Nonlinear Stage Of Development Research Articles

Quasilinear Simulation of the Development of Weibel Turbulence in Anisotropic Collisionless Plasma

A spectral quasilinear approach to the problem of TEM-Weibel instability in an anisotropic collisionless plasma is developed, which takes into account only the integral nonlinear interaction of modes through the joint variation of the spatially averaged particle velocity distribution induced by these modes. Within this approximation, a closed system of equations is obtained for the one- and two-dimensional evolution of spatial modes (harmonics) of the distribution function of particles and the electromagnetic field under conditions when the plasma anisotropy axis, the wave vector, and the magnetic field of the modes are orthogonal to each other. The numerical solution of this system of equations is compared with the available results of one-dimensional analytical quasilinear theory in the region of its applicability, as well as with the results of two-dimensional simulation by the particle-in-cell method, which also takes into account the direct four-wave interaction of modes. It is established that in the simplest cases of one-dimensional and axially symmetric two-dimensional problems for a bi-Maxwellian plasma, quasilinear phenomena play the leading role at a quite long stage of nonlinear development of turbulence. It is noted that at a later stage of decay of turbulence and in a more general formulation of the problem, in particular, in the presence of an external magnetic field, the direct nonlinear interaction of modes can manifest itself along with quasilinear phenomena. Based on the analysis carried out, the contribution of certain nonlinear effects to the evolution of the spatial spectrum of Weibel turbulence is revealed, and the properties of this turbulence are studied, including the self-similar character and qualitatively different stages of the dynamics of unstable modes.

Plasma Crystal as a Time Crystal

The nonlinear development stage is studied of the instability of coupled waves in the planar plasma crystal in the external confining force field. It is assumed that the particles are located in plasma consisting of the Maxwellian electrons and the directed flow of cold ions. It is demonstrated that the effect of nonlinearity results either in acceleration of the instability development or in the formation of a time crystal consisting of one or three stable waves. In the space of external parameters, the domains of existence of different phases are constructed.

Nonlinear Dynamics of a Linear Dust Particle Chain

The nonlinear stage of development of the dust particle chain instabilities is studied. It is assumed that dust particles interact with the plasma consisting of Maxwellian electrons and a directed flow of cold ions. The truncated equations are derived describing the nonlinear dynamics of the standing wave in the range of parameters close to the threshold for the development of the instability of coupled waves. It is shown that, in a narrow range of the ion flow Mach numbers, the instability of coupled waves becomes saturated in the weakly nonlinear stage of its development. In the remaining cases, the instability development becomes explosive. The nonlinear dynamics is also considered of the aperiodic period-doubling instability, which always becomes saturated in the weakly nonlinear stage of its development.

Solitons in the Einstein universe

We show that equations of Newtonian hydrodynamics and gravity with Einstein's cosmological constant included admit gravitostatic wave solutions propagating in the background of Einstein's static Universe. In the zero pressure limit these waves exist at an average matter density exceeding that of Einstein's Universe. They have the form of a lattice of integrable density singularities localized at the maxima of the gravitational potential. These singularities are steady-state counterparts of the so-called Zeldovich pancakes (ZP), interim wall-like structures appearing at nonlinear stages of development of gravitational instability. As the average matter density decreases, the period of the ZP lattice increases diverging at the density of Einstein's Universe. Solitary wave solutions are found at exactly the density of Einstein's Universe, and at a slightly larger density the wave may be viewed as a lattice of well-separated ZP solitons.

Experimental Study on Vertical Shear Behaviors of an Immersion Joint with Steel Shear Keys

The vertical shear behaviors of an immersion joint with steel shear keys subjected to multidirectional loads are investigated in this paper. An experiment of an immersion joint model is carried out. Two kinds of compression–shear tests of the joint are considered in this experiment. The first kind of compression–shear test applies a specific vertical shear load and five different levels of longitudinal compressive loads on the joint. An additional compression–vertical shear destruction test is also conducted under the minimum longitudinal compressive load, wherein the vertical shear load is incrementally increased until failure of the joint. The other kind of compression–shear test is a bidirectional shear test, in which both the longitudinal compressive load and the transverse shear load are fixed, and the vertical shear load is gradually increased until reaching a target value. The results show that the shear force–displacement curves of the joint in any loading case can be divided into two stages: a non-linear development stage and a quasi-linear development stage. The vertical shear stiffness of the joint is found to increase with increasing longitudinal compressive load, and the existence of a transverse shear load enhances this effect. The ultimate shear capacity of the joint is smaller than the sum of the shear capacities of all vertical steel keys. In addition, the failure of the joint appears at the shear key on one sidewall of the joint.

