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At least 415 records · Page 23
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

RELAP-7 Application and Enhancement for FLEX Strategies and ATF Behavior under Extended Loss of AC Power Conditions

This report summarizes the results of a three-year research project sponsored by the U.S. Department of Energy (DOE) Nuclear Energy University Program (NEUP) to enhance and apply the RELAP-7 code by adding and improving several important components (e.g., a mechanistic Reactor Core Isolation Cooling (RCIC) system model) for thermal hydraulic studies of LWRs under ELAP conditions and evaluating the time available for transition to portable FLEX equipment. The project team included University of Massachusetts–Lowell, The Ohio State University, Texas A&M University, Idaho National Laboratory and Oak Ridge National Laboratory. In the Fukushima accident, it was found that the RCIC system played a crucial role in delaying core meltdown by almost three days in Fukushima Daiichi Unit 2, because of self-regulated operation of the steam driven RCIC turbine-pump injection system. Steam flow in the convergent-divergent nozzles of the RCIC Terry turbine is two-phase non-equilibrium transonic flow with homogenous nucleation condensation. To more accurately predict the dynamic process and behavior of the transonic compressible steam flow, a one-dimensional transient two-phase analytical model is presented. A simplified four-fluid model was employed in the present work with the consideration of four separate fluid fields: vapor, liquid film, entrained droplets and condensed droplets. The mass, momentum and energy interactions between the fluids were considered and modeled. An extended seven-equation non-equilibrium critical flow model was developed to obtain the critical pressure and velocities of each phase at the nozzle throat. To predict the wetness in the divergent section, a mechanistic nucleation condensation model was integrated in the nozzle analysis model, considering the generation and consequent growth of droplets. The governing differential equations on a staggered grid were discretized using the second-order Lax-Wendroff scheme with a flux limiter, and the Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm was employed to solve the discrete linear system. To demonstrate the predictability and reliability of the physical models and the numerical method proposed in the present work, three representative nozzles were modeled and simulated. The results show good agreement with the available experimental data, even for condensation shock. Then, the 1D nozzle model was employed to obtain nozzle flow tables of the Terry turbine nozzle for different working pressures which can cover the operation pressure range of the RCIC system. A mechanistic RCIC turbine-pump system model was developed and implemented in the system code TRACE to simulate dynamic responses of the RCIC system under Beyond Design Basis Accident (BDBA) conditions. The turbine-pump governing equations are based on the control volume approach of the angular momentum balance. The physics based mechanistic RCIC model was developed using the TRACE control system components (i.e., signal variables, control blocks, and tables), and incorporated into a TRACE boiling water reactor (BWR) model. The TRACE model in this report has a detailed nodalization of the reactor pressure vessel (RPV), and all of the major flow paths and system components, including the safety relief valves (SRVs) and the containment suppression pool and drywell. Based on the nozzle flow tables generated from the 1D nozzle model developed, the turbine drive torque can be calculated from table lookup. Since the detailed specifications of the RCIC pump are unavailable, the homologous curves for a Bingham pump were used in the current pump component. A station black-out (SBO) accident test problem was selected to demonstrate the TRACE RCIC model. The short-term SBO simulations were performed for two cladding materials: Zircaloy and FeCrAl, to demonstrate the effect of the accident tolerant fuel cladding on fuel heat-up under BDBA accident conditions. The wetwell plays a vital safety role in SBO and other BWR accident scenarios in that it can reduce containment pressure and supply additional core make-up water. The suppression pool temperature distribution has a very large impact on both RPV and containment pressure. Thus, another novel contribution of the project comes mainly from an improved, systems-level wetwell model which can capture buoyancy-induced thermal stratification effects due to steam injection and condensation. A two-zone stratified wetwell model has been implemented in RELAP-7 and some results from that model are presented. This wetwell model is capable of simulating thermal stratification due to a low steam mass injection rate. With a low mass flow rate, the model assumes that all the steam condenses within the pipe and the resulting plume can be approximated with a purely buoyant, heat-source driven model. The wetwell model developed with these assumptions is adequate to simulate slow transients such as extended SBO transients.

