2.1 Definitions

Assume that the termination of the pulsar wind, which separates the cold magnetized incoming wind from the hot shocked wind, is corrugated. Potential sources of corrugation will be addressed in § 4. For simplicity, we neglect effects of spherical symmetry, which are not important to the present discussion, and therefore consider a planar shock, moving at velocity $\overline{{\it\beta}_{\text{f}}}$ along the $x$ -direction relative to the downstream plasma, i.e. relative to the nebula. The (uncorrugated) shock surface is defined by

(2.1) $$\begin{eqnarray}\displaystyle \overline{{\it\Phi}}(x)=x-\overline{{\it\beta}_{\text{f}}}ct=0, & & \displaystyle\end{eqnarray}$$

with corresponding shock normal four-vector:

(2.2) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{\ell }}_{{\it\mu}}=\frac{\partial _{{\it\mu}}\overline{{\it\Phi}}}{|\partial _{{\it\alpha}}\overline{{\it\Phi}}\partial ^{{\it\alpha}}\overline{{\it\Phi}}|^{1/2}}=(-\overline{{\it\gamma}_{\text{f}}}\overline{{\it\beta}_{\text{f}}},\overline{{\it\gamma}_{\text{f}}},0,0), & & \displaystyle\end{eqnarray}$$

where $\overline{{\it\gamma}_{\text{f}}}\equiv (1-\overline{{\it\beta}_{\text{f}}}^{2})^{-1/2}$ denotes the bulk Lorentz factor of the shock front relative to downstream.

Corrugation can be described by a perturbation of the shock surface:

(2.3) $$\begin{eqnarray}\displaystyle {\it\Phi}(x)=\overline{{\it\Phi}}(x)-{\it\delta}X(\boldsymbol{x}_{\bot },t)=0, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{x}_{\bot }\equiv (y,z)$ represents the coordinates in the (uncorrugated) shock front plane. For simplicity, consider for the purpose of this subsection a corrugation on a single length-scale characterized by a wavenumber $\boldsymbol{k}_{\bot }=(k_{y},k_{z})$ defined in the uncorrugated shock front plane, with harmonic behaviour at frequency ${\it\omega}_{\boldsymbol{k}}$ :

(2.4) $$\begin{eqnarray}\displaystyle {\it\delta}X(\boldsymbol{x}_{\bot },t)={\it\delta}X_{\boldsymbol{k}}\text{e}^{-\text{i}{\it\omega}_{\boldsymbol{k}}t+\text{i}\boldsymbol{k}_{\bot }\cdot \boldsymbol{x}_{\bot }}. & & \displaystyle\end{eqnarray}$$

The frequency ${\it\omega}_{\boldsymbol{k}}$ is directly related to $\boldsymbol{k}=(k_{x},\boldsymbol{k}_{\bot })$ and to the nature of the wave inducing the corrugation of the shock front (Lemoine, Ramos & Gremillet Reference Lemoine, Ramos and Gremillet2016); $k_{x}$ represents here the $x$ -wavenumber of that mode. At a relativistic shock, one typically has ${\it\omega}_{\boldsymbol{k}}\sim |\boldsymbol{k}|$ , hence this scaling is retained in the following.

In the linear approximation, the first-order perturbation of the shock normal is written:

(2.5) $$\begin{eqnarray}\displaystyle {\it\delta}\boldsymbol{\ell }_{{\it\mu}}=-\frac{\partial _{{\it\mu}}{\it\delta}X}{|\partial _{{\it\alpha}}\overline{{\it\Phi}}\partial ^{{\it\alpha}}\overline{{\it\Phi}}|^{1/2}}+\frac{\partial _{{\it\mu}}\overline{{\it\Phi}}\partial _{{\it\beta}}{\it\delta}X\partial ^{{\it\beta}}\overline{{\it\Phi}}}{|\partial _{{\it\alpha}}\overline{{\it\Phi}}\partial ^{{\it\alpha}}\overline{{\it\Phi}}|^{3/2}} & & \displaystyle\end{eqnarray}$$

or, for the above single-wave corrugation pattern,

(2.6) $$\begin{eqnarray}\displaystyle {\it\delta}\boldsymbol{\ell }_{\boldsymbol{k}{\it\mu}}=(\text{i}\overline{{\it\gamma}_{\text{f}}}^{3}{\it\omega}_{\boldsymbol{k}}{\it\delta}X_{\boldsymbol{k}}/c,-\text{i}\overline{{\it\gamma}_{\text{f}}}^{3}\overline{{\it\beta}_{\text{f}}}{\it\omega}_{\boldsymbol{k}}{\it\delta}X_{\boldsymbol{k}}/c,-\text{i}k_{y}\overline{{\it\gamma}_{\text{f}}}{\it\delta}X_{\boldsymbol{k}},-\text{i}k_{z}\overline{{\it\gamma}_{\text{f}}}{\it\delta}X_{\boldsymbol{k}}). & & \displaystyle\end{eqnarray}$$

A strongly corrugated shock is such that the above linear approximation breaks down, i.e. $|{\it\delta}\ell |\gtrsim 1$ , which translates into

(2.7) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\overline{{\it\gamma}_{\text{f}}}|k_{\bot }{\it\delta}X_{\boldsymbol{k}}|\gtrsim 1\\ \overline{{\it\gamma}_{\text{f}}}^{2}|{\it\omega}_{\boldsymbol{k}}{\it\delta}X_{\boldsymbol{k}}|\gtrsim 1\end{array}\right\}. & & \displaystyle\end{eqnarray}$$

Both conditions express the fact that the departure from the unperturbed shock surface becomes larger than the perpendicular wavelength in the shock front rest frame. For a moderately magnetized shock wave, with ${\it\sigma}_{1}\lesssim 3$ , one has $\overline{{\it\gamma}_{\text{f}}}\sim 1$ , hence a shock front at the onset of nonlinear corrugation is such that $|k{\it\delta}X|\sim 1$ (with ${\it\delta}X$ expressed in the downstream rest frame as previously).

