Weak Turbulence Limit Research Articles

Universality and Nonuniversality in Distributed Nuclear Burning in Homogeneous Isotropic Turbulence.

Nuclear burning plays a key role in a wide range of astrophysical stellar transients, including thermonuclear, pair instability, and core collapse supernovae, as well as kilonovae and collapsars. Turbulence is now understood to also play a key role in these astrophysical transients. Here, we demonstrate that turbulent nuclear burning may lead to large enhancements above the uniform background burning rate, since turbulent dissipation gives rise to temperature fluctuations, and in general the nuclear burning rates are highly sensitive to temperature. We derive results for the turbulent enhancement of the nuclear burning rate under the influence of strong turbulence in the distributed burning regime in homogeneous isotropic turbulence, using probability distribution function methods. We demonstrate that the turbulent enhancement obeys a universal scaling law in the limit of weak turbulence. We further demonstrate that, for a wide range of key nuclear reactions, such as C^{12}(O^{16},α)Mg^{24} and 3-α, even relatively modest temperature fluctuations, of the order of 10%, can lead to enhancements of 1-3 orders of magnitude in the turbulent nuclear burning rate. We verify the predicted turbulent enhancement directly against numerical simulations, and find very good agreement. We also present an estimation for the onset of turbulent detonation initiation, and discuss implications of our results for stellar transients.

Turbulence-level dependence of cosmic ray parallel diffusion

ABSTRACT Understanding the transport of energetic cosmic rays belongs to the most challenging topics in astrophysics. Diffusion due to scattering by electromagnetic fluctuations is a key process in cosmic ray transport. The transition from a ballistic to a diffusive-propagation regime is presented in direct numerical calculations of diffusion coefficients for homogeneous magnetic field lines subject to turbulent perturbations. Simulation results are compared with theoretical derivations of the parallel diffusion coefficient’s dependences on the energy and the fluctuation amplitudes in the limit of weak turbulence. The present study shows that the widely used extrapolation of the energy scaling for the parallel diffusion coefficient to high turbulence levels predicted by quasi-linear theory does not provide a universally accurate description in the resonant-scattering regime. It is highlighted here that the numerically calculated diffusion coefficients can be polluted for low energies due to missing resonant interaction possibilities of the particles with the turbulence. Five reduced-rigidity regimes are established, which are separated by analytical boundaries derived in this work. Consequently, a proper description of cosmic ray propagation can only be achieved by using a turbulence-level-dependent diffusion coefficient and can contribute to solving the Galactic cosmic ray gradient problem.

How turbulence fronts induce plasma spin-up.

A calculation which describes the spin-up of toroidal plasmas by the radial propagation of turbulence fronts with broken parallel symmetry is presented. The associated flux of parallel momentum is calculated by using a two-scale direct-interaction approximation in the weak turbulence limit. We show that fluctuation momentum spreads faster than mean flow momentum. Specifically, the turbulent flux of wave momentum is stronger than the momentum pinch. The scattering of fluctuation momentum can induce edge-core coupling of toroidal flows, as observed in experiments.

Alfvén wave collisions, the fundamental building block of plasma turbulence. II. Numerical solution

This paper presents the numerical verification of an asymptotic analytical solution for the nonlinear interaction between counterpropagating Alfvén waves, the fundamental building block of astrophysical plasma turbulence. The analytical solution, derived in the weak turbulence limit using the equations of incompressible MHD, is compared to a nonlinear gyrokinetic simulation of an Alfvén wave collision. The agreement between these methods signifies that the incompressible solution satisfactorily describes the essential dynamics of the nonlinear energy transfer, even under the weakly collisional plasma conditions relevant to many astrophysical environments.

A turbulent mixing Reynolds stress model fitted to match linear interaction analysis predictions

To predict the evolution of turbulent mixing zones developing in shock tube experiments with different gases, a turbulence model must be able to reliably evaluate the production due to the shock–turbulence interaction. In the limit of homogeneous weak turbulence, ‘linear interaction analysis’ (LIA) can be applied. This theory relies on Kovasznay's decomposition and allows the computation of waves transmitted or produced at the shock front. With assumptions about the composition of the upstream turbulent mixture, one can connect the second-order moments downstream from the shock front to those upstream through a transfer matrix, depending on shock strength. The purpose of this work is to provide a turbulence model that matches LIA results for the shock–turbulent mixture interaction. Reynolds stress models (RSMs) with additional equations for the density–velocity correlation and the density variance are considered here. The turbulent states upstream and downstream from the shock front calculated with these models can also be related through a transfer matrix, provided that the numerical implementation is based on a pseudo-pressure formulation. Then, the RSM should be modified in such a way that its transfer matrix matches the LIA one. Using the pseudo-pressure to introduce ad hoc production terms, we are able to obtain a close agreement between LIA and RSM matrices for any shock strength and thus improve the capabilities of the RSM.

