Recent advances in the stability theory of toroidal plasmas

1981 ◽  
Vol 300 (1456) ◽  
pp. 559-567
Keyword(s):  
Stability Theory ◽  
Short Wavelength ◽  
Linear Stability Theory ◽  
Long Wavelength ◽  
Toroidal Plasmas ◽  
Toroidal Plasma ◽  
Magnetically Confined ◽  
Eikonal Representation ◽  
The Stability ◽  
The Magnetic Field

Many of the most persistent instabilities of a magnetically confined plasma have short wavelength perpendicular to the magnetic field but long wavelength parallel to it. Such instabilities are difficult to treat in a toroidal system because the simple eikonal representation of short wavelength oscillations X (r) = Y (r) with 8 1 proves to be incompatible with the other requirements of toroidal periodicity and long parallel wavelength (which would require B >VS = 0). A new method of representing perturbations in a torus will be outlined. By using this, the two-dimensional stability problem posed by an axisymmetric toroidal equilibrium can be reduced to that of solving a one-dimensional eigenvalue equation. This technique essentially completes the linear stability theory of magnetohydrodynamic modes in a toroidal plasma, and is also applicable to the investigation of micro-instabilities that are described by the Vlasov-Maxwell equations.

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

1976 ◽  
Vol 78 (2) ◽  
pp. 355-383 ◽  
Author(s):  
H. Fasel
Keyword(s):  
Finite Difference ◽  
Stability Theory ◽  
Stokes Equations ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Small Periodic

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

Stability of the Prandtl model for katabatic slope flows

2019 ◽  
Vol 865 ◽  
Author(s):  
Cheng-Nian Xiao ◽  
Inanc Senocak
Keyword(s):  
Linear Stability ◽  
Stability Theory ◽  
Slope Angle ◽  
Transverse Mode ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Vortical Flow ◽  
Longitudinal Modes ◽  
Prandtl Model ◽  
The Stability

We investigate the stability of the Prandtl model for katabatic slope flows using both linear stability theory and direct numerical simulations. Starting from Prandtl’s analytical solution for uniformly cooled laminar slope flows, we use linear stability theory to identify the onset of instability and features of the most unstable modes. Our results show that the Prandtl model for parallel katabatic slope flows is prone to transverse and longitudinal modes of instability. The transverse mode of instability manifests itself as stationary vortical flow structures aligned in the along-slope direction, whereas the longitudinal mode of instability emerges as waves propagating in the base-flow direction. Beyond the stability limits, these two modes of instability coexist and form a complex flow structure crisscrossing the plane of flow. The emergence of a particular form of these instabilities depends strongly on three dimensionless parameters, which are the slope angle, the Prandtl number and a newly introduced stratification perturbation parameter, which is proportional to the relative importance of the disturbance to the background stratification due to the imposed surface buoyancy flux. We demonstrate that when this parameter is sufficiently large, then the stabilising effect of the background stratification can be overcome. For shallow slopes, the transverse mode of instability emerges despite meeting the Miles–Howard stability criterion of $Ri>0.25$. At steep slope angles, slope flow can remain linearly stable despite attaining Richardson numbers as low as $3\times 10^{-3}$.

First-principle study of the electronic structure and optical property of Nb-doped anatase TiO2

2014 ◽  
Vol 28 (17) ◽  
pp. 1450091
Author(s):  
Q. Y. Hou ◽  
Q. L. Liu ◽  
C. W. Zhao ◽  
Y. Zhang
Keyword(s):  
Electronic Structure ◽  
Optical Property ◽  
Band Gap ◽  
Absorption Edge ◽  
Short Wavelength ◽  
Density Functional ◽  
Functional Theory ◽  
Long Wavelength ◽  
The Density Functional Theory ◽  
The Stability

The absorption edge shifted to long wavelength direction and short wavelength direction of two opposite experimental conclusions have been reported, when the band-gap and absorption spectra of Nb -doped anatase TiO 2 were studied. In order to solve this contradiction, the electronic structure and the optical property of Nb heavy doped anatase TiO 2 have been studied by the first-principles plane-wave ultrasoft pseudopotential method based on the density functional theory with +U method modification. The calculated results indicate that the higher the Nb -doping is, the higher the total energy is, the worse the stability is, the higher the formation energy is, the more difficult the doping is, the wider the optical band-gap is, the more obvious the absorption edge shifting to short wavelength direction is, the lower the absorptivity and the reflectivity is, which is in agreement with the experimental results. The reasonable interpretation of the contradiction has been reported in this paper, too.

