Two-dimensional Turbulence Theory Research Articles

Self-similarity and the direct (enstrophy) cascade in forced two-dimensional fluid turbulence

In the absence of large-scale coherent structures, a widely used statistical theory of two-dimensional turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha ={-3}$ , in the inertial range associated with the direct cascade. A power-law scaling is also observed in the presence of coherent structures, but the scaling exponent becomes fractal and often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory that sheds new light on the relationship between the spatial and temporal structure of the large-scale flow and the scaling of small-scale structures representing filamentary vorticity. Specifically, we find hyperbolic regions of the large-scale flow to play a key role in the flux of enstrophy between scales. Small-scale vorticity in these regions can be described by dynamically self-similar solutions of the Euler equation, which explains the power-law scaling. Furthermore, we find that correlations between different hyperbolic regions are responsible for the emergence of fractal scaling exponents.

New numerical methods for calculating statistical equilibria of two-dimensional turbulent flows, strictly based on the Miller–Robert–Sommeria theory

New numerical methods are proposed for the mixing entropy maximization problem in the context of Miller–Robert–Sommeria’s (MRS) statistical mechanics theory of two-dimensional turbulence, particularly in the case of spherical geometry. Two of the methods are for the canonical problem; the other is for the microcanonical problem. The methods are based on the original MRS theory and thus take into account all Casimir invariants. Compared to the methods proposed in previous studies, our new methods make it easier to detect multiple statistical equilibria and to search for solutions with broken zonal symmetry. The methods are applied to a zonally symmetric initial vorticity distribution which is barotropically unstable. Two statistical equilibria are obtained, one of which has a wave-like structure with zonal wavenumber 1, and the other has a wave-like structure with zonal wavenumber 2. While the former is the maximum point of the mixing entropy, the wavenumber 2 structure of the latter is nearly the same as the structure that appears in the end state of the time integration of the vorticity equation. The new methods allow for efficient computation of statistical equilibria for initial vorticity distributions consisting of many levels of vorticity patches without losing information about all the conserved quantities. This means that the statistical equilibria can be obtained from an arbitrary initial vorticity distribution, which allows for the application of statistical mechanics to interpret a wide variety of flow patterns appearing in geophysical fluids.

Assessment of relative dispersion in the Gulf of Tonkin using numerical modeling and HF radar observations of surface currents

Particle pair statistics from synthetic drifter trajectories reconstructed from realistic, high-resolution numerical simulations (SYMPHONIE model) and HF radar velocity measurements are used to investigate the dispersion properties in the Gulf of Tonkin (GoT). This study takes an approach based on two-particles statistics providing the relative dispersion, relative diffusivity and Finite Size Lyapunov Exponent (FSLE) estimates. In the GoT, the relative dispersion follows the predictions from the theory of two-dimensional turbulence with two inertial subranges identified in the kinetic energy spectrum with the spectral slopes −5/3 and −3. The time evolution of dispersion shows an exponential growth during 5–8 days, followed by a power law regime during the next 6–20 days. Fixed-length indicators from the relative diffusivity and the FSLE reveal a local dispersion at large and intermediate scales (above Rossby radius of deformation) and non-local dispersion in sub-mesoscale range (below Rossby radius of deformation). The effect of river runoff on the local hydrodynamics and dispersion processes is assessed using the numerical model simulations without river discharge. The results show that in the model, the river plume, when present, highly impacts the Lagrangian statistics. High gradients of buoyancy reinforce the sub-mesoscale circulation in a large region along the Vietnamese coast and modify the scales and intensity of turbulent dispersion. However, a clear change of dispersion regime in the sub-mesoscale range is not identified, suggesting that the mesoscale circulation in the GoT largely governs particle spreading even at small scale.

Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions

We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Lévy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Lévy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence.

