Mechanical picture of the linear transient growth of vortical perturbations in incompressible smooth shear flows

2016 ◽  
Vol 1 (4) ◽  
Author(s):  
George Chagelishvili ◽  
Jan-Niklas Hau ◽  
George Khujadze ◽  
Martin Oberlack
Keyword(s):  
Shear Flows ◽  
Transient Growth

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

2014 ◽  
Vol 80 (5) ◽  
pp. 667-685
Author(s):  
D. Gogichaishvili ◽  
G. Chagelishvili ◽  
R. Chanishvili ◽  
J. Lominadze
Keyword(s):  
Shear Flows ◽  
Alfven Waves ◽  
Wave Number ◽  
Alfvén Waves ◽  
Transient Growth ◽  
Mean Velocity ◽  
Linear Dynamics ◽  
Linear Shear ◽  
Basic Physics ◽  
Cascade Processes

Our goal is to gain new insights into the physics of wave overreflection phenomenon in magnetohydrodynamic (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, U0=(Sy,0,0), that contain two different types of Alfvén waves. These waves are reduced to pseudo- and shear-Alfvén waves when wavenumber along Z-axis equals zero (i.e. when kz=0). Therefore, for simplicity, we labeled these waves as: P-Alfvén and S-Alfvén 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.

scholarly journals Sustaining turbulence in spectrally stable shear flows – interplay of linear transient growth and nonlinear transverse cascade

2019 ◽  
Vol 82 ◽  
pp. 423-434
Author(s):  
D. Gogichaishvili ◽  
G. Mamatsashvili ◽  
W. Horton ◽  
G. Chagelishvili
Keyword(s):  
Shear Flows ◽  
Nonlinear Process ◽  
Transient Growth ◽  
Fourier Space ◽  
Nonlinear Perturbations ◽  
Nonlinear Processes ◽  
2D And 3D ◽  
Fourier Harmonics ◽  
Nonlinear State

We analyze the sustaining mechanism of nonlinear perturbations/turbulence in spectrally stable smooth shear flows. The essence of the sustenance is a subtle interplay of linear transient growth of Fourier harmonics and nonlinear processes. In spectrally stable shear flows, the transient growth of perturbations is strongly anisotropic in spectral (k-)space. This, in turn, leads to anisotropy of nonlinear processes ink-space and, as a result, the main (new) nonlinear process appears to be not a direct/inverse, but rather a transverse/angular redistribution of harmonics in Fourier space referred to as the nonlinear transverse cascade. It is demonstrated that nonlinear state is sustained owing to the interplay of the linear nonmodal growth and the transverse cascade. The possibility of such course of events has been described ink-space byG. Chagelishvili, J.-P. Zahn, A. Tevzadze and J. Lominadze, A&A, 402, 401 (2003)that reliably exemplifies the well-known bypass scenario of subcritical turbulence in spectrally stable shear flows. We present selected results of the simulations performed in different (HD and MHD; 2D and 3D; plane and Keplerian) shear flows to demonstrate the transverse cascade in action.

scholarly journals Transient Dynamics of Nonsymmetric Perturbations versus Symmetric Instability in Baroclinic Zonal Shear Flows

2010 ◽  
Vol 67 (9) ◽  
pp. 2972-2989 ◽  
Author(s):  
G. R. Mamatsashvili ◽  
V. S. Avsarkisov ◽  
G. D. Chagelishvili ◽  
R. G. Chanishvili ◽  
M. V. Kalashnik
Keyword(s):  
Shear Flows ◽  
Stable Stratification ◽  
Linear Operators ◽  
Vertical Shear ◽  
Transient Dynamics ◽  
Transient Growth ◽  
Coriolis Parameter ◽  
Linear Dynamics ◽  
Wide Range ◽  
Symmetric Instability

