Long-time Propagation Research Articles

On a similarity solution for lock-release gravity currents affected by slope, drag and entrainment

We consider the long-time propagation of a Boussinesq inertia–buoyancy (large-Reynolds- number) gravity current released from a lock over a downslope of angle $\gamma$ , affected by entrainment and drag. We show that the shallow-water (depth-averaged) equations with a Benjamin-type front-jump condition admit a similarity solution $x_N(t) = K t^{2/3}$ while $h, \phi, u$ change like $t$ to the power of $2/3, -4/3, -1/3$ , respectively; here $x_N, h, \phi, u$ and $t$ are the position of the nose (distance from backwall), thickness, concentration of dense fluid, velocity and time, respectively, and K is a constant. Assuming that $\gamma$ and the coefficients of entrainment and drag are constant, we derive an analytical exact solution for the similarity profiles and show that $K \propto (\tan \gamma )^{1/3}$ ; the driving of the slope is balanced by entrainment and/or drag. The predicted $t^{2/3}$ propagation is in agreement with previously published experimental data but a conclusive quantitative assessment of the present theory cannot be performed due to various uncertainties (discussed in the paper) that must be resolved by future work.

Influence of confinement on the spreading of bacterial populations.

The spreading of bacterial populations is central to processes in agriculture, the environment, and medicine. However, existing models of spreading typically focus on cells in unconfined settings—despite the fact that many bacteria inhabit complex and crowded environments, such as soils, sediments, and biological tissues/gels, in which solid obstacles confine the cells and thereby strongly regulate population spreading. Here, we develop an extended version of the classic Keller-Segel model of bacterial spreading via motility that also incorporates cellular growth and division, and explicitly considers the influence of confinement in promoting both cell-solid and cell-cell collisions. Numerical simulations of this extended model demonstrate how confinement fundamentally alters the dynamics and morphology of spreading bacterial populations, in good agreement with recent experimental results. In particular, with increasing confinement, we find that cell-cell collisions increasingly hinder the initial formation and the long-time propagation speed of chemotactic pulses. Moreover, also with increasing confinement, we find that cellular growth and division plays an increasingly dominant role in driving population spreading—eventually leading to a transition from chemotactic spreading to growth-driven spreading via a slower, jammed front. This work thus provides a theoretical foundation for further investigations of the influence of confinement on bacterial spreading. More broadly, these results help to provide a framework to predict and control the dynamics of bacterial populations in complex and crowded environments.

Solving the time-dependent Schrödinger equation by combining smooth exterior complex scaling and Arnoldi propagator

We develop a highly efficient scheme for numerically solving the three-dimensional time-dependent Schrödinger equation of the single-active-electron atom in the field of laser pulses by combining smooth exterior complex scaling (SECS) absorbing method and Arnoldi propagation method. Such combination has not been reported in the literature. The proposed scheme is particularly useful in the applications involving long-time wave propagation. The SECS is a wonderful absorber, but its application results in a non-Hermitian Hamiltonian, invalidating propagators utilizing the Hermitian symmetry of the Hamiltonian. We demonstrate that the routine Arnoldi propagator can be modified to treat the non-Hermitian Hamiltonian. The efficiency of the proposed scheme is checked by tracking the time-dependent electron wave packet in the case of both weak extreme ultraviolet (XUV) and strong infrared (IR) laser pulses. Both perfect absorption and stable propagation are observed.

Explicit high-order exponential time integrator for discontinuous Galerkin solution of Maxwell's equations

In this paper, exponential propagator (EP) is introduced to the discontinuous Galerkin finite element method (DG-FEM) for solving transient electromagnetic problems. As a kind of explicit time-stepping method, the proposed EP has fourth order accuracy in time and requires only one storage unit for each of the degrees-of-freedom. It provides an attractive alternative to the high-order low-storage explicit Runge-Kutta (LSERK) approach, which is the most popular time-marching scheme currently used in high-fidelity simulation of Maxwell's equations with DG-FEM. When centered numerical flux is utilized, the discretized Maxwell's system in lossless region is symplectic structure-preserving (energy conserving), and thus very suitable for accurately computing the long-time propagation of waves in non-dissipative medium, or simulating the motion of charged particles in electromagnetic fields. Our method can also handle upwind numerical flux at the sacrifice of symplectic-conserving virtue for higher accuracy in spatial discretization and spurious inhibition. A major contribution of this work is that, the application range of this scheme is extended to lossy media such as conductors and particularly unsplit perfectly matched layers in the framework of DG-FEM. This greatly facilitates the treatment of open boundary conditions in scattering and/or radiation problems. Extensive numerical examples, including the analysis of photonic crystals, dielectric coated perfect conducting cylinder, propagation of waves in zero-index materials, and corrugated O-type Cherenkov oscillators etc., are presented to demonstrate its stability, accuracy, and performance. All the numerical experiments show that the presented method is superior to the LSERK scheme when centered numerical flux is adopted.

