Entropy Cascade Research Articles

Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence.

We consider a nearly collisionless plasma consisting of a species of "test particles" in one spatial and one velocity dimension, stirred by an externally imposed stochastic electric field-a kinetic analog of the Kraichnan model of passive advection. The mean effect on the particle distribution function is turbulent diffusion in velocity space-known as stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant C_{2}∝∫∫dxdv〈f^{2}〉-a quantity that here plays the role of (negative) entropy of the distribution function. We find that C_{2} is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of C_{2} in Fourier space, with k and s denoting spatial and velocity wave numbers, respectively. In our model, the nonlinearity in the evolution equationfor the spectrum turns into a fractional Laplacian operator in k space, leading to anomalous diffusion. Whereas even the linear phase mixing alone would lead to a constant flux of C_{2} to high s (towards the collisional dissipation range) at every k, the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of C_{2} is to both high k and high s simultaneously. Integrating over velocity (spatial) wave numbers, the k-space (s-space) flux of C_{2} is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the (k,s) plane, with power-law asymptotics at large k and s. Our model is fully analytically solvable, but the asymptotic scalings of the spectrum can also be found via a simple phenomenological theory whose key assumption is that the cascade is governed by a "critical balance" in phase space between the linear and nonlinear timescales. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one.

Evidence of entropy cascade in collisionless magnetized plasma turbulence

The turbulence of collisionless magnetized plasmas, as observed in space, astrophysical, and magnetically confined fusion plasmas, has attracted considerable interest for a long-time. The entropy cascade in collisionless magnetized plasmas is a theoretically proposed dynamics comparable to the Kolmogorov energy cascade in fluid turbulence. Here, we present evidence of an entropy cascade in laboratory plasmas by direct visualization of the entropy distribution in the phase space of turbulence in laboratory experiments. This measurement confirms the scaling laws predicted by the gyrokinetic theory with the dual self-similarity hypothesis, which reflects the interplay between the position and velocity of ions by perpendicular nonlinear phase mixing. This verification contributes to our understanding of turbulent heating in the solar corona, accretion disks, and magnetically confined fusion plasmas.

Fully developed relativistic turbulence

We use a simple model consisting of energy-momentum tensor conservation and a Maxwell-Cattaneo equation for its viscous part to study nonlinear phenomena in a real relativistic fluid. We focus on new types of behavior without nonrelativistic equivalents, such as an entropy cascade driven by fluctuations in the tensor degrees of freedom of the theory. We write down the von K\'arm\'an-Howarth equations for this kind of turbulence, and consider the correlations corresponding to fully developed turbulence.

Nonlinear fluctuations in relativistic causal fluids

In the Second Order Theories (SOT) of real relativistic fluids, the non-ideal properties of the flows are described by a new set of dynamical tensor variables. In this work we explore the non-linear dynamics of those variables in a conformal fluid. Among all possible SOTs, we choose to work with the Divergence Type Theories (DTT) formalism, which ensures that the second law of thermodynamics is fulfilled non-perturbatively. The tensor modes include two divergence-free modes which have no analog in theories based on covariant generalizations of the Navier-Stokes equation, and that are particularly relevant because they couple linearly to a gravitational field. To study the dynamics of this irreducible tensor sector, we observe that in causal theories such as DTTs, thermal fluctuations induce a stochastic stirring force, which excites the tensor modes while preserving energy momentum conservation. From fluctuation-dissipation considerations it follows that the random force is Gaussian with a white spectrum. The irreducible tensor modes in turn excite vector modes, which back-react on the tensor sector, thus producing a consistent non-linear, second order description of the divergence-free tensor dynamics. Using the Martin-Siggia-Rose (MSR) formalism plus the Two-Particle Irreducible Effective Action (2PIEA) formalism, we obtain the one-loop corrected equations for the relevant two-point correlation functions of the model: the retarded propagator and the Hadamard function. The overall result of the self-consistent dynamics of the irreducible tensor modes at this order is a depletion of the spectrum in the UV sector, which suggests that tensor modes could sustain an inverse entropy cascade.

