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We study a single-particle continuous-time quantum walk on a one-dimensional spatial lattice with complex nearest neighbour (NN) and next-nearest neighbour (NNN) hopping amplitudes. We show that the model allows for controlled information transfer and directional biasing without a biased initial state. Specifically, we show that for an unbiased initial state, complex couplings lead to chiral propagation and a causal cone structure asymmetric about the origin. We provide a hydrodynamic description for the dynamics in the large space–time limit. We obtain a global ‘quasi-stationary state’ which can be described in terms of the local quasi-particle densities satisfying Euler type of hydrodynamic equations and characterized by an infinite set of conservation laws. Further, we show that higher-order hydrodynamic equations can be used to describe the anomalous sub-diffusive scaling near the extremal fronts. The long-time behaviour for any complex NNN hopping with a nonzero real component is similar to that of purely real hopping; at a critical coupling strength, there is a Lifshitz transition where the topology of the causal structure changes from a regime with one causal cone to a regime with two nested (asymmetric) causal cones. For purely imaginary NNN hopping, there is a transition from one causal cone to a regime with two partially overlapping cones which can be attributed to the existence of degenerate maximal fronts. Both the nature of the Lifshitz transition and the scaling behaviour at the critical coupling are different in the two cases. We also discuss possible experimental realizations of such a model. [ABSTRACT FROM AUTHOR]

The 3D structure of a fire front propagating through a homogeneous porous solid-fuel layer was studied numerically at laboratory and field scales. At laboratory scale, wind-tunnel fires propagating through laser-cut cardboard fuel were numerically reproduced, while at field scale, simulations of grassland fires with quasi-infinite fire front were carried out for different wind speeds. These simulations were performed using FIRESTAR3D, based on a multiphase formulation that includes the main physical phenomena governing fire behavior. An unsteady RANS approach and a Large Eddy Simulation (LES) approach were used to simulate the reactive turbulent flow, whereas turbulent combustion was modeled using Eddy Dissipation Concept (EDC). Unlike other 3D wildfire tools available in the community, such as FIRETEC and WFDS, the model is based on an implicit, low-Mach number resolution of the governing equations, and makes no empirical assumptions in the resolution of the radiative transfer equation. The comparison with the experimental data concerned mainly the Rate of Spread (ROS) of fire, the fireline intensity, the flame-zone depth, and the wavelength characterizing the crest-and-trough structure of the fire front along the transverse direction. Particular attention was drawn to the similitude in the fire front dynamics between small and large scales. In order to highlight the physical mechanisms responsible for this dynamics, a dimensional analysis was carried out by introducing Byram's convection number N C based on the fireline intensity and Froude's numbers Fr based on the characteristic wavelength of the fire-front structure. The analysis shows that all the results (wind-tunnel fires and grassland fires, experimental and numerical) collapsed on a single scaling law in the form F r = N C − 2 / 3. [ABSTRACT FROM AUTHOR]

The Changjiang River discharges huge amounts of nutrients, which cause frequent harmful algal blooms off its estuary. By analyzing historical and in situ observational data, this study aimed to determine the distribution and physical controls of these blooms. The results indicated that phytoplankton bloomed within the 30–50 m isobaths south of the Changjiang River mouth, with several high-value centers. South of the river mouth, the river plume front propagated along the coast as bottom-trapped river plume, which reached the bottom along the 25–30 m isobaths. The algae bloomed between the locations of the bottom and surface fronts. The surface front determined the seaward limit of the nutrient-rich river-influenced water and set the outer boundary of algal blooming; the latter separated the stratified (hence less turbid) offshore surface water from the well-mixed thus turbid nearshore surface water, forming the shoreward limit of the algal blooming due to high turbidity. The surface plume front was relatively unstable compared with the bottom. The surface plume front relaxes around the bathymetric irregularities, which enlarges the distance between the plume's surface and bottom fronts, providing a larger area favorable to algal blooming. Consequently, several high chlorophyll-a centers developed around the bathymetric irregularities. East and northeast of the river mouth, the river plume floated above ambient seawater as a surface-trapped river plume. A subsurface chlorophyll-a maximum was observed above the pycnocline in the far-field plume area, which descended gradually to ~ 10 m in the offshore direction. The associated dynamical mechanism was also discussed. [ABSTRACT FROM AUTHOR]

The dynamics of a curved front in a plane between two stable phases with equal potentials is modeled via two-dimensional fractional in time partial differential equation. A closed equation governing a slow motion of a small-curvature front is derived and applied for two typical examples of the potential function. Approximate axisymmetric and non-axisymmetric solutions are obtained. [ABSTRACT FROM AUTHOR]

