2.1. Incompressible MHD

Before exploring relativistic Alfvénic turbulence, it is important to first review some of the fundamental knowledge learned from Newtonian turbulence to provide a framework for discussing the relativistic limit. This discussion is far from exhaustive, because incompressible MHD turbulence is an exceptionally broad and deep field.

We will begin our discussion by presenting some of the basic properties of incompressible turbulence. Incompressible MHD is the basis from which Newtonian turbulence theories have been derived. Incompressible MHD assumes $v \ll c_s$, where $c_s$ is the sound speed. In other words, compressive fluctuations are carried away from the source at essentially infinite speed, an assumption that is not applicable for relativistic systems. This assumption implies $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {v} = 0$, leading to the incompressible MHD equations

(2.1)\begin{gather} \frac{\partial \boldsymbol{v}}{\partial t} + \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{v} ={-}\boldsymbol{\nabla} P/\rho_0 + \boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{B}/\rho_0, \end{gather}

(2.2)\begin{gather}\frac{\partial \boldsymbol{B}}{\partial t} + \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{B} = \boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{v}, \end{gather}

(2.3)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{B} = 0, \end{gather}

where $P$ is the total (thermal plus magnetic) pressure and $\rho _0$ is the equilibrium mass density.

For turbulence analysis, the incompressible MHD equations are typically cast in Elsasser form (Elsasser Reference Elsasser1950) by adding and subtracting the momentum and induction equations to arrive at

(2.4)\begin{gather} \frac{\partial \boldsymbol{z}^\pm}{\partial t} \mp \boldsymbol{v}_A \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{z}^\pm{=}{-}\boldsymbol{z}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{z}^\pm{-}\boldsymbol{\nabla} P / \rho_0, \end{gather}

(2.5)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{z}^\pm{=} 0, \end{gather}

where the magnetic field has been separated into equilibrium and fluctuating parts, $\boldsymbol {B} = B_0 \hat {\boldsymbol {z}} + \delta \boldsymbol {B}$, $\boldsymbol {v}_A = \boldsymbol {B}_0 / \sqrt {4 {\rm \pi}\rho _0}$ and $\boldsymbol {z}^\pm = \boldsymbol {v} \pm \delta \boldsymbol {B} / \sqrt {4 {\rm \pi}\rho _0}$ are the Elsasser fields. Because the Elsasser fields are divergenceless, this is a closed system of equations (Montgomery Reference Montgomery1982). To obtain an equation for the pressure, one can simply take the divergence of (2.4) to find

(2.6)\begin{equation} \nabla^2 P/\rho_0 ={-}\boldsymbol{\nabla} \boldsymbol{\cdot} \left( \boldsymbol{z}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{z}^\pm\right). \end{equation}

The Elsasser equations, (2.4), are the progenitor equations to describe Alfvénic turbulence, and one can discern many important points about turbulence simply from the form of the equations. First, the terms on the left-hand side are linear, while those on the right are nonlinear. Therefore, linearizing the equations is as simple as setting the right-hand side to zero. By linearizing, one can immediately see that the system supports two propagating linear wave modes with $\omega = \pm k_\parallel v_A$: (i) Alfvén waves with $\boldsymbol {z}^\pm$ polarized in the $\hat {\boldsymbol {z}} \times \widehat {\boldsymbol {k}}^\pm$ direction and (ii) pseudo-Alfvén waves, the incompressible limit of magnetosonic slow modes, with polarization $\widehat {\boldsymbol {k}}^\pm \times (\hat {\boldsymbol {z}} \times \widehat {\boldsymbol {k}}^\pm )$, where $\widehat {\boldsymbol {k}}^\pm$ corresponds to the unit vector along the wavevector of $\boldsymbol {z}^\pm$. We adopt the convention that $\omega \ge 0$, and the sign of $k_\parallel$ determines the propagation direction. Note that the incompressible assumption orders the magnetosonic fast mode out of the system by imposing $c_s \rightarrow \infty$. Thus, one can interpret the Elsasser field $\boldsymbol {z}^+$ ($\boldsymbol {z}^-$) as describing the evolution of Alfvén or pseudo-Alfvén waves propagating down (up) the equilibrium magnetic field.

Next, we turn our focus to the primary nonlinear term, $\boldsymbol {z}^\mp \boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {z}^\pm$. The most immediate point one can see from the form of this term is that the nonlinearity only survives if both $\boldsymbol {z}^+$ and $\boldsymbol {z}^-$ are non-zero, i.e. the nonlinearity is one that does not involve self-interaction of an Alfvén wave with itself but its interaction with an oppositely propagating Alfvén wave. If either Elsasser field is zero, then the opposite Elsasser variable is an exact solution. For instance, if $\boldsymbol {z}^- = 0$, then $\boldsymbol {z}^+(x,y,z+v_A t)$ is an exact, nonlinear solution representing an arbitrary-amplitude Alfvén or pseudo-Alfvén wave propagating in the $-\hat {\boldsymbol {z}}$ direction.

Physically, the counter-propagating waves shear one another when they interact through this nonlinear term, and the shearing leads to a transfer of energy to smaller scales. For simplicity, let us focus on the case of counter-propagating Alfvén waves and examine the nonlinear term in more detail. In Fourier space, the nonlinear interaction term for a $\boldsymbol {z}^+$ wave distorted by a $\boldsymbol {z}^-$ wave is

(2.7)\begin{equation} \boldsymbol{z}^- \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{z}^+{\propto} \left[\left(\hat{\boldsymbol{z}} \times \widehat{\boldsymbol{k}}^- \right) \boldsymbol{\cdot}\widehat{\boldsymbol{k}}^+\right] \left(\hat{\boldsymbol{z}} \times \widehat{\boldsymbol{k}}^+ \right) = \left[\hat{\boldsymbol{z}} \boldsymbol{\cdot} \left( \widehat{\boldsymbol{k}}^-{\times} \widehat{\boldsymbol{k}}^+ \right)\right] \left(\hat{\boldsymbol{z}} \times \widehat{\boldsymbol{k}}^+ \right), \end{equation}

where we have used the polarization properties of the Alfvén waves to write $\boldsymbol {z}^\pm \propto \hat {\boldsymbol {z}} \times \widehat {\boldsymbol {k}}^\pm$. Therefore, for the nonlinear interaction of counter-propagating Alfvén waves to be non-zero, $\hat {\boldsymbol {z}} \boldsymbol {\cdot } (\widehat {\boldsymbol {k}}^- \times \widehat {\boldsymbol {k}}^+ ) \ne 0$, i.e. variation is required in both directions perpendicular to the equilibrium field. An alternative way to state this requirement is that the waves must be polarized with respect to one another in the plane perpendicular to the equilibrium field so that the perpendicular wavevector components are not collinear.

