Density Fluctuation–Mach Number Scaling in Compressible, High Plasma Beta Turbulence: In Situ Space Observations and High-Reynolds Number Simulations

Abstract Understanding the nature of compressible fluctuations in a broad range of turbulent plasmas, from the intracluster medium to the solar wind, has been an active field of research in the past decades. Theoretical frameworks for weakly compressible magnetohydrodynamical turbulence in an inhomogeneous background magnetic field predict a linear scaling of the normalized mass density fluctuation ( δρ / ρ 0 ), as a function of the turbulent Mach number ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:math> ), <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>δ</mml:mi> <mml:mi>ρ</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>∝</mml:mo> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:math> . However, so far, the scaling relation has been tested only using moderate to low plasma beta ( β ≲ 1) solar wind observational data, where the compressibility is weak δρ / ρ 0 ∼ 0.1. Here, we combine NASA’s Magnetospheric Multiscale Mission data in Earth’s magnetosheath, where β ∼ 10 is high, and β ∼ 1/8 highly compressible magnetohydrodynamic turbulence simulations at unprecedented resolutions. Both show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>δ</mml:mi> <mml:mi>ρ</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>∝</mml:mo> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:math> holds across a broad range of δρ / ρ 0 , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:math> , and β , demonstrating that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>δ</mml:mi> <mml:mi>ρ</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>∝</mml:mo> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:math> is a robust compressible turbulence relation, going beyond the asymptotics of the weakly compressible theory. We discuss the findings in the context of understanding the nature of strongly compressible turbulent fluctuations and the driving parameter in astrophysical and space plasmas.