Linear Marginal Stability Research Articles

Deriving the slip-front propagation velocity with slip-dependent and slip-velocity-dependent friction laws via the use of the linear marginal stability hypothesis.

We investigate analytically and numerically the determining factors of the slip front propagation (SFP) velocity. The slip front has two forms characterized by an intruding or extruding front. We assume a one-dimensional viscoelastic medium on a rigid and fixed substrate, and we employ the friction law depending on the slip and slip velocity. Despite this dependency potentially being nonlinear, we use the linear marginal stability hypothesis, which linearizes the governing equationfor the slip, to investigate the intruding and extruding front velocities. The analytically obtained velocities are found to be consistent with the numerical computation where we assume the friction law depends nonlinearly on both the slip and slip velocity. This implies that the linearized friction law is sufficient to capture the dominant features of SFP behavior.

Mean flow evolution of saturated forced shear flows in polytropic atmospheres

In stellar interiors shear flows play an important role in many physical processes. So far helioseismology provides only large-scale measurements, and so the small-scale dynamics remains insufficiently understood. To draw a connection between observations and three-dimensional DNS of shear driven turbulence, we investigate horizontally averaged profiles of the numerically obtained mean state. We focus here on just one of the possible methods that can maintain a shear flow, namely the average relaxation method. We show that although some systems saturate by restoring linear marginal stability this is not a general trend. Finally, we discuss the reason that the results are more complex than expected.

Testing linear marginal stability in stratified shear layers

We use two-dimensional direct numerical simulations of Boussinesq stratified shear layers to investigate the influence of the minimum gradient Richardson number $Ri_{m}$ on the early time evolution of Kelvin–Helmholtz instability to its saturated ‘billow’ state. Even when the diffusion of the background velocity and density distributions is counterbalanced by artificial body forces to maintain the initial profiles, in the limit as $Ri_{m}\rightarrow 1/4$, the perturbation growth rate tends to zero and the saturated perturbation energy becomes small. These results imply, at least for such canonical inflectional stratified shear flows, that ‘marginally unstable’ flows with $Ri_{m}$ only slightly less than 1/4 are highly unlikely to become ‘turbulent’, in the specific sense of being associated with significantly enhanced dissipation, irreversible mixing and non-trivial modification of the background distributions without additional externally imposed forcing.

IMPLICATIONS OF A NON-MODAL LINEAR THEORY FOR THE MARGINAL STABILITY STATE AND THE DISSIPATION OF FLUCTUATIONS IN THE SOLAR WIND

The non-modal approach for a linearized system differs from a normal mode analysis by following the temporal evolution of some perturbed equilibria, and therefore includes transient effects. We employ a non-modal approach for studying the stability of a bi-Maxwellian magnetized plasma using the Landau fluid model, which we briefly describe. We show that bi-Maxwellian stable equilibria can support transient growth of some physical quantities, and we study how these transients behave when an equilibrium approaches its marginally stable condition. This is relevant to anisotropic plasma, that are often observed in the solar wind with a temperature anisotropy close to values that can trigger a kinetic instability. The results obtained with a non-modal approach are relevant to a re-examination of the concept of linear marginal stability. Moreover, we discuss the topic of the dissipation of turbulent fluctuations, suggesting that the non-modal approach should be included in future studies.

Front propagation into unstable metal nanowires

Long, cylindrical metal nanowires have recently been observed to form and be stable for seconds at a time at room temperature. Their stability and structural dynamics is well described by a continuum model, the nanoscale free-electron model, which predicts cylinders in certain intervals of radius to be linearly unstable. In this paper, I study how a small, localized perturbation of such an unstable wire grows exponentially and propagates along the wire with a well-defined front. The front is found to be pulled and forms a coherent pattern behind it. It is well described by a linear marginal stability analysis of front propagation into an unstable state. In some cases, nonlinearities of the wire dynamics are found to trigger an invasive mode that pushes the front. Experimental procedures that could lead to the observation of this phenomenon are suggested.

Linear instability of magnetic Taylor-Couette flow with Hall effect.

