Statistical symmetries and its impact on new decay modes and integral invariants of decaying turbulence

Classical decay laws of isotropic turbulence usually derived from the von Kármán–Howarth equation are essentially based on two paradigms. First, scaling symmetries of space and time, both tracing back to the Navier–Stokes equations in the limit of large Reynolds numbers (or r?η), give rise to a temporal power-law decay for the turbulent kinetic energy and at the same time an algebraic growth of the integral length scale at an exponent that is uniquely coupled to the latter energy decay. Second, global invariants such as Birkhoff or Loitsianskii integrals determine the exponent of both power laws. We presently show that this class of decay laws may be considerably extended considering the entire set of multi-point correlation equations that admit a much wider class of symmetries. It was recently shown that these new symmetries are of paramount importance, e.g. in deriving the logarithmic law of the wall being an analytic solution of the multi-point equations. For the present case, it is particularly an additional scaling group, which we call statistical scaling group, that gives rise to two additional families of ‘canonical’ decay laws including those with an exponential characteristic for both the kinetic energy and the integral length scale. Finally, a second rather generic group admitted by all linear differential equations corresponding to the superposition principle induces an infinite set of scaling laws of rather complex form that may match rather generic initial conditions. All scaling laws are analyzed in the light of the above-mentioned integral invariants that have been further extended in the present contribution to an exponential-type invariant.     

Singular behavior of 1 P 1 + - quarkonium and positronium annihilation decays and relevance of relative energy

Using a four dimensional approach, we show that the singularities for small gluon momenta, which arise in the usual three dimensional treatment of the annihilation decay, disappear if all poles in the relative energy are taken into account correctly in the integration. We obtain an explicit formula for the decay width which involves a non-locality originating from the kinetic energy. We calculate not only the familiar logarithmic dependence on the binding energy, but also the constant to be added to the logarithm. The logarithmic term agrees with previous values in the literature. In QCD the constant turns out to be quite small, but only because there is an almost perfect cancellation between the tree graph and a non-abelian loop graph which contributes to the decay amplitude to the same order.Received: 31 March 2004, Revised: 5 May 2004, Published online: 2 July 2004     

The role of different linear and non-linear channels of relaxation in scintillator non-proportionality

The non-proportional dependence of a scintillator's light yield on primary particle energy is believed to be influenced crucially by the interplay of non-linear kinetic terms in the radiative and non-radiative decay of excitations versus locally deposited excitation density. A calculation of energy deposition, −dE/dx, along the electron track for NaI is presented for an energy range from several electron-volt to 1 MeV. Such results can be used to specify an initial excitation distribution, if diffusion is neglected. An exactly solvable two-channel (exciton and hole(electron)) model containing 1st and 2nd order kinetic terms is constructed and used to illustrate important features seen in non-proportional light-yield curves, including a dependence on pulse shaping (detection gate width).     

Cross-scale nonlinear coupling and plasma energization by Alfvén waves

We present a new channel for the nonlocal transport of wave energy from the large (MHD) scales to the small (kinetic) scales generated by the resonant decay of MHD Alfvén waves into kinetic Alfvén waves. This process does not impose any restriction on the wave numbers or frequencies of initial MHD waves, which makes it superior compared to the mechanisms of spectral transport studied before. Because of dissipative properties of the nonlinearly driven kinetic Alfvén waves, the decay leads to plasma heating and particle acceleration, which is observed in a variety of space and astrophysical plasmas. Two examples in the solar corona and the terrestrial magnetosphere are briefly discussed.     

Classical trajectories studies of DIET from calcium fluoride

We investigate desorption of positive ions resulting from the repulsive environment created by core-hole Auger decay from relaxed CaF₂ surfaces. The molecular dynamics simulations in the lamina geometry (with two-dimensional ion-lattice summation) is used. For both (011) and (111) surfaces the simulation with changed charge without providing additional kinetic energy does not lead to the ejection of F⁺ ion due to the lattice rearrangement and trapping of the ion. We also assume that the positive ion gains a substantial amount of kinetic energy at the onset of simulations, crudely mimicking ion-stimulated desorption. For the (011) surface the results are extremely sensitive to the size of the considered system, in sharp contrast to the ejection of positive ions from alkali halides. For a 384 ion system, ejection occurs if the kinetic energy, equal to 0.25 eV or more, is delivered to the F⁺ ion at the start of the simulation. For a 768 ion system ejection occurs only for the initial kinetic energy of 4 eV. This result is probably caused by inadequate classical potential and lack of full convergence of the two-dimensional Ewald summation scheme for a highly disordered system. For the (111) surface with 1536 ions in the cell, ejection occurs for an initial kinetic energy of 0.4 eV.     

