Finite-amplitude Perturbations Research Articles

Three-dimensional localized Rayleigh–Bénard convection in temperature-dependent viscosity fluids

The stability range of localized three-dimensional convective cells in Rayleigh–Bénard convection is determined across a broad range of viscosity contrasts between the boundaries of the fluid layer, for both free-slip and no-slip boundary conditions. The localized convective cell is generated by a finite-amplitude initial perturbation at subcritical Rayleigh numbers. It appears as a radially symmetric upwelling surrounded by nearly stagnant fluid, which can be characterized as an extremely weak plume. The parameter range in which three-dimensional localized upwellings are stable is slightly larger than that found for two-dimensional rolls. With free-slip boundaries, the lowest viscosity contrast at which the three-dimensional system can exhibit localization is approximately 35, about four times lower than for two-dimensional rolls. The wide range of conditions under which localization occurs in three-dimensional systems due to temperature-dependent viscosity further emphasizes its importance for the understanding of processes within the interiors of planetary bodies and for industrial applications.

On a priori bounding the growth of thermal instability waves

We have previously shown that the nonlinear growth of a finite-amplitude perturbation to a basic state given by a baroclinic zonal flow on the β-plane in a thermal quasigeostrophic reduced-gravity model can be a priori bounded. In this note, we show that, unlike we stated earlier, Lyapunov stability can be proved even when buoyancy varies linearly with the meridional coordinate. In addition to rectifying our earlier results, we expand them by deriving an instability saturation bound by making use of the existence of such a class of Lyapunov-stable basic states. This bound can be smaller than that one we estimated before, reinforcing our previous conclusions. We also present a numerical test of the accuracy of the derived bound.

A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

Onset of convection in internally heated, temperature-dependent, power-law viscosity fluids at large viscosity contrasts

We use two-dimensional numerical simulations to investigate the finite-amplitude onset of convection in internally heated, infinite Prandtl number fluids with strongly temperature-dependent, power-law viscosity. We focus on the stagnant-lid regime which is relevant to planetary interiors. We find that convection can occur in both the usual, widespread convection planform, where the convection cells form in the entire layer, as well as the localized convection planform characterized by a single upwelling surrounded by a nearly stationary fluid. The critical Rayleigh number for the onset of convection by finite-amplitude perturbations is nearly independent of the convection planform in a broad range of the spacings between upwellings, from of the order of the depth of the layer to up to infinity. A simple heuristic analysis suggests scaling relationships which fit the numerical results in the stagnant-lid regime for an arbitrary power-law exponent, n>1. The results of this study provide new fluid dynamical constrains on the threshold for convection in planetary interiors.

Formation of multiple BGK-like structures in the time-asymptotic state of collisionless Vlasov-Poisson plasmas.

The time-asymptotic state of a finite-amplitude perturbation in a collisionless and Maxwellian plasma is typically represented as a steady state of two nonlinearly superposed, counterpropagating Bernstein-Greene-Kruskal (BGK) modes. Using high-resolution Vlasov-Poisson simulations, we show that the plasma evolves self-consistently into a time-asymptotic state of multiple vortexlike structures that gradually fill the phase space and reduce filamentation. This occurs without the need for external forcing or the presence of an energetic plasma population. This finding suggests that the time-asymptotic regime of the plasma is rather akin to a nonlinear superposition of multiple BGK-like modes associated with nearly constant phase-speed waves. The electric field and the space-averaged particle distribution function exhibit a power-law broad spectrum, which is consistent with an energy cascade towards smaller scales in both position and velocity spaces.

