Wave Transport Equations Research Articles

Traveling Wave Method

A survey of the development of the traveling wave method for one-dimensional media is presented. The main results and changes in the statement of the problem of representing solutions of linear systems of partial differential equations in terms of “traveling waves” (more precisely, in terms of a system of wave transport equations) are presented. It is shown that as the study of systems becomes more complicated, the problem of representing the solution by the traveling wave method turns out to be applicable not only for hyperbolic systems but also for systems containing (even implicitly) both parabolic and elliptic components and thereby approaches the general problem of decomposition of an arbitrary system of linear equations into a system of first-order equations with a main part of the canonical type and with a subordinate linear part.

Numerical solutions of Schrödinger wave equation and Transport equation through Sinc collocation method

Numerical solutions of Schrödinger wave equation and Transport equation through Sinc collocation method

Monitoring the sedimentary budget and dislocated boulders in western Greece – results since 2008

AbstractDislocated boulders are one sign of high‐energy wave impacts on coasts. These high‐energy impacts, caused by severe storms or tsunamis, can trigger initial cracking and transport of boulders. Monitoring of these boulders, as well as the associated coastal sites is important in distinguishing between gradual coastal processes and high‐energy events. Western Greece is a seismically active area, where tsunamis and high‐energetic storms might occur and such past events are documented by historic and geoscientific research, making it an ideal location for monitoring dislocated boulders. Since 2008, monitoring of eight different coastal sites in this region was conducted by terrestrial laser scanning and photogrammetric approaches, with low‐cost unmanned aerial vehicles. The re‐use of similar surveying points in following years, allowed highly accurate monitoring. Point clouds derived from these methods were evaluated for change detection by point cloud comparisons. The data were also used to establish accurate three‐dimensional models of dislocated boulders (n = 70). The determined boulder volumes of these accurate three‐dimensional models were incorporated in wave transport equations and wave decay curves, and compared with monitoring results. A comprehensive overview of dislocated boulders in western Greece is presented. Three‐dimensional boulder reconstruction is compared to an approach which uses a tape‐based measuring of boulder axes, with the tape‐based measurement showing a mean overestimation of mass by 32%. Accurate monitoring over time by both methods, is achieved by using fixed networks of reference points. Changes for each site over time, detected by direct point cloud comparisons, are fit to the possible inundation calculated by wave decay curves based on computed minimum wave heights for boulder transport. Both storm and tsunami waves may have initiated movement from the cliff edge and further transport is also possible. However, boulders showed no further movement from their current position in the area for the time period of this study.

Influence of atomization quality modulation on flame dynamics in a hypergolic rocket engine

For the numerical evaluation of the thermoacoustic stability of rocket engines often hybrid methods are applied, which separate the computation of wave propagation in the combustor from the analysis of the flame response to acoustic perturbations. Closure requires a thermoacoustic feedback model which provides the heat release fluctuation in the source term of the employed wave transport equations. The influence of the acoustic fluctuations in the combustion chamber on the heat release fluctuations from the modulation of the atomization of the propellants in a hypergolic upper stage rocket engine is studied. Numerical modeling of a single injector provides the time mean reacting flow field. A network of transfer functions representing all aspects relevant for the feedback model is presented. Analytical models for the injector admittances and for the atomization transfer functions are provided. The dynamics of evaporation and combustion are studied numerically and the numerical results are analyzed. An analytical approximation of the computed flame transfer function is combined with the analytical models for the injector and the atomization quality to derive the feedback model for the wave propagation code. The evaluation of this model on the basis of the Rayleigh index reveals the thermoacoustic driving potential originating from the fluctuating spray quality.

Transport equations for lower hybrid waves in a turbulent plasma

We consider the limits of validity of ray tracing and ray diffusion equations, for short wavelength waves propagating in a turbulent plasma background. We derive an improved transport equation for the electric field autocorrelation function, where first order diffraction effects associated with these waves are included. We apply this description to the case of lower hybrid (LH) waves propagating in non-stationary plasma where density perturbations can occur due to drift wave turbulence, as well as magnetic field perturbations due to MHD turbulence. This is relevant to the problem of LH current drive.