If the incoming upstream flow is strongly magnetized, i.e. ${\it\sigma}_{1}\gtrsim 3$ , $\overline{{\it\gamma}_{\text{f}}}$ becomes larger than unity. However, the mean shock velocity ${\it\beta}_{\text{f}}$ along the shock normal can be strongly modified by corrugation; as discussed in the following, in particular, the generation of turbulence in the nonlinearly corrugated shock transition can reduce the actual ${\it\beta}_{\text{f}}$ to sub-relativistic values, implying ${\it\gamma}_{\text{f}}\sim 1$ . It is therefore speculated that $|k{\it\delta}X|\sim 1$ remains a valid threshold for nonlinear corrugation in the large magnetization regime ${\it\sigma}_{1}\gtrsim 3$ .

Actually, one could potentially envisage even larger values of ${\it\delta}X$ ; however, nonlinear back-reaction would likely limit ${\it\delta}X$ to the above threshold of nonlinearity, hence this value is retained in the following.

Interestingly, the condition $|k{\it\delta}X_{\boldsymbol{k}}|\gtrsim 1$ is compatible with a very small amplitude $|{\it\delta}X|$ , as measured relative to the scale of the termination shock (noted $R$ ), provided $|kR|\gg 1$ . In other words, the shock may be strongly corrugated on short spatial scales which are not observable at large distances, but which remain macroscopic compared to the thickness of the actual shock transition. The latter requirement is not essential for the present scenario, but the MHD description that this model uses can only be applied on scales much larger than the shock thickness, which is set by kinetic physics. To provide quantitative estimates, consider the case of the Crab nebula: if electrons are inflowing through the shock with a Lorentz factor ${\it\gamma}_{\text{w}}=10^{4}{\it\gamma}_{\text{w},4}$ (relative to the downstream-nebula rest frame), the typical gyroradius of these particles in the downstream magnetic field (strength $B_{d}\sim 0.1~\text{mG}$ ), $r_{\text{g}}\sim 10^{11}{\it\gamma}_{\text{w},4}~\text{cm}$ sets the typical thickness of the shock transition layer. Therefore, corrugation is envisaged here on all scales $k^{-1}$ larger than the above and smaller than $R\sim 3\times 10^{17}~\text{cm}$ . As discussed in § 4, this range of scales encompasses the gyroradii of all accelerated particles, even the highest-energy ones, indicating that corrugation may exert a strong influence on their kinematics.

Figure 1. Various quantities plotted at $t=0$ as a function of $y$ , the coordinate along the direction $\boldsymbol{y}$ , which is both perpendicular to the shock normal ( $\boldsymbol{x}$ ) and to the direction of the background magnetic field ( $\boldsymbol{z}$ ), in a case of nonlinear corrugation described by (2.8), assuming no rippling along $\boldsymbol{z}$ . Upper sinusoidal dotted line: $-\boldsymbol{\ell }_{0}/\boldsymbol{\ell }_{1}$ describing the spatial behaviour of the temporal component of the shock normal; the average value over $y$ , i.e. 0.7, represents the average velocity of the shock front relative to downstream. Lower sinusoidal dotted line: $-\boldsymbol{\ell }_{2}/\boldsymbol{\ell }_{1}$ , describing the rippling of the shock front in the $\boldsymbol{y}$ direction. Horizontal dashed curve: (relativistic) Alfvén three-velocity of waves in the downstream plasma, whose modulations are at too small an amplitude to emerge on this figure. Solid (blue) line: $y$ -component of the downstream flow three-velocity (on the shock front). Dashed (blue) thick line: $x$ -component of the downstream flow three-velocity. Finally, the arrows indicate the direction of the downstream three-velocity in the $(y,x)$ plane (for this arrow representation, the ordinate axis should be understood as indicating the $x$ -direction, while the abscissa points into the $y$ -direction).

2.2 A particular nonlinear solution

In the nonlinear regime where the above perturbative description breaks down, it is possible to extract an analytical solution of the shock crossing equations in a 2D limit, in which all quantities remain unperturbed along the direction of the background magnetic field (taken to be $\boldsymbol{z}$ here), see Lemoine et al. (Reference Lemoine, Ramos and Gremillet2016). The downstream quantities characterizing the state of the plasma can then be expressed at any time, on the corrugated shock front, in terms of the shock normal. The magnitude of the shock corrugation amplitude, which depends on the past history of the accumulated flow, is related through a non-trivial differential relation to the shock normal, see (2.2) and (2.3) in particular. The present description does not attempt to describe this corrugation amplitude but to describe the state of the downstream plasma on the shock front.

An example is shown in figure 1, which assumes an upstream magnetization parameter ${\it\sigma}_{1}=1$ , a relative Lorentz factor between up- and down-stream ${\it\gamma}_{1}=100$ , and a shock normal four-vector arbitrarily set to:

(2.8) $$\begin{eqnarray}\displaystyle \boldsymbol{\ell }_{{\it\mu}}=\{-0.98+0.42\sin (t-y),1.40,0.42\sin (t-y),0\}. & & \displaystyle\end{eqnarray}$$

In the downstream rest frame, this four-normal describes a mean shock velocity of ${\it\beta}_{\text{f}}\simeq 0.7$ (corresponding to the unperturbed shock crossing conditions for the above values of the magnetization and flow velocity), with harmonic corrugation slightly below the onset of the nonlinear limit.