From examination of the linear and nonlinear terms of the Elsasser equation, we have gleaned three crucial facts for turbulence: (i) the system supports two linear waves modes, both of which require $k_\parallel \ne 0$ to propagate; (ii) the nonlinearity requires counter-propagating fluctuations; and (iii) the nonlinearity requires that the fluctuations be relatively polarized with one another in the plane perpendicular to the equilibrium field. Thus, to fully capture the physics of the turbulent cascade one requires all three dimensions (Tronko, Nazarenko & Galtier Reference Tronko, Nazarenko and Galtier2013; Howes Reference Howes2015). This requirement to retain all three dimensions to capture Alfvénic turbulence persists in both full MHD and kinetic plasmas (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Howes Reference Howes2015).

Finally, we highlight one other important fact about turbulence, which can be seen by examining the Elsasser energy equation, obtained by taking the dot product of $\boldsymbol {z}^\pm$ with (2.4) and integrating over all of space. Assuming periodic boundary conditions or that the fields vanish at infinity, one obtains

(2.8)\begin{equation} \int \textrm{d}^3 \boldsymbol{r} \frac{\textrm{d} |\boldsymbol{z}^\pm|^2}{\textrm{d} t} = 0. \end{equation}

Equation (2.8) implies that there is no exchange of energy between the upward- and downward-propagating fluctuations, even during nonlinear interactions (Maron & Goldreich Reference Maron and Goldreich2001; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). The collisions of counter-propagating fluctuations are therefore elastic: one wave packet can scatter another, but the individual energies of the $z^+$ and $z^-$ fluctuations do not change.

2.2. The connection to RMHD

As noted in § 1, Alfvénic turbulence proceeds in an anistropic fashion as it cascades to smaller scales, eventually leading to a state in which $k_\parallel \ll k_\perp$ (and $\delta B / B_0 \ll 1$) regardless of the initial isotropy of the plasma at the outer scale. This scale-by-scale anisotropic turbulence cascade has been well observed in simulations (Cho & Vishniac Reference Cho and Vishniac2000; Maron & Goldreich Reference Maron and Goldreich2001; Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012) and in situ solar wind observations (Horbury, Forman & Oughton Reference Horbury, Forman and Oughton2008; Wicks et al. Reference Wicks, Horbury, Chen and Schekochihin2010; Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012; Chen Reference Chen2016). Thus, it would be advantageous to consider an ordering framework that leverages this fact. Fortunately, the minimal ordering assumptions for RMHD are anisotropic fluctuations ($k_\parallel \sim \epsilon k_\perp$), small-amplitude fluctuations relative to the background (e.g. $\delta B \sim \epsilon B_0$) and characteristic time scales $\omega \sim k_\parallel v_A$, where $\epsilon \ll 1$. Note that early derivations of RMHD (Kadomtsev & Pogutse Reference Kadomtsev and Pogutse1974; Strauss Reference Strauss1976) further assumed strong magnetization, implying plasma $\beta \ll 1$. However, more recent derivations (Schekochihin & Cowley Reference Schekochihin and Cowley2007; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009) have demonstrated that the RMHD equations are valid for arbitrary plasma $\beta$, assuming a homogeneous background. Conveniently, the equations of RMHD are essentially identical to (2.4), with the exception that the gradients in the nonlinear terms reduce to gradients perpendicular to the equilibrium magnetic field.

Much like the incompressible MHD equations, the fast wave is ordered out of RMHD. In the solar wind, this is well justified and supported by observations, which show that fast modes generally compose a small fraction of the solar wind (Tu & Marsch Reference Tu and Marsch1994; Howes et al. Reference Howes, Bale, Klein, Chen, Salem and TenBarge2012; Klein et al. Reference Klein, Howes, TenBarge, Bale, Chen and Salem2012). The RMHD equations have gained prominence as the preferred set of equations for describing Alfvénic turbulence because they have a few favourable properties compared with incompressible MHD, despite their close similarity. First, the RMHD equations are a rigorous set of equations for describing collisional or collisionless Alfvénic physics at scales large compared to the ion gyroradius, $k_\perp \rho _i \ll 1$ (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). Second, in the anisotropic limit of RMHD, the plus and minus Alfvén mode, plus and minus slow (or pseudo-Alfvén) mode and the lone entropy mode cascades decouple. In other words, there are five independent cascade channels in RMHD, and the energy in each channel is independently conserved. Thus, the five channels do not exchange energy with one another, analogous to the two independent channels described above for incompressible MHD. The Alfvénic cascade is described by the perpendicular vector components of (2.4), the slow mode cascade by the parallel component and the entropy mode cascade by the pressure balance condition (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). Formally, the RMHD equations are only valid for the slow and entropy modes in the collisional limit, because these modes are subject to collisionless wave–particle interactions via Landau (Landau Reference Landau1946) or Barnes (Barnes Reference Barnes1966) damping, even for scales $k_\perp \rho _i \ll 1$. To describe these modes in the collisionless limit, one can instead use kinetic RMHD, which evolves the perpendicular dynamics using conventional RMHD and the parallel dynamics using the reduced ion kinetic equation (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). Third, in the anisotropic limit of RMHD, the Alfvén waves are not affected by the slow or entropy modes, and the slow and entropy modes do not cascade on their own. The slow and entropy modes behave as passive scalars, which can only be cascaded to small scales by interacting with Alfvén waves. This fact also implies that slow and entropy fluctuations will not be generated in situ in a purely Alfvénic RMHD turbulence cascadeFootnote ³.

Finally, before moving forward to discuss compressible MHD turbulence, we note that RMHD is equally valid for describing weak and strong (critically balanced) turbulence (Galtier & Chandran Reference Galtier and Chandran2006). Comparing the strength of the nonlinear and linear terms in RMHD, we find

(2.9)\begin{equation} \chi = \frac{\boldsymbol{z}^\mp\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{z}^\pm}{\boldsymbol{v}_A \boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{z}^\pm} \sim \frac{z^\mp k_\perp}{v_A k_\parallel} \sim \frac{k_\perp \delta B }{k_\parallel B_0}. \end{equation}

The final expression is the ratio of asymptotically ordered quantities in RMHD and is therefore unordered. In other words, RMHD can equally well describe weak ($\chi \ll 1$) and critically balanced ($\chi \sim 1$) turbulence.