Fujiwhara interaction of tropical cyclone scale vortices using a weighted residual collocation method

The fundamental interaction between tropical cyclones was investigated through a series of water tank experiements by Fujiwhara [20, 21, 22]. However, a complete understanding of tropical cyclones remains an open research challenge although there have been numerous investigations through measurments with aircrafts/satellites, as well as with numerical simulations. This article presents a computational model for simulating the interaction between cyclones. The proposed numerical method is presented briefly, where the time integration is performed by projecting the discrete system onto a Krylov subspace. The method filters the large scale fluid dynamics using a multiresolution approximation, and the unresolved dynamics is modeled with a Smagorinsky type subgrid scale parameterization scheme. Numerical experiments with Fujiwhara interactions are considered to verify modeling accuracy. An excellent agreement between the present simulation and a reference simulation at Re = 5000 has been demonstrated. At Re = 37440, the kinetic energy of cyclones is seen consolidated into larger scales with concurrent enstrophy cascade, suggesting a steady increase of energy containing scales, a phenomena that is typical in two-dimensional turbulence theory. The primary results of this article suggest a novel avenue for addressing some of the computational challenges of mesoscale atmospheric circulations.

Revisiting two-dimensional turbulence by Lattice Boltzmann Method

In this paper, Lattice Boltzmann Method (LBM) is used to investigate the freely-decaying two-dimensional (2D) turbulence. The main purpose focuses on recent mathematical results about a central assumption in the theory of two-dimensional turbulence proposed by Batchelor. The numerical results found an approximate exponent of energy spectrum E(k) ∼ k−3.5 at the high mesh resolutions and an approximate decaying exponent of enstrophy ‹ω› ∼ T−2/3. In addition, an approximate exponent of integral length scale l ∼ T−1/2 is observed. These results validate the recent new results of 2D turbulence and are different from the results derived by Batchelor's 2D model.

On the properties of decaying two-dimensional turbulence: Characteristic features of the spectra and coherent structures

The main results of the theory of two-dimensional turbulence are presented. The conventional approach based on the Karman-Howarth equation is used to describe anomalous properties (sharply differing from those inherent in usual three-dimensional turbulence) of two-dimensional turbulent motion of an incompressible fluid, in particular, energy transfer across the spectrum from larger toward smaller wave numbers and the formation of coherent structures, whose origin is associated with the fact that the spectrum attains the form of a δ-function. A uniform method is proposed for obtaining self-similar spectra of two-dimensional turbulence in inertial ranges. The problem of turbulent diffusion of a passive tracer in the two-dimensional case is also considered. It turns out that the corresponding quantities, as well as those related to the dynamic characteristics of motion, can exhibit anomalous properties under certain conditions. The relationship between the results and experimental data is discussed. In particular, the experimentally observed inversion of the spectra of two-dimensional turbulence in the atmosphere is explained.

Revisiting Batchelor's theory of two-dimensional turbulence

Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a χ2/3k−1 enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation χ in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes.We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing χ in the limit Re → ∞. Our proposal is supported by high Reynolds number simulations which confirm that χ decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy 〈ω2〉/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing χ, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum χ2/3k−1 with 〈ω2〉 k−1 (ln Re)−1).

Decaying turbulence in soap films: energy and enstrophy evolution

This experimental study of quasi-two-dimensional grid turbulence in gravity-driven soap-film flow focuses on the differences between the behavior of the flow and the theoretical picture of two-dimensional turbulence. A previously unattainable quality of velocity field acquisition facilitates simultaneous measurement of velocity field features in the scale range spanning over two orders of magnitude. The highly-resolved flow field data are analyzed statistically in terms of velocity structure functions, as well as energy and enstrophy averages at different downstream positions. We find the rate of decay of these averages to be quantifiably greater than the predictions of the two-dimensional turbulence theory. This increased decay is likely to be the manifestation of the extra dissipation mechanism present in soap-film flows and prominent on the larger scales—air drag. The structure function analysis confirms the notion.

Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit

Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re. Here, by assuming power-law spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as Re → ∞, the dissipation approaches zero no slower than (lnRe) -1/2 . The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation - but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

Multiresolution analysis for 2D turbulence. Part 1: Wavelets vs cosine packets, a comparative study

