Marginal Stability Criterion Research Articles

On the speed of propagation in Turing patterns for reaction–diffusion systems

This study investigates transient wave dynamics in Turing pattern formation, focusing on waves emerging from localised disturbances. While the traditional focus of diffusion-driven instability has primarily centred on stationary solutions, considerable attention has also been directed towards understanding spatio-temporal behaviours, particularly the propagation of patterning from localised disturbances. We analyse these waves of patterning using both the well-established marginal stability criterion and weakly nonlinear analysis with envelope equations. Both methods provide estimates for the wave speed but the latter method, in addition, approximates the wave profile and amplitude. We then compare these two approaches analytically near a bifurcation point and reveal that the marginal stability criterion yields exactly the same estimate for the wave speed as the weakly nonlinear analysis. Furthermore, we evaluate these estimates against numerical results for Schnakenberg and CDIMA (chlorine dioxide–iodine–malonic acid) kinetics. In particular, our study emphasises the importance of the characteristic speed of pattern propagation, determined by diffusion dynamics and a complex relation with the reaction kinetics in Turing systems. This speed serves as a vital parameter for comparison with experimental observations, akin to observed pattern length scales. Furthermore, more generally, our findings provide systematic methodologies for analysing transient wave properties in Turing systems, generating insight into the dynamic evolution of pattern formation.

Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain.

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equationwith periodic boundary conditions on isotropically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; in contrast with a growing domain, the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e., phase slips that fail to develop. Then, we study how propagating fronts are influenced by a time-dependent domain. We identify three types of pulled fronts: homogeneous, pattern spreading, and Eckhaus fronts. By following the linear dynamics, we derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural "asymptotic" velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution-an effect present when the order parameter is conserved-increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains. The study provides general insight into the effects of domain growth on pattern onset, pattern transitions, and front propagation in systems across different scientific fields.

Stability Characteristics of Planar Rivlin–Ericksen Fluid Interface With Mass and Heat Transfer

Abstract The interface of viscous-Rivlin-Ericksen fluids is analyzed through the linear theory of stability analysis when mass and heat is transferring across the interface. The Rivlin-Ericksen fluid lies in the upper region while the lower region of the interface contains viscous fluid. The gravitational acceleration destabilizes the top-heavy arrangement and interface instability is governed by Rayleigh–Taylor instability. The two-dimensional interface is considered, and the viscous potential flow theory is employed to establish the relationship between perturbation's growth and wave number. This relationship is analyzed, and the perturbation's growth is plotted for various flow parameters. A marginal stability condition is obtained, and it is given in terms of heat transport coefficient Λ and wave number. The marginal stability criterion is analyzed using the well-known Newton–Raphson method. The heat and mass transfer phenomenon drives the unstable interface toward stability. It is pointed out that the viscoelastic coefficient λo influences the interface to be stable while the thickness of the viscoelastic fluid makes the interface unstable. Atwood numbers and Weber numbers show destabilizing behavior.

Dendrite tip selection during isothermal free growth in multi-component alloys: Marginal stability theories and insights from phase-field simulations

The principal length scales associated with dendritic solidification, which being the dendritic tip-radius (Rtip) and the primary dendritic arm spacing, are functions of the relative solutal interdiffusivities of the different components in a multi-component alloy. In this paper, we firstly derive marginal stability based theories to predict Rtip during isothermal free growth in any generic multi-component alloy for any generic diffusivity matrix. Here, we extend the Ivanstov solution of the diffusion problem in the liquid with a parabolic solid-liquid interface and propose the closure conditions between the Rtip and the selected dendrite tip velocity (V) based on modification of the classical marginal stability criteria for multi-component situations. Additionally, we derive the constant Rtip2V from phase-field simulations and using the Ivantsov parabola as the approximate interface shape compute the Rtip and the V, which we term as “Approximate microsolvability prediction (AMP)”. Thereafter, we compare our predictions of Rtip,V as well as phase compositions with independent phase-field simulation results for different choices of the diffusivity matrix, where we show that the analytical theories based upon the marginal stability theory are unable to accurately predict variations of the dendrite tip velocity V and the Rtip. We thereby utilize the phase-field simulation results to investigate the tip selection in multi-component alloys and derive an appropriate solvability constant σ∗. We find that the σ∗ in a multi-component alloy is a function of the diffusivity ratios of the components for the case of a diagonal diffusivity matrix.

