TURBULENT RECONNECTION IN RELATIVISTIC PLASMAS AND EFFECTS OF COMPRESSIBILITY

Magnetic reconnection is known as a process responsible for very efficient magnetic field dissipation in many plasma phenomena. In particular, it is expected to play an important role in the acceleration of relativistic outflow in high energy astrophysical phenomena accompanying Poynting-dominated plasmas, such as relativistic jets (Blandford & Znajek 1977; Begelman et al. 1984; Komissarov et al. 2007; Kino et al. 2015), pulsar wind (Kennel & Coroniti 1984a, 1984b; Lyubarsky & Kirk 2001; Kirk & Skjæraasen 2003), and gamma-ray bursts (GRBs) (Lyutikov & Blandford 2003; Zhang & Yan 2011). However, the classical theory of magnetic reconnection (Parker 1957; Sweet 1958) predicts that magnetic reconnection becomes very slow in high magnetic Reynolds number plasmas (R_m ∼ 10¹⁰), and fails to explain the observed dissipation timescale in space and astrophysical phenomena. To solve this problem, a lot of effort has gone into finding a fast-reconnection process that does not depend on the value of resistivity. Using the equation of continuity, the reconnection rate can be expressed as

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{{\rm{in}}}}{{c}_{{\rm{A}}}}=\displaystyle \frac{{\rho }_{{\rm{s}}}}{{\rho }_{{\rm{in}}}}\displaystyle \frac{{v}_{{\rm{s}}}}{{c}_{{\rm{A}}}}\displaystyle \frac{\delta }{L},\end{eqnarray} \tag{ 1 }$$

where the subscripts in and s indicate the inflow and outflow region, respectively, ${v}_{{\rm{in}}},{v}_{{\rm{s}}}$ are the inflow and outflow velocity, respectively, c_A is the Alfvén velocity, ρ is the mass density, δ is the sheet thickness, and L is the sheet length. This equation shows that fast reconnection processes can be obtained by increasing the density ratio: ${\rho }_{{\rm{s}}}/{\rho }_{{\rm{in}}},$  (Brunel et al. 1982), the outflow velocity: ${v}_{{\rm{s}}}/{c}_{{\rm{A}}},$ and the aspect ratio of sheets: δ/L (Biskamp 1986; Shibata & Tanuma 2001; Loureiro et al. 2007; Bhattacharjee et al. 2009; Uzdensky et al. 2010; Takamoto 2013; Sironi & Spitkovsky 2014).

Turbulence has been considered to be a key process that can accelerate magnetic field annihilation (Matthaeus & Lamkin 1985; Eyink 2011; Takamoto et al. 2012; Higashimori et al. 2013). In particular, many astrophysical objects are considered to be high Reynolds number plasma, and it is natural to assume those plasma are in a turbulent state.⁵ It was theoretically suggested that strong Alfvénic turbulence also increases the sheet aspect ratio, and the reconnection rate becomes independent of the resistivity (Lazarian & Vishniac 1999, henceforth LV99). LV99 predicts the following expression of reconnection rate:

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{{\rm{in}}}}{{c}_{{\rm{A}}}}\simeq \mathrm{min}\left[{\left(\displaystyle \frac{L}{l}\right)}^{1/2},{\left(\displaystyle \frac{l}{L}\right)}^{1/2}\right]{\left(\displaystyle \frac{{v}_{l}}{{c}_{{\rm{A}}}}\right)}^{2},\end{eqnarray} \tag{ 2 }$$

where l and v_l are the energy injection scale and the velocity dispersion of turbulence at the injection scale, respectively. This was examined using magnetohydrodynamics (MHD) simulation (Kowal et al. 2009). However, the numerical work was limited only in the non-relativistic incompressible regime with plasma β larger than unity, and its applicability to relativistic Poynting-dominated plasma with relativistic turbulence was unclear, which is very important in the context of high energy astrophysical phenomena (Lyutikov & Lazarian 2013; Kadowaki et al. 2015).

In this paper, we extend the previous work on relativistic plasma to include both matter- and Poynting-dominated plasma. We also investigate the effects of compressibility on reconnection rate. In Section 2 we introduce the numerical setup and the method for the turbulence injection. The numerical result is presented in Section 3, and its theoretical explanation is presented in Section 4. Their implications for some high energy astrophysical phenomena are discussed in Section 5. Section 6 summarizes our conclusions.

We modeled the evolution of a current sheet in a turbulent flow using three-dimensional resistive relativistic magnetohydrodynamics (RRMHD). The initial current sheet was modeled by the relativistic Harris sheet (Hoh 1966; Kirk & Skjæraasen 2003) whose magnetic field is expressed as

$$\begin{eqnarray}&&{\boldsymbol{B}}={B}_{0}\mathrm{tanh}[z/\lambda ]{{\boldsymbol{e}}}_{x}+{B}_{{\rm{G}}}{{\boldsymbol{e}}}_{y},\end{eqnarray} \tag{ 3 }$$

where λ is the half-thickness of the initial sheet, and B₀ and B_G are the reconnecting magnetic field and guide field component, respectively. The pressure inside of the sheet is assumed to satisfy the pressure balance, and the upstream pressure is determined by the magnetization parameter $\sigma \equiv {B}^{2}/4\pi \rho {{hc}}^{2}$, where $h=1+({\rm{\Gamma }}/({\rm{\Gamma }}-1))(p/\rho {c}^{2})$ is the specific enthalpy of relativistic ideal gas, where ${\rm{\Gamma }}=4/3,$ and $p,\rho ,c$ are the gas pressure, mass density, and the light velocity, respectively. The initial temperature is assumed to be uniform, ${\rm{\Theta }}\equiv {k}_{{\rm{B}}}T/{{mc}}^{2}=1,$ where k_B and m are the Boltzmann constant and particle rest mass, respectively.