Small matrix modular path integral: iterative quantum dynamics in space and time.

The modular path integral (MPI) formulation for one-dimensional extended systems, such as spin arrays or molecular aggregates, allows evaluation of spin- or exciton-vibration dynamics with effort that scales linearly with the number of units. This work presents a small matrix decomposition of the modular path integral (SMatMPI), which eliminates tensor storage and enables iterative long-time propagation.

Dark-antidark spinor solitons in spin-1 Bose gases

We consider a one-dimensional trapped spin-1 Bose gas and numerically explore families of its solitonic solutions, namely antidark-dark-antidark (ADDAD), as well as dark-antidark-dark (DADD) solitary waves. Their existence and stability properties are systematically investigated within the experimentally accessible easy-plane ferromagnetic phase by means of a continuation over the atom number as well as the quadratic Zeeman energy. It is found that ADDADs are substantially more dynamically robust than DADDs. The latter are typically unstable within the examined parameter range. The dynamical evolution of both of these states is explored and the implication of their potential unstable evolution is studied. Some of the relevant observed possibilities involve, e.g., symmetry-breaking instability manifestations for the ADDAD, as well as splitting of the DADD into a right- and a left-moving dark-antidark pair with the anti-darks residing in a different component as compared to prior to the splitting. In the latter case, the structures are seen to disperse upon long-time propagation.

A finite volume WENO scheme for immiscible inviscid two-phase flows

This article presents a two-phase finite volume (FV) method based on the concept of equilibrium volume fraction for simulating inviscid two-phase flows. A reduced three-equation hyperbolic system is adopted as the governing equations, which is closed with a barotropic equation of state for both phases. A modified weighted essentially non-oscillatory (WENO) scheme is proposed for reconstructing the fluid state at cell-interfaces, which can avoid strictly pressure or velocity oscillations near material interfaces in the test-case of the advection of an isolated interface, while retaining the convergence order of the WENO scheme in 1D and 2D cases. Besides, with the generalized Riemann invariants (GRI), the eigenstructure of the hyperbolic system is analyzed and an approximate Riemann flux solver is proposed based on the linearization of the GRI. The semi-discrete system is explicitly integrated in time by means of the classical 4th-order Runge-Kutta (RK4) scheme. The proposed scheme is validated in a series of two-phase test-cases, such as advection of a two-phase vortex, linear sloshing, long-time wave propagation and dam-break flows, for which good agreements are obtained with the references.

Modelling Leaky Waves in Planar Dielectric Waveguides

Experimentally observed leaky modes of a dielectric waveguide are characterised by a weak tunnelling of the light through the waveguide and its long-time propagation along the waveguide. Traditional mathematical models of leaky waveguide modes meet some contradictions resolved using additional considerations. We propose a model of leaky modes in a waveguide free from the above contradictions, akin to the quantum mechanical model of the “pseudo-stable” Gamow-Siegert states. By separating variables, from the complete problem for plane inhomogeneous waves we obtain a non-self-adjoint Sturm-Liouville problem to determine the complex coefficient of the phase delay of the studied mode. The solution of the complete wave problem determines the propagation cone for the leaky mode of the waveguide, inside which there are no contradictions. Thus, solution is in qualitative agreement with experimental data.

