly challenged by the new turbulent damping rates obtained here, and will be addressed in detail in a following work (hereafter Paper II).

This paper is organized as follows. In Section 2 the Elsässer formalism and the definitions of the timescales and nonlinear parameter are introduced. In Section 3 the formalism is employed to derive general expressions for the turbulent damping rates, whose scaling are then explicitly derived in Section 4 for various turbulence regimes and within different theories of MHD turbulence. Additionally, some considerations about the feedback of CR-driven AWs on pre-existing fluctuations and possible phenomenological models for the CR-modified background turbulence spectrum are discussed in Section 5. Finally, a summary and discussion of the results is provided in Section 6. It is worth noting that this work is meant to primarily provide a rigorous, general formalism for deriving the turbulent damping rates of CR-driven AW packets, as well as some criteria for the possible relevance of CR feedback on pre-existing fluctuations. A more extensive discussion about different damping rates, the role of coherent structures and compressible turbulence, as well as the implications for specific astrophysical systems will be the focus of Paper II.

2. Setting the stage: The Elsässer formalism

The magnetohydrodynamic (MHD) equations for an incompressible plasma with mass density ρ₀, viscosity ν, and resistivity η, can be conveniently expressed in terms of the Elsässer variables ${\mathit{z}}^{\pm }=\mathit{u}\pm \mathit{B}/\sqrt{4\pi {\rho }_0}=\mathit{u}\pm {\mathit{v}}_{\mathrm{A}}$ (Elsässer 1950), where u is the fluid velocity, B is the magnetic field, and v_A denotes the Alfvén-speed vector associated with B. The incompressible MHD equations in terms of z^± read as

$$\begin{array}{c}\hfill \frac{\partial {\mathit{z}}^{\pm }}{\partial t}+({\mathit{z}}^{\mp }·\mathbf{\nabla }){\mathit{z}}^{\pm }=-\frac{\mathbf{\nabla }{P}_{\mathrm{tot}}}{{\rho }_0}+{\mu }_{+}{\mathrm{\nabla }}^2{\mathit{z}}^{\pm }+{\mu }_{-}{\mathrm{\nabla }}^2{\mathit{z}}^{\mp },\end{array}$$(1)

$$\begin{array}{c}\hfill \mathbf{\nabla }·{\mathit{z}}^{\pm }=0,\end{array}$$(2)

where P_tot = P_th + B²/8π is the sum of the thermal and magnetic pressure, and μ_± = (ν ± η)/2. Here we assume ν = η for simplicity, so that μ₊ = η and μ₋ = 0. By splitting the variables into a background quantity (denoted by a “0” in subscript¹ ) and purely transverse fluctuations; in other words, ${\mathit{z}}^{\pm }={\mathit{z}}_0^{\pm }+\delta {\mathit{z}}_{\perp }^{\pm }$, where ${\mathit{z}}_0^{\pm }=\pm {\mathit{B}}_0/\sqrt{4\pi {\rho }_0}=\pm {\mathit{v}}_{\mathrm{A},0}$ is the Alfvén speed associated with the background magnetic field B₀, and $\delta {\mathit{z}}_{\perp }^{\pm }=\delta {\mathit{u}}_{\perp }\pm \delta {\mathit{B}}_{\perp }/\sqrt{4\pi {\rho }_0}$ the fluctuating Elsässer fields, equation (1) rewrites as

$$\begin{array}{c}\hfill (\frac{\partial }{\partial t}\mp \underset{{\omega }_{\mathrm{lin}}^{\pm }\sim k_{\Vert }^{\pm }{v}_{\mathrm{A},0}}{\underbrace{v_{\mathrm{A},0}{\mathrm{\nabla }}_{\Vert }}}+\underset{{\omega }_{\mathrm{nl}}^{\pm }\sim k_{\perp }^{\pm }\delta z_{\perp }^{\mp }}{\underbrace{\delta {\mathit{z}}_{\perp }^{\mp }·{\mathbf{\nabla }}_{\perp }}}-\underset{{\omega }_{\mathrm{diss}}^{\pm }\sim \eta {k^{\pm }}^2}{\underbrace{\eta {\mathrm{\nabla }}^2}})\delta {\mathit{z}}_{\perp }^{\pm }=-\frac{\mathbf{\nabla }\delta P_{\mathrm{tot}}}{{\rho }_0},\end{array}$$(3)

where the parallel and perpendicular directions are defined with respect to B₀ for the global equations (but are later defined with respect to a scale-dependent mean field ⟨B⟩_k in a turbulent environment); we also mention that the term ∇δP_tot/ρ₀ in practice contributes just as a multiplicative factor (in Fourier space) to the nonlinear term (and associated timescale) on the left-hand side². One important feature of the formulation in (3) is that it explicitly shows that the nonlinear term $(\delta {\mathit{z}}_{\perp }^{\mp }·{\mathbf{\nabla }}_{\perp })\delta {\mathit{z}}_{\perp }^{\pm }$ is due only to the interaction of counter-propagating Alfvén-wave packets, $\delta {\mathit{z}}_{\perp }^{+}$ being transverse fluctuations propagating at the Alfvén speed v_A, 0 in the direction of B₀, while $\delta {\mathit{z}}_{\perp }^{-}$ are fluctuations propagating at the same speed in the direction of −B₀.

From (3) one can define a nonlinear parameter χ, which measures the strength of nonlinear effects with respect to the linear propagation term, namely

$$\begin{array}{c}\hfill {\chi }^{\pm }\sim \frac{|(\delta {\mathit{z}}_{\perp }^{\mp }·{\mathbf{\nabla }}_{\perp })\delta {\mathit{z}}_{\perp }^{\pm }|}{|(v_{\mathrm{A},0}{\mathrm{\nabla }}_{\Vert })\delta {\mathit{z}}_{\perp }^{\pm }|}\sim \frac{{\omega }_{\mathrm{nl}}^{\pm }}{{\omega }_{\mathrm{lin}}^{\pm }}\sim \frac{{\tau }_{\mathrm{A}}^{\pm }}{{\tau }_{\mathrm{nl}}^{\pm }}\sim \frac{k_{\perp }^{\pm }\delta z_k^{\mp }}{k_{\Vert }^{\pm }{v}_{\mathrm{A},0}},\end{array}$$(4)

and involves the wave-vector components $(k_{\Vert }^{\pm },k_{\perp }^{\pm })$ of the evolving fluctuation $\delta {\mathit{z}}_{\perp }^{\pm }$ and the amplitude $\delta {\mathit{z}}_{\perp }^{\mp }$ of the counter-propagating fluctuation that induces the nonlinearities on $\delta {\mathit{z}}_{\perp }^{\pm }$. In the following this parameter plays a central role to estimate the nonlinear cascade rate (or turbulent damping) of an Alfvén-wave packet interacting with background fluctuations. In particular, to obtain the correct turbulent damping rate it is necessary to make a careful distinction between the nonlinear parameter χ^z, which characterizes counter-propagating pre-existing fluctuations, and the nonlinear parameter χ^w, which describes the interaction between the AW packet and background turbulence.

3. Turbulent damping of an Alfvén-wave packet

We consider here an Alfvén-wave packet that is injected in an environment filled with pre-existing Alfvénic turbulence. We let δw be the initial Elsässer variable of the packet, and ${\lambda }_{\perp }^w$ and ${\lambda }_{\Vert }^w$ respectively its wavelength perpendicular to and parallel to a mean-magnetic field ⟨B⟩_λ^w at such scales³, (the corresponding wave vectors being $k_{\perp }^w\sim 1/{\lambda }_{\perp }^w$ and $k_{\Vert }^w\sim 1/{\lambda }_{\Vert }^w$). The Alfvénic fluctuations populating the turbulent background are characterized by certain scale-dependent relations for their Elsässer amplitude $\delta z_{{\lambda }_{\perp }^z}^{\pm }$, their wavelength anisotropy ${\lambda }_{\Vert }^{z,\pm }/{\lambda }_{\perp }^{z,\pm }$ (for which the corresponding wave vectors can be denoted as $k_{\perp }^{z,\pm }\sim 1/{\lambda }_{\perp }^{z,\pm }$ and $k_{\Vert }^{z,\pm }\sim 1/{\lambda }_{\Vert }^{z,\pm }$), and, if allowed, for the alignment angle between $\delta {\mathit{z}}_{{\lambda }_{\perp }^z}^{+}$ and $\delta {\mathit{z}}_{{\lambda }_{\perp }^z}^{-}$(i.e., $sin{\theta }_{{\lambda }_{\perp }^{z,\pm }}^z$). It is now instructive to derive the general relations first, leaving the explicit scaling belonging to different turbulence theories for later. Hereafter we consider the case of balanced background turbulence, and thus drop the ± superscript everywhere for simplicity of notation.

