Convergence of the point vortex method for 2-D vortex sheet

We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

     

The conditions for the symmetric instability of the vortex motions of an ideal stratified liquid

The problem of the symmetric instability of the steady-state motions of an incompressible ideal liquid which is stratified with respect to its density is investigated in the case of two types of motion, axially symmetric and with translational symmetry. It is shown that the sufficient condition for stability obtained in [1] using a variational method (the direct Lyapunov method) for the motions under consideration is closely related to the extremal nature of their energy; stable motions are characterized by a conditional minimum of the energy. A minimum of the energy holds in the class of states for which a potential vortex, expressed in terms of the Lagrangian invariants, angular momentum and density, is represented by the same function of these invariants. Conditions for instability are formulated and estimates of the increase in the kinetic energy of perturbations are given.     

An eigenvalue search method using the Orr–Sommerfeld equation for shear flow

A physically-based computational technique was investigated which is intended to estimate an initial guess for complex values of the wavenumber of a disturbance leading to the solution of the fourth-order Orr–Sommerfeld (O–S) equation. The complex wavenumbers, or eigenvalues, were associated with the stability characteristics of a semi-infinite shear flow represented by a hyperbolic-tangent function. This study was devoted to the examination of unstable flow assuming a spatially growing disturbance and is predicated on the fact that flow instability is correlated with elevated levels of perturbation kinetic energy per unit mass. A MATLAB computer program was developed such that the computational domain was selected to be in quadrant IV, where the real part of the wavenumber is positive and the imaginary part is negative to establish the conditions for unstable flow. For a given Reynolds number and disturbance wave speed, the perturbation kinetic energy per unit mass was computed at various node points in the selected subdomain of the complex plane. The initial guess for the complex wavenumber to start the solution process was assumed to be associated with the highest calculated perturbation kinetic energy per unit mass. Once the initial guess had been approximated, it was used to obtain the solution to the O–S equation by performing a Runge–Kutta integration scheme that computationally marched from the far field region in the shear layer down to the lower solid boundary. Results compared favorably with the stability characteristics obtained from an earlier study for semi-infinite Blasius flow over a flat boundary.     

Cascade of energy in turbulent flows

A starting point for the conventional theory of turbulence [12–14] is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes [19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade of energy, a fundamental mechanism used to explain the Kolmogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rigorous manner, by working directly with the Navier–Stokes equations and stationary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some discussions and partly intuitive proofs appear in the literature. See, e.g., [1,2,10,11,16,17,21], and [22]. It is noteworthy that a mathematical framework can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier–Stokes equations. A similar result concerning the transfer of energy is valid in space dimension two. Here, however, due to vorticity constraints not present in the three-dimensional case, such energy transfer is accompanied by a similar transfer of enstrophy to higher modes. Moreover, at low wave numbers, in the spectral region below that of injection of energy, an inverse (from high to low modes) transfer of energy (as well as enstrophy) takes place. These results are directly related to the mechanisms of direct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy [1,15]. In a forthcoming article [9] we will discuss conditions for the actual existence of the inertial range in dimension three.     

A coastal ocean model intercomparison study for a three-dimensional idealised test case

Several coastal ocean models have been used to compute the circulation on the Northwest European Continental Shelf. Five of them, developed within the European Union, are compared in the scope of an idealised three-dimensional test case, dealing with the geostrophic adjustment of a freshwater cylinder. As the central eddy adjusts, unstable baroclinic vortices start to grow. All the models are able to produce such unstable vortices. However, two of them produce an order-two instability, which is in accordance with a previous laboratory experiment, while the others exhibit an order-four instability. Using a simple scaling analysis, it is seen that the azimuthal wavenumber depends on the ratio of the kinetic energy to the available potential energy. It appears that the discrepancy in the azimuthal wavenumber is mainly due to the effect of the discretisation of the horizontal advection of momentum which could produce significant decrease of the total kinetic energy.     

Anomalous energy transport in the FPU‐β chain

We consider the energy current correlation function for the FPU‐β lattice. For small nonlinearity one can rely on kinetic theory. The issue reduces then to a spectral analysis of the linearized collision operator. We prove thereby that, on the basis of kinetic theory, the energy current correlations decay in time as t^?3/5. It follows that the thermal conductivity is anomalous, increasing as N^2/5 with the system size N. © 2008 Wiley Periodicals, Inc.     

