**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Instability

Summary

In dynamical systems instability means that some of the outputs or internal states increase with time, without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.
In structural engineering, a structural beam or column can become unstable when excessive compressive load is applied. Beyond a certain threshold, structural deflections magnify stresses, which in turn increases deflections. This can take the form of buckling or crippling. The general field of study is called structural stability.
Atmospheric instability is a major component of all weather systems on Earth.
Instability in control systems
Stabilizability
In the theory of dynamical systems, a state variable in a system is said to be unstable if it evolves without bounds. A system itself is said to be unstable if at least one of its state variables is unstable.
In continuous time control theory, a system is unstable if any of the roots o

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related concepts (10)

Plasma (physics)

Plasma () is one of four fundamental states of matter, characterized by the presence of a significant portion of charged particles in any combination of ions or electrons. It is the most abundant form

Rayleigh–Taylor instability

The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an interface between two fluids of different densities which occurs when the lighter f

Kelvin–Helmholtz instability

The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference ac

Related courses (54)

Related publications (100)

ME-435: Aeroelasticity and fluid-structure interaction

La réponse statique et dynamique des systèmes couplés fluide-structure résultant de l'excitation indépendante, l'excitation induite par le mouvement, et l'excitation induite par les instabilités des fluides.

PHYS-423: Plasma I

Following an introduction of the main plasma properties, the fundamental concepts of the fluid and kinetic theory of plasmas are introduced. Applications concerning laboratory, space, and astrophysical plasmas are discussed throughout the course.

ME-466: Instability

This course focuses on the physical mechanisms at the origin of the transition of a flow from laminar to turbulent using the hydrodynamic instability theory.

Loading

Loading

Loading

Related people (103)

Related units (29)

Related lectures (129)

Microturbulence driven by plasma instabilities is in most cases the dominant cause of heat and particle loss from the core of magnetic confinement fusion devices and therefore presents a major challenge in achieving burning plasma conditions. The role of passing electron dynamics in turbulent transport driven by ion-scale microinstabilities, in particular Ion Temperature Gradient (ITG) and Trapped Electron Mode (TEM) instabilities, has been given relatively little attention. In first approximation, these particles, which are highly mobile along the confining magnetic field, are assumed to respond adiabatically to the low frequency ion-scale modes. However, near mode rational surfaces (MRSs), the non-adiabatic response of passing electrons becomes important and can no longer be neglected.
This non-adiabatic electron response actually has a destabilising effect and leads to generation of fine-structures located at the MRSs of each eigenmode. This thesis focuses on the effects of non-adiabatic response of passing electrons in tokamak core turbulence.
One such effect of non-adiabatic passing electrons that is of particular interest to this work is the self-interaction mechanism. It is essentially a process by which a microinstability eigenmode that is extended along the direction parallel to the magnetic field interacts non-linearly with itself, in turn generating E x B zonal flows. Unlike the usual picture of zonal flow drive in which microinstability eigenmodes coherently amplify the flow via modulational instabilities, the self-interaction drive of zonal flows from these eigenmodes are uncorrelated with each other. In the case of ITG driven turbulence, using novel statistical diagnostic methods, it is shown that the associated shearing rate of the fluctuating zonal flows therefore reduces as more toroidal modes are resolved in the simulation. In simulations accounting for the full toroidal domain, such an increase in the density of toroidal modes corresponds in fact to an increase in the system size, leading to a finite system size effect that is distinct from the other better known system size effects such as profile shearing or finite radial extend of the unstable region.
The study of non-adiabatic passing electron dynamics is pursued further to include more reactor relevant conditions such as collisions and background shear flow. It is found that, with increasing collisionality, electrons behave more adiabatic-like, especially the trapped electrons away from MRSs, thereby leading to a decrease in the growth rate of ITG eigenmodes. Furthermore, the shortened electron mean free path in presence of collisions leads to a radial broadening of the fine-structures at the MRS of corresponding eigenmodes. In nonlinear simulations, the turbulent flux levels decrease with increasing collisionality, as a result of the reduced drive from the less unstable ITG eigenmodes. The radial width of the fine structures at MRSs is found to reduce with increasing collisionality as a result of reduced nonlinear modification of the eigenmodes in turbulence simulations. A study of the effect of collisions on the self-interaction mechanism reveals that for physically relevant values of collisionality, the effect of self-interaction is still significant. A preliminary study of the effect of background E x B flow shear shows that the fine-structures associated with the non-adiabatic passing electron response persist even with finite background flow.