While propagating, the AW packet interacts nonlinearly only with counter-propagating Alfvénic fluctuations of the background. In terms of Elsässer variables, the nonlinear interaction is described by the term (δz ⋅ ∇) δw. The strength of this nonlinear interaction can then be determined by comparing the above nonlinear term with the term describing its linear propagation (v_A, 0 ⋅ ∇)δw. The interaction between the AW packet and the pre-existing fluctuations is described by the nonlinear parameter of the packet

$$\begin{array}{c}\hfill {\chi }^w\sim \frac{{\tau }_{\mathrm{A}}^w}{{\tau }_{\mathrm{nl}}^w}\sim \frac{({\lambda }_{\Vert }^w/v_{\mathrm{A},0})}{({\lambda }_{\perp }^w/\delta z_{{\lambda }_{\perp }^w})}\sim \left(\frac{{\lambda }_{\Vert }^w}{{\lambda }_{\perp }^w}\right)\left(\frac{\delta z_{{\lambda }_{\perp }^w}}{v_{\mathrm{A},0}}\right),\end{array}$$(5)

where local-in-scale interactions are assumed, so that $\delta z_{{\lambda }_{\perp }^z}$ is substituted with $\delta z_{{\lambda }_{\perp }^w}$ in the timescale associated with the nonlinear interaction between the AW packet and pre-existing turbulence: ${\tau }_{\mathrm{nl}}^w\sim {\lambda }_{\perp }^w/\delta z_{{\lambda }_{\perp }^z}\sim {\lambda }_{\perp }^w/\delta z_{{\lambda }_{\perp }^w}$. A parameter χ^w ≳ 1 means strong nonlinear interactions, while χ^w <  1 denotes the weakly nonlinear regime. We note that the parameter in (5) is different from the nonlinear parameter that characterizes background turbulence (i.e., ${\chi }^z\sim {\tau }_{\mathrm{A}}^z/{\tau }_{\mathrm{nl}}^z\sim ({\lambda }_{\Vert }^z/{\lambda }_{\perp }^z)(\delta z_{{\lambda }_{\perp }^z}/v_{\mathrm{A},0}))$⁴, and we point out that, while background fluctuations can have ${\lambda }_{\perp }^z\sim {\lambda }_{\perp }^w$ at scale ${\lambda }_{\perp }^z\sim {\lambda }_{\perp }^w$ (strong pre-existing turbulence), the condition χ^w ≳ 1 does not necessarily hold at these same scales.

It should be noted that χ^w is not only proportional to the amplitude of background fluctuations at the scale ${\lambda }_{\perp }^w$ (i.e., $\delta z_{{\lambda }_{\perp }^w}/v_{\mathrm{A},0}$), but it also depends on the AW packet’s propagation angle with respect to the mean magnetic field ⟨B⟩_λ^w at that scale: ${\lambda }_{\Vert }^w/{\lambda }_{\perp }^w\sim k_{\perp }^w/k_{\Vert }^w=tan{\mathrm{\Theta }}_{\mathit{kB}}^w$, where ${\mathrm{\Theta }}_{\mathit{kB}}^w$ is the angle between k^w and ⟨B⟩_{λ^w ∼ 1/k^w}. As a result, if the amplitude δz₀ of background fluctuations at injection scale ℓ₀ is such that δz₀/v_A, 0 ≲ 1, then strong nonlinear interactions at scales ${\lambda }_{\perp }^w\ll {\ell }_0$ (where $\delta z_{{\lambda }_{\perp }^w}\ll \delta z_0$) require ${\lambda }_{\Vert }^w/{\lambda }_{\perp }^w\sim v_{\mathrm{A},0}/\delta z_{{\lambda }_{\perp }^w}\gg 1$. This regime is thus achieved only by quasi-perpendicular AW packets. In critically balanced pre-existing turbulence, for instance, fluctuations obey the relation $\delta z_{{\lambda }_{\perp }^z}/v_{\mathrm{A},0}\sim {\lambda }_{\Vert }^z({\lambda }_{\perp }^z)/{\lambda }_{\perp }^z$. Therefore, assuming local-in-scale interactions (${\lambda }_{\perp }^z\sim {\lambda }_{\perp }^w$), the condition ${\lambda }_{\Vert }^w/{\lambda }_{\perp }^w\sim v_{\mathrm{A},0}/\delta z_{{\lambda }_{\perp }^w}$ means that an AW packet undergoes strong nonlinear interactions (and thus severe turbulent damping) only if its wave vector matches the anisotropy of background turbulence associated with the perpendicular scale ${\lambda }_{\perp }^w$ (i.e., ${\lambda }_{\Vert }^w\approx {\lambda }_{\Vert }^z({\lambda }_{\perp }^w))$. Therefore, the nonlinear interaction between a quasi-parallel AW (characterized by ${\lambda }_{\Vert }^w/{\lambda }_{\perp }^w\ll 1$), and anisotropic pre-existing turbulence (characterized by ${\lambda }_{\perp }^z/{\lambda }_{\Vert }^z\ll 1$) is always weak: χ^w ≪ 1.

This can be shown explicitly by considering the quasi-parallel propagation limit, which is the case of interest for CR-generated AW packets. In this case, the propagation can only be as parallel as the external turbulence allows, meaning that the propagation angle cannot be smaller than the amount $\delta b_{{\lambda }_{\perp }^w}/{⟨B⟩}_{{\lambda }^w}\sim \delta z_{{\lambda }_{\perp }^w}/v_{\mathrm{A},0}$ because of the field-line distortions induced by pre-existing turbulent fluctuations $\delta b_{{\lambda }_{\perp }^w}$ over the wavelength ${\lambda }_{\perp }^w$ of the packet. Therefore, the quasi-parallel (q∥) propagation limit is set by

$$\begin{array}{c}\hfill {\frac{{\lambda }_{\Vert }^w}{{\lambda }_{\perp }^w}|}_{\min }=\frac{{\lambda }_{\Vert }^{w,\mathrm{q}\Vert }}{{\lambda }_{\perp }^{w,\mathrm{q}\Vert }}\sim \frac{\delta z_{{\lambda }_{\perp }^{w,\mathrm{q}\Vert }}}{v_{\mathrm{A},0}}.\end{array}$$(6)

Hence, the associated nonlinear parameter in this limit is

$$\begin{array}{c}\hfill {\chi }^{w,\mathrm{q}\Vert }\sim {\left(\frac{\delta z_{{\lambda }_{\perp }^{w,\mathrm{q}\Vert }}}{v_{\mathrm{A},0}}\right)}^2.\end{array}$$(7)