42 ENGINEERING↗
DOE OSTI2015

How important are electron beams in driving chromospheric evaporation in the 2014 March 29 flare?

We present high spatial resolution observations of chromospheric evaporation in the flare SOL2014-03-29T17:48. Interface Region Imaging Spectrograph observations of the Fe xxi λ1354.1 line indicate evaporating plasma at a temperature of 10 MK along the flare ribbon during the flare peak and several minutes into the decay phase with upflow velocities between 30 and 200 km s{sup −1}. Hard X-ray (HXR) footpoints were observed by the Ramaty High Energy Solar Spectroscopic Imager for two minutes during the peak of the flare. Their locations coincided with the locations of the upflows in parts of the southern flare ribbon but the HXR footpoint source preceded the observation of upflows in Fe xxi by 30–75 s. However, in other parts of the southern ribbon and in the northern ribbon, the observed upflows were not coincident with an HXR source in time or space, most prominently during the decay phase. In this case evaporation is likely caused by energy input via a conductive flux that is established between the hot (25 MK) coronal source, which is present during the whole observed time-interval, and the chromosphere. The presented observations suggest that conduction may drive evaporation not only during the decay phase but also during the flare peak. Electron beam heating may only play a role in driving evaporation during the initial phases of the flare.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

The physics of spontaneous parity-time symmetry breaking in the Kelvin–Helmholtz instability

Abstract We show that the dynamical system of an inviscid fluid with velocity shear admits parity-time (PT) symmetry, which provides a physical explanation of the well-known observation that the spectrum of the perturbation eigenmodes of the system is symmetric with respect to the real axis. It is found that the Kelvin–Helmholtz instability is triggered when and only when the PT symmetry is spontaneously broken. The analysis of PT symmetry also reveals that the relative phase between parallel velocity and pressure perturbations needs to be locked at π /2 when the instability is suppressed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Elucidating the Microscale Behavior and Phase Separation Kinetics of Thermally Responsive Ionic Liquid–Water Mixtures

Thermally responsive ionic liquids (ILs) exhibit liquid-liquid phase separation into a water-rich (WR) and ionic-liquid-rich (ILR) phase when heated above a lower critical solution temperature (LCST). This phase behavior has been leveraged for applications ranging from forward osmosis (FO) desalination, where the IL acts as a draw solute, to refrigeration and dehumidification cycles, where the IL acts as a liquid desiccant. While significant effort has been devoted to characterizing the thermodynamic and thermophysical properties of LCST ILs, their phase separation kinetics have not been investigated. In this work, we describe the macroscale phase separation kinetics (phase separation time) by gleaning insight into the microscale colloidal behavior of aqueous mixtures of four different materials, P 4444 TFA (tetrabutylphosphonium-2,4-trifluoroacetate), P 4444 DMBS (tetrabutylphosphonium-2,4-dimethyl-benzenesulfonate), N 4444 Sal (tetrabutylammonium salicylate), and P 4444 Sal (tetrabutylphosphonium salicylate) as a function of IL concentration at a separation temperature of 70 °C. We report the discontinuous microscale size distributions for each material and correlate their theoretical settling velocities to experimental phase separation times. The results indicate that a simple Stokes' law model can predict the phase separation time within reasonable accuracy. Overall, this work lays the foundation for understanding the micro- to macroscale phase separation behavior and kinetics of LCST ILs for various water-energy applications.

LCST↗
DOE OSTI2021

Sequentially prepared Mo-V-Based SCR catalyst for simultaneous Hg 0 oxidation and NO reduction

