Arnol'd Nonlinear Stability Theorems and their Application to the Atmosphere and Oceans

Within the context of atmospheric and oceanic fluid dynamicsthe problems of nonlinear stability and instability, particularlythe Arnol'd second type nonlinear stability, are surveyed.The stability criteria obtained by means of the energy-Casimirand energy-Lagrange methods are presented for a varietyof models, the estimates for various generalized perturbationenergy and enstrophy are given. Potential applications of thesecriteria are shown in the estimation of bounds on the perturbationenergy and enstrophy, in the diagnostic study of the persistence orbreakdown of jet flows in the middle and high latitudes, and in theverification of the validity of the tangent linear model in bothatmospheric dynamics and oceanography.Some further research results are also highlighted.     

Baroclinic nonlinear exchanges of energy and potential enstrophy in the quasi-geostrophic two-layer model

Abstract

Merilees and Warn's (1975) nonlinear interaction analysis of two-dimensional nondivergent flow is extended to examine the quasi-geostrophic two-layer model. Two sets of triads exist in this model (Salmon, 1978). The purely barotropic triads are the same as the triads examined by Merilees and Warn. Baroclinic-barotropic triads are found to exchange more energy or potential enstrophy with smaller or larger scales depending on the scale of motion as compared with the internal Rossby deformation radius and the relative wavenumber position of baroclinic and barotropic components.     

Theory and application of nonlinear river dynamics

A theoretical model for river evolution including riverbed formation and meandering pattern formation is presented in this paper.Based on nonlinear mathematic theory,the nonlinear river dynamic theory is set up for river dynamic process.Its core content includes the stability and tropism characteristics of flow motion in river and river selves＇ evolution.The stability of river dynamic process depends on the response of river selves to the external disturbance,if the disturbance and the resulting response will eventually attenuate,and the river dynamics process can be restored to new equilibrium state,the river dynamic process is known as stable;otherwise,the river dynamic process is unstable.The river dynamic process tropism refers to that the evolution tendency of river morphology after the disturbance.As an application of this theory,the dynamical stability of the constant curvature river bend is calculated for its coherent vortex disturbance and response.In addition,this paper discusses the nonlinear evolution of the river peristaltic process under a large-scale disturbance,showing the nonlinear tendency of river dynamic processes,such as river filtering and butterfly effect.     

Penetrative convection and multi-component diffusion in a porous medium

Linear instability and nonlinear energy stability analyses are developed for the problem of a fluid-saturated porous layer stratified by penetrative thermal convection and two salt concentrations. Unusual neutral curves are obtained, in particular non-perfect `heart-shaped' oscillatory curves that are disconnected from the stationary neutral curve. These curves show that three critical values of the thermal Rayleigh number may be required to fully describe the linear stability criteria. As the penetrative effect is increased, the oscillatory curves depart more and more from a perfect heart shape. For certain values of the parameters it is shown that the minima on the oscillatory and stationary curves occur at the same Rayleigh number but different wavenumbers, offering the prospect of different types of instability occurring simultaneously at different wavenumbers. A weighted energy method is used to investigate the nonlinear stability of the problem and yields unconditional results guaranteeing nonlinear stability for initial perturbations of arbitrary sized amplitude.     

Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves

Abstract

We consider the mixing of passive tracers and vorticity by temporally fluctuating large scale flows in two dimensions. In analyzing this problem, we employ modern developments stemming from properties of Hamiltonian chaos in the particle trajectories; these developments generally come under the heading “chaotic advection” or “Lagrangian turbulence.” A review of the salient properties of this kind of mixing, and the mathematics used to analyze it, is presented in the context of passive tracer mixing by a vacillating barotropic Rossby wave. We then take up the characterization of subtler aspects of the mixing. It is shown the chaotic advection produces very nonlocal mixing which cannot be represented by eddy diffusivity. Also, the power spectrum of the tracer field is found to be k ^{? l} at shortwaves—precisely as for mixing by homogeneous, isotropic two dimensional turbulence,—even though the physics of the present case is very different. We have produced two independent arguments accounting for this behavior.