The evolution of the plasma is calculated using a three-dimensional RRMHD scheme developed by Takamoto & Inoue (2011), which solves the full RRMHD equations in a conservative fashion using the constrained transport algorithm. This allows us to treat the mass density, momentum, energy, and divergence of the magnetic field to be conserved within the machine round-off error. The resistivity, η, was assumed to be constant, typically $\eta /{Lc}={10}^{-4}.$ We followed the simulation setup similar to that used in Kowal et al. (2009). The numerical box is assumed to be $[-L/2,L/2]\times [0,L]\times [-L,L]$, where L = 20 λ.⁶ Note that the z-direction size of the numerical box is two times larger than the x-, y-direction in order to reduce the influence of turbulence on the reconnection inflow around the z-boundaries. We divided the numerical box into the homogeneous numerical cells with size: Δ = L/512. The time step size is set as: Δt = 0.1 Δ/c. We set the periodic boundary condition in the y-direction, and the free boundary condition the x- and z-direction.

In our model we drive turbulence using a similar method described by Mac Low (1999). We add a divergence-free 3-velocity field, $\delta {\boldsymbol{v}}$, and an electric field determined consistently at the injected velocity⁷ at time intervals ${\rm{\Delta }}{t}_{{\rm{inj}}}$ in a box region located around the current sheet: $[{-l}_{x},{l}_{x}]\times [0,L]\times [{-l}_{z},{l}_{z}]$, where l_x, l_z are scale lengths that are sufficiently larger than the injected turbulence eddy scale; Δt_inj is chosen to be shorter than the eddy turnover time at the injection scale l : ${\rm{\Delta }}{t}_{{\rm{inj}}}=l/4\pi {v}_{{\rm{inj,0}}}$, where ${v}_{{\rm{inj,0}}}=0.15c$ is a typical injection velocity in this study. We note that the dynamics of the turbulence becomes insensitive to the injection time interval as long as the injection time interval is around the eddy turnover time at the injection scale. Following Inoue et al. (2011) and Takamoto et al. (2012), the velocity field is described as: $\gamma \delta {v}^{i}={\sum }_{{\boldsymbol{k}}}P(k)\mathrm{sin}({\boldsymbol{k}}\cdot {\boldsymbol{x}}+{\phi }_{{\boldsymbol{k}}}^{i})$, where γ is the Lorentz factor of the injected velocity, i covers {x, y, z}, and ${\phi }_{{\boldsymbol{k}}}^{i}$ is a random phase. The one-dimensional power spectrum of the velocity field is assumed flat, ${k}^{2}P(k)\propto {k}^{0}.$ The initial-field perpendicular Fourier components ${k}_{\perp }=\sqrt{{k}_{y}^{2}+{k}_{z}^{2}}$ are chosen in a shell extending from ${k}_{l}-{\rm{\Delta }}k$ to ${k}_{l}+{\rm{\Delta }}k$, where ${k}_{l}L/2\pi =16,{\rm{\Delta }}{kL}/2\pi =3.$ Note that this scale size is a little larger than the initial sheet scale, and can be well-resolved by our present resolution. The parallel-field wave number ${k}_{| | }={k}_{x}$ is determined by ${k}_{| | }={k}_{\perp }{v}_{{\rm{inj}}}/{c}_{{\rm{A}}}$, where ${v}_{{\rm{inj}}}=\sqrt{\langle \delta {v}^{2}\rangle }$ is the root-mean-square velocity. Since the injected turbulent velocity does not immediately follow the critical balance condition (Goldreich & Sridhar 1995), the turbulence at the injected scale is weak (Galtier et al. 2000), which transits into the strong turbulence around the sheet width scale because of the energy cascade (Perez & Boldyrev 2008; Verdini & Grappin 2012; Meyrand et al. 2015). The weak MHD turbulence cascades the wave energy only perpendicular to the magnetic field, and the turbulence strength, $\chi \equiv {\tau }_{{\rm{A}}}/{\tau }_{{\rm{NL}}}\simeq {k}_{\perp }{v}_{\lambda }/{k}_{| | }{c}_{{\rm{A}}},$ gradually increases up towards unity, which results in the strong turbulence in the sheet (Verdini & Grappin 2012; Meyrand et al. 2015). Note that the injected velocity, v_inj, is different from the velocity at the injection scale, v_l, which was first pointed out in LV99. At the injection scale the weak MHD turbulence theory gives the following the energy cascade rate: ${\epsilon }_{{\rm{inj}}}\sim {v}_{l}^{2}/{\tau }_{{\rm{NL}}\;{\rm{}}}\sim {v}_{l}^{4}{l}_{| | }/{l}_{\perp }^{2}{c}_{{\rm{A}}}$. where ${\tau }_{{\rm{NL}}}\sim {({l}_{\perp }/{v}_{l})}^{2}/({l}_{| | }/{c}_{{\rm{A}}})$ is the distortion time of Alfvén wave packets. Combining this with the injected power: ${v}_{{\rm{inj}}}^{2}/{\rm{\Delta }}{t}_{{\rm{inj}}},$ we obtain