The influence of the Hall effect on the linear marginal stability of a molecular hydrodynamic Taylor-Couette flow in the presence of an axial uniform magnetic field is considered. The Hall effect leads to the situation that the Taylor-Couette flow becomes unstable for any ratio of the angular velocities of the inner and outer cylinders. The instability, however, does not exist for both signs of the axial magnetic field B0. For positive shear dOmega/dR the Hall instability exists for negative Hartmann number and for negative shear dOmega/dR the Hall instability exists for positive Hartmann number. For negative shear, of course, the Hall instability combines with the magnetorotational instability, resulting in a rather complex bifurcation diagram. In this case the critical magnetic Reynolds numbers with Hall effect are much lower than without Hall effect. In order to verify the presented shear-Hall instability at the laboratory with experiments using liquid metals, one would need rather large magnetic fields ( approximately 10(7) G).

Linear and nonlinear marginal stability for fronts of hyperbolic reaction diffusion equations.

We study traveling fronts of equations of the form u(tt)+phi(u)u(x)=u(xx)+f(u). A criterion for the transition from linear to nonlinear marginal stability is established for positive functions phi(u) and for any reaction term f(u) for which the usual parabolic reaction diffusion equation u(t)=u(xx)+f(u) admits a front. As an application, we treat reaction diffusion systems with transport memory.

Noise-induced fronts

Departament d’Estructura i Constituents de la Mate`ria, Universitat de Barcelona, Avenida Diagonal 647, 08028 Barcelona, Spain~Received 1 September 1998!A simple model is introduced that exhibits a noise-induced front propagation and where the noise entersmultiplicatively. The invasion of the unstable state is studied, both theoretically and numerically. A goodagreement is obtained for the mean value of the order parameter and the mean front velocity using theanalytical predictions of the linear marginal stability analysis. @S1063-651X~98!14912-7#PACS number~s!: 05.40.2aI. INTRODUCTION

Stability of Walking Vector Solitons

The stability of two-parameter families of walking vector solitons of coupled nonlinear Schr\odinger equations in investigated. It is shown that all known, lowest-order soliton types, namely, slow, fast, vector in phase, and vector out of phase are dynamically stable in certain regions of the parameter space. The condition of linear marginal stability of the solitons is not necessarily given by an explicit geometric criterion, because soliton instability mediated by the existence of complex eigenvalues of the corresponding Lyapunov operator is found to occur also.

Three basic issues concerning interface dynamics in nonequilibrium pattern formation

In these lecture notes, we discuss at an elementary level three themes concerning interface dynamics that play a role in pattern forming systems: (i) We briefly review three examples of systems in which the normal growth velocity is proportional to the gradient of a bulk field which itself obeys a Laplace or diffusion type of equation (solidification, viscous fingers and streamers), and then discuss why the Mullins–Sekerka instability is common to all such gradient systems. (ii) Secondly, we discuss how underlying an effective interface or moving boundary description of systems with smooth fronts or transition zones, is the assumption that the relaxation time of the appropriate order parameter field(s) in the front region is much smaller than the time scale of the evolution of interfacial patterns. Using standard arguments we illustrate that this is generally so for fronts that separate two (meta)stable phases: in such cases, the relaxation is typically exponential, and the relaxation time in the usual models goes to zero in the limit in which the front width vanishes. (iii) We finally summarize recent results that show that so-called “pulled” or “linear marginal stability” fronts which propagate into unstable states have a very slow universal power-law relaxation. This slow relaxation makes the usual “moving boundary” or “effective interface” approximation for problems with thin fronts, like streamers, impossible.

Speed of fronts of generalized reaction-diffusion equations

Recent work on generalized diffusion equations has given analytical and numerical evidence that, as in the standard reaction-diffusion equation, most initial conditions evolve into a traveling wave which corresponds to a minimum speed front joining a stable to an unstable state. We show that this minimal speed derives from a variational principle; from this we recover linear constraints on the speed (the linear marginal stability value) and provide upper and lower bounds on the speed. This enables us to characterize the functions for which linear marginal stability holds and also to provide a tool to calculate the speed when the marginal value does not predict its correct value.