Molybdenum (Mo)-vanadium (V)-based selective catalytic reduction (SCR) catalyst synthesized by the sequential impregnation of Mo and W followed by V was investigated for simultaneous elemental mercury (Hg 0 ) oxidation and nitrogen oxide (NO) reduction in an existing SCR unit with respect to different TiO 2 phases, calcination temperatures, flue gas constituents, gas velocities, and reaction temperatures. Anatase phase TiO 2 and the calcination temperatures of 400 and 500 degrees C resulted in similar to 69% Hg 0 oxidation at 10 ppmv HCl and 350 degrees C. The high calcination temperature of 700 degrees C resulted in TiO 2 phase transformation from anatase to rutile and agglomeration. The modified SCR catalyst prepared with the impregnation sequence of Mo and W followed by V using anatase TiO 2 and calcination temperature 500 degrees C showed similar to 99% Hg 0 oxidation and 87% NO reduction conversions at an NH 3 /NO molar ratio of 0.9 under 350 degrees C and 5,000 hr -1 space velocity in typical sub-bituminous and lignite coal simulated flue gases. Finally, the effects of these parameters and conditions were further investigated using various characterization techniques including BET, TEM, XRD, NH 3 TPD and XAFS.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Quantifying hydrate solidification front advancing using method of characteristics

Here, we develop a one-dimensional analytical solution based on the method of characteristics to explore hydrate formation from gas injection into brine-saturated sediments within the hydrate stability zone. Our solution includes fully coupled multiphase and multicomponent flow and the associated advective transport in a homogeneous system. Our solution shows that hydrate saturation is controlled by the initial thermodynamic state of the system and changed by the gas fractional flow. Hydrate saturation in gas-rich systems can be estimated by 1 – cl 0 /cl e when Darcy flow dominates, where cl 0 is the initial mass fraction of salt in brine, and cl e is the mass fraction of salt in brine at three-phase (gas, liquid, and hydrate) equilibrium. Hydrate saturation is constant, gas saturation and gas flux decrease, and liquid saturation and liquid flux increase with the distance from the gas inlet to the hydrate solidification front. The total gas and liquid flux is constant from the gas inlet to the hydrate solidification front and decreases abruptly at the hydrate solidification front due to gas inclusion into the hydrate phase. The advancing velocity of the hydrate solidification front decreases with hydrate saturation at a fixed gas inflow rate. This analytical solution illuminates how hydrate is formed by gas injection (methane, CO 2 , ethane, propane) at both the laboratory and field scales.

58 GEOSCIENCES↗
DOE OSTI2017

Failure of Taylor's hypothesis in the atmospheric surface layer and its correction for eddy-covariance measurements

Taylors’ frozen turbulence hypothesis suggests that all turbulent eddies are advected by the mean streamwise velocity, without changes in their properties. This hypothesis has been widely invoked to compute Reynolds’ averaging using temporal turbulence data measured at a single point in space. However, in the atmospheric surface layer, the exact relationship between convection velocity and wavenumber k has not been fully revealed since previous observations were limited by either their spatial resolution or by the sampling length. Using Distributed Temperature Sensing (DTS), acquiring turbulent temperature fluctuations at high temporal and spatial frequencies, we computed convection velocities across wavenumbers using a phase spectrum method. We found that convection velocity decreases as k –1/3 at the higher wavenumbers of the inertial subrange instead of being independent of wavenumber as suggested by Taylor's hypothesis. We further corroborated this result using large eddy simulations. Applying Taylor's hypothesis thus systematically underestimates turbulent spectrum in the inertial subrange. As a result, a correction is proposed for point-based eddy-covariance measurements, which can improve surface energy budget closure and estimates of CO 2 fluxes.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Resolving Anodic Current and Temperature Distributions in a Polymer Electrolyte Membrane Water Electrolysis Cell Using a Pseudo-Two-Phase Computational Fluid Dynamics Model

Expanding upon our prior experimental work, we constructed a three-dimensional model of a polymer electrolyte membrane water electrolyzer using computational fluid dynamics. We applied the assumption of pseudo-two-phase flow, the flow of two phases with equal velocity. Experimental data were used to obtain parameters and to determine the conditions under which this model was valid. Anodic distributions of current density, temperature, liquid saturation, and relative humidity were obtained at various flow rates. The overall current density and temperature difference from inlet to outlet at the anode agreed strongly with experimental measurements under most circumstances. This verification allowed us to further examine the apparent gas coverage calculated from experimental and model temperature data. Results suggested a low liquid saturation and low relative humidity at the anode due to the consumption of liquid water and water vapor. However, we questioned the accuracy of the pseudo-two-phase assumption at low water feed rates. We concluded that the model was applicable to systems with liquid water feed rates greater than 0.6 ml min -1 cm -2 . Therefore, it is a fair screening method that can advise which operating conditions lead to excessive temperatures or drying at the anode, thereby promoting the longevity of the membrane and catalyst.

