We investigate the transition from forward to inverse energy cascade in turbulent flows in thin layers, varying the functional form of the forcing and the thickness of the layer. We show that, as the forcing function becomes more three-dimensional, the inverse cascade is suppressed and the critical height hc, where the transition occurs, is decreased. We study the dependence of this critical height on a parameter r which measures the dimensionality of the forcing and thus construct a phase space diagram in the parameter space r-h. We discuss the effect of Reynolds number and domain size.
We demonstrate that like in the forward cascade of three dimensional turbulence that displays intermittency (lack of self-similarity) due to the concentration of energy dissipation in a small set of fractal dimension less than three, the inverse cascade of two-dimensional turbulence can also display lack of self-similarity and intermittency if the energy injection is constrained in a fractal set of dimension less than two. A series of numerical simulations of two dimensional turbulence are examined, using different forcing functions of the same forcing length-scale but different fractal dimens…
Using direct numerical simulation of hydrodynamic turbulence with helicity forcing applied at all scales, a near-maximum helical turbulent state is obtained, with an inverse energy cascade at scales larger than the energy forcing scale and a forward helicity cascade at scales smaller than the energy forcing scale. In contrast to previous studies using decimated triads, our simulations contain all possible triads. By computing the shell-to-shell energy fluxes, we show that the inverse energy cascade results from weakly non-local interactions among homochiral triads. Varying the helicity injecti…
We demonstrate that systems with a parameter-controlled inverse cascade can exhibit critical behavior for which at the critical value of the control parameter the inverse cascade stops. In the vicinity of such a critical point standard phenomenological estimates for the energy balance will fail since the energy flux towards large length scales becomes zero. We demonstrate these concepts using the computationally tractable model of two-dimensional magneto-hydrodynamics in a periodic box. In the absence of any external magnetic forcing the system reduces to hydrodynamic fluid turbulence with an …
This experimental study analyzes the relationship between the dimensionality of turbulence and the upscale or downscale nature of its energy transfers. We do so by forcing low-$Rm$ magnetohydrodynamic (MHD) turbulence in a confined channel, while precisely controlling its dimensionality by means of an externally applied magnetic field. We first identify a specific lengthscale $\hat{l}_\perp^c$ that separates smaller three-dimensional structures from larger quasi-two-dimensional ones. We then show that an inverse energy cascade of horizontal kinetic energy along horizontal scales is always obse…
We study the statistics of free-surface turbulence at large Reynolds numbers produced by direct numerical simulations in a fluid layer at different thickness with fixed characteristic forcing scale. We observe the production of a transient inverse cascade, with a duration which depends on the thickness of the layer, followed by a transition to three-dimensional turbulence initially produced close to the bottom, no-slip boundary. By switching off the forcing, we study the decaying turbulent regime and we find that it cannot be described by an exponential law. Our results show that boundary cond…
We first summarize briefly several properties concerning the dynamics of two-dimensional (2D) turbulence, with an emphasis on the inverse cascade of energy to the largest accessible scale of the system. In order to study a similar phenomenon in three-dimensional (3D) turbulence undergoing strong solid-body rotation, we test a previously developed Large Eddy Simulation (LES) model against a high-resolution direct numerical simulation of rotating turbulence on a grid of $3072^3$ points. We then describe new numerical results on the inverse energy cascade in rotating flows using this LES model an…
In this work we investigate, by means of direct numerical simulations, how rotation affects the bi-dimensionalization of a turbulent flow. We study a thin layer of fluid, forced by a two-dimensional forcing, within the framework of the "split cascade" in which the injected energy flows both to small scales (generating the direct cascade) and to large scale (to form the inverse cascade). It is shown that rotation reinforces the inverse cascade at the expense of the direct one, thus promoting bi-dimensionalization of the flow. This is achieved by a suppression of the enstrophy production at larg…
