Nonlinear Instability Research Articles

The onset distribution of rotating m,n=2,1 tearing modes and its consequences on the stability of high-confinement-mode plasmas in DIII-D

Analysis of a multi-scenario database of over 13 000 DIII-D H-mode discharges shows that the m,n=2,1 magnetic islands are dominantly pressure gradient driven, stochastically triggered non-linear instabilities at all edge safety factor ( q95 ) values. The instability onset time closely follows the exponential distribution in intermediate and high q95 scenarios and is characterized by near constant onset rate (λ), in accordance with Poisson-point processes. This implies that the plasmas are operated in marginally stable conditions, characterized by a small threshold for instability growth and variations in the trigger amplitude and/or the stabilizing mechanisms with temporally uniform random distribution in this database. While the majority of the tearing modes occur in the first current-profile relaxation time of the βN flattop, constant λ throughout the βN flattop shows that the tearing onset is insensitive to the evolution of the equilibrium current profile. In low q95 scenarios, where a large fraction of the plasmas are operated at low torque, λ increases over the course of the βN flattop, showing that these plasmas evolve toward more unstable conditions. The onset rate rapidly increases with βN , while it does not show a clear dependence on the current gradient at the mode rational surface. Overall, these observations support that the majority of the analyzed 2,1 tearing modes are non-linear, neoclassically driven instabilities and classical stability does not play a dominant role in their onset.

Nonlinear instability and scalar clouds of spherical exotic compact objects in scalar-Gauss-Bonnet theory

In this work, we present a new type of scalar clouds supported by spherically symmetric horizonless compact objects in the scalar-Gauss–Bonnet theory. Unlike the previous spontaneous scalarization that is triggered by the tachyonic instability, our scalarization arises from a nonlinear instability that is non-spontaneous. We explore two types of boundary conditions for the scalar field at the surface of the compact objects and find an infinite countable set of scalar clouds characterized by the number of nodes for both cases. Our study demonstrates that boundary conditions have a significant impact on the formation of scalar clouds. Specifically, for the Dirichlet boundary condition, scalarization is more likely to occur for compact objects with medium radii and becomes harder for ultra-compact and large ones. Conversely, for the Robin boundary condition, scalarization is easier for more compact objects.

Lorenz model of instability in porous media for van der Waals gas

The article is devoted to the study of the nonlinear instability of the van der Waals gas in a gap with a porous medium of finite thickness heated from below. A fixed temperature difference is set between the upper cold surface and the lower hot one. The Lorenz approach was used to solve the problem. During a numerical solution, criteria for monotonic instability Racr1 and oscillating instability Racr2 were obtained. The results of calculations of the criterion Racr3 for the appearance of a strange attractor in the phase space are presented, which can be interpreted as a criterion for the generation of undamped turbulent fluctuations. The nature of the dependences of the critical Rayleigh numbers on the physical properties of the gas, the porosity of the medium, and the parameters of the van der Waals equation of state Waa and Wab is analyzed.The phenomenon of instability (formation of vortices in liquids or gases) in the space between two horizontal boundaries, when the lower one is heated, is widespread in technical applications. Lorenz developed a non-linear approach to the study of this phenomenon. This approach makes it possible to determine the criteria for the emergence of vortices and their transformation. This article uses the Lorenz approach to obtain instability criteria for a real gas whose properties are described by the van der Waals equation of state. This gas is in a porous medium. This combination of gas and medium has a wide technological application in the food, chemical and other industries. That is why the present study focused on this problem.

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Nonlinear magnetoconvection of a Maxwell fluid in a porous layer with chemical reaction

ABSTRACT In this paper, the linear and nonlinear instability of magnetoconvection in a Darcy-Benard setup saturated by a Maxwell fluid with chemical reactions is studied. The governing non-dimension equations are solved using the normal modes, and we obtain the expressions for steady and oscillatory thermal Rayleigh numbers. The effects of different physical parameters such as the Damkohler number (), Hartmann number (), Solute Rayleigh number (), relaxation parameter (), Magnetic Prandtl number (), Lewis number () on stationary and oscillatory critical thermal Rayleigh numbers are presented and described. Enhancing the values of the Solute Rayleigh number and Lewis number makes the system unstable. Also, the Hartmann number and Damkohler number have a contrasting effect on stationary and oscillatory convection. Enhancing the value of the relaxation parameter makes the system more stable. In order to study heat transport by convection, the well-known equation, the Landau-Ginzburg equation, is derived.

