2 The model

2.1 Shock equations

In a plasma with a non-zero magnetic field, the motion of the particles along the magnetic field lines differs fundamentally from their motion perpendicular to the field lines. This points at the necessity of allowing different amounts of random energy for particle motions parallel and perpendicular to the field, respectively. One expects the distribution function to have an axial symmetry around the local magnetic field $\vec{B}$, if the Larmor radii of the particles are very small compared to the density scale in the perpendicular direction. Then the gas pressure is a tensor

$$\tens {P}=P_{\perp }\tens {I}+(P_{\parallel }-P_{\perp })\vec{B}\vec{B}/B^{2},$$ (2)

where $\tens {I}$ is the unit tensor. In a steady state, we can write all the macroscopic conservation laws in form (e.g., Boyd & Sanderson 1969)

$$\nabla \cdot \vec{\Phi }^{\alpha }=0$$ (3)

where $\vec{\Phi }^{\alpha }$ is one of the fluxes: the mass flux,

$$\vec{\Phi }^{m}=\rho \vec{u};$$ (4)

the momentum flux,

$$\vec{\Phi }^{p}=\rho \vec{u}\vec{u}+\tens {P}-\tens {T},$$ (5)

with

$$\tens {T}=\vec{B}\vec{B}/4\pi -(B^{2}/8\pi )\tens {I}$$ (6)

giving the electromagnetic stress tensor (neglecting small electric-field contributions); or energy flux

$$\vec{\Phi }^{E}=\frac{c}{4\pi }\vec{E}\times \vec{B}+\tens {... ...ot \vec{u}+\frac{1}{2}({\rm Tr\,}\tens {P}+\rho u^{2})\vec{u},$$ (7)

which holds for a non-relativistic gas with zero heat flux. Here, ${\rm Tr\,}\tens {P}=2P_{\perp }+P_{\parallel }$ is the trace of the pressure tensor.

We consider the conservation laws across a planar shock discontinuity. In addition to the macroscopic conservation laws, we use Maxwell's equations $\nabla \cdot \vec{B}=0$ and $c^{-1}\partial \vec{B}/\partial t+\nabla \times \vec{E}=0$, and Ohm's law in form $\vec{E}=-\vec{u}\times \vec{B}/c$ to arrive at jump conditions (see also Lyu & Kan 1986)

   

$$[B_{n}]^{2}_{1}=0,\, \, \, \, \, \left[ \vec{B}_{t}u_{n}-B_{n}\vec{u}_{t}\right] ^{2}_{1}$$ (8)

$$\left[ \rho u_{n}\right] ^{2}_{1}=0,\, \, \, \, \left[ \rho \vec{u}_{t}u_{n}-\phi \frac{B_{n}\vec{B}_{t}}{4\pi }\right] ^{2}_{1}$$ (9)

$$\left[ \rho u_{n}^{2}+P_{\parallel }+(2\phi -1)\frac{B^{2}_{t}}{8\pi }\right] ^{2}_{1}$$ (10)

 

$$\left[ \left\{ \frac{\rho }{2}u_{n}^{2}+\frac{5P_{\parallel }}{2}... ... u_{n}-\phi B_{n}\frac{\vec{u}_{t}\cdot \vec{B}_{t}}{4\pi }\right] ^{2}_{1} = 0$$ (11)

expressing the continuity of normal magnetic and tangential electric fields across the shock (Eq. (8)), and the conservation of mass and perpendicular momentum (Eq. (9)), parallel momentum (Eq. (10)), and energy (Eq. (11)). Here, subscripts n and t denote vector components normal and tangential to the planar shock and

$$\phi \equiv 1-\frac{P_{\parallel }-P_{\perp }}{B^{2}/4\pi },$$ (12)

is the firehose factor specifying the pressure anisotropies. Regarding the upstream values for all variables as known input parameters and the downstream values as unknowns, we have altogether 8 equations and 9 unknowns. To close the set, we simply take the downstream pressure to be isotropic, which should be reasonable at least for shocks with a compression ratio differing considerably from unity; for such shocks the transition layer is likely to produce short-scale magnetic turbulence that can lead to the isotropization of the particles just downstream the shock.

2.2 Alfvén-wave properties

Pressure anisotropies also modify the dispersion relation of Alfvén waves, which at long-wave-length limit becomes (e.g., Stix 1962)

$$\omega ^{2}=\phi V_{\rm A}^{2}k^{2}$$ (13)

for circularly polarized waves propagating parallel to the mean field. Here, the wave frequency $\omega$ and the wavenumber k are measured in the plasma frame. For positive values of $\phi$, Eq. (13) describes waves propagating at phase speed $\pm \phi ^{1/2}\vec{B}_{0}(4\pi \rho )^{-1/2}$ parallel (+) or anti-parallel (-) to the mean magnetic field, $\vec{B}_{0}$. For negative values of $\phi$, Eq. (13) describes the firehose instability: the magnetic fluctuations become group-standing and grow exponentially at rate $V_{\rm A}\vert\phi \vert^{1/2}\vert k\vert$. From now on we take $\phi \geq 0$, although the results could be generalized also to the firehose-unstable case. Note that the firehose unstable case can only be realized for $P_{\parallel }>B^{2}/4\pi$, so restricting the analysis to the stable cases is natural for plasmas with low $\beta$.

Combining Ohm's and Faraday's laws gives the relation

$$\delta \vec{u}^{\pm }=\mp \phi ^{1/2}\frac{\delta \vec{B}^{\pm }}{(4\pi \rho )^{1/2}}$$ (14)

between the velocity and magnetic-field perturbations associated with Alfvén waves propagating parallel (+) or anti-parallel (-) to the mean magnetic field. We assume that the turbulent magnetic field can be decomposed to two wave fields propagating in opposite directions,

$$\delta \vec{B}=\delta \vec{B}^{+}+\delta \vec{B}^{-},$$ (15)

yielding the equation

$$\delta \vec{u}=\phi ^{1/2}\frac{\delta \vec{B}^{-}-\delta \vec{B}^{+}}{(4\pi \rho )^{1/2}}\cdot$$ (16)