The notions of self-organised criticality (SOC) and turbulence are traditionally considered to be applicable to disjoint classes of phenomena. Nevertheless, scale-free burst statistics is a feature shared by turbulent as well as self-organised critical dynamics. It has also been suggested that another shared feature is universal non-gaussian probability density functions (PDFs) of global fluctuations. Here, we elucidate the unifying aspects through analysis of data from a laboratory dusty plasma monolayer. We compare analysis of experimental data with simulations of a two-dimensional (2D) many…
We demonstrate an inverse energy cascade in a minimal model of forced 2D quantum vortex turbulence. We simulate the Gross-Pitaevskii equation for a moving superfluid subject to forcing by a stationary grid of obstacle potentials, and damping by a stationary thermal cloud. The forcing injects large amounts of vortex energy into the system at the scale of a few healing lengths. A regime of forcing and damping is identified where vortex energy is efficiently transported to large length scales via an inverse energy cascade associated with the growth of clusters of same-circulation vortices, a Kolm…
Instabilities of fluid flows often generate turbulence. Using extensive direct numerical simulations, we study two-dimensional turbulence driven by a wavenumber-localised instability superposed on stochastic forcing, in contrast to previous studies of state-independent forcing. As the contribution of the instability forcing, measured by a parameter $γ$, increases, the system undergoes two transitions. For $γ$ below a first threshold, a regular large-scale vortex condensate forms. Above this threshold, shielded vortices (SVs) emerge within the condensate. At a second, larger value of $γ$, the c…
We report on the observation of surface gravity wave turbulence at scales larger than the forcing ones in a large basin. In addition to the downscale transfer usually reported in gravity wave turbulence, an upscale transfer is observed, interpreted as the inverse cascade of weak turbulence theory. A steady state is achieved when the inverse cascade reaches a scale in between the forcing wavelength and the basin size, but far from the latter. This inverse cascade saturation, which depends on the wave steepness, is probably due to the emergence of nonlinear dissipative structures such as sharp-c…
The Magneto-hydrodynamic (MHD) equations in the presence of a guiding magnetic field are investigated by means of direct numerical simulations. The basis of the investigation consists of 9 runs forced at the small scales. The results demonstrate that for a large enough uniform magnetic field the large scale flow behaves as a two dimensional (non-MHD) fluid exhibiting an inverse cascade of energy in the direction perpendicular to the magnetic field, while the small scales behave like a three dimensional MHD-fluid cascading the energy forwards. The amplitude of the inverse cascade is sensitive t…
A popular method of forcing the fluid in Direct Numerical Simulations of turbulence is to take the body force proportional to the projection of the velocity of the fluid onto its lowest Fourier modes, while keeping the injected external power constant. In this paper we perform a simple but rigorous analysis to establish bounds on the relationship between the energy dissipation rate and the resulting Reynolds number for this type of forcing. While this analysis cannot give detailed information of the energy spectrum, it does provide some indication of the balance of energy between the lower, di…
In this paper, the scaling property of the inverse energy cascade and forward enstrophy cascade of the vorticity filed $ω(x,y)$ in two-dimensional (2D) turbulence is analyzed. This is accomplished by applying a Hilbert-based technique, namely Hilbert-Huang Transform, to a vorticity field obtained from a $8192^2$ grid-points direct numerical simulation of the 2D turbulence with a forcing scale $k_f=100$ and an Ekman friction. The measured joint probability density function $p(C,k)$ of mode $C_i(x)$ of the vorticity $ω$ and instantaneous wavenumber $k(x)$ is separated by the forcing scale $k_f$ …
Dynamical and statistical properties of large scales in turbulence
Laboratoire de Physique de l’Ecole Normale Supérieure, CNRS, PSL Research University, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，法国国家科学研究中心，PSL研究大学，索邦大学，巴黎西岱大学）
A linear dynamical model for the development of the turbulent energy cascade was introduced in Apolinário \emph{et al} (J. Stat. Phys. \textbf{186}, 15 (2022)). This partial differential equation, randomly stirred by a forcing term which is smooth in space and delta-correlated in time, was shown to converge at infinite time towards a state of finite variance, without the aid of viscosity. Furthermore, the spatial profile of its solution gets rough, with the same regularity as a fractional Gaussian field. We here focus on the temporal behavior and derive explicit asymptotic predictions for the …