For the particular case of perturbations confined to the $y$ -direction, the equations of shock crossing lead to $B_{x|2}=B_{y|2}=0$ : the magnetic field retains only a $z$ -component, modulated along $\boldsymbol{y}$ . Note that $B_{x|2}$ and $B_{y|2}$ vanish on the corrugated shock surface, but not necessarily further downstream. One would need to follow the characteristics of the system to study the evolution of these two quantities downstream of the shock; in the linear limit at least, one can show that the outgoing wave modes develop a net ${\it\delta}B_{x|2}$ and ${\it\delta}B_{y|2}$ further downstream of the shock. Moreover, in a more general case with $k_{z}\neq 0$ , both components ${\it\delta}B_{x|2}$ and ${\it\delta}B_{y|2}$ would not vanish on the shock surface.

As illustrated by figure 1, the shock crossing equations imply the existence of significant velocity fluctuations along $x$ and $y$ , with different modulations. In turn, these generate sheared flows with non-vanishing ${\bf\beta}\times \boldsymbol{B}$ in the downstream, leading to the stretching and compression of magnetic field lines. These flow motions also generate convective electric fields, $\boldsymbol{E}=-{\bf\beta}\times \boldsymbol{B}$ , with $|\boldsymbol{E}|\sim |\boldsymbol{B}|$ because $|{\it\beta}|$ is close to unity. The resulting turbulence should thus be prone to dissipation and particle acceleration on short time scales.

2.3 Jump conditions at a corrugated shock front

As the upstream plasma inflows through the corrugated shock, the ordered magnetic energy is thus converted in part into turbulence modes. Assume that the MHD fluid immediately behind the shock can be described by an enthalpy density $w_{2}$ and pressure $p_{2}$ with equation of state $w_{2}\simeq 4p_{2}$ (relativistically hot fluid), and by a magnetic field $\boldsymbol{B}=\boldsymbol{B}_{2}+{\bf\delta}\boldsymbol{B}_{2}$ , with $\langle {\bf\delta}\boldsymbol{B}_{2}\rangle =0$ . The index $_{2}$ applies to downstream quantities; upstream quantities will be indexed with $_{1}$ . The average can be taken in the ensemble of realizations of the corrugation, or as usual, in the ergodic hypothesis, along the shock front plane.

On spatial scales much larger than the thickness of the shock, but much smaller than the corrugation amplitude $|{\it\delta}X|$ , the jump conditions at the shock are expressed through the integration of the conservation laws along the perturbed normal direction. However, the present discussion is rather concerned with computing the asymptotic behaviour of the downstream plasma, on a distance scale $\gg |{\it\delta}X|$ away from the shock. On such scales, the shock can be seen as a planar discontinuity, although the jump conditions should reflect the fact that turbulence has been generated in the transition layer. These jump conditions should thus be written in the downstream plasma rest frame as

(2.9) $$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle [nu^{{\it\mu}}\ell _{{\it\mu}}]=0\\ \displaystyle \text{}[\unicode[STIX]{x1D61B}^{{\it\mu}{\it\nu}}\ell _{{\it\mu}}]=0,\end{array} & & \displaystyle\end{eqnarray}$$

where the shock normal $\boldsymbol{\ell }_{{\it\mu}}=(-{\it\gamma}_{\text{f}}{\it\beta}_{\text{f}},{\it\gamma}_{\text{f}},0,0)$ as in (2.2) above. A distinction has to be made however in the notations: in (2.2), the overline symbols indicate that the solution applies in the absence of corrugation, while in the present case, corrugation is assumed to be present on small spatial scales, leading to the production of downstream MHD turbulence, so that the value of ${\it\beta}_{\text{f}}$ which is determined further below differs from $\overline{{\it\beta}_{\text{f}}}$ above.

The energy-momentum tensor in the ideal MHD approximation is written

(2.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D61B}^{{\it\mu}{\it\nu}} & = & \displaystyle \left(w+\frac{b_{{\it\alpha}}b^{{\it\alpha}}}{4{\rm\pi}}\right)u^{{\it\mu}}u^{{\it\nu}}+\left(p+\frac{b_{{\it\alpha}}b^{{\it\alpha}}}{8{\rm\pi}}\right){\it\eta}^{{\it\mu}{\it\nu}}-\frac{b^{{\it\mu}}b^{{\it\nu}}}{4{\rm\pi}}\end{eqnarray}$$

in terms of the magnetic four-vector $\boldsymbol{b}^{{\it\mu}}$ :

(2.11) $$\begin{eqnarray}\displaystyle \boldsymbol{b}^{{\it\mu}}=[u^{i}B_{i},(\boldsymbol{B}+u^{i}B_{i}\boldsymbol{u})/u^{0}], & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}^{{\it\mu}}$ represents the turbulent fluid four-velocity behind the shock, and latin indices represent spatial indices.

The average energy-momentum tensor of the turbulent downstream fluid is given by the correlators $\langle b^{{\it\alpha}}b^{{\it\beta}}\rangle$ , which can be calculated in full generality. In order to derive simple estimates however, the velocity field and the magnetic perturbations are assumed uncorrelated, i.e.

(2.12) $$\begin{eqnarray}\displaystyle \langle u^{{\it\alpha}}u^{{\it\beta}}B_{2i}B_{2j}\rangle =\langle u^{{\it\alpha}}u^{{\it\beta}}\rangle \langle B_{2i}B_{2j}\rangle & & \displaystyle\end{eqnarray}$$

and correlators involving odd powers of the turbulent fluid three-velocity are assumed to vanish; in particular, $\langle u^{i}\rangle =0$ by definition of the downstream rest frame. Finally, one may assume isotropic turbulence, meaning

(2.13) $$\begin{eqnarray}\displaystyle \langle {\it\delta}B_{2i}{\it\delta}B_{2j}\rangle ={\textstyle \frac{1}{3}}{\it\delta}_{ij}\langle {\it\delta}B_{2}^{2}\rangle . & & \displaystyle\end{eqnarray}$$

This assumption makes it possible to simplify the calculations, but it is not crucial to the present analysis, as discussed further on.