Abstract The linear dynamics of symmetric and nonsymmetric perturbations in unbounded zonal inviscid flows with a constant vertical shear of velocity, when a fluid is incompressible and density is stably stratified along the vertical and meridional directions, is investigated. A small–Richardson number Ri ≲ 1 and large–Rossby number Ro ≳ 1 regime is considered, which satisfies the condition for symmetric instability. Specific features of this dynamics are closely related to the nonnormality of linear operators in shear flows and are well interpreted in the framework of the nonmodal approach by tracing the linear dynamics of spatial Fourier harmonics (Kelvin modes) of perturbations in time. The roles of stable stratification, the Coriolis parameter, and vertical shear in the dynamics of perturbations are analyzed. Classification of perturbations into two types or modes—vortex (i.e., quasigeostrophic balanced motions) and inertia–gravity wave—is made according to the value of potential vorticity. The emerging picture of the (linear) transient dynamics for these two modes at Ri ≲ 1 and Ro ≳ 1 indicates that vortex mode perturbations are able to gain basic flow energy and undergo exponential transient amplification and in this process generate inertia–gravity waves. Transient growth of the vortex mode and, consequently, the effectiveness of the wave generation both increase with decreasing Ri and increasing Ro. This linear coupling of perturbation modes is, in general, specific to shear flows but is not fully appreciated yet. A parallel analysis of the transient dynamics of nonsymmetric perturbations versus symmetric instability is also presented. It is shown that the nonnormality-induced transient growth of nonsymmetric perturbations can prevail over the symmetric instability for a wide range of Ri and Ro. The current analysis suggests that the dynamical activity of fronts and jet streaks at Ri ≲ 1 and Ro ≳ 1 should be determined by nonsymmetric perturbations rather than by symmetric ones, as was accepted in earlier papers. It is noteworthy that the transient growth of perturbations is asymmetric in the wavenumber space—the constant phase plane of maximally amplified perturbations is inclined in a direction northeast to the zonal one and the inclination angle is different for different Ri and Ro.

scholarly journals Stochastic parametric resonance in shear flows

2005 ◽  
Vol 12 (6) ◽  
pp. 871-876 ◽  
Author(s):  
F. J. Poulin ◽  
M. Scott
Keyword(s):  
Shear Flow ◽  
Shear Flows ◽  
Parametric Instability ◽  
Mathieu Equation ◽  
Cape Cod ◽  
Transient Growth ◽  
Constant Frequency ◽  
Cape Cod Bay ◽  
Linearized System ◽  
Time Periodic

Abstract. Time-periodic shear flows can give rise to Parametric Instability (PI), as in the case of the Mathieu equation (Stoker, 1950; Nayfeh and Mook, 1995). This mechanism results from a resonance between the oscillatory basic state and waves that are superimposed on it. Farrell and Ioannou (1996a, b) explain that PI occurs because the snap-shots of the velocity profile are subject to transient growth. If the flows were purely steady the transient growth would subside and not have any long lasting effect. However, the coupling between transient growth and the time variation of the basic state create PI. Mathematically, transient growth, and therefore PI, are due to the nonorthogonal eigenspace in the linearized system. Poulin et al. (2003) studied a time-periodic barotropic shear flow that exhibited PI, and thereby produced mixing at the interface between Potential Vorticity (PV) fronts. The instability led to the formation of vortices that were stretched. A later study of an oscillatory current in the Cape Cod Bay illustrated that PI can occur in realistic shear flows (Poulin and Flierl, 2005). These studies assumed that the basic state was periodic with a constant frequency and amplitude. In this work we study a shear flow similar to that found in Poulin et al. (2003), but now where the magnitude of vorticity is a stochastic variable. We determine that in the case of stochastic shear flows the transient growth of perturbations of the snapshots of the basic state still generate PI.

Strong echo effect and nonlinear transient growth in shear flows

1998 ◽  
Vol 10 (6) ◽  
pp. 1398-1404 ◽  
Author(s):  
J. Vanneste ◽  
P. J. Morrison ◽  
T. Warn
Keyword(s):  
Shear Flows ◽  
Transient Growth ◽  
Nonlinear Transient ◽  
Echo Effect
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