REMARKS ON THE NONLINEAR INSTABILITY OF INCOMPRESSIBLE EULER EQUATIONS

In this paper, we consider the nonlinear instability of incompressible Euler equations. If a steady density is non-monotonic, then the smooth steady state is a nonlinear instability. First, we use variational method to find a dominant eigenvalue which is important in the construction of approximate solutions, then by energy technique and analytic method, we obtain the dynamical instability result.

NONLINEAR INSTABILITY OF WAVETRAIN UNDER INFLUENCES OF SHEAR CURRENT WITH VARYING VORTICITY AND AIR PRESSURE

Under the assumption of weak shear current with varying vorticity in water and weak air pressure the Zakharov theory is extended to include the effects of vorticity and air pressure on the modulation of water waves. This new equation is used to examine the influence of current and wind on the Benjamin-Feir sideband instability and long-time evolution of wavetrain. As strength of the current increases the bandwidth is found broadened, and the maximum growth rate of sidebands decreased. Periodic solution of sidebands in the presence of current is indicated, which means that shear current does not affect the downshift of wave spectrum peak. Energy input by imposing the air pressure leads to the enhancement of the lower sideband, which is in agreement with the finding of Hara and Mei (1991).

Investigation of influence of local cooling/heating on nonlinear instability of high-speed boundary layer with direct numerical simulations

The influence of local cooling/heating on two types of nonlinear instabilities of the high-speed boundary layer,namely,the First and Second Mode Oblique Breakdown(FMOB and SMOB),is studied using direct numerical simulations.Local cooling and heating are performed at the weak and strong nonlinear stages of the two types of nonlinear instabilities.It is found that for the FMOB,local cooling at the weak nonlinear region will suppress the increase of the fundamental mode,leading to transition delay.Opposite to local cooling,local heating at the weak nonlinear region of the FMOB will promote the growth of the fundamental mode,resulting in the occurrence of more upstream transition onset.However,if local cooling and heating are performed at the strong nonlinear region,the influence of both local cooling and heating on the FMOB can be neglected.Remarkably,both local heating and cooling can delay the SMOB for different mechanisms.Performing local cooling at the weak nonlinear region of the SMOB,the low amplitude of higher spanwise wavenumber steady mode caused by local cooling lies behind transition delay.When local cooling is set at the strong nonlinear region,the low amplitude of harmonic modes around the cooling area can cause transition delay.Additionally,local heating will suppress the SMOB for the slowing amplification rate of various modes caused by the local heating at both the weak and strong nonlinear stages of the SMOB.

Nonlinear Instability of Phillips Model

We investigate the instability of two-layer Phillips model in this paper, which is a prototypical geophysical fluid model. Using the results of Guo and Strauss et al, we obtained linear instability implies nonlinear instability provided the linearized system has an exponentially growing solution.

NONLINEAR INSTABILITY OF EQUILIBRIUM SOLUTION FOR THE GINZBURG-LANDAU EQUATION

We study the nonlinear instability of plane wave solutions to a GinzburgLandau equation with derivatives. We Show that, under some condition in coefficient of the equation, these waves are unstable.

The Impact of Nonlinear Stability and Instability on the Validity of the Tangent Linear Model

The impact of nonlinear stability and instability on the validity of tangent linear model (TLM) is investigated by comparing its results with those produced by the nonlinear model (NLM) with the identical initial perturbations. The evolutions of different initial perturbations superposed on the nonlinearly stable and unstable basic flows are examined using the two-dimensional quasi-geostrophic models of double periodic-boundary condition and rigid boundary condition. The results indicate that the valid time period of TLM, during which TLM can be utilized to approximate NLM with given accuracy, varies with the magnitudes of the perturbations and the nonlinear stability and instability of the basic flows. The larger the magnitude of the perturbation is, the shorter the valid time period. The more nonlinearly unstable the basic flow is, the shorter the valid time period of TLM. With the double—periodic condition the valid period of the TLM is shorter than that with the rigid—boundary condition. Key words Nonlinear stability and instability - Tangent linear model (TLM) - Validity This work was supported by the National Key Basic Research Project “Research on the Formation Mechanism and Prediction Theory of Severe Synoptic Disasters in China” (No.G1998040910) and the National Natural Science Foundation of China (No.49775262 and No.49823002).

NONLINEAR DYNAMIC INSTABILITY OF DOUBLE-WALLED CARBON NANOTUBES UNDER PERIODIC EXCITATION

A multiple-elastic beam model based on Euler-Bernoulli-beam theory is presented to investigate the nonlinear dynamic instability of double-walled nanotubes. Taking the geometric nonlinearity of structure deformation, the effects of van der Waals forces as well as the non- coaxial curvature of each nested tube into account, the nonlinear parametric vibration governing equations are derived. Numerical results indicate that the double-walled nanotube （DWNT） can be considered as a single column when the van der Waals forces are sufficiently strong. The stiffness of medium could substantially reduce the area of the nonlinear dynamic instability region, in particular, the geometric nonlinearity can be out of account when the stiffness is large enough. The area of the principal nonlinear instability region and its shifting distance aroused by the nonlinearity both decrease with the increment of the aspect ratio of the nanotubes.