We then examine integrations of the unforced barotropic vorticity equation with initial conditions chosen to give a large scale streamline geometry similar to that analyzed in the passive case. It is found that vorticity mixing proceeds along lines similar to passive tracer mixing. Broad regions of homogenized vorticity ultimately surround the separatrices of the large scale streamline pattern, with vorticity gradients limited to nonchaotic regions (regions of tori) in the corresponding passive problem.

Vorticity in the chaotic zone takes the form of an arrangement of strands which become progressively finer in scale and progressively more densely packed; this process transfers enstrophy to small scales. Although the enstrophy cascade is entirely controlled by the large scale wave, the shortwave enstrophy spectrum ultimately takes on the classical k ^{? l} form. If one accepts that the enstrophy cascade is indeed mediated by chaotic advection, this is the expected behavior. The extreme form of nonlocality (in wavenumber space) manifest in this example casts some doubt on the traditional picture of enstrophy cascade in the Atmosphere, which is based on homogeneous two dimensional turbulence theory. We advance the conjecture that these transfers are in large measure attributable to large scale, low frequency, planetary waves.

Upscale energy transfers amplifying the large scale wave do indeed occur in the course of the above-described process. However, the energy transfer is complete long before vorticity mixing has gotten very far, and therefore has little to do with chaotic advection. In this sense, the vorticity involved in the enstrophy cascade is “fossil vorticity,” which has already given up its energy to the large scale.

We conclude with some speculations concerning statistical mechanics of two dimensional flow, prompted by our finding that flows with identical initial energy and enstrophy can culminate in very different final states. We also outline prospects for further applications of chaotic mixing in atmospheric problems.     

Nonlinear baroclinic equilibration at finite supercriticality

The nonlinear equilibration of finite amplitude baroclinic waves in Phillips two-layer model is investigated at finite supercriticality. The aims are to quantify the robustness and relevance of the nonlinear theory of Warn, Gauthier and Pedlosky (WGP) for the evolution of the developing baroclinic wave, and to assess the tightness of pseudomomentum and improved pseudoenergy bounds for disturbance amplitude and energy. A high-resolution numerical model is used to perform a parameter sweep in (β,?W)-space, where β is the inverse criticality of the initial flow, and W is the ratio of the channel width to the (internal) Rossby radius. At low supercriticalities, the main predictions of WGP are found to be accurate at short times, but at long times the fully nonlinear results are found to diverge from WGP's solution. The mechanism for equilibration involves the elimination of the lower layer potential vorticity (PV) gradient, but as the supercriticality increases this is achieved by the roll-up of a train of opposite-signed vortices, rather than by coarse-grain PV homogenization as in WGP. Peak wave amplitudes are typically ≈90% of the maximum attainable under the pseudomomentum bound. New formulae are given for the pseudoenergy bound on disturbance energy which, unlike the WGP solution and the pseudomomentum bound, have non-trivial dependence on W. A detailed assessment is made of the extent to which these bounds are attained.     

The nonlinear development of disturbances in a zonal shear flow

Abstract

A study is made of the nonlinear stability of a weakly supercritical zonal shear flow in the β-plane approximation. The dynamics of initially small disturbances are examined. The main nonlinear effects are associated with the rearrangement of the critical layer. It is shown that as the wave grows in amplitude, linear regimes of the critical layer (viscous and nonstationary) change over to a nonlinear regime while the exponential law of disturbance growth becomes a power-law.     

