A mostly linear model of transition to turbulence

Physics of Plasmas, 2000

The linear properties of an electromagnetic drift-wave model are examined. The linear system is non-normal in that its eigenvectors are not orthogonal with respect to the energy inner product. The non-normality of the linear evolution operator can lead to enhanced finite-time growth rates compared to modal growth rates. Previous work with an electrostatic drift-wave model found that nonmodal behavior is important in the hydrodynamic limit. Here, similar behavior is seen in the hydrodynamic regime even with the addition of magnetic fluctuations. However, unlike the results for the electrostatic drift-wave model, nonmodal behavior is also important in the adiabatic regime with moderate to strong magnetic fluctuations.

Physical Review E, 1998

The modal and nonmodal linear properties of the Hasegawa-Wakatani system are examined. This linear model for plasma drift waves is nonnormal in the sense of not having a complete set of orthogonal eigenvectors. A consequence of nonnormality is that finite-time nonmodal growth rates can be larger than modal growth rates. In this system, the nonmodal time-dependent behavior depends strongly on the adiabatic parameter and the time scale of interest. For small values of the adiabatic parameter and short time scales, the nonmodal growth rates, wave number, and phase shifts ͑between the density and potential fluctuations͒ are time dependent and differ from those obtained by normal mode analysis. On a given time scale, when the adiabatic parameter is less than a critical value, the drift waves are dominated by nonmodal effects while for values of the adiabatic parameter greater than the critical value, the behavior is that given by normal mode analysis. The critical adiabatic parameter decreases with time and modal behavior eventually dominates. The nonmodal linear properties of the Hasegawa-Wakatani system may help to explain features of the full system previously attributed to nonlinearity.

Physics of Plasmas, 2004

Recently it was found ͓Poedts et al., Phys. Plasmas 7, 3204 ͑2000͔͒ that dusty plasma flows host nonperiodic modes-shear-dust-acoustic ͑SDA͒ vortices. These modes, interlaced with dust-acoustic ͑DA͒ waves, are able to exchange energy with the ambient flow. In this paper it is studied how these processes evolve in differentially rotating and self-gravitating flows of dusty plasmas. It is found that the presence of the self-gravity and of Coriolis forces makes both SDA vortices and DA waves transiently unstable. It is argued that the transient shear instability could be important for the formation of the fine structure of planetary rings, for the dynamics of charged dust masses and transition to dust-acoustic turbulence in galactic gaseous disks.

Physical Review Letters, 2012

Sheared toroidal flows can cause bifurcations to zero-turbulent-transport states in tokamak plasmas. The maximum temperature gradients that can be reached are limited by subcritical turbulence driven by the parallel velocity gradient. Here it is shown that q/ǫ (magnetic field pitch/inverse aspect ratio) is a critical control parameter for sheared tokamak turbulence. By reducing q/ǫ, far higher temperature gradients can be achieved without triggering turbulence, in some instances comparable to those found experimentally in transport barriers. The zero-turbulence manifold is mapped out, in the zero-magnetic-shear limit, over the parameter space (γE, q/ǫ, R/LT ), where γE is the perpendicular flow shear and R/LT is the normalised inverse temperature gradient scale. The extent to which it can be constructed from linear theory is discussed.

Physics of Plasmas, 2011

First-principles numerical simulations are used to describe a transport bifurcation in a differentially rotating tokamak plasma. Such a bifurcation is more probable in a region of zero magnetic shear than one of finite magnetic shear because in the former case the component of the sheared toroidal flow that is perpendicular to the magnetic field has the strongest suppressing effect on the turbulence. In the zero-magnetic-shear regime, there are no growing linear eigenmodes at any finite value of flow shear. However, subcritical turbulence can be sustained, owing to the existence of modes, driven by the ion temperature gradient and the parallel velocity gradient, which grow transiently. Nonetheless, in a parameter space containing a wide range of temperature gradients and velocity shears, there is a sizeable window where all turbulence is suppressed. Combined with the relatively low transport of momentum by collisional (neoclassical) mechanisms, this produces the conditions for a bifurcation from low to high temperature and velocity gradients. A parametric model is constructed which accurately describes the combined effect of the temperature gradient and the flow gradient over a wide range of their values. Using this parametric model, it is shown that in this reduced-transport state, heat is transported almost neoclassically, while momentum transport is dominated by subcritical PVG turbulence. It is further shown that for any given input of torque, there is an optimum input of heat which maximises the temperature gradient. The parametric model describes both the behaviour of the subcritical turbulence (which cannot be modelled by the quasi-linear methods used in current transport codes) and the complicated effect of the flow shear on the transport stiffness. It may prove useful for transport modelling of tokamaks with sheared flows.