Linear stability theory of oscillatory Stokes layers

1974 ◽  
Vol 62 (4) ◽  
pp. 753-773 ◽  
Author(s):  
Christian Von Kerczek ◽  
Stephen H. Davis
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Stability Theory ◽  
Absolute Stability ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Full Time ◽  
Full Theory ◽  
The Stability

The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to bemore stablethan that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed.

scholarly journals Effect of helical coils on the magnetic field and the stability of a compact toroidal plasma

2000 ◽  
Vol 42 (10) ◽  
pp. 1105-1121 ◽  
Author(s):  
W A Cooper ◽  
T N Todd ◽  
S Allfrey ◽  
T C Hender ◽  
D C Robinson
Keyword(s):  
Magnetic Field ◽  
Toroidal Plasma ◽  
Helical Coils ◽  
The Stability ◽  
The Magnetic Field

Secondary instability of crossflow vortices: validation of the stability theory by direct numerical simulation

2007 ◽  
Vol 583 ◽  
pp. 229-272 ◽  
Author(s):  
GIUSEPPE BONFIGLI ◽  
MARKUS KLOKER
Keyword(s):  
Stability Theory ◽  
Three Dimensional ◽  
Amplitude Distribution ◽  
Detailed Comparison ◽  
Secondary Instability ◽  
Linear Stability Theory ◽  
Local Maxima ◽  
Instability Modes ◽  
The Stability ◽  
Secondary Instabilities

Detailed comparison of spatial direct numerical simulations (DNS) and secondary linear stability theory (SLST) is provided for the three-dimensional crossflow-dominated boundary layer also considered at the DLR-Göttingen for experiments and theory. Secondary instabilities of large-amplitude steady and unsteady crossflow vortices arising from one single primary mode have been analysed. SLST results have been found to be reliable with respect to the dispersion relation and the amplitude distribution of the modal eigenfunction in the crosscut plane. However, significant deviations have been found in the amplification rates, the SLST results being strongly dependent on the necessarily simplified representation of the primary state. The secondary instability mechanisms are shown to be local, i.e. robust with respect to violations of the periodicity assumption made in the SLST for the wall-parallel directions. Perturbations associated with different local maxima of the spanwise periodic eigenfunctions develop independently from each other interacting only with the primary vortices next to them. Characteristic structures induced by different secondary instability modes have been analysed and an analogy with the Kelvin–Helmholtz instability mechanism has been highlighted.

scholarly journals Gyrokinetic stability theory of electron–positron plasmas

2016 ◽  
Vol 82 (3) ◽  
Author(s):  
P. Helander ◽  
J. W. Connor
Keyword(s):  
Laboratory Experiments ◽  
Particle Flux ◽  
Debye Length ◽  
Closed Field ◽  
Stability Threshold ◽  
Electron Positron ◽  
Magnetically Confined ◽  
Closed Field Line ◽  
The Stability ◽  
The Magnetic Field

The linear gyrokinetic stability properties of magnetically confined electron–positron plasmas are investigated in the parameter regime most likely to be relevant for the first laboratory experiments involving such plasmas, where the density is small enough that collisions can be ignored and the Debye length substantially exceeds the gyroradius. Although the plasma beta is very small, electromagnetic effects are retained, but magnetic compressibility can be neglected. The work of a previous publication (Helander, Phys. Rev. Lett., vol. 113, 2014a, 135003) is thus extended to include electromagnetic instabilities, which are of importance in closed-field-line configurations, where such instabilities can occur at arbitrarily low pressure. It is found that gyrokinetic instabilities are completely absent if the magnetic field is homogeneous: any instability must involve magnetic curvature or shear. Furthermore, in dipole magnetic fields, the stability threshold for interchange modes with wavelengths exceeding the Debye radius coincides with that in ideal magnetohydrodynamics. Above this threshold, the quasilinear particle flux is directed inward if the temperature gradient is sufficiently large, leading to spontaneous peaking of the density profile.

