On nonlinear stability of the regular vortex systems on a sphere

We present the necessary and sufficient conditions for stability and instability of the stationary rotation of a system of n identical point vortices located at the same latitude on a sphere at vertices of a regular n-gon. We also examine stability of the equilibrium configuration of identical point vortices, situated at the vertices of a regular polyhedra. It is proved that vortex tetrahedron, octahedron, and icosahedron are stable, while vortex cube and dodecahedron are unstable.     

Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling

A two-dimensional nonlinear Schrödinger lattice with nonlinear coupling, modelling a square array of weakly coupled linear optical waveguides embedded in a nonlinear Kerr material, is studied. We find that despite a vanishing energy difference (Peierls-Nabarro barrier) of fundamental stationary modes the mobility of localized excitations is very poor. This is attributed to a large separation in parameter space of the bifurcation points of the involved stationary modes. At these points the stability of the fundamental modes is changed and an asymmetric intermediate solution appears that connects the points. The control of the power flow across the array when excited with plane waves is also addressed and shown to exhibit great flexibility that may lead to applications for power-coupling devices. In certain parameter regimes, the direction of a stable propagating plane-wave current is shown to be continuously tunable by amplitude variation (with fixed phase gradient). More exotic effects of the nonlinear coupling terms like compact discrete breathers and vortices, and stationary complex modes with nontrivial phase relations are also briefly discussed. Regimes of dynamical linear stability are found for all these types of solutions.     

Nonlinear dynamics of modulation instability in distributed resonators under external harmonic driving

We study the regimes of complex field dynamics upon modulation instability in distributed nonlinear resonators under external harmonic driving. Two regimes are considered: the regime of a nonlinear ring cavity, described by nonlinear Schrödinger equation (NLS) with a delayed boundary condition, and the regime of a one-dimensional Fabri-Perot cavity, described by a system of coupled NLS for the forward and backward waves. Theoretical stability analysis of stationary forced oscillations is carried out. The results of numerical simulation of transition to chaos with increasing input intensity are presented.     

Influence of harmonically varying normal load on steady-state behavior of a 2dof torsional system with dry friction

A nonlinear dry friction problem with harmonically varying normal load is formulated, in the context of a two-degree of freedom torsional system, since virtually all of the prior literature focuses on the topic of time-invariant normal load. First, pure stick, pure slip and stick-slip motions are computationally and analytically determined when excited by a sinusoidal torque, in the presence of harmonically varying saturation torque; mean terms are included in both. These analyses yield both transient and steady-state time histories under various conditions. Second, the effects of time-varying normal load on steady-state responses have been investigated and nonlinear spectral maps (including super-harmonics) are developed. Results show that the actuation system parameters could affect steady-state stick-slip motions in different ways over the lower and higher frequency regimes, as a result of time-delay in slip motions with respect to the torque excitation. In particular, the negative slope characteristics in the friction law exaggerate the stick-slip vibration problems, and it is the major cause of bifurcations and quasi-periodic or chaotic motions. Around the super-harmonic peak frequencies, the nonlinear system tends to lose stability as abrupt jumps in the spectral maps take place. An equivalent viscous damping model is considered to analytically investigate the instability mechanism. Further, the periodicity of the system response under harmonically varying actuation is conceptually by employing the harmonic balance method. Finally, steady-state behavior is examined for the nonlinear, time-varying dry friction problem.     

Modeling of nonlinear and non-stationary multi-vortex behavior of CDWs at nanoscales in restricted geometries of internal junctions

We report on studies of stationary states and their transient dynamic for an incommensurate charge density wave (ICDW) in a restricted geometry of two spatial dimensions. The model takes into account multiple fields in mutual nonlinear interactions: the amplitude and the phase of the complex order parameter, and distributions of the electric and chemical potentials, of the density and the current of normal carriers. We observed spontaneous formation of vortices (the ICDW dislocations), and followed events of their creation and the subsequent evolution. The vortices appear when the voltage across, or the current through, the sample exceed a threshold. The number of vortices remnant in the reconstructed stationary state increases stepwise – in agreement with experiments, while a much greater number of vortices appears during the intermediate transient states. The vortex core concentrates the electric dipole leading to sharp drops of the electric and chemical potentials across the core. That can lead to enhanced inter-layer tunneling making the core to be a self-tuned microscopic tunneling junction. The results are applied to experiments on nano-fabricated mesa-junctions. They also appeal to modern efforts of the field-effect transformations in correlated electronic systems.     

Transitional regimes of penetrative convection in a plain layer

Penetrative convection in a plain layer of water is considered for the interval of temperatures containing the point of density maximum. When the unstable and stable layers are equal in the static conductive state, the development of convective instability is investigated from the formation of steady structures to chaotic motion. Scales of the periodicity cell for regimes with a strong nonlinear effect were chosen with particular attention. Specifics are shown for steady structures with small-scale vortices near upper boundaries and for periodic motions with synchronized oscillations of the lower parts of vertically elongated profiles of temperature that move symmetrically. With the increase of supercriticality, the motion loses reflection symmetry, becomes doubly periodic, and finally becomes quasi-periodic before transition to chaotic motion. Domains of hysteresis are investigated for which motions with different structure and heat fluxes coexist, and it is illustrated by the dependence of the Nusselt number on supercriticallity.     

