The Kelvin–Helmholtz Instability at the Boundary of Relativistic Magnetized Jets

Abstract

We study the linear stability of a planar interface separating two fluids in relative motion, focusing on conditions appropriate for the boundaries of relativistic jets. The jet is magnetically dominated, whereas the ambient wind is gas-pressure-dominated. We derive the most general form of the dispersion relation and provide an analytical approximation of its solution for an ambient sound speed much smaller than the jet Alfvén speed v _A , as appropriate for realistic systems. The stability properties are chiefly determined by the angle ψ between the wavevector and the jet magnetic field. For ψ = π /2, magnetic tension plays no role, and our solution resembles the one of a gas-pressure-dominated jet. Here, only sub-Alfvénic jets are unstable ( $0\lt {M}_{e}\equiv (v/{v}_{{\rm{A}}})\cos \theta \lt 1$ , where v is the shear velocity and θ the angle between the velocity and the wavevector). For ψ = 0, the free energy in the velocity shear needs to overcome the magnetic tension, and only super-Alfvénic jets are unstable ( $1\lt {M}_{e}\lt \sqrt{{(1+{{\rm{\Gamma }}}_{w}^{2})/[1+({v}_{{\rm{A}}}/c)}^{2}{{\rm{\Gamma }}}_{w}^{2}]}$ , with Γ _w the wind adiabatic index). Our results have important implications for the propagation and emission of relativistic magnetized jets.

Keywords

• Plasma astrophysics
• Plasma jets
• High energy astrophysics
• Relativistic jets
• Magnetohydrodynamics
• Relativistic fluid dynamics