Mean dynamics and transition to turbulence in oscillatory channel flow

The mean dynamics in oscillatory channel flow is examined to investigate the dynamical mechanisms underlying the transition to turbulence in oscillatory wall-bounded flow. The analyses employ direct numerical simulation data acquired at three Stokes Reynolds numbers: $Re_{s}=648$, 801 and 1009, where the lower $Re_{s}$ flow is transitional over the entire cycle and the two higher $Re_{s}$ flows exhibit flow characteristics similar to steady turbulent wall-bounded flow during part of the cycle. The flow evolution over a half-period of oscillation for all three $Re_{s}$ is as follows: near-wall streamwise velocity streaks develop during the early accelerating portion of the cycle; then at some later point in the cycle that depends on $Re_{s}$, the near-wall streaks breakdown (demarking the onset of the nonlinear development stage), and the near-wall Reynolds stress grows explosively; the Reynolds stress remains elevated for part of the cycle before diminishing (yet remaining finite) during the late decelerating portion of the cycle. This process is then repeated indefinitely. The present findings demonstrate that transition to turbulence occurs when the nonlinear development stage begins during the accelerating portion of the cycle. This crucially leads to the diminishing importance of the centreline momentum source, the emergence of a locally accelerating/decelerating internal layer centred about the edge of the Stokes layer and the wall-normal rearrangement of the mean forces prior to the start of the decelerating portion of the cycle. The rearrangement of mean forces culminates in a four layer structure in the mean balance of forces. This is significant on a number of accounts since empirical and theoretical evidence suggests that the formation of a four layer structure is an important characteristic of a self-similar hierarchal structure that underlies logarithmic dependence of the mean velocity profile in steady turbulent wall-bounded flows (Klewicki et al., J. Fluid Mech., vol. 638, 2009, pp. 73–93). When the nonlinear development stage begins during the decelerating portion of the cycle (i.e. at $Re_{s}=648$), a four layer structure is not observed in the mean balance of forces and the flow remains weakly transitional over the entire cycle.

Experimental observation of the bipolar super-regular breathers dynamics induced by modulation instability from arbitrary perturbed background

The solitons dynamics induced by more practical and complex forms of modulation instability (MI) have still not been revealed. In this paper, the solitons dynamics induced by MI with randomly perturbed initial conditions are observed experimentally. The mark of the MI development from linear stage to nonlinear development stage is the appearance of particular classes of super-regular breathers with bipolar complex oscillation structure. The interaction of the multiple sets of opposite quasi-degenerate four wave mixing processes can be treated as a sort of “turbulence” appearing as a result of nonlinear development of MI. Our results open up possibilities to explore the MI that develops from some kind of arbitrary perturbed background.

Instability Development in Shear Flow with an Inflection–Free Velocity Profile and Thin Pycnocline

Weakly stratified flows of the class under study have a wide 3D spectrum of the most unstable waves with very close growth rates and phase velocities so that their individual critical layers merge into a common one. The analysis of evolution equations for those waves has shown that throughout a weakly nonlinear stage of development their amplitudes grow explosively. During the first (three-wave) phase, the most rapidly growing are low-frequency waves whereas at the next phase, when numerous and diverse higher-order wave interactions come into play, the growth of highfrequency waves is accelerated and they overtake low-frequency waves. The results obtained are illustrated by numerical calculations for some ensembles of waves.

Spatio-temporal instabilities in large aperture lasers

The paper is devoted to the theoretical investigation of spatiotemporal dynamics of large aperture class B lasers. The linear stability analysis of spatially homogeneous generation is carried out. It is shown that in such lasers the wave instability is realized, the bifurcation diagrams are constructed. The numerical simulation of spatiotemporal dynamics of the laser confirms the linear analysis results. The nonlinear stage of development of the perturbation demonstrates switching between different spatial modes.