Cascades of temperature and entropy fluctuations in compressible turbulence

Cascades of temperature and entropy fluctuations are studied by numerical simulations of stationary three-dimensional compressible turbulence with a heat source. The fluctuation spectra of velocity, compressible velocity component, density and pressure exhibit the $-5/3$ scaling in an inertial range. The strong acoustic equilibrium relation between spectra of the compressible velocity component and pressure is observed. The $-5/3$ scaling behaviour is also identified for the fluctuation spectra of temperature and entropy, with the Obukhov–Corrsin constants close to that of a passive scalar spectrum. It is shown by Kovasznay decomposition that the dynamics of the temperature field is dominated by the entropic mode. The average subgrid-scale (SGS) fluxes of temperature and entropy normalized by the total dissipation rates are close to 1 in the inertial range. The cascade of temperature is dominated by the compressible mode of the velocity field, indicating that the theory of a passive scalar in incompressible turbulence is not suitable to describe the inter-scale transfer of temperature in compressible turbulence. In contrast, the cascade of entropy is dominated by the solenoidal mode of the velocity field. The different behaviours of cascades of temperature and entropy are partly explained by the geometrical properties of SGS fluxes. Moreover, the different effects of local compressibility on the SGS fluxes of temperature and entropy are investigated by conditional averaging with respect to the filtered dilatation, demonstrating that the effect of compressibility on the cascade of temperature is much stronger than on the cascade of entropy.

Cascades and Dissipative Anomalies in Nearly Collisionless Plasma Turbulence

We develop first-principles theory of kinetic plasma turbulence governed by the Vlasov-Maxwell-Landau equations in the limit of vanishing collision rates. Following an exact renormalization-group approach pioneered by Onsager, we demonstrate the existence of a "collisionless range" of scales (lengths and velocities) in 1-particle phase space where the ideal Vlasov-Maxwell equations are satisfied in a "coarse-grained sense". Entropy conservation may nevertheless be violated in that range by a "dissipative anomaly" due to nonlinear entropy cascade. We derive "4/5th-law" type expressions for the entropy flux, which allow us to characterize the singularities (structure-function scaling exponents) required for its non-vanishing. Conservation laws of mass, momentum and energy are not afflicted with anomalous transfers in the collisionless limit. In a subsequent limit of small gyroradii, however, anomalous contributions to inertial-range energy balance may appear due both to cascade of bulk energy and to turbulent redistribution of internal energy in phase space. In that same limit the "generalized Ohm's law" derived from the particle momentum balances reduces to an "ideal Ohm's law", but only in a coarse-grained sense that does not imply magnetic flux-freezing and that permits magnetic reconnection at all inertial-range scales. We compare our results with prior theory based on the gyrokinetic (high gyro-frequency) limit, with numerical simulations, and with spacecraft measurements of the solar wind and terrestrial magnetosphere.

Performance assessment of cascade control loops with non-Gaussian disturbances using entropy information

Cascade control is one of the routinely used control strategies in industrial processes because it can dramatically improve the performance of single-loop control, reducing both the maximum deviation and the integral error of the disturbance response. Currently, many control performance assessment methods of cascade control loops are developed based on the assumption that all the disturbances are subject to Gaussian distribution. However, in the practical condition, several disturbance sources occur in the manipulated variable or the upstream exhibits nonlinear behaviors. In this paper, a general and effective index of the performance assessment of the cascade control system subjected to the unknown disturbance distribution is proposed. Like the minimum variance control (MVC) design, the output variances of the primary and the secondary loops are decomposed into a cascade-invariant and a cascade-dependent term, but the estimated ARMA model for the cascade control loop based on the minimum entropy, instead of the minimum mean squares error, is developed for non-Gaussian disturbances. Unlike the MVC index, an innovative control performance index is given based on the information theory and the minimum entropy criterion. The index is informative and in agreement with the expected control knowledge. To elucidate wide applicability and effectiveness of the minimum entropy cascade control index, a simulation problem and a cascade control case of an oil refinery are applied. The comparison with MVC based cascade control is also included.

Convection in anelastic models of the Earth’s liquid core

The effects of adiabatic cooling on the convection in the anelastic model of the liquid core of the Earth are considered. It is shown that even minor adiabatic cooling causes significant changes in the pattern of the convection, shifting the peak in the convection intensity to the inner part of the core. Just as in the Boussinesq model, both direct and inverse kinetic energy cascades are simultaneously present, and the direct cascade of entropy is observed.

Experimental Verification of Entropy Cascade in Two-Dimensional Electrostatic Turbulence in Magnetized Plasma

The wave number spectrum (one-dimensional spectrum) of electrostatic potential fluctuations at sub-Larmor scales was measured in two-dimensional (2D) electrostatic turbulence in laboratory magnetized plasma. The spectrum at scales k([perpendicular])ρ(i)>1, where k([perpendicular]) and ρ(i) are the fluctuation wave number perpendicular to the magnetic field and ion Larmor radius, respectively, supports the existence of the k(-10/3) inertial range of the entropy cascade induced by nonlinear phase mixing. This indicates agreement with a theoretical prediction [A. A. Schekochihin et al., Plasma Phys. Controlled Fusion 50, 124024 (2008)] and the result of a 2D gyrokinetic simulation [T. Tatsuno et al., Phys. Rev. Lett. 103, 015003 (2009)]. The cutoff wave numbers of the spectrum, above which the entropy cascade is smeared by collisions, observed in this experiment were consistent with those in the theory.