$$\begin{eqnarray}&&{v}_{{\rm{inj}}}\gtrsim \sqrt{\displaystyle \frac{{\rm{\Delta }}{t}_{{\rm{inj}}}{l}_{| | }}{{c}_{{\rm{A}}}{l}_{\perp }^{2}}}\;{v}_{l}^{2}\propto {v}_{l}^{2},\end{eqnarray} \tag{ 4 }$$

where the inequality resulted from the excitation of compression modes. In other words, v_inj is related to the injected power by some external force or free energy; on the other hand, v_l is the velocity resulted from the energy cascade of the weak MHD turbulence. We observed this relation in our simulations, and assumed this in the following (see also Lazarian & Vishniac 1999; Kowal et al. 2009).

Figure 1 is a snapshot of the gas pressure and magnetic field lines in the turbulent sheet in the highly magnetized case: σ = 5. Different from the Sweet–Parker sheet, this sheet is highly stochastic due to the turbulence, which induces a lot of reconnection points in the sheet and drives a fast reconnection process. Figure 2 shows the observed reconnection rates, v_R, which are measured using a method proposed by Kowal et al. (2009) (see Equation (13) in this paper); this allows us to measure the effective value of ${E}_{y}/{B}_{0}$ in the three-dimensional case, which provides us with a reconnection inflow velocity less contaminated by turbulent flows than the direct measure of inflow velocity, v_z.⁸ The left panel shows the reconnection rate with respect to the injected turbulent velocity in various kinds of magnetized plasmas. This shows that the turbulent reconnection rate shows three characteristic behaviors depending on the injected turbulent velocity ${v}_{{\rm{inj}}}/{c}_{{\rm{A}}}:$ (1) an increasing region following LV99; (2) a saturation region giving maximum rate; and (3) a decreasing region. When the injected turbulent velocity is sufficiently small, incompressible approximation can be applied, and the reconnection rate grows, following Equation (2). On the other hand, when the injected velocity becomes comparable to the Alfvén velocity, turbulence becomes compressible and the reconnection rate deviates from the incompressible theory. Interestingly, the injection velocity, ${v}_{{\rm{inj}}}/{c}_{{\rm{A}}}$, at the maximum rate becomes smaller as the magnetization parameter increases. We will discuss the relation of this tendency to the compressible effects in the next section. Note that the error bar in the panel seems to decrease with σ. We consider this to be because the kinetic energy of turbulence becomes smaller compared to the magnetic field energy as the magnetization parameter σ increases. The right panel of Figure 2 is the reconnection rate with respect to a different Lundquist number. It shows the reconnection rate is independent of the Lundquist number, and is determined by the turbulent strength. Note that the obtained maximum reconnection rate is very fast, ${v}_{R}/{c}_{{\rm{A}}}\sim 0.05,$ and is comparable to the relativistic Petschek reconnection rate (Lyubarsky 2005). This maximum reconnection rate also indicates that it will be possible to reach around 0.1–0.2 if the injection scales are comparable to the sheet length, as indicated by Equation (2).⁹

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Figure 3 is the guide field dependence of the reconnection rate in the case of ${\rm{\Delta }}{v}_{{\rm{inj}}}=0.15{c}_{{\rm{A}}}$ and ${\sigma }_{R}=5$, where σ_R is the magnetization parameter determined by the reconnecting magnetic field component B₀. We fixed the reconnected magnetic field and added the guide field component. As was reported by Kowal et al. (2009), the reconnection rate becomes independent of the guide field strength even in a relativistic Poynting-dominated plasma. In the case of ${B}_{{\rm{G}}}/{B}_{0}=1,$ the timescale necessary for reaching the steady state becomes five times longer than the no guide field case.

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4.1. Sheet Density

The obtained reconnection rate in Figure 2 shows an interesting behavior owing to compressibility, which cannot be explained by the incompressible theory, as shown in Equation (2). In the following, we give an explanation for the saturation and depression of the reconnection rate in the high Alfvén Mach number regime. Equation (1) indicates the compressible effects can be divided into two parts: (1) the density ratio between the sheet and inflow region ${\rho }_{{\rm{s}}}/{\rho }_{{\rm{in}}};$ (2) the decrease of the sheet width $\delta /L.$ Note that δ is the actual sheet thickness determined by the turbulence, which is different from the initial thickness, λ.