25 ENERGY STORAGE↗
DOE OSTI2012

Coupling between interface and velocity perturbations in the weakly nonlinear Rayleigh-Taylor instability

Weakly nonlinear (WN) Rayleigh-Taylor instability (RTI) initiated by single-mode cosinusoidal interface and velocity perturbations is investigated analytically up to the third order. Expressions of the temporal evolutions of the amplitudes of the first three harmonics are derived. It is shown that there are coupling between interface and velocity perturbations, which plays a prominent role in the WN growth. When the 'equivalent amplitude' of the initial velocity perturbation, which is normalized by its linear growth rate, is compared to the amplitude of the initial interface perturbation, the coupling between them dominates the WN growth of the RTI. Furthermore, the RTI would be mitigated by initiating a velocity perturbation with a relative phase shift against the interface perturbation. More specifically, when the phase shift between the interface perturbation and the velocity perturbation is {pi} and their equivalent amplitudes are equal, the RTI could be completely quenched. If the equivalent amplitude of the initial velocity perturbation is equal to the initial interface perturbation, the difference between the WN growth of the RTI initiated by only an interface perturbation and by only a velocity perturbation is found to be asymptotically negligible. The dependence of the WN growth on the Atwood numbers and the initial perturbation amplitudes is discussed. In particular, we investigate the dependence of the saturation amplitude (time) of the fundamental mode on the Atwood numbers and the initial perturbation amplitudes. It is found that the Atwood numbers and the initial perturbation amplitudes play a crucial role in the WN growth of the RTI. Thus, it should be included in applications where the seeds of the RTI have velocity perturbations, such as inertial confinement fusion implosions and supernova explosions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2013

Crystal Growth Simulations To Establish Physically Relevant Kinetic Parameters from the Empirical Kolmogorov-Johnson-Mehl-Avrami Model

A series of simulations was performed to enable interpretation of the material and physical significance of the parameters defined in the Kolmogorov, Johnson and Mehl, and Avrami (KJMA) rate expression commonly used to describe phase boundary controlled reactions of condensed matter. The parameters k, n, and t 0 are shown to be highly correlated, which if unaccounted for seriously challenge mechanistic interpretation. It is demonstrated that rate measurements exhibit an intrinsic uncertainty without precise knowledge of the location and orientation of nucleation with respect to the free volume into which it grows. More significantly, it is demonstrated that the KJMA rate constant k is highly dependent on sample size. However, under the simulated conditions of slow nucleation relative to crystal growth, sample volume and sample anisotropy correction affords a means to eliminate the experimental condition dependence of the KJMA rate constant, k, producing the material-specific parameter, the velocity of the phase boundary, v pb .

Dill, Eric D.↗
DOE OSTI2013

The Hamiltonian structure and Euler-Poincare formulation of the Vlasov-Maxwell and gyrokinetic systems

We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in H. Cendra et al., [J. Math. Phys. 39, 3138 (1998)]. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models, and Casimir type stability methods.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2014

1FGL J0523.5–2529: A NEW PROBABLE GAMMA-RAY PULSAR BINARY

We report optical photometric and Southern Astrophysical Research spectroscopic observations of an X-ray source found within the localization error of the Fermi Large Area Telescope unidentified γ-ray source 1FGL J0523.5–2529. The optical data show periodic flux modulation and radial velocity variations indicative of a binary with a 16.5 hr period. The data suggest a massive non-degenerate secondary (≳ 0.8 M {sub ☉}), and we argue the source is likely a pulsar binary. The radial velocities have good phase coverage and show evidence for a measurable eccentricity (e = 0.04). There is no clear sign of irradiation of the secondary in either photometry or spectroscopy. The spatial location out of the Galactic plane and γ-ray luminosity of the source are more consistent with classification as a recycled millisecond pulsar than as a young pulsar. Future radio timing observations can confirm the identity of the primary and further characterize this interesting system.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Effect of the volume fraction gradient on the phase interaction force model for disperse two-phase flows

