Nonlinear Growth Rate Research Articles

Latitudinal Mapping of Chorus Waves Growth Rates Based on Multi‐Spacecraft Wave and Plasma Measurements

AbstractWhistler mode chorus waves are one of the most intense and important electromagnetic emissions in the Earth's radiation belt, where these waves are responsible for electron acceleration to relativistic energies and for energetic electron precipitation into the Earth's atmosphere. This study reports a unique multi‐spacecraft observation event that allowed for the first time to verify theoretical models of chorus wave generation. The growth rates derived from the multi‐spacecraft observations of chorus waves Cluster WBD electric field waveforms collected during the crossing of the equatorial source region were compared to the theoretical linear and nonlinear growth rates calculated using simultaneous PEACE electron measurements. We have revealed that the growth rate estimated from wave measurements is substantially higher than the estimated linear growth rate but is quite close to the estimated nonlinear growth rate. A notable wave amplification is found up to 5 degrees of magnetic latitude and higher.

Dynamics of an amensalism system with strong Allee effect and nonlinear growth rate in deterministic and fluctuating environment

Dynamics of an amensalism system with strong Allee effect and nonlinear growth rate in deterministic and fluctuating environment

Modelling dynamics between free-living amoebae and bacteria.

Free-living amoebae (FLA) serve as hosts for a variety of endosymbionts, which are microorganisms that reside and multiply within the FLA. Some of these endosymbionts pose a pathogenic threat to humans, animals, or both. The symbiotic relationship with FLA not only offers these microorganisms protection but also enhances their survival outside their hosts and assists in their dispersal across diverse habitats, thereby escalating disease transmission. This review is intended to offer an exhaustive overview of the existing mathematical models that have been applied to understand the dynamics of FLA, especially concerning their interactions with bacteria. An extensive literature review was conducted across Google Scholar, PubMed, and Scopus databases to identify mathematical models that describe the dynamics of interactions between FLA and bacteria, as published in peer-reviewed scientific journals. The literature search revealed several FLA-bacteria model systems, including Pseudomonas aeruginosa, Pasteurella multocida, and Legionella spp. Although the published mathematical models account for significant system dynamics such as predator-prey relationships and non-linear growth rates, they generally overlook spatial and temporal heterogeneity in environmental conditions, such as temperature, and population diversity. Future mathematical models will need to incorporate these factors to enhance our understanding of FLA-bacteria dynamics and to provide valuable insights for future risk assessment and disease control measures.

Wave–particle interactions in tokamaks

Transport consequences of the wave–particle interactions in the quasilinear plateau (QP) regime are presented. Eulerian approach is adopted to solve the drift kinetic equation that includes the physics of the nonlinear trapping (NT) and QP regimes. The localization of the perturbed distribution simplifies the test particle collision operator. It is shown that a mirror force like term responsible for the flattening of the distribution in the NT regime is subdominant in the QP regime, and controls the transition between these two regimes. Transport fluxes, flux-power relation, and nonlinear damping or growth rate are all calculated. There is no explicit collision frequency dependence in these quantities; however, the width of the resonance does. Formulas that join the asymptotic results of these two regimes to facilitate thermal and energetic particle transport, and nonlinear wave evolution of a single mode are presented.