We begin by discussing the density ratio. Figure 4 plots ${\rho }_{{\rm{s}}}/{\rho }_{{\rm{in}}}$ with respect to the injected turbulence velocity in the matter- and the Poynting-dominated cases $\sigma =0.04$ and 5, respectively. They show that the density ratio decreases linearly with the turbulent strength. This can be understood from the conservation of energy flux:

$$\begin{eqnarray}&&{\rho }_{{\rm{in}}}{h}_{{\rm{in}}}{c}^{2}\left(1+\sigma \right){v}_{{\rm{in}}}L+{\rho }_{{\rm{in}}}(1+2{h}_{{\rm{in}}}\sigma ){\epsilon }_{{\rm{inj}}}{l}_{x}{l}_{z}\\ &&=\left({\rho }_{{\rm{s}}}{h}_{{\rm{s}}}{c}^{2}{\gamma }_{{\rm{s}}}^{2}+\displaystyle \frac{{B}_{{\rm{s}}}^{2}}{4\pi }\right){v}_{{\rm{s}}}\delta .\end{eqnarray} \tag{ 5 }$$

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We assumed a non-relativistic inflow, ${\gamma }_{{\rm{in}}}=1.$ The first and third terms are the energy flux in the inflow and outflow region, respectively. Note that the second term in the left-hand side of the equation expresses the kinetic and electric field energy of the injected turbulence; the turbulent components in the sheet is neglected because we use a sub-Alfvénic turbulence whose kinetic energy is small compared to the other terms. Using the pressure balance: ${p}_{{\rm{s}}}={p}_{{\rm{in}}}+{B}_{{\rm{in}}}^{2}/8\pi {\gamma }_{{\rm{in}}}^{2},$ the steady state condition: ${{cE}}_{y}={B}_{{\rm{in}}}{v}_{{\rm{in}}}={B}_{{\rm{s}}}{v}_{{\rm{s}}},$ and the equation of continuity, Equation (1), we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{{\rm{s}}}}{{\rho }_{{\rm{in}}}}=\displaystyle \frac{(2{h}_{{\rm{in}}}\sigma +4{\rm{\Theta }}){\gamma }_{{\rm{s}}}}{{h}_{{\rm{in}}}(1+\sigma )-{\gamma }_{{\rm{s}}}}\left[1-\displaystyle \frac{1+2{h}_{{\rm{in}}}\sigma }{(2{h}_{{\rm{in}}}\sigma +4{\rm{\Theta }}){\gamma }_{{\rm{s}}}^{2}}\displaystyle \frac{{\epsilon }_{{\rm{inj}}}}{\delta }\displaystyle \frac{{l}_{x}{l}_{z}}{{v}_{{\rm{s}}}{c}^{2}}\right],\end{eqnarray} \tag{ 6 }$$

where we neglected a small term proportional to ${(\delta /L)}^{2}\ll 1$ resulting from ${B}_{{\rm{s}}}^{2}/4\pi .$ Note that the denominator, ${h}_{{\rm{in}}}(1+\sigma )-{\gamma }_{{\rm{s}}},$ is always positive because ${\gamma }_{{\rm{s}}}\lesssim {\gamma }_{{\rm{A}}}=\sqrt{1+\sigma }$, where ${\gamma }_{{\rm{A}}}$ is the Lorentz factor of the Alfvén velocity in the upstream region. Since LV99 predicts $\delta \propto L({v}_{{\rm{inj}}}/{c}_{{\rm{A}}})$ (see Equations (2) and (4)), the second term in Equation (6) becomes proportional to: ${\epsilon }_{{\rm{inj}}}/{{Lv}}_{{\rm{inj}}}\propto {v}_{{\rm{inj}}};$ and finally, we obtain the following form of the relation: ${\rho }_{{\rm{s}}}/{\rho }_{{\rm{in}}}=\alpha (1-\beta {v}_{{\rm{inj}}}/{c}_{{\rm{A}}})$, which qualitatively reproduces the linear dependence of the density ratio on the injected turbulence strength, v_inj, in Figure 4.¹⁰

The decrease of the sheet density can be explained as follows: when the turbulence energy injection rate is small, ${v}_{{\rm{inj}}}\ll {c}_{{\rm{A}}},$ it increases the sheet width as predicted by LV99. However, the increase of the sheet width is proportional to $| {v}_{{\rm{inj}}}| \propto {\epsilon }_{{\rm{inj}}}^{1/2},$ and the turbulence energy injection rate, ${\epsilon }_{{\rm{inj}}}$, cannot be absorbed into the sheet width expansion as indicated by the second term in Equation (5). In this case, in order to keep the energy flux conservation, the system reduces the inflow velocity, v_in, on the left-hand side of Equation (5), and this results in the decrease of the sheet density compared to the inflow density as indicated in Equation (1). In other words, this is because the second term on the left-hand side increases with ${\epsilon }_{{\rm{inj}}}$ but right-hand side only increases with $\delta \propto {\epsilon }_{{\rm{inj}}}^{1/2}.$ 

Note that Equation (6) indicates that the density in sheets becomes negative when a too strong turbulence is injected, which does not occur in real situations. This is prevented by including neglected terms in Equation (5). In particular, as is discussed in Section 4.2, energy flux escaping as compressible waves cannot be neglected as the injected turbulent Alfvén Mach number approaches unity.