Non-local susceptibility of the wire medium in the spatial domain considering material boundaries

We show that the non-local susceptibility for a non-translationally invariant homogenized wire medium is, modulo a constant, given by a simple Green function related to the material geometry. We also show that two previous methods for solving wave interaction problems for bounded wire media (wave expansion method and transport equation) are equivalent to each other, and to a third method involving particle reflection at the boundary. We discuss the importance of the dead layer or virtual interface, and find it to be analogous to the excitonic semiconductor case. Several examples are provided to clarify the material.

TERRESTRIAL LASER SCANNING FOR COASTAL GEOMORPHOLOGIC RESEARCH IN WESTERN GREECE

Abstract. We used terrestrial laser scanning (TLS) for (i) accurate volume estimations of dislocated boulders moved by high-energy impacts and for (ii) monitoring of annual coastal changes. In this contribution, we present three selected sites in Western Greece that were surveyed during a time span of four years (2008-2011). The Riegl LMS-Z420i laser scanner was used in combination with a precise DGPS system (Topcon HiPer Pro). Each scan position and a further target were recorded for georeferencing and merging of the point clouds. For the annual detection of changes, reference points for the base station of the DGPS system were marked. Our studies show that TLS is capable to accurately estimate volumes of boulders, which were dislocated and deposited inland from the littoral zone. The mass of each boulder was calculated from this 3D-reconstructed volume and according density data. The masses turned out to be considerably smaller than common estimated masses based on tape-measurements and according density approximations. The accurate mass data was incorporated into wave transport equations, which estimate wave velocities of high-energy impacts. As expected, these show smaller wave velocities, due to the incorporated smaller mass. Furthermore, TLS is capable to monitor annual changes on coastal areas. The changes are detected by comparing high resolution digital elevation models from every year. On a beach site, larger areas of sea-weed and sandy sediments are eroded. In contrast, bigger gravel with 30-50 cm diameter was accumulated. At the other area with bigger boulders and a different coastal configuration only slightly differences were detectable. In low-lying coastal areas and along recent beaches, post-processing of point clouds turned out to be more difficult, due to noise effects by water and shadowing effects. However, our studies show that the application of TLS in different littoral settings is an appropriate and promising tool. The combination of both instruments worked well and the annual positioning procedure with own survey point is precose for this purpose.

Simulation and prediction of phenolic compounds fate in Songhua River, China

The demand for numerical models to simulate and predict the transport and fate of organic pollutants in the environment is a reflection of certain restrictions of experimental research. Various phenolic pollutants have been detected at different concentration levels in the Songhua River, China. A multimedia fate model is coupled with the kinematic wave transport equations to accurately and simply characterize dynamic water flow, velocity and depth. The resulting model is applied to describe the temporal and spatial behavior of eight phenol compounds in the Songhua River. The predictive power of this model is evaluated by statistical tests using field observations during different hydrological seasons of the year 2007. The goodness-of-fit of model prediction for phenolic contaminants is, in general, agreeable. For one focal compound, the pathways and mass fluxes from the river system to the surrounding environment are also evaluated. Uncertainty analysis is performed by Monte Carlo stimulation to judge the influence of variability of input parameters on modeled results. The results indicate that the model has the ability to provide decision makers with valuable reference data for consideration of water pollution controls. Model simulation indicates that phenolic pollution in the river during the low-flow period of 2009 is remarkably reduced when compared to the same period of 2007.

Large boulders along the south-eastern Ionian coast of Sicily: Storm or tsunami deposits?