The strongly nonlinear regime can thus be achieved only at scales λ where the AW packet interacts with pre-existing super-Alfvénic fluctuations (δz_λ/v_A, 0 >  1). In this regime, the concept of quasi-parallel propagation in (6) does not apply because at scales where δb_λ^w/⟨B⟩_λ^w >  1, the distinction between ${\alpha }_{\perp }^w$ and ${\lambda }_{\Vert }^w$ is lost and ${\lambda }_{\perp }^w\sim {\lambda }_{\Vert }^w\sim \lambda$ holds; hence, χ^w ∼ δz_λ/v_A, 0 ≳ 1. However, even for external turbulence that is injected with super-Aflvénic amplitude (i.e., δz₀/v_A, 0 ≈ M_A, 0 >  1 at scale ℓ₀) the fluctuation amplitude decreases with decreasing scale. As a result, the nonlinear interaction between the AW and external fluctuations becomes weak at small-enough scales (i.e., at scales ${\lambda }_{\perp }^w<{\ell }_{\mathrm{A}}={\ell }_0/M_{\mathrm{A},0}^3\ll {\ell }_0$) for which the initially super-Alfvénic fluctuations become sub-Alfvénic, $\delta z_{{\lambda }_{\perp }^w}/v_{\mathrm{A},0}<1$, and anisotropic (see Appendix A). Another way to visualize this is by rewriting (7) as

$$\begin{array}{c}\hfill {\chi }^{w,\mathrm{q}\Vert }\sim {\left(\frac{{\lambda }_{\perp }^z}{{\lambda }_{\Vert }^z}\right)}^2{({\chi }^z)}^2,\end{array}$$(8)

which is ≪χ^z for anisotropic background fluctuations, ${\lambda }_{\perp }^z\ll {\lambda }_{\Vert }^z$, and thus χ^w, q∥ ≪ 1 even if pre-existing turbulence is critically balanced (χ^z ∼ 1).

The difference between the intrinsic nonlinear parameter of background fluctuations (χ^z) and the nonlinear parameters of a quasi-parallel Alfvén wave interacting with those fluctuations (χ^w) for explicit MHD turbulent scaling in different regimes is reported the last two columns of Table 1.

Having clarified the packet’s nonlinear regimes that have to be considered in terms of background turbulence, we can now estimate the associated rate of turbulent damping. This means estimating the timescale over which an AW packet undergoes a cascade process due to its nonlinear interaction with pre-existing turbulent fluctuations. The cascade time of the packet is given by ${\tau }_{\mathrm{casc}}^w\sim {({\tau }_{\mathrm{nl}}^w)}^2/{\tau }_{\mathrm{A}}^w\sim {\tau }_{\mathrm{nl}}^w/{\chi }^w$ for the weak regime (χ^w <  1), while it is ${\tau }_{\mathrm{casc}}^w\sim {\tau }_{\mathrm{nl}}^w$ in the strongly nonlinear case (χ^w ≳ 1, which we recall also implies that ${\lambda }_{\Vert }^w\sim {\lambda }_{\perp }^w\sim {\lambda }^w$; see discussion after equation (7)).

As a result, the turbulent damping rate is

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\mathrm{turb}}^w\sim \frac{1}{{\tau }_{\mathrm{casc}}^w}\sim \{\begin{array}{ccc}\left(\frac{{\lambda }_{\Vert }^w}{{\lambda }_{\perp }^w}\right){\left(\frac{{\lambda }_{\perp }^w}{{\ell }_0}\right)}^{-1}{\left(\frac{\delta z_{{\lambda }_{\perp }^w}}{v_{\mathrm{A},0}}\right)}^2\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill \mathrm{if}{\chi }^w<1\\ \hfill & & \\ {\left(\frac{{\lambda }^w}{{\ell }_0}\right)}^{-1}\left(\frac{\delta z_{{\lambda }^w}}{v_{\mathrm{A},0}}\right)\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill \mathrm{if}{\chi }^w\gtrsim 1,\end{array}\end{array}$$(9)

which reduces to

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\mathrm{turb}}^{w,\mathrm{q}\Vert }\sim {\left(\frac{{\lambda }_{\perp }^{w,\mathrm{q}\Vert }}{{\ell }_0}\right)}^{-1}{\left(\frac{\delta z_{{\lambda }_{\perp }^{w,\mathrm{q}\Vert }}}{v_{\mathrm{A},0}}\right)}^3\frac{v_{\mathrm{A},0}}{{\ell }_0}\end{array}$$(10)

for quasi-parallel propagation and χ^w, q∥ <  1. We recall that χ^w should not be identified with the nonlinear parameter χ^z that describes the strength of background turbulence. Therefore, when χ^w <  1 the turbulent damping rate of an AW packet is nonlinear with respect to the background-fluctuation amplitude and depends on the propagation angle of the wave, becoming a third-order quantity of the pre-existing turbulent amplitude in the quasi-parallel limit.

We conclude by highlighting that in (10) there is a factor ${\lambda }_{\perp }^{-1}$ in front of the $\delta z_{{\lambda }_{\perp }}^3$ term. Therefore, a turbulent perpendicular scaling for $\delta z_{{\lambda }_{\perp }}\propto {\lambda }_{\perp }^{\alpha }$ with a spectral index α >  1/3 will produce a turbulent damping rate of quasi-parallel AW packets that decreases with decreasing scale. That would be the case of weak Alfvénic turbulence, as in Galtier et al. (2000), or the tearing-mediated regime, as in Boldyrev & Loureiro (2017) and Mallet et al. (2017). On the other hand, fluctuations that scale with α <  1/3 result in a damping rate that increases with decreasing ${\lambda }_{\perp }^w$. This would be the case of critically balanced strong Alfvénic turbulence with scale-dependent alignment, as in Boldyrev (2006). Finally, for a Kolmogorov-like perpendicular scaling $\delta z_{{\lambda }_{\perp }}\propto {\lambda }_{\perp }^{1/3}$, as in critically balanced strong Alfvénic turbulence without dynamic alignment (Goldreich & Sridhar 1995), we can expect that the turbulent damping of quasi-parallel AW packets becomes scale-independent (i.e., ${\mathrm{\Gamma }}_{\mathrm{GS}95}^{w,\mathrm{q}\Vert }\sim \mathrm{const}$).

4. Turbulent damping with explicit MHD scalings

The injection-scale Alfvénic Mach number is defined as the ratio M_A, 0 ≈ δz₀/v_A, 0, where δz₀ is the fluctuation amplitude at injection scale ℓ₀, and determines the cascading regimes of background fluctuations. The Lunquist number at injection scale, S₀ = ℓ₀ v_A, 0/η, is related to the system’s resitivity η and determines the dissipation scale of turbulent fluctuations; we recall that here we have assumed ν = η. If we assume isotropic injection, the nonlinear parameter of background turbulence at scale ℓ₀ indeed corresponds to the injection-scale Alfvénic Mach number, ${\chi }_0^z\approx M_{\mathrm{A},0}$. If M_A, 0 <  1, the turbulence is called sub-Alfvénic: it starts as an anisotropic weak cascade that transitions into critically balanced strong turbulence at smaller scales (still anisotropic, but in a different fashion). Trans-Alfvénic turbulence (M_A, 0 ≈ 1) consists of an anisotropic strong cascade of critically balanced fluctuations at all scales. When M_A, 0 >  1 (large-amplitude injection), turbulence is called super-Alfvénic and fluctuations initially undergo an isotropic (“hydrodynamic-like”) cascade until sub-Alfvénic amplitudes are attained at smaller scales, and turbulence becomes critically balanced and anisotropic. Then, if scale-dependent (dynamic) alignment of fluctuations is allowed in the critical-balance range, an additional transition to a different regime of anisotropic, strong turbulence can occur at even smaller scales due to magnetic reconnection (if the injection-scale Lundquist number S₀ is large enough; see Section 4.2). In the following we only summarize the relevant scaling of background turbulent fluctuations in the different ranges. These scaling relations, along with the intrinsic nonlinear parameter χ^z of background fluctuations and the nonlinear parameter χ^w of a quasi-parallel AW propagating through such background turbulence, are also shown in Table 1 for convenience. A more detailed derivation of these ranges and of the associated scaling is provided in Appendix A.

The turbulent damping rates in this section were derived as follows. First, we employed the known perpendicular scaling of background turbulence, δz_λ⊥, and locality of interactions (${\lambda }_{\perp }^z\sim {\lambda }_{\perp }^{w,\mathrm{q}\Vert }$) in (10) to obtain the turbulent damping rate as a function of the packet’s perpendicular wavelength, ${\mathrm{\Gamma }}_{\mathrm{turb}}^{w,\mathrm{q}\Vert }({\lambda }_{\perp }^{w,\mathrm{q}\Vert })$. Then we used the quasi-parallel condition in (6) to retrieve the scaling of ${\mathrm{\Gamma }}_{\mathrm{turb}}^{w,\mathrm{q}\Vert }$ with respect to the parallel wavelength ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }$.