As a brief aside, it is worthwhile at this point to provide a physical interpretation of critical balance, and justify why we have chosen to ignore the case $\chi > 1$. Physically, critical balance amounts to equating the linear, propagation time, $\tau _A = l_\parallel / v_A$, with the nonlinear (or ‘eddy turnover’) time, $\tau _{NL} \sim l_\perp / u_\perp$. The case $\tau _A \ll \tau _{NL}$ ($\chi \ll 1$) is the weak turbulence case, which will eventually, at sufficiently small scales, transition to $\tau _A \sim \tau _{NL}$ because the weak turbulence cascade does not produce a cascade in the direction parallel to the equilibrium field: $l_\parallel$ is fixed at the outer scale. Thus, in weak turbulence $\tau _A$ remains fixed while $\tau _{NL}$ will decrease with scale. The case $\tau _A \gg \tau _{NL}$ corresponds essentially to creating two-dimensional structures perpendicular $\boldsymbol {B}_0$ and is not sustainable at small scales separated from any external forcing. A fluctuation can only remain two-dimensional if it is causally connected, but for a system with an equilibrium $B_0$, the maximum parallel length over which a fluctuation can be coherent is $l_\parallel \sim \tau _{NL} v_A$, i.e. $l_\parallel / v_A \sim \tau _A \sim \tau _{NL}$. Thus, if a system is driven such that $\tau _A \gg \tau _{NL}$, it will rapidly relax back to critical balance, $\tau _A \sim \tau _{NL}$, at small scales. It is also worth noting that the critical balance condition, or predictions that follow from critical balance, has been observed in simulations (Cho & Vishniac Reference Cho and Vishniac2000; Maron & Goldreich Reference Maron and Goldreich2001; Perez & Boldyrev Reference Perez and Boldyrev2008; TenBarge & Howes Reference TenBarge and Howes2012; Mallet et al. Reference Mallet, Schekochihin and Chandran2015; Mallet & Schekochihin Reference Mallet and Schekochihin2017), in situ solar wind observations (Horbury et al. Reference Horbury, Forman and Oughton2008; Chen et al. Reference Chen, Wicks, Horbury and Schekochihin2010; Wicks et al. Reference Wicks, Horbury, Chen and Schekochihin2010; Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012; Chen Reference Chen2016) and laboratory experiments (Ghim et al. Reference Ghim, Schekochihin, Field, Abel, Barnes, Colyer, Cowley, Parra, Dunai and Zoletnik2013)Footnote ⁴.

Although the current discussion is focused on RMHD, which requires a strong equilibrium magnetic field, the preceding discussion concerning critical balance can equally well apply to small scales in a system without an equilibrium magnetic field. Alfvénic fluctuations at a scale $l$ propagate along the total, local magnetic field at position $\boldsymbol {x}_0$, $\boldsymbol {B}_0^{local} (\boldsymbol {x}_0) = \boldsymbol {B}_0 + \sum _{l' \gtrsim l} \delta \boldsymbol {B}_{l'}(\boldsymbol {x}_0)$. Since the propagation and nonlinear times both decrease with scale, the fluctuations with $l' \gtrsim l$ are approximately static relative to small-scale turbulence fluctuations, and those with $l' \lesssim l$ are more rapid and will average to zero. Therefore, small-scale Alfvénic turbulence always sees an approximately constant, local mean magnetic field.

2.3. Compressible MHD

Given the complexity of incompressible turbulence, it is somewhat unsurprising that compressible MHD turbulence has received less attention. The first important point about compressible turbulence is the nature of the fast mode cascade. The fast mode propagates isotropically ($\omega _F \propto k$), and therefore the fast mode turbulence cascade is also isotropic, as confirmed in numerical simulations of compressible MHD (Cho & Lazarian Reference Cho and Lazarian2002,  Reference Cho and Lazarian2003). In the weak limit, Chandran (Reference Chandran2005), Luo & Melrose (Reference Luo and Melrose2006) and Chandran (Reference Chandran2008) have examined the wave kinetic equation in detail to explore the couplings between the Alfvén and fast modes. For quasi-parallel fluctuations ($k_\parallel > k_\perp$), the frequency and wavevector matching conditions permit any of the following three-wave interactions on approximately equal footing: $\mathrm {A}+\mathrm {A}\rightarrow \mathrm {A}$, $\mathrm {A}+\mathrm {A}\rightarrow \mathrm {F}$, $\mathrm {A}+\mathrm {F}\rightarrow \mathrm {F}$ and $\mathrm {F}+\mathrm {F}\rightarrow \mathrm {F}$. However, in the obliquely propagating limit ($k_\parallel \lesssim k_\perp$), the cascades decouple, leaving only $\mathrm {A}+\mathrm {A}\rightarrow A$ and $\mathrm {F}+\mathrm {F}\rightarrow \mathrm {F}$, because in this limit, the frequency of the fast modes exceeds significantly that of the Alfvén modes, making the frequency matching conditions involving mixed wave modes impossible. Although the wavevector and frequency matching conditions are broadened as the turbulence becomes stronger, one still expects the turbulence to be dominated by interactions that are nearby in scale (Kolmogorov Reference Kolmogorov1941; Frisch Reference Frisch1995; Howes, TenBarge & Dorland Reference Howes, TenBarge and Dorland2011), and thus also nearby in wavevector and frequency space. Considering these facts combined with the isotropic cascade of fast modes and the anisotropic cascade of Alfvénic modes, it is expected that the cascades decouple at small scales, regardless of the initial wavevector distribution. In other words, regardless of the way a system is driven or initialized, the Alfvénic portion of the cascade will eventually decouple and behave the same as the RMHD cascade described in the preceding section. Further, moderate-amplitude fast modes rapidly form shocks as they propagate, and in weakly collisional plasmas fast modes are moderately to strongly damped via resonant wave–particle interactions (Landau Reference Landau1946; Barnes Reference Barnes1966) for a wide range of plasma parameters (Klein et al. Reference Klein, Howes, TenBarge, Bale, Chen and Salem2012). Therefore, in the Newtonian limit, focusing only on the Alfvénic turbulence cascade is, generally, well justified.

