Decaying capillary wave turbulence under broad-scale dissipation

2015 ◽  
Vol 780 ◽  
Author(s):  
Yulin Pan ◽  
Dick K. P. Yue
Keyword(s):  
Energy Transfer ◽  
Energy Dissipation ◽  
Energy Flux ◽  
Power Law ◽  
Time Moment ◽  
Capillary Waves ◽  
Inertial Range ◽  
Turbulence Theory ◽  
Weak Turbulence ◽  
Spectral Slope

We study the freely decaying weak turbulence of capillary waves by direct numerical solution of the primitive Euler equations. By introducing a small amount of wave dissipation, measured by the viscosity magnitude ${\it\gamma}_{0}$, we are able to recover phenomena observed in experiments that are not described by weak-turbulence theory (WTT), including the exponential modal decay and time variation of the width and power-law spectral slope ${\it\alpha}$ of the inertial range. In contrast to WTT, this problem also involves non-constant inter-modal energy transfer across the inertial range, which imposes a difficulty in quantifying and measuring the energy flux $P$ associated with a certain power-law spectrum. We propose an effective and novel way to evaluate $P$ in such cases by physically considering the unsteady effects of the spectrum and variation of the inter-modal energy transfer. Our results show the fundamental difference between the energy flux $P$ and the total energy dissipation rate ${\it\Gamma}$, which is due to significant energy dissipation within the inertial range. This settles the previous debate on the measurement of $P$ which assumes the equivalence of the two. Based on our numerical data, we obtain a general form of the time-evolving inertial-range spectrum, where the parameters involved are functions of ${\it\gamma}_{0}$ only. The value of the spectral slope ${\it\alpha}$ at each time moment in the decay, however, is found to be uniquely related to the spectral magnitude at that time and irrespective of ${\it\gamma}_{0}$, in the range we consider. This physically reveals the dominant effect of nonlinear wave interaction in forming the power-law spectrum within the inertial range. The evolutions of the inertial-range energy are shown to be predicted by analytical integration of the evolving spectra for different values of ${\it\gamma}_{0}$.

Local energy flux and subgrid-scale statistics in three-dimensional turbulence

1998 ◽  
Vol 366 ◽  
pp. 1-31 ◽  
Author(s):  
VADIM BORUE ◽  
STEVEN A. ORSZAG
Keyword(s):  
Energy Transfer ◽  
Energy Dissipation ◽  
Energy Flux ◽  
Dissipation Rate ◽  
Three Dimensional ◽  
Energy Dissipation Rate ◽  
Inertial Range ◽  
Local Energy ◽  
Subgrid Scale ◽  
Velocity Gradients

Statistical properties of the subgrid-scale stress tensor, the local energy flux and filtered velocity gradients are analysed in numerical simulations of forced three-dimensional homogeneous turbulence. High Reynolds numbers are achieved by using hyperviscous dissipation. It is found that in the inertial range the subgrid-scale stress tensor and the local energy flux allow simple parametrization based on a tensor eddy viscosity. This parametrization underlines the role that negative skewness of filtered velocity gradients plays in the local energy transfer. It is found that the local energy flux only weakly correlates with the locally averaged energy dissipation rate. This fact reflects basic difficulties of large-eddy simulations of turbulence, namely the possibility of predicting the locally averaged energy dissipation rate through inertial-range quantities such as the local energy flux is limited. Statistical properties of subgrid-scale velocity gradients are systematically studied in an attempt to reveal the mechanism of local energy transfer.

scholarly journals On energetics and inertial-range scaling laws of two-dimensional magnetohydrodynamic turbulence

2012 ◽  
Vol 703 ◽  
pp. 238-254 ◽  
Author(s):  
Luke A. K. Blackbourn ◽  
Chuong V. Tran
Keyword(s):  
Energy Transfer ◽  
Kinetic Energy ◽  
Energy Flux ◽  
Scaling Laws ◽  
Magnetic Energy ◽  
Inertial Range ◽  
Two Dimensional ◽  
Magnetohydrodynamic Turbulence ◽  
Direct Energy ◽  
Energy Equipartition