4.2. Compressible Turbulence Effects

LV99 obtained the following relation: $\delta /L\propto {({v}_{l}/{c}_{{\rm{A}}})}^{2}\propto {v}_{{\rm{inj}}}/{c}_{{\rm{A}}}$ using the incompressible MHD turbulence cascade law. In this paper, we treated compressible MHD turbulence, so it is expected that the above relation should be modified. More precisely, the LV99's relation can be rewritten as:

$$\begin{eqnarray}&&\displaystyle \frac{\delta }{L}\sim {\left(\displaystyle \frac{2{\epsilon }_{{\rm{inj}}}l}{{c}_{{\rm{A}}}^{3}}\right)}^{1/2}{\rm{min}}\left[{\left(\displaystyle \frac{L}{l}\right)}^{1/2},{\left(\displaystyle \frac{l}{L}\right)}^{1/2}\right],\end{eqnarray} \tag{ 7 }$$

and substituting ${\epsilon }_{{\rm{inj}}}\sim {v}_{l}^{4}/2{{lc}}_{{\rm{A}}}$ recovers Equation (2). Hence, if we find an expression of the energy injection rate, ${\epsilon }_{{\rm{inj}}}$, including compressible effects, Equation (7) may give us a new expression of the sheet width. Recently, Banerjee & Galtier (2013) obtained an exact relation of the energy cascade rate in the non-relativistic isothermal MHD turbulence. In the strong background average magnetic field limit, the relation reduces to:

$$\begin{eqnarray}&&-4\epsilon ={\rm{\nabla }}\cdot {\boldsymbol{F}}+{B}_{0}^{2}S,\end{eqnarray} \tag{ 8 }$$

 where the divergence ${\rm{\nabla }}$ is performed on the correlation length, which plays the role of the eddy scale length, ${\boldsymbol{F}}$ is the energy flux vector including compressible effects with order of ${B}_{0}^{2},$ and S is a source or sink term due to the compressible effects. This indicates that the compressible effects cannot be neglected in the strong background magnetic field, and the energy cascade rate should be redefined as an effective mean total energy cascade rate: ${\epsilon }_{{\rm{eff}}}\equiv \epsilon +{B}_{0}^{2}S/4,$ which will give us the necessary correction term in Equation (7).¹¹ Performing the Taylor expansion of ${\epsilon }_{{\rm{eff}}}$ in ${v}_{{\rm{inj}}}/{c}_{{\rm{A}}}\lt 1$ up to the second order, the corrected sheet width can be written as:

$$\begin{eqnarray}&&\displaystyle \frac{\delta }{L}\simeq \mathrm{min}\left[{\left(\displaystyle \frac{L}{l}\right)}^{1/2},{\left(\displaystyle \frac{l}{L}\right)}^{1/2}\right]\left[\displaystyle \frac{{v}_{{\rm{inj}}}}{{c}_{{\rm{A}}}}-{C}_{2}{\left(\displaystyle \frac{{v}_{{\rm{inj}}}}{{c}_{{\rm{A}}}}\right)}^{2}\right],\end{eqnarray} \tag{ 9 }$$

where C₂ is a coefficient resulting from the expansion. Figure 5 shows the reconnection rates with various kinds of magnetization parameters: $\sigma =0.04,0.5,1,5$, which are fitted by two functions; one uses Equation (9) with the density ratio, Equation (6): $f({v}_{{\rm{inj}}}/{c}_{{\rm{A}}})={C}_{1}({\rho }_{{\rm{s}}}/{\rho }_{{\rm{in}}})[{v}_{{\rm{inj}}}/{{c}_{{\rm{A}}}-{C}_{2}({v}_{{\rm{inj}}}/{c}_{{\rm{A}}})}^{2}],$ whereas the other function only takes into account the density ratio and uses the original LV99 sheet width, Equation (2): $g({v}_{{\rm{inj}}}/{c}_{{\rm{A}}})={C}_{1}({\rho }_{{\rm{s}}}/{\rho }_{{\rm{in}}}){v}_{{\rm{inj}}}/{c}_{{\rm{A}}}.$ Note that C₁ and C₂ describe coefficients independent of the injection velocity, ${v}_{{\rm{inj}}}$, indicated by Equation (9). As can be seen, they are well-reproduced only by function f when C₂ is near unity in all the cases.