3.1. Relativistic Elsasser equations

To begin our exploration of weak turbulence in relativistic, magnetically dominated plasmas, we start from an equation set that: (i) makes direct contact with the Newtonian Elsasser equations, (2.4), and (ii) highlights the fundamental role of Alfvén wave collisions in establishing the turbulence cascade. To this end, we turn to Chandran, Foucart & Tchekhovskoy (Reference Chandran, Foucart and Tchekhovskoy2018), who present a set of Elsasser-like equations for general relativistic MHD, including an inhomogeneous background. Here, we outline the derivation and consider only the special relativistic form with a homogeneous background. From this point forward, we employ Lorentz–Heaviside units and all speeds are normalized to the speed of light, $c=1$, and we assume a fixed Minkowski metric with signature $\eta ^{\mu \nu } = \text {diag}\{-1,1,1,1\}$. The ideal relativistic MHD equations are mass conservation

(3.1)\begin{equation} \partial_\nu(\rho u^\nu) = 0, \end{equation}

the stress energy equation

(3.2)\begin{equation} \partial_\nu T^{\mu \nu} = 0 \end{equation}

and the induction equation

(3.3)\begin{equation} \partial_\nu (b^\mu u^\nu - b^\nu u^\mu) = 0, \end{equation}

where $\rho$ is the rest-mass density, $u^\mu = (\gamma,\gamma \boldsymbol {v})$ is the four-velocity and $\gamma = 1 / \sqrt {1 - v^2}$ is the Lorentz factor. Furthermore,

(3.4)\begin{equation} T^{\mu \nu} = \mathcal{E} u^\mu u^\nu + \left(p + \frac{b^2}{2}\right) \eta^{\mu \nu} - b^\mu b^\nu \end{equation}

is the stress-energy tensor,

(3.5)\begin{equation} b^\mu = \frac{1}{2}\varepsilon^{\mu \nu \kappa \lambda} u_\nu F_{\kappa \lambda} = \left(\gamma (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{B}), \frac{1}{\gamma} \left[B^i + \gamma^2 (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{B}) v^i \right]\right) \end{equation}

is the magnetic field four-vector, $b^2 = b^\mu b_\mu$, $F_{\kappa \lambda }$ is the Faraday tensor, $\mathcal {E} = h + b^2$, $h = \rho (1 + \epsilon ) + p$ is the enthalpy density, $\varepsilon$ is the specific internal energy and $p$ is the gas pressure. Repeated indices indicate summation, Greek indices span the four-dimensional spacetime (0 to 3), while Latin indices (1 to 3) correspond to the spatial directions in a suitably chosen $3+1$ foliation of spacetime. We adopt an ideal equation of state $p = \rho \epsilon (\varGamma -1)$, with an adiabatic index $\varGamma = 4/3$ appropriate for relativistic plasmas. For this equation of state and adiabatic index, $h = \rho + 4 p$.

An Elsasser-like formulation of the relativistic MHD equations can be achieved by simply multiplying (3.3) by $\pm \sqrt {\mathcal {E}}$, adding the result to (3.2) and dividing the sum by $\mathcal {E}$, yielding

(3.6)\begin{equation} \partial_\nu (z^\mu_\pm z^\nu_\mp{+} \varPi \eta^{\mu \nu}) + \left(\frac{3}{4} z^\mu_\pm z^\nu_\mp{+} \frac{1}{4} z^\mu_\mp z^\nu_\pm{+} \varPi \eta^{\mu \nu}\right) \frac{\partial_\nu \mathcal{E}}{\mathcal{E}} = 0, \end{equation}

where

(3.7)\begin{equation} z^\mu_\pm{=} u^\mu \pm \frac{b^\mu}{\sqrt{\mathcal{E}}} \end{equation}

are the relativistic Elsasser four-vector fields and

(3.8)\begin{equation} \varPi = \frac{2p + b^2}{2\mathcal{E}} = \frac{1}{2} - \frac{2 p + \rho}{2\mathcal{E}}. \end{equation}

Much like in non-relativistic MHD, (3.6) represents the evolution of upward- and downward-propagating Elsasser fields; however, (3.6) represents the fully compressible system and is thus not completely analogous to (2.4). In the relativistic limit, one cannot simply go to the incompressible limit typically used to derive incompressible MHD, because the maximum speed is the speed of light, and thus compressible fluctuations cannot be ordered in such a way as to instantaneously carry information away from a source. We can, however, consider a limit similar to that of RMHD to isolate the Alfvénic fluctuations described by (3.6). Such a limit, in which fluctuations are highly elongated relative to a mean magnetic field, is reasonable to consider for many astrophysical plasmas which often have strong mean fields, e.g. black hole accretion disks and jets (Narayan, Igumenshchev & Abramowicz Reference Narayan, Igumenshchev and Abramowicz2003), coronae (Chandran et al. Reference Chandran, Foucart and Tchekhovskoy2018; Yuan, Blandford & Wilkins Reference Yuan, Blandford and Wilkins2019) and magnetar magnetospheres (Parfrey, Beloborodov & Hui Reference Parfrey, Beloborodov and Hui2012). Further, as we show in § 4, relativistic Alfvénic turbulence proceeds in the same anisotropic sense as Newtonian turbulence, eventually leading to highly anisotropic, small-amplitude fluctuations at small scales, regardless of the outer-scale conditions. Additionally, many astrophysical systems are characterized by exceptionally large inertial rangesFootnote ⁵, vastly exceeding the three- to four-order-of-magnitude-wide inertial range observed in the solar wind near Earth (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2008; Bruno & Carbone Reference Bruno and Carbone2013; Kiyani, Osman & Chapman Reference Kiyani, Osman and Chapman2015; Chen Reference Chen2016).

3.2. Relativistic RMHD

To derive a set of Elsasser equations for RRMHD, we begin by separating the fluid into mean (background) quantities and fluctuating quantities of the form

(3.9)\begin{equation} b^\mu = \langle b^\mu \rangle + \delta b^\mu. \end{equation}

We also construct an average fluid rest frame in which $\langle u^i\rangle$ vanishesFootnote ⁶. We define $\lambda$ to be the correlation length transverse to $\langle B^i \rangle$, and $L$ to be the characteristic scale of the background, i.e. the length scale parallel to $\langle B^i \rangle$. As in RMHD, we assume $\lambda / L \sim \mathcal {O}\left (\epsilon \right )$, where $\epsilon \ll 1$. In addition to the anisotropy assumption, we also assume that the characteristic frequency is of the order of the Alfvén frequency, $\partial _t \sim \langle B^i\rangle \partial _i \sim k_\parallel v_A$, and that the fluctuations are ordered small:

(3.10)\begin{gather} \frac{\sqrt{\delta u^2}}{v_A} \equiv \frac{\sqrt{\delta u^\mu \delta u_\mu}}{v_A} \sim \sqrt{\delta b^2} \sim \mathcal{O}\left(\epsilon\right), \end{gather}

(3.11)\begin{gather}\frac{\delta \rho}{\langle\rho\rangle} \sim \frac{\delta p}{\langle p\rangle} \sim \mathcal{O}\left(\epsilon^2\right), \end{gather}

where

(3.12)\begin{equation} v_A^i = \frac{\langle b^i \rangle}{\sqrt{\langle\mathcal{E}\rangle}} \end{equation}

and $v_A = \sqrt {v_A^i v_{Ai}}$. We further assume that all background quantities, e.g. $\langle B^i\rangle = \boldsymbol {B}_0, \langle \rho \rangle \equiv \rho _0,\langle p\rangle \equiv p_0$, are constant with no background inhomogeneities.