Some Problems in Nonlinear Dynamic Instability and Bifurcation Theory for Engineering Structures

In civil engineering, the nonlinear dynamic instability of structures occurs at a bifurcation point or a limit point. The instability at a bifurcation point can be analyzed with the theory of nonlinear dynamics, and that at a limit point can be discussed with the theory of elastoplasticity. In this paper, the nonlinear dynamic instability of structures was treated with mathematical and mechanical theories. The research methods for the problems of structural nonlinear dynamic stability were discussed first, and then the criterion of stability or instability of structures, the method to obtain the bifurcation point and the limit point, and the formulae of the directions of the branch solutions at a bifurcation point were elucidated. These methods can be applied to the problems of nonlinear dynamic instability of structures such as reticulated shells, space grid structures, and so on. Key words nonlinear dynamic instability - engineering structures - non-stationary nonlinear system - bifurcation point - instability at a bifurcation point - limit point MSC 2000 74K25 Project supported by the Science Foundation of Shanghai Municipal Commission of Education (Grant No. 02AK04), the Science Foundation of Shanghai Municipal Commission of Science and Technology (Grant No. 02ZA14034)

NONLINEAR MHD KELVIN-HELMHOLTZ INSTABILITY IN A PIPE

Nonlinear MHD Kelvin-Helmholtz(K-H)instability in a pipe is treated with the deriva- tive expansion method in the present paper The linear stability problem was discussed in the past by Chandrasekhar(1961)and Xu et al.(1981).Nagano(1979)discussed the nonlinear MHD K-H instability with infinite depth.He used the singular perturbation method and extrapolated the ob- tained second order modifier of amplitude vs.frequency to seek the nonlinear effect on the instability growth rate γ.However,in our view,such an extrapolation is inappropriate.Because when the instabili- ty sets in,the growth rates of higher,order terms on the right hand side of equations will exceed the cor- responding secular producing terms,so the expansion will still become meaningless even if the secular producing terms are eliminated.Mathematically speaking,it's impossible to derive formula(39) when γ_0~2 is negative in Nagano's paper.Moreover,even as early as γ_0~2→O^+,the expansion be- comes invalid because the 2nd order modifier γ_2(in his formula(56))tends to infinity.This weak- ness is removed in this paper,and the result is extended to the case of a pipe with finite depth.

Application of the Conditional Nonlinear Optimal Perturbations Method in a Theoretical Grassland Ecosystem

Using a simplified nonlinearly theoretical grassland ecosystem proposed by Zeng et al.,we study the sensitivity and nonlinear instability of the grassland ecosystem to finiteamplitude initial perturbations with the approach of conditional nonlinear optimal perturbation （CNOP）.The results show that the linearly stable grassland （desert or latent desert） states can turn to be nonlinearly unstable with finite amplitude initial perturbations.When the precipitation is between the two bifurcation points,a large enough finite amplitude initial perturbation can induce a transition between the grassland statethe desert state or the latent desert.

物理生成的长峰不规则波的非高斯统计特性

An experimental study is presented on the non-Gaussian statistics of random unidirectional laboratory wave fields described by JONSWAP spectra.Relationships between statistical parameters indicative of the occurrence of largeamplitude waves are discussed in the context of the initial steepness of the waves combined with the effect of spectral peakedness.The spatial evolution of the relevant statistical and spectral parameters and features is also considered.It is demonstrated that over the distance the spectra exhibit features typical for developing nonlinear instabilities,such as spectral broadening and downshift of the peak,along with lowering of the high-frequency tail and decrease of the peak magnitude.The wave fields clearly show an increase of third-order nonlinearity with the distance,which can be significant,depending on the input wave environment.The steeper initial conditions,however,while favouring the occurrence of extremely large waves,also increase the chances of wave breaking and loss of energy due to dissipation,which results in lower extreme crests and wave heights.The applied Miche-Stokes-type criteria do confirm that some of the wave extremes exceed the limiting individual steepness.Eventually,this result agrees with the observation that the largest number of abnormal waves is recorded in sea states with moderate steepness.

Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory

By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material(FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.

Three dimensional simulations of penetrative convection in a porous medium with internal heat sources

The problem of penetrative convection in a fluid saturated porous medium heated internally is analysed. The linear instability theory and nonlinear energy theory are derived and then tested using three dimensions simulation.Critical Rayleigh numbers are obtained numerically for the case of a uniform heat source in a layer with two fixed surfaces. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three dimensional simulation. Our results show that the linear threshold accurately predicts the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.