Gyrokinetic Simulations of Solar Wind Turbulence from Ion to Electron Scales

A three-dimensional, nonlinear gyrokinetic simulation of plasma turbulence resolving scales from the ion to electron gyroradius with a realistic mass ratio is presented, where all damping is provided by resolved physical mechanisms. The resulting energy spectra are quantitatively consistent with a magnetic power spectrum scaling of k(-2.8) as observed in in situ spacecraft measurements of the "dissipation range" of solar wind turbulence. Despite the strongly nonlinear nature of the turbulence, the linear kinetic Alfvén wave mode quantitatively describes the polarization of the turbulent fluctuations. The collisional ion heating is measured at subion-Larmor radius scales, which provides evidence of the ion entropy cascade in an electromagnetic turbulence simulation.

Studying anomalous scaling and heat transport of turbulent thermal convection using a dynamical model

Turbulent thermal convection is a well-studied problem with various issues of interest. In this paper, we review our work which shows the nature and origin of anomalous scaling and heat transport in the limit of very strong thermal forcing, can be gained by studying a dynamical model, known as shell model, of homogeneous turbulent thermal convection in which buoyancy acts directly at most scales. Specifically, we have obtained two results. The first result is that when buoyancy acts directly at most scales such that the dynamics are governed by a cascade of entropy, the scaling behavior is described by Bolgiano and Obukhov scaling plus corrections that are due to the variations of the local entropy transfer rate. This result indicates the validity of the extension of refined similarity hypothesis to turbulent thermal convection. The second result is that when buoyancy is acting directly at most scales, a damping term acting on the largest scale, which has to be added for the system to achieve stationarity, plays a crucial role in heat transport, and that the heat transport depends on the strength of thermal forcing in the same manner as that predicted for the ultimate state of very strong thermal forcing. With our interpretation of the damping term representing the effect of the boundaries, this result indicates that in the ultimate state of turbulent thermal convection, when buoyancy is acting at most scales, boundaries would play a significant role in heat transport.

Nonlinear Phase Mixing and Phase-Space Cascade of Entropy in Gyrokinetic Plasma Turbulence

Electrostatic turbulence in weakly collisional, magnetized plasma can be interpreted as a cascade of entropy in phase space, which is proposed as a universal mechanism for dissipation of energy in magnetized plasma turbulence. When the nonlinear decorrelation time at the scale of the thermal Larmor radius is shorter than the collision time, a broad spectrum of fluctuations at sub-Larmor scales is numerically found in velocity and position space, with theoretically predicted scalings. The results are important because they identify what is probably a universal Kolmogorov-like regime for kinetic turbulence; and because any physical process that produces fluctuations of the gyrophase-independent part of the distribution function may, via the entropy cascade, result in turbulent heating at a rate that increases with the fluctuation amplitude, but is independent of the collision frequency.

Inertial range turbulence in kinetic plasmas

The transfer of turbulent energy through an inertial range from the driving scale to dissipative scales in a kinetic plasma followed by the conversion of this energy into heat is a fundamental plasma physics process. A theoretical foundation for the study of this process is constructed, but the details of the kinetic cascade are not well understood. Several important properties are identified: (a) The conservation of a generalized energy by the cascade; (b) the need for collisions to increase entropy and realize irreversible plasma heating; and (c) the key role played by the entropy cascade—a dual cascade of energy to small scales in both physical and velocity space—to convert ultimately the turbulent energy into heat. A strategy for nonlinear numerical simulations of kinetic turbulence is outlined. Initial numerical results are consistent with the operation of the entropy cascade. Inertial range turbulence arises in a broad range of space and astrophysical plasmas and may play an important role in the thermalization of fusion energy in burning plasmas.

Dynamical aspect of entropy transfer in free convection turbulence

From a dynamical perspective, entropy transfer processes are investigated in two-dimensional free convection turbulence in comparison with an entropy cascade picture based on the coupled dynamics of the T vorticity $\mathit{\ensuremath{\chi}}\ensuremath{\equiv}({\ensuremath{\partial}}_{y}T,\ensuremath{-}{\ensuremath{\partial}}_{x}T)$ and the velocity gradient tensor. Typical entropy transfer processes are observed in direct numerical simulations. For these processes, two characteristic times, the transfer time and the staying time are determined: the former time obeys a Bolgiano-Obukhov (BO) time scaling corresponding to the eddy turnover time in energy cascade. It is suggested that this typical transfer process is not an elementary process of cascade but a dynamical manifestation of intermittency. To examine the meaning of the characteristic times of the typical entropy transfer process, a shell model is constructed based on the entropy cascade picture. By this model, it is shown numerically that typical entropy transfer processes are regarded as the fluctuations satisfying a dynamical similarity. This similarity proved by perturbation analysis requires naturally that the transfer time should obey the BO time scaling.