Controlling the breakup of spiralling jets: results from experiments, nonlinear simulations and linear stability analysis

We experimentally and numerically study the dynamics of a liquid jet issued from a rotating orifice, whose breakup is regulated by a vibrating piezo element. The helical trajectory of the spiralling jet yields fictitious forces varying along the jet whose longitudinal projections stretch and thin the jet, affecting the growth of perturbations. We show that by quantifying these fictitious forces, one can estimate the jet intact length and size distribution of drops formed at jet breakup. The presence of the locally varying fictitious forces may render high-frequency perturbations, that would otherwise be stable in the abscence of stretching, unstable, as observed similarly in the case of straight jets stretching under gravity. The perturbation amplitude then dictates how strong the perturbation is coupled to the jet compared with random noise that is inherently present in any experimental set-up. In the present study we exploit the slenderness of the jet to separate the calculation of the base flow and the growth of perturbations. The fictitious forces calculated from the base flow trajectory are then used in a nonlinear slender-jet model, which treats the spiralling jet as a quasi-straight jet with locally varying body forces. We show both experimentally and numerically that jet breakup characteristics (e.g. intact length and drop size distribution) can be controlled by finite-amplitude perturbations created by mechanically induced pressure modulations. Finally, we revisit the integrated net gain approach developed for straight jets under gravity and we provide simple analogous relations for spiralling jets.

The unpredictable nature of bubble evolution

Unpredictable dynamics arising from a sensitivity to initial conditions is commonly associated with chaos. We demonstrate how similar unpredictability manifests in a nonlinear system that possesses a large number of long-term outcomes, namely the propagation of an air bubble within a viscous fluid-filled channel. The system under investigation supports various stable states of single-bubble propagation. In addition, bubbles can readily break up during their propagation. Upon subjecting steadily-propagating bubbles to finite-amplitude perturbations in the form of localised channel constrictions, we identify localised regions of the driving flow rate for which the resulting evolutions are unpredictable. Visibly-indistinguishable bubbles are observed to evolve towards a multitude of long-term outcomes, including each of the stable states available to the initial bubble and various states of permanently-changed bubble topology. By combining high-precision experimental results with simulations of a depth-averaged lubrication model of the system, we determine that this behaviour is driven by a sensitive dependence on initial conditions within the vicinity of an unstable periodic orbit.

Laminar–turbulent intermittency in pipe flow for an Herschel–Bulkley fluid: Radial receptivity to finite-amplitude perturbations

We investigate the laminar-to-turbulent transition for non-Newtonian Herschel–Bulkley fluids that exhibit either a shear-thinning or shear-thickening behavior. The reduced-order model developed in this study also includes the effect of yield-stress for the fluid. Within our model framework, we investigate how the Newtonian dynamics change when significant non-Newtonian effects are considered either via the flow index n or the yield-stress τ0 or both. We find that an increase in τ0 as well as a decrease in n lead to a delayed transition if a perturbation of the given turbulent intensity is injected at various radial locations. As the radial position of the injection for the perturbation is varied in this study, our reduced-order model allows for the investigation of the flow receptivity to the finite-amplitude perturbations and to their radial position of inception. We observe that, for a given mean flow profile, the same perturbation becomes more prone to induce turbulence the closer it approaches the wall because of its initial amplitude being relatively higher with respect to the local mean flow. An opposite trend is found when the perturbation amplitude is rescaled on the local mean flow.

Stability and bifurcation of dynamic contact lines in two dimensions

The moving contact line between a fluid, liquid and solid is a ubiquitous phenomenon, and determining the maximum speed at which a liquid can wet/dewet a solid is a practically important problem. Using continuum models, previous studies have shown that the maximum speed of wetting/dewetting can be found by calculating steady solutions of the governing equations and locating the critical capillary number,$Ca_{{crit}}$, above which no steady-state solution can be found. Below$Ca_{{crit}}$, both stable and unstable steady-state solutions exist and if some appropriate measure of these solutions is plotted against$Ca$, a fold bifurcation appears where the stable and unstable branches meet. Interestingly, the significance of this bifurcation structure to the transient dynamics has yet to be explored. This article develops a computational model and uses ideas from dynamical systems theory to show the profound importance of the unstable solutions on the transient behaviour. By perturbing the stable state by the eigenmodes calculated from a linear stability analysis it is shown that the unstable branch is an ‘edge’ state that is responsible for the eventual dynamical outcomes and that the system can become transient when$Ca< Ca_{{crit}}$due to finite-amplitude perturbations. Furthermore, when$Ca>Ca_{{crit}}$, we show that the trajectories in phase space closely follow the unstable branch.

How far below the critical Rayleigh number can convection occur in temperature-dependent viscosity fluids?