Then one writes

(2.14) $$\begin{eqnarray}\displaystyle \langle b^{{\it\alpha}}b^{{\it\beta}}\rangle =\langle b_{(0)}^{{\it\alpha}}b_{(0)}^{{\it\beta}}\rangle +\langle b_{(1)}^{{\it\alpha}}b_{(1)}^{{\it\beta}}\rangle , & & \displaystyle\end{eqnarray}$$

where $b_{(0)}^{{\it\alpha}}$ does not contain any ${\it\delta}B_{i}$ component, while $b_{(1)}^{{\it\alpha}}$ does not contain any $B_{i}$ term. Assuming that the average background field lies along $\boldsymbol{z}$ , one readily obtains

(2.15) $$\begin{eqnarray}\displaystyle \langle b_{(0)}^{{\it\alpha}}b_{(0)}^{{\it\beta}}\rangle ={\it\delta}_{0}^{{\it\alpha}}{\it\delta}_{0}^{{\it\beta}}\langle u_{z}^{2}\rangle B_{2}^{2}+{\it\delta}_{i}^{{\it\alpha}}{\it\delta}_{j}^{{\it\beta}}\left[\left\langle \frac{1}{{\it\gamma}^{2}}\right\rangle {\it\delta}^{i3}{\it\delta}^{j3}+2\left\langle \frac{u_{z}^{2}}{{\it\gamma}^{2}}\right\rangle {\it\delta}^{i3}{\it\delta}^{j3}+\left\langle \frac{u^{i}u^{j}u_{z}^{2}}{{\it\gamma}^{2}}\right\rangle \right]B_{2}^{2}\quad & & \displaystyle\end{eqnarray}$$

and

(2.16) $$\begin{eqnarray}\displaystyle \langle b_{(1)}^{{\it\alpha}}b_{(1)}^{{\it\beta}}\rangle ={\it\delta}_{0}^{{\it\alpha}}{\it\delta}_{0}^{{\it\beta}}\frac{1}{3}\langle u^{2}\rangle \langle {\it\delta}B_{2}^{2}\rangle +{\it\delta}_{i}^{{\it\alpha}}{\it\delta}_{j}^{{\it\beta}}\frac{1}{3}\left[\left\langle \frac{1}{{\it\gamma}^{2}}\right\rangle {\it\delta}^{ij}+2\left\langle \frac{u^{i}u^{j}}{{\it\gamma}^{2}}\right\rangle +\left\langle \frac{u^{i}u^{j}u^{2}}{{\it\gamma}^{2}}\right\rangle \right]\langle {\it\delta}B_{2}^{2}\rangle . & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

A key observation here is that these correlators do not share the same dependence. To proceed further and obtain a more tractable expression leading to a simple estimate of the modified jump conditions, assume that the turbulence in the downstream rest frame is moderately relativistic; this is actually ensured if the corrugation is mildly nonlinear and ${\it\sigma}_{1}$ not large compared to unity, as discussed in Lemoine et al. (Reference Lemoine, Ramos and Gremillet2016). The following expressions thus retain only the leading-order terms in powers of $u$ in the above correlators, which then reduce to standard non-relativistic expressions for magnetized turbulence:

(2.17) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\langle b_{(0)}^{{\it\alpha}}b_{(0)}^{{\it\beta}}\rangle \approx {\it\delta}_{3}^{{\it\alpha}}{\it\delta}_{3}^{{\it\beta}}B_{2}^{2}\\ \langle b_{(1)}^{{\it\alpha}}b_{(1)}^{{\it\beta}}\rangle \approx {\it\delta}_{i}^{{\it\alpha}}{\it\delta}_{j}^{{\it\beta}}{\it\delta}^{ij}{\textstyle \frac{1}{3}}\langle {\it\delta}B_{2}^{2}\rangle .\end{array}\right\} & & \displaystyle\end{eqnarray}$$

It should be understood that this set of approximations is intended to show in a quantitative way how corrugation affects the jump conditions at the MHD shock. Qualitatively speaking however, the key point is that the correlators of the turbulence modes contained in $\langle b^{{\it\alpha}}b^{{\it\beta}}\rangle$ differ from those of the average background field.

With the above approximations, the non-zero downstream energy-momentum tensor components can be written:

(2.18) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D61B}_{2}^{00}\approx W_{2}-P_{2}\\ \displaystyle \unicode[STIX]{x1D61B}_{2}^{11}\approx P_{2}-\frac{1}{3}\frac{\langle {\it\delta}B_{2}^{2}\rangle }{4{\rm\pi}}\\ \displaystyle \unicode[STIX]{x1D61B}_{2}^{22}\approx P_{2}-\frac{1}{3}\frac{\langle {\it\delta}B_{2}^{2}\rangle }{4{\rm\pi}}\\ \displaystyle \unicode[STIX]{x1D61B}_{2}^{33}\approx P_{2}-\frac{B_{2}^{2}}{4{\rm\pi}}-\frac{1}{3}\frac{\langle {\it\delta}B_{2}^{2}\rangle }{4{\rm\pi}}\end{array}\right\}, & & \displaystyle\end{eqnarray}$$

with $W_{2}\equiv w_{2}+B_{2}^{2}/(4{\rm\pi})+\langle {\it\delta}B_{2}^{2}\rangle /(4{\rm\pi})$ a generalized enthalpy density and $P_{2}\equiv p_{2}+B_{2}^{2}/(8{\rm\pi})+\langle {\it\delta}B_{2}^{2}\rangle /(8{\rm\pi})$ a generalized pressure.