Instability in Parallel Flows Revisited

An example of an unstable inviscid plane parallel shear flow with classical boundary conditions is presented. The complete unstable spectrum is exhibited using techniques of continued fractions for the shear flow with profile U ( y )=cos  m y . For such flows spectral instability implies nonlinear instability. A three-dimensional generalization is discussed.     

The water bag model and gyrokinetic applications

Predicting turbulent transport in nearly collisionless fusion plasmas requires to solve kinetic (or more precisely gyrokinetic) equations. In spite of considerable progress, several pending issues remain; although more accurate, the kinetic calculation of turbulent transport is much more demanding in computer resources than fluid simulations. An alternative approach is based on a water bag representation of the distribution function which is not an approximation but rather a special class of initial conditions allowing to reduce the full kinetic Vlasov equation into a set of hydrodynamic equations while keeping its kinetic character. This model has been applied to gyrokinetic modelling with very encouraging results. The instability threshold for ITG instability is found to be very close to the results obtained from continuous Maxwellian distribution, even for only 10 bags.     

Nonlinear and linear instability of the Rossby-Haurwitz wave

As is known, the large-scale dynamics of barotropic atmosphere can approximately be described by the nonlinear barotropic vorticity equation. It is also well known that the Rossby-Haurwitz (RH) waves, being exact solutions to this equation, represent one of the main features of meteorological fields. Therefore the stability properties of the RH wave are of considerable interest for deeper understanding of the low-frequency variability of the atmosphere. Many works has been devoted to the barotropic instability of flows on a beta-plane and a sphere. However, mathematically, the nonlinear stability problem of the RH wave is still far from its complete solution. Indeed, some of the stability results have been obtained numerically, and hence, contain calculation errors. Severe truncation of perturbations used in the spectral stability analysis, though leads to interesting and useful conclusions, does not allow obtaining comprehensive results. The weak point of some analytical nonlinear instability studies consists in using inappropriate norms for perturbations. It should also be noted that a necessary condition for the linear instability of the RH wave was obtained only recently (Skiba, 2000). In the present work, the nonlinear stability of the RH wave in an ideal incompressible fluid on a rotating sphere is analytically studied. Let H(n) be a subspace of homogeneous spherical polynomials of degree n. Mathematically, a RH wave of degree n is the sum of a super-rotating flow of subspace H(1) and a homogeneous spherical polynomial of subspace H(n). First, we derive a conservation law for arbitrary RH-wave perturbations which asserts that any perturbation evolves in such a way that its kinetic energy E(t) and enstrophy q(t) decrease, remain constant or increase simultaneously. The law is used to divide all the perturbations into three invariant sets depending on the value of their mean spectral number k(t)=q(t)/E(t) introduced by Fjortoft (1953). These sets are denoted as M where k(t)¡n(n+1) (large-scale perturbations), N where k(t)¿n(n+1) (small-scale perturbations), and Z where k(t)=n(n+1) (boundary surface between the sets M and N). Note that Z includes one more invariant set, namely, the subspace H(n). The existence of invariant sets of perturbations allows us to study the RH wave instability in each set separately. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Instabilities and pattern formation in low temperature plasmas

This paper is concerned with a dynamical systems analysis of an instability occurring in a cylindrical high frequency plasma discharge. Stationary, spatially periodic modulations of the plasma density and temperature have been observed in experiments and reproduced in kinetic simulations. A macroscopic model is proposed in order to determine parameter ranges for which these plasma “striations” may form. The model consists of two nonlinear, coupled partial differential equations which describe ambipolar diffusion of electrons and ions and electron energy transport. We emphasize the importance of a particular thermoelectric transport coefficient, usually neglected in fluid models of plasma discharges, which proves essential in describing the pattern formation mechanism.     

Stability in the Rotating Bénard Problem and Its Optimal Lyapunov Functions

Stability analysis of the rotating Bénard problem gives a spectral instability threshold of the purely conducting solution that can be expressed as a critical Rayleigh number R ² depending on the Taylor number T ². The definition of a functional which can be used to prove Lyapunov stability up to the threshold of spectral instability (optimal Lyapunov function) is an important step forward both, for a proof of nonlinear stability and for the investigation of the basin of attraction of the equilibrium. In previous works a Lyapunov function was found, but its optimality could be proven only for small T ². In this work we describe the reason why this happens, and provide a weaker definition of Lyapunov function which allows to prove that, for the linearized system, the spectral instability threshold is also the Lyapunov stability threshold for every value of T ².     

