Localized Initial Perturbation Research Articles

Three-dimensional propagation of kink wave trains in solar coronal slabs

ABSTRACT Impulsively excited wave trains are of considerable interest in solar coronal seismology. To our knowledge, however, it remains to examine the three-dimensional (3D) dispersive propagation of impulsive kink waves in straight, field-aligned, symmetric, low-beta, slab equilibria that are structured only in one transverse direction. We offer a study here, starting with an analysis of linear oblique kink modes from an eigenvalue problem perspective. Two features are numerically found for continuous and step structuring alike, one being that the group and phase velocities may lie on opposite sides of the equilibrium magnetic field ($\boldsymbol {B}_0$), and the other being that the group trajectories extend only to a limited angle from $\boldsymbol {B}_0$. We justify these features by making analytical progress for the step structuring. More importantly, we demonstrate by a 3D time-dependent simulation that these features show up in the intricate interference patterns of kink wave trains that arise from a localized initial perturbation. In a plane perpendicular to the direction of inhomogeneity, the large-time slab-guided patterns are confined to a narrow sector about $\boldsymbol {B}_0$, with some wavefronts propagating toward $\boldsymbol {B}_0$. We conclude that the phase and group diagrams lay the necessary framework for understanding the complicated time-dependent behaviour of impulsive waves.

Numerical simulation of solitary gravity waves on deep water with constant vorticity

We present a numerical study of essentially nonlinear dynamics of surface gravity waves on deep water with constant vorticity using governing equations in conformal coordinates. The dispersion relation of surface gravity waves on shear flow is known to have two branches; one of which is weakly dispersive for long waves. Weakly nonlinear evolution of the waves of this branch can be described by the Benjamin–Ono equation, which is integrable and has soliton and multi-soliton solutions. Currently, the extent to which the properties of such solitary waves obtained within the weakly nonlinear model are preserved in the exact Euler equations is unknown. We investigate the behavior of this class of solitary waves without the restrictive assumption of weak nonlinearity by using the exact Euler equations. The evolution of localized initial perturbations leading to the formation of single or multiple solitary waves is modeled, and the properties of finite-amplitude solitary waves are discussed. We show that within the framework of the exact equations, two-soliton collisions are almost elastic, but in contrast to solutions of the Benjamin–Ono equation, the waves receive a phase shift as a result of the interaction.

Boundary layer collapses described by the two-dimensional intermediate long-wave equation

We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter D. In the limits of large and small D, the 2d-ILW equation respectively tends to the 2d Benjamin-Ono and 2d Zakharov-Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude a and width σ, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane (a, σ) for various values of D. The amplitude threshold a increases as D and σ decrease and tends to infinity at D → 0. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all D except D = 0. But the equation itself has not been proved for small D. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.

Formation of the Shock Pressure Regime in the Form of a Traveling Wave in Nematic Twisted Cells

The paper presents a numerical study, within a nonlinear generalization of the classical Ericksen–Leslie theory, of a new nonlinear mechanism for the formation of the regime of shock pressure on the boun-ding walls of a microsized twisted nematic cell (TNC), realized in the form of the kinklike traveling wave $$\mathcal{P}$$(z – $${v}t$$) under the effect of the externally applied electric field and induced by a strong local distribance of the director field in the form of a Gaussian (normal) distribution. The mechanisms responsible for the formation of the traveling wave of shock pressure propagating in the TNC from one of its boundaries to another are studied, and it is shown how the magnitude of the electric field and the shape of the localized initial perturbation of the director field affect the similarity of the traveling wave to the kinklike wave. Studies of the dynamic relaxation of the director field in TNCs also showed that, at temperatures exceeding the temperature for the nematic–smectic A (NA) TNA phase transition by several dozen mK, fluctuations in the order parameter of the emerging smectic phase suppress the effect of the electric field and favor the singular behavior of the azimuthal anchoring energy density as T → TNA.