As in Newtonian RMHD, in RRMHD, the fast wave will be ordered out of the system, since $\omega _F \sim k \gg \omega \sim k_\parallel$. Relativistic RMHD will describe Alfvén and pseudo-Alfvén fluctuations; however, we are not interested in the pseudo-Alfvén waves, which have fluctuations parallel to the background magnetic field and are a separate, passive, cascade channel. Therefore, we also assume $B_{0i} \delta z^i_\pm = \mathcal {O}\left (\epsilon ^2\right )$ to remove the pseudo-Alfvén modes. With all of the above assumptions, we note that $\gamma \sim 1 + \mathcal {O}\left (\epsilon ^2\right )$, $\partial _i \delta u^i \sim \partial _\mu \delta z^\mu _\pm \sim \langle b^t\rangle \sim \delta b^t \sim \delta u^t / v_A \sim \mathcal {O}\left (\epsilon ^2\right )$.

Applying the above reduced assumptions to the relativistic Elsasser equation (3.6), we arrive at

(3.13)\begin{equation} v^\nu_{A\mp} \partial_\nu \delta z^\mu_\pm{=}{-}\delta z^\nu_\mp \partial_\nu \delta z^\mu_\pm{-} \partial_\nu (\delta \varPi \eta^{\mu \nu}), \end{equation}

where to lowest non-zero order

(3.14)\begin{gather} v_{A\pm}^\mu = \left(1,\pm\frac{\boldsymbol{B}_0}{\sqrt{\mathcal{E}_0}}\right) \equiv (1,\pm \boldsymbol{v}_A), \end{gather}

(3.15)\begin{gather}\delta z^\mu_\pm{=} \delta u^\mu \pm \frac{\delta b^\mu}{\sqrt{\mathcal{E}_0}} = \left(0,\delta \boldsymbol{v}_\perp{\pm} \frac{\delta \boldsymbol{B}_\perp}{\sqrt{\mathcal{E}_0}}\right), \end{gather}

(3.16)\begin{gather}\delta \varPi ={-}\frac{2 \delta p + \delta \rho}{2 \mathcal{E}_0} + \frac{2 p_0 + \rho_0}{2 \mathcal{E}_0}\frac{\delta \mathcal{E}}{\mathcal{E}_0}, \end{gather}

$\mathcal {E}_0 = 4 p_0 + \rho _0 + B_0^2$ and $\delta \mathcal {E} = 4\delta p + \delta \rho + 2 B_0 \delta B_\parallel$. In three-vector form, the equation set is particularly simple and similar to the Newtonian Elsasser equations:

(3.17)\begin{equation} \frac{\partial \delta \boldsymbol{z}_{{\perp}{\pm}}}{\partial t} \mp \boldsymbol{v}_A \boldsymbol{\cdot} \boldsymbol{\nabla} \delta \boldsymbol{z}_{{\perp}{\pm}} ={-} \delta \boldsymbol{z}_{{\perp}{\mp}} \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp \delta \boldsymbol{z}_{{\perp}{\pm}} - \boldsymbol{\nabla}_\perp \delta \varPi. \end{equation}

The terms of the left-hand side of (3.13) are linear, while those on the right-hand side are nonlinear. Due to the assumptions regarding pressure and parallel fluctuations, the only relevant components of (3.13) are the two components transverse to the mean magnetic field, as expressed in (3.17). The other two components, time-like and parallel, are one order higher in the ordering parameter, $\epsilon$. Importantly, the parallel fluctuations do not appear at lowest order in the nonlinear $\boldsymbol {E} \times \boldsymbol {B}$ term in the perpendicular, Alfvén wave equations, (3.13) and (3.17), and they therefore do not cascade the Alfvén waves. However, the primary nonlinear term for the parallel fluctuations is of the form $\delta \boldsymbol {z}_{\perp _\pm } \boldsymbol {\cdot } \boldsymbol {\nabla }_\perp \delta z_\parallel$. Thus, the parallel fluctuations are passively scattered/mixed by the Alfvén waves, much like in Newtonian RMHD turbulence (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). Note that as in the Newtonian RMHD equations, the system is closed by taking the divergence of (3.17), or four-divergence of (3.13), to find an equation for $\delta \varPi$.

The reduced relativistic Elsasser equations, (3.17), for RRMHD are identical in form to the Newtonian RMHD equations, and at this point the standard approach to solve for the Alfvén dynamics would be to take the curl of (3.17) to eliminate the pressure term and solve for the Elsasser potentials rather than the Elsasser fields. Indeed, this is the approach we will also take; however, we note that it is possible to simplify the system even further, because there is a straightforward limit to consider in relativistic plasmas. This final simplification to consider is to remove the pressure fluctuations by assuming that we are in the magnetically dominated regime, $\sigma = b^2 / h \gg 1$. In this limit, $\langle \varPi \rangle = 1/2$, and $\delta \varPi \sim \mathcal {O}\left (\epsilon ^2\right ) / \sigma \ll \mathcal {O}\left (\epsilon ^2\right )$. Therefore, we arrive at the magnetically dominated RRMHD equations:

(3.18)\begin{equation} v^\nu_{A\mp} \partial_\nu \delta z^\mu_\pm{=}{-}\delta z^\nu_\mp \partial_\nu \delta z^\mu_\pm. \end{equation}

In three-vector form, the equation set is particularly simple:

(3.19)\begin{equation} \frac{\partial \delta \boldsymbol{z}_{{\perp}{\pm}}}{\partial t} \mp \boldsymbol{\nabla}_\parallel \delta \boldsymbol{z}_{{\perp}{\pm}} ={-} \delta \boldsymbol{z}_{{\perp}{\mp}} \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp \delta \boldsymbol{z}_{{\perp}{\pm}}. \end{equation}

Being in the $\sigma \gg 1$ limit, (3.18) and (3.19) are essentially the reduced analogue of the force-free limit of the ideal MHD equations (Gruzinov Reference Gruzinov1999; Komissarov Reference Komissarov2002). In this limit, $v_A = c = 1$.