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The above discussion assumes that the turbulence becomes compressible. To confirm the validity of this assumption, we performed the Helmholtz decomposition and the MHD wave mode decomposition of the velocity field. The analyses were performed using a background obtained by new runs without current sheets but using the same setup. This is because the current sheets introduce an inhomogeneous background that makes it very difficult to perform the above decomposition. The left panel of Figure 6 is the ratio between the compressible and incompressible velocity components: $\langle {v}_{{\rm{c}}}^{2}\rangle$ and $\langle {v}_{{\rm{i}}}^{2}\rangle ,$ respectively, in terms of the Alfvén Mach number of the injected turbulence velocity dispersion, ${v}_{{\rm{inj}}}/{c}_{{\rm{A}}},$ using various magnetization parameters σ. As expected, this shows that in all cases the compressible component increases with the turbulent strength. We also note that the maximum reconnection rate in Figure 2 is obtained when $\langle {v}_{{\rm{c}}}^{2}\rangle /\langle {v}_{{\rm{i}}}^{2}\rangle \sim 0.4.$ Interestingly, Figure 6 shows that the compressible component increases with the magnetization of the plasma similarly to non-relativistic MHD turbulence (Cho & Lazarian 2002). We consider this to be due to the ${B}_{0}^{2}$ factor in Equation (8), which indicates that the compressible effects becomes more important as the background magnetic field increases. Unfortunately, Equation (8) is a result in the case of the non-relativistic MHD turbulence, and its relativistic extension is the subject of our future work. The right panel of Figure 6 is the ratio of the fast MHD wave to the Alfvén wave power. The wave decomposition was performed in the Fourier space, assuming the linear dispersion relation of relativistic MHD plasma to be applicable to decompose waves in the turbulence, similarly to the non-relativistic case by Cho & Lazarian (2002). It shows a very similar behavior obtained by the Helmholtz decomposition. Interestingly, we also found that the slope of the ratio, ${({V}_{f}/{V}_{{\rm{A}}})}^{2},$ is proportional to $\sqrt{\sigma },$ so that it can be written as: ${({V}_{f}/{V}_{{\rm{A}}})}^{2}\propto \sqrt{\sigma }{v}_{{\rm{inj}}}/{c}_{{\rm{A}}}.$ Equation (9) can effectively be derived by considering the wave decomposition. The turbulent reconnection theory in LV99 considers that MHD turbulence results in a wider current sheet because of the wandering motion of the magnetic field driven by Alfvén waves. Hence, ${\epsilon }_{{\rm{inj}}}$ in Equation (7) is equivalent to the Alfvén wave power, ${V}_{{\rm{A}}}^{2},$ where V_A is the Alfvén wave component of the velocity. In the compressible regime, a part of the injected energy is distributed into the fast wave, and ${\epsilon }_{{\rm{inj}}}$ in Equation (7) should be rewritten as:

$$\begin{eqnarray}&&{V}_{{\rm{A}}}^{2}\simeq {v}_{{\rm{inj}}}^{2}-{V}_{f}^{2}\sim {v}_{{\rm{inj}}}^{2}-{V}_{{\rm{A}}}^{2}\sqrt{\sigma }{v}_{{\rm{inj}}}/{c}_{{\rm{A}}},\end{eqnarray} \tag{ 10 }$$

where V_f is the fast wave component of the velocity. This reduces to

$$\begin{eqnarray}&&\sqrt{{V}_{{\rm{A}}}^{2}}\sim {v}_{{\rm{inj}}}\left[1-\displaystyle \frac{\sqrt{\sigma }}{2}\displaystyle \frac{{v}_{{\rm{inj}}}}{{c}_{{\rm{A}}}}\right],\end{eqnarray} \tag{ 11 }$$

where we assume ${v}_{{\rm{inj}}}/{c}_{{\rm{A}}}\lt 1.$ This reproduces the dependence of Equation (11) on ${v}_{{\rm{inj}}}.$ ¹² However, we cannot find the dependence of C₂ on $\sqrt{\sigma }$ indicated in Equation (11). We consider this may be due to the effect of the inhomogeneous background structure and the back reaction from the tearing instability, which are not taken into account to obtain Figure 6.

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In this section we discuss applications to high energy astrophysical phenomena, that is, the Crab pulsar wind nebula, relativistic jets, and GRBs. We estimate the necessary spatial and temporal scales for explaining those phenomena, and compare them with the actual observational indications. In those phenomena, it is natural to consider that the driving source of turbulence depends on the phenomena and that the resulting turbulence strength is different. However, there are still a lot of theoretical and observational uncertainties in those phenomena, such as the precise spatial distribution of the magnetic field strength, the particle number density, and the particle composition, which is necessary to estimate synchrotron radiation flux. Identifying each turbulent process is beyond the scope of this paper. Hence, in the following, we only use the values of the reconnection rate to estimate those scales, so that the discussions can be generally applied for another dissipation mechanism, such as collisionless reconnection or plasmoid-chains.

1. Crab Pulsar Wind Nebula

In the case of the Crab pulsar wind nebula, the wind region is filled with current sheets (striped wind). It is known that the magnetic field cannot be completely dissipated in the wind region (Lyubarsky & Kirk 2001; Kirk & Skjæraasen 2003). One way to avoid this problem is to assume that the magnetic field dissipates just behind the termination shock  (Lyubarsky 2003; Pétri & Lyubarsky 2007; Sironi & Spitkovsky 2011). If assuming high-σ upstream flow, the downstream of the termination shock is still a relativistic flow with a Lorentz factor $\sqrt{\sigma }.$ In the downstream rest frame, the sheet separation is around $\pi \sqrt{\sigma }{r}_{{\rm{LC}}}$, where ${r}_{{\rm{LC}}}=c/{\rm{\Omega }}$ is the light cylinder radius, and Ω is the rotation period of the Crab pulsar. The necessary timescale to dissipate the magnetic field between the sheets by reconnection with reconnection rate v_in is $\pi \sqrt{\sigma }{r}_{{\rm{LC}}}/{v}_{{\rm{in}}}.$ In the termination shock rest frame, which would be equivalent to the observer's frame, the additional Lorentz factor $\sqrt{\sigma }$ is multiplied to the dissipation timescale, and it becomes

$$\begin{eqnarray}&&{\tau }_{{\rm{dissip,PWN}}}=\pi \sigma {r}_{{\rm{LC}}}/{v}_{{\rm{in}}}.\end{eqnarray} \tag{ 12 }$$

