Further results on multifractality in shell models

Bolgiano-Obukhov spectrum and mixing efficiency in stably stratified turbulence

In this paper, using a shell model, we simulate highly turbulent stably stratified flow for weak to moderate stratification at unitary Prandtl number. We investigate the energy spectra and fluxes of velocity and density fields. We observe that for moderate stratification, in the inertial range, the kinetic energy spectrum E_{u}(k) and the potential energy spectrum E_{b}(k) show dual scaling-Bolgiano-Obukhov scaling [E_{u}(k)∼k^{-11/5} and E_{b}(k)∼k^{-7/5}] for k<k_{B}, where k_{B} is the Bolgiano wave number, and Kolmogorov scaling (∼k^{-5/3}) for k>k_{B}. In addition, we find that the mixing efficiency η_{mix} varies as η_{mix}∼Ri for weak stratification, whereas η_{mix}∼Ri^{1/3} for moderate stratification, where Ri is the Richardson number.

Efficient calculation of fractal properties via the Higuchi method

Higuchi's method of determining fractal dimension is an important, well-used, research tool that, compared to many other methods, gives rapid, efficient, and robust estimations for the range of possible fractal dimensions. One major shortcoming in applying the method is the correct choice of tuning parameter (k _max); a poor choice can generate spurious results, and there is no agreed upon methodology to solve this issue. We analyze multiple instances of synthetic fractal signals to minimize an error metric. This allows us to offer a new and general method that allows determination, a priori, of the best value for the tuning parameter, for a particular length data set. We demonstrate its use on physical data, by calculating fractal dimensions for a shell model of the nonlinear dynamics of MHD turbulence, and severe acute respiratory syndrome coronavirus 2 isolate Wuhan-Hu-1 from the family Coronaviridae.

Dissipation-range fluid turbulence and thermal noise

We revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equation of Landau and Lifschitz is more appropriate. In particular, our analysis implies that both the exponentially decaying energy spectrum and the far-dissipation-range intermittency predicted by Kraichnan for deterministic Navier-Stokes will be generally replaced by Gaussian thermal equipartition at scales just below the Kolmogorov length. Stochastic shell model simulations at high Reynolds numbers verify our theoretical predictions and reveal furthermore that inertial-range intermittency can propagate deep into the dissipation range, leading to large fluctuations in the equipartition length scale. We explain the failure of previous scaling arguments for the validity of deterministic Navier-Stokes equations at any Reynolds number and we provide a mathematical interpretation and physical justification of the fluctuating Navier-Stokes equation as an "effective field theory" valid below some high-wave-number cutoff Λ, rather than as a continuum stochastic partial differential equation. At Reynolds number around a million, comparable to that in Earth's atmospheric boundary layer, the strongest turbulent excitations observed in our simulation penetrate down to a length scale of about eight microns, still two orders of magnitude greater than the mean free path of air. However, for longer observation times or for higher Reynolds numbers, more extreme turbulent events could lead to a local breakdown of fluctuating hydrodynamics.

Nested polyhedra model of isotropic magnetohydrodynamic turbulence

A nested polyhedra model has been developed for magnetohydrodynamic turbulence. Driving only the velocity field at large scales with random, divergence-free forcing results in a clear, stationary k^{-5/3} spectrum for both kinetic and magnetic energies. Since the model naturally effaces disparate scale interactions, does not have a guide field, and avoids injecting any sign of helicity by random forcing, the resulting three-dimensional k spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. It was also observed that the timescale for the equipartition offset with those of the smallest scales shows a k^{-1/2} scaling.