Mean dynamics of transitional boundary-layer flow

The dynamical mechanisms underlying the redistribution of mean momentum and vorticity are explored for transitional two-dimensional boundary-layer flow at nominally zero pressure gradient. The analyses primarily employ the direct numerical simulation database of Wu & Moin (J. Fluid Mech., vol. 630, 2009, p. 5), but are supplemented with verifications utilizing subsequent similar simulations. The transitional regime is taken to include both an instability stage, which effectively generates a finite Reynolds stress profile, −ρuv(y), and a nonlinear development stage, which progresses until the terms in the mean momentum equation attain the magnitude ordering of the four-layer structure revealed by Wei et al. (J. Fluid Mech., vol. 522, 2005, p. 303). Self-consistently applied criteria reveal that the third layer of this structure forms first, followed by layers IV and then II and I. For the present flows, the four-layer structure is estimated to be first realized at a momentum thickness Reynolds number Rθ = U∞ θ/ν ≃ 780. The first-principles-based theory of Fife et al. (J. Disc. Cont. Dyn. Syst. A, vol. 24, 2009, p. 781) is used to describe the mean dynamics in the laminar, transitional and four-layer regimes. As in channel flow, the transitional regime is marked by a non-negligible influence of all three terms in the mean momentum equation at essentially all positions in the boundary layer. During the transitional regime, the action of the Reynolds stress gradient rearranges the mean viscous force and mean advection profiles. This culminates with the segregation of forces characteristic of the four-layer regime. Empirical and theoretical evidence suggests that the formation of the four-layer structure also underlies the emergence of the mean dynamical properties characteristic of the high-Reynolds-number flow. These pertain to why and where the mean velocity profile increasingly exhibits logarithmic behaviour, and how and why the Reynolds stress distribution develops such that the inner normalized position of its peak value, ym+, exhibits a Reynolds number dependence according to $y_m^+ {\,\simeq\,} 1.9 \sqrt{\delta^+}$.

Mean dynamics of transitional channel flow

The redistribution of mean momentum and vorticity, along with the mechanisms underlying these redistribution processes, is explored for post-laminar flow in fully developed, pressure driven, channel flow. These flows, generically referred to as transitional, include an instability stage and a nonlinear development stage. The central focus is on the nonlinear development stage. The present analyses use existing direct numerical simulation data sets, as well as recently reported high-resolution molecular tagging velocimetry measurements. Primary considerations stem from the emergence of the effects of turbulent inertia as represented by the Reynolds stress gradient in the mean differential statement of dynamics. The results describe the flow evolution following the formation of a non-zero Reynolds stress peak that is known to first arise near the critical layer of the most unstable disturbance. The positive and negative peaks in the Reynolds stress gradient profile are observed to undergo a relative movement toward both the wall and centreline for subsequent increases in Reynolds number. The Reynolds stress profiles are shown to almost immediately exhibit the same sequence of curvatures that exists in the fully turbulent regime. In the transitional regime, the outer inflection point in this profile physically indicates a localized zone within which the mean dynamics are dominated by inertia. These observations connect to recent theoretical findings for the fully turbulent regime, e.g. as described by Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) and Klewicki, Fife & Wei (J. Fluid Mech., vol. 638, 2009, p. 73). In accord with momentum equation analyses at higher Reynolds number, the present observations provide evidence that a logarithmic mean velocity profile is most rapidly approximated on a sub-domain located between the zero in the Reynolds stress gradient (maximum in the Reynolds stress) and the outer region location of the maximal Reynolds stress gradient (inflection point in the Reynolds stress profile). Overall, the present findings provide evidence that the dynamical processes during the post-laminar regime and those operative in the high Reynolds number regime are connected and describable within a single theoretical framework.

Rayleigh-Taylor instability of viscoplastic liquid

The Rayleigh-Taylor instability of viscoplastic medium (VPM) is considered, which medium corresponds to real VPM or to the high-temperature limit of highly elastic-viscoplastic fluids. An effective rheological model is provided by the Bingham-Shvedov model (BSM) which is similar to the dry friction model in the mechanics of deformable solid. The main feature of BSM is the existence of yield stress corresponding to dry friction limit. Well-known in the literature is the solution for round pipe flow of BSM, in the case of which the presence of yield stress causes the emergence of a moving core at the center of flow. It is the prime objective of this study to construct a numerical model of three-dimensional flow of BSM in the gravity field, which is accompanied by a nonlinear stage of development of instability of VPM, and to analyze the correlation between the development of instability and yield stress.