This study investigates the size, shape, position, pre-transport setting and long-axis orientation of 175 boulders found along the south-eastern Ionian coast of Sicily. These megaclasts were deposited by the sea or moved from the same rocky coast to the shore platform and distributed within 60 m of the shoreline. Most boulders are rectangular to ellipsoidal, with sharp, broken edges. They are calcarenite fragments up to 27 t in mass. Some of the boulders (up to 32 m from shore) were observed to have moved after strong winter storms occurring during December 2008–January 2009, whereas the positions of many boulders were unchanged. We use wave transport equations jointly with statistical analysis of boulders in order to determine the extreme events – geological or meteorological – responsible for these singular accumulations. Using the significant wave height and period of maximum observed storms and historical tsunamis (1693 and 1908 events) along the Ionian coast of Sicily, we have estimated the approximate transport distance of boulders by these waves at the coast. Results show that the largest storm waves were probably responsible for the current distribution of most boulders. However, since the size and horizontal displacement distance of some boulders scattered randomly along the shoreline at distance > 40 m cannot be explained by storm wave action, we suggest that they are likely deposited by tsunamis. Radiocarbon dating performed on three of these anomalous boulders, at a distance > 40 m from the shoreline, gave an age suggesting that two of them were likely deposited by the 1693 or 1908 tsunamis, whereas the third was emplaced by a tsunami that occurred after 650–930 A.D. We show that both wave processes (storm and tsunami) can deposit boulders. We find that the significant difference between storm and tsunami waves is not the breaking wave height but the wave period that influences attenuation, and then the flooding distance. Hence, if the boulder-transport limit of the largest storm waves in the past at the study area is estimated, then it is possible to recognize that the boulders deposited far beyond this limit, because of their size and shape, were not transported by storm waves, but could have been emplaced by waves with longer period, such as the tsunamis.

AN EFFICIENT APPROXIMATION OF THE CORONAL HEATING RATE FOR USE IN GLOBAL SUN-HELIOSPHERE SIMULATIONS

The origins of the hot solar corona and the supersonically expanding solar wind are still the subject of debate. A key obstacle in the way of producing realistic simulations of the Sun-heliosphere system is the lack of a physically motivated way of specifying the coronal heating rate. Recent one-dimensional models have been found to reproduce many observed features of the solar wind by assuming the energy comes from Alfven waves that are partially reflected, then dissipated by magnetohydrodynamic turbulence. However, the nonlocal physics of wave reflection has made it difficult to apply these processes to more sophisticated (three-dimensional) models. This paper presents a set of robust approximations to the solutions of the linear Alfven wave reflection equations. A key ingredient to the turbulent heating rate is the ratio of inward to outward wave power, and the approximations developed here allow this to be written explicitly in terms of local plasma properties at any given location. The coronal heating also depends on the frequency spectrum of Alfven waves in the open-field corona, which has not yet been measured directly. A model-based assumption is used here for the spectrum, but the results of future measurements can be incorporated easily. The resulting expression for the coronal heating rate is self-contained, computationally efficient, and applicable directly to global models of the corona and heliosphere. This paper tests and validates the approximations by comparing the results to exact solutions of the wave transport equations in several cases relevant to the fast and slow solar wind.

TRANSPORT EQUATION FOR MHD TURBULENCE: APPLICATION TO PARTICLE ACCELERATION AT INTERPLANETARY SHOCKS

The aim of the present paper is to unify the various transport equations for turbulent waves that are used in different areas of space physics. Here, we mostly focus on the magnetohydrodynamic turbulence, in particular the Alfvturbulence. The applied methods, however, are general and can be extended to other forms of turbulence, for example the acoustic turbulence, or Langmuir plasma waves. With minor modifications, the derivations followed here can be extended for relativistic motions, thus making it possible to apply them to the wave transport in astrophysical objects with high plasma speeds (radiojets), or strong gravity (black hole surroundings).

