We present a numerical study of turbulence in Bose-Einstein condensates within the 3D Gross-Pitaevskii equation. We concentrate on the direct energy cascade in forced-dissipated systems. We show that behavior of the system is very sensitive to the properties of the model at the scales greater than the forcing scale, and we identify three different universal regimes: (1) a non-stationary regime with condensation and transition from a four-wave to a three-wave interaction process when the largest scales are not dissipated, (2) a steady weak wave turbulence regime when largest scales are dissipated with a friction-type dissipation, (3) a state with a scale-by-scale balance of the linear and the nonlinear timescales when the large-scale dissipation is a hypo-viscosity.
We present a statistical analysis of the two-point vorticity probability density of the vorticity field generated in the inverse cascade of stationary two-dimensional turbulence.
Helicity, as one of only two inviscid invariants in three-dimensional turbulence, plays an important role in the generation and evolution of turbulence. From the traditional viewpoint, there exists only one channel of helicity cascade similar to that of kinetic energy cascade. Through theoretical analysis, we find that there are two channels in helicity cascade process. The first channel mainly originates from vortex twisting process, and the second channel mainly originates from vortex stretching process. By analysing the data of direct numerical simulations of typical turbulent flows, we fin…
Markovian properties of a discrete random multiplicative cascade model of log-normal type are discussed. After taking small-scale resummation and breaking of the ultrametric hierarchy into account, qualitative agreement with Kramers-Moyal coefficients, recently deduced from a fully developed turbulent flow, is achieved.
The Log-Poisson phenomenological description of the turbulent energy cascade is evoked to discuss high-order statistics of velocity derivatives and the mapping between their probability distribution functions at different Reynolds numbers. The striking confirmation of theoretical predictions suggests that numerical solutions of the flow, obtained at low/moderate Reynolds numbers can play an important quantitative role in the analysis of experimental high Reynolds number phenomena, where small scales fluctuations are in general inaccessible from direct numerical simulations.
Here we show a procedure to generate an analytical structure producing a cascade that scales as the energy spectrum in isotropic homogeneous turbulence. We obtain a function that unveils a non-self-similar fractal at the origin of the cascade. It reveals that the backbone underlying $-5/3$ cascades is formed by deterministic nested polynomials with parameters tuned in a Hopf bifurcation critical point. The cascade scaling is exactly obtainable (not by numerical simulations) from deterministic low dimensional nonlinear dynamics. Consequently, it should not be exclusive for fluids but also prese…
We study statistical properties of turbulent inverse cascades in a class of nonlinear models describing a scalar field transported by a two-dimensional incompressible flow. The class is characterized by a linear relation between the transported field and the velocity, and include several cases of physical interest, such as Navier-Stokes, surface quasi-geostrophic and Charney-Hasegawa-Mima equations. We find that some statistical properties of the inverse turbulent cascades in such systems are conformal invariant. In particular, the zero-isolines of the scalar field are statistically equivalent…
As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analogue (enstrophy) in the inviscid case. With a large-scale forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency and self-similarity in the dynamical system structure.
The multiplier statistics of discrete and continuous nonconservative multiplicative cascade models, employed to describe the energy cascade in fully developed turbulence, is investigated. It is found to be indistinguishable due to small-scale resummation and restoration of spatial homogeneity. Only the subclass of positively skewed weight distributions, associated to the cascade generator, qualifies to describe observed unconditional as well as conditional multiplier distributions qualitatively; popular log-Poisson and log-stable weight distributions do not share this property.