Stability of thin fluid films characterised by a complex form of effective disjoining pressure

We discuss instabilities of fluid films of nanoscale thickness, with a particular focus on films where the destabilising mechanism allows for linear instability, metastability, and absolute stability, depending on the mean film thickness. Our study is motivated by nematic liquid crystal films; however, we note that similar instability mechanisms, and forms of the effective disjoining pressure, appear in other contexts, such as the well-studied problem of polymeric films on two-layered substrates. The analysis is carried out within the framework of the long-wave approximation, which leads to a fourth-order nonlinear partial differential equation for the film thickness. Within the considered formulation, the nematic character of the film leads to an additional contribution to the disjoining pressure, changing its functional form. This effective disjoining pressure is characterised by the presence of a local maximum for non-vanishing film thickness. Such a form leads to complicated instability evolution that we study by analytical means, including the application of marginal stability criteria, and by extensive numerical simulations that help us develop a better understanding of instability evolution in the nonlinear regime. This combination of analytical and computational techniques allows us to reach novel understanding of relevant instability mechanisms, and of their influence on transient and fully developed fluid film morphologies. In particular, we discuss in detail the patterns of drops that form as a result of instability, and how the properties of these patterns are related to the instability mechanisms.

Limiting energy spectrum of an electron radiation belt

AbstractTo determine the Kennel‐Petschek limiting particle flux in a planetary radiation belt in a fully relativistic regime, without assuming a predetermined form for the particle energy distribution, has been a long‐standing challenge in space physics. In this paper, for the case of whistler mode wave‐electron interaction, we meet this challenge. The limiting flux is determined by a steady state marginal stability criterion in which a convective wave gain condition is applied over all frequencies for which wave growth occurs. This condition produces an integral equation for the trapped flux. We find that in the relativistic regime the limiting electron energy spectrum varies asymptotically as 1/E, for large energy E, just as in the nonrelativistic case. However, the scaling coefficient in the relativistic case is twice that in the nonrelativistic result. We compare numerical solutions for the limiting spectra with measured energetic electron spectra at Jupiter.

Gravitational stability and mass estimation of stellar disks

AbstractWe estimate masses of galaxy disks using the marginal gravitational stability criterion and compare them with the photometric disk‐mass evaluations. The comparison reveals that stellar disks of most spiral galaxies we have considered cannot be substantially overheated (at least within several radial scale lengths) and are therefore unlikely to have experienced a significant merging event in their history. However, for a substantial part of S0‐type galaxies, their stellar velocity dispersion is well in excess of the gravitational stability threshold suggesting a major merger event in the past. For four low‐surface‐brightness galaxies, we find the disk masses corresponding to the marginal stability condition to be significantly higher than one may expect from their brightness. Either their disks are dynamically overheated, or they contain a large amount of non‐luminous matter. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Inertial instability of intense stratified anticyclones. Part 1. Generalized stability criterion

AbstractThe stability of axisymmetric vortices to inertial perturbations is investigated by means of linear stability analysis, taking into account stratification, vertical eddy viscosity, as well as finite depth of the flow. We consider different types of circular barotropic vortices in a linearly stratified shallow layer confined with rigid lids. For the simplest case of the Rankine vortex we develop an asymptotic analytic dispersion relation and a marginal stability criterion, which compares well with numerical results. This is a further generalization to the well-known generalized Rayleigh criterion, which is only valid for non-dissipative and non-stratified eddies. Unlike the Rayleigh criterion, it predicts that intense anticyclones may be stable even with a core region of negative absolute vorticity, and that the dissipation and stratification work together to stabilize the flow. Numerical analysis reveals that the stability diagrams for various types of vortices are almost identical in the Rossby, Burger and Ekman parameter space. This allows extension of our analytical solutions for the Rankine vortex to a wide variety of vortices. Furthermore, we show that a more suitable parameter for the intensity of the vortex is the vortex Rossby number, while for the inviscid case it is the local normalized vorticity. These predictions are in agreement with laboratory experiments presented in part 2 (J. Fluid Mech., vol. 732, 2013, pp. 485–509).

Solidification fronts in supercooled liquids: How rapid fronts can lead to disordered glassy solids

We determine the speed of a crystallization (or, more generally, a solidification) front as it advances into the uniform liquid phase after the system has been quenched into the crystalline region of the phase diagram. We calculate the front speed by assuming a dynamical density functional theory (DDFT) model for the system and applying a marginal stability criterion. Our results also apply to phase field crystal (PFC) models of solidification. As the solidification front advances into the unstable liquid phase, the density profile behind the advancing front develops density modulations and the wavelength of these modulations is a dynamically chosen quantity. For shallow quenches, the selected wavelength is precisely that of the crystalline phase and so well-ordered crystalline states are formed. However, when the system is deeply quenched, we find that this wavelength can be quite different from that of the crystal, so the solidification front naturally generates disorder in the system. Significant rearrangement and aging must subsequently occur for the system to form the regular well-ordered crystal that corresponds to the free energy minimum. Additional disorder is introduced whenever a front develops from random initial conditions. We illustrate these findings with simulation results obtained using the PFC model.