The widely accepted theory of two-dimensional turbulence predicts a direct downscale enstrophy cascade with an energy spectrum behaving like $k^{-3}$ and an inverse upscale energy cascade with a $k^{-5/3}$ decay. Nevertheless, this theory is in fact an idealization valid only in an infinite domain in the limit of infinite Reynolds numbers, and is almost impossible to reproduce numerically. A more complete theoretical framework for the two-dimensional turbulence has been recently proposed by Tung et al . This theory seems to be more consistent with experimental observations, and numerical simulations than the classical one developed by Kraichnan, Leith and Batchelor.Multiresolution methods like the wavelet packets or the cosine packets, well known in signal decomposition, can be used for the 2D turbulence analysis. Wavelet or cosine decompositions are more and more used in physical applications and in particular in fluid mechanics. Following the works of M. Farge et al , we present a numerical and qualitative study of a two-dimensional turbulence fluid using these methods. The decompositions allow to separate the fluid in two parts which are analyzed and the corresponding energy spectra are computed. In the first part of this paper, the methods are presented and the numerical results are briefly compared to the theoretical spectra predicted by the both theories. A more detailed study, using only wavelet packets decompositions and based on numerical and experimental data, will be carried out and the results will be reported in the second part of the paper. A tentative of physical interpretation of the different components of the flow will be also proposed.

On the double cascades of energy and enstrophy in two dimensional turbulence. Part 2. Approach to the KLB limit and interpretation of experimental evidence

This paper is concerned with three interrelated issues on our proposal of double cascades intended to serve as a more realistic theory of two-dimensional turbulence. We begin by examining the approach to the KLB limit. We present improved proofs of the result by Fjortoft. We also explain why in that limit the subleading downscale energy cascade and upscale enstrophy cascade are hidden in the energy spectrum. Then we review the experimental evidence from numerical simulations concerning the realizability of the energy and enstrophy cascade. The inverse energy cascade is found to be affected by the presense of a particular solution, and the downscale enstrophy cascade is not robust. In particular, while it is possible to have either the upscale range or the downscale range with suitable choice of dissipations, the dual cascade of KLB does not appear to be realizable, not even approximately. Finally, we amplify the hypothesis that the energy spectrum of the atmosphere reflects a combined downscale cascade of energy and enstrophy. The possibility of the downscale helicity cascade is also considered.

Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets.

The Kolmogorov-Batchelor-Kraichnan (KBK) theory of two-dimensional turbulence is generalized for turbulence on the surface of a rotating sphere. The energy spectrum develops considerable anisotropy; a steep -5 slope emerges in the zonal direction, while in all others the classical KBK scaling prevails. This flow regime in robust steady state is reproduced in simulations with linear drag. The conditions favorable for this regime may be common for giant planets' atmospheric circulations; the same steep spectra are found in their observed zonal velocity profiles and utilized to explain their basic characteristics.

MHD turbulence measurements in a sodium channel flow exposed to a transverse magnetic field

The influence of a static transverse magnetic field on a turbulent sodium flow in a channel with a rectangular cross-section is investigated. The turbulence has been forced by mechanical means employing a grid of cylindrical bars. Electric potential probes have been used to determine the longitudinal component of the liquid velocity. The experiments cover a wide range of the non-dimensional parameters Hartmann number ( Ha⩽3000) and the magnetic interaction parameter ( N⩽800). Measurements of the turbulence intensity as a function of the magnetic interaction number N will be presented and discussed. The measured spectra are essentially different from those predicted according to the theory of two-dimensional turbulence.

WKB theory for rapid distortion of inhomogeneous turbulence

A WKB method is used to extend RDT (rapid distortion theory) to initially inhomogeneous turbulence and unsteady mean flows. The WKB equations describe turbulence wavepackets which are transported by the mean velocity and have wavenumbers which evolve due to the mean strain. The turbulence also modifies the mean flow and generates large-scale vorticity via the averaged Reynolds stress tensor. The theory is applied to Taylor's four-roller flow in order to explain the experimentally observed reduction in the mean strain. The strain reduction occurs due to the formation of a large-scale vortex quadrupole structure from the turbulent spot confined by the four rollers. Both turbulence inhomogeneity and three-dimensionality are shown to be important for this effect. If the initially isotropic turbulence is either homogeneous in space or two-dimensional, it has no effect on the large-scale strain. Furthermore, the turbulent kinetic energy is conserved in the two-dimensional case, which has important consequences for the theory of two-dimensional turbulence. The analytical and numerical results presented here are in good qualitative agreement with experiment.