The scaling of the turbulent damping rate for different background cascades and the corresponding range of scales where that is valid are given in Table 2, along with an explicit comparison with the rates available in the existing literature (i.e., from Farmer & Goldreich 2004, hereafter FG04, and from Lazarian 2016, hereafter L16). The behavior of these damping rates and the comparison with previous estimates for two choices of sub-Alfvénic and super-Alfvénic injection (M_A, 0 = 0.1, 10) and for S₀ = 10¹⁴ are also shown in Figure 1, for convenience. We note that previous results not only overestimate the damping rate by a factor that could be several orders of magnitude, but in most cases they also obtain a completely different result on how this damping rate depends upon the packet’s parallel wavelength ${\lambda }_{\Vert }^w$.

Fig. 1. Normalized turbulent damping rate $\frac{{\ell }_0}{v_{\mathrm{A},0}}{\mathrm{\Gamma }}_{\mathrm{turb}}^{w,\mathrm{q}\Vert }$ for quasi-parallel AW packets with normalized parallel wavelength ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }/{\ell }_0$, in a background plasma with Lunquist number S0 = 1014 and different turbulent regimes (see Appendix A). Solid lines represent damping rates derived in this work (equations (11), (13), (14), and (16)), while dashed lines report the damping rates in Lazarian (2016) for reference. General expressions for transition scales and damping-rate values are reported on the right and upper axes. Left: damping rates in sub-Alfvénic turbulence (MA, 0 = 0.1). Right: damping rates in super-Alfvénic turbulence (MA, 0 = 10).

4.1. Magnetohydrodynamic turbulence without scale-dependent alignment

We first consider the classic picture in which dynamic alignment of turbulent fluctuations does not occur. In this case, we have three possible regimes for background turbulence:

• [W0] A weak anisotropic cascade with fluctuation scaling $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{W}0)}/v_{\mathrm{A},0}\sim M_{\mathrm{A},0}{({\lambda }_{\perp }^z/{\ell }_0)}^{1/2}$ that only generates smaller perpendicular scales (i.e., ${\lambda }_{\Vert }^z\sim {\ell }_0\approx \mathrm{const}.$) and transitions into a strong cascade at the critical balance (CB) scale ${\lambda }_{\perp ,\mathrm{CB}}^z\sim M_{\mathrm{A},0}^2{\ell }_0$. This cascade is realized in the range of scales ${\lambda }_{\perp ,\mathrm{CB}}^z\lesssim {\lambda }_{\perp }^z\lesssim {\ell }_0$, and only for sub-Alfvénic injection (M_A, 0 <  1).

• [K41] A strong, isotropic (hydrodynamic-like) cascade characterized by the scaling $\delta z_{{\lambda }^z}^{(\mathrm{K}41)}/v_{\mathrm{A},0}\sim M_{\mathrm{A},0}({\lambda }^z/{\ell }_0{)}^{1/3}$. These fluctuations attain sub-Alfvénic amplitudes, becoming anisotropic and critically balanced, at a scale ${\ell }_{\mathrm{A}}\sim M_{\mathrm{A},0}^{-3}{\ell }_0$. This cascade is realized at scales ℓ_A ≲ λ^z ≲ ℓ₀, and only for super-Alfvénic injection (M_A, 0 >  1).

• [GS95] A strong anisotropic cascade of critically balanced fluctuations with perpendicular scaling $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{GS}95)}\propto {({\lambda }_{\perp }^z)}^{1/3}$. This type of cascade is realized either for trans-Alfvénic injection (M_A, 0 ≈ 1), or when cascading fluctuations of the two regimes above reach the scale ${\lambda }_{\perp ,CB}^z$ and ℓ_A, respectively. For trans-/sub-Alfvénic injection (M_A, 0 ≲ 1), the dependence on M_A, 0 of the scaling is $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{GS}95)}/v_{\mathrm{A},0}\sim M_{\mathrm{A},0}^{4/3}{({\lambda }_{\perp }^z/{\ell }_0)}^{1/3}$, and the cascade achieves dissipation at a scale ${\lambda }_{\perp ,\min }^{z(\mathrm{subA})}/{\ell }_0\sim M_{\mathrm{A},0}^{-1}{S}_0^{-3/4}$. For super-Alfvénic injection (M_A, 0 >  1), the fluctuation scaling with M_A, 0 is linear (i.e., $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{GS}95)}/v_{\mathrm{A},0}\sim M_{\mathrm{A},0}{({\lambda }_{\perp }^z/{\ell }_0)}^{1/3}$), and the dissipation scale is given by ${\lambda }_{\perp ,\min }^{z(\mathrm{supA})}/{\ell }_0\sim {(M_{\mathrm{A},0}{S}_0)}^{-3/4}$. Here S₀ = ℓ₀ v_A, 0/η is the Lundquist number at injection scale.

By using (7) we can verify that the nonlinear interaction between a quasi-parallel AW packets with wavelength ${\lambda }_{\perp }^{w,\mathrm{q}\Vert }$ and the pre-existing turbulence is weak, χ^w, q∥ ≪ 1, in the range of scales where the cascade of background fluctuations is either weak (W0) or critically balanced (GS95). This means χ^w, q∥ ≪ 1 at scales ${\lambda }_{\perp }^{w,\mathrm{q}\Vert }<{\ell }_0$ and ${\lambda }_{\perp }^{w,\mathrm{q}\Vert }<{\ell }_{\mathrm{A}}$ respectively for sub-Alfvénic and super-Alfvénic injection (see Table 1 for the explicit scaling of χ^w, q∥ in these different regimes). Hence, the cascade time ${\tau }_{\mathrm{casc}}^{w,\mathrm{q}\Vert }\sim {\tau }_{\mathrm{nl}}^w/{\chi }^{w,\mathrm{q}\Vert }$ of the AW packets for these cases is not just the nonlinear time ${\tau }_{\mathrm{nl}}^w$, and the turbulent damping rate is given by (10). In Farmer & Goldreich (2004), for instance, the nonlinear time ${\tau }_{\mathrm{nl}}^z$ instead of ${\tau }_{\mathrm{casc}}^{w,\mathrm{q}\Vert }$ was used to compute the turbulent damping rate. In a subsequent work by Lazarian (2016), the nonlinear parameter of background turbulence χ^z was used instead of χ^w, q∥ ≪ χ^z to compute a cascade time ${\tau }_{\mathrm{nl}}^w/{\chi }^z$. This resulted in an estimated timescale for turbulent damping that was notably shorter than the actual cascade time that should be used. Taking properly into account the difference between χ^z and χ^w, q∥ thus changes significantly the effectiveness of turbulent damping in pre-existing turbulence with respect to all these previous estimates (see Table 2 for the generic case, or Figure 1 for two specific examples of sub- and super-Alfvénic injection regimes).