In this work, the effects of the particle volume fraction gradient on fluid-particle interactions are studied. The phase interaction force is decomposed into three terms. For the first term, namely the symmetrized force density, we present theoretical reasoning and numerical evidence to assume that it is independent of the particle volume fraction gradient. The second term is the particle volume fraction gradient times a newly introduced diffusion stress. The third term is the divergence of the particle-fluid-particle (PFP) stress. If this assumption of independence of the particle volume fraction gradient for the first term can be verified, to the first order of the ratio of the mean distance between particles to the macroscopic lengthscale, all three terms can be studied and modeled in flows with uniform particle distributions. Models thus obtained are applicable to statistically inhomogeneous flows, with the second and third terms accounting for statistical inhomogeneity. To verify this assumption, numerical simulations of flows passing fixed arrays of particles are performed. Both uniform and nonuniform particle volume fractions are studied and compared for disperse multiphase flows with the particle Reynolds numbers ranging from 1 to 100, and particle volume fraction ranging from 1% to 26% in statistically steady states. It is found that the symmetrized force (first) term can be well approximated by the drag force obtained from studies of uniform flows. The diffusion stress is positive along the flow direction and negative in the directions perpendicular to the flow. In the case of moving particles, this stress could potentially cause particle aggregation in the flow direction and dispersion in the directions perpendicular to the flow. Finally, the diffusion stress is only important when there is a volume fraction gradient, while the PFP stress can be important in inhomogeneous flows with either nonuniform particle concentrations or nonuniform average relative velocities between the phases.

42 ENGINEERING↗
DOE OSTI2014

Structure analysis of turbulent liquid phase by POD and LSE techniques

In this paper, vortical structures and turbulence characteristics of liquid phase in both single liquid phase and two-phase slug flow in pipes were studied. Two dimensional velocity vector fields of liquid phase were obtained by Particle image velocimetry (PIV). Two cases were considered one single phase liquid flow at 80 l/m and second slug flow by introducing gas at 60 l/m while keeping liquid flow rate same. Proper orthogonal decomposition (POD) and Linear stochastic estimation techniques were used for the extraction of coherent structures and analysis of turbulence in liquid phase for both cases. POD has successfully revealed large energy containing structures. The time dependent POD spatial mode coefficients oscillate with high frequency for high mode numbers. The energy distribution of spatial modes was also achieved. LSE has pointed out the coherent structured for both cases and the reconstructed velocity fields are in well agreement with the instantaneous velocity fields.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

CFD-DEM study of bubble properties in a cylindrical fluidized bed of Geldart Group D particles and comparison with prior MRI data

Numerical simulations of a freely bubbling cylindrical fluidized bed are carried out using a coupled computational fluid dynamics and discrete element method (CFD-DEM) model and compared to recent experimental data. The experiments were conducted using high-resolution and high-frequency magnetic resonance imaging providing high-fidelity data of the bubbling within a central 10 mm slice of the bed. Qualitatively, we nd more regular (i.e., less chaotic) structures in the simulated beds than observed experimentally. Quantitatively, however, the bubble diameter and number of bubbles as a function of height within the bed is predicted well by the base model. Unfortunately, the regularity in the simulations manifests as a considerable discrepancy in the speed of the (dense) emulsion phase. The simulated velocity probability distribution functions show an accumulation of low-speed regions and deficiency of high-speed regions. A simple parametric study of the base model is also carried out considering many of the most common CFD-DEM modeling parameters. It is found that the fluid grid size, geometry resolution, transfer kernel and drag law did not have a significant effect on bubble or particle dynamics.