The shock jump conditions for energy and momentum fluxes are then expressed as

(2.19) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle -{\it\beta}_{\text{f}}(W_{2}-P_{2})=({\it\beta}_{1}-{\it\beta}_{\text{f}}){\it\gamma}_{1}^{2}W_{1}+{\it\beta}_{\text{f}}P_{1}\\ \displaystyle P_{2}-\frac{1}{3}\frac{\langle {\it\delta}B_{2}^{2}\rangle }{4{\rm\pi}}=({\it\beta}_{1}-{\it\beta}_{\text{f}}){\it\gamma}_{1}^{2}{\it\beta}_{1}W_{1}+P_{1}\end{array}\right\}. & & \displaystyle\end{eqnarray}$$

The above equations are written in the downstream rest frame, so that ${\it\beta}_{1}$ corresponds to the velocity of upstream relative to downstream. Neglecting $P_{1}$ in front of $P_{2}$ (or, alternatively ${\it\gamma}_{1}^{2}W_{1}$ ) and assuming a hot relativistic plasma downstream, $p_{2}=w_{2}/4$ , one obtains easily

(2.20) $$\begin{eqnarray}\displaystyle {\it\beta}_{1}{\it\beta}_{\text{f}}=-\frac{\displaystyle w_{2}+\frac{B_{2}^{2}}{2{\rm\pi}}+\frac{\langle {\it\delta}B_{2}^{2}\rangle }{6{\rm\pi}}}{\displaystyle 3w_{2}+\frac{B_{2}^{2}}{2{\rm\pi}}+\frac{\langle {\it\delta}B_{2}^{2}\rangle }{2{\rm\pi}}}. & & \displaystyle\end{eqnarray}$$

This equation reduces to the standard result ${\it\beta}_{\text{f}}\rightarrow 1/3$ in the limits ${\it\beta}_{1}\rightarrow -1$ (ultra-relativistic limit), $B_{2}\rightarrow 0$ and $\langle {\it\delta}B_{2}^{2}\rangle ^{1/2}\rightarrow 0$ (hydrodynamic shock), e.g. Kirk & Duffy (Reference Kirk and Duffy1999). In the highly magnetized and uncorrugated case, meaning $w_{2}\ll B_{2}^{2}/(4{\rm\pi})$ and $\langle {\it\delta}B_{2}^{2}\rangle ^{1/2}\rightarrow 0$ , one also recovers ${\it\beta}_{1}{\it\beta}_{\text{f}}\simeq -1$ , indicating that the shock moves away from downstream at a relativistic velocity. As discussed by Kennel & Coroniti (Reference Kennel and Coroniti1984a ), the mismatch between this solution and the general morphology of the Crab nebula points towards a smaller than unity magnetization parameter behind the termination shock.

More interestingly, if $\langle {\it\delta}B_{2}^{2}\rangle ^{1/2}$ is not negligible compared to $B_{2}$ , one finds a solution with a shock moving away from downstream at sub-relativistic velocities, independently of how strongly magnetized the flow initially was. Consider for instance the case of equipartition $\langle {\it\delta}B_{2}^{2}\rangle ^{1/2}\sim B_{2}$ and $w_{2}\ll B_{2}^{2}/(4{\rm\pi})$ , which is effectively what one expects if corrugation is mildly nonlinear. Then ${\it\beta}_{\text{f}}\simeq 2/3$ for ${\it\beta}_{1}\simeq -1$ . Other interesting limits are $\langle {\it\delta}B_{2}^{2}\rangle \gg B_{2}^{2}$ (strong corrugation), leading to ${\it\beta}_{\text{f}}\simeq 1/3$ as in a pure hydrodynamical shock; or $\langle {\it\delta}B_{2}^{2}\rangle \sim B_{2}^{2}\sim 4{\rm\pi}w_{2}$ , leading to ${\it\beta}_{\text{f}}\simeq 0.5$ .

These particular solutions emerge whenever the field line tension of the turbulence can contribute to the $xx$ component of $\unicode[STIX]{x1D61B}^{{\it\mu}{\it\nu}}$ , i.e. whenever the correlators $\langle b^{{\it\mu}}b^{{\it\nu}}\rangle$ possess a non-trivial $xx$ component. As discussed here and in the previous section, such a component could arise from the shock corrugation directly or from the nonlinear processing of the magnetic modulations induced by shock corrugation. The following subsections also argue that the partial dissipation of such turbulence would preheat the pairs and thus lead to a solution with a moderate shock velocity relative to downstream (corresponding to the last case of equipartition discussed above).

3.1 Collisionless damping of hydromagnetic waves

Even if $w_{2}\ll B_{2}^{2}/(4{\rm\pi}),\langle {\it\delta}B_{2}^{2}\rangle /(4{\rm\pi})$ immediately downstream of the shock, various dissipative effects will transfer energy from the magnetized turbulence to the particles, thereby decreasing the magnetization of the plasma to values of order unity (see below). This section discusses the collisionless damping of magnetosonic modes at the Landau resonance; stochastic particle acceleration will be discussed in § 3.2; other processes, such as turbulent reconnection, are of course plausible sources of dissipation.

The collisionless damping of hydromagnetic waves in a relativistic plasma has been discussed by Barnes & Scargle (Reference Barnes and Scargle1973). At the Landau resonance, they find

(3.1) $$\begin{eqnarray}\displaystyle \text{Im}\,{\it\omega}\simeq \frac{{\rm\pi}}{8}\text{Re}\,{\it\omega}\frac{{\it\delta}B_{k}^{2}}{4{\rm\pi}W_{{\it\delta}B_{k}}}\sin ^{2}{\it\theta}|w_{r}|(1-w_{r}^{2})^{2}{\it\Theta}(1-w_{r}){\it\sigma}^{-1}, & & \displaystyle\end{eqnarray}$$

where ${\it\omega}$ corresponds to the wave frequency, $W_{{\it\delta}B_{k}}\sim {\it\delta}B_{k}^{2}/(4{\rm\pi})$ to the wave energy density, ${\it\theta}$ to the angle between $\boldsymbol{k}$ and $\boldsymbol{B}_{2}$ , ${\it\Theta}$ to the Heaviside function, and ${\it\sigma}$ to the magnetization, i.e. the ratio between the magnetic energy density and the electron energy density. Finally,

(3.2) $$\begin{eqnarray}\displaystyle w_{r}\equiv \frac{\text{Re}\,{\it\omega}}{kc\cos {\it\theta}} & & \displaystyle\end{eqnarray}$$

is the resonance parameter.