Hybrid time-domain model for ship motions in nonlinear extreme waves using HOS method

HOS (High-Order Spectral) method incorporated with a 3-D time-domain ship motion model based on the weakly nonlinear assumption has been proposed and applied to compute motions of a ship advancing with constant speed in various nonlinear waves. In the proposed approach, HOS method is employed to simulate arbitrary incident wave systems described in earth-based coordinate, including regular, irregular waves and focused waves and resolve nonlinear wave-wave interactions up to an arbitrary order. Its fast convergence and high efficiency allow modelization of long-time nonlinear propagation of predetermined input wave spectra and simulation of realistic sea states and extreme events while keeping reasonable computational efforts. Nonlinear F–K (Froude-krylov) forces and restoring forces are calculated and evaluated over the instantaneous wetted surface to capture the strong nonlinear features of incident wave field simulated by HOS method. The equations of time-domain ship motions in nonlinear waves are established. This hybrid model benefits from the efficiency of HOS method to solve nonlinear free surface boundary conditions for incident wave, in the same breath, the time-domain model employed here is computationally simple and of great practical significance.To illustrate the feasibility and accuracy of this method, comparisons to experiment data have been conducted in the case of regular head waves and gain satisfactory agreement. Furthermore, the nonlinearities of focused waves as well as irregular waves and the subsequent nonlinear effects to ship motions are investigated. It is shown that the nonlinearity of the incident waves plays a significant role for the extreme wave occurrence and results in a larger F–K force and a remarkable larger heave motion.

High-Order Symplectic Compact Finite-Different Time-Domain Algorithm for Guide-Wave Structures

Based on higher order symplectic integrator temporal derivatives and compact spatial derivatives, a new high-order symplectic compact finite-difference time-domain (FDTD) algorithm is proposed. The numerical stability criteria and dispersion relation are derived analytically. Compared with the traditional compact FDTD algorithm, the proposed algorithm can significantly reduce the numerical dispersion error and can be made nearly independent of the maximum limitation of the Courant–Friedrich–Levy law. By means of the numerical examples, the proposed algorithm has the benefit of conserving energy for long-time propagation, which can be used to enhance the computational efficiency for the full-wave-analysis of guided-wave structures.

A study of the stability properties of Sagdeev solutions in the ion-acoustic regime using kinetic simulations

The Sagdeev pseudo-potential approach has been employed extensively in theoretical studies to determine large-amplitude (fully) nonlinear solutions in a variety of multi-species plasmas. Although these solutions are repeatedly considered as solitary waves (and even solitons), their temporal stability has never been proven. In this paper, a numerical study of the Vlasov-Poisson system is made to follow their temporal evolution in the presence of numerical noise and thereby test their long-time propagation stability. Considering the ion-acoustic regime, both constituents of the plasma, i.e., electrons and ions are treated following their distribution functions in these sets of fully-kinetic simulations. The findings reveal that the stability of the Sagdeev solution depends on a combination of two parameters, i.e., velocity and trapping parameter. It is shown that there exists a critical value of trapping parameter for both fast and slow solutions which separates stable from unstable solutions. In the case of stable solutions, it is shown that these nonlinear structures can propagate for long periods, which confirms their status as solitary waves. Stable solutions are reported for both Maxwellian and Kappa distribution functions. For unstable solutions, it is demonstrated that the instability causes the Sagdeev solution to decay by emitting ion-acoustic wave-packets on its propagation trail. The instability is shown to take place in a large range of velocities and even for Sagdeev solutions with a velocity much higher than the ion-sound speed. Besides, in order to validate our simulation code, two precautionary measures are taken. First, the well-known effect of the ion dynamics on a stationary electron hole solution is presented as a benchmarking test of the approach. Second, In order to verify the numerical accuracy of the simulations, the conservation of energy and entropy is presented.

Solitons in a modified discrete nonlinear Schr\xf6dinger equation

We study the bulk and surface nonlinear modes of a modified one-dimensional discrete nonlinear Schrödinger (mDNLS) equation. A linear and a modulational stability analysis of the lowest-order modes is carried out. While for the fundamental bulk mode there is no power threshold, the fundamental surface mode needs a minimum power level to exist. Examination of the time evolution of discrete solitons in the limit of strongly localized modes, suggests ways to manage the Peierls-Nabarro barrier, facilitating in this way a degree of soliton steering. The long-time propagation of an initially localized excitation shows that, at long evolution times, nonlinear effects become negligible and as a result, the propagation becomes ballistic. The qualitative similarity of the results for the mDNLS to the ones obtained for the standard DNLS, suggests that this kind of discrete soliton is an robust entity capable of transporting an excitation across a generic discrete medium that models several systems of interest.