The decadally modulating eddy field in the upstream Kuroshio Extension and its related mechanisms

Both the level of the high-frequency eddy kinetic energy（HF-EKE） and the energy-containing scale in the upstream Kuroshio Extension（KE） undergo a well-defined decadal modulation, which correlates well with the decadal KE path variability. The HF-EKE level and the energy-containing scales will increase with unstable KE path and decrease with stable KE path. Also the mesoscale eddies are a little meridionally elongated in the stable state, while they are much zonally elongated in the unstable state. The local baroclinic instability and the barotropic instability associated with the decadal modulation of HF-EKE have been investigated. The results show that the baroclinic instability is stronger in the stable state than that in the unstable state, with a shorter characteristic temporal scale and a larger characteristic spatial scale. Meanwhile, the regional-averaged barotropic conversion rate is larger in the unstable state than that in the stable state. The results also demonstrate that the baroclinic instability is not the dominant mechanism influencing the decadal modulation of the mesoscale eddy field, while the barotropic instability makes a positive contribution to the decadal modulation.

NONLINEAR BAROTROPIC INSTABILITY INCLUDING FRICTION DISSIPATION, THERMAL DRIVING AND LARGE TOPOGRAPHY

From a nonlinear quasi-geostrophic barotropic vorticity equation including frictional dissipation, thermal driving and large topography used by Charney in investigation of the multiple flow equilibria and blocking, using the Serrin-Joseph energy method and the variational principle, we found the nonlinear barotropic stability criteria of the zonal basic flow with the total energy, total enstrophy and their linear combination respectively, and compared the criteria with Charney's results.

LINEAR AND NONLINEAR BAROTROPIC INSTABILITY

Based on a non-frictional and non-divergent nonlinear barotropic vorticity equation and its solutions of travelling waves,the criteria for linear and nonlinear barotropic instability are gained respectively at an equilibrium point of the equation on a phase plane.The linear and nonlinear analytical solutions to instability waves are also found.The computational results show that if their amplitudes are equal at the initial time,the amplitude increments of nonlinear instable barotropic wave are always less than those of linear instable barotropic wave. The nonlinear effects can slow down the exponential growth of linear instability.The time needed for making the amplitude double that of initial time by instabilities,is about 6h for linear instability and about 18h for nonlinear instability,the latter is in agreement with the observations in the real atmosphere.

Nonlinear Saturation Amplitude in Classical Planar Richtmyer–Meshkov Instability

The classical planar Richtmyer–Meshkov instability(RMI) at a fluid interface supported by a constant pressure is investigated by a formal perturbation expansion up to the third order,and then according to definition of nonlinear saturation amplitude(NSA) in Rayleigh–Taylor instability(RTI),the NSA in planar RMI is obtained explicitly.It is found that the NSA in planar RMI is affected by the initial perturbation wavelength and the initial amplitude of the interface,while the effect of the initial amplitude of the interface on the NSA is less than that of the initial perturbation wavelength.Without marginal influence of the initial amplitude,the NSA increases linearly with wavelength.The NSA normalized by the wavelength in planar RMI is about 0.11,larger than that corresponding to RTI.

Study on the dynamic behavior of herringbone gear structure of marine propulsion system powered by double-cylinder turbines

Propulsion systems powered by double-cylinder turbines(DCT)are widely used in large-scale ships.However,the nonlinear instability leads to hidden dangers associated with the safe operation,and there is a lack of theoretical and systematic research on this problem.Based on the gear transmission principle and non-Newtonian thermal elastohydrodynamic lubrication(EHL)theory,a torsional model of a two-stage herringbone system forced by unsymmetrical load is established.The nonlinear and time-varying factors of meshing friction,meshing stiffness,and gear pair backlash are included in the model,and multiple meshing states,including single-and double-sided impact are studied.New nonlinear phenomena of the dynamic system are explored and the effects of the unsymmetrical load on the system stability are quantified.The results indicate that the stability of the gear system is improved,and that the back-sided impact gradually disappears with the increases of load ratio between the two inputs and the input load value.Furthermore,it is found that the gear pairs on the low-load side experience more severe vibration than those on the high-load side.Finally,the stability of the gear pairs decreases along the power transmission path of the multistage gear system.The results of this research will be useful when making predictions of the stability of such systems and in the optimization of the load parameters.

LONG VALID TIME ENERGY PERFECT CONSERVATIVE FIDELITY SPECTRAL SCHEMES OF BAROTROPIC PRIMITIVE EQUATIONS

In accordance with a new compensation principle of discrete computations,the traditional meteo- rological global (pseudo-) spectral schemes of barotropic primitive equation (s) are transformed into perfect energy conservative fidelity schemes,thus resolving the problems of both nonlinear computa- tional instability and incomplete energy conservation,and raising the computational efficiency of the traditional schemes. As the numerical tests of the new schemes demonstrate,in solving the problem of energy conser- vation in operational computations,the new schemes can eliminate the (nonlinear) computational in- stability and,to some extent even the (nonlinear) computational diverging as found in the traditional schemes,Further contrasts between new and traditional schemes also indicate that,in discrete opera- tional computations,the new scheme in the case of nondivergence is capable of prolonging the valid in- tegral time of the corresponding traditional scheme,and eliminating certain kind of systematical com- putational“climate drift”,meanwhile increasing its computational accuracy and reducing its amount of computation.The working principle of this paper is also applicable to the problem concerning baroclin- ic primitive equations.