Progress in the Self-Similar Turbulent Flame premixed combustion model

This paper is devoted to premixed combustion modeling in turbulent flow. First, we briefly remind the main features of the Self-Similar Turbulent Flame model that was more extensively developed in a former paper. Then, we carefully describe some improvements of the model. The determination of the turbulent flame velocity is based on the observed self-similarity of the turbulent flame and uses the local flame brush width as a fundamental parameter, which must be retrieved. With respect to the former version, we now derive more rigorously how the density variation has to be taken into account in the width retrieving function. We reformulate the diffusion term as a classical flux divergence term. We enforce the compatibility of the model for the limit of weak turbulence. We include a contracting effect of the source term, thus allowing to give a stationary mono-dimensional asymptotic solution with a finite width. We also include in a preliminary form, a stretch factor, which proves to be useful for controlling the flame behavior close to the flame holder and near the walls. The model implementation in the Star-CD CFD code is then tested on three different flame configurations. Finally, we shortly discuss the model improvements and the simulation results.

Validation of a Turbulent Flame Speed Model across Combustion Regimes

A flame speed expression proposed recently is studied in detail and extensively validated in this study. This expression has no adjustable parameters, and its constants are closely tied to the physics of scalar mixing at small scales. In the weak turbulence limit, the flame speed expression recovers a linear dependence of the turbulent to laminar flame speed ratio, , on the normalized turbulence rms velocity, , in accordance with Damköhler's classical result. However, in the limit of intense turbulence, Damköhler's result gives a square-root dependence, while the new expression gives a combination of linear and square-root terms. Predictions of the new expression are compared to a wide range of experimental data of S T from various flame configurations and conditions, with the same values of the model constants. The quantitative comparisons are found to be very good with experimental data beyond the usually restricted range of of existing models. Predictions of S T for high-pressure turbulent flames, up to 3 MPa, also compare acceptably well with experimental measurements. The flame speed is found to increase when the mean curvature of the flame brush is positive, and this dependence seems to be linear.

Electromagnetic turbulence driven by the mixed mode instability

In continuation of a previous work, numerical results are presented, concerning relativistically counterstreaming plasmas. Here, the relativistic mixed mode instability evolves through and beyond the linear saturation, well into the nonlinear regime. Besides confirming earlier findings that wave power initially peaks on the mixed mode branch, it is observed that during late time evolution, wave power is transferred to other wave numbers. It is argued that the isotropization of power in wavenumber space may be a consequence of weak turbulence. Further, some modifications to the ideal weak turbulence limit is observed. Development of almost isotropic predominantly electrostatic—partially electromagnetic—turbulent spectra holds relevance when considering the spectral emission signatures of the plasma, namely, bremsstrahlung—partially magnetobremsstrahlung (synchrotron radiation and jitter radiation)—from relativistic shocks in astrophysical jets and from shocks in gamma-ray bursts and active galactic nuclei.

Characterization of zonal flow generation in weak electrostatic turbulence

The influence of the diamagnetic Kubo number, which is proportional to the diamagnetic drift velocity, on the zonal flow generation by an anisotropic stochastic electrostatic potential is considered from a semi-analytic point of view. The analysis is performed in the weak turbulence limit and as an analytical tool the decorrelation trajectory method is used. It is shown that the fragmentation of the drift wave structures (a signature of the zonal flow generation) is influenced not only by the anisotropy parameter and the electrostatic Kubo number as expected, but also by the diamagnetic Kubo number. Global Lagrangian averages of characteristic quantities are calculated and interpreted.

Anisotropic fluxes and nonlocal interactions in magnetohydrodynamic turbulence

We investigate the locality or nonlocality of the energy transfer and the spectral interactions involved in the cascade for decaying magnetohydrodynamic (MHD) flows in the presence of a uniform magnetic field B at various intensities. The results are based on a detailed analysis of three-dimensional numerical flows at moderate Reynolds numbers. The energy transfer functions, as well as the global and partial fluxes, are examined by means of different geometrical wave number shells. On the one hand, the transfer functions of the two conserved Elsässer energies E+ and E- are found local in both the directions parallel (k|| direction) and perpendicular (kperpendicular direction) to the magnetic guide field, whatever the B strength. On the other hand, from the flux analysis, the interactions between the two counterpropagating Elsässer waves become nonlocal. Indeed, as the B intensity is increased, local interactions are strongly decreased and the interactions with small k|| modes dominate the cascade. Most of the energy flux in the kperpendicular direction is due to modes in the plane at k||=0, while the weaker cascade in the k|| direction is due to the modes with k||=1. The stronger magnetized flows tend thus to get closer to the weak turbulence limit, where three-wave resonant interactions are dominant. Hence, the transition from the strong to the weak turbulence regime occurs by reducing the number of effective modes in the energy cascade.