On the stiffness of the sand pile profile

The existence of a critical gradient is an important feature of many transport models. [M. Kotschenreuther et al., Phys. Plasmas 2, 2381 (1995)]. However, fundamental differences exist in the dynamics near marginal stability, depending on whether the transport phenomena are controlled by strict (linear) marginal stability or by a self-organized criticality. One of the most striking differences is in the stiffness of the profiles. In this paper, a sand pile model is used to gain some basic understanding of the stiffness of the profile under different dynamics.

Universal Algebraic Relaxation of Fronts Propagating into an Unstable State and Implications for Moving Boundary Approximations

We analyze the relaxation of fronts propagating into unstable states. While ``pushed'' fronts relax exponentially like fronts propagating into a metastable state, ``pulled'' or ``linear marginal stability'' fronts relax algebraically. As a result, for thin fronts of this type, the standard moving boundary approximation fails. The leading relaxation terms for velocity and shape are of order $1/t$ and $1/{t}^{3/2}$. These universal terms are calculated exactly with a new systematic analysis that unifies various heuristic approaches to front propagation.

Velocity Selection for Propagating Fronts in Superconductors.

Using the time-dependent Ginzburg-Landau equations we study the propagation of planar fronts in superconductors, which would appear after a quench to zero applied magnetic field. Our numerical solutions show that the fronts propagate at a unique speed which is controlled by the amount of magnetic flux trapped in the front. For small flux the speed can be determined from the linear marginal stability hypothesis, while for large flux the speed may be calculated using matched asymptotic expansions. At a special point the order parameter and vector potential are dual, leading to an exact solution which is used as the starting point for a perturbative analysis.

Validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation.

We consider the problem of the speed selection mechanism for the one-dimensional nonlinear diffusion equation ${u}_{t}={u}_{\mathrm{xx}}+f(u)$. It has been rigorously shown by Aronson and Weinberger that for a wide class of functions $f$, sufficiently localized initial conditions evolve in time into a monotonic front which propagates with speed ${c}^{*}$ such that $2\sqrt{{f}^{\ensuremath{'}}(0)}\ensuremath{\le}{c}^{*}<2\sqrt{supop(\frac{f(u)}{u})}$. The lower value ${c}_{L}=2\sqrt{{f}^{\ensuremath{'}}(0)}$ is that predicted by the linear marginal stability speed selection mechanism. We derive a new lower bound on the speed of the selected front, this bound depends on $f$ and thus enables us to assess the extent to which the linear marginal selection mechanism is valid.

Chaotic step bunching during crystal growth

We study a simple model for step bunching during crystal growth by propagating a disturbance into an unstable system of equidistant steps. The system exhibits a wide range of different bunching modes as a function of the initial step spacing, leading to distinctive spatial patterns: periodic (with subharmonic bifurcations), chaotic and intermittent. Linear and nonlinear marginal stability theory gives extremely accurate predictions of the velocity of the propagating front. One of the bifurcations is identified as a transition from a regime where linear marginal stability applies to a nonlinear marginal stability regime. The location of this bifurcation is determined accurately.

Gravitational instabilities of thin liquid layers: dynamics of pattern selection

We present recent results obtained in the study of the instabilities induced by gravity on liquid layers hanging below solid surfaces, in situation of perfect wetting (without contact lines). Two cases are discussed: two-dimensional films of constant volume, and films continuously supplied with “fresh” liquid by an external source. In the first case, we show experimental evidences suggesting that the formation of a two-dimensional structure within the film arises by means of front propagation mechanisms. In the simplest situation, propagation of a one-dimensional structure (“roll” formation), the measured front speed is close to that deduced from the linear marginal stability theory. In the second case, the liquid film is hanging below an horizontal overflowing half-cylinder. Depending on the rate of supply, different regimes are observed (dripping, arrays of parallel jets, triangular sheets). The arrays of drops and jets exhibit interesting spatio-temporal phase dynamics: oscillations, pairing or nucleation of cells, forced tilt waves.