Velocity increments over a distance r and turbulent energy dissipation on a box of size r are well described by the multifractal models of fully developed turbulence. These quantities and models however, do not involve time-correlations and therefore are not a detailed test of the dynamics of the turbulent cascade. If the time development of the turbulent cascade, in the inertial range, is related to the lifetime of the eddies at different length scales, the time correlations may be described by a stochastic process on a tree with jumping kernels which are a function of the ultrametric (tree) …
A model of scalar turbulent advection in compressible flow is analytically investigated. It is shown that, depending on the dimensionality $d$ of space and the degree of compressibility of the smooth advecting velocity field, the cascade of the scalar is direct or inverse. If $d>4$, the cascade is always direct. For small enough degree of compressibility, the cascade is direct again. Otherwise it is inverse, i.e. very large scales are excited. The dynamical hint for the direction of the cascade is the sign of the Lyapunov exponent for particles separation. Positive Lyapunov exponents are assoc…
Cascade models based on dynamical complex networks are proposed as models of turbulent energy cascade. Taking a simple shell model as the initial regular lattice with only nearest neighbor interactions, small world network models are constructed by adding or replacing some of the existing local interactions by nonlocal ones. The models are then evolved over time, both by solving for the shell variable for velocity using an arbitrary network generalization of the shell model evolution and by rewiring the network each time from the original lattice in regular time intervals. This results in a mo…
We use momentum transfer arguments to predict the friction factor $f$ in two-dimensional turbulent soap-film flows with rough boundaries (an analogue of three-dimensional pipe flow) as a function of Reynolds number Re and roughness $r$, considering separately the inverse energy cascade and the forward enstrophy cascade. At intermediate Re, we predict a Blasius-like friction factor scaling of $f\propto\textrm{Re}^{-1/2}$ in flows dominated by the enstrophy cascade, distinct from the energy cascade scaling of $\textrm{Re}^{-1/4}$. For large Re, $f \sim r$ in the enstrophy-dominated case. We use …
Turbulence is characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. Recently, new experimental, numerical and theoretical works have revealed that many turbulent configurations deviate from the ideal 3D/2D isotropic cases characterized by the presence of a strictly direct/inverse energy cascade, respectively. We review recent works from a unified point of view and we present a classification of all known transfer mechanisms. Beside the classical cases of direct and inverse casca…
Inverse Energy Cascade in Turbulent Taylor-Couette Flows
Faculty of Mechanical Engineering, Zhejiang Sci-Tech University（浙江理工大学机械工程学院）；School of Mechanical Engineering, Zhejiang University of Water Resources and Electric Power（浙江水利水电学院机械工程学院）；School of Mechanical Engineering, Hangzhou Dianzi University（杭州电子科技大学机械工程学院）
One-point time-series measurements limit the observation of three-dimensional fully developed turbulence to one dimension. For one-dimensional models, like multiplicative branching processes, this implies that the energy flux from large to small scales is not conserved locally. This then renders the random weights used in the cascade curdling to be different from the multipliers obtained from a backward averaging procedure. The resulting multiplier distributions become solutions of a fix-point problem. With a further restoration of homogeneity, all observed correlations between multipliers in …
We develop a fundamental approach to a turbulent constitutive law for the 2D inverse cascade, based upon a convergent multi-scale gradient (MSG) expansion. To first order in gradients we find that the turbulent stress generated by small-scale eddies is proportional not to strain but instead to `skew-strain,' i.e. the strain tensor rotated by $45^\circ.$ The skew-strain from a given scale of motion makes no contribution to energy flux across eddies at that scale, so that the inverse cascade cannot be strongly scale-local. We show that this conclusion extends a result of Kraichnan for spectral t…
Dissipating of disorder quantum vortices in an annular two-dimensional Bose-Einstein condensate can form a macroscopic persistent flow of atoms. We propose a protocol to create persistent flow with high winding number based on a double concentric ring-shaped configuration. We find that a sudden geometric quench of the trap from single ring-shape into double concentric ring-shape will enhance the circulation flow in the outer ring-shaped region of the trap when the initial state of the condensate is with randomly distributed vortices of the same charge. The circulation flows that we created are…
We consider the dynamics of a spherically symmetric massless scalar field coupled to general relativity in Anti--de Sitter spacetime in the small-amplitude limit. Within the context of our previously developed two time framework (TTF) to study the leading self-gravitating effects, we demonstrate the existence of two new conserved quantities in addition to the known total energy $E$ of the modes: The particle number $N$ and Hamiltonian $H$ of our TTF system. Simultaneous conservation of $E$ and $N$ implies that weak turbulent processes undergo dual cascades (direct cascade of $E$ and inverse ca…
An inverse cascade - energy transfer to progressively larger scales - is a salient feature of two-dimensional turbulence. If the cascade reaches the system scale, it creates a coherent flow expected to have the largest available scale and conform with the symmetries of the domain. In a doubly periodic rectangle, the mean flow with zero total momentum was therefore believed to be unidirectional, with two jets along the short side; while for an aspect ratio close to unity, a vortex dipole was expected. Using direct numerical simulations, we show that in fact neither the box symmetry is respected…
This paper analyses the turbulent energy cascade from the perspective of statistical mechanics, and relates inter-scale energy fluxes to statistical irreversibility and information-entropy production. The microscopical reversibility of the energy cascade is tested by constructing a reversible 3D turbulent system using a dynamic model for the sub-grid stresses. This system, when reversed in time, develops a sustained inverse cascade towards the large scales, evidencing that the characterisation of the inertial energy cascade must consider the possibility of an inverse regime. This experiment is…
Is there really such a thing as weak turbulence? Here we analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence: the mutual information between modes is shown to be stationary and small in equilibrium but grows linearly with time in weak turbulence. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the …
We study numerically a Constantin-Lax-Majda-De Gregorio model generalized by Okamoto, Sakajo and Wunsch, which is a model of fluid turbulence in one dimension with an inviscid conservation law. In the presence of the viscosity and two types of the large-scale forcings, we show that turbulent cascade of the inviscid invariant, which is not limited to quadratic quantity, occurs and that properties of this model's turbulent state are related to singularity of the inviscid case by adopting standard tools of analyzing fluid turbulence.
