Instability Of Parallel Shear Flows Research Articles

How shear increments affect the flow production branching ratio in CSDX

The coupling of turbulence-driven azimuthal and axial flows in a linear device absent magnetic shear (Controlled Shear Decorrelation Experiment) is investigated. In particular, we examine the apportionment of Reynolds power between azimuthal and axial flows, and how the azimuthal flow shear affects axial flow generation and saturation by drift wave turbulence. We study the response of the energy branching ratio, i.e., ratio of axial and azimuthal Reynolds powers, PzR/PyR, to incremental changes of azimuthal and axial flow shears. We show that increasing azimuthal flow shear decreases the energy branching ratio. When axial flow shear increases, this ratio first increases but then decreases to zero. The axial flow shear saturates below the threshold for parallel shear flow instability. The effects of azimuthal flow shear on the generation and saturation of intrinsic axial flows are analyzed. Azimuthal flow shear slows down the modulational growth of the seed axial flow shear, and thus reduces intrinsic axial flow production. Azimuthal flow shear reduces both the residual Reynolds stress (of axial flow, i.e., ΠxzRes) and turbulent viscosity (χzDW) by the same factor |⟨vy⟩′|−2Δx−2Ln−2ρs2cs2, where Δx is the distance relative to the reference point where ⟨vy⟩=0 in the plasma frame. Therefore, the stationary state axial flow shear is not affected by azimuthal flow shear to leading order since ⟨vz⟩′∼ΠxzRes/χzDW.

Negative viscosity from negative compressibility and axial flow shear stiffness in a straight magnetic field

Negative compressibility ion temperature gradient (ITG) turbulence in a linear plasma device controlled shear de-correlation experiment can induce a negative viscosity increment. However, even with this negative increment, we show that the total axial viscosity remains positive definite, i.e., no intrinsic axial flow can be generated by pure ITG turbulence in a straight magnetic field. This differs from the case of electron drift wave turbulence, where the total viscosity can turn negative, at least transiently. When the flow gradient is steepened by any drive mechanism, so that the parallel shear flow instability (PSFI) exceeds the ITG drive, the flow profile saturates at a level close to the value above which PSFI becomes dominant. This saturated flow gradient exceeds the PSFI linear threshold, and grows with ∇Ti0 as |∇V∥|/|k∥cs|∼|∇Ti0|2/3/(k∥Ti0)2/3. This scaling trend characterizes the effective stiffness of the parallel flow gradient.

On the spectral instability of parallel shear flows

This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number R→∞. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.

Parallel Kelvin-Helmholtz instability in edge plasma

In the scrape-off layer (SOL) of tokamaks, the flow acceleration due to the presence of limiter or divertor plates rises the plasma velocity in a sonic regime. These high velocities imply the presence of a strong shear between the SOL and the core of the plasma that can possibly trigger some parallel shear flow instability. The existence of these instabilities, denoted as parallel Kelvin-Helmholtz instability in some works [1, 2] have been investigated theoretically in [3] using a minimal model of electrostatic turbulence composed of a mass density and parallel velocity equations. This work showed that the edge plasma around limiters might indeed be unstable to this type of parallel shear flow instabilities. In this work, we perform 3D simulations of the same simple mathematical model to validate an original finite volume numerical method aimed to the numerical study of edge plasma. This method combines the use of triangular unstructured meshes in the poloidal section and structured meshes in the toroidal direction and is particularly suited to the representation of the real complex geometry of the vacuum chamber of a tokamak.The numerical results confirm that in agreement with the theoretical expectations as well as with other numerical methods, the sheared flows in the SOL are subject to parallel Kelvin-Helmholtz instabilities. However, the growth rate of these instabilities is low and these computations require both a sufficient spatial resolution and a long simulation time. This makes the simulation of parallel Kelvin-Helmholtz instabilities a demanding benchmark.

Irreversible compressible work and available potential energy dissipation in turbulent stratified fluids

Although it plays a key role in the theory of stratified turbulence, the concept of available potential energy (APE) dissipation has remained until now a rather mysterious quantity, owing to the lack of rigorous result about its irreversible character or energy conversion type. Here, we show by using rigorous energetics considerations rooted in the analysis of the Navier–Stokes for a fully compressible fluid with a nonlinear equation of state that the APE dissipation is an irreversible energy conversion that dissipates kinetic energy into internal energy, exactly as viscous dissipation. These results are established by showing that APE dissipation contributes to the irreversible production of entropy, and by showing that it is a part of the work of expansion/contraction. Our results provide a new interpretation of the entropy budget, that leads to a new exact definition of turbulent effective diffusivity, which generalizes the Osborn–Cox model, as well as a rigorous decomposition of the work of expansion/contraction into reversible and irreversible components. In the context of turbulent mixing associated with parallel shear flow instability, our results suggests that there is no irreversible transfer of horizontal momentum into vertical momentum, as seems to be required when compressible effects are neglected, with potential consequences for the parameterizations of momentum dissipation in the coarse-grained Navier–Stokes equations.