To arrive at this simple form, it may seem that we have rather seriously brutalized the relativistic MHD equations by applying a series of restrictive asymptotic orderings: (i) $k_\parallel \ll k_\perp$; (ii) $\omega \sim k_\parallel v_A$; (iii) small-amplitude fluctuations with constant (mean) backgrounds; (iv) second-order parallel and pressure fluctuations; and (v) magnetically dominated, $\sigma \gg 1$. The reduced assumptions (i)–(iii) and (v) are consistent with one another, i.e. they can all be obtained by assuming the system is strongly magnetized. In the Newtonian limit, the fast mode can be eliminated by assuming the system is incompressible; however, the finite speed of light prevents easy elimination of the fast mode (Takamoto & Lazarian Reference Takamoto and Lazarian2017) without assuming the fluctuations are anisotropic ($k_\parallel \lesssim k_\perp$). To remove the slow mode, we assumed that parallel and pressure fluctuations are second-order quantities when deriving the reduced relativistic Elsasser equations. Moving to the magnetically dominated regime produces the same result, since in the $\sigma \rightarrow \infty$ limit, the slow wave is also ordered out of the system. Thus, despite the myriad assumptions to achieve a simple set of equations for describing relativistic Alfvénic turbulence, they are all self-consistent. This set of assumptions is also relevant to many astrophysical systems, such as magnetars (Li & Beloborodov Reference Li and Beloborodov2015; Li et al. Reference Li, Zrake and Beloborodov2019; Yuan et al. Reference Yuan, Levin, Bransgrove and Philippov2020b), glitches affecting radio emission from pulsar magnetospheres (Bransgrove, Beloborodov & Levin Reference Bransgrove, Beloborodov and Levin2020; Yuan et al. Reference Yuan, Beloborodov, Chen and Levin2020a) and X-ray-emitting coronae around black hole accretion disks (Thompson & Blaes Reference Thompson and Blaes1998; Chandran et al. Reference Chandran, Foucart and Tchekhovskoy2018). We also note that the most important assumption is wavevector anisotropy, $k_\parallel \ll k_\perp$, which naturally arises as the Alfvénic turbulence cascades to small scales.

3.3. Connection to and comparison with Newtonian RMHD solutions

Equation (3.17) will form the basis for the following analysis in § 4, because it represents the simplest set of equations for describing Alfvénic turbulence in magnetized, relativistic environments and has a form that is nearly identical to the Newtonian RMHD Elsasser equations. Thus, the system shares many properties with the Newtonian RMHD system of equations, some of which can be seen immediately: (i) the system supports linear Alfvén modes, which require $k_\parallel \ne 0$ to propagate; (ii) the nonlinearity requires counter-propagating fluctuations; and (iii) the nonlinearity requires that the fluctuations be polarized with respect to each other in the perpendicular plane. Further, the wave kinetic equations for the system coincide with the kinetic equations for shear Alfvén waves derived in Galtier et al. (Reference Galtier, Nazarenko, Newell and Pouquet2000) in the incompressible MHD limit and Boldyrev & Perez (Reference Boldyrev and Perez2009) in the RMHD limit neglecting imbalanced interactions, i.e. non-zero cross helicity. As demonstrated in those papers, the wave kinetic equation evolves the spectral energy $\textrm {e}^{\pm } = \langle |\delta \boldsymbol {z}_\perp ^\pm (\boldsymbol {k})|^2 \rangle$ and can be expressed as

(3.20)\begin{equation} \frac{\partial \,\textrm{e}^\pm(\boldsymbol{k})}{\partial t} = \int M_{k,pq} \textrm{e}^\mp(\boldsymbol{q}) [\textrm{e}^\pm(\boldsymbol{p}) - \textrm{e}^\pm(\boldsymbol{k})] \delta(q_\parallel)d_{k,pq}, \end{equation}

where $M_{k,pq} = ({\rm \pi} / v_A) (\boldsymbol {k}_\perp \times \boldsymbol {q}_\perp )^2 (\boldsymbol {k}_\perp \boldsymbol {\cdot } \boldsymbol {p}_\perp )^2 / (k_\perp q_\perp p_\perp )^2$ and $d_{k,pq} = \delta (\boldsymbol {k} - \boldsymbol {p} - \boldsymbol {q})d^3p \ d^3q$. Since the kinetic equation is unchanged from RMHD, we also know that the weak turbulence solutions: (i) are dominated by three-wave interactions and (ii) lead to an energy spectrum of the form $f(k_\parallel ) k_\perp ^{-2}$, where $f(k_\parallel )$ is set by external forcing or initial conditions. In the following sections, we explore in detail the three-wave interaction of Alfvén waves first analytically via heuristic solutions through third order, and then numerically to demonstrate the decoupling of the fast mode from the Alfvén waves in the obliquely propagating (reduced) limit. Note that although we employ the RRMHD equations (3.17) to derive the turbulence solutions in the following section, the Alfvén solutions are identical for the $\sigma \rightarrow \infty$ limit of the RRMHD equations (3.19).

5.1. Simulation description

To confirm the analytical results of §§  3 and 4, we consider the nonlinear interaction between two perpendicularly polarized Alfvén waves that counter-propagate in a three-dimensional, periodic domain along a uniform guide field $\boldsymbol {B_0} = B_0 \boldsymbol {\hat {z}}$ using the general relativistic MHD code BHAC (Porth et al. Reference Porth, Olivares, Mizuno, Younsi, Rezzolla, Moscibrodzka, Falcke and Kramer2017; Olivares et al. Reference Olivares, Porth, Davelaar, Most, Fromm, Mizuno, Younsi and Rezzolla2019; Ripperda et al. Reference Ripperda, Bacchini, Porth, Most, Olivares, Nathanail, Rezzolla, Teunissen and Keppens2019a, ). The recently added force-free limit for the resistive MHD code BHAC employs the numerical scheme of Ripperda et al. (Reference Ripperda, Bacchini, Porth, Most, Olivares, Nathanail, Rezzolla, Teunissen and Keppens2019a, ) and damps force-free violations $E>B$ and $\boldsymbol {E} \boldsymbol {\cdot } \boldsymbol {B} \neq 0$ on resistive time scalesFootnote ⁷. We employ both a periodic cubic domain with $L_{\perp }=L_x=L_y=L_{\parallel } = L_z = 2{\rm \pi}$ and a periodic elongated domain with $L_{\parallel } = 10 L_\perp = 20{\rm \pi}$, with resolution $N_x = N_y = N_z = 256$ and $N_z = 2560$ cells for the elongated case. Initially, we set a gas-to-magnetic-pressure ratio of $\beta =2p/B_0^2=0.02$ and magnetization $\sigma _{cold} \equiv b^2 / \rho \in [0.01;0.1;1;10;100]$ (corresponding to $\sigma \in [0.01;0.1;1;7;20]$), where we vary the density, $\rho$, but maintain a constant guide field, $\boldsymbol {B_0} = \boldsymbol {\hat {z}}$, and pressure $p=0.01$ with adiabatic index $\varGamma =4/3$ for an ideal relativistic gas. In the force-free case, $\beta \rightarrow 0$, $\sigma \rightarrow \infty$ and $v_A \rightarrow 1$.