A quantification model for apoptosis in mouse embryos in the early stage of fetation

Apoptosis is the most important inducement and modulator for embryos in the early stage of fetation, i.e. after the 8-cell stage, mostly the morula and blastula stage, to proceed to the stage of nonlinear development. Using a two-photon laser scanning microscopy (TPLSM) system, we obtained 3-dimensional (3D) fluorescent images of preimplantation mouse embryos. A model for quantification was established. The statistical results for the spatial location of apoptosis bodies in embryos was obtained following image processing, as well as investigation of the kinetics of apoptosis. It was found that most (70%) apoptosis occurred in the trophectoderm, and the departure between the centroid and geometric center of embryos had a step transition when embryos developed into the 32-cell stage, which was consistent with the theoretical prediction that the blastocele would induce a symmetry break of the distribution of cells in embryos.

X and Y waves in the spatiotemporal Kerr dynamics of a self-guided light beam

The nonlinear stage of development of the spatiotemporal instability of the monochromatic Townes beam in a medium with self-focusing nonlinearity and normal dispersion is studied by analytical and numerical means. Small perturbations to the self-guided light beam are found to grow into two giant, splitting Y pulses featuring shock fronts on opposite sides. Each shocking pulse amplifies a co-propagating X wave, or dispersion- and diffraction-free linear wave mode of the medium, with super-broad spectrum.

Numerical simulation of fine structure in the Io plasma torus produced by the centrifugal interchange instability

The Io plasma torus as a whole has a radial width scale ∼1 RJ, much larger than the width of the localized Io plasma source (∼1 RIo ≈ RJ/39). One of the most prominent features of the Io torus observed by the Voyager spacecraft and Earth‐based instruments is the “ribbon” structure near Io's orbit. The stability properties of this narrower ribbon structure embedded within the larger torus have been investigated by the Rice Convection Model for Jupiter. Four initial plasma distributions having different radial widths are each represented by 82 longitudinally symmetric edges establishing 41 levels of the flux tube mass content η with the peak η value at Io's orbit. The same initial perturbation is put on each of these edges and is subjected to centrifugal interchange. Our simulations produce regularly spaced long, thin fingers moving outward from the outer edges. It is shown that the azimuthal width of the interchange convection cells (the distance between outflowing fingers in the nonlinear stage of development) is proportional to the radial width scale of the initial distribution that produced them. The constant of proportionality is ≈0.5. Since the exponential growth rate is essentially proportional to the azimuthal wave number of the disturbance and hence is inversely proportional to its azimuthal width, the ribbon‐scale interchange structures grow faster than torus‐scale interchange structures.

Nonlinear mechanisms of sound generation in a perturbed parallel jet flow

An initial value problem with relevance to jet noise is investigated. A plane parallel jet flow is subjected to a spatially localized initial disturbance and is then left to evolve according to the two-dimensional compressible Navier–Stokes equations. The hydrodynamic response is in the form of a convecting vortex packet. The Ffowcs Williams–Hawkings approach is formulated in the time domain and used to extrapolate from the simulated near field to the acoustic far field. The predominant downstream sound radiation comes from an early stage of nonlinear development of the vortex packet. Two simplified models to account for the radiation are introduced, based on nonlinear mode interactions on a prescribed base flow. The first uses two sets of linearized Euler equations, coupled via the inviscid Lilley–Goldstein acoustic analogy. This formulation separates the linear sound field from the sound field driven by nonlinear interactions; qualitative agreement of the latter with the Navier–Stokes computations demonstrates the importance of nonlinear interactions. The second model uses combinations of linear inviscid eigenmodes to drive the sound field, which allows extraction of the dominant mode interactions responsible for the observed radiation pattern. The results indicate that a difference-wavenumber nonlinear interaction mechanism dominates sound radiation from subsonic instability modes in shear flows.