Parallel shear flow instability in the tokamak edge

The transition between the core and scrape-off layer of a tokamak corresponds to a marked momentum shear layer, owing to sheath acceleration on limiters which drives near-sonic flows along the plasma magnetic field in the scrape-off layer, and a parallel shear flow instability can possibly be triggered. The possibility of this instability driven by the velocity gradient is investigated numerically, using a minimum model of particle and parallel momentum transport in the edge of a tokamak, in a computational domain modelling a limiter plasma with background turbulence modelled as an effective diffusion. It is found that unstable regions can exist in the vicinity of a limiter, in agreement with experimental findings, when momentum radial transport – and therefore coupling between SOL and core flows – is sufficiently weak. Instability is reinforced by core rotation, and is found to be maximum downstream of the limiter (with respect to the core plasma flow).

Effects of a poloidally asymmetric ionization source on toroidal drift wave stability and the generation of sheared parallel flow

The effects of a poloidally asymmetric ionization source on both dissipative toroidal drift wave stability and the generation of mean sheared parallel flow are examined. The first part of this work extends the development of a local model of ionization-driven drift wave turbulence [Phys. Fluids B 4, 877 (1992)] to include the effects of magnetic shear and poloidal source asymmetry, as well as poloidal mode coupling due to both magnetic drifts and the source asymmetry. Numerical and analytic investigation confirm that ionization effects can destabilize collisional toroidal drift waves. However, the mode structure is determined primarily by the magnetic drifts, and is not overly effected by the poloidal source asymmetry. The ionization source drives a purely inward particle flux, which can explain the anomalously rapid uptake of particles which occurs in response to gas puffing. In the second part of this work, the role poloidal asymmetries in both the source and turbulent particle diffusion play in the generation of sheared mean parallel flow is examined. Analysis indicates that predictions of sonic parallel shear flow [v∥(r)∼cs] are an unphysical result of the assumption of purely parallel flow (i.e., v⊥=0) and the neglect of turbulent parallel momentum transport. Results indicate that the flow produced is subcritical to the parallel shear flow instability when diamagnetic effects are properly considered.

Three-dimensional linear instability of parallel shear flows

The three-dimensional linear instability of parallel shear flows is investigated through simple geometrical constructions based on the classical Squire transformation. It is shown that if a profile is unstable for a finite Reynolds number and some value of (total) horizontal wavenumber, this profile is also unstable for all larger Reynolds numbers and the same wavenumber. In particular, the direction of the unstable mode tends toward the perpendicular as the Reynolds number increases, thus providing the viscous/shear balance required for resistive instability. The example of plane Poiseuille flow is discussed in detail, and the results compared with classical viscous and inviscid stability criteria.

Instability and wave over-reflection in stably stratified shear flow

We reexamine the related problems of instability of parallel shear flows and over-reflection of internal waves at a critical level, concentrating on the stratified case. Our primary aim is to delineate the specific aspects of a flow that permit overreflection and instability. A related and partly realized aim is to develop a mechanistic ‘picture’ of how over-reflection and instability work. In the course of this study we have also uncovered some new results concerning the instability of stratified shear flows – showing how regions of enhanced static stability and enhanced damping can destabilize otherwise stable flows.For the scattering of steady plane waves, we show that, of the conditions found by Lindzen & Tung (1978), in the unstratified case, only the existence of wave-propagation regions above and below the critical level is always necessary for over-reflection (at least in the absence of damping), although a trapping region around the critical level and a reflecting surface bounding the upper wave region may play crucial roles in some cases. Our results suggest that the role of the upper wave region may be to allow a wave flux through the critical level. Moreover, we show numerically that the effect of an upper wave region can be mimicked by a region of localized damping which leads to over-reflection as well.We also consider an initial-value problem, using numerical methods. When a wave is incident on the incident level, the reflection and transmission coefficients grow smoothly to their final values. The rate of growth depends on the flow parameters, but there is some evidence to suggest there is a characteristic timescale involved that depends only on the shear (and not on wave travel time). This fits a mechanistic picture of over-reflection and instability that we describe, in which the essential part is a kinematic interaction between wave and mean flow at the critical level, depending only on shear.

The inviscid nonlinear instability of parallel shear flows

In this paper we assume the existence of a nonlinear boundary layer centred on the critical point, and explore its effect on the development of unstable parallel shear flows. A velocity matching condition derived in a qualitative discussion suggests a growth of harmonics which differs from that predicted by previous theories; however, the prediction is in excellent agreement with experimental data. A hyperbolic-tangent velocity profile, subjected to perturbations with wavenumbers and frequencies close to marginal values, is then chosen as a mathematical model of the nonlinear development, both temporal and spatial instability growth being considered.A singularity in the analysis which has been treated in previous theories by the introduction of viscosity is dealt with in the present work by the introduction of a growth boundary layer. The asymptotics are non-uniform and the time-dependent solution does not resemble the steady viscous solutions, even as the growth rate tends to zero. The theory suggests that the instability will develop as a series of temporally growing spiral vortices, a description differing from that of a cat's-eye pattern predicted by existing theories, but in accord with experimental and field observations.

Finite-amplitude instability of parallel shear flows

A formal expansion method for analysis of the non-linear development of an oblique wave in a parallel flow is presented. The present approach constitutes an extension and modification of the method of Stuart and Watson. Results are obtained for plane Poiseuille flow, and for a combination of plane Poiseuille and plane Couette flow. The Poiseuille flow exhibits finite-amplitude subcritical instability, and relatively weak but finite disturbances markedly reduce the critical Reynolds number. The combined flow, which becomes stable to infinitesimal disturbances at all Reynolds numbers when the Couette component is sufficiently great, remains unstable to finite disturbances.