Instability of H1-stable periodic peakons for the higher-order μ-Camassa-Holm equation

Considered in this paper is a higher-order μ-Camassa-Holm equation. As a prototype, this higher-order μ-Camassa-Holm equation can be viewed as a generalization of the μ-Camassa-Holm equation. Periodic peaked waves of this higher-order μ-Camassa-Holm equation have been established to be H1-orbitally stable. By utilizing the method of characteristics, we prove the nonlinear instability of peaked perturbations to the H1-orbitally stable periodic peakons. Furthermore, a delicate analysis is employed to show that small initial W1,∞(S1) perturbations of the above periodic peakons lead to the blow-up phenomenon in finite time in the nonlinear evolution of the higher-order μ-Camassa-Holm equation.

Rayleigh–Taylor instability of 3D inhomogeneous incompressible Euler equations with damping in a horizontal slab

In this paper, we consider the Rayleigh–Taylor instability of three-dimensional inhomogeneous incompressible Euler equations with damping in a horizontal slab. We show that if the steady density profile is non-monotonous along the height, then the Euler system with damping is nonlinearly unstable around the given steady state. In this article, we develop a new variational structure to construct the growing mode solution, and overcome the difficulty in proving the sharp exponential growth rate by exploiting the structures in linearized Euler equations. Then combined with error estimates and a standard bootstrapping argument, we finish the nonlinear instability.

Two-fluid reconnection jets in a gravitationally stratified atmosphere

Context. Density decreases exponentially with height in the gravitationally stratified solar atmosphere, and therefore collisional coupling between the ionized plasma and the neutrals also decreases. Reconnection is a process observed at all heights in the solar atmosphere. Aims. Here, we investigate the role of collisions between ions and neutrals in the reconnection process occurring at various heights in the atmosphere. Methods. We performed simulations of magnetic reconnection induced by a localized resistivity in a gravitationally stratified atmosphere, in which we varied the height of the initial reconnection X-point. We compared a magnetohydrodynamic (MHD) model and two two-fluid configurations: one in which the collisional coupling was calculated from local plasma parameters, and another in which the coupling was decreased so that collisional effects would be enhanced. The latter setup has a more representative solar collisionality regime. Results. Simulations in a stratified atmosphere show similar structures in MHD and two-fluid simulations, with strong coupling. However, when collisional effects are increased to attain representative parameter regimes, we find a nonlinear runaway instability, which separates the plasma-neutral densities across the current sheet (CS). With increased collisional effects, the initial decoupling in velocity heats the neutrals and this sets up a nonlinear feedback loop, according to which neutrals migrate outside the CS, replacing charged particles that accumulate toward the center of the CS. Conclusions. The reconnection rate has a maximum value of around 0.1 for both reconnection heights, and is consistent with the locally enhanced resistivity used in all three models. The early-stage plasmoid formation observed near the end of our simulations is influenced by the outflow from the primary reconnection point, rather than by collisions. We synthesized optically thin emission for both MHD and two-fluid models, which can show a very different evolution when the charged-particle density is used instead of the total density. Our simulations have relevance for observed plasmoid features associated with chromospheric to low-coronal flare events.

Weakly nonlinear instability analysis of triple diffusive convection under internal heat generator and modulated boundaries

A nonlinear instability analysis of triple diffusive convection under the time-dependent heat and mass transfer boundary conditions in the presence of internal heat source is evaluated in this study. On various physical parameters, the momentary behavior of both Sherwood and Nusselt number profiles is examined. In the geometry, we have considered two parallel infinite horizontal plates acting gravity vertically downward z-direction. By using the weakly nonlinear analysis, the Ginzburg–Landau equation is generated for the rate of heat and mass transport. Here, we have considered the temperature and concentration of two solutes. The temperature and first concentration of the solute at the lower plate are higher than the upper plate, while the second concentration of the solute at the upper plate is higher compared to that of the lower plate. According to the different modulation, we have considered four cases based on the phase angle of the modulations. The convective heat and mass transports are measured as a function of the Nusselt number (Nu) and Sherwood number (Sh1 and Sh2) for both the concentration. From the results, it is found that the first Lewis number increases all the considered profiles, while Ri increases the Nusselt number profile only. The principal discovery elucidated by this article resides in the observation that the internal heat source, subject to modulated boundaries, maintains the convective instability if different solutes are used from both ends.