During this timescale, the current sheets propagate

$$\begin{eqnarray}{l}_{{\rm{dissip}}} & = & c{\tau }_{{\rm{dissip,PWN}}}=\pi \sigma {r}_{{\rm{LC}}}c/{v}_{{\rm{in}}}\\ & = & 5\times {10}^{-5}[\mathrm{pc}]\left(\displaystyle \frac{2\pi {r}_{{\rm{LC}}}}{{10}^{9}[\mathrm{cm}]}\right)\left(\displaystyle \frac{\sigma }{{10}^{4}}\right){\left(\displaystyle \frac{{v}_{{\rm{in}}}/c}{0.1}\right)}^{-1},\end{eqnarray} \tag{ 13 }$$

which is sufficiently short compared to the Crab pulsar wind nebula scale size (∼1 [pc]), and the turbulent reconnection can be one of the possible dissipation mechanisms in order to solve the σ-problem. The indicated value in Equation (13) is still too small to be resolved by X-ray (e.g., by the Chandra telescope). However, it may be possible to be observed by a future mission if σ is larger than 10⁴ and the reconnection rate, ${v}_{{\rm{in}}}/c$, is smaller than 0.1.

2. Relativistic Jets

In this case, although there is no proof of the existence of current sheets in the observed jets, we assume a dynamo process such that the magneto-rotational instability (MRI) in their accretion disk results in current sheets in jets (Barkov & Baushev 2011). Assuming the separation in the fluid comoving frame as $\bar{l},$ the reconnection timescale can be written as: ${\bar{\tau }}_{{\rm{dissip}}}=\bar{l}/{v}_{{\rm{in}}}$ in the fluid comoving frame. If we assume the reversing of the magnetic field direction occurs every one-Kepler rotation time at a radius r, the timescale in the blackhole rest frame can be written as, ${\tau }_{{\rm{rot}}}\sim \pi {r}_{M}{(2r/{r}_{M})}^{3/2}/c$, where r_M is the Schwarzshild radius. Hence, $\bar{l}$ can be written as, ${{\rm{\Gamma }}}_{{\rm{jet}}}c{\tau }_{{\rm{rot}}}.$ Since the dissipation timescale in the central blackhole comoving frame, ${{\rm{\Gamma }}}_{{\rm{jet}}}{\bar{\tau }}_{{\rm{dissip}}},$ should be less than the jet propagation time, ${l}_{{\rm{jet}}}/c,$ we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{{\rm{in}}}}{c}\gtrsim \displaystyle \frac{\bar{l}}{{l}_{{\rm{jet}}}}{{\rm{\Gamma }}}_{{\rm{jet}}}\sim \pi {{\rm{\Gamma }}}_{{\rm{jet}}}^{2}\left(\displaystyle \frac{{r}_{M}}{{l}_{{\rm{jet}}}}\right){\left(2\displaystyle \frac{r}{{r}_{M}}\right)}^{3/2}\\ &&\sim \;\left\{\begin{array}{l}1.9\times {10}^{-3}{\left(\displaystyle \frac{{l}_{{\rm{jet}}}}{60\;[{\rm{pc}}]}\right)}^{-1}{\left(\displaystyle \frac{r}{3{r}_{M}}\right)}^{3/2}\\ \times \;\left(\displaystyle \frac{{r}_{M}}{{10}^{-4}\;[\mathrm{pc}]}\right){\left(\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{jet}}}}{5}\right)}^{2}(\mathrm{radio}),\\ 3.3\times {10}^{-4}{\left(\displaystyle \frac{{l}_{{\rm{jet}}}}{0.5\;[{\rm{kpc}}]}\right)}^{-1}{\left(\displaystyle \frac{r}{3{r}_{M}}\right)}^{3/2}\\ \times \;\left(\displaystyle \frac{{r}_{M}}{{10}^{-4}\;[\mathrm{pc}]}\right){\left(\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{jet}}}}{6}\right)}^{2}({\text{}}{HST}),\end{array}\right.\end{eqnarray} \tag{ 14 }$$

where the upper and lower values are based on the radio and HST data of M87 (Rieger & Aharonian 2012). Here we assume the reversing of the magnetic field direction occurs at the innermost secure radius $r=3{r}_{M}.$ This indicates that both observation results can be explained by turbulent reconnection.

3. Gamma Ray Bursts

It is suggested that GRBs can be explained by the Poynting-dominated plasma model, and Zhang & Yan (2011) suggested a model called the ICMART model that can explain GRBs, including prompt emission spectral curves (Zhang & Zhang 2014). In the model, the reconnection rate assumes a relativistic value; the minimal value is around 0.1c. As indicated in Figure 2, this value of reconnection rate will be obtained by turbulent reconnection in the Poynting-dominated plasma. The model also discussed a possible role of reconnection outflow assuming Alfvénic velocity, ${\gamma }_{{\rm{A}}}\sim \sqrt{1+\sigma }.$ In our calculations, a relativistic outflow was observed only locally and intermittently; the averaged outflow velocity was a sub-relativistic velocity up to ∼0.3c even in the high-σ regime, which is commonly seen in the relativistic MHD reconnection resulting from the tearing instability. If we see a smaller scale, the collisionless plasma regime will appear when the outflow velocity is the Alfvén velocity in the high-σ regime (Liu et al. 2015).