Noninterferometric characterization of partially coherent scalar wave fields and application to scattered light

We report on the application of a simple propagation-based phase-space tomographic technique to the determination of characteristic projections through the mutual optical intensity and the generalized radiance of a scalar, quasi-monochromatic partially coherent wave field. This method is applied to the reconstruction of the coherence functions of an initially spatially coherent optical wave field that has propagated through a suspension of polystyrene microspheres. As anticipated, we see that the field separates into a ballistic, or unscattered, component and a scattered component with a much shorter coherence length. Good agreement is obtained between experimental results and the results of a model based on a wave-transport equation.

Transport equations for waves in randomly perturbed Hamiltonian systems, with application to Rossby waves

Transport equations for linear waves in randomly perturbed media are derived for a general class of Hamiltonian systems. These equations govern the evolution in position–wavevector space of the energy density of the waves; they describe the modulations caused by large-scale inhomogeneities of the medium, and the scattering due to weak random perturbations with spatial scales comparable to the wavelengths. Particular attention is paid to the conservative properties of the transport equations inherited from the Hamiltonian structure of the systems considered. The general results are used to derive transport equations for Rossby waves propagating in a two-layer model with random topography.

On the Generation, Propagation, and Reflection of Alfven Waves from the Solar Photosphere to the Distant Heliosphere

We present a comprehensive model of the global properties of Alfven waves in the solar atmosphere and fast solar wind. Linear non-WKB wave transport equations are solved from the photosphere to 4 AU, and for wave periods ranging from 3 seconds to 3 days. We derive a radially varying power spectrum of kinetic and magnetic energy fluctuations for waves propagating in both directions along a superradially expanding magnetic flux tube. This work differs from previous models in 3 major ways. (1) In the chromosphere and low corona, the successive merging of flux tubes on granular and supergranular scales is described using a 2D magnetostatic model of a network element. Below a critical merging height the waves are modeled as thin-tube kink modes, and we assume that all of the kink-mode wave energy is transformed into volume-filling Alfven waves above the merging height. (2) The frequency spectrum of horizontal motions is specified only at the photosphere based on prior analyses of G-band bright point kinematics. Everywhere else the amplitudes of outward and inward propagating waves are computed with no free parameters. We find that the wave amplitudes in the corona agree well with off-limb nonthermal line widths. (3) Nonlinear turbulent damping is applied to the results of the linear model using a phenomenological loss term. A single choice for the normalization of the turbulent outer-scale length produces both the right amount of damping at large distances (to agree with in situ measurements) and the right amount of heating in the extended corona (to agree with empirical wind acceleration models). In the corona, the modeled heating rate differs by more than an order of magnitude from a rate based on isotropic Kolmogorov turbulence.

Electron beam acceleration and potential formation induced by the Compton scattering of extraordinary waves

High-energy or relativistic electron beam acceleration along and across a magnetic field, and the generation of an electric field transverse to the magnetic field, both induced by the Compton scattering of almost perpendicularly propagating extraordinary waves, are investigated theoretically based on kinetic wave equations and transport equations. Compton scattering occurs via nonlinear Landau damping of two extraordinary waves interacting nonlinearly with the electron beam, satisfying the resonance condition of ω k -ω k '-(κ⊥-κ'⊥)v d -(k∥-k'∥)v b = mω ce (m = 0,±1), where v b and v d are the parallel and perpendicular velocities of the electron beam, respectively. The transport equations can be derived from the single-particle theory and also from Vlasov-Maxwell equations. The transport equations show that two extraordinary waves accelerate the electron beam in the k direction (k=k-k'). Simultaneously, an intense cross-field electric field E 0 =B 0 × v d /c is generated via the dynamo effect owing to the perpendicular drift of the electron beam to satisfy the generalized Ohm's law, which means that this cross-field electron drift is identical to the E x B drift. The single-particle theory is very useful for an easy and straightforward understanding of the physical mechanism of the electron beam acceleration and the generation of cross-field electric field, although the rigorously exact transport equations are derived from Vlasov-Maxwell equations.