We perform experiments to study the inverse cascade regime of gravity wave turbulence on the surface of a fluid. Surface waves are forced at an intermediate scale corresponding to the gravity-capillary wavelength. In response to this forcing, waves at larger scales are observed. The spectrum of their amplitudes exhibits a frequency-power law at high enough forcing. Both observations are ascribed to the upscale wave action transfers of gravity wave turbulence. The spectrum exponent is close to the value predicted by the weak turbulence theory. The spectrum amplitude is found to scale linearly w…
We investigate the critical transition from an inverse cascade of energy to a forward energy cascade in a two-dimensional magneto-hydrodynamic flow as the ratio of magnetic to mechanical forcing amplitude is varied. It is found that the critical transition is the result of two competing processes. The first process is due to hydrodynamic interactions, cascades the energy to the large scales. The second process couples small scale magnetic fields to large scale flows transferring the energy back to the small scales via a non-local mechanism. At marginality the two cascades are both present and …
Here we report the first evidence of the inverse energy cascade in a flow dominated by 3D motions. Experiments are performed in thick fluid layers where turbulence is driven electromagnetically. It is shown that if the free surface of the layer is not perturbed, the top part of the layer behaves as quasi-2D and supports the inverse energy cascade regardless of the layer thickness. Well below the surface the cascade survives even in the presence of strong 3D eddies developing when the layer depth exceeds half the forcing scale. In a bounded flow at low bottom dissipation, the inverse energy cas…
Classically, large-scale forced turbulence is characterized by a transfer of energy from large to small scales via nonlinear interactions. We have investigated the changes in this energy transfer process in broad-band forced turbulence where an additional perturbation of flow at smaller scales is introduced. The modulation of the energy dynamics via the introduction of forcing at smaller scales occurs not only in the forced region but also in a broad range of length-scales outside the forced bands due to non-local triad interactions. Broad-band forcing changes the energy distribution and energ…
We solve the Navier-Stokes equations with two simultaneous forcings. One forcing is applied at a given large-scale and it injects energy. The other forcing is applied at all scales belonging to the inertial range and it injects helicity. In this way we can vary the degree of turbulence helicity from non helical to maximally helical. We find that increasing the rate of helicity injection does not change the energy flux. On the other hand the level of total energy is strongly increased and the energy spectrum gets steeper. The energy spectrum spans from a Kolmogorov scaling law $k^{-5/3}$ for a …
The restoration of symmetries is one of the most fascinating properties of turbulence. We report a study of the emergence of isotropy in the Gross-Pitaevskii model with anisotropic forcing. Inspired by recent experiments, we study the dynamics of a Bose-Einstein condensate in a cylindrical box driven along the symmetry axis of the trap by a spatially uniform force. We introduce a measure of anisotropy $A(k,t)$ defined on the momentum distributions $n(\boldsymbol{k},t)$, and study the evolution of $A(k,t)$ and $n(\boldsymbol{k},t)$ as turbulence proceeds. As the system reaches a steady state, t…
MHD turbulence at low Magnetic Reynolds number is experimentally investigated by studying a liquid metal flow in a cubic domain. We focus on the mechanisms that determine whether the flow is quasi-2D, 3D or in any intermediate state. To this end, forcing is applied by injecting a DC current $I$ through one wall of the cube only, to drive vortices spinning along the magnetic field. Depending on the intensity of the externally applied magnetic field, these vortices extend part or all of the way through the cube. Driving the flow in this way allows us to precisely control not only the forcing int…
Transition from a split to a forward kinetic energy cascade system is explored in the context of rotating turbulence using direct numerical simulations with a three-dimensional isotropic random force uncorrelated with the velocity field. Our parametric study covers confinement effects in large aspect ratio domains and a broad range of rotation rates. Results indicate that for fixed geometrical dimensions the Rossby number acts as a control parameter, whereas for a fixed Rossby number the product of the domain size along the rotation axis and forcing wavenumber governs the amount of energy that…