4.1.1. Sub- and trans-Alfvénic turbulence (M_A, 0 ≤ 1) without dynamic alignment

In sub-Alfvénic background turbulence without dynamic alignment, a quasi-parallel AW packet with normalized parallel wavelength ${\widehat{\lambda }}_{\Vert }^w={\lambda }_{\Vert }^{w,\mathrm{q}\Vert }/{\ell }_0$ is subjected to the following turbulent damping rate:

[M_A, 0 <  1, no dynamic alignment]

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\mathrm{subA}}^{w,\mathrm{q}\Vert }\sim \{\begin{array}{ccc}{M}_{\mathrm{A},0}^{8/3}{\left({\widehat{\lambda }}_{\Vert }^w\right)}^{1/3}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill M_{\mathrm{A},0}^4\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim M_{\mathrm{A},0}\\ \hfill & & \\ M_{\mathrm{A},0}^4\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill {\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{subA})}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim M_{\mathrm{A},0}^4.\end{array}\end{array}$$(11)

Here ${\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{subA})}\sim {(M_{\mathrm{A},0}{\widehat{\lambda }}_{\perp ,\min }^{z(\mathrm{subA})})}^{4/3}\sim S_0^{-1}$ is the minimum packet wavelength that is effectively subjected to turbulent damping, with ${\widehat{\lambda }}_{\perp ,\min }^{z(\mathrm{subA})}\sim M_{\mathrm{A},0}^{-1}{S}_0^{-3/4}$ being the normalized dissipation scale of the turbulence. The ranges of ${\widehat{\lambda }}_{\Vert }^w$ in (11) were determined according to the quasi-parallel condition in (6) and, assuming local interactions ${\lambda }_{\perp }^{w,\mathrm{q}\Vert }\sim {\lambda }_{\perp }^z$, employing the ${\lambda }_{\perp }^z$ range of validity for each turbulent regime (see Appendix A).

The trans-Alfvénic regime is trivially obtained from (11) when the initial weak cascade does not occur:

[M_A, 0 ≃ 1, no dynamic alignment]

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\mathrm{transA}}^{w,\mathrm{q}\Vert }\sim \frac{v_{\mathrm{A},0}}{{\ell }_0}{({\widehat{\lambda }}_{\perp ,\min }^z)}^{4/3}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim 1.\end{array}$$(12)

Here ${\widehat{\lambda }}_{\perp ,\min }^z\sim S_0^{-3/4}$ is the (normalized) dissipation scale of GS95 turbulence in the trans-Alfvénic regime.

The damping rates in (11) differ from the values previously derived in the literature (see Table 2) because here the nonlinear parameter of the AW packet is properly taken into account (see Table 1). We can verify that the turbulent damping rate in the W0 range of sub-Alfvénic turbulence (i.e., equation (46) in Lazarian 2016) can be recovered if the nonlinear parameter χ^z of background turbulence is employed instead of the nonlinear parameter χ^w of the AW packet. Analogously, the result in the GS95 range of sub-Alfvénic turbulence in Eq. (34) of the same paper is recovered by assuming strong interactions between the quasi-parallel AW packet and background fluctuations (i.e., identifying χ^w with χ^z ∼ 1 at those scales). However, given the expression for χ^w in (5), the assumption χ^w ∼ 1 would require ${\lambda }_{\Vert }^w/{\lambda }_{\perp }^w\sim v_{\mathrm{A},0}/\delta z_{{\lambda }_{\perp }^w}\gg 1$, which is inconsistent with the quasi-parallel limit ${\lambda }_{\Vert }^w/{\lambda }_{\perp }^w\ll 1$. The same argument applies when comparing the damping rate for the trans-Alfvénic case in (12) with Eq. (9) in Farmer & Goldreich (2004).

It is important to note that the results obtained here strongly change the effectiveness of the turbulent damping of CR-generated Alfvén-wave packets. Depending on the Alfvénic Mach number M_A, 0 and on the Lunquist number S₀, the damping rates in (11) and (12) can be several orders of magnitude lower than the previously derived rates, those usually employed in CR studies (see Table 2 and the left panel in Figure 1). In particular, we can see that the damping rate of an AW packet interacting with pre-existing weak turbulence is at least a factor M_A, 0 lower than previously estimated (i.e., at scale ${\lambda }_{\Vert ,\max }^{w,\mathrm{q}\Vert }\sim M_{\mathrm{A},0}{\ell }_0$, when this difference is at its minimum, then it increases even further due to the different dependence on ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }$). When the packet starts to interact with strong turbulence (i.e., for ${\lambda }_{\Vert ,\mathrm{CB}}^{w,\mathrm{q}\Vert }\sim M_{\mathrm{A},0}^4{\ell }_0$), the damping rate becomes at least a factor $M_{\mathrm{A},0}^4\ll 1$ lower than has been derived in the literature (a difference that, again, increases even further with decreasing packet’s parallel wavelength due to the radically different wavelength dependence of ${\mathrm{\Gamma }}_{\mathrm{GS}95}^{w,\mathrm{q}\Vert }$ in (11) and (12) with respect to the results in Farmer & Goldreich 2004 and in Lazarian 2016). This is also true for trans-Alfvénic injection (M_A, 0 ≃ 1), in which case the damping rate in (12) would be the same as that in the literature, only at scales ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }\sim {\ell }_0$, and then the two results would rapidly diverge with decreasing wavelength of the quasi-parallel AW packet at ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }<{\ell }_0$. Finally, the maximum difference between the damping rate obtained here and those found in the literature is achieved at the minimum wavelength for which this damping mechanism is effective; at ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }\sim {\lambda }_{\Vert ,\min }^w$ the actual damping rate is a factor $\sim S_0^{-1/2}{M}_{\mathrm{A},0}^2\ll 1$ lower than the results in Farmer & Goldreich (2004) and in Lazarian (2016); in astrophysical systems this factor can represent many orders of magnitude since S₀ can be extremely large, for example larger than 10²⁰ (Priest & Forbes 2007). We also point out that in sub-Alfvénic turbulence the ordering $S_0\gg M_{\mathrm{A},0}^{-4}$ is implied in order to have a significant GS95 range (see Appendix A).

4.1.2. Super-Alfvénic turbulence (M_A, 0 >  1) without dynamic alignment

When the injection regime of background fluctuations is super-Alfvénic, an AW packet is instead subjected to a turbulent damping given by

[M_A, 0 >  1, no dynamic alignment]

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\mathrm{supA}}^{w,\mathrm{q}\Vert }\sim \{\begin{array}{ccc}{M}_{\mathrm{A},0}{\left({\widehat{\lambda }}^w\right)}^{-2/3}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill M_{\mathrm{A},0}^{-3}\lesssim {\widehat{\lambda }}^w\lesssim 1\\ \hfill & & \\ M_{\mathrm{A},0}^3\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill {\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{supA})}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim M_{\mathrm{A},0}^{-3},\end{array}\end{array}$$(13)

where the damping rate in the range $M_{\mathrm{A},0}^{-3}\lesssim {\widehat{\lambda }}^w\lesssim 1$ is obtained from the χ^w ≳ 1 part of (9), and the shortest wavelength affected by turbulent damping is ${\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{supA})}\sim M_{\mathrm{A},0}{({\widehat{\lambda }}_{\perp ,\min }^{z(\mathrm{supA})})}^{4/3}\sim S_0^{-1}$. We also point out that the isotropic normalized wavelength ${\widehat{\lambda }}^w={\lambda }^w/{\ell }_0$ enters the damping rate in the range ℓ_A ≲ λ^w ≲ ℓ₀ (i.e., where the AW packet interacts with hydrodynamic-like pre-existing turbulence).

In the range $M_{\mathrm{A},0}^{-3}\lesssim {\widehat{\lambda }}^w\lesssim 1$, the packet’s nonlinear parameter is larger than unity (i.e., χ^w ≈ δz_λ/v_A, 0, and δz_λ/v_A, 0 >  1 at those scales; see Table 1). Thus, the result obtained above for this range of scales agrees with the corresponding result provided in equation (55) of Lazarian (2016). This is a consequence of the fact that the quasi-parallel condition (6) does not apply in the range ${\ell }_0\gtrsim \lambda \gtrsim {\ell }_{\mathrm{A}}\approx M_{\mathrm{A},0}^{-3}{\ell }_0$ and the distinction between ${\lambda }_{\Vert }^w$ and ${\lambda }_{\perp }^w$ is lost. As a result, for local and isotropic interactions (meaning λ^w ∼ λ^z ∼ λ), there is no difference between the expressions for χ^w and for χ^z (see Table 1). On the other hand, at smaller scales (λ^w ≲ ℓ_A), we recover a distinction between ${\lambda }_{\Vert }^w$ and ${\lambda }_{\perp }^w$ because background fluctuations become sub-Alfvénic and anisotropic (i.e., δz_λ⊥/v_A, 0 <  1 and ${\lambda }_{\perp }^z\ll {\lambda }_{\Vert }^z$), and the quasi-parallel condition (6) does apply again, affecting χ^w. Therefore, a quasi-parallel AW packet with ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }<{\ell }_{\mathrm{A}}$ experiences a weak nonlinear interaction with background turbulence (i.e., ${\chi }_{{\lambda }_{\perp }}^{w,\mathrm{q}\parallel }\approx (\delta z_{{\lambda }_{\perp }}/v_{\mathrm{A},0}{)}^2\ll 1$), while background turbulence is critically balanced, ${\chi }_{{\lambda }_{\perp }}^z\sim 1$ (cf. equation (8) in Section 3 and Table 1). Hence, the two nonlinear parameters need not be confused at scales below ℓ_A, and this is why the turbulent damping rate of quasi-parallel AWs that we obtain for this range of scales is again different from equation (52) of Lazarian (2016).