Spatial and spectral evolution of turbulence

Spreading of turbulence as a result of nonlinear mode couplings and the associated spectral energy transfer is studied. A derivation of a simple two-field model is presented using the weak turbulence limit of the two-scale direct interaction approximation. This approach enables the approximate overall effect of nonlinear interactions to be written in the form of Fick’s law and leads to a coupled reaction-diffusion system for turbulence intensity. For this purpose, various classes of triad interactions are examined, and the effects that do not lead to spreading are neglected. It is seen that, within this framework, large scale, radially extended eddies are the most effective structures in promoting spreading of turbulence. Thus, spectral evolution that tends toward such eddies facilitates spatial spreading. Self-consistent evolution of the background profile is also considered, and it is concluded that the profile is essentially slaved to the turbulence in this phase of rapid evolution, as opposed to the case of avalanches, where it is the turbulence intensity that would be slaved to the evolving profile. The characteristic quantity describing the evolving background profile is found to be the mean “potential vorticity” (PV). It is shown that the two-field model with self-consistent mean PV evolution can be reduced to a single Fisher-like turbulence intensity transport equation. In addition to the usual nonlinear diffusion term, this equation also contains a “pinch” of turbulence intensity. It is also noted that internal energy spreads faster than kinetic energy because of the respective spectral tendencies of these two quantities.

Nonlinear Diffusive Shock Acceleration with Magnetic Field Amplification

We introduce a Monte Carlo model of nonlinear diffusive shock acceleration that allows for the generation of large-amplitude magnetic turbulence, i.e., ΔB B0, where B0 is the ambient magnetic field. The model is the first to include strong wave generation, efficient particle acceleration to relativistic energies in nonrelativistic shocks, and thermal particle injection in an internally self-consistent manner. We find that the upstream magnetic field B0 can be amplified by large factors and show that this amplification depends strongly on the ambient Alfven Mach number. We also show that, in the nonlinear model, large increases in B do not necessarily translate into a large increase in the maximum particle momentum a particular shock can produce, a consequence of high-momentum particles diffusing in the shock precursor where the large amplified field converges to the low ambient value. To deal with the field growth rate in the regime of strong fluctuations, we extend to strong turbulence a parameterization that is consistent with the resonant quasi-linear growth rate in the weak turbulence limit. We believe our parameterization spans the maximum and minimum range of the fluctuation growth, and within these limits we show that the nonlinear shock structure, acceleration efficiency, and thermal particle injection rates depend strongly on the yet to be determined details of wave growth in strongly turbulent fields. The most direct application of our results will be to estimate magnetic fields amplified by strong cosmic-ray modified shocks in supernova remnants.

Simulating the phase fluctuations produced by many stars in the weak atmospheric turbulence limit

We propose a method allowing for a simulation of cross-correlated turbulence-induced phase fluctuations produced by many stars is proposed. It turns out that for weak-turbulence conditions which are important for many applications, it is possible to suggest a relatively simple and fast simulation procedure. Both the variation of C 2 n along a propagation path and an effect of finite outer scale of the turbulence are considered. The validity of the simulation is verified by comparing theoretical and simulated results, and it is shown that there is good agreement takes place even for big separations between the observation points. Adaptive optics, guide stars, speckle interferometry, long-base interferometry, imaging through the turbulence, as well as simulations of the image distortions of extended objects are among the possible applications of this method.

Anomalous transport in plasmas

AbstractThe application of stochastic methods to the problem of particle and energy transport in turbulent plasmas is briefly reviewed. The “classical” Corrsin approximation is shown to be valid only in the limit of weak turbulence. The recently developed method of decorrelation trajectories is applicable over the whole range of turbulence intensities and yields the correct asymptotic behavior in the limit of very strong turbulence (subdiffusion). © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Anisotropic fluid turbulence in the interstellar medium and solar wind

The interstellar medium and solar wind is permeated by a magnetic field that renders magnetohydrodynamic turbulence anisotropic. In the classic work of Iroshnikov [Astron. Zh. 40, 742 (1963)] and Kraichnan [Phys. Fluids 8, 1385 (1965)], it is assumed that the turbulence is isotropic, and an inertial range energy spectrum that scales as k−3/2 is deduced based on the nonlinear interaction of Alfvén wave packets. Much insight can be gained by analysis and high-resolution numerical simulations of such interactions. In the weak-turbulence limit in which three-wave interactions dominate, analytical and high-resolution numerical results based on random scattering of shear-Alfvén waves propagating parallel to a large-scale magnetic field demonstrate an anisotropic energy spectrum that scales as k⊥−2. Even in the absence of a background magnetic field, when the energy spectrum is globally isotropic, anisotropy is found to develop with respect to the local magnetic field. The two-dimensional case is studied by means of simulations and phenomenological arguments. Despite the presence of local anisotropy, we obtain the Iroshnikov–Kraichnan spectrum, rather than the Kolmogorov spectrum. The same techniques are used to study turbulence in electron magnetohydrodynamics where whistler waves mediate the energy cascade, and comparisons are made with turbulence in magnetohydrodynamics.

