Effect of coupling and linear transformation of waves in shear flows

Abstract

A new linear mechanism of reciprocal transformation of waves and a corresponding energy transfer in shear flows is discovered. The effect is demonstrated on the simplest example -the two-dimensional waves in unbounded, parallel hydromagnetic flow with uniform velocity shear. The phenomenon discovered is of importance for magnetohydrodynamics and fusion plasma devices and for various terrestrial and astrophysical shear flows. Grasping the result became possible thanks to the nonmodal analyses of the perturbation evolution in the flow. ͓S1063-651X͑96͒11605-6͔PACS number͑s͒: 52.35. Bj, 03.40.Kf, 47.20.Ft Oscillatory systems are prevalent in the nature and play an important role in the wide variety of physical processes. It should be emphasized that: ͑i͒ these kinds of dynamical systems usually consist of coupled oscillators and have several degrees of freedom; ͑ii͒ parameters of the oscillators in most cases are not constant but vary slowly in time ͓1͔. The processes taking place in such systems are immensely complex and their study is applicable not only to the various branches of physics, but also to chemistry, biology, sociology, and other sciences. Investigation of these two key aspects of dynamics, closely related to each other in some physical situations, has become one of the most important interdisciplinary problems in applied mathematics ͓1-3͔. Hydrodynamical and plasma flows constitute such oscillatory systems, where coupling, as a rule, is associated with nonlinear processeswave decay processes ͓4͔, which ensures the mutual transformation of different wave modes. In plasma physics, a linear coupling phenomenon is also known: mutual transformation of different kinds of plasma waves arising due to a spatial inhomogeneity of a medium. For example, existence of the density inhomogeneity induces coupling between magnetohydrodynamic ͑MHD͒ oscillations ͓5͔ or the transformation of compressional-type waves into electromagnetic-type waves propagating across a density discontinuity ͓6͔, etc.In the present paper, we describe a mechanism of the linear reciprocal transformation of wave modes arising in flows due to a velocity inhomogeneity. The effect is demonstrated for the simplest example: MHD waves in twodimensional ͑2D͒, compressible, magnetized, unbounded parallel flow with uniform velocity shear ͑plane, magnetized Couette flow͒. The result is obtained by means of the nonmodal approach applied to the study of the evolution of small-scale perturbations in the flow. This effect makes more diverse the variety of processes taking place in shear flows. Under a traditional modal analysis, this phenomenon was not noticed for the following reason: the linear operators arising in shear flows ͑as becomes apparent in a number of recent contributions ͓7-13͔͒ are non-normal. It results in a set of nonorthogonal eigenfunctions strongly interfering with each other. That is why analysis of distinct eigenfunction evolution, performed in the framework of the modal approach ͑without adequate consideration of the interference͒, ...

Spatial aspect of wave transformations in astrophysical flows

“…The linear phase of the temporal evolution of the system is governed by the following set of partial differential equations (Chagelishvili et al 1996):…”

“…Therefore, before addressing in detail the concrete situations pertaining to the different astrophysical contexts described above, we think that an essential step is to consider an ideal case in which we can examine whether mode transformation can indeed be observed also in real space and for which we can develop the tools for analysing the process. This is the aim of the present paper and in the next section we describe a simple 2-D MHD setup, corresponding to Chagelishvili et al (1996) where SITs were originally identified, which allows as to study mutual transformations of SMWs and FMWs. We write down the system of partial differential equations for the perturbations of physical variables and describe briefly the "nonmodal" evolution of the amplitudes of individual Fourier harmonics.…”

“…Originally, this effect was discovered for a magnetohydrodynamic (MHD), two-dimensional (2-D) shear flow: it was shown that the shear couples the slow magnetosonic waves (SMWs) and the fast magnetosonic waves (FMWs) and that under favorable conditions, namely, when the speed of sound C S and the Alfvén speed C A are of the same order of magnitude, the coupling leads to the efficient reciprocal transformations of these waves (Chagelishvili et al 1996).…”

“…The phenomenon of Shear Induced wave Transformations (SITs) (Chagelishvili et al 1996), is a common feature of flows, sustaining n > 1 mode of wave motion. Until now this "nonmodal" phenomenon was described only in terms of the time evolution of individual Fourier harmonics of perturbations in the space of wave numbers (k-space).…”

Linear coupling of acoustic and cyclotron waves in plasma flows

“…Another important and interesting issue, related with the propagation of waves in flows, is related with the ability of the velocity shear to couple waves [10] and to ensure their mutual transformations. Shear-induced wave couplings exist in hydrodynamic systems (coupling of sound waves and internal gravity waves [11]), in MHD (both electron-proton and electron-positron plasmas), where velocity shear couples all three MHD wave modes (Alfvén waves, slow magnetosonic waves and fast magnetosonic waves) [12], and in dusty plasmas [5].…”