As for the sub-Alfvénic case discussed earlier, we note that also for this M_A, 0 >  1 regime the result in (13) implies a drastic change in the effectiveness of turbulent damping for CR-generated Alfvén-wave packets. While our result agrees with the turbulent damping rate found in the literature for the range of scales ℓ_A ≲ λ^w ≲ ℓ₀, the corresponding rate at smaller scales, ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }<{\ell }_{\mathrm{A}}$, can be several orders of magnitude smaller than that usually employed in CR studies (see Table 2 and the right panel in Figure 1). The damping rate in (13) indeed rapidly diverges from the one given in Lazarian (2016) with decreasing wavelength of the quasi-parallel AW packet when ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }<{\ell }_{\mathrm{A}}$, reaching its maximum difference at ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }\sim {\lambda }_{\Vert ,\min }^w$, where the actual damping rate in (13) is a factor $\sim S_0^{-1/2}{M}_{\mathrm{A},0}^{3/2}\ll 1$ lower than the value provided in the literature (see the explicit comparison in Table 2). We also point out that for super-Alfvénic injection, the ordering $S_0\gg M_{\mathrm{A},0}^3$ is implied in order to have a significant GS95 range (see Appendix A).

4.2. Magnetohydrodynamic turbulence with dynamic alignment

The classic picture presented above is now extended to the case in which counter-propagating Elsässer fields $\delta {\mathit{z}}_{{\lambda }_{\perp }}^{+}$ and $\delta {\mathit{z}}_{{\lambda }_{\perp }}^{-}$ (or, in a similar way, velocity and magnetic-field fluctuations, δu_λ⊥ and δb_λ⊥) tend to align with each other in a scale-dependent fashion (Boldyrev 2006). This dynamic alignment not only modifies the fluctuation scaling and anisotropy by inducing a weakening of the nonlinear interaction, but can also open the possibility of a reconnection-mediated regime at small scales (still within the MHD range of scales, not in the kinetic regime; see, e.g., Boldyrev & Loureiro 2017; Mallet et al. 2017). In this section we consider the case when such a scale-dependent alignment occur only in critically balanced turbulent fluctuations⁵, χ^z ∼ 1. In this case, in addition to the (W0) and (K41) regimes of the previous Section 4.1, one can have two additional regimes for background turbulence (see also Table 1):

• [B06] An anisotropic, strong cascade of critically balanced and dynamically aligned fluctuations that replaces the GS95 regime. In this case, the fluctuation alignment angle decreases with decreasing scale so that $sin{\theta }_{{\lambda }_{\perp }}^z\propto {({\lambda }_{\perp }^z/{\ell }_0)}^{1/4}$. For sub- and trans-Alfvénic injection (M_A, 0 ≤ 1), the perpendicular scaling of turbulent fluctuations at scales ${\lambda }_{\perp }^z\lesssim {\lambda }_{\perp ,\mathrm{CB}}^z$, turns out to be $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{B}06)}/v_{\mathrm{A},0}\sim M_{\mathrm{A},0}^{3/2}{({\lambda }_{\perp }^z/{\ell }_0)}^{1/4}$. When $S_0\gg M_{\mathrm{A},0}^{-4}$, this cascade can further turn into a reconnection-mediated regime below a transition scale ${\lambda }_{\perp ,\ast }^{z(\mathrm{subA})}/{\ell }_0\sim M_{\mathrm{A},0}^{-2/7}{S}_0^{-4/7}$. For super-Alfvénic injection (M_A, 0 >  1), turbulent fluctuations at scales ${\lambda }_{\perp }^z\lesssim {\ell }_{\mathrm{A}}$ follow instead a perpendicular scaling given by $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{B}06)}/v_{\mathrm{A},0}\sim M_{\mathrm{A},0}^{3/4}{({\lambda }_{\perp }^z/{\ell }_0)}^{1/4}$. In this super-Alfvénic regime, a transition to reconnection-mediated turbulence may occur at a scale ${\lambda }_{\perp ,\ast }^{z(\mathrm{supA})}/{\ell }_0\sim M_{\mathrm{A},0}^{-9/7}{S}_0^{-4/7}$ if $S_0\gg M_{\mathrm{A},0}^3$. If $S_0\lesssim M_{\mathrm{A},0}^{-4}$ and $S_0\lesssim M_{\mathrm{A},0}^3$ respectively in the sub-Alfvénic and super-Alfvénic regime, the dissipation scale for a given regime is larger than the corresponding transition scale and the (B06) cascade does not transition into the tearing-mediated regime. When this is case, the dissipation scale is achieved at ${\lambda }_{\perp ,\min }^{z(\mathrm{subA})}/{\ell }_0\sim {(M_{\mathrm{A},0}{S}_0)}^{-2/3}$ in the trans-/sub-Alfvénic regime, or at ${\lambda }_{\perp ,\min }^{z(\mathrm{supA})}/{\ell }_0\sim M_{\mathrm{A},0}^{-1}{S}_0^{-2/3}$ for super-Alfvénic injection.

• [TMT] A strong anisotropic cascade of critically balanced and dinamically (mis-)aligned fluctuations that are generated by magnetic-reconnection processes. In this case, fluctuations scale as $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{TMT})}\propto S_0^{1/5}{({\lambda }_{\perp }^z)}^{3/5}$ and are subjected to a scale-dependent mis-alignment given by $sin{\theta }_{{\lambda }_{\perp }^z}^z\propto {({\lambda }_{\perp }^z/{\ell }_0)}^{-4/5}$. For sub- and trans-Alfvénic injection (M_A, 0 ≤ 1), the perpendicular scaling of tearing-mediated turbulent fluctuations is given by $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{TMT})}/v_{\mathrm{A},0}\sim S_0^{1/5}{M}_{\mathrm{A},0}^{8/5}{({\lambda }_{\perp }^z/{\ell }_0)}^{3/5}$, while in the super-Alfvénic regime (M_A, 0 >  1) they scale as $\delta z_{{\lambda }_{\perp }^z}^{(\mathrm{TMT})}/v_{\mathrm{A},0}\sim S_0^{1/5}{M}_{\mathrm{A},0}^{6/5}{({\lambda }_{\perp }^z/{\ell }_0)}^{3/5}$. In this regime the dissipation scale is the same as for the GS95 cascade (i.e., ${\lambda }_{\perp ,\min }^{z(\mathrm{subA})}/{\ell }_0\sim M_{\mathrm{A},0}^{-1}{S}_0^{-3/4}$ for trans-/sub-Alfvénic turbulence, or ${\lambda }_{\perp ,\min }^{z(\mathrm{supA})}/{\ell }_0\sim {(M_{\mathrm{A},0}{S}_0)}^{-3/4}$ for super-Alfvénic injection).

We can verify that the nonlinear interaction between a quasi-parallel AW packet and the anisotropic turbulent fluctuations populating the background is also weak for these cascades (i.e., ${\chi }_{{\lambda }_{\perp }}^{w,\mathrm{q}\parallel }<1$), except for the case of super-Alfvénic injection at scales λ^w ≳ ℓ_A, where instead ${\chi }_{\lambda }^w\sim {\chi }_{\lambda }^z>$ holds (see Table 1).