Instability of Liquid Film with Odd Viscosity over a Non-Uniformly Heated and Corrugated Substrate.

The effect of odd viscosity on the instability of liquid film along a wavy inclined bottom with linear temperature variation is investigated. By utilizing the long-wave approximation, the non-linear evolution equation of the free surface is derived. By applying the normal mode method, the linear instability of thin film flow is investigated. With the help of multi-scale analysis methods, the weakly non-linear instability of thin film flow is also investigated. The results reveal that the Marangoni effect caused by non-uniform temperature distribution promotes the instability of the liquid film, while the odd viscosity has a stabilizing effect. In addition, for a positive local inclination angle θ, an increase in bottom steepness ζ inhibits the instability of the liquid film flow. In contrast, with a negative local inclination angle θ, increased bottom steepness ζ promotes the instability of the liquid film flow. The results of the temporal linear instability analysis and the weakly non-linear instability analysis have been substantiated through numerical simulations of the non-linear evolution equations.

Marangoni stability of a thin liquid film falling down above or below an inclined thick wall with slip

AbstractThe linear and nonlinear instability of a thin liquid film flowing down above or below (Rayleigh-Taylor instability) an inclined thick wall with finite thermal conductivity are investigated in the presence of slip at the wall-liquid interface. A nonlinear evolution equation for the free surface deformation is obtained under the lubrication approximation. The curves of linear growth rate, maximum growth rate and critical Marangoni number are calculated. When the film flows below the wall it will be subjected to destabilizing and stabilizing Marangoni numbers. It is found that from the point of view of the linear growth rate the flow destabilizes with slip in a wavenumber range. However slip stabilizes for larger wavenumbers up to the critical (cutoff) wavenumber. From the point of view of the maximum growth rate flow slip may stabilize or destabilize increasing the slip parameter depending on the magnitude of the Marangoni and Galilei numbers. Explicit formulas were derived for the intersections (the wavenumber for the growth rate and the Marangoni number for the maximum growth rate) where slip changes its stabilizing and destabilizing properties. From the numerical solution of the nonlinear evolution equation of the free surface profiles, it is found that slip may suppress or stimulate the appearance of subharmonics depending on the magnitudes of the selected parameters. In the same way, it is found that slip may increase or decrease the nonlinear amplitude of the free surface deformation. The effect of the thickness and finite thermal conductivity of the wall is also investigated.

Experimental tweaking of symmetry breaking in recurrent nonlinear modulational instability

Experimental tweaking of symmetry breaking in recurrent nonlinear modulational instability

Bifurcation of coherent vortex flow in a magnetic island through nonlinear parity instability

The topology of the vortex flow associated with the magnetic island plays a significant role in modulating the turbulent transport near the magnetic island. In this paper, self-consistent nonlinear simulations of multi-scale interactions among large scale tearing mode, vortex flow, and small scale ion temperature-gradient (ITG) mode are numerically investigated based on the five-field Landau-fluid model. We found that the coherent vortex flow in a magnetic island has different parities in the nonlinear quasi-steady state, and this can be described by a theoretical framework—nonlinear parity instability. In the ITG stable case, the structure of the vortex flow bifurcates from tearing parity to twisting parity, which is characterized by modulational parity instability, modeled by a four-wave nonlinear coupling process. In the ITG unstable case, the vortex flow stays in tearing parity without parity bifurcation, and the energy is transferred from the twisting parity modes to the tearing parity modes. The impact of the parity instability on the magnetic island width is discussed as well.