We use 2D numerical simulations to study the onset of convection below the critical Rayleigh number in a basally-heated fluid with temperature-dependent viscosity. The existence of convection below the critical Rayleigh number has previously been demonstrated in variable-viscosity fluids. Under these, subcritical conditions, convection can be initiated by finite-amplitude perturbations and can assume a number of different steady-state planforms. We find the absolute minimum critical Rayleigh number and aspect ratio for the onset of subcritical convection in a broad range of viscosity contrasts. For various planetary bodies, the threshold for the onset of convection by finite-amplitude perturbations is about 40% below the critical Rayleigh number. The lowest viscosity contrast at which subcritical convection can occur is 1.3 × 102. The existence of subcritical regime extends the range of conditions under which convection can occur in planetary interiors and expands the assortment of possible convective planforms that can explain various surface features on planetary bodies.

Устойчивость адвективного течения во вращающемся слое проводящей жидкости, помещенной в постоянное однородное магнитное поле

An exact solution of the Navier-Stokes equations in the Boussinesq approximation is presented. It describes advective flow in a flat rotating layer of an incompressible conducting fluid, on the horizontal boundaries of which a linear temperature distribution is set. The rotation axis is perpendicular to the fluid layer. The layer is placed in a constant homogeneous magnetic field oriented opposite to the force of gravity. The behavior of the flow velocity and temperature depending on the Taylor and Hartmann numbers is investigated. The stability of advective flow under normal perturbations is investigated within the framework of linear theory using a new numerical method. The problem is reduced to a one-dimensional system of partial differential equations with unknown functions depending on time and the vertical coordinate, which is solved using the grid method. Calculations have shown that a weak magnetic field increases the stability of the flow without changing the oscillatory character of instability. For the nonlinear formulation of the problem, the finite-amplitude perturbations in the supercritical region near the minima of neutral curves are studied numerically. Calculations have shown that thermal spots moving in opposite directions appear near the upper and lower boundaries of the layer. Near the horizontal boundaries of the layer forms a system of running helical vortices, arising in the region with unstable temperature stratification. The magnetic field strength perturbation has a spiral shape.

Сплошносредная модель биоконвекции с центростремительной силой

The paper is devoted to mathematical modeling of the unusual collective behavior of emperor penguins in winter. Recent observations of flocks of penguins of this species in their natural habitat have shown that penguins use various forms of adaptive behavior that help them survive in theconditions of the Antarctic winter. Despite the fact that each penguin is well protected from cold, these sea-birds are periodically forced to gather in dense groups, within which the temperature necessary for survival is maintained. Observations have revealed the effect of a sudden liquefaction of a dense huddle of penguins, accompanied by a macroscopic circulation of birds from the edge of the group to its center and back. This paper is the first to assert that the observed phenomenon is bioconvection, which should be interpreted as a second-order phase transition. Moreover, the circulation of birds in a dense group can be explained by the mechanism of thermal convection, with the desire of penguins to move toward the temperature maximum playing the role of an effective force field. In the continuum approximation, we propose a mathematical model of the phenomenon, which is reduced to the equation of a self-gravitating porous disk saturated with an incompressible fluid that generates heat. We derive the governing equations in the Darcy-Boussinesq approximation and formulate a nonlinear boundary value problem. An exact solution of the linearized problem for infinitesimal perturbations of the base state is obtained, and the critical values of the control parameter for the onset of bioconvection are calculated. For finite-amplitude perturbations, the boundary value problem is solved using a finite-difference method. We show that in the case of axial symmetry, the most dangerous is the four-vortex motion. In the case of wind breaking the symmetry of the problem, the two-vortex circulation of penguins becomes the most dangerous. The numerical results obtained based on our theoretical developments are compared with observations of penguins in the Antarctic.