As discussed in Barnes & Scargle (Reference Barnes and Scargle1973), the presence of the Heaviside function limits the damping coefficient of waves by defining a critical angle beyond which Landau damping vanishes. The real frequency of magnetosonic waves propagating at an angle ${\it\theta}$ to the magnetic field is given by

(3.3) $$\begin{eqnarray}\displaystyle \text{Re}\,{\it\omega}=\frac{kc}{\sqrt{2}}\{{\it\beta}_{+}^{2}+{\it\beta}_{A}^{2}c_{s}^{2}\cos ^{2}{\it\theta}\pm [({\it\beta}_{+}^{2}+{\it\beta}_{A}^{2}c_{s}^{2}\cos ^{2}{\it\theta})^{2}-4{\it\beta}_{A}^{2}c_{s}^{2}\cos ^{2}{\it\theta}]^{1/2}\}^{1/2}, & & \displaystyle\end{eqnarray}$$

with the plus sign pertaining to fast magnetosonic waves, and the minus sign to slow magnetosonic waves; ${\it\beta}_{+}^{2}\equiv {\it\beta}_{A}^{2}+c_{s}^{2}-{\it\beta}_{A}^{2}c_{s}^{2}$ in terms of the (relativistic) Alfvén velocity ${\it\beta}_{A}$ and sound velocity $c_{s}$ . Assuming an isotropic bath of waves, one can calculate the average damping coefficient, relatively to the mode wavenumber, as

(3.4) $$\begin{eqnarray}\displaystyle \frac{\langle \text{Im}\,{\it\omega}\rangle }{kc}\simeq \frac{{\rm\pi}}{16}{\it\sigma}^{-1}\int \text{d}{\it\theta}\sin ^{3}{\it\theta}\cos {\it\theta}w_{r}^{2}(1-w_{r}^{2})^{2}{\it\Theta}(1-w_{r}). & & \displaystyle\end{eqnarray}$$

For fast magnetosonic waves in a strongly magnetized plasma ( ${\it\sigma}\gg 1$ ), the fast magnetosonic wave phase velocity approaches unity, so that the critical angle ${\it\theta}_{c}\sim 0$ , implying $\langle \text{Im}\,{\it\omega}/kc\rangle \approx 0$ ; those waves are effectively undamped in the highly magnetized regime. At more moderate magnetization, damping may become appreciable however: at ${\it\sigma}=1$ for instance, $\langle \text{Im}\,{\it\omega}/kc\rangle \approx 10^{-4}$ .

For slow magnetosonic waves, however, one finds typically

(3.5) $$\begin{eqnarray}\displaystyle \langle \text{Im}\,{\it\omega}\rangle \simeq 10^{-2}kc{\it\sigma}^{-1} & & \displaystyle\end{eqnarray}$$

at ${\it\sigma}\gtrsim 1$ .

The above should be considered as a lower limit to $\text{Im}\,{\it\omega}$ since the above neglects Landau-synchrotron damping effects, as well as other dissipative effects. It nevertheless makes it possible to set an upper limit on the damping length associated with the shock corrugation: ${\it\lambda}_{\text{diss.}}\simeq {\it\beta}_{\text{f}}c\langle \text{Im}\,{\it\omega}\rangle ^{-1}$ .

In the more realistic case of corrugation spread over a broad range of $k$ -modes, the above indicates that the short-scale modes (large $k$ ) will dissipate on short length-scales $\propto k^{-1}$ . Turbulence on the large scales may dissipate through cascading to shorter scales, followed by damping; given that fluctuations are mildly relativistic behind the shock front if corrugation is mildly non-relativistic, as discussed above, the typical time scale of eddy cascading may not be much larger than $(kc)^{-1}$ , implying an efficient damping of slow magnetosonic turbulence. The general picture that characterizes nonlinear shock corrugation is thus the generation of a turbulent layer behind the shock front, a part of which dissipates on a small length-scale ${\it\lambda}_{\text{diss.}}\sim O(k_{\text{peak}}^{-1})$ , $k_{\text{peak}}$ denoting the mode on which the maximum turbulent power is concentrated.

At such a corrugated shock front, $\langle {\it\delta}B_{2}^{2}\rangle \sim B_{2}^{2}$ and slow magnetosonic waves carry a significant fraction of the turbulence magnetic energy density. Therefore, on a distance scale ${\it\lambda}_{\text{diss.}}$ , a fraction ${\it\eta}_{s}$ , with ${\it\eta}_{s}$ not far below unity, of the magnetic energy density has been dissipated into particles, reducing the magnetization from

(3.6) $$\begin{eqnarray}\displaystyle {\it\sigma}_{2<}=\frac{{\it\delta}B_{2}^{2}+B_{2}^{2}}{4{\rm\pi}w_{2}} & & \displaystyle\end{eqnarray}$$

immediately downstream of the shock, to

(3.7) $$\begin{eqnarray}\displaystyle {\it\sigma}_{2>}=\frac{(1-{\it\eta}_{s}){\it\sigma}_{2<}}{1+{\it\eta}_{s}{\it\sigma}_{2<}} & & \displaystyle\end{eqnarray}$$

beyond the dissipation layer. Consequently, if ${\it\sigma}_{2<}\gtrsim 1$ and ${\it\eta}_{s}\gtrsim 1/{\it\sigma}_{2<}$ , the magnetization is reduced to values ${\it\sigma}_{2>}\simeq (1-{\it\eta}_{s})/{\it\eta}_{s}$ , of order unity, independently of how high the magnetization initially was.