We note that a tearing-mediated range emerges either when $S_0\gg M_{\mathrm{A},0}^{-4}$ and $S_0\gg M_{\mathrm{A},0}^3$ for sub-Alfvénic and super-Alfvénic turbulence injection, respectively (see Appendix A). Even admitting a wide range of values for the injection-scale Alfvénic Mach number M_A, 0, it seems reasonable to assume that these conditions would be met quite easily in many astrophysical systems. This is because the turbulent plasmas hosted by these environments are typically very weakly collisional, and thus characterized by large Lundquist numbers (see, e.g., Priest & Forbes 2007; Ji & Daughton 2011, and references therein). Nevertheless, for a TMT range to exist, 3D anisotropy of turbulent fluctuations is required. Hence, scale-dependent (dynamic) alignment is absolutely necessary. How and under what circumstances dynamic alignment occurs is still largely unexplored and matter of ongoing debate (see, e.g., Schekochihin 2022; Cerri et al. 2022, and references therein).

4.2.1. Sub- and trans-Alfvénic turbulence (M_A, 0 ≤ 1) with dynamic alignment

A quasi-parallel AW packet with normalized parallel wavelength ${\widehat{\lambda }}_{\Vert }^w={\lambda }_{\Vert }^{w,\mathrm{q}\Vert }/{\ell }_0$ injected in pre-existing sub-Alfvénic turbulence for which dynamic alignment of critically balanced fluctuations occurs is subjected to the following turbulent damping rate:

[M_A, 0 <  1, with dynamic alignment]

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\mathrm{subA}}^{w,\mathrm{q}\Vert }\sim \{\begin{array}{ccc}{M}_{\mathrm{A},0}^{8/3}{\left({\widehat{\lambda }}_{\Vert }^w\right)}^{1/3}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill M_{\mathrm{A},0}^4\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim M_{\mathrm{A},0}\\ \hfill & & \\ M_{\mathrm{A},0}^{24/5}{\left({\widehat{\lambda }}_{\Vert }^w\right)}^{-1/5}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill {\widehat{\lambda }}_{\Vert ,\ast }^{w(\mathrm{subA})}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim M_{\mathrm{A},0}^4\\ \hfill & & \\ M_{\mathrm{A},0}^4{\left(S_0{\widehat{\lambda }}_{\Vert }^w\right)}^{1/2}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill {\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{subA})}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim {\widehat{\lambda }}_{\Vert ,\ast }^{w(\mathrm{subA})}.\end{array}\end{array}$$(14)

Here ${\widehat{\lambda }}_{\Vert ,\ast }^{w(\mathrm{subA})}\sim S_0^{1/5}{(M_{\mathrm{A},0}{\widehat{\lambda }}_{\perp ,\ast }^{z(\mathrm{subA})})}^{8/5}\sim S_0^{-5/7}{M}_{\mathrm{A},0}^{8/7}$ is the wavelength below which the AW packet interacts with background fluctuations in the TMT regime, while ${\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{subA})}\sim S_0^{1/5}{(M_{\mathrm{A},0}{\widehat{\lambda }}_{\perp ,\min }^{z(\mathrm{subA})})}^{8/5}\sim S_0^{-1}$ is the shortest wavelength at which the turbulent damping is effective.

The trans-Alfvénic regime is obtained from the above case (i.e., when there is no W0 range):

[M_A, 0 ≃ 1, with dynamic alignment]

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\mathrm{transA}}^{w,\mathrm{q}\Vert }\sim \{\begin{array}{ccc}{\left({\widehat{\lambda }}_{\Vert }^w\right)}^{-1/5}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill S_0^{-5/7}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim 1\\ \hfill & & \\ {\left(S_0{\widehat{\lambda }}_{\Vert }^w\right)}^{1/2}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill S_0^{-1}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim S_0^{-5/7}.\end{array}\end{array}$$(15)

Here we have explicitly written the transition and dissipation scales: ${\widehat{\lambda }}_{\Vert ,\ast }^{w(\mathrm{transA})}\sim S_0^{1/5}{({\widehat{\lambda }}_{\perp ,\ast }^{z(\mathrm{transA})})}^{8/5}\sim S_0^{-5/7}$ and ${\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{transA})}\sim S_0^{1/5}{({\widehat{\lambda }}_{\perp ,\min }^{z(\mathrm{transA})})}^{8/5}\sim S_0^{-1}$, respectively.

One can see that when it comes to the interaction of the AW packet with anisotropic background fluctuations, including dynamic alignment in the picture changes the behavior of the turbulent damping rate significantly with respect to the classic scenario (cf. Table 2). In the range of scales for which the packet interacts with critically balanced turbulence (i.e., ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }\lesssim {\lambda }_{\Vert ,\mathrm{CB}}^{w,\mathrm{q}\Vert }$), the damping rate due to this nonlinear interaction is always higher than the corresponding rate obtained without dynamic alignment (cf. Eqs. (11) and (12) and Table 2; see also the left panel of Figure 1 for an immediate visual example). This can be understood by considering that dynamic alignment means a shallower perpendicular spectrum of background fluctuations (−3/2 instead of −5/3), and thus at any scale ${\lambda }_{\perp }^z<{\lambda }_{\perp ,\mathrm{CB}}^z$ there is more turbulent power to nonlinearly damp the AW packet.

In general, if a CR-driven Alfvén-wave is injected in a background of sub-Alfvénic turbulence with dynamic alignment, now the damping rate interestingly exhibits two breaks that separate the three distinct regimes available in this scenario (contrary to the single break that would be present without dynamic alignment). This is a consequence of the new tearing-mediated regime that is only possible when a scale-dependent alignment takes place, and is well summarized in Tables 1 and 2 (see also Appendix A). The first break is the same as in turbulence without dynamic alignment, and it occurs for wavelengths interacting with background fluctuations at the transition scale between weak and strong turbulence (${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }\sim {\lambda }_{\Vert ,\mathrm{CB}}^{w,\mathrm{q}\Vert }\sim M_{\mathrm{A},0}^4{\ell }_0$). The second break instead emerges when the wavelength corresponds to a scale for which the AW packet starts to interact with tearing-mediated turbulence (${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }\sim {\lambda }_{\Vert ,\ast }^{w(\mathrm{subA})}\sim S_0^{-5/7}{M}_{\mathrm{A},0}^{8/7}{\ell }_0$). In astrophysical situations for which this damping mechanism is the main process that determines the efficiency of CR confinement, these breaks could leave a signature at the corresponding energies in the propagated spectrum of these cosmic particles (see Section 6 for a brief discussion about the values of M_A, 0 and S₀ for which these breaks in the damping rate could be responsible for the features that are observed in the propagated CR spectrum).

4.2.2. Super-Alfvénic turbulence (M_A, 0 >  1) with dynamic alignment

When background fluctuations are injected with M_A, 0 >  1 and dynamic alignment of critically balanced turbulent fluctuations takes place, an AW packet undergoes turbulent damping with the following rate:

[M_A, 0 >  1, with dynamic alignment]

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\mathrm{supA}}^{w,\mathrm{q}\Vert }\sim \{\begin{array}{ccc}{M}_{\mathrm{A},0}{\left({\widehat{\lambda }}^w\right)}^{-2/3}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill M_{\mathrm{A},0}^{-3}\lesssim {\widehat{\lambda }}^w\lesssim 1\\ \hfill & & \\ M_{\mathrm{A},0}^{12/5}{\left({\widehat{\lambda }}_{\Vert }^w\right)}^{-1/5}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill {\widehat{\lambda }}_{\Vert ,\ast }^{w(\mathrm{supA})}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim M_{\mathrm{A},0}^{-3}\\ \hfill & & \\ M_{\mathrm{A},0}^3{\left(S_0{\widehat{\lambda }}_{\Vert }^w\right)}^{1/2}\frac{v_{\mathrm{A},0}}{{\ell }_0}\hfill & & \hfill {\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{supA})}\lesssim {\widehat{\lambda }}_{\Vert }^w\lesssim {\widehat{\lambda }}_{\Vert ,\ast }^{w(\mathrm{supA})}.\end{array}\end{array}$$(16)

Here ${\widehat{\lambda }}_{\Vert ,\ast }^{w(\mathrm{supA})}\sim S_0^{1/5}{M}_{\mathrm{A},0}^{6/5}{({\widehat{\lambda }}_{\perp ,\ast }^{z(\mathrm{supA})})}^{8/5}\sim S_0^{-5/7}{M}_{\mathrm{A},0}^{-6/7}$, and the shortest wavelength for turbulent damping to be effective is ${\widehat{\lambda }}_{\Vert ,\min }^{w(\mathrm{supA})}\sim S_0^{1/5}{M}_{\mathrm{A},0}^{6/5}{({\widehat{\lambda }}_{\perp ,\min }^{z(\mathrm{supA})})}^{8/5}\sim S_0^{-1}$.