Формирование режима ударного давления в форме бегущей волны в нематических твистовых ячейках

Abstract The paper presents a numerical study, within a nonlinear generalization of the classical Ericksen–Leslie theory, of a new nonlinear mechanism for the formation of the regime of shock pressure on the boun-ding walls of a microsized twisted nematic cell (TNC), realized in the form of the kinklike traveling wave $$\mathcal{P}$$ ( z – $${v}t$$ ) under the effect of the externally applied electric field and induced by a strong local distribance of the director field in the form of a Gaussian (normal) distribution. The mechanisms responsible for the formation of the traveling wave of shock pressure propagating in the TNC from one of its boundaries to another are studied, and it is shown how the magnitude of the electric field and the shape of the localized initial perturbation of the director field affect the similarity of the traveling wave to the kinklike wave. Studies of the dynamic relaxation of the director field in TNCs also showed that, at temperatures exceeding the temperature for the nematic–smectic A (NA) T _NA phase transition by several dozen mK, fluctuations in the order parameter of the emerging smectic phase suppress the effect of the electric field and favor the singular behavior of the azimuthal anchoring energy density as T → T _NA.

Nonlinear flavor development of a two-dimensional neutrino gas

We present a numerical survey of the nonlinear flavor development of dense neutrino gases. This study is based on the stationary, two-dimensional ($x$ and $z$), two-beam, monochromatic neutrino line model with a periodic boundary condition along the $x$ direction. Similar to a previous work, we find that small-scale flavor structures can develop in a neutrino gas even if the physical conditions are nearly homogeneous along the $x$ axis initially. The power diffusion from the large-scale to small-scale structures increases with the neutrino density and helps to establish a semi-exponential dependence of the magnitudes of the Fourier moments on the corresponding wave numbers. The overall flavor conversion probabilities in the neutrino gases with small initial sinusoidal perturbations reach certain equilibrium values at large distances which are mainly determined by the neutrino-antineutrino asymmetry. Similar phenomena also exist in a neutrino gas with a localized initial perturbation, albeit only inside an expanding flavor conversion region. Our work suggests that a statistical treatment may be possible for the collective flavor oscillations of a dense neutrino gas in a multi-dimensional environment.

L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves

Assuming spectral stability conditions of periodic reaction-diffusion waves u¯(x), we consider L1(R)-nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. Lp(R)-nonlinear stability of such waves has been studied in Johnson et al. (2013) for p≥2 by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in Jung, (2012) and Jung and Zumbrun (2016), we extend Lp(R)-nonlinear stability (p≥2) in Johnson et al. (2013) to L1(R)-nonlinear stability. More precisely, we obtain L1(R)-estimates of modulated perturbations u~(x-ψ(x,t),t)-u¯(x) of u¯ with a phase function ψ(x,t) under small initial perturbations consisting of localized initial perturbations u~(x-h0(x),0)-u¯(x) and nonlocalized initial modulations h0(x)=ψ(x,0).

Oscillation structure of localized perturbations in modulationally unstable media.

We characterize the properties of the asymptotic stage of modulational instability arising from localized perturbations of a constant background, including the number and location of the individual peaks in the oscillation region. We show that, for long times, the solution tends to an ensemble of classical (i.e., sech-shaped) solitons of the focusing nonlinear Schrödinger equation (as opposed to the various breatherlike solutions of the same equation with a nonzero background). We also confirm the robustness of the theoretical results by comparing the analytical predictions with careful numerical simulations with a variety of initial conditions, which confirm that the evolution of modulationally unstable media in the presence of localized initial perturbations is indeed described by the same asymptotic state.

Phonon Excitation and Energy Redistribution in Phonon Space for Energy Dissipation and Transport in Lattice Structure with Nonlinear Dispersion

We first propose fundamental solutions of wave propagation in dispersive chain subject to a localized initial perturbation in the displacement. Analytical solutions are obtained for both second order nonlinear dispersive chain and homogenous harmonic chain using stationary phase approximation. Solution is also compared with numerical results from molecular dynamics (MD) simulations. Locally dominant phonon modes (k-space) are introduced based on these solutions. These locally defined spatially and temporally varying phonon modes k(x, t) are critical to the concept of the local thermodynamic equilibrium (LTE). Wave propagation accompanying with the nonequilibrium dynamics leads to the excitation of these locally defined phonon modes. It is found that the system energy is gradually redistributed among these excited phonons modes (k-space). This redistribution process is only possible with nonlinear dispersion and requires a finite amount of time to achieve a steady state distribution. This time scale is dependent on the spatial distribution (or frequency content) of the initial perturbation and the dispersion relation. Sharper and more concentrated perturbation leads to a faster energy redistribution and dissipation. This energy redistribution generates localized phonons with various frequencies that can be important for phonon-phonon interaction and energy dissipation in nonlinear systems. Depending on the initial perturbation and temperature, the time scale associated with this energy distribution can be critical for energy dissipation compared to the Umklapp scattering process. Ballistic type of heat transport along the harmonic chain reveals that at any given position, the lowest mode (k = 0) is excited first and gradually expanding to the highest mode (kmax(x,t)), where kmax(x,t) can only asymptotically approach the maximum mode kB of the first Brillouin zone (kmax(x,t) → kB). No energy distributed into modes with kmax(x,t) < k < kB demonstrates that the local thermodynamic equilibrium cannot be established in harmonic chain. Energy is shown to be uniformly distributed in all available phonon modes k ≤ kmax(x,t) at any position with heat transfer along the harmonic chain. The energy flux along the chain is shown to be a constant with time and proportional to the sound speed (ballistic transport). Comparison with the Fourier' law leads to a time-dependent thermal conductivity that diverges with time.

Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source

We construct asymptotic solutions to the wave equation with velocity rapidly oscillating against a smoothly varying background and with localized initial perturbations. First, using adiabatic approximation in the operator form, we perform homogenization that leads to a linearized Boussinesq-type equation with smooth coefficients and weak “anomalous” dispersion. Then, asymptotic solutions to this and, as a consequence, to the original equations are constructed by means of a modified Maslov canonical operator; for initial perturbations of special form, these solutions are expressed in terms of combinations of products of the Airy functions of a complex argument. On the basis of explicit formulas obtained, we analyze the effect of fast oscillations of the velocity on the solution fronts and solution profiles near the front.

Localized subcritical convective cells in temperature-dependent viscosity fluids

Numerical simulations of infinite Prandtl number convection in the stagnant lid regime of temperature-dependent viscosity convection demonstrate the existence of spatially localized, stable convective cells below the critical Rayleigh number (subcritical convection). These solutions are in stark contrast to the usual, supercritical, convective planforms, where convective cells form in the entire layer. The isolated cell has a shape of an axisymmetric dome with an upwelling at the center and thus appears as a very weak plume. Formation of these structures requires subcritical conditions and a localized initial temperature perturbation but does not require any spatial heterogeneity in the material properties or the heat flux. When several localized plumes form, they tend to attract to each other and form stable clusters. This type of subcritical convection may play a role in the formation and longevity of localized features on planetary bodies, including the crustal dichotomy and Tharsis region on Mars and the asymmetric pattern of volcanism on Mercury.

Numerical simulation of dispersive Alfvén wave turbulence in auroral plasmas affecting the Earth's communications

This paper presents the numerical simulation of the model equation governing the nonlinear dynamics of dispersive Alfvén waves (DAWs) in low-β plasmas. When the nonlinearity arises due to the ponderomotive force and Joule heating-driven density perturbations, the model equation turns out to be a modified nonlinear Schrödinger (MNLS) equation. The nonlinear MNLS is used to investigate the evolution of a finite-amplitude localized initial perturbation. The initial perturbation is of the form of a sinusoidal pulse on a DAW having a Gaussian wavefront. The localized structures of DAWs with sidebands are obtained. Also, we have studied the power spectra of magnetic fluctuations with different scaling laws. These turbulent structures may be responsible for particle acceleration in the aurora region and may affect the Earth's communication.

Generalized nonlinear Schrödinger equation as a model for turbulence, collapse, and inverse cascade

A two-dimensional generalized cubic nonlinear Schrödinger equation with complex coefficients for the group dispersion and nonlinear terms is used to investigate the evolution of a finite-amplitude localized initial perturbation. It is found that modulation of the latter can lead to sideband formation, wave condensation, collapse, turbulence, and inverse energy cascade, although not all together and not in that order.