Effect of the ponderomotive force caused by Alfvén waves on a background plasma in the dayside magnetosphere

The effect of the ponderomotive force on the background plasma modification near magnetic holes, which form at the dayside magnetospheric boundary under the action of the solar wind, has been studied. It was shown that this effect results in a substantial increase in a nonlinear plasma density disturbance. The dependence of the ponderomotive force on the magnetospheric parameters (the magnetic longitude, distance from the Earth’s surface, ratio of the wave frequency to the proton gyrofrequency, and ionospheric ion cyclotron wave amplitude) has been studied. Nonlinear plasma density disturbances will be maximal in the region of magnetic holes, which are located in the dayside magnetosphere at λ ~ 0°?30° geomagnetic longitudes (λ = 0° corresponds to noon), where the effect of the solar wind pressure is maximal. A similar effect is also observed in the dependence of a nonlinear plasma density disturbance on other magnetospheric parameters.     

Dynamics of equilibrium upset and electromagnetic energy transformation in the geomagnetotail: A theory and simulation using particles. 2. Numerical simulation using particles

The process of rapid geomagnetotail reconfiguration after the equilibrium upset (substorm activation) has been studied using the numerical simulation. The decisive reconfiguration feature consists in the spontaneous formation of nonlinear kinetic small-scale structures in the position of the initial current sheet (CS). Substantially different reconfiguration channels are possible depending on the value of the magnetic field component normal to CS: the dominating process can be the formation of a collisionless slow magnetosonic shock or a forced kinetic current sheet (FKCS). The first option is presented in this part of the work. During the late stages, the process is independent of the type of an initial disturbance; the disturbance intensity is spontaneously formed at a certain final level. Far from CS, the disturbance has the form of a rarefaction wave propagating from CS in background plasma. The reconfiguration results in a large-scale transformation of electromagnetic energy into the energy of plasma flows and heat.     

Analysis and control of vibration of ropes in a high-rise elevator under earthquake excitation

In this research, the vibration of elevator ropes, including the main rope and compensation rope are investigated simultaneously in a high-rise elevator system under earthquake excitation. Moreover, the paper presents a new control method to restrain the sway of both ropes. This study considers varying rope lengths during elevator operation which cause other system parameters such as natural frequency, and damping ratio to be time-variant variables. The dynamics of the ropes are analyzed by solving the governing non-stationary, nonlinear equation numerically. In order to mitigate the vibration of ropes in several motion conditions, particularly upwards movement, downward movement, stopped at the lowest position, and stopped at the highest position, an active equipment is installed at the compensation sheave. The stability of the system using the controller is analyzed at four states: without disturbance and static car, without disturbance and mobile car, including disturbance and static car, and including disturbance and mobile car. The efficiency of the controller used for dampening the vibration of elevator ropes is validated by numerical simulation results.     

Theoretical investigation of large-amplitude waves in granular soils

A theoretical investigation of plane waves in granular soils is presented. Dynamic equations are derived with the use of the hypoplasticity theory for granular materials. For numerical calculations the material parameters of Karlsruhe sand are used. Wave speeds as slopes of characteristics of the dynamic equations are calculated for various stresses and densities. It is shown that under certain conditions the dynamic equations lose hyperbolicity and the initial boundary value problem thus becomes ill-posed. Two types of ill-posedness are found, known as flutter ill-posedness and stationary discontinuity. The latter is shown to arise at higher shear stress than the former. A comparison is made between dynamic ill-posedness and stability of static equilibrium. With the use of the second-order work stability criterion it is found that the dynamic equations lose hyperbolicity when the static equilibrium under a dead load is still stable. Numerical solutions to the problem of propagation of boundary disturbance in a half-space are obtained. Owing to dilatancy and contractancy of the granular material, a purely transverse disturbance induces a longitudinal component of velocity in the wave, and vice versa.     