Interfacial instabilities driven by co-directional rarefaction and shock waves

We report the first experiments on hydrodynamic instabilities of a single-mode light/heavy interface driven by co-directional rarefaction and shock waves. The experiments are conducted in a specially designed rarefaction-shock tube that enables the decoupling of interfacial instabilities caused by these co-directional waves. After the impacts of rarefaction and shock waves, the interface evolution transitions into Richtmyer–Meshkov unstable states from Rayleigh–Taylor (RT) stable states, which is different from the finding in the previous case with counter-directional rarefaction and shock waves. A scaling method is proposed, which effectively collapses the RT stable perturbation growths. An analytical theory for predicting the time-dependent acceleration and density induced by rarefaction waves is established. Based on the analytical theory, the model proposed by Mikaelian (Phys. Fluids, vol. 21, 2009, p. 024103) is revised to provide a good description of the dimensionless RT stable behaviour. Before the shock arrival, the unequal interface velocities, caused by rarefaction-induced uneven vorticity, result in a V-shape-like interface. The linear growth rate of the amplitude is insensitive to the pre-shock interface shape, and can be well predicted by the linear superposition of growth rates induced by rarefaction and shock waves. The nonlinear growth rate is higher than that of a pure single-mode case, which can be predicted by the nonlinear models (Sadot et al., Phys. Rev. Lett., vol. 80, 1998, pp. 1654–1657; Dimonte & Ramaprabhu, Phys. Fluids, vol. 22, 2010, p. 014104).

Nonlinear structure and growth rate of solitary waves for extended Zakharov-Kuznetsov equation in plasma having (r, q) distribution electrons

The Reductive Perturbation Technique (RPT) is used to investigate ion acoustic waves (IAWs) in magnetized plasmas made up of electrons obeying the generalized form of ( r,q ) distribution. The analysis makes use of the Extended Zakharov-Kuznetsov (EZK) equation and the hydrodynamic theory of plasmas. The study investigates the effects of electron flatness ( r ) at low energy and superthermal ( q ) at high energy on the characteristics of IAWs, and the solutions to the EZK equation are plotted for different values of the plasma characteristics, such as the total ion cyclotron frequency, the ion to electron temperature ratio, and the ion-specific heat. The research goes into great detail to estimate the instability threshold and growth rate of these waves by applying a small-k perturbation technique. Overall, this investigation contributes to the better understanding of ion acoustic structures in space plasmas.

Global Regulation of Feedforward Nonlinear Systems: A Logic-Based Switching Gain Approach.

In this article, we investigate the global regulation problem for a class of feedforward nonlinear systems. Notably, the systems under consideration allow unknown input-output-dependent nonlinear growth rates, which has not been considered in existing works. A novel logic-based switching (LBS) gain approach is proposed to counteract the system uncertainties and nonlinearities. The key idea of the proposed approach is that whenever the Lyapunov function does not decrease as desired, the switching mechanism is activated so that a new gain is adopted. Furthermore, a tanh-type speed-regulation function is embedded into the switching mechanism for the first time to improve the convergence speed and transient performance. Then, a switching adaptive output feedback (SAOF) controller, which is of concise form and low complexity, is proposed based on the developed switching mechanism. It is shown that the objective of global regulation is achieved with faster convergence speed and better transient performance under the proposed controller. Moreover, it is also shown that the proposed control approach can be extended with the enhanced switching mechanism to deal with feedforward nonlinear systems with external disturbances. Finally, numerical examples are presented to demonstrate the effectiveness and advantages of our approach in comparison with the existing approaches.

Numerical study of the development of premixed flame interface induced by shock wave: A parameterization study of inhomogeneity of medium and chemical reaction

The evolution of a perturbed flame interface induced by shock waves occurs frequently in natural phenomena and in engineering applications. In practice, spatial inhomogeneity can exist and have a significant impact on the interface development in the real world. On the other hand, the effect of the chemical reaction on the flame interface evolution is still not clear. Therefore, in the present study, the evolution of a novel configuration of interaction between a planar premixed flame interface and a straight incident shock wave in inhomogeneous medium is studied numerically by solving the two-dimensional, time-dependent, reactive, and compressible Navier–Stokes equations with a high-order numerical scheme. A series of numerical cases are designed such that the amplitude of density perturbation σ, representing the effect of medium inhomogeneity, and the activation energy Ea, representing the effect of chemical reaction, can be varied independently. The result shows a unique dual-spike structure of flame interface during the interaction in inhomogeneous medium, which is different from the conventional spike–bubble structure during the interaction in homogeneous medium. For the range of σ and Ea considered, when Ea is fixed, σ is found to appreciably affect the growth of mixing zone area on the flame interface. This is because larger σ leads to stronger vortices and subsequently results in a nonlinear growth rate in the area of mixing zone. In contrast, when σ is fixed, the impact of Ea on the growth of mixing zone area on the flame interface is mainly manifested as the local consumption of reactants and changes of the flame thickness. Similarly, a high reactivity (small Ea) results in a nonlinear growth rate of the mixing zone area. Finally, for the cases in the range in which the growth rate of mixing zone area is linear, an empirically mathematic model, involved in activation energy (Ea) and inhomogeneity (σ) as independent variables, is proposed to reasonably predict the growth rate of mixing zone area in the reactive and inhomogeneous medium.