Existence and stability analysis for the Carleman kinetic system

A global existence theorem for the discrete Carleman system in the Sobolev class W ^1,2 is proved by the Leray-Schauder topological degree method, which was not previously applied to discrete kinetic equations. The instability of the nonequilibrium steady flow on a bounded interval is established in the linear approximation.     

A stabilized Hermite spectral method for second‐order differential equations in unbounded domains

A stabilized Hermite spectral method, which uses the Hermite polynomials as trial functions, is presented for the heat equation and the generalized Burgers equation in unbounded domains. In order to overcome instability that may occur in direct Hermite spectral methods, a time‐dependent scaling factor is employed in the Hermite expansions. The stability of the scheme is examined and optimal error estimates are derived. Numerical experiments are given to confirm the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

非对称开采时矿柱失稳的尖点突变模型

针对非对称开采时矿柱稳定性问题建立了一个简化的力学模型．基于势能原理,应用尖点突变理论对矿柱成为非稳定系统进行了探讨,导出了失稳的充要条件、矿柱变形突跳量和能量释放表达式,为定量研究其失稳问题奠定了基础．结果表明:系统的失稳不仅与其所受载荷有关,而且与其内部刚度分配有关,当相对刚度值越大,所承受的临界载荷也越大,越不容易失稳．反之,越容易失稳,且失稳时所释放的能量越大,危害也越大．给出了算例,其计算结果可为安排开采顺序、合理布置采场等提供依据．     

The dynamic stability of an elastic column

Lyapunov's second method is used to investigate the stability of the rectilinear equilibrium modes of a non-linearly elastic thin rod (column) compressed at its end. Stability here is implied relative to certain integral characteristics, of the type of norms in Sobolev spaces; the analysis is carried out for all values of the problem parameter except the bifurcation values.

The realm of problems connected with the Lagrange-Dirichlet equilibrium stability theorem and its converse involves specific difficulties when considered in the infinite-dimensional case: stability in infinite-dimensional systems is investigated relative to certain integral characteristics such as norms /1/, and as the latter may be chosen with a certain degree of arbitrariness, different choices may result in different stability results. On the other hand, there is no relaxation of any of the difficulties encountered in the case of a finite number of degrees of freedom.

We shall consider a certain natural mechanical system with a finite number of degrees of freedom. If the first non-trivial form of the potential energy expansion is positive-definite, the equilibrium position is stable. A similar statement has been proved for infinitely many dimensions as well /1–3/, using Lyapunov's direct method, and the total energy may play the role of the Lyapunov function.

The situation with respect to instability is more complex. In the finite-dimensional case, if the first non-trivial form of the potential energy expansion may take negative values, instability may be demonstrated in many cases by means of a function proposed by Chetayev in /4/. A general theorem has been proved /1/ for instability in infinitely many dimensions, relying on an analogue of Chetayev's function. Such functions have also been used /5, 6/ to prove the instability of equilibrium in specific linear systems with an infinite number of degrees of freedom.

However, Chetayev's functions /4/ are not suitable tools to prove the instability of equilibrium in most non-linear systems. Another “Chetayev function”, which is actually a perturbed form of Chetayev's original function from /4/, has been proposed /7/, and it has been used to prove instability when the equilibrium position is an isolated critical point of the first non-trivial form of the potential energy expansion.

The majority of problems concerning the onset of instability of equilibrium configurations of elastic systems have been considered from a quasistatic point of view (see, e.g., /8, 9/). Problems of elastic stability and instability were considered in a dynamical setting in /2, 5/, where stability was investigated by Lyapunov's direct method. However, most of the results obtained in this branch of the field concern linear systems, and there are extremely few publications dealing with the onset of instability in non-linear elastic systems using Lyapunov's direct method. This is because in an unstable elastic system the quadratic part of the potential energy may change sign, and therefore the analogues of Chetayev's function from /4/ are not usually suitable for solving these problems. Dynamic instability has been studied or a specific non-linearly elastic system /10/, with the fact of instability established by using an analogue of the Chetayev function from /7/.