Physics of Fluids, 2008

The role of non-normality and nonlinearity in thermoacoustic interaction in a Rijke tube is investigated in this paper. The heat release rate of the heating element is modeled by a modified form of King's law. This fluctuating heat release from the heating element is treated as a compact source in the one-dimensional linear model of the acoustic field. The temporal evolution of the acoustic perturbations is studied using the Galerkin technique. It is shown that any thermoacoustic system is non-normal. Non-normality can cause algebraic growth of oscillations for a short time even though the eigenvectors of the system could be decaying exponentially with time. This feature of non-normality combined with the effect of nonlinearity causes the occurrence of triggering, i.e., the thermoacoustic oscillations decay for some initial conditions whereas they grow for some other initial conditions. If a system is non-normal, then there can be large amplification of oscillations even if the excited frequency is far from the natural frequency of the system. The dependence of transient growth on time lag and heater positions is studied. Such amplifications ͑pseudoresonance͒ can be studied using pseudospectra, as eigenvalues alone are not sufficient to predict the behavior of the system. The geometry of pseudospectra can be used to obtain upper and lower bounds on the growth factor, which provide both necessary and sufficient conditions for the stability of a thermoacoustic system.

Astronomy and Astrophysics, 2008

Aims. We present a qualitative analysis of key (but yet unappreciated) linear phenomena in stratified hydrodynamic Keplerian flows: (i) the occurrence of a vortex mode, as a consequence of strato-rotational balance, with its transient dynamics; (ii) the generation of spiral-density waves (also called inertia-gravity or gΩ waves) by the vortex mode through linear mode coupling in shear flows. Methods. Non-modal analysis of linearized Boussinesq equations were written in the shearing sheet approximation of accretion disk flows. Results. It is shown that the combined action of rotation and stratification introduces a new degree of freedom, vortex mode perturbation, which is in turn linearly coupled with the spiral-density waves. These two modes are jointly able to extract energy from the background flow, and they govern the disk dynamics in the small-scale range. The transient behavior of these modes is determined by the non-normality of the Keplerian shear flow. Tightly leading vortex mode perturbations undergo substantial transient growth, then, becoming trailing, inevitably generate trailing spiral-density waves by linear mode coupling. This course of events-transient growth plus coupling-is particularly pronounced for perturbation harmonics with comparable azimuthal and vertical scales, and it renders the energy dynamics similar to the 3D unbounded plane Couette flow case. Conclusions. Our investigation strongly suggests that the so-called bypass concept of turbulence, which has been recently developed by the hydrodynamic community for spectrally stable shear flows, can also be applied to Keplerian disks. This conjecture may be confirmed by appropriate numerical simulations that take the vertical stratification and consequent mode coupling into account in the high Reynolds number regime.

Physical Review Letters, 1997

Physics of Plasmas, 1997

Evolution of three-dimensional magnetohydrodynamic ͑MHD͒ waves ͓fast magnetosonic ͑FMW͒, slow magnetosonic ͑SMW͒ and Alfvén waves͔ is studied in unbounded parallel flows with uniform shear of velocity and uniform magnetic field directed along the flow. The energy exchange between the MHD waves and background flow is explored. This process is noticeably different for each type of wave and is characterized by the unusual ͑algebraic͒ behavior of the linear amplification processes. Another novelty is shown in the wave linear evolution process-the coupling of MHD waves and their mutual transformations are originated in a limited time interval for a wide range of systems ͑flow and waves͒ parameters. Significant transformation of Alfvén waves into FMW may take place ͑depending on the parameters of the system͒ if the former has been initially generated in shear flow. It is possible to reveal these results by employing the nonmodal linear approach which has been extensively used in the study of evolution of perturbations in shear flows since the beginning of the 1990s. The change in the understanding of flow turbulence due to the coupling of the MHD modes is discussed. Namely, the usual consideration of just Alfvén wave turbulence in some astrophysical flows is not always sufficient for a complete analysis-not only should Alfvén waves be ''ingredients'' of turbulence, but magnetosonic waves, as well. In this case MHD turbulence should be of a ''mixed'' type.