Nested polyhedra model of turbulence

A discretization of the wave-number space is proposed, using nested polyhedra, in the form of alternating dodecahedra and icosahedra that are self-similarly scaled. This particular choice allows the possibility of forming triangles using only discretized wave vectors when the scaling between two consecutive dodecahedra is equal to the golden ratio and the icosahedron between the two dodecahedra is the dual of the inner dodecahedron. Alternatively, the same discretization can be described as a logarithmically spaced (with a scaling equal to the golden ratio), nested dodecahedron-icosahedron compounds. A wave vector which points from the origin to a vertex of such a mesh, can always find two other discretized wave vectors that are also on the vertices of the mesh (which is not true for an arbitrary mesh). Thus, the nested polyhedra grid can be thought of as a reduction (or decimation) of the Fourier space using a particular set of self-similar triads arranged approximately in a spherical form. For each vertex (i.e., discretized wave vector) in this space, there are either 9 or 15 pairs of vertices (i.e., wave vectors) with which the initial vertex can interact to form a triangle. This allows the reduction of the convolution integral in the Navier-Stokes equation to a sum over 9 or 15 interaction pairs, transforming the equation in Fourier space to a network of "interacting" nodes that can be constructed as a numerical model, which evolves each component of the velocity vector on each node of the network. This model gives the usual Kolmogorov spectrum of k^{-5/3}. Since the scaling is logarithmic, and the number of nodes for each scale is constant, a very large inertial range (i.e., a very high Reynolds number) with a much lower number of degrees of freedom can be considered. Incidentally, by assuming isotropy and a certain relation between the phases, the model can be used to systematically derive shell models.

A spatiotemporal tree model for turbulence in dispersed phase multiphase flows: Energy dissipation rate behavior in single particle and binary particles arrays

In this article, a spatiotemporal dynamical system model (tree model) is utilized for investigating the features of forced and unforced turbulence in a dispersed phase two-phase system. The tree model includes a variable for spatial dimension in addition to variables of wavenumber and time, which display both spatial and temporal intermittencies. The focus of this paper is to study the turbulence modulation due to the presence of rigid particles. The study considers particles with the sizes of 32, 64, and 128 times the Kolmogorov length scale. Specifically, the study of the energy dissipation rate (EDR) at the particle-fluid interface is considered. Two models, namely, A and B with different types of interaction connections between nearby shells, are used first to compare the results of the particle-laden case with decaying turbulence. The number of tree connections in the model is found to affect the amount of augmentation of EDR near the particle surface. Model B is studied further with different sizes of particles in forced turbulence cases and compared to the unladen case with the same parameters. Also, the model expression is studied in the forced turbulence case of dual particles separated by given distances. The results of spatiotemporal shell models provide new approach of handling high Reynolds turbulence in dispersed phase multiphase systems.

Dynamic multiscaling in magnetohydrodynamic turbulence

We present a study of the multiscaling of time-dependent velocity and magnetic-field structure functions in homogeneous, isotropic magnetohydrodynamic (MHD) turbulence in three dimensions. We generalize the formalism that has been developed for analogous studies of time-dependent structure functions in fluid turbulence to MHD. By carrying out detailed numerical studies of such time-dependent structure functions in a shell model for three-dimensional MHD turbulence, we obtain both equal-time and dynamic scaling exponents.

Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence

A detailed systematic derivation of a logarithmically discretized model for two-dimensional turbulence is given, starting from the basic fluid equations and proceeding with a particular form of discretization of the wave-number space. We show that it is possible to keep all or a subset of the interactions, either local or disparate scale, and recover various limiting forms of shell models used in plasma and geophysical turbulence studies. The method makes no use of the conservation laws even though it respects the underlying conservation properties of the fluid equations. It gives a family of models ranging from shell models with nonlocal interactions to anisotropic shell models depending on the way the shells are constructed. Numerical integration of the model shows that energy and enstrophy equipartition seem to dominate over the dual cascade, which is a common problem of two-dimensional shell models.

Multiscaling in superfluid turbulence: A shell-model study

We examine the multiscaling behavior of the normal- and superfluid-velocity structure functions in three-dimensional superfluid turbulence by using a shell model for the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations. Our 3D-HVBK shell model is based on the Gledzer-Okhitani-Yamada shell model. We examine the dependence of the multiscaling exponents on the normal-fluid fraction and the mutual-friction coefficients. Our extensive study of the 3D-HVBK shell model shows that the multiscaling behavior of the velocity structure functions in superfluid turbulence is more complicated than it is in fluid turbulence.