Statistical features of homogeneous, isotropic, two-dimensional turbulence is discussed on the basis of a set of direct numerical simulations up to the unprecedented resolution $32768^2$. By forcing the system at intermediate scales, narrow but clear inertial ranges develop both for the inverse and for direct cascades where the two Kolmogorov laws for structure functions are, for the first time, simultaneously observed. The inverse cascade spectrum is found to be consistent with Kolmogorov-Kraichnan prediction and is robust with respect the presence of an enstrophy flux. The direct cascade is …
The effects of large scale mechanical forcing on the dynamics of rotating turbulent flows are studied by means of numerical simulations, varying systematically the nature of the mechanical force in time. We demonstrate that the statistically stationary solutions of these flows depend on the nature of the forcing mechanism. Rapidly enough rotating flows with a forcing that has a persistent direction relatively to the axis of rotation bifurcate from a non-helical state to a helical state despite the fact that the forcing is non-helical. We find that the nature of the mechanical force in time and…
Rapidly rotating turbulent flow is characterized by the emergence of columnar structures that are representative of quasi-two dimensional behavior of the flow. It is known that when energy is injected into the fluid at an intermediate scale $L_f$, it cascades towards smaller as well as larger scales. In this paper we analyze the flow in the \textit{inverse cascade} range at a small but fixed Rossby number, {$\mathcal{R}o_f \approx 0.05$}. Several {numerical simulations with} helical and non-helical forcing functions are considered in periodic boxes with unit aspect ratio. In order to resolve t…
We study the energy transfer properties of three dimensional homogeneous and isotropic turbulence where the non-linear transfer is altered in a way that helicity is made sign-definite, say positive. In this framework, known as homochiral turbulence, an adapted eddy-damped quasi-normal Markovian (EDQNM) closure is derived to analyze the dynamics at very large Reynolds numbers, of order $10^5$ based on the Taylor scale. In agreement with previous findings, an inverse cascade of energy with a kinetic energy spectrum like $\propto k^{-5/3}$ is found for scales larger than the forcing one. Conjoint…
In this paper, we present an overview of concepts and data concerning inverse cascades of excitation towards scales larger than the forcing scale in a variety of contexts, from two-dimensional fluids and wave turbulence, to geophysical flows in the presence of rotation and stratification. We briefly discuss the role of anisotropy in the occurrence and properties of such cascades. We then show that the cascade of some invariant, for example the total energy, may be transferred through nonlinear interactions to both the small scales and the large scales, with in each case a constant flux. This i…
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.
When the symmetries of homogenous isotropic turbulent flows are broken, different sets of modes with different physical roles emerge. In particular, choosing a forcing which puts more weight on one or the other of these sets may result in different statistics for the energy transfers. We use the general method of computing a partition function restricted to a portion of phase space to study analytically these different statistics. We illustrate this method in the case of parity symmetry breaking, measured by helicity. It is shown that when helicity is sign definite at all scales, an inverse ca…
We briefly review helicity dynamics, inverse and bi-directional cascades in fluid and magnetohydrodynamic (MHD) turbulence, with an emphasis on the latter. The energy of a turbulent system, an invariant in the non-dissipative case, is transferred to small scales through nonlinear mode coupling. Fifty years ago, it was realized that, for a two-dimensional fluid, energy cascades instead to larger scales, and so does magnetic helicity in three-dimensional MHD. However, evidence obtained recently indicates that in fact, for a range of governing parameters, there are systems for which their ideal i…
A non-local cascade model for anisotropic MHD turbulence in the presence of a guiding magnetic field is proposed. The model takes into account that (a) energy cascades in an anisotropic manner and as a result a different estimate for the cascade rate in the direction parallel and perpendicular to the guiding field is made. (b) the interactions that result in the cascade are between different scales. Eddies with wave numbers $k_\|$ and $k_\perp$ interact with eddies with wave numbers $q_\|,q_\perp$ such that a resonance condition between the wave numbers $q_\|,q_\perp$ and $k_\|,k_\perp$ holds.…
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.