Thermodynamic calculations using reverse Monte Carlo: Simultaneously tuning multiple short-range order parameters for 2D lattice adsorption problem.

Lattice simulations are an important class of problems in crystalline solids, surface science, alloys, adsorption, absorption, separation, catalysis, to name a few. We describe a fast computational method for performing lattice thermodynamic calculations that is based on the use of the reverse Monte Carlo (RMC) technique and multiple short-range order (SRO) parameters. The approach is comparable in accuracy to the Metropolis Monte Carlo (MC) method. The equilibrium configuration is determined in 5-10 Newton-Raphson iterations by solving a system of coupled nonlinear algebraic SRO growth rate equations. This makes the RMC-based method computationally more efficient than MC, given that MC typically requires sampling of millions of configurations. The technique is applied to the interacting 2D adsorption problem. Unlike grand canonical MC, RMC is found to be adept at tackling geometric frustration, as it is able to quickly and correctly provide the ordered c(2 × 2) adlayer configuration for Cl adsorbed on a Cu (100) surface.

Global Stabilizability Theorems on Discrete-Time Nonlinear Uncertain Systems

This paper focuses on the a basic class of discrete-time nonlinear systems with multiple unknown parameters. We claim that such a system is stabilizable if its nonlinear growth rate is dominated by a polynomial rule. This rule cannot be relaxed in general since it becomes a necessary and sufficient condition when the system has a polynomial form [10]. We further prove that the concerned stabilizable system is possible to grow exponentially fast. Meanwhile, optimality and closed-loop identification are also discussed herein.

Bifurcation analysis in a diffusive predator–prey system with nonlinear growth rate and protection zone

Bifurcation analysis in a diffusive predator–prey system with nonlinear growth rate and protection zone

Dynamics of a predator–prey model with nonlinear growth rate and B–D functional response

In this paper, a predator–prey diffusive model, subject to homogeneous Dirichlet boundary conditions, with Beddington–DeAngelis functional response and nonlinear growth rate on the predator is proposed and the well posedness of its solution is systematically studied. Taking the capture rate m as the main parameter, we mainly investigate the existence, stability and exact number of positive solutions when m is large and other parameters meet a certain range of conditions. Meanwhile, some numerical simulations are applied to illustrate the analytical results. The main techniques used in this paper include the fixed point index theory, the super-sub solution method, the Lyapunov-Schmidt reduction procedure and the perturbation principle.

The existence and uniqueness of solutions of a nonlinear toxin-dependent size-structured population model

In this paper, we study a toxin-mediated size-structured population model with nonlinear reproduction, growth, and mortality rates. By using the characteristic method and the contraction mapping argument, we establish the existence-uniqueness of solutions to the model. We also prove the continuous dependence of solutions on initial conditions.