Inverse energy cascade in nonlocal helical shell models of turbulence

Following the exact decomposition in eigenstates of helicity for the Navier-Stokes equations in Fourier space [F. Waleffe, Phys. Fluids A 4, 350 (1992)], we introduce a modified version of helical shell models for turbulence with nonlocal triadic interactions. By using both an analytical argument and numerical simulation, we show that there exists a class of models, with a specific helical structure, that exhibits a statistically stable inverse energy cascade, in close analogy with that predicted for the Navier-Stokes equations restricted to the same helical interactions. We further support the idea that turbulent energy transfer is the result of a strong entanglement among triads possessing different transfer properties.

Enhancement of intermittency in superfluid turbulence

We consider the intermittent behavior of superfluid turbulence in (4)He. Because of the similarity in the nonlinear structure of the two-fluid model of superfluidity and the Euler and Navier-Stokes equations, one expects the scaling exponents of the structure functions to be the same as in classical turbulence for temperatures close to the superfluid transition T(λ) and also for T << T(λ). This is not the case when the densities of normal and superfluid components are comparable to each other and mutual friction becomes important. Using shell model simulations, we propose that in this situation there exists a range of scales in which the effective exponents indicate stronger intermittency. We offer a bridge relation between these effective and the classical scaling exponents. Since this effect occurs at accessible temperatures and Reynolds numbers, we propose that experiments should be conducted to further assess the validity and implications of this prediction.

Three-dimensional turbulent relative dispersion by the Gledzer-Ohkitani-Yamada shell model

We study pair dispersion in a three-dimensional incompressible high Reynolds number turbulent flow generated by Fourier transforming the dynamics of the Gledzer-Ohkitani-Yamada (GOY) shell model into real space. We show that GOY shell model can successfully reproduce both the Batchelor and the Richardson-Obukhov regimes of turbulent relative dispersion. We also study how the crossover time scales with the initial separations of a particle pair and compare it to the prediction by Batchelor.

Anomalous scaling exponents in nonlinear models of turbulence

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by "statistically preserved structures" which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.

Scaling law of plasma turbulence with nonconservative fluxes

It is shown that in the presence of anisotropic kinetic dissipation existence of the scale invariant power law spectrum of plasma turbulence is possible. The obtained scale invariant spectrum is not associated with the constant flux of any physical quantity. Application of the model to the high frequency part of the solar wind turbulence is discussed.

Drag reduction by polymer additives in decaying turbulence

We present results from a systematic numerical study of decaying turbulence in a dilute polymer solution by using a shell-model version of the finitely extensible nonlinear elastic and Peterlin equations. Our study leads to an appealing definition of the drag reduction for the case of decaying turbulence. We exhibit several new results, such as the potential-energy spectrum of the polymer, hitherto unobserved features in the temporal evolution of the kinetic-energy spectrum, and characterize intermittency in such systems. We compare our results with the Gledzer-Ohkitani-Yamada shell model for fluid turbulence.

Varieties of dynamic multiscaling in fluid turbulence

We show that different ways of extracting time scales from time-dependent velocity structure functions lead to different dynamic-multiscaling exponents in fluid turbulence. These exponents are related to equal-time multiscaling exponents by different classes of bridge relations, which we derive. We check this explicitly by detailed numerical simulations of the Gledzer-Ohkitani-Yamada shell model for fluid turbulence. Our results can be generalized to any system in which both equal-time and time-dependent structure functions show multiscaling.

Decay of magnetohydrodynamic turbulence from power-law initial conditions

We derive relations for the decay of the kinetic and magnetic energies and the growth of the Taylor and integral scales in unforced, incompressible, homogeneous, and isotropic three-dimensional magnetohydrodynamic (3DMHD) turbulence with power-law initial energy spectra. We also derive bounds for the decay of the cross and magnetic helicities. We then present results from systematic numerical studies of such decay both within the context of a MHD shell model and direct numerical simulations of 3DMHD. We show explicitly that our results about the power-law decay of the energies hold for times t< t*, where t* is the time at which the integral scales become comparable to the system size. For t< t*, our numerical results are consistent with those predicted by the principle of "permanence of large eddies."

