Nature and dynamics of overreflection of Alfvén waves in MHD shear flows

Our goal is to gain new insights into the physics of wave overreflection phenomenon in MHD nonuniform/shear flows changing the existing trend/approach of the phenomenon study. The performed analysis allows to separate from each other different physical processes, grasp their interplay and, by this way, construct the basic physics of the overreflection in incompressible MHD flows with linear shear of mean velocity, ${\bf U}_0=(Sy,0,0)$, that contain two different types of Alfv${\rm \acute{e}}$n waves. These waves are reduced to pseudo- and shear shear-Alfv${\rm \acute{e}}$n waves when wavenumber along $Z$-axis equals zero (i.e., when $k_z=0$). Therefore, for simplicity, we labelled these waves as: P-Alfv${\rm \acute{e}}$n and S-Alfv${\rm \acute{e}}$n waves (P-AWs and S-AWs). We show that: (1) the linear coupling of counter-propagating waves determines the overreflection, (2) counter-propagating P-AWs are coupled with each other, while counter-propagating S-AWs are not coupled with each other, but are asymmetrically coupled with P-AWs; S-AWs do not participate in the linear dynamics of P-AWs, (3) the transient growth of S-AWs is somewhat smaller compared with that of P-AWs, (4) the linear transient processes are highly anisotropic in wave number space, (5) the waves with small streamwise wavenumbers exhibit stronger transient growth and become more balanced, (6) maximal transient growth (and overreflection) of the wave energy occurs in the two-dimensional case -- at zero spanwise wavenumber. To the end, we analyze nonlinear consequences of the described anisotropic linear dynamics -- they should lead to an anisotropy of nonlinear cascade processes significantly changing their essence, pointing to a need of revisiting the existing concepts of cascade processes in MHD shear flows.