Spot formation in three-dimensional Yukawa liquid

Dynamics of a three-dimensional (3D) plane Couette flow (PCF), which subjected to a 3D finite amplitude particle velocity perturbation, is addressed using 3D “classical first principles” molecular dynamics simulation with screened Coulomb potential or a Yukawa potential as the inter-particle interaction. Such systems are often realized in complex plasmas and charged colloids. Parameters are chosen such that the system is a Yukawa liquid whose kinematic viscosity is a time-dependent function of the particle correlation strength Γ controlled by shear heating. This feature is found to facilitate a unique quench study of the Reynolds number Re as a function of time for fixed system size and fixed flow speed. For small cross-sectional aspect ratios ∼20, starting from Re ∼ 1211-717, a laminar 3D PCF initial condition is shown to become unstable to localized 3D finite amplitude perturbation for various increasing amplitude strengths, clearly demonstrating the formation of a turbulent spot. This spot is found to spread in time into the otherwise laminar regions, a signature of subcriticality or co-existence of laminar and turbulent regions in PCF in a 3D Yukawa liquid. It is shown unambiguously that the range of interaction of Yukawa potential determines the nature of spot formation and its dynamics. At long range, a qualitative similarity of our results to those found in turbulent spots of PCF in conventional hydrodynamics is discussed. Our findings may have ramifications for a wide range of physical systems that exhibit sub-critical transition to turbulence.

A simple scenario of the laminar breakdown in liquid metal flows

In the article, authors present a numerical method for modelling a laminar-turbulent transition in magnetohydrodynamic flows. The small magnetic Reynolds number approach is considered. Velocity, pressure and electrical potential are decomposed to the sum of state values and finite amplitude perturbations. A solver based on the Nektar++ framework is described. The authors suggest using small-length local perturbations as a transition trigger. They can be imposed by blowing or by electrical enforcing. The stability of the Hartmann flow and the flow in the bend are considered as examples. Tables 4, Figs 19, Refs 28.

Paradoxical phase response of gamma rhythms facilitates their entrainment in heterogeneous networks.

The synchronization of different γ-rhythms arising in different brain areas has been implicated in various cognitive functions. Here, we focus on the effect of the ubiquitous neuronal heterogeneity on the synchronization of ING (interneuronal network gamma) and PING (pyramidal-interneuronal network gamma) rhythms. The synchronization properties of rhythms depends on the response of their collective phase to external input. We therefore determine the macroscopic phase-response curve for finite-amplitude perturbations (fmPRC) of ING- and PING-rhythms in all-to-all coupled networks comprised of linear (IF) or quadratic (QIF) integrate-and-fire neurons. For the QIF networks we complement the direct simulations with the adjoint method to determine the infinitesimal macroscopic PRC (imPRC) within the exact mean-field theory. We show that the intrinsic neuronal heterogeneity can qualitatively modify the fmPRC and the imPRC. Both PRCs can be biphasic and change sign (type II), even though the phase-response curve for the individual neurons is strictly non-negative (type I). Thus, for ING rhythms, say, external inhibition to the inhibitory cells can, in fact, advance the collective oscillation of the network, even though the same inhibition would lead to a delay when applied to uncoupled neurons. This paradoxical advance arises when the external inhibition modifies the internal dynamics of the network by reducing the number of spikes of inhibitory neurons; the advance resulting from this disinhibition outweighs the immediate delay caused by the external inhibition. These results explain how intrinsic heterogeneity allows ING- and PING-rhythms to become synchronized with a periodic forcing or another rhythm for a wider range in the mismatch of their frequencies. Our results identify a potential function of neuronal heterogeneity in the synchronization of coupled γ-rhythms, which may play a role in neural information transfer via communication through coherence.

Subcritical asymmetric Rayleigh breakup of a charged drop induced by finite amplitude perturbations in a quadrupole trap.

The breakup pathway of Rayleigh fission of a charged drop is unequivocally demonstrated by continuous, high-speed imaging of a drop levitated in an AC quadrupole trap. The experimental observations consistently exhibited asymmetric, subcritical Rayleigh breakup with an upward (i.e., opposite to the direction of gravity) ejection of a jet from the levitated drop. These experiments supported by numerical calculations show that the gravity induced downward shift of the equilibrium position of the drop in the trap causes significant, large amplitude shape oscillations superimposed over the center-of-mass oscillations. The shape oscillations result in sufficient deformations to act as triggers for the onset of instability below the Rayleigh limit (a subcritical instability). The concurrently occurring, center-of-mass oscillations, which are out of phase with the applied voltage, are shown to lead to an asymmetric breakup such that the Rayleigh fission occurs upwards via the ejection of a jet at the pole of the deformed drop. As an important application, it follows by inference that the nanodrop generation in electrospray devices will occur, more as a rule rather than as an exception, via asymmetric, subcritical Rayleigh fission events of microdrops due to inherent directionality provided by the external electric fields.