Recently, Lazarian & Medvedev (2015) proposed a new scenario for the GRBs powered by turbulent reconnection based on kink instability of relativistically magnetized jets (Mizuno et al. 2012). In this model, the authors considered a turbulence induced by the kink instability, and applied it to the turbulent reconnection model. They showed that their model can provide a good fit to the dynamics of GRBs. Note that this model may also be able to be applied to active galactic nucleus (AGN) jets, which may result in a different condition from Equations (14).

4. General Remarks

Finally, we give several comments which can be applied generally to any phenomenon if turbulent reconnection works. First, our results, such as in Figure 2, indicate that the maximum reconnection rate is obtained when ${v}_{{\rm{inj}}}/{c}_{{\rm{A}}}\sim 0.2.$ Assuming the Alfvénic mode, $\delta B/{B}_{0}\sim \delta {v}_{\perp }/{c}_{{\rm{A}}},$ where B₀ and $\delta B$ are the background and fluctuation of the magnetic field, respectively, $\delta {v}_{\perp }$ is the fluctuation velocity perpendicular to B₀, we can expect that the turbulent reconnection is efficient if an observed magnetic fluctuation of some phenomena is around $\delta B\sim 0.2{B}_{0}.$ In addition, we can expect that such a low magnetic fluctuation will allow a high degree of polarization of synchrotron radiation observed by Coburn & Boggs (2003). Second, it is well known that magnetic reconnetion is also related to particle acceleration. For example, many Particle-in-cell simulations indicate that the energy spectrum index obtained in relativistic collisionless pair–plasma reconnection is around −1.5  (Zenitani & Hoshino 2001; Bessho & Bhattacharjee 2012; Sironi & Spitkovsky 2014; Guo et al. 2015). On the other hand, de Gouveia dal Pino & Lazarian (2005) found that the turbulent reconnection is also a location of the first-order Fermi process, and they obtained a little softer energy spectrum index, −2.5. The non-thermal particle energy index is directly related to the observed synchrotron spectrum, and the above energy spectrum index may be a useful tool to determine the location of turbulent reconnection. Applications of the turbulent reconnection model to some AGNs are provided in Kadowaki et al. (2015), Khiali et al. (2015), and Singh et al. (2015). A more comprehensive discussion of recent observations is given in Lazarian et al. (2015).

In this paper we investigated the turbulent reconnection in relativistic plasmas from the matter- dominated to Poynting-dominated cases using the relativistic resistive MHD model. The results show that the turbulence can enhance magnetic reconnection even in relativistic plasmas, and can be a candidate for a fast reconnection process. We found that the reconnection rate in turbulence shows the following three characteristic phases depending on the velocity of the injected turbulence: (1) LV99 region (incompressible turbulence); (2) saturation region giving maximum rate; and (3) reducing due to the compressibility. Saturation occurs when the compressible component becomes dominant, typically around ${v}_{{\rm{c}}}^{2}/{v}_{{\rm{i}}}^{2}\gtrsim 0.4$ at which the maximum reconnection rate is about 0.05. This shows that the LV99 expressions for incompressible fluid should be modified to account for compressibility as we have done in Equations (6) and (9). Interestingly, Banerjee & Galtier (2013) showed that dilatation of fluid, ${\rm{\nabla }}\cdot {\boldsymbol{v}}\gt 0,$ reduces the effective energy cascade rate, ${\epsilon }_{{\rm{eff}}}.$ All of our numerical results showed dilation, and the reconnection rates are indeed reduced. This indicates that the turbulent reconnection rate may become larger than LV99's prediction if compression of the turbulence occurs, such as the MHD turbulence driven by collisions of magnetized blobs (Inoue et al. 2011; Deng et al. 2015).

Finding a fast reconnection process is one of the most important topics in plasma physics, and a considerable number of studies have been conducted on it for a long time. Turbulence is a very general process in high-Reynolds number plasmas, so turbulent reconnection can appear in many kinds of phenomena, such as astrophysical phenomena, nuclear fusion, and laser plasma. In particular, our work investigated the extension of turbulent reconnection to relativistic plasma with compressible turbulence, which allows us to apply this process to many high energy astrophysical phenomena, such as flares in pulsar wind nebulae, GRBs, and relativistic jets.

We would like to thank John Kirk, Sébastien Galtier, and Supratik Banerjee for many fruitful comments and discussions. We also would like to thank our anonymous referee for a lot of fruitful comments on our paper. Numerical computations were carried out on the Cray XC30 at Center for Computational Astrophysics, CfCA, of the National Astronomical Observatory of Japan. Calculations were also carried out on SR16000 at YITP at Kyoto University. This work is supported in part by the Postdoctoral Fellowships for Research Abroad program by the Japan Society for the Promotion of Science No. 20130253 and also by the Research Fellowship for Young Scientists (PD) by the Japan Society for the Promotion of Science No. 20156571 (M.T.). One of the authors (A. Lazarian) is supported by NSF AST 1212096. This work is also supported in part by Grants-in-Aid for Scientific Research from the MEXT of Japan, 15K05039 (T.I.).