A simple analytical expression for the power spectrum of cascading Alfvén waves in the solar wind

Alfven wave transport in the solar wind, including non-linear spectral energy transfer, is studied. We present numeri- cal solutions of wave transport using a diusive flux function previously introduced for spectral energy transfer, and compare it with the analytical solution obtained for a convective flux function. The two models of cascading produce very similar behavior of a power spectrum initially of 1= f -form at the solar surface, provided that the cascading constants are tuned to produce the same spectral flux in the inertial range. We present an analytical expression for the power spectrum of the diusively-cascading Alfven waves in the solar wind derived from a solution of the wave transport equation and show that it compares well with the exact solutions. Our expression enables (semi) analytical evaluation of the cyclotron heating rate, the wave pressure gradient, and the energetic-particle mean free path related to the Alfven waves in the corona and solar wind.

The Effect of the Wave Propagation Scheme on Nearshore Wave Predictions

WAM (acronym for WAve Model) is a third-generation wave model used to compute the spectra of random short-crested wind-generated waves on Eulerian grids and is one of the most extensively tested and widely used wave models in the world. Waves can be propagated on a spherical or a Cartesian grid. For large ocean regions, waves must be propagated on a sphere to account for the curvature of the Earth. For coastal and nearshore regions, smaller nests with higher mesh resolution are used. As the size of the nests becomes smaller, the differences between a spherical grid and a Cartesian grid for the same nest become smaller and one would assume that the differences in the computed results would also be small. However, one should not assume that this behavior is automatically recovered in numerical wave models owing to the different nature of spherical and Cartesian grids — the wave transport equation when written in spherical coordinates is not divergence-free whereas it is divergence-free when written in Cartesian coordinates! Therefore, the transition of spherical methods to divergence-free needs to be numerically verified. The purpose of this numerical study was to examine results obtained with the spherical and Cartesian wave propagation options in the WAM code for a nearshore zone where depth effects are important. This study found that Cartesian wave propagation in the WAM code produced more accurate nearshore wave predictions than spherical wave propagation!

Radiative transport for elastic waves

Radiative transport equations for elastic waves in random media can be derived from first principles in a certain regime of physical parameters. They account for both longitudinal and transverse waves, as well as polarization. Transport boundary conditions can also be derived from first principles. An important application of transport theory is to the analysis of the deep coda of elastic waves, where some rather interesting phenomena of energy distribution emerge. A reference to work in this area is by Ryzhik et al. [Wave Motion 24, 327–370 (1996)]. Another reference, more directly related to seismics, is Ryzhik et al. [Bull. Seismol. Soc. Am. 86, 1107–1115 (1996)].

An acoustic wave equation for anisotropic media

A wave equation, derived using the acoustic medium assumption for P-waves in transversely isotropic (TI) media with a vertical symmetry axis (VTI media), yields a good kinematic approximation to the familiar elastic wave equation for VTI media. The wavefield solutions obtained using this VTI acoustic wave equation are free of shear waves, which significantly reduces the computation time compared to the elastic wavefield solutions for exploding‐reflector type applications. From this VTI acoustic wave equation, the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a TI medium are derived. These equations, based on the acoustic assumption (shear wave velocity = 0), are much simpler than their elastic counterparts, yet they yield an accurate description of traveltimes and geometrical amplitudes. Numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex TI models.

Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations

Many transport equations, such as the neutron transport, radiative transfer, and transport equations for waves in random media, have a diffusive scaling that leads to the diffusion equations. In many physical applications, the scaling parameter (mean free path) may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes within one problem, and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter. In an earlier work [Jin, Pareschi, and Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439] we handled this numerical problem for discrete-velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws [Jin and Xin, Comm. Pure Appl. Math., 48 (1995), pp. 235--277]. Such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit, yet explicitly implementable, schemes that work with high resolution uniformly with respect to the relaxation parameter. In this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities, when one uses the even and odd parities of the transport equation.