Inelastic interaction of ultrashort pulses of polarized light with an ensemble of isolated quantum dots

A system of equations is formulated describing the evolution of a slowly varying envelope of an arbitrarily polarized ultrashort pulse of electromagnetic radiation in a medium with its resonant properties determined by an ensemble of isolated quantum dots. It is assumed that the concentration of quantum dots is small and that the whole system is equivalent to a gas of resonant four-level atoms. Particular solutions are found that correspond to the propagation of a stationary optical pulse. It is shown by numerical solution of the generalized truncated Maxwell-Bloch equations that steady-state propagation is possible only for circularly polarized light pulses, whereas the pulses of arbitrary polarization either decay and experience the dispersion-related broadening or are converted into circularly polarized solitary waves.     

Vortex stretching as a mechanism for quantum kinetic energy decay

A pair of perturbed antiparallel quantum vortices, simulated using the three-dimensional Gross-Pitaevskii equations, is shown to be unstable to vortex stretching. This results in kinetic energy K(?ψ) being converted into interaction energy E(I) and eventually local kinetic energy depletion that is similar to energy decay in a classical fluid, even though the governing equations are Hamiltonian and energy conserving. The intermediate stages include the generation of vortex waves, their deepening, multiple reconnections, the emission of vortex rings and phonons, and the creation of an approximately -5/3 kinetic energy spectrum at high wave numbers. All of the wave generation and reconnection steps follow from interactions between the two original vortices. A four vortex example is given to demonstrate that some of these steps might be general.     

Particles production in expanding color flux tube

This article generalizes Schwinger’s mechanism for particle production in the time dependent field volume in the quasi-static approximation. Solution of DGLAP equations in the double leading log approximation for a small x gluon distribution function was used to derive the new formula for initial chromofield energy density. This initial chromofield energy is distributed among color neutral clusters or strings. These strings are stretched by the receding nucleus. From the proposed mechanism of string fragmentation or color field decay, the new formula for the rapidity spectrum of produced partons was derived. The text was submitted by the author in English.     

Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation

Shallow water magnetohydrodynamic (MHD) theory describing incompressible flows of plasma is generalized to the case of compressible flows. A system of MHD equations is obtained that describes the flow of a thin layer of compressible rotating plasma in a gravitational field in the shallow water approximation. The system of quasilinear hyperbolic equations obtained admits a complete simple wave analysis and a solution to the initial discontinuity decay problem in the simplest version of nonrotating flows. In the new equations, sound waves are filtered out, and the dependence of density on pressure on large scales is taken into account that describes static compressibility phenomena. In the equations obtained, the mass conservation law is formulated for a variable that nontrivially depends on the shape of the lower boundary, the characteristic vertical scale of the flow, and the scale of heights at which the variation of density becomes significant. A simple wave theory is developed for the system of equations obtained. All self-similar discontinuous solutions and all continuous centered self-similar solutions of the system are obtained. The initial discontinuity decay problem is solved explicitly for compressible MHD equations in the shallow water approximation. It is shown that there exist five different configurations that provide a solution to the initial discontinuity decay problem. For each configuration, conditions are found that are necessary and sufficient for its implementation. Differences between incompressible and compressible cases are analyzed. In spite of the formal similarity between the solutions in the classical case of MHD flows of an incompressible and compressible fluids, the nonlinear dynamics described by the solutions are essentially different due to the difference in the expressions for the squared propagation velocity of weak perturbations. In addition, the solutions obtained describe new physical phenomena related to the dependence of the height of the free boundary on the density of the fluid. Self-similar continuous and discontinuous solutions are obtained for a system on a slope, and a solution is found to the initial discontinuity decay problem in this case.     

Statistical emission of large fragments: A general theoretical approach

A theory for the statistical emission of large fragments is developed. In analogy with the fission saddle point, a ridge line in the potential energy surface is defined which controls the decay width of the system into any two given fragments. The normal modes at the ridge are separated into three classes: decay modes, amplifying modes, and non-amplifying modes. Amplifying modes are those whose thermal fluctuations are amplified and lead to a broadening of the kinetic energy distribution. Analytical expressions for the kinetic energy distributions are developed for various combinations of amplifying and non-amplifying modes. The limit for large amplifications is a Gaussian kinetic energy distribution. The limit for no amplification is a Maxwellian-like distribution. Thus the formalism comprehends the fission decay on one hand and the neutron evaporation on the other. The angular distributions are evaluated in terms of the ridge line principal moments of inertia. A general analytical expression has been derived which predicts, correctly in both limits, the angular distributions of the evaporated neutrons and of the fission fragments.     

Granular cooling of hard needles

We have developed a kinetic theory of hard needles undergoing binary collisions with loss of energy due to normal and tangential restitution. In addition, we have simulated many particle systems of granular hard needles. The theory, based on the assumption of a homogeneous cooling state, predicts that granular cooling of the needles proceeds in two stages: An exponential decay of the initial configuration to a state where translational and rotational energies take on a time independent ratio (different from unity), followed by an algebraic decay of the total kinetic energy of approximately t(-2). The simulations support the theory very well for low and moderate densities. For higher densities, we have observed the onset of the formation of clusters and shear bands.     