This paper presents one more example of a study of dynamic instability crried out for a non-linearly elastic system by Lyapunov's direct method.     

On non-standard finite difference models of reaction–diffusion equations

Reaction–diffusion equations arise in many fields of science and engineering. Often, their solutions enjoy a number of physical properties. We design, in a systematic way, new non-standard finite difference schemes, which replicate three of these properties. The first property is the stability/instability of the fixed points of the associated space independent equation. This property is preserved by non-standard one- and two-stage theta methods, presented in the general setting of stiff or non-stiff systems of differential equations. Schemes, which preserve the principle of conservation of energy for the corresponding stationary equation (second property) are constructed by non-local approximation of nonlinear reactions. Assembling of theta-methods in the time variable with energy-preserving schemes in the space variable yields non-standard schemes which, under suitable functional relation between step sizes, display the boundedness and positivity of the solution (third property). A spectral method in the space variable coupled with a suitable non-standard scheme in the time variable is also presented. Numerical experiments are provided.     

Fast spectral solutions of the double-gyre problem in a turbulent flow regime

Several semi-analytical models are considered for a double-gyre problem in a turbulent flow regime for which a reference fully numerical eddy-resolving solution is obtained. The semi-analytical models correspond to solving the depth-averaged Navier–Stokes equations using the spectral Galerkin approach. The robustness of the linear and Smagorinsky eddy-viscosity models for turbulent diffusion approximation is investigated. To capture essential properties of the double-gyre configuration, such as the integral kinetic energy, the integral angular momentum, and the jet mean-flow distribution, an improved semi-analytical model is suggested that is inspired by the idea of scale decomposition between the jet and the surrounding flow.     

HIGH-ORDER WENO SIMULATIONS OF THREE-DIMENSIONAL RESHOCKED RICHTMYER-MESHKOV INSTABILITY TO LATE TIMES:DYNAMICS,DEPENDENCE ON INITIAL CONDITIONS,AND COMPARISONS TO EXPERIMENTAL DATA

The dynamics of the reshocked multi-mode Richtmyer-Meshkov instability is investigated using 513×257 2three-dimensional ninth-order weighted essentially nonoscillatory shock-capturing simulations.A two-mode initial perturbation with superposed random noise is used to model the Mach 1.5 air/SF6 Vetter-Sturtevant shock tube experiment. The mass fraction and enstrophy isosurfaces,and density cross-sections are utilized to show the detailed flow structure before,during,and after reshock.It is shown that the mixing layer growth agrees well with the experimentally measured growth rate before and after reshock.The post-reshock growth rate is also in good agreement with the prediction of the Mikaelian model.A parametric study of the sensitivity of the layer growth to the choice of amplitudes of the short and long wavelength initial interfacial perturbation is also presented.Finally,the amplification effects of reshock are quantified using the evolution of the turbulent kinetic energy and turbulent enstrophy spectra,as well as the evolution of the baroclinic enstrophy production,buoyancy production,and shear production terms in the enstrophy and turbulent kinetic transport equations.     

Instability induced by cross-diffusion in reaction–diffusion systems

In this paper the instability of the uniform equilibrium of a general strongly coupled reaction–diffusion is discussed. In unbounded domain and bounded domain the sufficient conditions for the instability are obtained respectively. The conclusion is applied to the ecosystem, it is shown that cross-diffusion can induce the instability of an equilibrium which is stable for the kinetic system and for the self-diffusion–reaction system.     

Calculating Thomson's Spectral Multitapers by Inverse Iteration

Abstract

Spectral estimation using a set of orthogonal tapers is becoming widely used and appreciated in scientific research. It produces direct spectral estimates with more than 2 df at each Fourier frequency, resulting in spectral estimators with reduced variance. Computation of the orthogonal tapers from the basic defining equation is difficult, however, due to the instability of the calculations—the eigenproblem is very poorly conditioned. In this article the severe numerical instability problems are illustrated and then a technique for stable calculation of the tapers—namely, inverse iteration—is described. Each iteration involves the solution of a matrix equation. Because the matrix has Toeplitz form, the Levinson recursions are used to rapidly solve the matrix equation. FORTRAN code for this method is available through the Statlib archive. An alternative stable method is also briefly reviewed.