To answer the question whether a cascade of energy exists or not in turbulence, we propose a set of correlation functions able to test if there is an irreversible transfert of energy, step by step, from large to small structures. These tests are applied to real Eulerian data of a turbulent velocity flow, taken in the wind grid tunnel of Modane, and also to a prototype model equation for wave turbulence. First we demonstrate the irreversible character of the flow by using multi-time correlation function at a given point of space. Moreover the unexpected behavior of the test function leads us to…
Inviscid invariants of flow equations are crucial in determining the direction of the turbulent energy cascade. In this work we investigate a variant of the three dimensional Navier-Stokes equations that shares exactly the same ideal invariants (energy and helicity) and the same symmetries (under rotations, reflexions and scale transforms) as the original equations. It is demonstrated that the examined system displays a change in the direction of the energy cascade when varying the value of a free parameter which controls the relative weights of the triadic interactions between different helic…
We develop an expansion of the turbulent stress tensor into a double series of contributions from different scales of motion and different orders of space-derivatives of velocity, a Multi-Scale Gradient (MSG) expansion. The expansion is proved to converge to the exact stress, as a consequence of the locality of cascade both in scale and in space. Simple estimates show, however, that the convergence rate may be slow for the expansion in spatial gradients of very small scales. Therefore, we develop an approximate expansion, based upon an assumption that similar or `coherent' contributions to tur…
The Ocean's turbulent energy cycle has a paradox; large-scale eddies under the control of Earth's rotation primarily transfer kinetic energy (KE) to larger scales via an inverse cascade, while a transfer to smaller scales is needed to accomplish dissipation. It has been argued, using numerical simulations, that fronts, waves and other turbulent structures can produce a forward cascade of KE toward dissipation scales. However, this forward cascade and its coexistence with known inverse cascade were not confirmed in observations. Here we present the first evidence of a dual KE cascade in the Oce…
We present a theoretical framework for plasma turbulence in astrophysical plasmas (solar wind, interstellar medium, galaxy clusters, accretion disks). The key assumptions are that the turbulence is anisotropic with respect to the mean magnetic field and frequencies are low compared to the ion cyclotron frequency. The energy injected at the outer scale scale has to be converted into heat, which ultimately cannot be done without collisions. A KINETIC CASCADE develops that brings the energy to collisional scales both in space and velocity. Its nature depends on the physics of plasma fluctuations.…
Turbulence theory is usually concerned with the statistical moments of the velocity or its fluctuations. One could also analyze the implicit probability distributions. This is the purview of information theory. Here we use information theory, specifically the conditional entropy, to analyze (quasi-)2D turbulence. We recast Richardson's "eddy hypothesis" that large eddies break up into small eddies in time in the language of information theory. In addition to confirming Richardson's idea, we find that self-similarity and turbulent length scales reappear naturally. Not surprisingly, we also find…
The Total Ozone Mapping Spectrometer (TOMS) database is subjected to an analysis in terms of the Karhunen-Loeve (KL) empirical eigenfunctions. The concentration variance spectrum is transformed into a wavenumber spectrum, $% E_c(k)$. In terms of wavenumber $E_c(k)$ is shown to be $O(k^{-2/3})$ in the inverse cascade regime, $O(k^{-2})$ in the enstrophy cascade regime with the spectral {\it knee} at the wavenumber of barotropic instability.The spectrum is related to known geophysical phenomena and shown to be consistent with physical dimensional reasoning for the problem. The appropriate Reynol…
