Rapid distortion theory for homogeneous shear-driven MHD turbulence
Authors: Safonov Sergei , Petrosyan Arakel 
Published: March 2020

&lt;p&gt;&lt;span&gt;&lt;span&gt;In this report we study statistical properties of astrophysical turbulent plasma flows &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;subjected to large scale velocity shear and an external magnetic field&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; using Rapid Distortion Theory (RDT). &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;The problem of shear-driven turbulence arises in several important physical systems, such as&lt;/span&gt;&lt;/span&gt; &lt;span&gt;&lt;span&gt;the &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;solar wind &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;and ionized atmospheres of exoplanets&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;.&lt;/span&gt;&lt;/span&gt; &lt;span&gt;&lt;span&gt;Rapid distortion theory is a linearization method for Reynolds-averaged Navier-Stockes equations. Its&lt;/span&gt;&lt;/span&gt; &lt;span&gt;&lt;span&gt;main&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; assumption is that the turbulence responds to the external distortion &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;by velocity shear&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; so fast, that inertial &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;forces &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;result in a negligible change in velocity &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;field statistics at small time scales&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;. This allows to linearize equations and to derive equations for second moments of turbulence. We apply RDT &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;approach&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; to incompressible &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;homogeneous &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;MHD &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;turbulence&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; distorted with &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;an &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;external magnetic field and &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;a &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;linear velocity shear in cases of rotating and non-rotating plasma. It is shown that even with a strong nonlinearity many properties of turbulence can be qualitatively studied using a linear theory. A closed system of linear equations &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;is derived&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; for &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;e&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;nergy, helicity and polarization &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;of velocity and magnetic field correlations&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;. &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;Structural analysis is conducted showing the change of energy distribution between components of spectral tensor of turbulence. &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;Development of initially isotropic turbulence and transition to anisotropy are studied. &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;Model e&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;quations for fluid, current and cross helicity are derived. Differences in cases of rotating and non-rotating flows are discussed. This work was supported by the Russian Foundation for Basic Research (project no. 19-02-00016).&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;