Spatializing Random Measures: Doubly Indexed Processes and the Large Deviation Principle

The main theorem is the large deviation principle for the doubly indexed sequence of random measures $$W_{r,q}(dx \times dy) \doteq \theta(dx) \otimes \textstyle\sum\limits_{k=1}^{2^r} 1_{D_{r,k}}(x) L_{q,k} (dy).$$ Here $\theta$ is a probability measure on a Polish space $\mathscr{X},\{D_{r,k}k=1,\ldots,2^r\}$ is a dyadic partition of $\mathscr{X}$ (hence the use of $2^r$ summands) satisfying $\theta\{D_{r,k}\}= 1/2^r$ and $L_{q,1}L_{q,2},\ldots , L_{q,2_r}$ is an independent, identically distributed sequesnce of random probability measures on a Ploish space $ \mathscr{Y}$ such that $\{L_{q,k}q\in \mathbb{N}\}$ satisfies the large deviation principle with a convex rate function. A number of related asymptotic results are also derived. The random measures $W_{ r,q}$ have important applications to the statistical mechanics of turbulence. In a companion paper, the large deviation principle presented here is used to give a rigorous derivation of maximum entropy principles arising in the well-known Miller–Robert theory of two-dimensional turbulence as well as in a modification of that theory recently proposed by Turkington.

The kinetic theory of point vortices

The statistical dynamics of a set of point vortices is investigated. The method of N.N. Bogolubov (1965) is employed and the BBGKY hierarchy is derived. It is shown that most physical quantities of interest, e.g. the velocity field, the pressure, and the interaction energy, are given in terms of a few reduced distribution functions. We also show how the introduction of the vortex dipole chaos assumption permits one to derive a kinetic equation for a ‘gas’ of vortex dipoles. The analysis of the observations of two-dimensional turbulent flows, made by experiments and direct numerical simulations, reveals that the kinetic theory of point vortex dipoles can be used, in conjunction with a scaling theory, to provide a hybrid theory of two-dimensional turbulence with no adjustable parameters.

Renormalization group in the theory of two-dimensional turbulence: Instability of the fixed point with respect to weak anisotropy

A statistical model of fully developed turbulence in two-dimensional space is considered by means of the renormalization group method in the weak anisotropy approximation. It is shown that the corresponding fixed point of the renormalization group equations is not infrared stable. Hence, the weak, anisotropy approximation is not valid for describing two-dimensional turbulence.

A critical review on the dynamics of Jovian atmospheres.

Some basic theoretical issues of the Jovian atmospheric dynamics are discussed. Those include the depth of the motions, thermal convection both with deep and shallow configurations, the two-dimensional turbulence theory and its relevance. The refractive index analysis is suggested to interpret the coherent structures like Jupiter's Great Red Spot in terms of Charney-Drazin constraint based on the observations. A possibility of Jovian data assimilation is proposed as a future direction.

Spectra and energy transfer in stably stratified turbulence

The influence of stabilizing buoyancy forces on the spectral characteristics and spectral energy transfer of grid-generated turbulence was studied in a ten-layer closed-loop stratified water channel. The results are compared to the limiting ideal cases of the three-dimensional turbulence and two-dimensional turbulence theories. The velocity power spectra evolve from a classical isotropic shape to a shape of almost k−2 after the suppression of the net vertical mixing. This final spectral shape is rather different from the k−3 to k−4 predicted by the theory of two-dimensional turbulence and could result from the interaction between small-scale internal waves and quasi-two-dimensional turbulent structures as well as some Doppler shift of advected waves. Several lengthscales are derived from the cospectra of the vertical velocity and density fluctuations and compared with the buoyancy, overturning and viscous lengthscales measured in previous studies, e.g. Stillinger, Helland & Van Atta (1983) and Itsweire, Helland & Van Atta (1986). The smallest turbulent scale, defined when the buoyancy flux goes to zero, can be related to the peak of the cospectra of the buoyancy flux. This new relationship can be used to provide a measure of the smallest turbulent scale in cases where the buoyancy flux never goes to zero, i.e. a growing turbulent stratified shear flow. Finally, the one-dimensional energy transfer term computed from the bispectra shows evidence of a reverse energy cascade from the small scales to the large scales far from the grid where buoyancy forces dominate inertial forces. The observed reverse energy transfer could be produced by the development of quasi-two-dimensional eddies as the original three-dimensional turbulence collapses.