Unstable periodic solutions embedded in a shell model turbulence

An approach to intermittency of a shell model turbulence is proposed from the viewpoint of dynamical systems. We detected unstable solutions of the Gledzer-Ohkitani-Yamada shell model and studied their relation to turbulence statistics. One of the solutions has an unstable periodic orbit (UPO), which shows an intermittency where the scaling exponents of the structure function have a nonlinear dependence on its order, quite similar to that of turbulence solution at the same parameter values. The attractor in the phase space is found to be well approximated by a continuous set of solutions generated from the UPO through a one-parameter phase transformation, which implies that the intermittency of the shell model turbulence is described by this UPO.

Critical "dimension" in shell model turbulence

We investigate the Gledzer-Ohkitani-Yamada (GOY) shell model within the scenario of a critical dimension in fully developed turbulence. By changing the conserved quantities, one can continuously vary an "effective dimension" between d=2 and d=3. We identify a critical point between these two situations where the flux of energy changes sign and the helicity flux diverges. Close to the critical point the energy spectrum exhibits a turbulent scaling regime followed by a plateau of thermal equilibrium. The corrections due to intermittency persist close to the critical point. We identify scaling laws and perform a rescaling argument to derive a relation between the critical exponents. We further discuss the distribution function of the energy flux.

Quasisolitons and asymptotic multiscaling in shell models of turbulence

A variation principle is suggested to find self-similar solitary solutions (referred to as solitons) of shell model of turbulence. For the Sabra shell model the shape of the solitons is approximated by rational trial functions with relative accuracy of O(10(-3)). It is found how the soliton shape, propagation time t(n) (from a shell n to shells with n --> infinity), and the dynamical exponent z(0) (which governs the time rescaling of the solitons in different shells) depend on parameters of the model. For a finite interval of z the author discovered quasisolitons which approximate with high accuracy corresponding self-similar equations for an interval of times from -infinity to some time in the vicinity of the peak maximum or even after it. The conjecture is that the trajectories in the vicinity of the quasisolitons (with continuous spectra of z) provide an essential contribution to the multiscaling statistics of high-order correlation functions, referred to in the paper as an asymptotic multiscaling. This contribution may be even more important than that of the trajectories in the vicinity of the exact soliton with a fixed value z(0). Moreover there are no solitons in some regions of the parameters where quasisolitons provide a dominant contribution to the asymptotic multiscaling.

Fluctuation-response relation in turbulent systems

We address the problem of measuring time properties of response functions (Green functions) in Gaussian models (Orszag-McLaughin) and strongly non-Gaussian models (shell models for turbulence). We introduce the concept of halving-time statistics to have a statistically stable tool to quantify the time decay of response functions and generalized response functions of high order. We show numerically that in shell models for three-dimensional turbulence response functions are inertial range quantities. This is a strong indication that the invariant measure describing the shell-velocity fluctuations is characterized by short range interactions between neighboring shells.

Self-similar fluctuation and large deviation statistics in the shell model of turbulence

Both static and dynamic multiscalings of fluctuations of energy flux and energy dissipation rate in the Gledzer-Ohkitani-Yamada (GOY) shell model of turbulence are numerically investigated. We compute the large deviation rate function of energy flux not only in the inertial range (IR) but also around the crossover between the inertial range and the dissipation range (DR). The rate function in IR exists to be concave, which assures the applicability of the Legendre transformation with the anomalous scaling exponents that have been investigated so far, and turns out to be independent of the Reynolds number. On the contrary, near the crossover scale, an intermediate dissipation range (IMDR) scaling is observed with the rate function in IMDR, which is accounted with the argument on dissipation scale fluctuation dominated by the energy flux fluctuation in the inertial range. Furthermore, to study the difference between IR intermittency and DR intermittency, we compute finite time-averaged quantities of energy flux and energy dissipation rate and investigate their multiscaling behavior. The difference observed in terms of their dynamic multiscaling is discussed.