This paper reports an experimental investigation of premixed propane and methane-air flames propagating freely downwards in tubes 1.5 m long and with diameters 54 and 94 mm. The tubes are opened at the upper end and closed at the bottom end with an acoustic damper that eliminates thermo-acoustic instability. Igniting the flame either on the tube axis or close to the wall, two regimes of propagation are distinguished by correlating the flame speed and the radius of curvature at the flame tip. The characteristic lengths are then related to the cut-off wavelengths estimated from linear theories and compared to previous results of Michelson–Sivashinsky simulations. [ABSTRACT FROM AUTHOR]

Fluid displacement fronts in porous media exhibit a peculiar duality; the seemingly regular macroscopic motion of the front is propelled by numerous and irregular pore scale interfacial jumps. These pore scale events shape emergent front morphology, affect phase entrapment behind a front, and are likely important for colloidal mobilization and solute dispersion at the front. We present an experimental study focusing on drainage fluid front invasion dynamics through sintered glass beads using a high-speed camera and rapid capillary pressure measurements to resolve pore scale invasion events over a wide range of boundary conditions (flow rates and gravitational influences). We distinguished three types of 'pores': geometrical pores deduced from image analyses; individual pore invasion volumes imaged during displacement; and pore volumes deduced from capillary pressure fluctuations during constant withdrawal rates. The resulting pore volume distributions were remarkably similar for slow drainage rates. Invaded pore volumes were not affected by gravitational forces, however with increased viscous forces (higher displacement rates) the fraction of small invaded volumes increased. Capillary pressure fluctuations were exponentially distributed in agreement with findings from previous studies. The distribution of pressure fluctuations exhibited a distinct cutoff concurrent with the onset of simultaneous invasion events. The study highlights the different manifestation of 'pores' and their sensitivity to external (macroscopic) boundary conditions. The remarkable similarity of geometrical and pressure-deduced pore spaces offers opportunities for deducing pore size distribution dynamically. [ABSTRACT FROM AUTHOR]

The response of a dynamical flame model to imposed acoustic accelerations is studied analytically and numerically. Through linear stability analyses, two analytical approximations for the primary and the parametric stability boundaries are found. The approximation for the primary instability boundary is accurate for any periodic accelerations, in the limit of large acoustic frequencies. The critical acoustic amplitudeuafor Landau–Darrieus instability suppression is identified and found to depend only on the density contrast and the shape of the periodic acoustic stimuli. The proposed model evolution equation is next integrated numerically with various imposed acoustic accelerations; the primary and parametric flame responses are identified. It is shown analytically and numerically that in the presence of a fully developed, yet weakened by acoustics, Landau–Darrieus (or primary) instability the wrinkle amplitude and the mean flame speed oscillate at the same frequency as the acoustic stimuli; the threshold for suppression of primary instability by acoustic forcing is determined exactly. Increasing the acoustic amplitude allows the flame to respond parametrically to the acoustics. This response is characterised by troughs and crests interchanging their roles while the mean flame speed again oscillates with the same frequency as the acoustic stimuli and at twice that of wrinkle amplitude oscillations. [ABSTRACT FROM PUBLISHER]

Ecosystem regime shifts are regarded as abrupt global transitions from one stable state to an alternative stable state, induced by slow environmental changes or by global disturbances. Spatially extended ecosystems, however, can also respond to local disturbances by the formation of small domains of the alternative state. Such a response can lead to gradual regime shifts involving front propagation and the coalescence of alternative-state domains. When one of the states is spatially patterned, a multitude of intermediate stable states appears, giving rise to step-like gradual shifts with extended pauses at these states. Using a minimal model, we study gradual state transitions and show that they precede abrupt transitions. We propose indicators to probe gradual regime shifts, and suggest that a combination of abrupt-shift indicators and gradual-shift indicators might be needed to unambiguously identify regime shifts. Our results are particularly relevant to desertification in drylands where transitions to bare soil take place from spotted vegetation, and the degradation process appears to involve step-like events of local vegetation mortality caused by repeated droughts. [ABSTRACT FROM AUTHOR]

Abstract: Non-equilibrium front propagation in a two-dimensional network modelling wildfire propagation was studied. The model includes deterministic long-range interactions due to radiation and a time weighting procedure. Three weight-dependent propagation regimes were found: dynamical, static, and non-propagative. The dynamical regime shows saturation for small weight values and a percolation transition area depending on the weight and size of the interaction domain. From the scaling interface exponents, the model seems to belong to the dynamical percolation universality class. In the limit of static regime it belongs to the random deposition class. [Copyright &y& Elsevier]