Combined action of different mechanisms of convective instability in multilayer systems

The nonlinear regimes of convection in a system of three immiscible viscous fluids are investigated by the finite-difference method. We study new phenomena caused by direct and indirect interaction of thermocapillary and buoyancy (Rayleigh and anticonvective) instability mechanisms. Two variants of heating-from below and from above-are considered. The interfaces are assumed to be flat. We focus on nonlinear evolution of steady and oscillatory motions and selection of stable convective structures depending on the parameters of systems. The influence of the lateral boundary conditions is also investigated. A classification of different variants of interaction between Rayleigh and thermocapillary instability mechanisms is presented, and several typical examples are studied. Specifically, we considered six different configurations where the Rayleigh convection arises mainly in a definite layer, and the thermocapillary convection appears mainly near the definite interface. Also, the case where both interfaces are active and alternatively play a dominant role is investigated. Some configurations of interaction between anticonvective and thermocapillary instability mechanisms are considered.     

相邻双侧边盖驱动方腔流动的三维线性整体稳定性

文章考察了相邻双侧边盖驱动方腔流动（即上壁面向右运动和左侧壁面向下运动）的三维线性整体稳定性.首先,采用Taylor-Hood有限元方法并经由Newton迭代过程计算得到双侧边盖驱动方腔流动的二维稳态基本流.其次,Taylor-Hood有限元在Chebyshev Gauss配置点上进行离散,同时Gauss配置点也可以用于线性稳定性方程的高阶有限差分格式离散.然后,离散得到的矩阵形式的广义特征值问题可以结合shift-and-invert算法采用隐式重启Arnoldi方法计算.最后,通过对线性稳定性方程特征值的计算,发现了一个最不稳定的驻定模态和两对对称行波模态.最不稳定的三维驻定模态的临界Reynolds数为Re_c=261.5,远远小于二维不稳定的临界Reynolds数Re_c2d=1 061.7.通过画出这3类三维不稳定模态的流向扰动速度和扰动涡量的空间等值面图像,可以发现不稳定扰动位于稳态基本流的两个主涡区域,因此可以认为主涡区域是三维扰动失稳的主要能量来源地.      

Improved algorithm for solving nonlinear parabolized stability equations

Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations(NPSE) approach has been widely used to study the stability and transition mechanisms. However,it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation(DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers.     

高斯变迹布拉格光纤光栅中的调制不稳定性

利用激光脉冲在光纤光栅中传播时所遵守的相干耦合非线性薛定谔方程,研究了激光脉冲在高斯变迹布拉格光纤光栅中传输时，在反常色散区和正常色散区所产生的调制不稳定性.结果表明在反常色散区和正常色散区都能产生调制不稳定性；在反常色散区，当输入功率达到一定数值时，产生明显的有规律的增益谱；在正常色散区，在产生调制不稳定性功率区域，调制不稳定性存在并从给定值一直持续到无穷；并且，在反常色散区和在正常色散区，增益谱都受到高斯变迹函数的制约. 关键词： 高斯变迹 布拉格光栅 调制不稳定性 增益     

Competing mechanisms of plasma transport in inhomogeneous configurations with velocity shear: the solar-wind interaction with earth's magnetosphere

Two-dimensional simulations of the Kelvin-Helmholtz instability in an inhomogeneous compressible plasma with a density gradient show that, in a transverse magnetic field configuration, the vortex pairing process and the Rayleigh-Taylor secondary instability compete during the nonlinear evolution of the vortices. Two different regimes exist depending on the value of the density jump across the velocity shear layer. These regimes have different physical signatures that can be crucial for the interpretation of satellite data of the interaction of the solar wind with the magnetospheric plasma.     

Dominant vortices in impinging jet flows

A novel method of flow visualization by dye was used in conjunction with numerical solutions to investigate the formation stages of large stationary vortical motions located in the “trajectory bend” centers of impinging jet flows. The vortices dominate the flowfield and were found to have profound influence on the wall transport phenomena. Depending on the value of Reynolds number, four regimes were identified with different flow character.     

Study of the Nonlinear Dynamics of a Traveling-Wave-Tube Oscillator with Delayed Feedback

We present the results of numerical simulations of the nonlinear dynamics of a traveling-wave-tube (TWT) oscillator with delayed feedback. Basic properties of stationary single-frequency oscillation regimes are considered, and the onset of self-modulation is studied in detail. Various route-to-chaos scenarios corresponding to successively increasing values of the beam current are simulated numerically. It is shown that the basic scenario is a quasi-periodic route to chaos, while the beam deceleration in strongly nonlinear regimes causes transitions via intermittency to regimes based on modes with higher frequencies. Competition between these two scenarios leads to a complex picture of regular and chaotic self-modulation regimes in the parameter space. Such a behavior is typical of distributed electron–wave self-excited oscillators with delayed feedback.     

