Energy Cascade Process Research Articles

A dynamical model for turbulence. V. The effect of rotation

We study turbulence undergoing rapid rotation. We show that there exist two different regimes divided by the new number N=K/νΩ. N<1: rotation is so strong that it suppresses the energy cascade process and no inertial regime can develop. In the freely decaying case, only viscosity operates: an initially isotropic three-dimensional (3-D) turbulence remains so and never tends toward a 2-D state. N>1: the energy cascade, though inhibited, allows an inertial regime in which 3-D and a quasi-2-D state exist in equilibrium. The latter is restricted to a narrow band of values δkz∼kΩ−1k2, where kΩ=(Ω3ε−1)1/2. For k>kΩ, the Kolmogorov spectrum sets in, while for k<kΩ, the spectrum exhibits a new form, E(k)∼(εΩ)1/2k−2. A freely decaying turbulence within a time scale τΩ∼(Ωε−1)1/2L tends to a 2-D–3-D regime: K∼t−n while horizontal length scales grow as tm, where (m,n) are half the values of the Ω=0 case. As for the vertical length scales, one grows as the horizontal scales, while the other grows much faster, demonstrating the existence of the 2-D mode. The nonlinear interactions, though weakened, are the main cause of two-dimensionalization, which, however, is not of the Proudman–Taylor type since the latter requires negligible nonlinearities. We also derive the dynamic equation for the dissipation rate ε. All the results are shown to be in agreement with numerical simulation and experimental data.

On the spectrum of thermospheric gravity waves observed by the Super Dual Auroral Radar Network

The spectrum of acoustic gravity waves in the thermosphere observed using the Super Dual Auroral Radar Netowrk HF radar network is presented and various features are discussed. HF radars are sensitive to medium‐scale gravity waves in the thermosphere that interact with the F‐region ionization. The interaction of waves in the neutral atmosphere with the ionization is discussed to determine the relationship of the observed spectrum to the gravity wave spectrum. The observations showed that (1) the observed spectrum shows peaks at frequencies that correspond to quasi‐monochromatic wave packets observed in the time series; (2) the spectrum often shows multiple peaks at harmonic frequencies; and (3) the spectrum nearly always shows a background level with a power law decrease having a slope of −5/3. It was also found that the overall level of the spectrum was elevated when gravity waves were present, possibly indicating an energy cascade process from the quasi‐monochromatic wave to the power law background.

On a discrete dynamical model for local turbulence

Works towards an explanation of the observed experimental deviation of K41 exponent law are generally based on an energy cascade process. An alternative approach, based on the analysis of turbulent sequences, allows us to construct a sequence model where this deviation has a dynamical origin. We finally show why we were forced to go beyond the markovian model and incorporate into the model the frequent long flights, an intermittent effect also present in the experimental data.

Fine-Scale Structure of High-Reynolds-Number Turbulence.

Fine-scale structure and local isotropy were experimentally investigated for turbulence fields of turbulence Reynolds numbers of Rλ=80∼393. Waveform analysis was performed for velocity fluctuations independently in the low wavenumber range, the inertial subrange and the viscous dissipation range and also for the time derivative of the velocity fluctuations. For Rλ>200, the turbulence fluctuations were random in the low wavenumber range but intermittent in the other two wavenumber ranges. Probability density analysis showed that the flatness factor was about 3 in the low wavenumber range and the intermittency was stronger in the viscous dissipation range that in the inertial subrange. The flatness factor of the time derivative of the velocity fluctuations monotonically increased with Rλ and reached about 7.08 at Rλ=393. We introduced the anisotropic tensor spectrum. The scale of the maximum isotropic eddy, lG, determined the upper limit of the inertial subrange where the turbulence eddies smaller than lG attained local isotropy and universal equilibrium in the energy cascade process.

Lagrangian drifter paths and length scales in the tropical Pacific warm pool from 1990 to 1991: with application of fractal techniques

Abstract. This paper presents an analysis of WOCE/TOGA surface drifter paths and its interpretation in conjunction with the west Pacific warm pool water motion. Our interest here lies in the existence of scale invariance in the observed data sets. The analysis proceeds by detecting scale invariance in the drifter paths data, and interpreting the invariance in terms of the statistical second order moment. The range of constant scaling exponent was found to be between 5 days and 10 days, and this range corresponded with the "long tail" of the temporal correlation function in the zonal direction. Velocity covariances in both the zonal and meridional directions were computed, and corresponding diffusivities were 8100 m2/sec meridionally and 41000 m2/sec zonally. Considering the existence of large scale mean flow, it is thought that self-similar energy cascade processes associated with constant scaling exponent may be responsible for the anomalous zonal diffusivity, while the meridional diffusivity may be approximated by ordinary Brownian processes. We suggest that the scale invariance of the WOCE/TOGA surface drifter paths may be a manifestation of energy cascade processes from large scale mean flow to smaller scale irregular flow that is represented by fractional Brownian motion in the zonal direction.