scholarly journals Strongly driven surface-global kinetic ballooning modes in general toroidal geometry

2018 ◽  
Vol 84 (3) ◽  
Author(s):  
A. Zocco ◽  
K. Aleynikova ◽  
P. Xanthopoulos
Keyword(s):  
Magnetic Flux ◽  
Flux Tube ◽  
Equilibrium Model ◽  
Global Effect ◽  
Toroidal Plasmas ◽  
Plasma Fusion ◽  
Magnetically Confined ◽  
Label Dependence ◽  
The Stability ◽  
Toroidal Geometry

Kinetic ballooning modes in magnetically confined toroidal plasmas are investigated putting emphasis on specific stellarator features. In particular, we propose a Mercier criterion which is purposely designed to allow for direct comparison with local flux-tube gyrokinetics simulations. We investigate the influence on the marginal frequency of the mode of a magnetic curvature which is inhomogeneous on the magnetic flux surface due to the fieldline-label dependence. This is a typical (surface) global effect present in non-axisymmetry. Finally, we propose an artificial equilibrium model that explicitly retains the fieldline-label dependence in the magnetic drift, and analyse the stability of the system by introducing a representation of the perturbations similar to the flux-bundle model of Sugamaet al. (Plasma Fusion Res., vol. 7, 2012, 2403094). The coupling of flux bundles is shown to have a stabilising effect on the most unstable local flux-tube mode.

Stability of obliquely propagating plane solitons of the Zakharov–Kuznetsov equation

1995 ◽  
Vol 53 (1) ◽  
pp. 63-73 ◽  
Author(s):  
M. A. Allen ◽  
G. Rowlands
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Excellent Agreement ◽  
Higher Order ◽  
Long Wavelength ◽  
First Order ◽  
Subsequent Elimination ◽  
The Stability ◽  
The Magnetic Field ◽  
Secular Terms

By determining to first order the growth rate of a small, long-wavelength, perturbation to a Zakharov–Kuznetsov plane soliton moving at an angle α to the magnetic field, it has been found that such solitons are unstable for α α0( ∼ 38 °). To determine the stability for angles greater than α0, one needs the growth rate to higher order. The conventional approach generates a second-order growth rate that is singular at α = α0. We rigorously obtain an expression that is bounded at this point, by developing a method in which exponentially secular terms that arise are regrouped before their subsequent elimination. We then show that these solitons are unstable for all α, although the growth rate is small for α>α and goes to zero as α→½π. The relevant linearized equation is solved numerically, and excellent agreement between analytical and numerical results is obtained.

Salt fingers at low Rayleigh numbers

2002 ◽  
Vol 452 ◽  
pp. 25-37 ◽  
Author(s):  
R. KRISHNAMURTI ◽  
Y.-H. JO ◽  
A. STOCCHINO
Keyword(s):  
Linear Stability ◽  
Laboratory Study ◽  
Stability Theory ◽  
Stability Boundary ◽  
Linear Stability Theory ◽  
Salt Fingers ◽  
Linear Profiles ◽  
The Stability ◽  
Rayleigh Numbers

This is a laboratory study of salt fingers at low Rayleigh numbers. We report on the stability boundary in the (RS, RT)-plane (where RS and RT are the salt and heat Rayleigh numbers respectively), the wavenumber of the observed fingers, and the planform. In this low RS, RT range, fingers have width comparable to their height, as predicted by linear stability theory. The planform appears to be close-packed polygonal cells when they are formed on curved profiles of temperature and salinity. However, the planform is distinctly rolls when care is taken to approximate linear profiles.

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