Analysis for Free Dendritic Growth Model Applicable to Nondilute Alloys

A steady-state free dendritic growth model applicable to concentrated alloys was proposed as an extension of Wang et al.’s model.[14] The present model adopted a realistic thermodynamic model to replace the Baker–Cahn equation and included a generalized marginal stability criterion and a nondilute solute trapping model to completely eliminate the dilute alloy limitation. Comparative analysis shows that Wang et al.’s model is a very close approximation to the present model at low undercoolings for dilute alloys. However, the difference appears at high undercoolings even for dilute alloys. Furthermore, the difference of the model predictions for both models increases with nominal composition of alloys due to the inherent limitation of dilute alloys in Wang et al.’s model. A comparison with the experimental data for Cu70Ni30 alloy demonstrates the applicability of the present model to nondilute alloys.

Modeling constrained dendrite growth in rapidly directional solidification

Incorporating the effect of nonlinear liquidus and solidus, a constrained dendrite growth model was established for rapidly directional solidification, where both the interface and the bulk liquid are under local non-equilibrium conditions. The effect of nonlinear liquidus and solidus was first introduced into the interface kinetics, based on which the marginal stability criterion was then derived. By this way, not only the temperature dependent equilibrium partition coefficient ke, equilibrium liquidus slope ml and equilibrium solidus slope ms but also the derivation of ke with temperature are considered. The model is more physically realistic than the works in which the effect of nonlinear liquidus and solidus is not incorporated into the interface kinetics and hence the temperature dependent ms is not embodied. Applying the model to rapidly directional solidification of Ag–15 wt% Cu alloy, the effects of nonlinear liquidus and solidus and local non-equilibrium on the constrained dendrite growth were studied.

Substrate-mediated pattern formation in monolayer transfer: a reduced model

The formation of regular stripe patterns during the transfer of surfactant monolayers from water surfaces onto moving solid substrates can be understood as a phase decomposition process under the influence of the effective molecular interaction between the substrate and the monolayer, also called substrate-mediated condensation (SMC). To describe this phenomenon, we propose a reduced model based on an amended Cahn–Hilliard equation. A combination of numerical simulations and continuation methods is employed to investigate stationary and time-periodic solutions of the model and to determine the resulting bifurcation diagram. The onset of spatiotemporal pattern formation is found to result from a homoclinic and a Hopf bifurcation at small and large substrate speeds, respectively. The critical velocity corresponding to the Hopf bifurcation can be calculated by means of the marginal stability criterion for pattern formation behind propagating fronts. In the regime of low transfer velocities, the stationary solutions exhibit snaking behavior.

Study of a high-order-mode gyrotron traveling-wave amplifier

Physics and performance issues of a TE01-mode gyrotron traveling-wave amplifier are studied in theory. For a high order mode, absolute instabilities on neighboring modes at the fundamental and higher cyclotron harmonic frequencies impose severe constraints to the device capability. Methods for their stabilization are outlined, on the basis of which the performance characteristics are examined in a multidimensional parameter space under the marginal stability criterion. The results demonstrate the viability of a high-order-mode traveling-wave amplifier and provide a roadmap for design tradeoffs among power, bandwidth, and efficiency. General trends are observed and illustrated with specific examples.

Coexistence of Flexibility and Stability of Proteins: An Equation of State

We consider a recently suggested “equation of state” for natively folded proteins, and verify its validity for a set of about 5800 proteins. The equation is based on a fractal viewpoint of proteins, on a generalization of the Landau-Peierls instability, and on a marginal stability criterion. The latter allows for coexistence of stability and flexibility of proteins, which is required for their proper function. The equation of state relates the protein fractal dimension , its spectral dimension , and the number of amino acids N. Using structural data from the protein data bank (PDB) and the Gaussian network model (GNM), we compute and for the entire set and demonstrate that the equation of state is well obeyed. Addressing the fractal properties and making use of the equation of state may help to engineer biologically inspired catalysts.