Atomic physics effects on dissipative toroidal drift wave stability

The effects of atomic physics processes such as ionization, charge exchange, and radiation on the linear stability of dissipative drift waves are investigated in toroidal geometry, both numerically and analytically. For typical Tokamak Fusion Test Reactor (TFTR) [Plasma Physics and Controlled Nuclear Fusion Research, 1986 (IAEA, Vienna, 1987), Vol. 1, p. 51] and Texas Experimental Tokamak (TEXT) [Nucl. Technol. Fusion 1, 479 (1981)] edge parameters, overall linear stability is determined by the competition between the destabilizing influence of ionization and the stabilizing effect due to the electron temperature gradient. An analytical expression for the linear marginal stability condition, ηcrite, is derived. The instability is most likely to occur at the extreme edge of tokamaks with a significant ionization source and a steep electron density gradient.

Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence.

In an earlier paper, we developed a general physical picture for the linear-marginal-stability mechanism governing the dynamics of front propagation into linearly unstable states. The main conclusion from this approach and the expressions for the resulting front velocity are similar to those obtained along different lines for the space-time evolution of instabilities in plasma physics and fluid dynamics with the so-called pinch-point analysis (a special type of saddle-point analysis). However, as stressed by Ben-Jacob et al. [Physica 14D, 348 (1985)], it is known from the work of Aronson and Weinberger [in Partial Differential Equations and Related Topics, edited by J. A. Goldstein (Springer, Heidelberg, 1975); Adv. Math. 30, 33 (1978)] on a class of simple model equations that exceptions can occur to the linear-marginal-stability velocity selection. In this paper, we generalize these observations and incorporate such exceptions into our general picture of front propagation into unstable states. We show that a breakdown of linear marginal stability occurs if the linear-marginal-stability front profile becomes unstable against a particular nonlinear ``invasion mode.''If this happens, a larger front speed is selected at a point at which the front profile is now marginally stable against this nonlinear invasion mode. We therefore refer to this as the nonlinear-marginal-stability mechanism. (Ben-Jacob et al. call it case-II marginal stability.) We present the results of detailed numerical studies that support our identification of the nonlinear-marginal-stability mechanism, and present the first examples of it for fronts in pattern-forming systems. In the neighborhood of a transition from linear to nonlinear marginal stability, the wavelength of the pattern generated by the front is only weakly dependent on the nonlinearities. We also analyze front propagation properties close to the threshold for instability at a pitchfork bifurcation. We conclude that linear marginal stability generally holds near a continuous transition (corresponding to a supercritical or forward bifurcation point), while front propagation close to a first-order transition (corresponding to a subcritical or inverted-bifurcation point) is generally governed by nonlinear marginal stability. These results are of importance for recent applications of the various approaches in fluid dynamics and other fields.Finally, we derive an expression for the rate of convergence of the front velocity to its asymptotic value. For the class of equations studied by Aronson and Weinberger, our expression reduces to a rigorous result by Bramson [Mem. Am. Math. Soc. 285, 1 (1983)], but it differs from the one often quoted in the pinch-point or saddle-point analysis. We argue that the latter one is only valid in a limited region of space, and show how to extend the usual analysis to arrive at our result. Several experimental systems to which our results are relevant are discussed.

Two-dimensional nonlinear processes associated with ‘type 1’ irregularities in the equatorial electrojet

It is suggested that the stabilization mechanism of type 1 irregularities in the equatorial electrojet has two aspects: (1) the turbulence tends to inhibit the formation of large currents and to maintain the plasma in a state relatively ‘close’ to linear marginal stability, and (2) in this state, energy is transferred by two-dimensional nonlinear wave-coupling processes from the linearly growing modes to linearly damped ones propagating in other directions. Energy transfer processes can thus provide detailed stabilization over the entire spectrum. It is argued that (1) the theory provides a plausible interpretation of the as yet puzzling experimental result that the irregularities appear more or less simultaneously over a substantial range of radar elevation angles and (2) the fact that their phase velocities are close to the sound speed and often completely independent of the direction of propagation is perhaps a consequence of mode coupling, but definite prediction here requires numerical calculations.