Subgrid-scale modeling suggested by a two-scale DIA

Subgrid-scale (SGS) modeling in large eddy simulation is studied using the results of a two-scale DIA about the effects of helicity controlling the energy cascade process. Such effects are closely related to vorticity effects on the SGS Reynolds stress that are lacking in the SGS-viscosity representation. These results are applied to the improvement of the SGS models of the Smagorinsky type. Discussion is also made about the relationship with the present two-equation model with the Smagorinsky model and the one-equation model based on the SGS energy.

Subgrid-scale modeling suggested by a two-scale DIA

Subgrid-scale (SGS) modeling in large eddy simulation is studied using the results of a two-scale DIA about the effects of helicity controlling the energy cascade process. Such effects are closely related to vorticity effects on the SGS Reynolds stress that are lacking in the SGS-viscosity representation. These results are applied to the improvement of the SGS models of the Smagorinsky type. Discussion is also made about the relationship with the present two-equation model with the Smagorinsky model and the one-equation model based on the SGS energy.

Bridging between eddy-viscosity-type and second-order turbulence models through a two-scale turbulence theory.

A gap between eddy-viscosity-type and second-order models is bridged using the results of a two-scale direct-interaction approximation developed for the study of turbulent shear flows. This work provides a method for incorporating the findings from models of eddy-viscosity-type into second-order models, and vice versa. Specifically, the effect of helicity controlling energy-cascade processes is incorporated into a second-order model. Then, a higher-order eddy-viscosity-type expression for the Reynolds stress is derived through the application of an iterative approximation to the second-order model. The latter result is tested in a turbulent rotating channel flow and its usefulness is confirmed. Effects of flow trajectory are also discussed in the context of the effect of an adverse pressure gradient on the isotropic eddy viscosity.

Amplification and Meridional Confinement of Stationary and Quasi-stationary Eddies in a Two-Layer Model

Abstract The problem of resonance and meridional confinement of planetary-scale waves is investigated in the framework of a quasigeostrophic, two-layer, β-plane model with lateral sponge layers. It is shown that the model resonant wavenumber is meridionally confined despite the predictions of the linear barotropic theory based on the time-mean zonal wind. As a consequence, the fully nonlinear model exhibits a resonant response to orographic forcing significantly higher than that predicted by the linearized (baroclinic) version of the model itself.The model response to a localized (zonally and meridionally) orography is also studied and found to be consistent with previous results observed in spherical geometry. The significance of this result and the mechanisms leading to meridional confinement are discussed. It is suggested that nonlinearity and the inverse energy cascade process are responsible for the observed, enhanced wave confinement.

Analysis of impedance‐gas pressure characteristics of a tuning‐fork‐type quartz resonator with the rough surfaces by using fractal model of turbulence

AbstractThe impedance vs. gas pressure characteristics of a tuning‐fork‐type quartz resonator with rough surfaces have been studied experimentally and theoretically. On the quartz resonator with rough surfaces, the flow of the gas is separated so that the laminar flow is changed into a disturbed state of fluid considered as turbulence. It is assumed that this turbulence induces the surface friction resistance larger than that of a smooth surface. The change of impedance due to the turbulence can be formulated by introduction of the fractal model of the turbulence describing the energy cascade process into the mixing length theory by Prandtl. The obtained theoretical formula for the impedance variation is proportional to the square power of the pressure and agrees well with the experiments.

An identification of Energy Cascade in Turbulence by Orthonormal Wavelet Analysis

Orthonormal wavelet expansion method is applied to an analysis of atmospheric turbulence data which shows more than two decades of the inertial subrange spectrum. The result of the orthonor­ mal wavelet analysis of the turbulence data is discussed in comparison with those of an artificial random noise. The local wavelet spectra of turbulence show a characteristic structure. which is absent in the artificial random noise and is identified with the trace of the energy cascade process. The higher· order structure function of velocity. obtained by the wavelet analysis. shows the intermit· tent structure of the flow field. In 1941 Kolmogorov proposed a universal theory of fluid turbulence/) in which every statistical quantity concerning the inertial subrange of flow field is assumed to be determined only by the energy dissiation rate. According to this theory, the average of the n-th order of velocity difference between two points separated by spatial distance r is proportional to r13. This prediction has been repeatedly examined in accurate experiments in high Reynolds number flows, and it is now widely accepted that as far as lower order of velocity difference is concerned, the r-dependence agrees well with the Kolmogorov theory. In particular, experimental forms of the energy spectrum of the velocity field, which corresponds to n=2, coincide with the Kolmogorov form of k- 5/3 • However, it has been repeatedly confirmed that higher order of the velocity difference has a statistical property different from Kolmogorov's prediction, so that the n-th order structure function of the velocity field, when normalized by the second order of the velocity difference, shows a clear r-dependence. This fact implies that the energy cascade process in the inertial subrange has an unnegligible deviation from the Kolmogorov picture. The deviation is reflected in the shape of the probability distribution function of the velocity difference, which has longer tail for smaller distance r. This deviation is often called intermittency, and its characterization is regarded as one of the central problems of fluid turbulence. The fact that intermittency becomes more prominent at smaller scales indicates that the intermittency is an essential part of the energy cascade process. However, this cascade process itself, sometimes called Richardson cascade, has been only a matter of theoretical consideration, and its characteristic structure has not yet been clearly captured in experiments or in numerical simula­ tions.

