Linear and nonlinear processes in magnetohydrodynamic shear flows : their special nature, interplay and consequences
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In the 1990s it was rigorously revealed by the hydrodynamic community non-normal nature of operators of the modal analysis of linear processes in shear flows – so, called “shear flow non-normality”. The non-normality ensures finite-time, or transient, growth of perturbations in spectrally stable shear flows that is essentially anisotropic in spectral space and, in turn, leads to anisotropy of nonlinear processes in spectral space – the dominant nonlinear process turns out to be not a direct/inverse, but a transverse cascade, that is, a transverse/angular redistribution of perturbation harmonics in spectral space. The investigation of the linear transient growth and nonlinear transverse cascade in magnetohydrodynamic (MHD) unbounded constant shear flows – their special nature, interplay and consequences – is the goal of the present thesis. Initially, in Chapter 2, is investigated in detail the transient linear dynamics in the form of overreflection of pseudo- and shear-Alfv´en waves (P-AWs and S-AWs) in spectrally stable MHD plane constant shear flow. 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. Further, in the thesis is investigated MHD turbulence in spectrally stable two dimensional (2D) plane shear flows (Chapter 3) and three dimensional (3D) Keplerian disk flows, threaded by a non-zero azimuthal (Chapter 4) and vertical (Chapter 5) net magnetic flux. In order to gain a deeper insight into the underlying dynamical balances and sustaining mechanism, we performed a set of numerical simulations in the shearing box model and based on the simulation data, analyzed in detail the turbulence dynamics in Fourier space. 2D MHD plane shear flow, considered in Chapter 3, is spectrally stable, so the turbulence is subcritical by nature and hence it can be energetically supported just by a transient growth mechanism due to shear flow non-normality. We focus on analysis of the character of nonlinear processes and the underlying self-sustaining scheme of the turbulence, i.e., on the interplay between linear transient growth and nonlinear processes, in the spectral plane. The study, being concerned with a new type of energy-injecting process for turbulence - the transient growth - represents an alternative to the main trends of magnetohydrodynamic (MHD) turbulence research. In the case of Keplerian disk flows with a net azimuthal field, classical exponential/modal instabilities are absent and linear growth of perturbations (shearing waves) has a transient nature, also referred to as nonmodal growth. Particularly, in the case of disk flows with azimuthal field and rotation, magnetorotational instability (MRI), being only available source of energy for turbulence, has transient nature and by itself, cannot ensure a long-term sustenance of the perturbations, i.e. is imperfect in this sense. A necessary positive nonlinear feedback is required to regenerate new transiently growing modes. In other words, the role of nonlinearity becomes crucial: it lies at the heart of the sustenance of turbulence. The detailed analysis of the dynamics in apextral/Fourier space, allows to demonstrate existence of the positive feedback. Specifically, main novelties of the findings are the following: I. The nonmodal growth process is strongly anisotropic in Fourier space that, in turn, leads to anisotropy of nonlinear processes in this space. As a result, the main nonlinear process appears to be not an usual direct/inverse, but rather a new type of transverse/angular redistribution of perturbation modes in Fourier space, when their wavevector mainly changes orientation during nonlinear mode interactions – nonlinear transverse cascade. II. Both the linear nonmodal growth and nonlinear transverse cascade mainly operate at large length scales, comparable to the box/system size. Consequently, the central, small wavenumber area of Fourier space, is crucial in the turbulence sustenance process and is called the vital area. III. The turbulence is sustained by a subtle interplay of the linear nonmodal growth (transient MRI in the case of Keplerian disks) and the nonlinear transverse cascade. Analyzing this interplay, it is revealed the basic subcycle of the sustenance scheme that clearly shows synergy of the linear and nonlinear processes in the self-organization of the magnetized flow system. In the case of net vertical field, there is exponential growth of axisymmetric channel modes, and hence no deficit in energy supply. Due to this, the role of the transverse cascade in the turbulence sustenance is not as crucial as in the case of the azimuthal field. But it still shapes the dynamics, sets the saturation level, and determines the overall“design” of the net vertical field MRI-turbulence. In particular, it accounts for the transfer of energy among the “building blocks” of this turbulence: the axisymmetric channel mode, zonal flow, to a broad spectrum of nonaxisymmetric (parasitic) modes. The analyzed refined interplay of linear and nonlinear processes should be relevant to the understanding of subcritical turbulence in other sheared and magnetized complex environmental and engineering flows (e.g., sheared E×B plasma fusion flows).