The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks

We present a series of high-resolution, three-dimensional hydrodynamics simulations of a gravitationally unstable solar nebula model. The influences of both azimuthal grid resolution and the treatment of thermal processes on the origin and evolution of gravitational instabilities are investigated. In the first set of simulations, we vary the azimuthal resolution for a locally isothermal simulation, doubling and quadrupling the resolution used in a previous study; the largest number of grid points is (256, 256, 64) in cylindrical coordinates (r, , z). At this resolution, the disk breaks apart into a dozen short-lived condensations. Although our previous calculations underresolved the number and growth rate of clumps in the disk, the overall qualitative, but fundamental, conclusion remains: fragmentation under the locally isothermal condition in numerical simulations does not in itself lead to the survival of clumps to become gaseous giant protoplanets. Since local isothermality represents an extreme assumption about thermal processes in the disk, we also present several extended simulations in which heating from an artificial viscosity scheme and cooling from a simple volumetric cooling function are applied to two different models of the solar nebula. The models are differentiated primarily by disk temperature: a high-Q model generated directly by our self-consistent field equilibrium code and a low-Q model generated by cooling the high-Q model in a two-dimensional version of our hydrodynamics code. Here, high-Q and low-Q refer to the minimum values of the Toomre stability parameter Q in each disk, Qmin = 1.8 and 0.9, respectively. Previous simulations, by ourselves as well as others, have focused on initial states that are already gravitationally unstable, i.e., models similar to the low-Q model. This paper presents for the first time the numerical evolution of an essentially stable initial equilibrium state (the high-Q model) to a severely unstable one by cooling. The additional heating and cooling are applied to each model over the outer half of the disk or the entire disk. The models are subject to the rapid growth of a four-armed spiral instability; the subsequent evolution of the models depends on the thermal behavior of the disk. The cooling function tends to overwhelm the heating included in our artificial viscosity prescription, and as a result the spiral structure strengthens. The spiral disturbances transport mass at prodigious rates during the early nonlinear stages of development and significantly alter the disk's vertical surface. Although dense condensations of material can appear, their character depends on the extent of the volumetric cooling in the disk. In the simulation of the high-Q model with heating and cooling applied throughout the disk, thin, dense rings form at radii ranging from 1 to 3 AU and steadily increase in mass; later companion formation may occur in these rings as cooling drives them toward instability. When heating and cooling are applied only over the outer radial half of the disk, however, a succession of single condensations appears near 5 AU. Each clump has roughly the mass of Saturn, and some survive a complete orbit. Since the clumps form near the artificial boundary in the treatment of the disk gas physics, the production of a clump in this case is a numerical artifact. Nevertheless, radially abrupt transitions in disk gas characteristics, for example, in opacity, might mimic the artificial boundary effects in our simulations and favor the production of stable companions in actual protostellar and protoplanetary disks. The ultimate survival of condensations as eventual stellar or substellar companions to the central star is still largely an open question.

Stability of Couette flow of an ideal fluid with free boundaries

The linear and nonlinear stage of development of instability of Couette flow with two free boundaries is studied. It is established that instability occurs only for long waves, and the critical wave number is computed. In the presence of surface-tension forces, instability is preserved only at Weber numbersWe ≤ 1/3.

Nonlinear interaction of a large vortex with a boundary layer

The nonlinear problem of boundary layer instability under the influence of a plane vortex is investigated for high Reynolds numbers. The vortex occupies the entire thickness of the boundary layer and has a longitudinal dimension of the order of the Tollmien-Schlichting wavelength. The initial vortex is rapidly swept away by the flow, inducing a Stokes layer near the surface of the plate. Expanding, this layer reaches the dimensions of the viscous sublayer of free interaction theory, where wave packet generation takes place. In the case in question a feature of the nonlinear stage of development of the disturbances is the formation of a concentrated vortex, which arises in the Stokes layer and grows rapidly, whereas the wave packet propagated ahead of it remains linear. From the calculations there emerges a tendency for the new vortex to be formed above the wail, whereas the maximum vorticity of the vortex generated in the Stokes layer corresponds to the wall itself.

Statistical characteristics of polarization of pulse signals in vertical probing of the ionosphere

In the model of equatorial bubbles (2)-(6), the effects of plasma diamagnetism are taken into account. The results of numerical experiments show that the gradient plasma drift and related diamagnetic currents have no marked effect on the dynamics and structure of RayleighTaylor inhomogeneities greatly extended along the field lines, for both the linear and nonlinear stages of development. The action of the gradient drift is manifested only in the formation of the structure of the current cells. For the boundary conditions (7) and (8), eliminating the local frequency of solutions of system (2)-(6), the diamagnetic current in the region of integration forms a structure in the shape of Greek letter g. This current flows into the region occupied by the inhomogeneity in the west, flows around it in a clockwise direction, and flows out in the east. Disregarding the diamagnetism, the current lines are practically parallel to the y-axis.