Digital‐Image‐Correlation for Simulating Cyclic Local Buckling in Steel Beams

AbstractNonlinear continuum finite element (CFE) analyses rely on accurate multiaxial constitutive law formulations along with reliable imperfection patterns for simulating nonlinear geometric instabilities in steel members under mechanical loading. Validations of these models usually rely on conventional measurements of macroscopic parameters (e.g., displacement and strain field). It is common to use global deformation characteristics (e.g., deduced moment versus chord rotation) for model validation purposes of steel members exhibiting inelastic cyclic local buckling. The reason is that measurements from conventional strain gauge readings are not deemed to be reliable well before the onset of local buckling. This is potentially problematic when the accurate prediction of local strain demands is essential. This paper discusses the experimental results from a steel beam instrumented with a digital image correlation (DIC) system and how optical measurements acquired from DIC can be used to benchmark a CFE model representation of this beam. It is shown that while different CFE model types simulate accurately the moment rotation relationship of the steel beam, local strain demand predictions could vary considerably between model types. The results suggest that DIC measurements can inform the selection of proper imperfection patterns to be used in CFE models for the reliable estimation of inelastic strain demands.

Nonlinear dynamic characteristics of self-excited thermoacoustic instabilities in premixed swirling flames

The nonlinear behavior of thermoacoustic oscillation for premixed swirl flames fueled with CH4/air mixtures is experimentally studied by changing the equivalence ratio Φ, combustor inlet bulk velocity Uin and swirl number S. The nonlinear dynamic characteristics of the system are analyzed by the methods of phase space reconstruction, recurrence plot, Poincaré section and 0–1 test for chaos. Five kinds of flame modes are observed in the combustion system with the change of Φ, Uin, and S, which are lean/rich blow-off, stable combustion, quasi-periodic oscillation, and limit cycle oscillation. Subcritical Hopf bifurcation of the oscillation is captured either by changing Φ or Uin, and Φ and Uin are found to have similar effects on the flame mode transition under specific operating conditions. It is observed that the appearance of flame in the outer circulation zone (ORZ) and the enhancement of its burning intensity are matched with the occurrence of quasi-periodic and limit cycle oscillations, respectively. It is examined that the lean/rich blow-off limits and oscillation range are extended with a larger S, and the stable combustion regime is wider with a smaller S. This study contributes to understanding the nonlinear thermoacoustic instability of swirl flames under broad flame conditions, which helps to develop effective instability prediction and control methods in stable, efficient, and low-emission gas turbines.

Geometric tolerances of welded connections with inelastic panel zones

AbstractPrequalified beam‐to‐column connections in steel moment resisting frames usually concentrate inelastic deformations in the steel beam ends. As such, nonlinear geometric instabilities (i.e., local buckling) may occur in these regions at modest lateral drift demands, leading to potentially high residual deformations. Previous research has shown that welded connections featuring inelastic panel zones exhibit a stable hysteretic behaviour. In this case, residual deformations in steel beams are often not visually detectable after inelastic cyclic straining. This paper contrasts the residual deformations of steel beams as part of welded connections with elastic and inelastic panel zones through continuum finite element analyses. The employed finite element model is thoroughly validated with available test data. Besides the actual hysteretic response of the connections, the primary focus of the present work is on a number of geometric tolerance indicators, such as the web concavity, flange out‐of‐square and beam axial shortening. These are strongly correlated to the potential for reuse of structural steel members. The results suggest that connection designs featuring inelastic panel zones delay beam local buckling relative to their strong panel zone counterparts, achieving acceptable residual deformations even at lateral drift demands higher than 4% when considering the geometric tolerances of international standards.

Thermohaline convection of a Casson fluid in a porous layer: Linear and non-linear stability analyses

The study investigates the thermosolutal convection of a Casson fluid in a horizontal layer that is heated and salted from below. Both linear and non-linear analyses are performed using the method of normal modes to solve the governing equations. Interestingly, the study demonstrates that the linear and non-linear stability thresholds coincide. To solve the differential eigenvalue problem for linear theory, a one-term Galerkin approach is employed. Meanwhile, for the eigenvalue problem of non-linear instability, a numerical solution is obtained using the bvp4c routine in MATLAB. The results reveal some important findings. First, the Casson parameter is shown to destabilize the flow, leading to instability. However, the Darcy number and solutal Rayleigh number are found to have a stabilizing effect on the system. Furthermore, the study develops a weakly non-linear theory using multiple scale analysis to investigate heat and mass transport, offering valuable insight into these transport phenomena within the context of the system under consideration.