In this way, a magnetized relativistic corrugated shock wave can efficiently dissipate the incoming magnetic energy into the shock, leading to a near hydrodynamical shock with a magnetization of order unity beyond ${\it\lambda}_{\text{diss.}}$ .

3.2 Phenomenological model of particle pre-acceleration

The acceleration of particles in a bath of magnetized turbulence can be described in a phenomenological way through a Fokker–Planck equation for the distribution function $f(\boldsymbol{p},t)$ of particlesFootnote ¹ :

(3.8) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}f=\frac{1}{p^{2}}\frac{\partial }{\partial p}\left(p^{2}D_{pp}\frac{\partial }{\partial p}f\right)-\frac{1}{p^{2}}\frac{\partial }{\partial p}({\dot{p}}p^{2}f)-\frac{f}{{\it\tau}_{\text{esc}}}+q(p), & & \displaystyle\end{eqnarray}$$

where $q$ models the injection of particles into the dissipation region, ${\it\tau}_{\text{esc}}$ the escape time scale out of the dissipation region and ${\dot{p}}$ characterizes systematic energy gains/losses. Dissipation is characterized by the diffusion coefficient in momentum space

(3.9) $$\begin{eqnarray}\displaystyle D_{pp}=\frac{\langle {\rm\Delta}p^{2}\rangle }{2{\rm\Delta}t}. & & \displaystyle\end{eqnarray}$$

The injection function takes the form $q(p)={\dot{n}}/(4{\rm\pi}p_{0}^{2}){\it\delta}(p-p_{0})$ , with ${\dot{n}}$ the density of particles per unit time inflowing into the shock, as measured in the downstream rest frame; the injection momentum $p_{0}$ is related to the shock Lorentz factor through $p_{0}\simeq {\it\gamma}_{1}mc$ .

Ignoring systematic energy gains/losses and considering a stationary state, standard techniques (e.g. Schlickeiser Reference Schlickeiser1984) make it possible to solve the above equation for various momentum dependences of $D_{pp}$ and ${\it\tau}_{\text{esc}}$ . In the problem at hand, escape presumably takes place through advection at velocity ${\it\beta}_{\text{f}}$ , as the dissipation region is confined to a distance ${\it\lambda}_{\text{diss.}}$ from the shock front, which itself moves away at velocity ${\it\beta}_{\text{f}}$ . Thus ${\it\tau}_{\text{esc}}\propto p^{0}$ ; as for the momentum diffusion coefficient, it is written

(3.10) $$\begin{eqnarray}\displaystyle D_{pp}=\frac{p_{0}^{2}}{{\it\tau}_{s,0}}\left(\frac{p}{p_{0}}\right)^{2+{\it\alpha}}, & & \displaystyle\end{eqnarray}$$

where ${\it\tau}_{s,0}$ characterizes the typical acceleration time scale at momentum $p_{0}$ . For ${\it\alpha}=0$ , corresponding to the simplest scaling with an interaction time in the turbulence that does not depend on momentum, one finds

(3.11) $$\begin{eqnarray}\displaystyle f(p) & = & \displaystyle c_{1}\left(\frac{p}{p_{0}}\right)^{-(3/2)+(1/2)\sqrt{9+4{\it\tau}_{s,0}/{\it\tau}_{\text{e sc }}}}{\it\Theta}(p_{0}-p)\nonumber\\ \displaystyle & & \displaystyle +\,c_{2}\left(\frac{p}{p_{0}}\right)^{-(3/2)-(1/2)\sqrt{9+4{\it\tau}_{s,0}/{\it\tau}_{\text{e sc }}}}{\it\Theta}(p-p_{0}),\end{eqnarray}$$

where $c_{1}$ and $c_{2}$ are two integration constants related to ${\dot{n}}$ , $p_{0}$ , ${\it\tau}_{s,0}$ and ${\it\tau}_{\text{esc}}$ . The asymptotic behaviour at large momenta is thus a power-law $f(p)\propto p^{-s_{f}}$ with index

(3.12) $$\begin{eqnarray}\displaystyle s_{f}={\textstyle \frac{3}{2}}+{\textstyle \frac{1}{2}}\sqrt{9+4{\it\tau}_{s,0}/{\it\tau}_{\text{e sc }}}, & & \displaystyle\end{eqnarray}$$

which depends on how fast escape balances acceleration. One should keep in mind that this index is that of the distribution function in the acceleration zone, so that the index of the distribution function per momentum interval $\text{d}N/\text{d}p\propto p^{-s}$ in this acceleration zone is $s=s_{f}-2$ . As to the distribution of escaping particles, i.e. those that eventually populate the nebula, its index is in principle modified by the escape rate, i.e.

(3.13) $$\begin{eqnarray}\displaystyle \frac{\text{d}N_{\text{esc}}}{\text{d}p}\propto \frac{1}{{\it\tau}_{\text{esc}}}\frac{\text{d}N}{\text{d}p} & & \displaystyle\end{eqnarray}$$

but since ${\it\tau}_{\text{esc}}\propto p^{0}$ here, the index remains unchanged.

The above phenomenological model indicates that the dissipation of the turbulence produced by corrugation leads to a power-law with index $s$ comprised between $1$ and $2$ if ${\it\tau}_{s,0}\sim {\it\tau}_{\text{esc}}$ . The development of this power-law does not remain unbounded, because most of the energy is then carried by particles of maximum momentum, if $s<2$ . Thus, in the absence of significant energy losses, one expects that the dissipation process saturates at a momentum such that a significant fraction of the turbulence energy density has been dissipated into the particles, i.e. such that the backreaction of the dissipation process on the turbulence energy density cannot be ignored. This takes place on a length scale ${\it\lambda}_{\text{diss.}}$ which, accounting for all dissipative processes, characterizes the width of the layer beyond which a fraction of order unity of the magnetized turbulence energy has been dumped into the particles.

