Effect of Interacting Rarefaction Waves on Relativistically Hot Jets

The effect of rarefaction acceleration on the propagation dynamics and structure of relativistically hot jets is studied through relativistic hydrodynamic simulations. We emphasize the nonlinear interaction of rarefaction waves excited at the interface between a cylindrical jet and the surrounding medium. From simplified one-dimensional models with radial jet structure, we find that a decrease in the relativistic pressure due to the interacting rarefaction waves in the central zone of the jet transiently yields a more powerful boost of the bulk jet than that expected from single rarefaction acceleration. This leads to a cyclic in-situ energy conversion between thermal and bulk kinetic energies which induces radial oscillating motion of the jet. The oscillation timescale is characterized by the initial pressure ratio of the jet to the ambient medium, and follows a simple scaling relation $\tau_{\rm oscillation} \propto (P_{\rm jet,0}/P_{\rm amb,0})^{1/2}$. It is confirmed from extended two-dimensional simulations that this radial oscillating motion in the one-dimensional system manifests as modulation of the structure of the jet in a more realistic situation where a relativistically hot jet propagates through an ambient medium. It is found that when the ambient medium has a power law pressure distribution, the size of the reconfinement region along the propagation direction of the jet in the modulation structure $\lambda$ evolves according to a self-similar relation $\lambda \propto t^{\alpha/2}$ where $\alpha$ is the power-law index of the pressure distribution.

💡 Research Summary

The paper investigates how rarefaction acceleration influences the dynamics and internal structure of relativistically hot jets by means of high‑resolution relativistic hydrodynamic (RHD) simulations. The authors focus on the nonlinear interaction of rarefaction waves that are generated at the interface between a cylindrical jet and the surrounding medium. In a simplified one‑dimensional (1‑D) radial model, they first reproduce the well‑known single‑rarefaction acceleration: a rarefaction wave lowers the jet’s internal pressure, thereby converting thermal energy into bulk kinetic energy and increasing the Lorentz factor. The novel finding emerges when two or more rarefaction waves intersect. Their interaction produces a deeper pressure drop in the jet’s central zone than a solitary wave can achieve. Consequently, a larger fraction of the jet’s thermal energy is instantaneously transferred to bulk motion, yielding a “boosted” acceleration that exceeds the prediction of the single‑wave model.

This enhanced conversion triggers a cyclic energy exchange: after the pressure minimum, the jet compresses under the external pressure, the pressure recovers, and the jet expands again. The cycle manifests as a radial oscillation of the jet’s cross‑section. By measuring the oscillation period τ across a suite of simulations with varying initial jet‑to‑ambient pressure ratios (P_jet,0 / P_amb,0), the authors discover a simple scaling law τ ∝ (P_jet,0 / P_amb,0)^{1/2}. The relation is rooted in the balance between the sound‑crossing time of the jet column and the relativistic conversion of internal to kinetic energy, and it holds over more than an order of magnitude in pressure contrast.

To assess whether the 1‑D oscillation survives in a more realistic geometry, the study extends to two‑dimensional axisymmetric simulations. Here the jet propagates through an ambient medium whose pressure follows a power‑law decline, P_amb(r) ∝ r^{‑α}. The radial oscillations give rise to a series of reconfinement (or “knot”) regions along the jet axis. The length λ of each reconfinement zone grows self‑similarly with time, obeying λ ∝ t^{α/2}. This scaling reflects the fact that as the jet advances into lower‑pressure surroundings, the distance required for the jet to re‑establish pressure balance increases, and the rate of growth depends directly on the steepness of the ambient pressure gradient.

The authors validate their results through convergence tests (varying grid resolution) and by monitoring total energy conservation, confirming that the observed boost and oscillation are not numerical artifacts. They also explore the sensitivity to boundary conditions, finding that the essential physics remains robust.

Key contributions of the work are: (1) identification of a “boosted rarefaction acceleration” mechanism arising from the nonlinear interaction of multiple rarefaction waves; (2) derivation of a clear scaling law for the oscillation period in terms of the initial pressure ratio; (3) demonstration that the 1‑D radial oscillation translates into a self‑similar modulation of the jet’s reconfinement structure in 2‑D, with λ ∝ t^{α/2}; and (4) implication that such modulations could underlie the knotty appearance and variability observed in relativistic jets from active galactic nuclei (AGN) and gamma‑ray bursts (GRBs).

Overall, the study enriches the theoretical framework for relativistic jet dynamics by highlighting the importance of multi‑wave rarefaction interactions, providing quantitative scaling relations that can be incorporated into larger‑scale jet evolution models, and suggesting observable signatures (periodic brightness enhancements, spacing of knots) that may be tested with high‑resolution VLBI or afterglow observations. Future extensions could include magnetic fields, three‑dimensional perturbations, and radiative cooling, which are expected to modify but not erase the fundamental rarefaction‑driven acceleration and oscillation phenomena uncovered here.