Thermally enhanced tearing in solar current sheets: Explosive reconnection with plasmoid-trapped condensations

Context. Thermal instability plays a major role in condensation phenomena in the solar corona (e.g., for coronal rain and prominence formation). In flare-relevant current sheets, tearing instability may trigger explosive reconnection and plasmoid formation. However, how the two instabilities influence the disruption of current concentrations in the solar corona has received less attention to date. Aims. We explore how the thermal and tearing modes reinforce each other in the fragmentation of a current sheet in the solar corona through an explosive reconnection process, characterized by the formation of plasmoids which interact and trap condensing plasma. Methods. We used a resistive magnetohydrodynamic simulation of a 2D current layer, incorporating the non-adiabatic effects of optically thin radiative energy loss and background heating using the open-source code MPI-AMRVAC. Multiple levels of adaptive mesh refined grids are used to achieve a high resolution to resolve the fine structures during the evolution of the system. Results. Our parametric survey explores different resistivities and plasma-β to quantify the instability growth rate in the linear and nonlinear regimes. We note that for dimensionless resistivity values within 10−4 − 5 × 10−3, we obtain explosive behavior where thermal instability and tearing behavior reinforce each other. This is clearly below the usual critical Lundquist number range of pure resistive explosive plasmoid formation. We calculate the mean growth rate for the linear phase and different nonlinear phases of the evolution. The nonlinear growth rates follow a weak power-law dependency with resistivity. The fragmentation of the current sheet and the formation of the plasmoids in the nonlinear phase of the evolution due to the thermal and tearing instabilities are obtained. The formation of plasmoids is found for the Lundquist number (SL) range 4.6 × 103 − 2.34 × 105. We quantify the temporal variation in the plasmoid numbers and the density filling factor of the plasmoids for different physical conditions. We also find that the maximum plasmoid numbers scale as SL0.223.Within the nonlinearly coalescing plasmoid chains, localized cool condensations gather, realizing density and temperature contrasts similar to coronal rain or prominences.

Self-consistent amplitude profile of ducted VLF transmitter signal due to resonant interaction with energetic electrons in the magnetosphere

Of the various sources of whistler waves in the magnetosphere, the signals from terrestrial VLF transmitters are apparently the simplest for theoretical analysis. The role of these signals in the acceleration and pitch-angle scattering of energetic electrons in the Earth’s radiation belts has not been fully studied quantitatively, despite many decades of relevant research. The main problem in studying this issue is that to calculate the acceleration and pitch-angle scattering of particles, it is necessary to know the spatial profile of the signal amplitude. This is especially critical for phase-trapped particles, for which the energy exchange with the wave is the most significant. At the same time, the wave amplitude profile itself is determined by the interaction of the wave with resonant particles. Thus, we are dealing with a self-consistent problem, which is described by a nonlinear system of equations - a kinetic equation for the distribution function of resonant particles with a self-consistent field, and an equation for the wave field, in which the nonlinear and non-local growth rate is determined by the distribution function of resonant particles. In this paper, we develop an approach to solving this system of equations and the corresponding physical problem using the method of successive approximations.

Asymmetries in the ENSO phase space

El Niño Southern Oscillation (ENSO) dynamics are best described by the recharge oscillator model, in which the eastern tropical Pacific sea surface temperatures (T) and subsurface heat content (thermocline depth; h) have an out-of-phase relationship. This defines a 2-dimensional phase space diagram between T and h. In an idealized, stochastically forced damped oscillator, the mean phase space diagram should be a perfectly symmetrical circle with a clockwise propagation over time. However, the observed phase space shows strong asymmetries. In this study we illustrate how the ENSO phase space can be used to discuss the phase-dependency of ENSO dynamics. A normalized spherical coordinate system allows the definition of phase-depending ENSO growth rates and phase transition speeds. Based on these we discuss the implications of the observed asymmetries with regards to the dynamics and predictability of ENSO; with a particular focus on the variations in the growth rate and coupling of ENSO along the oscillation cycle. Using linear and non-linear recharge oscillator models we will show how dynamics and noise are driving ENSO at different phases of the ENSO cycles. The results illustrate that the ENSO cycle with positive phase transitions is present in all phases but has strong variations in its strength. Much of these variations result from presenting the ENSO phase space with estimates of h based on the iso-thermal depth, that is not ideal as it is not out-of-phase with T. Future work should address how h can be estimated better, including aspects such as the vertical temperature gradients and the meridional or zonal range. We further illustrated that a non-linear growth rate of T can explain most of the observed non-linear phase space characteristics.