3.3 Fermi acceleration

One may also expect the development of a relativistic Fermi process in the above conditions. As a note of caution, however, the following discussion remains qualitative and further work would be needed to establish this statement on solid grounds.

At a relativistic shock of moderate to large magnetization, the Fermi process is inhibited because of the superluminal nature of the magnetic configuration. Lemoine, Pelletier & Revenu (Reference Lemoine, Pelletier and Revenu2006) and Pelletier, Lemoine & Marcowith (Reference Pelletier, Lemoine and Marcowith2009) have discussed in some detail what prevents Fermi cycles in such a configuration but it may be useful for the present discussion to recall the salient points. Consider a planar relativistic magnetized shock, with some turbulence upstream of the shock, laid on a scale ${\it\lambda}$ assumed much larger than the typical gyroradius $r_{\text{g}}$ of accelerated particles in the downstream rest frame. A key point is that the accelerated particles only probe a region of size $r_{\text{g}}$ during their Fermi cycles in such turbulence: those that probe a deeper region downstream of the shock are actually unable to return to the shock because of the superluminal nature of the shock wave: in order to do so, particles would need to diffuse across the magnetic field at an effective velocity larger than ${\it\beta}_{\text{f}}$ . Thus, on the length scale $r_{\text{g}}\ll {\it\lambda}$ probed by the particles, the turbulence appears as an essentially coherent magnetic field. One may then show that incoming particles can execute at most 1.5 Fermi cycles up $\rightarrow$ down $\rightarrow$ up $\rightarrow$ down in this configuration before being advected downstream, away from the shock. Those particles that are able to return once to the shock are those whose momentum is oriented relative to the magnetic field in such a way as to authorize a bounce on the downstream magnetic field, pushing them back across the shock; but, for a near coherent upstream magnetic field, this can happen only once for any particle.

For this reason, at a steady planar shock front, it has been proposed that particle acceleration was associated with the development of intense micro-turbulence on a scale $\ll r_{\text{g}}$ , in the shock precursor (Lemoine et al. Reference Lemoine, Pelletier and Revenu2006; Pelletier et al. Reference Pelletier, Lemoine and Marcowith2009). This point of view has been validated by particle-in-cell numerical simulations which observe the concomitant development of micro-turbulence and of particle acceleration (e.g. Spitkovsky Reference Spitkovsky2008; Martins et al. Reference Martins, Fonseca, Silva and Mori2009; Sironi, Spitkovsky & Arons Reference Sironi, Spitkovsky and Arons2013).

However, a crucial point of the previous argument is that the direction of the coherent magnetic field line in the shock front plane is conserved through the crossing of the shock. If this direction were randomized through some process, then particles could bounce on the downstream magnetic field with a non-zero probability at any Fermi cycle and return to the shock. This bounce would be similar to an isotropization of the particle directions downstream, i.e. similar to a fast isotropic scattering process. Therefore, it would lead to the development of a Fermi process as modelled by early test-particle Monte Carlo simulations (which implicitly assumed the non-conservation of the direction of the magnetic field in the shock front plane) (e.g. Bednarz & Ostrowski Reference Bednarz and Ostrowski1998; Kirk et al. Reference Kirk, Guthmann, Gallant and Achterberg2000; Achterberg et al. Reference Achterberg, Gallant, Kirk and Guthmann2001; Lemoine & Pelletier Reference Lemoine and Pelletier2003), with an index $s\simeq 2.2$ for $\text{d}N/\text{d}p\propto p^{-s}$ .

Returning to the corrugated shock front, if $\langle {\it\delta}B_{2}^{2}\rangle ^{1/2}\sim B_{2}$ on a scale $k^{-1}$ , corrugation precisely does the above: even in the absence of upstream turbulence, the in-plane direction of the downstream magnetic field is randomized on a scale $k^{-1}$ by the generation of turbulence at the corrugated shock. This should therefore lead to an efficient Fermi process for particles of gyroradius $r_{\text{g}}\sim k^{-1}$ , although this should admittedly be explicitly demonstrated by dedicated numerical simulations. Interestingly, if corrugation sustains acceleration in the above way, the typical acceleration time scale is then expected of order $r_{\text{g}}/c$ in the shock frame, as in the above test-particle simulations of the relativistic Fermi process.

How this process affects particles of gyroradius $r_{\text{g}}\ll k^{-1}$ is not obvious. In a first approximation, one could expect Fermi acceleration to be inoperant in that range of gyroradii because those particles ‘see’ the turbulent field as an essentially coherent field. However, the scale $r_{\text{g}}$ is then also much smaller than the corrugation amplitude $|{\it\delta}X|$ , hence the time dependence of the corrugation, the rippled shock structure and the relativistic turbulence existing in this layer could help sustain acceleration. At the opposite extreme, $r_{\text{g}}\gg k^{-1}$ , the turbulence may sustain acceleration, as long as the scattering frequency $k^{-1}/r_{\text{g}}^{2}$ remains larger than the advection frequency $r_{\text{g},0}^{-1}$ in the background magnetic field (the index 0 meaning that $r_{\text{g},0}$ is to be calculated relatively to the background field).

In any case, one does not expect corrugation to occur on a single scale $k$ , but on a broad range of scales; in this case, the above argument suggests that acceleration should at least take place for all $r_{\text{g}}$ in gyroresonance with the inertial range provided $\langle {\it\delta}B_{2}^{2}\rangle ^{1/2}\sim B_{2}$ can be realized on those scales, i.e. provided corrugation is mildly nonlinear on all scales.

The overall picture becomes somewhat more complicated in the presence of dissipation downstream of the shock, in particular how stochastic energy gains interplay with systematic energy gains due to the Fermi process. However, given that the Fermi process only transfers a small fraction of the available energy to a small fraction of particles, one may expect that dissipation would build up the hard power-law until near equipartition with the magnetic field and that the Fermi power-law would develop at higher momenta, until it saturates due to energy losses.