Tearing-mediated Alfvénic Turbulence in a Relativistic Plasma

Abstract In astrophysical relativistic plasmas, Alfvénic turbulence exists across a wide range of scales. Similar to a nonrelativistic case, this turbulence can be influenced by the tearing instability. We argue that in an ultrarelativistic pair plasma the tearing effects become significant at the critical scale given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>∼</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>rel</mml:mi> </mml:msub> <mml:msup> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mo>⊥</mml:mo> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>rel</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo> <mml:mn>5</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:math> , where L ⊥ represents the outer scale of the turbulence and d rel the relativistic electron inertial scale. Here n = 1 corresponds to the Harris magnetic configurations, while n = 2 applies to localized (non-Harris) magnetic profiles of turbulence-generated current sheets. Below the critical scale, the energy spectrum of magnetic fluctuations steepens to W ( k ) dk ∝ k −(7 n +2)/3 n dk . However, this happens only when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mi>σ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>≪</mml:mo> <mml:msup> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mo>⊥</mml:mo> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>rel</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>/</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo> <mml:mn>5</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>˜</mml:mo> </mml:mrow> </mml:mover> </mml:math> is the magnetization parameter based on the magnetic fluctuations. If the opposite inequality holds, the Alfvénic cascade is instead affected by charge starvation at scales below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>ch</mml:mi> </mml:msub> <mml:mo>∼</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>rel</mml:mi> </mml:msub> <mml:msup> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>rel</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mo>⊥</mml:mo> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mover accent="true"> <mml:mi>σ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> , and the energy spectrum becomes steeper than W ( k ) dk ∝ k −3 dk . When d rel is replaced by its nonrelativistic equivalent, our results agree with the nonrelativistic analysis of N. F. Loureiro &amp; S. Boldyrev (2018). In the Harris case n = 1, they agree with the recent relativistic studies by I. Demidov &amp; Y. Lyubarsky.