We present new experimental results on the quenching dynamics of an extended thermo-convective system (a network array of approximately 100 convective oscillators) going through a secondary subcritical bifurcation. We characterize a dynamical phase transition through the nature of the domain walls (1D-fronts) that connect the basic multicellular pattern with the new oscillating one. Two different mechanisms of the relaxing dynamics at the threshold are characterized depending on the crossing rate of the quenched transition. From the analysis of fronts, we show that these mechanisms follow different correlation length scales ξ ~ μ-σ. Below a critical value μc, a slow response dynamics yields a spatiotemporal coherent front with weak coupling between oscillators. Above μc, for rapid quenches, defects are trapped at the front with a strong coupling between oscillators, similarly to the Kibble-Zurek mechanism in quenched phase transitions. These defects, pinned to the fronts, yield a strong decay of the correlation length. [ABSTRACT FROM AUTHOR]

Abstract: A number of recent studies suggest that many biological species follow a Lévy random walk in their search for food. Such a strategy has been shown to be more efficient than classical Brownian motion when resources are scarce. However, current diffusion–reaction models used to describe many ecological systems do not account for the superdiffusive spread of populations due to Lévy flights. We have developed a model to simulate the spatial spread of two species competing for the same resources and driven by Lévy flights. The model is based on the Lotka–Volterra equations and has been obtained by replacing the second-order diffusion operator by a fractional-order one. Consistent with previous known results, theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the population fronts and a power-law decay of the fronts'' leading tail. Depending on the skewness of the fractional derivative, we derive catch-up conditions for different types of fronts. Our results indicate that second-order diffusion–reaction models are not well-suited to simulate the spatial spread of biological species that follow a Lévy random walk as they are inclined to underestimate the speed at which these species propagate. [Copyright &y& Elsevier]

Abstract: A number of recent studies suggest that human and animal mobility patterns exhibit scale-free, Lévy-flight dynamics. However, current reaction-diffusion epidemics models do not account for the superdiffusive spread of modern epidemics due to Lévy flights. We have developed a SIR model to simulate the spatial spread of a hypothetical epidemic driven by long-range displacements in the infective and susceptible populations. The model has been obtained by replacing the second-order diffusion operator by a fractional-order operator. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the epidemic''s front and a power-law decay of the front''s leading tail. Our results indicate the potential of fractional-order reaction-diffusion models to represent modern epidemics. [Copyright &y& Elsevier]

In two-dimensional free-interface problems, the front dynamics can be modeled by single parabolic equations such as the Kuramoto-Sivashinsky equation (K-S). However, away from the stability threshold, the structure of the front equation may be more involved. In this paper, a generalized K-S equation, a nonlinear wave equation with a strong damping operator, is considered. As a consequence, the associated semigroup turns out to be analytic. Asymptotic convergence to K-S is shown, while numerical results illustrate the dynamics. [ABSTRACT FROM AUTHOR]

Abstract: We herein consider two-component reaction–diffusion systems with a specific bistable and odd symmetric nonlinearity, which have the bifurcation structure of pitchfork type traveling front solutions with opposite velocities. We introduce a spatial heterogeneity, for example, a Heaviside-like abrupt change at the origin in the space, into diffusion coefficients. Numerically, the responses of traveling fronts via the heterogeneity can be classified into four types of behavior depending on the strength of the heterogeneity, which, in the present paper, is represented by the height of the jump: passage, stoppage, and two types of reflection. The goal of the present paper is to reduce the PDE dynamics to finite-dimensional ODE systems on a center manifold and show the mathematical mechanism for producing the four types of response in the PDE systems using finite-dimensional ODE systems. The reduced ODE systems include the terms (referred to as heterogeneous perturbations) originating from the interaction between traveling front solutions and the heterogeneity, which is very important for determining the dynamics of the ODE systems. In the present paper, we succeed in calculating these heterogeneous perturbations exactly and explicitly. [Copyright &y& Elsevier]

In this paper, we seek similarity solutions to the shallow water (Saint-Venant) equations for describing the motion of a non-Boussinesq, gravity-driven current in an inertial regime. The current is supplied in fluid by a source placed at the inlet of a horizontal plane. Gratton and Vigo (1994) found similarity solutions to the Saint-Venant equations when a Benjamin-like boundary condition was imposed at the front (i.e., nonzero flow depth); the Benjamin condition represents the resisting effect of the ambient fluid for a Boussinesq current (i.e., a small-density mismatch between the current and the surrounding fluid). In contrast, for non-Boussinesq currents the flow depth is expected to be zero at the front in absence of friction. In this paper, we show that the Saint-Venant equations also admit similarity solutions in the case of non-Boussinesq regimes provided that there is no shear in the vertical profile of the streamwise velocity field. In that case, the front takes the form of an acute wedge with a straight free boundary and is separated from the body by a bore. [ABSTRACT FROM AUTHOR]