As in § 4, we prescribe a scale-free definition of characteristic wavelengths $(k_x/k_\perp, k_y/k_\perp,$ $k_z / k_\parallel )$, and initialize our Alfvén wave simulations with two overlapping, counter-propagating and perpendicularly polarized Alfvén waves described by the wavevectors $\boldsymbol {k}_{+}=(1,0,-1)$ and $\boldsymbol {k}_{-}= (0,1,1)$. The magnetic field is initialized through a vector potential $\boldsymbol {A} = (-B_0 y, 0, \delta B_\perp [\sin (k_\perp x+ k_\parallel z) + \sin (k_\perp y-k_\parallel z)])$, representing the initial counter-propagating Alfvén waves with frequency $\omega _0 = k_{\parallel } v_A$. The electric field is initialized as $\boldsymbol {E} = (v_A B_y, v_A B_x,0)$ such that the velocity is equal to the drift velocity, $\delta \boldsymbol {u}_\perp = \boldsymbol {E} \times \boldsymbol {B}/B^2$. Note that the overlapping Alfvén waves, in contrast to a single Alfvén wave, are not an exact force-free equilibrium due to a small second-order violation $\boldsymbol {E}_{\mp } \boldsymbol {\cdot } \boldsymbol {B}_{\pm } \neq 0$ between the fields of the two waves, which is damped on a short time scale in BHAC.

The strength of the nonlinearity is characterized by $\chi = k_{\perp } \delta B_{\perp } / k_{\parallel } B_0$. To maintain weak turbulence, we fix $\chi = 0.01$ in all of the following simulations. By fixing $\chi$, we expect to maintain self-similar behaviour as we explore elongated domains that approximate the reduced limit. Thus, in the cubic domain with $k_{\perp } = k_{\parallel }$, $\delta u_{\perp } / v_A = \delta B_\perp / B_0 = 0.01$, and in the elongated case with $k_{\perp } = 10 k_{\parallel }$, $\delta u_{\perp } / v_A = \delta B_\perp / B_0 = 0.001$.

5.2. Alfvén wave–Alfvén wave collisions

5.2.1. Cubic domain

In figure 1, we present the evolution of the mode amplitudes of the $B_\perp$ (Alfvénic) fluctuations in a cubic domain scanning $\sigma _{cold} \equiv b^2 / \rho \in [0.01;0.1;1;10;100;\infty ]$, presented in descending order of $\sigma$ from top-left to bottom-right. The initial, counter-propagating and perpendicularly polarized Alfvén waves are represented by the red lines in each panel, the $(0,1,-1)$ mode by dashed lines and the $(1,0,1)$ mode by dotted lines. These modes interact to produce a secondary, nonlinear, magnetic shear mode indicated by the green line, representing the $(1,1,0)$ mode. Note that this mode is purely oscillatory in time with $\omega = 2 \omega _0$, as found in § A.3. Finally, the secondary mode interacts with each of the primary Alfvén waves to produce at third order two higher $k_\perp$ Alfvén waves, but with the same $k_\parallel$ as the primary waves, where the $(1,2,-1)$ and $(2,1,1)$ modes are represented by the blue dashed and dotted curves. These dynamics are in agreement with the results of § A.4. These modes grow secularly in time, $B_{\perp 3} \propto t$, as indicated by the black line in each panel, which is a curve proportional to time and scaled by the final amplitude of $B_{\perp 3}$. The net result of this Alfvén wave–Alfvén wave collision is the anisotropic transfer of energy from large to small scales, which is governed by three-wave interactions.

Figure 1. Mode evolution of $B_{\perp }$ for a cubic domain with $\chi =0.01$ and $\sigma _{cold} \equiv b^2 / \rho \in [0.01;0.1;1;10;100;\infty ]$, presented in descending order of $\sigma$ from (a) to (f). The red lines correspond to the initial counter-propagating Alfvén modes: the $(0,1,-1)$ and $(1,0,1)$ modes are shown by dashed and dotted lines, respectively. Dashed green lines correspond to the second-order nonlinear shear mode $(1,1,0)$. The $(1,2,-1)$ and $(2,1,1)$ third-order, secularly growing modes are shown by dashed and dotted blue lines, respectively. The black line corresponds to a secular growth directly proportional to $t$ and scaled by the final amplitude of the third-order mode.

5.2.2. What about fast waves? Exploring the Alfvén wave–fast wave coupling

Our analytical analysis in the preceding sections purposefully neglected the fast wave by choosing the reduced limit. However, the fast wave may play an important role in releasing energy in strongly magnetized, relativistic, astrophysical plasmas, because the fast wave can travel across field lines (Li et al. Reference Li, Zrake and Beloborodov2019; Yuan et al. Reference Yuan, Levin, Bransgrove and Philippov2020b). For instance, Alfvén waves travel along magnetic fields, and on closed magnetic field lines in pulsar and magnetar magnetospheres their energy is confined to the magnetosphere, but fast waves can release their energy into the surrounding medium by propagating across field lines. For this reason, the coupling between the fast and Alfvén branches in relativistic MHD and its force-free limit has been explored in detail in recent years (Cho Reference Cho2005; Takamoto & Lazarian Reference Takamoto and Lazarian2016,  Reference Takamoto and Lazarian2017; Li et al. Reference Li, Zrake and Beloborodov2019). Through a numerical simulation study of relativistic turbulence, Takamoto & Lazarian (Reference Takamoto and Lazarian2016,  Reference Takamoto and Lazarian2017) find that the fast-to-Alfvén mode power scales as $(\delta u_f / \delta u_A)^2 \propto \sqrt {(1+\sigma )} \delta u_A / c_{f\perp }$ in the $\sigma \gtrsim 1$ limit, where $c_{f\perp }$ is the fast mode speed in the perpendicular plane. In the $\sigma \ll 1$ limit, Cho & Lazarian (Reference Cho and Lazarian2002,  Reference Cho and Lazarian2003) find that $(\delta u_F / \delta u_A)^2 \propto \delta u_A / c_{f\perp }$. Thus, for $\sigma \ll 1$, the fast and Alfvén branches decouple as the background magnetic field strength is increased; however, for $\sigma \gtrsim 1$, the coupling between the modes increases with $\sqrt {1 + \sigma }$, asymptotically approaching unity for large $\sigma$.