Physical Review E, 2010

Recent studies have suggested that in some cases transition can be triggered by some purely nonlinear mechanisms. Here we aim at verifying such an hypothesis, looking for a localized perturbation able to lead a boundary-layer flow to a chaotic state, following a nonlinear route. Nonlinear optimal localized perturbations have been computed by means of an energy optimization which includes the nonlinear terms of the Navier-Stokes equations. Such perturbations lie on the turbulent side of the laminar-turbulent boundary, whereas, for the same value of the initial energy, their linear counterparts do not. The evolution of these perturbations toward a turbulent flow involves the presence of streamwise-inclined vortices at short times and of hairpin structures prior to breakdown.

Astronomy & Astrophysics, 2004

We present here both analytical and numerical results of hydrodynamic stability investigations of rotationally supported circumstellar flows using the shearing box formalism. Asymptotic scaling arguments justifying the shearing box approximation are systematically derived, showing that there exist two limits which we call small shearing box (SSB) and large shearing box (LSB). The physical meaning of these two limits and their relationship to model equations implemented by previous investigators are discussed briefly. Two dimensional (2D) dynamics of the SSB are explored and shown to contain transiently growing (TG) linear modes, whose nature is first discussed within the context of linear theory. The fully nonlinear regime in 2D is investigated numerically for very high Reynolds (Re) numbers. Solutions exhibiting long-term dynamical activity are found and manifest episodic but recurrent TG behavior and these are associated with the formation and long-term survival of coherent vortices. The lifetime of this spatio-temporal complexity depends on the Re number and the strength and nature of the initial disturbance. The dynamical activity in finite Re solutions ultimately decays with a characteristic time increasing with Re. However, for large enough Re and appropriate initial perturbation, a large number of TG episodes recur before any viscous decay begins to clearly manifest itself. In cases where Re = ∞ nominally (i.e. any dissipation resulting only from numerical truncation errors), the dynamical activity persists for the entire duration of the simulation (hundreds of box orbits). Because the SSB approximation used here is equivalent to a 2D incompressible flow, the dynamics can not depend on the Coriolis force. Therefore, three dimensional (3D) simulations are needed in order to decide if this force indeed suppresses nonlinear hydrodynamical instability in rotationally supported disks in the shearing box approximation, and if recurrent TG behavior can still persist in three dimensions as well-possibly giving rise to a subcritical transition to long-term spatio-temporal complexity.

Theoretical and Computational Fluid Dynamics, 2015

The counter-propagating Rossby wave perspective to shear flow instability is extended here to the weakly nonlinear phase. The nonlinear action at a distance interaction mechanism between a pair of waves is identified and separated from the linear one. In the former, the streamwise velocity converges the far-field vorticity anomaly of the opposed wave, whereas in the latter, the cross-stream velocity advects the far-field mean vorticity. A truncated analytical model of two vorticity interfaces shows that higher harmonics generated by the nonlinear interaction act as a forcing on the nonnormal linear dynamics. Furthermore, an intrinsic positive feedback toward small-scale enstrophy results from the fact that higher harmonic pair of waves are generated in anti-phase configuration which is favored for nonnormal growth. Near marginal stability, the waves preserve their structure and numerical simulations of the weakly nonlinear interaction show wave saturation into finite amplitudes, in good agreement both with the fixed point solution of the truncated model, as well as with its corresponding weakly nonlinear Ginzburg-Landau amplitude equation. Keywords Counter-propagating Rossby waves • Linear nonnormal and nonlinear interactions • Geophysical fluid dynamics 1 Introduction The counter-propagating Rossby wave (CRW) is a concept that successfully rationalizes many aspects of linearized dynamics of shear flows. It was originally presented by Bretherton [1] in the context of atmospheric baroclinic instability, and after reviewed by Hoskins et al. [2], it received considerable attention as a conceptual tool to explain both baroclinic and barotropic instabilities (e.g., [3-11]). The essence of the Rossby wave counter-propagation mechanism and the modal and optimal nonmodal (nonnormal) CRW interaction are explained in [12] (hereafter HM05) and is summarized in Figs. 1 and 2 here. Communicated by W. Dewar.