Abstract: We investigate an optimal velocity car-following model for n cars on a circular single-lane road, where reaction-time delay of drivers is taken into account. The stability of the uniform flow equilibrium is studied analytically, while bifurcating periodic solutions for different wave numbers are investigated with numerical continuation techniques. This reveals that the periodic solution with the smallest wave number may be stable, and all other periodic solutions are unstable. As n is increased, periodic solutions develop stop- and go-fronts that correspond to rapid deceleration and acceleration between regions of uniformly flowing and stagnant traffic. In terms of the positions of all cars on the ring these fronts are associated with traffic jams. All traffic jams form a traffic pattern that evolves under time, due to slow motion of the fronts. The traffic pattern corresponding to the stable periodic motion of cars is the only stable one. However, we find that other periodic orbits may be unstable only so weakly that they give rise to transient traffic jams that may persist for long times. Eventually, such traffic jams either merge with one another or disperse, until the stable traffic pattern is reached. [Copyright &y& Elsevier]

We show that the bifurcation scenario in a high-dimensional system with interacting moving fronts can be related to the universal U-sequence which is known from the symbolic analysis of iterated one-dimensional maps. This connection is corroborated for a model of a semiconductor superlattice, which describes the complex dynamics of electron accumulation and depletion fronts. By a suitable Poincaré section we reduce the dynamics to a low-dimensional iterated map, for which in the most elementary case the bifurcation points can be determined analytically. [ABSTRACT FROM AUTHOR]

We propose and analyze a model for sharp fronts in porous media, aiming at an investigation of the capillary pressure. Using the notion of microlocal patterns we analyze the local behavior of the system. Depending on the structure of the local patterns we can derive upscaled equations that characterize the capillary pressure and include the hysteresis effect that is known from the physical system. [ABSTRACT FROM AUTHOR]

We use the FitzHugh–Nagumo (F–N) model to study front solutions in inhomogeneous media. Inhomogeneity is modeled by adding a space-dependent term, z(x), in the equations. Changes in z shift the cubic nullcline up and down in the phase-plane of the F–N model. This is used to mimic shifts in the cubic-shaped nullcline of a bidomain model of calcium waves in egg cells [Physica D, in press] caused by changes in the total calcium concentration. Conditions for the existence and stability of stationary front solutions for a general class of z functions are obtained analytically when the dynamics are piecewise linear. We show that spatial inhomogeneities cause pinning and oscillations of front solutions. This is best demonstrated when z(x) is a linear ramp. When the slope is large, a linear ramp breaks the translational invariance of the stationary front and stabilizes the front. The front becomes less stable as the slope decreases. At a critical slope, the front becomes unstable through a Hopf bifurcation beyond which oscillations in the front occur. This also occurs for nonlinear spatial inhomogeneities including a step increase with varying magnitude and steepness. This bifurcation has been found previously [Physica D 106 (1997) 270; Phys. Rev. E 55 (1997) 366] in a local analysis of the F–N model near a co-dimension two point (sometimes called a non-equilibrium Ising–Bloch (NIB) bifurcation) and with inhomogeneities of small magnitude. The analysis presented here applies globally in parameter space and is valid for inhomogeneities of arbitrary magnitude. We show that this bifurcation need not occur in conjunction with the NIB point. It is caused exclusively by spatial inhomogeneity. A rich variety of wave phenomena including front oscillation, front pinning, and front reflection are shown to occur in inhomogeneous media. [Copyright &y& Elsevier]

The formation and propagation of thermal fronts in a cylindrical medium that is undergoing microwave heating is studied in detail. The model consists of Maxwell's wave equation coupled to a temperature diffusion equation containing a bistable nonlinear term. [ABSTRACT FROM PUBLISHER]

Spatially localized oscillations in periodically forced systems are intriguing phenomena. They may occur in spatially homogeneous media (oscillons), but quite often emerge in heterogeneous media, such as the auditory system, where localized oscillations are believed to play an important role in frequency discrimination of incoming sound waves. In this paper, we use an amplitude-equation approach to study the spatial profile of the oscillations and the factors that affect it. More specifically, we use a variant of the forced complex Ginzburg–Landau (FCGL) equation to describe an oscillatory system below the Hopf bifurcation with space-dependent Hopf frequency, subject to both parametric and additive forcing. We show that spatial heterogeneity, combined with bistability of system states, results in spatial asymmetry of the localized oscillations. We further identify parameters that control that asymmetry, and characterize the spatial profile of the oscillations in terms of maximum amplitude, location, width and asymmetry. Our results bear qualitative similarities to empirical observation trends that have found in the auditory system. • The study is motivated by localized oscillations observed in the cochlea • Heterogenous oscillatory media under parametric and additive forcing are considered • Spatial profiles of localized oscillations are studied analytically and numerically • Factors controlling the asymmetry of the oscillations profile are identified • Relations to observations in the cochlea are discussed [ABSTRACT FROM AUTHOR]