Steady states of the Parker instability

ABSTRACT We study the linear properties, non-linear saturation, and a steady, strongly non-linear state of the Parker instability in galaxies. We consider magnetic buoyancy and its consequences with and without cosmic rays. Cosmic rays are described using the fluid approximation with anisotropic, non-Fickian diffusion. To avoid unphysical constraints on the instability (such as boundary conditions often used to specify an unstable background state), non-ideal magnetohydrodynamic equations are solved for deviations from a background state representing an unstable magnetohydrostatic equilibrium. We consider isothermal gas and neglect rotation. The linear evolution of the instability is in broad agreement with earlier analytical and numerical models; but we show that most of the simplifying assumptions of the earlier work do not hold, such that they provide only a qualitative rather than quantitative picture. In its non-linear stage the instability has significantly altered the background state from its initial state. Vertical distributions of both magnetic field and cosmic rays are much wider, the gas layer is thinner, and the energy densities of both magnetic field and cosmic rays are much reduced. The spatial structure of the non-linear state differs from that of any linear modes. A transient gas outflow is driven by the weakly non-linear instability as it approaches saturation.

Lightweight Pneumatically Elastic Backbone Structure with Modular Construction and Nonlinear Interaction for Soft Actuators.

There has been a growing need for soft robots operating various force-sensitive tasks due to their environmental adaptability, satisfactory controllability, and nonlinear mobility unique from rigid robots. It is of desire to further study the system instability and strongly nonlinear interaction phenomenon that are the main influence factors to the actuations of lightweight soft actuators. In this study, we present a design principle on lightweight pneumatically elastic backbone structure (PEBS) with the modular construction for soft actuators, which contains a backbone printed as one piece and a common strip balloon. We build a prototype of a lightweight (<80 g) soft actuator, which can perform bending motions with satisfactory output forces (∼20 times self-weight). Experiments are conducted on the bending effects generated by interactions between the hyperelastic inner balloon and the elastic backbone. We investigated the nonlinear interaction and system instability experimentally, numerically, and parametrically. To overcome them, we further derived a theoretical nonlinear model and a numerical model. Satisfactory agreements are obtained between the numerical, theoretical, and experimental results. The accuracy of the numerical model is fully validated. Parametric studies are conducted on the backbone geometry and stiffness, balloon stiffness, thickness, and diameter. The accurate controllability, operation safety, modularization ability, and collaborative ability of the PEBS are validated by designing PEBS into a soft laryngoscope, a modularized PEBS library for a robotic arm, and a PEBS system that can operate remote surgery. The reported work provides a further applicability potential of soft robotics studies.

Plasmaspheric High‐Frequency Whistlers as a Candidate Cause of Shock Aurora at Earth

AbstractAuroral brightening driven by interplanetary shocks on Earth's closed magnetic field lines is commonly attributed to the 0.1–10 keV electron precipitations by electron cyclotron harmonic waves and whistler‐mode chorus waves in the low‐density region. In contrast, we propose here a new candidate driver, high‐frequency whistlers in the high‐density plasmasphere. Theoretical calculations predict the comparable abilities of whistler‐mode waves with sufficiently high frequencies inside the plasmasphere and chorus waves outside to produce auroral electron precipitations. Such expected waves are further verified to exist following an interplanetary shock. The shock perpendicularly accelerated source electrons inside the plasmasphere and then produced strong high‐frequency whistler‐mode waves. These waves appeared as a hybrid of chorus and hiss in the frequency‐time spectrogram and probably resulted from a combination of linear and nonlinear instabilities. Our findings demonstrate that interplanetary shocks can promptly promote energy transfer toward the ionosphere from the inner high‐density magnetosphere.