[Paper Review] Kelvin-Helmholtz instability of relativistic jets - the transition from linear to nonlinear regime

M. Hanasz|arXiv (Cornell University)|Nov 23, 1997

Astrophysics and Cosmic Phenomena1 references3 citations

TL;DR

This paper investigates the nonlinear evolution of Kelvin-Helmholtz (KH) instability in relativistic jets, focusing on the transition from linear to nonlinear behavior. It shows that acoustic wave over-reflection at the jet boundary leads to shock formation primarily in the nonrelativistic external medium, while internal waves remain linear longer; nonlinear effects from the Lorentz factor limit boundary oscillations to amplitudes comparable to a fraction of the jet radius.

ABSTRACT

The observed wiggles and knots in astrophysical jets as well as the curvilinear motion of radio emitting features are frequently interpreted as signatures of the Kelvin-Helmholtz (KH) instability (eg. Hardee 1987). We investigate the KH instability of a hydrodynamic jet composed of a relativistic gas, surrounded by a nonrelativistic external medium and moving with a relativistic bulk speed. We show basic nonlinear effects, which become important for a finite amplitude KH modes. Since the KH instability in supersonic jets involves acoustic waves over-reflected on jet boundaries, the basic nonlinear effect relies on the steepening of the acoustic wave fronts, leading to the formation of shocks. It turns our that the shocks appear predominantly in the external nonrelativistic gas, while the internal acoustic waves remain linear for a much longer time. In addition, the external medium "hardens" as soon as the boundary oscillation velocity becomes comparable to the external sound speed. On the other hand, the amplification of internal waves due to the over-reflection is limited by a nonlinearity of the Lorentz $γ$ factor. This implies that the sidereal oscillations of the jet boundary, resulting from the K-H instability, are limited to very small amplitudes comparable to a fraction of the jet radius.

Motivation & Objective

• To understand the transition from linear to nonlinear behavior in Kelvin-Helmholtz (KH) instability within relativistic hydrodynamic jets.
• To analyze how nonlinear effects, particularly wave steepening and shock formation, influence the evolution of KH modes in jets with relativistic bulk flow.
• To investigate the role of acoustic wave over-reflection at the jet boundary and its nonlinear consequences in both internal and external media.
• To determine the amplitude limitations of jet boundary oscillations due to relativistic effects, especially the nonlinearity of the Lorentz factor.
• To clarify why observed jet features like wiggles and knots may arise from KH instability, despite nonlinear damping mechanisms.

Proposed method

• Modeling a relativistic jet as a hydrodynamic flow with a relativistic internal medium and a nonrelativistic external medium.
• Applying linear stability analysis to identify KH mode growth, followed by nonlinear analysis to track mode evolution.
• Tracking acoustic wave propagation and over-reflection at the jet boundary, focusing on wave steepening and shock formation.
• Using the relativistic Lorentz factor γ to model nonlinear corrections to wave amplification, particularly in the internal jet medium.
• Comparing the nonlinear evolution of waves in the internal (relativistic) and external (nonrelativistic) media to assess differential shock development.
• Evaluating the threshold at which the external medium effectively 'hardens' due to boundary velocity approaching the external sound speed.

Experimental results

Research questions

• RQ1 How do nonlinear effects modify the growth and evolution of Kelvin-Helmholtz modes in relativistic jets?
• RQ2 What determines the formation of shocks in the external medium versus the persistence of linear waves in the internal jet?
• RQ3 To what extent is the amplitude of jet boundary oscillations limited by relativistic nonlinearities in the Lorentz factor?
• RQ4 How does over-reflection of acoustic waves at the jet boundary contribute to nonlinear wave steepening and shock formation?
• RQ5 At what point does the external medium become effectively 'hardened' due to boundary motion approaching the sound speed?

Key findings

• Shocks form predominantly in the nonrelativistic external medium due to steepening of over-reflected acoustic waves, while internal waves remain linear for longer durations.
• The external medium 'hardens' when the jet boundary oscillation velocity approaches the external sound speed, limiting further wave growth.
• Nonlinear effects from the Lorentz factor γ limit the amplification of internal waves, restricting boundary oscillations to amplitudes comparable to a fraction of the jet radius.
• Over-reflection of acoustic waves at the jet boundary drives nonlinear wave steepening, but this effect is suppressed in the internal relativistic medium.
• The transition from linear to nonlinear regime is dominated by external medium dynamics, with internal wave behavior remaining weakly nonlinear for extended periods.
• The model explains why observed jet wiggles and knots are consistent with KH instability despite amplitude limitations from relativistic nonlinearities.