Non-local MHD turbulence

We consider an example of strongly non-local interaction in incompressible magnetohydrodynamic (MHD) turbulence which corresponds to the case where the Alfvén waves travelling in the opposite directions have essentially different characteristic wavelengths. We use two approaches to the dynamics of turbulent Alfvénic wavepackets: the first is a geometrical WKB theory [Phys. Lett. A 165 (1992) 330] and the second one is a three-wave kinetic equation derived for weakly turbulent waves [J. Plasma Phys., in press]. We show that these theories have a common limit of weak turbulence with scale separation in which they both predict the same Fokker–Planck equation for the wave power spectrum. In both cases the packet wavenumbers (and therefore the Lagrangian field-line separations) are allowed to experience order 1 changes. The WKB theory developed here formalises an intuitive geometrical argument of Goldreich and Sridhar [ApJ 485 (1997) 680] and allows one to see where such an intuition leads to a wrong conclusion about the inapplicability of the three-wave kinetic equation for order 1 wavepacket distortions. We show that the exponent of the constant flux non-local spectrum matches the value previously found for local turbulence at the boundary of the locality interval. The relationship between the WKB theory and the weak turbulence theory found in this paper for an ensemble of Alfvén waves seems to be general for three-wave systems.

Memory effects in plasma transport theory

The problem of anomalous transport due to drift wave turbulence (in a simpleslab geometry) in the limit of strong turbulence is treated using a recentlydeveloped approach which introduces a new type of approximation (the decorrelation trajectorymethod). A hybrid kinetic equation isderived for the average distribution function (the density profile):it appears to be a typical non-Markovian equation. In the limit ofweak turbulence, this equation can be Markovianized, and the known results ofquasilinear transport theory are recovered. In the opposite limit of verystrong turbulence the memory effects become quite important. As a result, thetransport process is governed by a truly non-Markovian diffusion equation ofthe type appearing in the theory of continuous time random walks(CTRW). The shape and the evolution in time of the density profile are thenprofoundly altered, as compared to the usual solution of the normal diffusionequation.

A simple theory of capillary–gravity wave turbulence

Employing a recently proposed ‘multi-wave interaction’ theory (Glazman 1992), inertial spectra of capillary-gravity waves are derived. This case is characterized by a rather high degree of nonlinearity and a complicated dispersion law. The absence of scale invariance makes this and some other problems of wave turbulence (e.g. nonlinear inertia-gravity waves) intractable by small-perturbation techniques, even in the weak-turbulence limit. The analytical solution obtained in the present work for an arbitrary degree of nonlinearity is shown to be in reasonable agreement with experimental data. The theory explains the dependence of the wave spectrum on wind input and describes the accelerated roll-off of the spectral density function in the narrow sub-range separating scale-invariant regimes of purely gravity and capillary waves, while the appropriate (long- and short-wave) limits yield power laws corresponding to the Zakharov-Filonenko and Phillips spectra.

The problem of flame propagation in a random velocity field: weak turbulence limit

The problem of thin flame front propagation with curvature-dependent speed in a weak turbulent flow has been considered, and its connection with classical problems in the physics of disordered systems such as polymers in a random medium, growing interfaces and n-body interaction problems has been discussed. By a path-integral approach, an explicit formula for the randomly moving flame front in terms of the fluctuating velocity field has been derived. The steepest-descent approximation has been used to find the random configuration of the flame surface in the limit of small Markstein diffusivity. New expressions for the turbulent burning velocity (the overall propagation rate of a flame surface subject to a random velocity field) involving the random velocity field along the Lagrangian trajectories have been derived.

Effective Geometric F-ront Dynamics for Premixed Turbulent Combustion With Separated Velocity Scales

Abstract We study the large scale effective flame front equations for premixed turbulent combustion with separated scale turbulent velocity fields recently formulated and derived by Majda and Souganidis. For the particular case of a steady incompressible velocity field consisting of a mean flow plus a small scale periodic shear we use analytical expressions and numerical quadrature to study the effective turbulent flame front velocity, its dependence on the turbulence intensity, and the role played by the mean flow. In general the dependence of the turbulent flame speed on the turbulence intensity is nonlinear, and the local logarithmic rate of growth can take a continuum of values, depending on the magnitudes of the mean and turbulent intensities. In the weak turbulence limit this dependence can be either linear or quadratic, depending on the magnitude and direction of the mean flow relative to the shear. In the strong turbulence limit the dependence is always linear. This behavior is documented for vari...