Evolution of three-dimensional coherent structures in compressible axisymmetric jet

A high order difference scheme is used to simulate the spatially developing compressible axisymmetric jet. The results show that the Kelvin-Helmholtz instability appears first when the jet loses its stability, and then with development of jet the increase in nonlinear effects leads to the secondary instability and the formation of the streamwise vortices. The evolution of the three-dimensional coherent structure is presented. The computed results verify that in axisymmetric jet the secondary instability and formation of the streamwise vortices are the important physical mechanism of enhancing the flow mixing and transition occurring.     

Nonlinear equation for curved nonstationary flames and flame stability

A time-dependent nonlinear equation for a nonstationary curved flame front of an arbitrary expansion coefficient is derived under the assumptions of a small but finite flame thickness and weak nonlinearity. On the basis of the derived equation, stability of two-dimensional curved stationary flames propagating in tubes with ideally adiabatic and slip walls is studied. The stability analysis shows that curved stationary flames become unstable for sufficiently wide tubes. The obtained stability limits are in a good agreement with the results of numerical simulations of flame dynamics and with semiqualitative stability analysis of curved stationary flames. Possible outcomes of the obtained instability at the nonlinear stage are discussed. The instability may result in extra wrinkles at a flame front close to the stability limits and in self-turbulization of the flame far from the limits. The self-turbulization can also be interpreted as a fractal structure. The fractal dimension of a flame front and velocity of a self-turbulized flame are evaluated.     

Modulation instability of homogeneous resonant excitation regimes of molecular J aggregates

We have performed a linear analysis of the stability of stationary states of homogeneous resonant excitation regimes of molecular J aggregates with coherent sustaining radiation. In the analysis, we have taken into account not only two-particle but also three-particle interactions of molecules, which are related to the exciton-exciton annihilation mechanism. Our detailed investigation of the stability of stationary states has allowed us to determine stability and instability regions and correlate them with the existence of bistability boundaries.     

Drift waves and counter rotating vortices in pair-ion plasmas

Linear dispersion relation has been found for drift and acoustic waves in pair-ion-electron plasmas. The stationary solution in the form of counter rotating vortices has been obtained in the presence of equilibrium potential profile. It is noticed that the speed of nonlinear structures is reduced with the increase of electrons concentration in pair-ion plasmas. Linear instability condition has also been found in the presence of shear flow. It is pointed out that the present results can be useful for future pair-ion plasma experiments.     

Generation of nonstationary Görtler vortices by localized surface nonuniformities. Receptivity coefficients

The mechanism of production of nonstationary Görtler vortices in a boundary layer on concave wall by surface nonuniformities (vibrations and roughness) has been experimentally examined. The nonuniformities were produced by a specially developed disturbance source. They were controlled, localized along the streamwise coordinate, and periodic over the span of the experimental model. Tests in a low-turbulence wind tunnel have proved that the disturbance source is an efficient means of experimental study of the receptivity and stability problem for boundary layers dominated by Görtler instability. The operation of the disturbance source leads to the production of small-amplitude nonstationary Görtler vortices (tenth or hundredth fractions of a per cent of the free-stream velocity) with predefined characteristics (frequency and spanwise wavelength). In our experiments, we quantitatively examined the problem of linear receptivity of boundary layer to surface nonuniformities in a broad range of frequencies for the most dangerous spanwise scales of Görtler vortices. The values of the amplitudes and phases of the receptivity coefficients were determined. The amplitudes proved to be much smaller in magnitude in comparison with the excitation of modes of hydrodynamic instabilities of other types (Tollmien-Schlichting waves and cross-flow-instability modes). It was found that, with increasing the frequency, the amplitudes of the receptivity coefficients showed a distinct growth while for high frequencies those amplitudes also exhibited a growth with the spanwise scale of perturbations, although for stationary surface roughness no effect due to this scale was observed. It was found that the dependences on frequency of the efficiency of the mechanisms of stability and receptivity showed opposing behaviors, were in competition, and could partially compensate each other, promoting, thus, the production of boundary-layer Görtler vortices in a broad range of frequencies.     

Slow dynamics in a turbulent von Kármán swirling flow

An experimental study of a turbulent von Kármán flow in a cylinder is presented. The mean flow is stationary up to a Reynolds number Re=10(4) where a bifurcation takes place. The new regime breaks some symmetries of the problem and becomes time dependent because of equatorial vortices moving with a precession movement. In the exact counterrotating case, a bistable regime appears and spontaneous reversals of the azimuthal velocity are registered. A three-well potential model with additive noise reproduces this dynamic. A regime of periodic response is observed when a very weak forcing is applied.     

Rayleigh convective instability in a cloud medium

The problem of convective instability of an atmospheric layer containing a horizontally finite region filled with a cloud medium is considered. Solutions exponentially growing with time, i.e., solitary cloud rolls or spatially localized systems of cloud rolls, have been constructed. In the case of axial symmetry, their analogs are convective vortices with both ascending and descending motions on the axis and cloud clusters with ring-shaped convective structures. Depending on the anisotropy of turbulent exchange, the scale of vortices changes from the tornado scale to the scale of tropical cyclones. The solutions with descending motions on the axis can correspond to the formation of a tornado funnel or a hurricane eye in tropical cyclones.