42 ENGINEERING↗
DOE OSTI2022

Crust and Upper Mantle Structure Beneath the Eastern United States

Abstract The Eastern United States (EUS) has a complex geological history and hosts several seismic active regions. We investigate the subsurface structure beneath the broader EUS. To produce reliable images of the subsurface, we simultaneously invert smoothed P‐wave receiver functions, Rayleigh‐wave phase and group velocity measurements, and Bouguer gravity observations for the 3D shear‐wave speed. Using surface‐wave observations (3–250 s) and spatially smoothed receiver functions, our velocity models are robust, reliable, and rich in detail. The shear‐wave velocity models fit all three types of observations well. The resulting velocity model for the eastern U.S. shows thinner crust beneath New England, the east coast, and the Mississippi Embayment (ME). A relatively thicker crust was found beneath the stable North America craton. A relatively slower upper mantle was imaged beneath New England, the east coast, and western ME. A comparison of crust thickness derived from our model against four recent published models shows first‐order consistency. A relatively small upper mantle low‐speed region correlates with a published P‐wave analysis that has associated the anomaly with a 75 Ma kimberlite volcanic site in Kentucky. We also explored the relationship between the subsurface structure and seismicity in the eastern U.S. We found that earthquakes often locate near regions with seismic velocity variations, but not universally. Not all regions of significant subsurface wave speed changes are loci of seismicity. A weak correlation between upper mantle shear velocity and earthquake focal mechanism has been observed.

58 GEOSCIENCES↗
DOE OSTI2014

Study of plasma start-up initiated by second harmonic electron cyclotron resonance heating on WEGA experiment

Although both 1st harmonic ordinary mode (O1) and 2nd harmonic extra-ordinary mode (X2) have been successfully used to initiate pre-ionization and breakdown in many devices, a complete theoretical model is still missing to explain the success of this method. Moreover, some experimental observations are not completely understood, such as what occurs during the delay time between the turn-on of ECRH power and first signals of density or light measurements. Since during this free period the ECRH power has to be absorbed by in-vessel components, it is of prime importance to know what governs this delay time. Recently, dedicated start-up experiments have been performed on WEGA, using a 28 GHz ECRH system in X2-mode. This machine has the interesting capability to be run also as a tokamak allowing comparative experiments between stellarator (ι/2π > 0) and tokamak (ι/2π = 0) configurations. Different scans in heating power, neutral gas pressure, and rotational transform (ι) show clearly that the start-up is a two step process. A first step following the turn-on of the ECRH power during which no measurable electron density (or just above the noise level in some cases), ECE and radiated power is detected. Its duration depends strongly on the level of injected power. The second step corresponds to the gas ionization and plasma expansion phase, with a velocity of density build-up and filling-up of the vessel volume depending mainly on pressure, gas and rotational transform. Moreover, an interesting scenario of ECRH pre-ionization without loop voltage in tokamak configuration by applying a small optimal vertical field is relevant for start-up assistance on future experiments like ITER. The results from this experimental parametric study are useful for the modeling of the start-up assisted by the second harmonic electron cyclotron resonance heating. The aim of this work is to establish predictive scenarios for both ITER and W7-X operation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Rotation-induced nonlinear wavepackets in internal waves

The long time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual formation of a localised wavepacket. Here this initial value problem is considered within the context of the Ostrovsky, or the rotation-modified Korteweg-de Vries (KdV), equation and a numerical method for obtaining accurate wavepacket solutions is presented. The flow evolutions are described in the regimes of relatively-strong and relatively-weak rotational effects. When rotational effects are relatively strong a second-order soliton solution of the nonlinear Schrödinger equation accurately predicts the shape, and phase and group velocities of the numerically determined wavepackets. It is suggested that these solitons may form from a local Benjamin-Feir instability in the inertia-gravity wave-train radiated when a KdV solitary wave rapidly adjusts to the presence of strong rotation. When rotational effects are relatively weak the initial KdV solitary wave remains coherent longer, decaying only slowly due to weak radiation and modulational instability is no longer relevant. Wavepacket solutions in this regime appear to consist of a modulated KdV soliton wavetrain propagating on a slowly varying background of finite extent.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