Effects of obstacles on shock-induced perturbation growth

Effects of obstacles on interface evolution and mixing width induced by Richtmyer–Meshkov instability are investigated experimentally and numerically. In the experiment, the soap film technique is adopted to create an initial interface whose shape is governed by constraint strips protruding into the flow field. By varying the heights of constraint strips protruding into the flow field, effects of obstacles on post-shock flow features are highlighted. First, the interaction of a planar shock with an unperturbed interface is investigated numerically and experimentally. The results show that the obstacles have negligible effects on the transmitted shock velocity, but they greatly increase the reflected shock velocity. The obstacles induce the non-uniform pressure and velocity fields behind the shock, which change the interface evolution and mixing width. Then, experiments of planar shock wave interacting with single-mode interfaces with different initial amplitudes are performed. Induced by the non-uniform post-shock flow, the experimental schlieren images indicate that the spike tip becomes flat, but its size increases in the spanwise direction and the volume of the bubble is reduced. The effects of obstacles are magnified as their heights increase and are more pronounced when the initial interface amplitudes are small. The linear and nonlinear growth rates obtained from experiments show that the obstacles inhibit the perturbation growth, which is partially caused by less kinetic energy the interface obtains from the shock due to the block by the obstacles.

Local and global behavior of solutions to 2 D-elliptic equation with exponentially-dominated nonlinearities

We study the family of blowup solutions to semilinear elliptic equations in two-space dimensions with exponentially-dominated nonnegative nonlinearities. Such a family admits an exclusion of the boundary blowup, finiteness of blowup points, and pattern formation. Then, Hamiltonian control of the location of blowup points, residual vanishing, and mass quantization arise under the estimate from below of the nonlinearity. Finally, if the principal growth rate of nonlinearity is exactly exponential and the residual part has a gap relative to this term, there is a locally uniform estimate of the solution which ensures its asymptotic non-degeneracy.

Energetic bounds on gyrokinetic instabilities. Part 2. Modes of optimal growth

We introduce modes of instantaneous optimal growth of free energy for the fully electromagnetic gyrokinetic equations. We demonstrate how these ‘optimal modes’ arise naturally from the free energy balance equation, allowing its convenient decomposition, and yielding a simple picture of energy flows. Optimal modes have a number of other favourable features, such as their low dimensionality, efficiency of computation and the fact that their growth rates provide a rigorous and ‘tight’ upper bound on both the nonlinear growth rate of energy, and the linear growth rate of traditional gyrokinetic (normal mode) instabilities. We provide simple closed-form solutions for the optimal growth rates in a number of asymptotic limits, and compare these with our previous bounds.

Nonlinear Wave Growth of Whistler‐Mode Hiss Emissions in a Uniform Magnetic Field

AbstractWe conduct electromagnetic particle simulations in a uniform magnetic environment to verify the nonlinear wave growth process of plasmaspheric hiss in the equatorial plasmasphere. The satisfaction of the separability criterion for coexisting multiple frequency waves in the initial stage of wavenumber‐time evolution declares that wave packets are coherent and capable of growing nonlinearly. Spatial and temporal evolutions of two typical modes located in wavenumber‐time evolution demonstrate the consistency among wave growths, frequency variations, and inhomogeneity factor S in coherent wave packets, showing that rising and falling tones occur at negative and positive S values, respectively, and an obvious wave growth happens in a reasonable range of S satisfying the second‐order resonance condition. Wave packets extracted from wave fields in space and time by a frequency band‐pass filter confirm a good agreement between the nonlinear theory and simulation results. The nonlinear growth rates of the extracted wave packets possess similar magnitudes to the growth rates of wave packets in the simulation, and they are much greater than the theoretical linear growth rate, indicating that the nonlinear process is essential in the generation of plasmaspheric hiss.