An extended morphological stability model for a planar interface incorporating the effect of nonlinear liquidus and solidus

Incorporating the effect of nonlinear liquidus and solidus, an extended morphological stability model for a planar interface was developed under local non-equilibrium conditions both at the solid/liquid interface and in the bulk liquid. The model was adopted to describe the cellular breakdown of the planar interface upon rapid solidification of Si–Sn alloys, and a quantitative agreement was obtained between the model predictions and the experimental results. It has been found that, independent of the effect of non-equilibrium liquid diffusion, two critical interface velocities V min and V max always occur, because of the retrograde characters of the Si–Sn phase diagram. If V min ⩽ V ⩽ V max, the so-called neutral stability curve can be evaluated in terms of the planar interface response function and the marginal stability criterion. Whereas if V > V max or V < V min, the calculation of the maximum solid solubility C S max ( V ) from the planar interface response function is performed. In comparison with the model assuming linear liquidus and solidus, which fails to predict the retrograde characters of the Si–Sn phase diagram both at low ( k → k e) and at high ( k → 1) interface velocities, the effect of nonlinear liquidus and solidus on the cellular breakdown of the planar interface is highlighted.

Quasilinear theory of interchange modes in a closed field line configuration

Two important issues for any magnetic fusion configuration are the maximum achievable values of β and energy confinement time when ideal magnetohydrodynamic (MHD) modes are excited. It is well known that the excitation of the MHD unstable modes typically can lead to violent restructuring of the plasma profiles. The particle and energy transport associated with these modes normally dominates all other transport mechanisms and can lead to plasma disruptions and a rapid loss of energy. This paper analytically investigates the transport of particle density, energy, and magnetic field due to the ideal MHD interchange mode in a closed-line system using the quasilinear approximation. The transport equations are derived for a static plasma in a hardcore Z-pinch configuration and generalized to an arbitrary axisymmetric toroidal closed poloidal field line configuration. It is shown that violation of the marginal stability criterion leads to rapid quasilinear transport that drives the pressure profile back to its marginal profile and forces the particle density to be inversely proportional to ∮dl∕B. The applicability of the quasilinear approximation is numerically tested for the hardcore Z-pinch magnetic configuration using a full nonlinear code.

Forbidden interval of propagation speed for exothermic chemical fronts

We consider the propagation of an exothermic chemical front toward an unstable steady state. The hydrodynamic equations are solved numerically for increasing values of the activation energy of the reaction which controls the reaction front speed. For a large speed, the marginal stability criterion of the isothermal case is recovered. For a small speed, we observe two well-separated traveling waves: a heat front is preceding the reaction front. We find analytically a forbidden speed interval where the hydrodynamical system does not admit stationary traveling solutions.

Dendritic solidification morphology viewed from the perspective of constructal theory

In this paper we focus on the application of the constructal theory in predicting the dendritic solid structure. First we analyse the marginal stability criterion from the perspective of the constructal principle. Having as a guiding principle the constructal law we have shown that among the whole range of possible dendrite tip radiuses predicted by the stability analysis the dendrite tip will choose the smallest one, that is a radius equal with the smallest perturbation wavelength leading to instabilities. We identify as well the existence of a competition between the diffusion controlled growth and the dendritic growth. Second, we develop a model for the secondary arm spacing. We identify a competition between the lateral diffusion controlled growth of a needle and the dendritic growth of lateral secondary arms. By analysing this competition we are able to characterize the sidebranching mechanism and to finally compute the secondary arm spacing. The result is in good agreement with the experimental results. Finally, the primary arm spacing is analysed from the perspective of the constructal law. The constructal law predicts that the only way the columnar tips can optimize the solidification process is to minimize the spacing between two adjacent tips, namely λ1. By quantifying the two mechanisms responsible for the selection of λ1, the dendrite division and the dendrite overgrown mechanisms, we were finally able to obtain a model for the primary arm spacing. This model is also validated against various experimental data.

The effect of kinetics on the interface stability under the non-equilibrium condition

The kinetic effect, in combination with the effect of relaxation, was considered in the interface stability criterion under the non-equilibrium solidification. Employing a fresh mathematical derivation, a more general marginal stability criterion was obtained by incorporating the non-equilibrium kinetics concluding the kinetic undercooling and the liquidus slope in the kinetic phase diagram. It was shown that the interface stability is correlated with the kinetic effect, i.e. by incorporation in the slope of non-equilibrium liquidus, but is independent of the kinetic undercooling.

Compressibility effect on magnetic-shear-localized ideal magnetohydrodynamic interchange instability

Eigenmode analysis of a magnetic-shear-localized ideal magnetohydrodynamic interchange instability in the presence of plasma compressibility indicates the marginal stability criterion (DI=1∕4) is not affected by the compressibility effects. Above the marginal stability criterion, plasma compressibility causes a significant reduction in the growth rate of an ideal interchange instability.