Orthonormal wavelet analysis of turbulence

The orthonormal wavelet expansion method is applied to data of atmospheric turbulence extending over more than two decades in Kolmogorov's inertial subrange. First, the orthonormal wavelet expansion is described and a numerical scheme for the expansion coefficients is proposed. Then, by the use of the orthonormal wavelets, several characteristics of turbulence are discussed in comparison with those of artificial, random noise. In particular, the local wavelet spectra of turbulence show a similarity structure, which is absent in the artificial random noise and is identified with the trace of the energy cascade process.

An identification of Energy Cascade in Turbulence by Orthonormal Wavelet Analysis

Orthonormal wavelet expansion method is applied to an analysis of atmospheric turbulence data which shows more than two decades of the inertial subrange spectrum. The result of the orthonormal wavelet analysis of the turbulence data is discussed in comparison with those of an artificial random noise. The local wavelet spectra of turbulence show a characteristic structure, which is absent in the artificial random noise and is identified with the trace of the energy cascade process. The higher-order structure function of velocity, obtained by the wavelet analysis, shows the intermittent structure of the flow field.

Symbiotic Relation between Planetary and Synoptic-Scale Waves

Abstract It is hypothesized that the low-frequency planetary scale waves and the high-frequency cyclone-scale waves in an equilibrated state of the atmosphere are symbiotically dependent upon one another. This is demonstrated with an analysis of a dissipative atmospheric model driven by a zonally symmetric forcing. Under a geophysically relevant parameter condition, the synoptic scale waves in the equilibrated state of this system intermittently extract a sufficient amount of energy from the modified instantaneous zonal flow to compensate not only for their own dissipative loss of energy but also for a net supply of energy to the planetary scale waves through the upscale energy cascade process. The planetary scale waves gain this energy in the barotropic form. The planetary scale waves, in turn, create localized strong baroclinic regions whereby the synoptic scale waves may preferentially intensify downstream of the model planetary jet streams. Such cyclone scale eddies collectively give rise to two model...

Orthonormal Wavelet Expansion and Its Application to Turbulence

Orthonormal wavelet expansion is applied to experimental data of turbulence. A direct relation is found between the wavelet spectrum and the Fourier spectrum. The orthonormal wavelet analysis with conditional sampling is applied to data of wind turbulence, yielding Kolmogorov's spectrum and the dissipation correlation with the intermittency exponent ,11'0.2. Fourier transform method is a fundamental and indispensable tool in data analysis since it enables us to decompose data into components with different scales. Many fundamental properties of physical systems have been described in terms of Fourier spectrum, that is, the amplitude of Fourier coefficients. However, since Fourier spectrum totally ignores the phase of each Fourier coefficient, it lacks infor­ mation about positions of local events which underlie the characteristics of the spectrum. The Fourier spectrum analysis therefore encounters difficulty in ana)yzing data in temporal or spatial intervals which include different kinds of local events. The method of scale analysis applicable even to such complicated situations should enable us to identify the origin of characteristics of the spectrum with local events occurring in physical space. This requirement would be satisfied at least partially by an expansion in terms of basis functions which are local both in physical and Fourier space, although the locality is limited in its extent by the uncertainty principle. In this paper, we adopt an expansion method' in terms of orthonormal wavelets (discrete wavelet analysis) as one of such types of expansion method. The orthonormal wavelet expansion is a discrete version of continuous wavelet analysis.!) The latter is an integral transform method with kernel functions obtained by translating and dilating a l~calized function (analyzing wavelet). The continuous wavelet transform of a square integrable function is an isometric transform between a Hilbert space (V space on Rn) and V space on a locally compact topological group (a group of translation and dilation) with its Haar measure: 2H ) The continuous wavelet is a useful tool especially for studying a singularity or a fractal structure of a given function. 5H ) In particular, the energy cascade process in fully-developed turbulence has been captured remarkably in such an analysis. 7 ) However, it is not very advantageous if one is interested in the energetic aspect because the kernel functions are not mutually orthogonal and no physically immediate meaning can be associated with the expansion coefficients.