Stability range of localized subcritical Rayleigh–Bénard convection in temperature-dependent viscosity fluids: Constraints from two-dimensional simulations

We use a finite element code CITCOM to determine the stability range of a unique planform of Rayleigh–Bénard convection in temperature-dependent viscosity fluids—a spatially localized upwelling. Such a planform can only exist at Rayleigh numbers below the critical Rayleigh number for the linear onset of convection. Its initiation requires a finite-amplitude perturbation. We perform numerical simulations in long two-dimensional rectangular boxes with free-slip boundary conditions, which are relevant to convection in planetary interiors. The main result is a stability map of spatially localized single upwellings. Localized upwellings can exist at viscosity contrasts higher than about 150. The range of Rayleigh numbers at which localized upwellings are stable increases with the viscosity contrast. At the largest viscosity contrast studied, 1010, localized upwellings are stable in the range between about 0.5 and 0.9 of the critical Rayleigh number. The horizontal scale of localized upwellings is on the order of the layer thickness and decreases with the viscosity contrast. Spatially localized upwellings can occur on icy satellites and small planets where the conditions are likely to be subcritical due to the small size of these bodies. Localized upwellings might have industrial applications as well, such as patterning of polymers.

Sensitivity of long periodic orbits of chaotic systems

The properties of long, numerically determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents, and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.

Nondestructive Prediction of the Buckling Load of Imperfect Shells

From soda cans to space rockets, thin-walled cylindrical shells are abundant, offering exceptional load carrying capacity at relatively low weight. However, the actual load at which any shell buckles and collapses is very sensitive to imperceptible defects and cannot be predicted, which challenges the of such structures. Consequently, probabilistic descriptions in terms of empirical design rules are used and designing reliable structures requires the use of conservative strength estimates. We introduce a nonlinear description where finite-amplitude perturbations trigger buckling. Drawing from the analogy between imperfect shells which buckle and imperfect pipe flow which becomes turbulent, we experimentally show that lateral probing of cylindrical shells reveals their strength nondestructively. A new ridge-tracking method is applied to commercial cylinders with a hole showing that when the location where buckling nucleates is known we can accurately predict the buckling load of each individual shell, within ±5%. Our study provides a new promising framework to understand shell buckling, and more generally, imperfection-sensitive instabilities.

Acoustic wave propagation through solar granulation: Validity of effective-medium theories, coda waves

Context. The frequencies, lifetimes, and eigenfunctions of solar acoustic waves are affected by turbulent convection, which is random in space and in time. Since the correlation time of solar granulation and the periods of acoustic waves (∼5 min) are similar, the medium in which the waves propagate cannot a priori be assumed to be time independent. Aims. We compare various effective-medium solutions with numerical solutions in order to identify the approximations that can be used in helioseismology. For the sake of simplicity, the medium is one dimensional. Methods. We consider the Keller approximation, the second-order Born approximation, and spatial homogenization to obtain theoretical values for the effective wave speed and attenuation (averaged over the realizations of the medium). Numerically, we computed the first and second statistical moments of the wave field over many thousands of realizations of the medium (finite-amplitude sound-speed perturbations are limited to a 30 Mm band and have a zero mean). Results. The effective wave speed is reduced for both the theories and the simulations. The attenuation of the coherent wave field and the wave speed are best described by the Keller theory. The numerical simulations reveal the presence of coda waves, trailing the ballistic wave packet. These late arrival waves are due to multiple scattering and are easily seen in the second moment of the wave field. Conclusions. We find that the effective wave speed can be calculated, numerically and theoretically, using a single snapshot of the random medium (frozen medium); however, the attenuation is underestimated in the frozen medium compared to the time-dependent medium. Multiple scattering cannot be ignored when modeling acoustic wave propagation through solar granulation.