Again, we note that the isotropic normalized wavelength ${\widehat{\lambda }}^w={\lambda }^w/{\ell }_0$ enters the damping rate in the range ℓ_A ≲ λ^w ≲ ℓ₀ (i.e., where the AW packet interacts with hydrodynamic-like pre-existing turbulence). In this range of scales, the result is unchanged with respect to the turbulent damping rate obtained without dynamic alignment (Section 4.1.2). At smaller scales ${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }\lesssim {\ell }_{\mathrm{A}}$, the quasi-parallel condition in (6) applies again and the damping rate depends explicitly on the normalized parallel wavelength ${\widehat{\lambda }}_{\Vert }^w={\lambda }_{\Vert }^{w,\mathrm{q}\Vert }/{\ell }_0$. In this regime, the turbulent damping rate differs significantly from the one obtained without dynamic alignment. When a scale-dependent alignment of fluctuations is taken into account, the turbulent damping is always much more effective than in the case obtained without dynamic alignment. This leads to a damping rate that can be higher by orders of magnitude with respect to that in (13), depending on the Alfvénic Mach number M_A, 0 and on the Lunquist number S₀ at injection scales (see Table 2 for a comparison of the scaling and the right panel of Figure 1 for and explicit graphic example). Finally, analogously to the case with M_A, 0 <  1, the damping rate also exhibits two breaks in a background of super-Alfvénic turbulence, if dynamic alignment can occur. The first break emerges for wavelengths comparable to the transition scale between hydrodynamic-like and critically balanced turbulence (${\lambda }^w\sim {\ell }_{\mathrm{A}}\sim M_{\mathrm{A},0}^{-3}{\ell }_0$). A second break occurs at wavelengths corresponding to the scale marking the transition between a dynamically aligned cascade and the tearing-mediated range (${\lambda }_{\Vert }^{w,\mathrm{q}\Vert }\sim {\lambda }_{\Vert ,\ast }^{w(\mathrm{supA})}\sim S_0^{-5/7}{M}_{\mathrm{A},0}^{-6/7}{\ell }_0$). For which values of M_A, 0 and S₀ these breaks in the damping rate could be responsible for the features that are observed in the propagated CR spectrum is briefly discussed in Section 6.

5. Feedback on background fluctuations

When deriving the scaling of ${\mathrm{\Gamma }}_{\mathrm{turb}}^{w,\mathrm{q}\Vert }$ in the previous section, we neglected the feedback that the injected Alfvén-wave packets could have on pre-existing turbulent fluctuations. In general, background fluctuations could also be affected by their nonlinear interaction with these AW packets, which we call feedback. It is therefore instructive to understand when this effect has to be taken into account. The relevance of this feedback can be estimated by comparing two timescales. The first is the intrinsic cascade time of background fluctuations, which is the timescale of the cascade process induced by pre-existing (counter-propagating) fluctuations on themselves, ${\tau }_{\mathrm{casc}}^{(z|z)}\sim {\tau }_{\mathrm{nl}}^{(z|z)}/{\chi }^{(z|z)}$ (or just ${\tau }_{\mathrm{casc}}^{(z|z)}\sim {\tau }_{\mathrm{nl}}^{(z|z)}$, if χ^(z|z) ≳ 1). The second is CR-induced cascade time, which is the timescale of the cascade that would be induced by the injected AW packets on the background fluctuations, ${\tau }_{\mathrm{casc}}^{(z|w)}\sim {\tau }_{\mathrm{nl}}^{(z|w)}/{\chi }^{(z|w)}$ (or just ${\tau }_{\mathrm{casc}}^{(z|w)}\sim {\tau }_{\mathrm{nl}}^{(z|w)}$, if χ^(z|w) ≳ 1).

Hereafter, the simpler superscript “z” is used instead of “(z|z)” for the sake of homogeneity of notation with the previous sections. Moreover, for the sake of clarity in the qualitative discussion that follows, the effect of dynamic alignment is not taken into account in this section. The nonlinear parameter χ^(z|w) describing the interaction of background fluctuations with CR-driven AW packets is ${\chi }_{{\lambda }_{\perp }}^{(z|w)}\sim ({\lambda }_{\Vert ,{\lambda }_{\perp }}^z/{\lambda }_{\perp }^z)(\delta w_{{\lambda }_{\perp }}/v_{\mathrm{A},0})\sim (\delta w_{{\lambda }_{\perp }}/\delta z_{{\lambda }_{\perp }}){\chi }_{{\lambda }_{\perp }}^z$, while the timescale associated with this nonlinear interaction is ${\tau }_{\mathrm{n}\mathrm{l},{\lambda }_{\perp }}^{(z|w)}\sim {\lambda }_{\perp }/\delta w_{{\lambda }_{\perp }}\sim (\delta z_{{\lambda }_{\perp }}/\delta w_{{\lambda }_{\perp }}){\tau }_{\mathrm{n}\mathrm{l},{\lambda }_{\perp }}^z$. The ratio of the two cascade timescales is thus given by

$$\begin{array}{c}\hfill \frac{{\tau }_{\mathrm{casc}}^{(z|w)}}{{\tau }_{\mathrm{casc}}^z}\sim \{\begin{array}{cc}{(\delta z_{{\lambda }_{\perp }}/\delta w_{{\lambda }_{\perp }})}^2\hfill & \text{if}\{\begin{array}{c}{\chi }_{{\lambda }_{\perp }}^z<1\text{and}{\chi }_{{\lambda }_{\perp }}^{(z|w)}\lesssim 1\\ \text{or}\\ {\chi }_{{\lambda }_{\perp }}^z\sim 1\text{and}{\chi }_{{\lambda }_{\perp }}^{(z|w)}<1\end{array}\hfill \\ \hfill & \\ (\delta z_{{\lambda }_{\perp }}/\delta w_{{\lambda }_{\perp }}){\chi }_{{\lambda }_{\perp }}^z\hfill & \text{if}{\chi }_{{\lambda }_{\perp }}^z<1\text{and}{\chi }_{{\lambda }_{\perp }}^{(z|w)}>1\hfill \\ \hfill & \\ (\delta z_{{\lambda }_{\perp }}/\delta w_{{\lambda }_{\perp }})\hfill & \text{if}{\chi }_{{\lambda }_{\perp }}^z\sim 1\text{and}{\chi }_{{\lambda }_{\perp }}^{(z|w)}\gtrsim 1\hfill \\ \hfill & \\ (\delta z_{\lambda }/\delta w_{\lambda }){(\delta w_{\lambda }/v_{\mathrm{A},0})}^{-1}\hfill & \text{if}{\chi }_{\lambda }^z>1\text{and}{\chi }_{\lambda }^{(z|w)}<1\hfill \\ \hfill & \\ (\delta z_{\lambda }/\delta w_{\lambda })\hfill & \text{if}{\chi }_{\lambda }^z>1\text{and}{\chi }_{\lambda }^{(z|w)}\gtrsim 1.\hfill \end{array}\end{array}$$(17)