Hydrodynamical Modeling Of Oceanic Vortices

Mesoscale coherent vortices are numerous in the ocean.Though they possess various structures in temperature and salinity,they are all long-lived, fairly intense and mostly circular. Thephysical variable which best describes the rotation and the density anomaly associated with coherent vortices is potential vorticity. It is diagnostically related to velocity and pressure, when the vortex is stationary. Stationary vortices can be monopolar (circular or elliptical) or multipolar; their stability analysis shows thattransitions between the various stationary shapes are possible when they become unstable. But stable vortices can also undergo unsteady evolutions when perturbed by environmental effects, likelarge-scale shear or strain fields, -effect or topography. Changes in vortex shapes can also result from vortex interactions. such as the pairing, merger or vertical alignment of two vortices, which depend on their relative polarities and depths. Such interactions transfer energy and enstrophy between scales, and are essential in two-dimensional and in geostrophic turbulence. Finally, in relation with the observations, we describe a few mechanisms of vortex generation.     

A one-dimensional turbulence model; Enstrophy cascades and small-scale intermittency in decaying turbulence

Abstract

Severe unidirectional Fourier truncation of the equations for 2-D incompressible flow leads to a system of three coupled PDEs in one space dimension with the same quadratic invariants as the original set (i.e. energy and enstrophy). Numerically generated equilibria for inviscid, truncated versions of the reduced system are well approximated by Kraichnan's energy-enstrophy equipartition spectra. Viscous calculations for decaying turbulence at moderate resolution (1024 degrees of freedom) also appear to be consistent with a direct, k ^?3, enstrophy cascading inertial range when the dissipation is small. Dissipation range intermittency in the form of spatially intermittent enstrophy dissipation with occasional strong bursts producing linear phase locking is also observed. In contrast to full 2-D simulations, no tendency towards the emergence of isolated, coherent vorticity structures is observed. The model consequently mimics some, but not all, of the properties of the full 2-D set.     

Nonlinear waves on a coupled density front

Abstract

It is shown that the inclusion of the nonlinear terms in the equations of motion of a coupled density front of zero potential vorticity results in wave solutions which merely propagate with time. The linear theory, on the other hand, predicts an exponential temporal growth. The nonlinear equation admits steady solutions representing standing waves whereas if the nonlinear terms are omitted no steady solutions exist. The general initial value problem is difficult to solve numerically since the linear problem is ill posed.

In addition we prove that the general similarity solution of the nonlinear equation tends to zero for large times, at any point in space, regardless of the initial condition.     

A Rosenbrock‐W method for real‐time dynamic substructuring and pseudo‐dynamic testing

A variant of the Rosenbrock‐W integration method is proposed for real‐time dynamic substructuring and pseudo‐dynamic testing. In this variant, an approximation of the Jacobian matrix that accounts for the properties of both the physical and numerical substructures is used throughout the analysis process. Only an initial estimate of the stiffness and damping properties of the physical components is required. It is demonstrated that the method is unconditionally stable provided that specific conditions are fulfilled and that the order accuracy can be maintained in the nonlinear regime without involving any matrix inversion while testing. The method also features controllable numerical energy dissipation characteristics and explicit expression of the target displacement and velocity vectors. The stability and accuracy of the proposed integration scheme are examined in the paper. The method has also been verified through hybrid testing performed of SDOF and MDOF structures with linear and highly nonlinear physical substructures. The results are compared with those obtained from the operator splitting method. An approach based on the modal decomposition principle is presented to predict the potential effect of experimental errors on the overall response during testing. Copyright © 2009 John Wiley & Sons, Ltd.     

Stability of the time‐domain analysis method including a frequency‐dependent soil–foundation system

A number of methods have been proposed that utilize the time‐domain transformations of frequency‐dependent dynamic impedance functions to perform a time‐history analysis. Though these methods have been available in literature for a number of years, the methods exhibit stability issues depending on how the model parameters are calibrated. In this study, a novel method is proposed with which the stability of a numerical integration scheme combined with time‐domain representation of a frequency‐dependent dynamic impedance function can be evaluated. The method is verified with three independent recursive parameter models. The proposed method is expected to be a useful tool in evaluating the potential stability issue of a time‐domain analysis before running a full‐fledged nonlinear time‐domain analysis of a soil–structure system in which the dynamic impedance of a soil–foundation system is represented with a recursive parameter model. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonlinear wave Packets of Equatorial Kelvin Waves