Reexamining the origin of the directionality of myosin V

The nature of the conversion of chemical energy to directional motion in myosin V is examined by careful simulations that include two complementary methods: direct Langevin Dynamics (LD) simulations with a scaled-down potential that provided a detailed time-resolved mechanism, and kinetic equations solution for the ensemble long-time propagation (based on information collected for segments of the landscape using LD simulations and experimental information). It is found that the directionality is due to the rate-limiting ADP release step rather than the potential energy of the lever arm angle. We show that the energy of the power stroke and the barriers involved in it are of minor consequence to the selectivity of forward over backward steps and instead suggest that the selective release of ADP from a postrigor myosin motor head promotes highly selective and processive myosin V. Our model is supported by different computational methods-LD simulations, Monte Carlo simulations, and kinetic equations solution-as well as by structure-based binding energy calculations.

Long-time behaviour and propagation of chaos for mean field kinetic particles

The trend to equilibrium in large time is studied for a large particle system associated to a Vlasov–Fokker–Planck equation in the presence of a convex external potential, without smallness restriction on the interaction. From this are derived uniform in time propagation of chaos estimates, which themselves yield in turn an exponentially fast convergence for the semi-linear equation itself. The approach is quantitative.

A theoretical study of the relaxation of a phenyl group chemisorbed to an RDX freestanding thin film.

Energy relaxation from an excited phenyl group chemisorbed to the surface of a crystalline thin film of α-1,3,5-trinitro-1,3,5-triazacyclohexane (α-RDX) at 298 K and 1 atm is simulated using molecular dynamics. Two schemes are used to excite the phenyl group. In the first scheme, the excitation energy is added instantaneously as kinetic energy by rescaling momenta of the 11 atoms in the phenyl group. In the second scheme, the phenyl group is equilibrated at a higher temperature in the presence of static RDX geometries representative of the 298 K thin film. An analytical model based on ballistic phonon transport that requires only the harmonic part of the total Hamiltonian and includes no adjustable parameters is shown to predict, essentially quantitatively, the short-time dynamics of the kinetic energy relaxation (∼200 fs). The dynamics of the phenyl group for times longer than about 6 ps follows exponential decay and agrees qualitatively with the dynamics described by a master equation. Long-time heat propagation within the bulk of the crystal film is consistent with the heat equation.

An operator expansion method for computing nonlinear surface waves on a ferrofluid jet

We present a new numerical method to simulate the time evolution of axisymmetric nonlinear waves on the surface of a ferrofluid jet. It is based on the reduction of this problem to a lower-dimensional computation involving surface variables alone. To do so, we describe the associated Dirichlet–Neumann operator in terms of a Taylor series expansion where each term can be efficiently computed by a pseudo-spectral scheme using the fast Fourier transform. We show detailed numerical tests on the convergence of this operator and, to illustrate the performance of our method, we simulate the long-time propagation and pairwise collisions of axisymmetric solitary waves. Both depression and elevation waves are examined by varying the magnetic field. Comparisons with weakly nonlinear predictions are also provided.

Finite-difference time dispersion transforms for wave propagation

The finite-difference (FD) wave equation is widely implemented in seismic imaging for oil exploration. But the numerical dispersion due to discretization of time and space derivatives can introduce severe numerical errors. We have investigated time dispersion, which might distort the phase and introduce severe artifacts to the data and images, especially for long-time propagation. We first studied precisely how the time dispersion was produced, and then we developed an efficient approach — the time dispersion transforms — to cope with time dispersion errors in wave propagation. The proposed forward time dispersion transform can predict or simulate the time dispersion errors, and it can be applied to remove time dispersion errors in reverse time migration. The inverse time dispersion transform can be used to eliminate time dispersion errors from synthetic data after modeling. The proposed method is general and works for low- and high-order time FD schemes. By using this method, large time steps were allowed in wave propagation; thus, we can remove the time dispersion errors at a negligible numerical cost, and the efficiency is not affected. To test the effectiveness and robustness of the proposed time dispersion transforms, we used isotropic and anisotropic examples, as well as modeling and imaging examples.

