The Kelvin–Helmholtz Instability at the Boundary of Relativistic Magnetized Jets
Anthony Chow, Jordy Davelaar, Michael Rowan, Lorenzo Sironi
Abstract
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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">A</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>cos</mml:mi> <mml:mi>θ</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:math> , 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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mo><</mml:mo> <mml:msqrt> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">A</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mi>c</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:msqrt> </mml:math> , with Γ w the wind adiabatic index). Our results have important implications for the propagation and emission of relativistic magnetized jets.