Optical fibre-grating pulse compression

Numerical simulation is used to consider non-linear pulse propagation in fibres and subsequent pulse compression in a dispersive delay line. It is shown that for small initial pulse powers the conventional non-linear Schrödinger equation (NSE) is quite accurate to describe the process of pulse propagation in fibres. In this case initially symmetrical pulses undergo squaring and spectral broadening in fibres, and frequency chirp is linearized over most of the pulse, while shapes of the pulse, spectrum and frequency chirp remain symmetrical at the output of the fibre. There is a certain optimum fibre lengthZ _opt which is determined by the initial pulse parameters and fibre characteristics for pulse compression in the dispersive delay line. When the fibre lengthZ>Z _opt the optical wave breaking effect distorts the linearity of the frequency chirp and thus deteriorates the quality of the compressed pulse. The region of NSE approximation accuracy is determined. It is demonstrated that at increase of the initial pulse power (initial pulse width makes no difference) the NSE approximation becomes inaccurate. So the pulse dynamics in the fibre were described by the modified NSE derived in the higher-order approximation of the method of slowly varying amplitudes from Maxwell's equations. In this case the shock wave appears at the trailing edge of the pulse, which accelerates the wave breaking process. This results in a decrease of the optimum fibre length and deterioration of compressed pulse parameters, compared with the NSE case. Spectral windowing of the extreme Stokes components of the pulse spectrum permits significant improvement in the quality of the compressed pulse. The main features of the compression of pulses with asymmetrical initial shape are also considered.     

Energy oscillations in a one-dimensional harmonic crystal on an elastic substrate

A one-dimensional harmonic crystal on an elastic substrate is considered as a stochastic system into which randomness is introduced through initial conditions. The use of the particle velocity and displacement covariances reduces the stochastic problem to a closed deterministic problem for statistical characteristics of particle pairs. An equation of rapid motion that describes oscillations of potential and kinetic energy components of the system has been derived and solved. The obtained solutions are used to determine the character and to estimate the time of decay of the transient process that brings the system to thermodynamic equilibrium.     

Investigation of anomalous very fast decay regimes in homogeneous isotropic turbulence

The emergence of anomalous fast decay regimes in homogeneous isotropic turbulence (HIT) decay is investigated via both theoretical analysis and eddy-damped quasi-normal Markovian simulations. The work provides new insight about a fundamental issue playing a role in HIT decay, namely the influence of non-standard shapes of the energy spectrum, in particular in the large energetic scale region. A detailed analysis of the kinetic energy spectrum E(k) and the non-linear energy transfer T(k) shows that anomalous decay regimes are associated with the relaxation of initial energy spectra which exhibit a bump at energetic scales. This feature induces an increase in the energy cascade rate, toward solutions with a smooth shape at the spectrum peak. Present results match observations reported in wind-tunnel experiments dealing with turbulence decay in the wake of grids and bluff bodies, including scaling laws for the dissipation parameter C_?. They also indicate that the ratio between the initial eddy turnover time and the advection time determines of how fast anomalous regimes relax toward classical turbulence free-decay. This parameter should be used for consistent data comparison and it opens perspectives for the control of multiscale effects in industrial applications.     

Numerical simulations of the aspherical collapse of laser and acoustically generated bubbles

The details of nonlinear axisymmetric oscillations and collapse of bubbles subject to large internal or external pressure disturbances, are studied via a boundary integral method. Weak viscous effects on the liquid side are accounted for by integrating the equations of motion across the boundary layer that is formed adjacent to the interface. Simulations of single-cavitation bubble luminescence (SCBL) and single-bubble sonoluminescence (SBSL) are performed under conditions similar to reported experimental observations, aiming at capturing the details of bubble collapse. It is shown that any small initial deviation from sphericity, modeled through a small initial elongation along the axis of symmetry, may result in the formation and impact of two counter-propagating jets during collapse of the bubble, provided the amplitude of the initial disturbance is large enough and the viscosity of the surrounding fluid is small enough. Comparison between simulations and experimental observations show that this is the case for bubbles induced via a nano-second laser pulse (SCBL) during a luminescence event. In a similar fashion, simulations show that loss of sphericity accompanied with jet formation and impact during collapse is also possible with acoustically trapped bubbles in a standing pressure wave (SBSL), due to the many afterbounces of the bubble during its collapse phase. In both cases jet impact occurs as a result of P(2) growth in the form of an afterbounce instability. When the sound amplitude is decreased or liquid viscosity is increased the intensity of the afterbounce is decreased and jet impact is suppressed. When the sound amplitude is increased jet formation is superceded by Rayleigh-Taylor instability. In the same context stable luminescence is quenched in experimental observations. In both SCBL and SBSL simulations the severity of jet impact during collapse is quite large, and its local nature quite distinct. This attests to the fact that it is an energy focusing mechanism whose importance in generating the conditions under which a luminescence event is observed should be further investigated.     