We establish Lagrangian formulae for energy conservation anomalies involving the discrepancy between short-time two-particle dispersion forward and backward in time. These results are facilitated by a rigorous version of the Ott-Mann-Gawȩdzki relation, sometimes described as a "Lagrangian analogue of the 4/5ths law". In particular, we prove that for any space-time $L^3$ weak solution of the Euler equations, the Lagrangian forward/backward dispersion measure matches on to the energy defect in the sense of distributions. For strong limits of $d\geq3$ dimensional Navier-Stokes solutions the defec…
The dimensionality of turbulence in fluid layers determines their properties. We study electromagnetically driven flows in finite depth fluid layers and show that eddy viscosity, which appears as a result of three-dimensional motions, leads to increased bottom damping. The anomaly coefficient, which characterizes the deviation of damping from the one derived using a quasi-two-dimensional model, can be used as a measure of the flow dimensionality. Experiments in turbulent layers show that when the anomaly coefficient becomes high, the turbulent inverse energy cascade is suppressed. In the oppos…
We study the influence of the energy spectrum on the extent of the cascade range of a passive tracer in turbulent flows. The interesting cases are when there are two different spectra over the potential range of the tracer cascade (in 2D when the tracer forcing is in the inverse energy cascade range, and in 3D when the Schmidt number Sc is large). The extent of the tracer cascade range is then limited by the width of the range for the shallower of the two energy spectra. Nevertheless, we show that in dimension $d=2,3$ the tracer cascade range extends (up to a logarithm) to $κ_{d\text{D}}^{p}$,…
We investigate the impact of a solid-body rotation $Ω_0$ on the large-scale dynamics of an incompressible magnetohydrodynamic turbulent flow in presence of a background magnetic field $\bf B_0$ and at low Rossby number. Three-dimensional direct numerical simulations are performed in a periodic box, at unit magnetic Prandtl number and with a forcing at intermediate wavenumber $k_f=20$. When $Ω_0$ is aligned with $\bf B_0$ (i.e. $θ\equiv \widehat{\left(Ω_{0}, \bf B_0 \right)} = 0$), inverse transfer is found for the magnetic spectrum at $k<k_f$. This transfer is stronger when the forcing excites…
Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: applications to fluid turbulence and generation of fractional Gaussian fields
In this paper we study analytically the viscous `sabra' shell model of energy turbulent cascade. We prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom, specifically: a finite dimensional global attractor and globally invariant inertial manifolds. Moreover, we establish the existence of exponentially decaying energy dissipation range for the sufficiently smooth forcing.