Effects of entropy inhomogeneity on density-temperature correlation in solar wind.

Compressive fluctuations in solar wind slow speed streams are studied by means of a magnetohydrodynamics (MHD) model, which represents the plasma in the vicinity of the heliospheric current sheet. The model contains a current sheet, as well as density and temperature variations, corresponding to a large scale modulation of the specific entropy. Alfvénic fluctuations are initially superimposed on the background equilibrium and compressive fluctuations are consequently generated during the time evolution. The resulting correlation between density and temperature fluctuations at various spatial scales is interpreted in terms of both generation of magnetosonic fluctuations and of an "entropy cascade." The latter phenomenon arises as a consequence of the interaction between the MHD turbulence and the underlying large scale entropy structure. In particular, it is responsible for anticorrelated density and temperature fluctuations detected at various scales. The results of the model are compared with the proton density-temperature correlation calculated during several crossings of solar wind slow speed streams by the Helios spacecraft. The model reproduces to a good extent the main observed features, in particular the dependence of the correlation coefficient on location (close to or far from the current sheet) and on the fluctuation scale. The results show that large scale inhomogeneities, in particular, that of specific entropy, are important ingredients in the dynamics of the MHD turbulence in slow speed streams.

Time Scalings of Entropy Transfer Process in Free Convection

To study one elementary process of entropy cascade in hard turbulence, we propose a shell model which governs the temperature gradient rotated through the angle \(+\frac{\pi}{2}, {\mib \chi} \equiv (\partial_y T, -\partial_x T)\), and the velocity gradient, i.e., strain rates. This model is based on the correspondence of the dynamics of χ in a two-dimensional free convection system to that of the vorticity in a three-dimensional Navier-Stokes system. We obtain the following results: 1) A steady solution which satisfies Bolgiano-Obukhov (BO) scaling in the viscous case exists. 2) After a small disturbance of large scale is added to the steady solution, time series of the modes in the inertial range become similar to each other under the proper scaling, and each mode has its own characteristic time predicted by BO scaling. 3) When the same disturbance is added to the null state, the modes also evolve similarly in the inertial range. The characteristic times of the modes are, however, explained by a dimensio...

Entropy cascade and temporal intermittency in a shell model for convective turbulence.

Entropy cascade and temporal intermittency in a shell model for convective turbulence.

MODE-COUPLING THEORY, DYNAMIC SCALING, AND TWO-DIMENSIONAL TURBULENCE

A self-consistent mode-coupling scheme, along with dynamic scaling ideas, is used to obtain a renormalized perturbation theory in the Eulerian framework from Wyld’s perturbation theory of the forced Navier-Stokes equation. For the force-correlation behaving as k−(d−4+y), the Kolmogorov and Kraichnan-Batchelor scaling spectra of two-dimensional turbulence for the inverse energy cascade, [Formula: see text] and the direct entropy cascade, [Formula: see text], are obtained for y=4 and y=6 respectively, including the logarithmic correction for the latter. Unlike the usual Eulerian formulations (e.g. the direct-interaction approximation), the theory is finite in the energy regime, while it becomes marginal in the enstrophy regime, leading to the logarithmic correction. Calculations yield C=6.447 and C′=1.923 at one-loop order, which are in exact agreement with those of field-theoretic renormalization group calculations [P. Olla, Phys. Rev. Lett. 67, 2465 (1991)]. However, a self-consistent treatment of the logarithmic scalings in E(k) and the inverse response-time yields a different value: C′=2.201. The theory is free of any external parameter; the choice of y(=4 or 6) is dictated by the condition of conserved transfer of energy or enstrophy.

Entropy cascade and energy inverse transfer in two-dimensional convective turbulence.

Two-dimensional system modeling a central region in an experimental box on the basis of the Boussinesq approximation is examined numerically. It is assumed that the effect of mixing and boundary layers on the central region is taken into account only as a forcing to the temperature dynamics. Neutrally stable stratification and homogeneity are also assumed. Entropy (${T}^{2}$) spectra are compared with ones obtained by experiments. It is shown that smaller scales are governed by the entropy cascade process and large-scale modes are induced through the energy inverse transfer.