The weakly nonlinear dynamics of packets of equatorial Kelvin waves is studied using singular perturbation theory applied to the shallow water wave equations. Within the limits of the perturbation theory, which is formally restricted to weak mean shear and weak nonlinearity, we derive a Nonlinear Schroedinger equation to describe the envelope of the wave packet. We find that nonlinearity has a defocusing effect so that coherent wave packets must owe their existence entirely to the generation mechanism rather than to nonlinear focusing of a broad initial disturbance.     

Numerical simulation on the evolution of cloud particles in 3-D convective cloud

A 3-D convective cloud model with compressible non-hydrostatic dynamics and the spectral bin microphysics of a 2-D slab-symmetric model has been used to simulate an observed supercell storm occurring on 29 June, 2000 near Bird City, Kansas, USA. The main objective of this paper is to study the evolution of particles in this convective storm with bin spectral microphysics scheme. Graupels form and grow through two mechanisms, deposition and riming, with the riming process dominant on top of the inflow and in the upper portion of main updraft. Over the outflow and during the developing and mature stages of the storm, graupel particles mainly grow through deposition with dominant unimodal spectra. Most fall out after growing up. Reducing initial relative humidity disturbance (increasing initial potential temperature disturbance) has negative impact on the formation and growth of graupels over the inflow (outflow). This study shows that large graupel and hail could be suppressed by altering the deposition and coalescence process over the inflow and main updraft. At different locations of the convective cells and with different initial humidity and potential temperature disturbance, the graupel formation and growth mechanisms are different, so as to the feasible hail suppression locations and methods. Supported by National Natural Science Foundation of China (Grant Nos. 40537034, 40805057), and Foundation of Key Laboratory of Meteorological Disaster of Ministry of Education in Nanjing University of Information Science & Technology (Grant No. KLME060202)     

Dynamics of equilibrium upset and electromagnetic energy transformation in the geomagnetotail: A theory and simulation using particles. 1. Evolution of configurations in an MHD approximation

The process of equilibrium disruption in the system with a current sheet (CS) under the conditions of small magnetic field component normal to CS, which is induced by an external disturbance, has been theoretically studied within the scope of MHD. In the geomagnetotail, this disturbance can be caused by a tearing instability developing in the more distant tail section, or by a ballooning instability in the tail nearest section, or by a rapid reconfiguration at the magnetopause during the disturbance passage in the solar wind. Locally, in a limited CS section, a longitudinal momentum balance is rapidly (on the Alfvén time scale) upset when a fast MHD disturbance, the form of which depends on the presence of CS, passes along the tail. The nonequilibrium temperature, which subsequently evolves through splitting of CS into several current structures, originates on a substantially larger (due to the smallness of the normal field component) time scale. Such a reconfiguration SPONTANEOUSLY develops after the initial equilibrium upset under the action of an external (weak) disturbance. During an analysis within the scope of MHD, this reconfiguration can be described as the well-known process with two pairs of nonlinear waves propagating in both directions from the central sheet plane at constant velocities: these are fast rarefaction waves and the following slow “switching-off” shocks. However, the kinetic theory reveals substantially different relaxation channels. These channels are studied in the second and third work sections, where the kinetic numerical simulation of the problem is presented and the results of this simulation are analyzed.     

Formation of density cavities with a nonstationary electric field in the zone of auroral field-aligned currents

The formation of small-scale density cavities with a nonstationary electric field, which are registered in the Earth’s auroral magnetosphere, has been analyzed. It has been indicated that cavities are probably initially caused by quasi-static field-aligned electric currents and currents of kinetic Alfvén waves exceeding threshold values. Urgent variants of the linear and nonlinear stages of density disturbance instabilities have been considered. The determined properties of the parameters of small-scale density cavities and nonstationary electric fields are in agreement with the known experimental data.