The energy pathways from propagating internal waves to the scales of irreversible mixing in the ocean are not fully described. In the ocean interior, the triadic resonant instability is an intrinsic destabilization process that may enhance the energy cascade away from topographies. The present study focuses on the integrated impact of mixing processes induced by a propagative normal mode-1 over long term experiments in an idealised setup. The internal wave dynamics and the evolution of the density profile are followed using the light attenuation technique. Diagnostics of the turbulent diffusiv…
We present results from two 1536^3 direct numerical simulations of rotating turbulence where both energy and helicity are injected into the flow by an external forcing. The dual cascade of energy and helicity towards smaller scales observed in isotropic and homogeneous turbulence is broken in the presence of rotation, with the development of an inverse cascade of energy now coexisting with direct cascades of energy and helicity. In the direct cascade range, the flux of helicity dominates over that of energy at low Rossby number. These cascades have several consequences for the statistics of th…
We generalize Kirchhoff's point vortex model of two-dimensional fluid motion to a rotor model which exhibits an inverse cascade by the formation of rotor clusters. A rotor is composed of two vortices with like-signed circulations glued together by an overdamped spring. The model is motivated by a treatment of the vorticity equation representing the vorticity field as a superposition of vortices with elliptic Gaussian shapes of variable widths, augmented by a suitable forcing mechanism. The rotor model opens up the way to discuss the energy transport in the inverse cascade on the basis of dynam…
Two dimensional turbulence has a remarkable tendency to self-organize into large, coherent structures, forming a mean flow. The purpose of this paper is to elucidate how these structures are sustained, and what determines them and the fluctuations around them. A recent theory for the mean flow will be reviewed. The theory assumes turbulence is excited by a forcing supported on small scales, and uses a linear shear model to relate the turbulent momentum flux to the mean shear rate. Extending the theory, it will be shown here that the relation between the momentum flux and mean shear is valid, a…
Systematic Evaluation of Stencil Configuration, Forcing Scheme, and Resolution Effects in the Stratified Taylor--Green Vortex: A Lattice Boltzmann Study
School of Atmospheric Sciences, Sun Yat-sen University（中山大学大气科学学院）
We investigate a model of thin layer turbulence that follows the evolution of the two-dimensional motions ${\bf u}_{_{2D}} (x,y)$ along the horizontal directions $(x,y)$ coupled to a single Fourier mode along the vertical direction ($z$) of the form ${\bf u}_q (x, y, z)=[v_x(x,y) \sin(qz), v_y(x,y)\sin(qz), v_z(x,y)\cos(qz)\, ]$, reducing thus the system to two coupled, two-dimensional equations. The reduced dimensionality of the model allows a thorough investigation of the transition from a forward to an inverse cascade of energy as the thickness of the layer $H=π/q$ is varied. Starting from …
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 dissipat…
Stratified flows forced by internal waves similar to those obtained in the Coriolis platform (LEGI, Grenoble, France) \cite{Savaro2020} are studied by pseudospectral triply-periodic simulations. The experimental forcing mechanism consisting in large oscillating vertical panels is mimicked by a penalization method. The analysis of temporal and spatiotemporal spectra reveals that the flow for the strongest forcing in the experiments is composed of two superposed large and quasi-steady horizontal vortices, of internal waves in box modes and of much weaker waves outside the modes. Spatial spectra …
It is well known that an inverse turbulent cascade in a finite ($2 π\times 2 π$) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random in space, and had a correlation length $l_f = 2π/k_f$ with $k_f$ ranging from $100$ to $12.5$. Previously, it was found that in the asymptotic limit of small-scale forcing, the vorticity exhibits the power-law behavior $Ω(r) = (3 ε/α)^{1/2} r^{-1}$, where $r$ is…
In two dimensional turbulence, vortex thinning process is one of the attractive mechanism to explain inverse energy cascade in terms of vortex dynamics. By direct numerical simulation to the two-dimensional Navier-Stokes equations with small-scale forcing and large-scale damping, Xiao-Wan-Chen-Eyink (2009) found an evidence that inverse energy cascade may proceed with the vortex thinning mechanism. The aim of this paper is to analyze the vortex-thinning mechanism mathematically (using the incompressible Euler equations), and give a mathematical evidence that large-scale vorticity gains energy …
We investigate experimentally three-dimensional (3D) hydrodynamic turbulence at scales larger than the forcing scale. We manage to perform a scale separation between the forcing scale and the container size by injecting energy into the fluid using centimetric magnetic particles. We measure the statistics of the fluid velocity field at scales larger than the forcing scale (energy spectra, velocity distributions, and energy flux spectrum). In particular, we show that the large-scale dynamics are in statistical equilibrium and can be described with an effective temperature, although not isolated …
