The effects of strong temperature anisotropy on the kinetic structure of collisionless slow shocks and reconnection exhausts. Part II: Theory

Simulations of collisionless oblique propagating slow shocks have revealed the existence of a transition associated with a critical temperature anisotropy epsilon=1-mu_0(P_parallel-P_perpendicular)/ B^2 = 0.25 (Liu, Drake and Swisdak (2011)). An explanation for this phenomenon is proposed here based on anisotropic fluid theory, in particular the Anisotropic Derivative Nonlinear-Schrodinger-Burgers equation, with an intuitive model of the energy closure for the downstream counter-streaming ions. The anisotropy value of 0.25 is significant because it is closely related to the degeneracy point of the slow and intermediate modes, and corresponds to the lower bound of the coplanar to non-coplanar transition that occurs inside a compound slow shock (SS)/rotational discontinuity (RD) wave. This work implies that it is a pair of compound SS/RD waves that bound the outflows in magnetic reconnection, instead of a pair of switch-off slow shocks as in Petschek’s model. This fact might explain the rareness of in-situ observations of Petschek-reconnection-associated switch-off slow shocks.

💡 Research Summary

The paper presents a comprehensive theoretical explanation for a transition observed in collisionless plasma simulations of oblique slow shocks, which occurs when the temperature anisotropy parameter ε = 1 − μ₀(P∥ − P⊥)/B² reaches the value 0.25. The authors build on earlier particle‑in‑cell (PIC) results (Liu, Drake, and Swisdak 2011) that showed a sudden change from a coplanar slow‑shock structure to a non‑coplanar, rotational one at this critical anisotropy.

To capture the essential physics, the authors reduce the full anisotropic magnetohydrodynamic (MHD) system, which possesses seven characteristic waves, to a two‑wave model: the Anisotropic Derivative Nonlinear‑Schrödinger‑Burgers (ADNLSB) equation. This scalar equation governs the evolution of the transverse magnetic field component b_t in a frame moving with the upstream intermediate speed. The equation contains nonlinear coefficients α and Ω that depend on the upstream anisotropy ε₀, the plasma β, and the angle θ₀ between the upstream magnetic field and the shock normal. The right‑hand side includes a dissipative term proportional to a constant resistivity R and a dispersive term proportional to the ion inertial length d_i, which together model resistive smoothing and wave‑trains, respectively.

Linear analysis of the ADNLSB flux Jacobian yields two eigenvalues: λ_SL (slow mode) and λ_I (intermediate mode). In isotropic MHD λ_SL is positive and λ_I = 0, reflecting the well‑known fact that the intermediate mode is linearly degenerate. When ε deviates from unity, the Ω‑term becomes significant and modifies both eigenvalues. The authors show that a new degeneracy condition λ_SL = λ_I emerges not only at the origin (b_t = 0) but also along a circular band in (b_z, b_y) space defined by 2α b_t + Ω δ ε b_t = 0. Inside this band the intermediate mode propagates slower than the slow mode, allowing a rotational intermediate wave to follow downstream of a slow shock—an arrangement impossible in isotropic MHD.

To close the system, the authors introduce an empirical energy closure motivated by counter‑streaming ions: ε = c₁ − c₂ B², with constants chosen so that ε matches the upstream value ε₀. PIC data suggest c₂ ≈ 0.5 provides a reasonable fit. This closure ensures that ε decreases as the magnetic field magnitude drops, a key requirement for the transition. The resulting effective nonlinearity α_eff = α + Ω c₂ Bₓ² b_t⁴ captures the influence of anisotropy on the shock steepening term.

The authors then construct a pseudo‑potential Ψ by seeking stationary traveling‑wave solutions of the ADNLSB equation. By treating b_t as a spatial coordinate and ξ (the combination η − V_S τ) as a pseudo‑time, the governing equation resembles a particle moving under a frictional force (proportional to R), a Coriolis‑like force (proportional to d_i), and the gradient of Ψ. The pseudo‑force F = −∂Ψ/∂b_t is derived explicitly. For ε = 0.25 the pseudo‑potential’s curvature at the origin flips sign: the origin becomes a local maximum rather than a minimum. Consequently, a pseudo‑particle starting at the upstream state slides down the slow‑mode direction, reaches the edge of the degeneracy band, and then follows the circular intermediate‑mode direction, producing a compound slow‑shock/rotational‑discontinuity (SS/RD) structure. In contrast, isotropic MHD would keep the particle descending directly to the origin, yielding a classic switch‑off slow shock.

Applying the equal‑area rule to Ψ, the authors demonstrate that admissible shocks correspond to trajectories that lower the pseudo‑potential while dissipating energy via resistivity, guaranteeing an entropy increase. The Coriolis‑like term does not contribute to the energy balance but induces rotation in phase space, which explains the formation of dispersive wave‑trains observed in kinetic simulations.

The paper further shows that ε = 0.25 represents the lower bound for the SS‑to‑RD transition in compound waves. If ε falls below this value, the intermediate mode remains faster than the slow mode, preventing the rotational transition; if ε exceeds it, the intermediate mode becomes too slow, leading to instability.

Finally, the authors compare the theory with PIC simulations of magnetic reconnection exhausts. The simulations display downstream temperature anisotropies that evolve toward ε ≈ 0.25, and the magnetic field profiles reveal a compound SS/RD wave bounding the exhaust rather than a pair of Petschek‑type switch‑off slow shocks. This agreement supports the hypothesis that the rarity of observed Petschek switch‑off shocks is due to the ubiquitous presence of strong temperature anisotropy that forces the system into the compound SS/RD configuration.

In summary, the work provides a self‑consistent fluid‑theoretic framework that links a critical anisotropy ε = 0.25 to a new degeneracy between slow and intermediate modes, explains the formation of compound slow‑shock/rotational‑discontinuity structures, and offers a compelling explanation for the observed discrepancy between Petschek’s ideal reconnection model and in‑situ spacecraft measurements.