Comment on "Two-Equation Model Consistent with Second Law"

T HE authors are to be congratulated for providing some light on the significance of the Second Law of Thermodynamics on two-equation turbulence models. The statements made by the Second Law on irreversibility are especially pertinent to turbulence and may in the future allow us a better understanding of the turbulent energy cascade and transition processes. The purpose of this Comment is to point out a few inconsistencies in the turbulence model presented in the above paper, which may help save practitioners of turbulence modeling from pointless programming. First, it is clear with reference to an earlier paper by Busnaina et al. that the value of C used was the usual 0.09, not 0.99. The lack of a low-Reynolds-number damping expression for the turbulent viscosity indicates that the model is not suitable for near-wall or other lowReynolds-number applications. A scrutiny of the proposed dissipation equation indicates that for the equation to balance in the nearwall region, C must equal half of C to insure a quadratic nearwall profile for turbulent kinetic energy, whereas the proposed constants' values were C = C = 1.92. In addition, since the authors were solving for dissipation itself rather than a dissipation variable, the presence of the turbulence kinetic energy k in the denominator of the two leading order terms

Wavelet analysis of turbulence reveals the multifractal nature of the Richardson cascade

THE central problem of fully developed turbulence is the energy-cascading process. It has resisted all attempts at a full physical understanding or mathematical formulation. The main reasons for this failure are related to the large hierarchy of scales involved, the highly nonlinear character inherent in the Navier–Stokes equations, and the spatial intermittency of the dynamically active regions. Richardson1 has described the interplay between large and small scales with the words 'Big whirls have little whirls which feed on their velocity ... and so on to viscosity", and the phenomenon so described is known as the Richardson cascade. This local interplay also forms the basis of a theory by Kolmogorov2. It was later realized that the cascade ought to be intermittent, and Mandelbrot3 has given a fractal description of the intermittency of the fine structure. A particular case that emphasizes the dynamical aspect of the fractal models is the β-model4, in which the flux of energy is transferred to only a fixed fraction β of the eddies of smaller scales. More recently, a multifractal model of the fine-scale intermittency has been introduced by Parisi and Frisch5; this accounts for the more complex cascading process suggested by the experimental data on inertial-range structure functions obtained by Anselmet et al.6. These statistical models are insufficient, however, because they cannot give any topological information about the intermittent structures in real space. Here we use the wavelet transform7 to analyse the velocity field of wind-tunnel turbulence at very high Reynolds numbers8. This 'space-scale' analysis is shown to provide the first visual evidence of the celebrated Richardson cascade1, and reveals in particular its fractal character3. The results also indicate that the energy-cascading process has remarkable similarities with the deterministic construction rules of non-homogeneous Cantor sets9,10.

Fluctuation-dissipation relation and anharmonic generation in driven CDW models

A stochastic theory for CDW current fluctuation is proposed based on the results of numerical simulation of Fukuyama-Lee-Rice model at finite temperature. It is shown that an empirical modelling of CDW current becomes valid when there are energy cascading process and strong mixing of modes. This empirical model can explain (i) the occurrence of a noise-induced transition (the change of profiles in the probability density function) for CDW current fluctuation near the onset point of the CDW running state; (ii) the existence of a generalized fluctuation- dissipation relation for the CDW current.

Temporal Intermittency in the Energy Cascade Process and Local Lyapunov Analysis in Fully-Developed Model Turbulence

Le processus de cascade d'energie est etudie numeriquement sur un modele scalaire de la turbulence tridimensionnelle entierement developpee. L'energie se propage a travers l'intervalle inerte par intermittence, comme des explosions alternant avec des periodes de repos (on revoit le point de vue de Siggia). Pendant la phase activee, le premier exposant de Lyapunov local oscille fortement et le support du premier vecteur de Lyapunov s'etend sur le sous-intervalle inerte

Cascade model of turbulence

A simple low-order dynamic system is proposed to simulate the energy cascade process of a fully developed hydrodynamic turbulence, and a series of numerical experiments is conducted. The main structural properties of this dynamic system are the same as those of the normalized Navier–Stokes equation. A quick estimation of its first Lyapunov exponent λ1 is made by calculating the chaos index at sampling points on trajectories. The large positive values (20–22) of λ1 indicate that the system is highly chaotic and that it has a strange attractor. After the transient period the dynamic system moves along its strange attractor, and a very wide inertial range is observed, in which the energy flux across the wavenumbers is a constant and equal to the energy dissipation rate. Moreover, the Kolmogorov − (5)/(3) spectrum is obtained as a statistical property of chaotic trajectories along the strange attractor.