Erratum to: Shallow-water solutions for gravity currents in non-rectangular cross-area channels with stratified ambient

We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate $$x$$ . The current is of constant density, $$\rho _c$$ , and the ambient has a linear stable stratification, from $$\rho _{b}$$ at the bottom $$z=0$$ to $$\rho _{o}$$ at the top $$z=H$$ . The cross-section of the channel is given by the quite general $$-f_1(z)\le y \le f_2(z)$$ for $$0 \le z \le H$$ . We develop a one-layer shallow-water (SW) formulation, and we implement it for the solution of a gravity current of fixed volume released from a lock (we focus on a “heavy” bottom current with $$\rho _{c}\ge \rho _{b}$$ ). The dependent variables are the position of the interface, $$h(x,t)$$ , and the speed (averaged over the area of the current), $$u(x,t)$$ , where $$t$$ is the time. The non-rectangular cross-section geometry enters the formulation via $$f(h)$$ and integrals of $$f(z), z f(z)$$ and $$z^2 f(z)$$ , where $$f(z) = f_1(z) + f_2(z)$$ is the width of the channel. For a given geometry $$f(z)$$ , the free input parameters are (1) the height ratio $$H/h_0$$ of ambient to lock; and (2) the stratification parameter $$S = (\rho _{b}- \rho _{o})/(\rho _{c}- \rho _{o})$$ . In general, the SW equations of motion are a hyperbolic PDE system which we solve by a finite-difference method. However, we show that the initial motion displays a “slumping” stage with constant speed of propagation; this can be calculated exactly by the method of characteristics. An analytical solution for the long-time self-similar propagation is also presented for $$S=1$$ and the power-law $$f(z) = bz^\alpha $$ cross section profile. Solutions are presented for various stratification and typical geometries: power-law ( $$f(z) = b z^\alpha $$ , where $$b, \alpha $$ are positive constants), power-law B ( $$f(z) = b( H-z)^\alpha $$ ), trapezoidal, and circle. In general, the increase of the stratification parameter $$S$$ causes a reduction of the speed of propagation, but the details depend on the geometry of the cross section. We also show that, upon a simple transformation, the solutions for the bottom (“heavy”) current can be used for the prediction of top (“light”) currents The present solution is a significant generalization of the classical analysis of the gravity current in a stratified ambient problem (including the “intrusion” situation when $$S=1$$ ). The classical formulation for a rectangular (or laterally unbounded) channel is now just a particular case, $$f(z) = $$ const., in the wide domain of cross-sections covered by the new model. The present solution is reduced to the non-stratified (homogeneous) ambient counterpart by setting $$S=0$$ . The flow under consideration is relevant to environmental and geophysical applications like ventilation of tunnels and discharges in ducts and rivers.

Uniform stable conformal convolutional perfectly matched layer for enlarged cell technique conformal finite-difference time-domain method**Project supported by the National Natural Science Foundation of China (Grant No. 61231003).

Based on conformal construction of physical model in a three-dimensional Cartesian grid, an integral-based conformal convolutional perfectly matched layer (CPML) is given for solving the truncation problem of the open port when the enlarged cell technique conformal finite-difference time-domain (ECT-CFDTD) method is used to simulate the wave propagation inside a perfect electric conductor (PEC) waveguide. The algorithm has the same numerical stability as the ECT-CFDTD method. For the long-time propagation problems of an evanescent wave in a waveguide, several numerical simulations are performed to analyze the reflection error by sweeping the constitutive parameters of the integral-based conformal CPML. Our numerical results show that the integral-based conformal CPML can be used to efficiently truncate the open port of the waveguide.

Two-dimensional reactive scattering with transmitted quantum trajectories

A simple and easy-to-implement method is presented for the study of time-dependent reaction dynamics by propagating an ensemble of transmitted quantum trajectories. During the trajectory evolution, reflected trajectories are gradually removed and all the remaining trajectories represent the transmitted subensemble. The removal process of reflected trajectories avoids numerical instabilities arising from node formation in the reactant region, and allows stable long-time propagation of transmitted trajectories. This method is applied to a two-dimensional model chemical reaction. Excellent computational results are obtained for the time-dependent reaction probabilities evaluated by the time integration of the probability flux. © 2014 Wiley Periodicals, Inc.