AbstractWe study two-dimensional magnetohydrodynamic turbulence, with an emphasis on its energetics and inertial-range scaling laws. A detailed spectral analysis shows that dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while anti-dynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo interacting triads are integral parts of the direct energy transfer, the anti-dynamo inverse flux partially neutralizes the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. This result is consistent with a qualitative prediction of energy transfer reduction due to Alfvén wave effects by the Iroshnikov–Kraichnan theory (which was originally formulated for magnetohydrodynamic turbulence in three dimensions). We numerically confirm the correlation between dynamo action and direct magnetic energy flux and investigate the applicability of quantitative aspects of the Iroshnikov–Kraichnan theory to the present case, particularly its predictions of energy equipartition and ${k}^{\ensuremath{-} 3/ 2} $ spectra in the energy inertial range. It is found that for turbulence satisfying the Kraichnan condition of magnetic energy at large scales exceeding total energy in the inertial range, the kinetic energy spectrum, which is significantly shallower than ${k}^{\ensuremath{-} 3/ 2} $, is shallower than its magnetic counterpart. This result suggests no energy equipartition. The total energy spectrum appears to depend on the energy composition of the turbulence but is clearly shallower than ${k}^{\ensuremath{-} 3/ 2} $ for $r\approx 2$, even at moderate resolutions. Here $r\approx 2$ is the magnetic-to-kinetic energy ratio during the stage when the turbulence can be considered fully developed. The implication of the present findings is discussed in conjunction with further numerical results on the dependence of the energy dissipation rate on resolution.

scholarly journals Direct Evidence of an Oceanic Inverse Kinetic Energy Cascade from Satellite Altimetry

2005 ◽  
Vol 35 (9) ◽  
pp. 1650-1666 ◽  
Author(s):  
Robert B. Scott ◽  
Faming Wang
Keyword(s):  
Kinetic Energy ◽  
Energy Flux ◽  
Direct Evidence ◽  
Inertial Range ◽  
Turbulence Theory ◽  
Kinetic Energy Density ◽  
Geostrophic Flow ◽  
Inverse Cascade ◽  
Barotropic Mode ◽  
Geostrophic Turbulence

Abstract Sea surface height measurements from satellites reveal the turbulent properties of the South Pacific Ocean surface geostrophic circulation, both supporting and challenging different aspects of geostrophic turbulence theory. A near-universal shape of the spectral kinetic energy flux is found and provides direct evidence of a source of kinetic energy near to or smaller than the deformation radius, consistent with linear instability theory. The spectral kinetic energy flux also reveals a net inverse cascade (i.e., a cascade to larger spatial scale), consistent with two-dimensional turbulence phenomenology. However, stratified geostrophic turbulence theory predicts an inverse cascade for the barotropic mode only; energy in the large-scale baroclinic modes undergoes a direct cascade toward the first-mode deformation scale. Thus if the surface geostrophic flow is predominately the first baroclinic mode, as expected for oceanic stratification profiles, then the observed inverse cascade contradicts geostrophic turbulence theory. The latter interpretation is argued for. Furthermore, and consistent with this interpretation, the inverse cascade arrest scale does not follow the Rhines arrest scale, as one would expect for the barotropic mode. A tentative revision of theory is proposed that would resolve the conflicts; however, further observations and idealized modeling experiments are needed to confirm, or refute, the revision. It is noted that no inertial range was found for the inverse cascade range of the spectrum, implying inertial range scaling, such as the established K−5/3 slope in the spectral kinetic energy density plot, is not applicable to the surface geostrophic flow.

Understanding discrete capillary-wave turbulence using a quasi-resonant kinetic equation

2017 ◽  
Vol 816 ◽  
Author(s):  
Yulin Pan ◽  
Dick K. P. Yue
Keyword(s):  
Kinetic Equation ◽  
Euler Equations ◽  
Power Law ◽  
Capillary Wave ◽  
Quantitative Measure ◽  
Energy Cascade ◽  
Turbulence Theory ◽  
Finite Domain ◽  
Spectral Slope ◽  
Infinite Domain

Experimental and numerical studies have shown that, with sufficient nonlinearity, the theoretical capillary-wave power-law spectrum derived from the kinetic equation (KE) of weak turbulence theory can be realized. This is despite the fact that the KE is derived assuming an infinite domain with continuous wavenumber, while experiments and numerical simulations are conducted in realistic finite domains with discrete wavenumbers for which the KE theoretically allows no energy transfer. To understand this, we first analyse results from direct simulations of the primitive Euler equations to elucidate the role of nonlinear resonance broadening (NRB) in discrete turbulence. We define a quantitative measure of the NRB, explaining its dependence on the nonlinearity level and its effect on the properties of the obtained stationary power-law spectra. This inspires us to develop a new quasi-resonant kinetic equation (QKE) for discrete turbulence, which incorporates the mechanism of NRB, governed by a single parameter $\unicode[STIX]{x1D705}$ expressing the ratio of NRB and wavenumber discreteness. At $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{0}\approx 0.02$, the QKE recovers simultaneously the spectral slope $\unicode[STIX]{x1D6FC}_{0}=-17/4$ and the Kolmogorov constant $C_{0}=6.97$ (corrected from the original derivation) of the theoretical continuous spectrum, which physically represents the upper bound of energy cascade capacity for the discrete turbulence. For $\unicode[STIX]{x1D705}<\unicode[STIX]{x1D705}_{0}$, the obtained spectra represent those corresponding to a finite domain with insufficient nonlinearity, resulting in a steeper spectral slope $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{0}$ and reduced capacity of energy cascade $C>C_{0}$. The physical insights from the QKE are corroborated by direct simulation results of the Euler equations.