Optimal wave packets in a boundary layer and initial phases of a turbulent spot

The three-dimensional global optimal dynamics of a flat-plate boundary layer is studied by means of an adjoint-based optimization in a spatial domain of long – but finite – streamwise dimension. The localized optimal initial perturbation is characterized by a pair of streamwise-modulated counter-rotating vortices, tilted upstream, yielding at the optimal time elongated streaks of alternating sign in the streamwise direction. This indicates that perturbations with non-zero streamwise wavenumber have a role in the transient dynamics of a boundary layer. A scaling law is provided, describing the variation of the streamwise modulation of the optimal initial perturbation with respect to the streamwise domain length and to the Reynolds number. For spanwise-extended domains, a near-optimal three-dimensional perturbation is extracted during the optimization process; it is localized also in the spanwise direction, resulting in a wave packet of elongated disturbances modulated in the spanwise and streamwise directions. The nonlinear evolution of the optimal and near-optimal perturbations is investigated by means of direct numerical simulations. Both perturbations are found to induce transition at lower levels of the initial energy than local optimal and suboptimal perturbations. Moreover, it is observed that transition occurs in a well-defined region of the convected wave packet, close to its centre, via a mechanism including at the same time oscillations of the streaks of both quasi-sinuous and quasi-varicose nature. Hairpin vortices are observed before transition; they have an active role in the breakdown of the streaks and result in a turbulent spot which spreads out in the boundary layer.

Localized edge states in plane Couette flow

The dynamics at the threshold of transition in plane Couette flow is investigated numerically in a large spatial domain for a certain type of localized initial perturbation, for Re between 350 and 1000. The corresponding edge state is an unsteady spotlike structure, localized in both streamwise and spanwise directions, which neither grows nor decays in size. We show that the localized nature of the edge state is numerically robust, and is not influenced by the size of the computational domain. The edge trajectory appears to transiently visit localized steady states. This suggests that basic spatiotemporally intermittent features of transition to turbulence, such as the growth of a turbulent spot, can be described as a dynamical system.

Evolution of initially localized perturbations in stratified ionized discs

A detailed solution of an initial value problem of a vertically localized initial perturbation in rotating magnetized vertically stratifled disk is presented. The appropriate linearized MHD equations are solved by employing the WKB approximation and the results are verifled numerically. The eigenfrequencies as well as eigenfunctions are explicitly obtained. It is demonstrated that the initial perturbation remains conflned within the disk. It is further shown that thin enough disks are stable but as their thickness grows increasing number of unstable modes participate in the solution of the initial value problem. However it is demonstrated that due to the localization of the initial perturbation the growth time of the instability is signiflcantly longer than the calculated inverse growth rate of the individual unstable eigenfunctions.

Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

Formation of structured islands by ion-beam erosion

Ion-beam-eroded semiconductor materials can generically show, depending on the experimental parameters and setup, a plethora of self-organized large-area surface patterns. However, as recently reported by Allmers et al. [J. Vac. Sci. Technol. B 24, 582 (2006)], localized structures such as islands covered with hexagonally arranged dots surrounded by a basically flat surface can also evolve. Extending a recently developed continuum approach for the evolution of surface morphologies by ion-beam erosion, we present two scenarios how such localized structures can be understood: (i) by ion-beam inhomogeneities or (ii) by finite localized initial surface perturbations.

Exact periodic and localized solutions of the equationh t =Δ inh

New exact regular solutions of the nonlinear-diffusion equation are found. Various types of evolution of certain classes of localized initial perturbations are described. We show that, when a localized distribution in the form of a ring is specified, the instantaneous occurrence of the singularity in its center results from the diffusive spreading.

Structures of dynamically unstable shear flows and their implications for shallow internal gravity waves part II: Nonlinear response

The nonlinear response of a dynamically unstable shear flow with critical level to an initial temperature anomaly is investigated using a nonlinear numerical model. Both nonconstant and constant shear profiles of the basic flow are considered. Effects of the solid lower boundary on the dynamically unstable, nonlinear flow are also studied. It is found that in a dynamically unstable, linear flow with a hyperbolic tangent wind profile, the updraft is tilted upshear. The result in consistent with that of a linear stability model (LC). The upshear tilt can be explained by the Orr mechanism (1907) and the energy argument proposed by LC. In a dynamically unstable, nonlinear flow, the updrafts produced by a sinusoidal initial temperature perturbation are stronger in the lower layer and are more compact and located further apart compared to the corresponding linear flow. In addition, the perturbed wave energy is slightly smaller than the linear case. It is found that the growth rate is smaller during the early stage and much larger during the later stage. For a localized initial temperature perturbation in a dynamically unstable flow, a stronger updraft with two compensated downdrafts are produced. Gravity waves are produced in a dynamically stable flow with both a hyperbolic tangent wind profile and a linear wind profile. For a linear shear flow with Richardson number less than 1/4, the disturbance grows in the early stage and then decays algebraically at later times, similar to that found in other linear theoretical studies. The influence of the solid lower boundary is to suppress the shear instability in a nonlinear flow with a hyperbolic tangent wind profile ofRi<1/4.