Outliers, extreme events, and multiscaling

Extreme events have an important role which is sometimes catastrophic in a variety of natural phenomena, including climate, earthquakes, and turbulence, as well as in manmade environments such as financial markets. Statistical analysis and predictions in such systems are complicated by the fact that on the one hand extreme events may appear as "outliers" whose statistical properties do not seem to conform with the bulk of the data, and on the other hand they dominate the tails of the probability distributions and the scaling of high moments, leading to "abnormal" or "multiscaling." We employ a shell model of turbulence to show that it is very useful to examine in detail the dynamics of onset and demise of extreme events. Doing so may reveal dynamical scaling properties of the extreme events that are characteristic to them, and not shared by the bulk of the fluctuations. As the extreme events dominate the tails of the distribution functions, knowledge of their dynamical scaling properties can be turned into a prediction of the functional form of the tails. We show that from the analysis of relatively short-time horizons (in which the extreme events appear as outliers) we can predict the tails of the probability distribution functions, in agreement with data collected in very much longer time horizons. The conclusion is that events that may appear unpredictable on relatively short time horizons are actually a consistent part of a multiscaling statistics on longer time horizons.

Fixed points, stability, and intermittency in a shell model for advection of passive scalars

We investigate the fixed points of a shell model for the turbulent advection of passive scalars introduced in Jensen, Paladin, and Vulpiani [Phys. Rev. A 45, 7214 (1992)]. The passive scalar field is driven by the velocity field of the popular Gledzer-Ohkitani-Yamada (GOY) shell model. The scaling behavior of the static solutions is found to differ significantly from Obukhov-Corrsin scaling straight theta(n) approximately k(-1/3)(n), which is only recovered in the limit where the diffusivity vanishes, D-->0. From the eigenvalue spectrum we show that any perturbation in the scalar will always damp out, i.e., the eigenvalues of the scalar are negative and are decoupled from the eigenvalues of the velocity. We estimate Lyapunov exponents and the intermittency parameters using a definition proposed by Benzi, Paladin, Parisi, and Vulpiani [J. Phys. A 18, 2157 (1985)]. The full model is found to be as chaotic as the GOY model, measured by the maximal Lyapunov exponent, but is more intermittent.

Symmetries, invariants, and cascades in a shell model of turbulence

Reduced wave number models of turbulence, namely shell models, show cascade processes and anomalous scaling of correlators which might be analogous to what is observed in Navier-Stokes (NS) turbulence. The scaling properties of the shell models depend on the specific symmetries and invariants of the models. A shell model is investigated. It is argued that this model might have a closer resemblance than the standard Gledzer-Ohkitani-Yamada model to the NS turbulence. The shell model investigated here coincides with the Sabra model proposed by L'vov et al. [Phys. Rev. E. 58, 1811 (1998)] for a specific choice of the free parameters of their model. For this choice of parameters, besides the energy and the "helicity," the model has a cubic inviscid invariant.

Large deviation statistics of the energy-flux fluctuation in the shell model of turbulence

The energy-flux fluctuation in the shell model of turbulence is numerically analyzed from the large deviation statistical point of view. We first observe that the rate function defined in the inertial range is independent of the Reynolds number. The rate function derived by the cascade model of the log-Poisson statistics turns out to be in good agreement with the present numerical result in the region where strong singularity of fluctuation exits. This fact may imply the universality as well as the robustness of the large deviation statistical quantities in turbulence.

Extended self-similarity and dissipation range dynamics of three-dimensional turbulence

We carry out a self-consistent calculation of the structure functions in the dissipation range using the Navier-Stokes equation. Combining these results with the known structures in the inertial range, we actually propose crossover functions for the structure functions that take one smoothly from the inertial to the dissipation regime. These crossover functions are shown to exhibit extended self-similarity properties consistent with experimental findings.

Numerical study of chaos based on a shell model

A shell model is introduced to study a turbulence driven by the thermal instability (Rayleigh-Benard convection). This model equation describes cascade and chaos in the strong turbulence with high Rayleigh number. The chaos is numerically studied based on this model. The characteristics of the turbulence are analyzed and compared with those of the Gledzer-Ohkitani-Yamada (GOY) model. Quantities such as a mean value of total fluctuation energy, it's standard deviation, time averaged wave spectrum, probability distribution function, frequency spectrum, the maximum instantaneous Lyapunov exponent, distribution of instantaneous Lyapunov exponents, are evaluated. The dependences of these quantities on the error of numerical integration are also examined. There is not a clear correlation between the numerical accuracy and the accuracy of these quantities, since the interaction between a truncation error and an intrinsic nonlinearity of the system exists. A finding is that the maximum Lyapunov exponent is insensitive to a truncation error. (c) 1999 American Institute of Physics.