To achieve reactor-relevant conditions in a tokamak plasma, auxiliary heating systems are required and can be realized by waves injected in the plasma that heat ions or electrons under certain conditions. Electron cyclotron resonant heating (ECRH) is a very flexible and robust technique featuring localized power deposition and current drive (CD) capabilities. Its fundamental principles such as damping on the cyclotron resonance are well understood and the application of ECRH is a proven and established tool; electron cyclotron current drive (ECCD) is regularly used to develop advanced scenarios and control magnetohydrodynamics (MHD) instabilities in the plasma by tailoring the current profile. There remain important open questions, such as the phase space dynamics, the observed radial broadening of the suprathermal electron distribution function (e.d.f.) and discrepancies in predicted and experimental CD efficiency. These are addressed in this thesis. One of its main goals is indeed to improve the understanding of wave-particle interaction in plasmas and current drive mechanisms. This was accomplished by combined experimental and numerical studies, strongly based on the conjunction of hard X-ray (HXR) bremsstrahlung measurements and Fokker-Planck modeling, characterizing the suprathermal electron population. The hard X-ray tomographic spectrometer (HXRS) diagnostic was purposely developed to perform these studies, in particular by investigating spatial HXR emission asymmetries in the co- and counter-current directions and within the poloidal plane. The system uses cadmium-telluride (CdTe) detectors and digital acquisition to store the complete time history of incoming photon pulses. An extensive study of digital pulse processing algorithms was performed and its consequent application allows the HXRS to handle high count rates in a noisy tokamak environment. Numerous other numerical tools were developed in the course of this thesis, among others to improve the time resolution by conditional averaging and to obtain local information with the general tomographic inversion (GTI) package. The interfaces of the comparatively new LUKE code and well-established CQL3D Fokker-Planck (F-P) code to the tokamak à configuration variable (TCV) data were refurbished and a detailed benchmarking of these two codes was performed for the first time. Indeed, the theory-predicted toroidal and poloidal emission asymmetries could be consistently verified by experiment and modeling in many cases, including scans of a variety of plasma and wave parameters. The effects of supra-thermal electron diffusion and radio frequency (RF) wave scattering, both resulting in a radial broadening of the HXR emission, were separated by a poloidal deposition location angle scan. Furthermore, previous results on anomalous diffusion and CD efficiency were reproduced with increased confidence arising from enhanced diagnostic specifications. The plasma response to electron cyclotron (EC) absorption and the role of quasi-linear effects were investigated using the coherent averaging capabilities of the HXRS. Several MHD instabilities can occur in the plasma center and better understanding of these modes and events is indispensable for their mitigation in order to prevent their negative effects on confinement and stability. Sawtooth crashes are such a major instability and localized at the q=1 surface. They can be described as the evolution of an internal m=1 kink mode leading to [...]

Open flows, such as wakes, jets, separation bubbles, mixing layers, boundary layers, etc., develop in domains where fluid particles are continuously advected downstream. They are encountered in a wide variety of situations, ranging from nature to technology. Such configurations are characterised by the development of strong instabilities resulting in observable unsteady dynamics. They can be categorised as oscillators which present intrinsic dynamics through self-sustained oscillations, or as amplifiers, which exhibit a strong sensitivity to external disturbances through extrinsic dynamics. Over the years, different linear and nonlinear approaches have been adopted to describe the dynamics of oscillators and amplifiers. However, a simplified physical description that accurately accounts for the nonlinear saturation of instabilities in oscillators as well as that of the response to disturbances in stable amplifier flows is still missing. In this thesis, this question is addressed by introducing a self-consistent semi-linear model. The model is formally constructed by a set of equations where the mean flow is coupled to a linear perturbation equation through the Reynolds stress. The full nonlinear fluctuating motion is thus approximated by a linear equation. The nonlinear dynamics of oscillators is studied in the cylinder wake, where the most unstable eigenmode of finite amplitude is coupled to the instantaneous mean flow for different oscillation amplitudes. This family of solutions provides an instantaneous mean flow evolution as a function of an equivalent slow time. A transient physical picture is formalised, wherein a harmonic perturbation grows and changes the amplitude, frequency, growth-rate and structure due to the modification of the instantaneous mean flow by the Reynolds stress forcing. Eventually this perturbation saturates when the flow is marginally stable. In contrast to standard linear stability analysis around the mean flow, the iterative solution of the model provides a priori an accurate prediction of the instantaneous amplitude, frequency and growth rate, as well as the flow fields, without resorting to any input from numerical or experimental data. Regarding noise amplifiers, the nonlinear saturation of the large linear amplification to external disturbances is studied in the framework of the receptivity analysis of the backward facing step flow. The self-consistent model is first introduced for harmonic forcing and later generalised to stochastic forcing by reformulating it conveniently in frequency domain. The results show an accurate prediction of the response energy as well as the flow fields. Hence, a similar picture is revealed, wherein the Reynolds stress dominates the saturation process. Despite the difference in the dynamics of the described flows, they share the same nonlinear saturation mechanism: the mean flow distortion.