scholarly journals Turbulence of capillary waves forced by steep gravity waves

2018 ◽  
Vol 850 ◽  
pp. 803-843 ◽  
Author(s):  
M. Berhanu ◽  
E. Falcon ◽  
L. Deike
Keyword(s):  
Nonlinear Wave ◽  
Power Law ◽  
Gravity Waves ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Turbulence Theory ◽  
Wave Turbulence ◽  
Small Scale ◽  
Turbulent Regime ◽  
Strong Nonlinearity

We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2–0.3), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from the large scale to the small scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small scale in a fast time scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.

Observations of Small-Scale Turbulence and Energy Dissipation Rates in the Cloudy Boundary Layer

2006 ◽  
Vol 63 (5) ◽  
pp. 1451-1466 ◽  
Author(s):  
Holger Siebert ◽  
Katrin Lehmann ◽  
Manfred Wendisch
Keyword(s):  
Boundary Layer ◽  
Energy Dissipation ◽  
Wind Velocity ◽  
Turbulence Theory ◽  
Horizontal Wind ◽  
Inertial Subrange ◽  
Intermittent Flow ◽  
Small Scale ◽  
Dissipation Rates ◽  
Wide Range

Abstract Tethered balloon–borne measurements with a resolution in the order of 10 cm in a cloudy boundary layer are presented. Two examples sampled under different conditions concerning the clouds' stage of life are discussed. The hypothesis tested here is that basic ideas of classical turbulence theory in boundary layer clouds are valid even to the decimeter scale. Power spectral densities S( f ) of air temperature, liquid water content, and wind velocity components show an inertial subrange behavior down to ≈20 cm. The mean energy dissipation rates are ∼10−3 m2 s−3 for both datasets. Estimated Taylor Reynolds numbers (Reλ) are ∼104, which indicates the turbulence is fully developed. The ratios between longitudinal and transversal S( f ) converge to a value close to 4/3, which is predicted by classical turbulence theory for local isotropic conditions. Probability density functions (PDFs) of wind velocity increments Δu are derived. The PDFs show significant deviations from a Gaussian distribution with longer tails typical for an intermittent flow. Local energy dissipation rates ɛτ are derived from subsequences with a duration of τ = 1 s. With a mean horizontal wind velocity of 8 m s−1, τ corresponds to a spatial scale of 8 m. The PDFs of ɛτ can be well approximated with a lognormal distribution that agrees with classical theory. Maximum values of ɛτ ≈ 10−1 m2 s−3 are found in the analyzed clouds. The consequences of this wide range of ɛτ values for particle–turbulence interaction are discussed.

Multiple harmonics of electron waves studied using weak turbulence theory in a two-dimensional formulation

2021 ◽  
Vol 28 (10) ◽  
pp. 102302
Author(s):  
E. C. Fonseca-Pongutá ◽  
L. F. Ziebell ◽  
R. Gaelzer
Keyword(s):  
Turbulence Theory ◽  
Weak Turbulence ◽  
Two Dimensional ◽  
Weak Turbulence Theory

Scaling Behavior of Dynamic Hysteresis in Hard PZT Bulk Ceramics under Influence of Compressive Stress

2008 ◽  
Vol 55-57 ◽  
pp. 281-284 ◽  
Author(s):  
N. Wongdamnern ◽  
Athipong Ngamjarurojana ◽  
Supon Ananta ◽  
Yongyut Laosiritaworn ◽  
Rattikorn Yimnirun
Keyword(s):  
Energy Dissipation ◽  
Electric Field ◽  
Mechanical Stress ◽  
Compressive Stress ◽  
Power Law ◽  
Field Amplitude ◽  
Electric Field Amplitude ◽  
Bulk Ceramic ◽  
Compressive Stresses ◽  
The Difference

Effects of electric field-amplitude and mechanical stress on hysteresis area were investigated in partially depoled hard PZT bulk ceramic. At any compressive stress, the hysteresis area was found to depend on the field-amplitude with a same set of exponents to the power-law scaling. Consequently, inclusion of compressive stresses into the power-law was also obtained in the form of < A – Aσ=0 > α E05.1σ1.19 which indicated the difference of the energy dissipation between the under-stress and stress-free conditions.

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