Neutron light output function and resolution investigation of the deuterated organic liquid scintillator EJ-315

In this paper we investigate the light response to fast neutrons and estimate the pulse height resolution of a deuterated liquid organic scintillator, EJ-315, considering the detector's non-linear light response to gamma-rays. Initially, collision data and a neutron beam trigger are recorded in coincidence mode, and incident neutron energy is calculated with a time-of-flight technique. Fast neutrons are further discriminated from gamma-ray background based on the scintillation material decay patterns using a pulse shape discrimination algorithm. A light response matrix composed of multiple neutron energy and their corresponding light outputs is derived. The pulse height resolution property of the EJ-315 is characterized utilizing the derivatives of the pulse height distributions with corrections of the measurements setup uncertainties. Additionally, the EJ-315's pulse height resolution is also characterized by comparing the smoothed derivatives of quasi-monoenergetic neutron pulse height distributions, given by the Peierls-formula-based analytic model, to match the measurement data. Results show rather consistent 10–13% pulse resolution for mono-energetic neutrons with kinetic energy above 2 MeV. The resolution decreases slightly with an increase in neutron energy indicating the improved resolution performance of EJ-315 in the higher energy events.     

von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion

A new approach to obtain the closure formulas for the von Kármán–Howarth and Corrsin equations is presented, which is based on the Lagrangian representation of the fluid motion, and on the Liouville theorem associated to the kinematics of a pair of fluid particles. This kinematics is characterized by the finite scale separation vector which is assumed to be statistically independent from the velocity field. Such assumption is justified by the hypothesis of fully developed turbulence and by the property that this vector varies much more rapidly than the velocity field. This formulation leads to the closure formulas of von Kármán–Howarth and Corrsin equations in terms of longitudinal velocity and temperature correlations following a demonstration completely different with respect to the previous works. Some of the properties and the limitations of the closed equations are discussed. In particular, we show that the times of evolution of the developed kinetic energy and temperature spectra are finite quantities which depend on the initial conditions.     

Pulse-induced Raman scattering

The kinetic equations are used to show that induced Raman scattering can reduce the pulse height and length, and also can split a single pulse into several short pulses of decreasing energy. General formulas are derived for the pulse characteristics.     

Non-diffusive ignition of a gaseous reactive mixture following time-resolved,spatially distributed energy deposition

Systematic asymptotic methods are applied to the compressible conservation and state equations for a reactive gas, including transport terms, to develop a rational thermomechanical formulation for the ignition of a chemical reaction following time-resolved, spatially distributed thermal energy addition from an external source into a finite volume of gas. A multi-parameter asymptotic analysis is developed for a wide range of energy deposition levels relative to the initial internal energy in the volume when the heating timescale is short compared to the characteristic acoustic timescale of the volume. Below a quantitatively defined threshold for energy addition, a nearly constant volume heating process occurs, with a small but finite internal gas expansion Mach number. Very little added thermal energy is converted to kinetic energy. The gas expelled from the boundary of the hot, high-pressure spot is the source of mechanical disturbances (acoustic and shock waves) that propagate away into the neighbouring unheated gas. When the energy addition reaches the threshold value, the heating process is fully compressible with a substantial internal gas expansion Mach number, the source of blast waves propagating into the unheated environmental gas. This case corresponds to an extremely large non-dimensional hot-spot temperature and pressure. If the former is sufficiently large, a high activation energy chemical reaction is initiated on the short heating timescale. This phenomenon is in contrast to that for more modest levels of energy addition, where a thermal explosion occurs only after the familiar extended ignition delay period for a classical high activation reaction. Transport effects, modulated by an asymptotically small Knudsen number, are shown to be negligible unless a local gradient in temperature, concentration or velocity is exceptionally large.     

Numerical modeling of ozone production in a pulsed homogeneousdischarge: a parameter study

The pulsed volume discharge is an alternative for the efficient generation of ozone in compact systems. This paper presents a parameter study of the reactions in this kind of homogeneous discharge by using a numerical model which solves plasma chemical kinetic rate and energy equations. Simulations are performed for 10^-9-10^-5 s single pulses and oxygen gas density in the range 10<300 K and normalized electric field of 1000 decreased. The maximum concentration is 3% at 10 amagat and 100 K. The results on ozone accumulation in multiple pulse discharges are presented. In contrast to the single pulse case, higher efficiency is achieved at lower gas density. This scaling can be explained by losses due to ion currents