In figure 2, we present the evolution of the highest-amplitude $B_\parallel$ modes for the same initial configuration as presented in figure 1. Here $B_\parallel$ serves as a proxy for the amplitude of compressible, fast mode fluctuations, since the Alfvén modes have negligible $B_\parallel$ components. As an additional test (not shown), we have projected the fluctuations onto the fast and Alfvén wave eigenfunctions and recovered quantitatively similar results.

Figure 2. Mode evolution of $B_{\parallel }$ for a cubic domain with $\chi =0.01$ and $\sigma _{cold} \equiv b^2 / \rho \in [0.01;0.1;1;10;100;\infty ]$, presented in descending order of $\sigma$ from (a) to (f). Line styles and colours are as defined in figure 1.

We first note that the $(0,1,-1)$ and $(1,0,1)$ $B_\parallel$ modes have zero amplitude at $t=0$ but do develop finite amplitude that is approximately independent of $\sigma$. These modes appear at a mixture of two or more frequencies, the most dominant of which is at the fast mode frequency, $\omega \sim \omega _f \sim k c_f \sim \sqrt {2} \omega _0$, and remain at the same or lower amplitude as the tertiary (blue) modes. The secondary mode, $(1,1,0)$ (green), is the highest-amplitude mode and is again composed of two or more frequencies, the most dominant of which are $\omega \sim \omega _f \sim k c_f \sim \sqrt {2} \omega _0$ and $\omega \sim \omega _0 / 2$. Unlike the Alfvén branch, the $(1,1,0)$ mode can be a linear fast mode, which is consistent with a component frequency being at the fast mode frequency. Finally, the tertiary modes (blue) again display secular growth with time, with a primary frequency given by the fast mode frequency $\omega \sim \omega _f \sim k c_f \sim \sqrt {6} \omega _0$. The tertiary mode also displays the most significant dependence on $\sigma$. For $\sigma \ll 1$, the amplitude is independent of $\sigma$.

5.2.3. Elongated domain

The scaling of the Alfvén–fast mode coupling discussed above focuses purely on the strength of the background magnetic field while neglecting any wavevector anisotropy that may naturally arise from a strong magnetic field. As noted in § 2.3, in the Newtonian, obliquely propagating limit ($k_\parallel \lesssim k_\perp$), the fast and Alfvén branch turbulent cascades decouple, and we expect an analogous decoupling to occur in the relativistic limit, since the fast and Alfvén wave frequencies remain well separated in the oblique limit. Thus, we now examine the results of an elongated box simulation with $L_\parallel = 10L_\perp$.

In figure 3(a,b) are presented the mode analysis results of a force-free simulation of obliquely propagating Alfvén waves, with $k_\perp = 10 k_\parallel$. The colour and line styles are as in the previous figures. Comparing the relative amplitudes between each of the $B_\perp$ modes with the relative amplitudes for the cubic domain case presented in figure 1(a), it is clear that the elongated domain does not change the Alfvénic $B_\perp$, cascade. However, the three parallel modes’ growth, amplitude and frequency in the elongated case compared with the cubic case in figure 2(a) are markedly different. First, no secular growth of parallel modes is apparent. If there is a secularly growing mode, it is sufficiently low in amplitude that it does not modulate or interfere with the non-growing, purely oscillatory modes. Second, the mode amplitudes relative to the Alfvén wave primaries are an order of magnitude smaller in the elongated case relative to the cubic domain, which is consistent with the compressible mode amplitudes scaling with the elongation factor. Third, in the elongated case, the first-order (red) and third-order (blue) parallel modes are dominated by low-frequencyoscillations that match the frequency of the Alfvénic secondary mode. The second-order (green) mode is much higher in frequency, well above even the fast mode frequency, suggesting nonlinear modes with multiple frequencies contribute to this component.

Figure 3. Mode evolution of $B_{\perp }$ and $B_{\parallel }$ with $\chi =0.01$. (a,b) Results for an Alfvén wave–Alfvén wave collision in an elongated domain, $L_z = 10 L_x$. (c–f) Results for an Alfvén wave–fast wave collision in a cubic domain (c,d) and elongated domain (e,f). The line style and colour are as in previous figures.

These results support our intuition that the Alfvén and fast mode cascades decouple as the wavevector anisotropy increases and extend the findings of Chandran (Reference Chandran2005) to the relativistic limit. In the case examined here, $\omega _A = k_\parallel \simeq 0.1 \omega _F = 0.1 k$, leading to a poor frequency matching condition required for the resonant weak turbulence interaction. Although the wave matching condition broadens in strong turbulence, the Alfvén and fast mode cascades will remain well separated in the limit $k_\parallel \ll k_\perp$.

5.3. Alfvén wave–fast wave collisions

In the Alfvén wave–Alfvén wave collision case, we have shown that the amplitude of the product fast modes decreases as the initial modes become anisotropic, indirectly demonstrating that the modes decouple. Here, we directly diagnose the coupling of the Alfvén and fast mode cascades by examining the collision between an Alfvén wave and a fast wave in both cubic and elongated domains. Unlike the Alfvén wave–Alfvén wave collision explored in the previous sections, there is not a canonical choice for the counter-propagating fast mode. Therefore, we choose a fiducial fast mode with $\boldsymbol {k} = (0,1,-1)$, and an Alfvén wave with $\boldsymbol {k} = (1,0,1)$. This choice is largely for consistency with the Alfvén wave–Alfvén wave collision; however, it also obeys reasonable wavevector and frequency matching conditions, and it has the same expected product modes as the Alfvén wave–Alfvén wave collision. Further, the secondary modes will still be $(1,1,0)$ and $(-1,1,-2)$, and for these initial modes in a cubic domain, the secondary $(1,1,0)$ mode can satisfy the linear fast mode dispersion relation. For these simulations, we continue the convention of fixing $\chi = 0.01$ to satisfy weak turbulence.