Physics of Plasmas, 2006

The nonlinear dynamics of coherent circular/elliptical cyclonic and anticyclonic vortices in plane flow with constant shear is investigated numerically using a dealiased Fourier pseudospectral code. The flow is asymptotically linearly stable, but is highly non-normal, allowing perturbations to gain energy transiently from the background shear flow. This linear transient growth interplays with nonlinear processes. In certain cases it is shown that the nonlinear feedback is positive, leading to self-sustaining coherent vortices. Self-sustaining coherent vortices exist where the vorticity is parallel to the mean flow vorticity ͑cyclonic rotation͒. The required nonlinear feedback is absent for small amplitude anticyclonic vortices. However, elliptical anticyclonic vortices become self-sustaining if the amplitude exceeds a threshold value. The self-sustaining of coherent vortices is similar to the subcritical, so-called bypass, transition to turbulence in shear flows. The common features are: transient linear growth; positive nonlinear feedback; and anisotropy of the linear and nonlinear phenomena ͑in contrast to isotropic Kolmogorov turbulence͒. A plasma laboratory experiment is suggested based on the results of this investigation.

Physics of Plasmas, 2004

Special features of magnetohydrodynamic waves linear dynamics in smooth shear flows are studied. Quantitative asymptotic and numerical analysis are performed for wide range of system parameters when basic flow has constant shear of velocity and uniform magnetic field is parallel to the basic flow. The special features consist of magnetohydrodynamic wave mutual transformation and over-reflection phenomena. The transformation takes place for arbitrary shear rates and involves all magnetohydrodynamic wave modes. While the over-reflection occurs only for slow magnetosonic and Alfvén waves at high shear rates. Studied phenomena should be decisive in the elaboration of the self-sustaining model of magnetohydrodynamic turbulence in the shear flows.

Physics of Plasmas, 2016

Our goal is to gain new insight into the physics of wave dynamics in ionospheric zonal shear flows. We study the shear flow non-normality induced linear coupling of planetary scale (slow) modified Rossby waves and westward propagating fast magnetized (Khantadze) waves using an approach different from the existing one to the linear wave dynamics. The performed analysis allows us to separate from each other different physical processes, grasp their interplay, and, by this way, construct the basic physics of the linear coupling of the slow and fast waves in an ionospheric zonal flow with linear shear of mean velocity, U0=(Sy,0). It should be noted from the beginning that we consider incompressible flow and the classified “slow” and “fast” waves are not connected with the similarly labeled magnetosonic waves in compressible heliosphere. We show that: the modified Rossby waves generate fast magnetized waves due to the coupling for a quite wide range of ionospheric and shear flow paramete...

Journal of Experimental and Theoretical Physics, 1997

The incompressible fluctuation background in laminar shear flows with a smooth velocity profile is investigated. Concrete calculations are performed for parallel Couette flow using nonmodal analysis of the linear dynamics of the disturbances. Nonmodal analysis makes it possible to grasp phenomena that could not be grasped in the early investigations, and thereby makes it possible to represent the fluctuation background in a completely new light: In incompressible shear flows the spatial spectral energy density of the fluctuation background is anisotropic, and furthermore in certain regions of wave-number space it is higher than that of the thermal noise. It is also shown that in the stationary state of the nonequilibrium system studied there exists a new, indirect channel for thermalization of the energy of the mean flowenergy is constantly transferred from the mean flow into the spatial Fourier harmonics of vortex pertubations and ultimately into heat. Possible manifestations of the fluctuation background described in this paper are listed.

Physical Review E, 1996

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. ͓S1063-651X͑96͒11605-6͔

Physical review. E, 2016

To understand the mechanism of the self-sustenance of subcritical turbulence in spectrally stable (constant) shear flows, we performed direct numerical simulations of homogeneous shear turbulence for different aspect ratios of the flow domain with subsequent analysis of the dynamical processes in spectral or Fourier space. There are no exponentially growing modes in such flows and the turbulence is energetically supported only by the linear growth of Fourier harmonics of perturbations due to the shear flow non-normality. This non-normality-induced growth, also known as nonmodal growth, is anisotropic in spectral space, which, in turn, leads to anisotropy of nonlinear processes in this space. As a result, a transverse (angular) redistribution of harmonics in Fourier space is the main nonlinear process in these flows, rather than direct or inverse cascades. We refer to this type of nonlinear redistribution as the nonlinear transverse cascade. It is demonstrated that the turbulence is ...

International Journal of Spray and Combustion Dynamics, 2016