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.

Numerical Investigation of Mechanisms Underlying Oceanic Internal Gravity Wave Power-Law Spectra

2020 ◽  
Vol 50 (9) ◽  
pp. 2713-2733
Author(s):  
Yulin Pan ◽  
Brian K. Arbic ◽  
Arin D. Nelson ◽  
Dimitris Menemenlis ◽  
W. R. Peltier ◽  
...  
Keyword(s):  
Power Law ◽  
Gravity Waves ◽  
High Frequency ◽  
Internal Gravity Wave ◽  
Numerical Data ◽  
Field Measurements ◽  
Internal Gravity Waves ◽  
Collision Integral ◽  
Turbulence Theory ◽  
Nonlinear Interactions

AbstractWe consider the power-law spectra of internal gravity waves in a rotating and stratified ocean. Field measurements have shown considerable variability of spectral slopes compared to the high-wavenumber, high-frequency portion of the Garrett–Munk (GM) spectrum. Theoretical explanations have been developed through wave turbulence theory (WTT), where different power-law solutions of the kinetic equation can be found depending on the mechanisms underlying the nonlinear interactions. Mathematically, these are reflected by the convergence properties of the so-called collision integral (CL) at low- and high-frequency limits. In this work, we study the mechanisms in the formation of the power-law spectra of internal gravity waves, utilizing numerical data from the high-resolution modeling of internal waves (HRMIW) in a region northwest of Hawaii. The model captures the power-law spectra in broad ranges of space and time scales, with scalings ω−2.05±0.2 in frequency and m−2.58±0.4 in vertical wavenumber. The latter clearly deviates from the GM76 spectrum but is closer to a family of induced-diffusion-dominated solutions predicted by WTT. Our analysis of nonlinear interactions is performed directly on these model outputs, which is fundamentally different from previous work assuming a GM76 spectrum. By applying a bicoherence analysis and evaluations of modal energy transfer, we show that the CL is dominated by nonlocal interactions between modes in the power-law range and low-frequency inertial motions. We further identify induced diffusion and the near-resonances at its spectral vicinity as dominating the formation of power-law spectrum.

scholarly journals Transition from geostrophic flows to inertia-gravity waves in the spectrum of a differentially heated rotating annulus experiment.

2020 ◽  
Author(s):  
Costanza Rodda ◽  
Uwe Harlander
Keyword(s):  
Gravity Waves ◽  
Large Scale ◽  
Small Scale ◽  
Turbulent Regime ◽  
Weakly Nonlinear ◽  
Geostrophic Flows ◽  
Balanced Flow ◽  
Energy And Momentum ◽  
Open Question ◽  
Inertia Gravity Waves

<p>Inertia-gravity waves (IGWs) are known to play an essential role in the terrestrial atmospheric dynamics as they can lead to energy and momentum flux when they propagate upwards. An open question is to which extent nearly linear IGWs contribute to the total energy and to flattening of the energy spectrum observed at the mesoscale.<br>In this work, we present an experimental investigation of the energy distribution between the large-scale balanced flow and the small-scale imbalanced flow. Weakly nonlinear IGWs emitted from baroclinic jets are observed in the differentially heated rotating annulus experiment. Similar to the atmospheric spectra, the experimental kinetic energy spectra reveal the typical subdivision into two distinct regimes with slopes <em>k</em><sup>-3</sup> for the large scales and <em>k<sup>-</sup></em><sup>5/3</sup> for smaller scales. By separating the spectra into a vortex and wave part, it emerges that at the largest scales in the mesoscale range the gravity waves observed in the experiment cause a flattening of the spectra and provide most of the energy. At smaller scales, our data analysis suggests a transition towards a turbulent regime with a forward energy cascade up to where dissipation by diffusive processes occurs.</p>

Capillary–gravity and capillary waves generated in a wind wave tank: observations and theories

1995 ◽  
Vol 289 ◽  
pp. 51-82 ◽  
Author(s):  
Xin Zhang
Keyword(s):  
Power Law ◽  
Gravity Waves ◽  
Water Surface ◽  
Wind Wave ◽  
Wind Waves ◽  
Capillary Waves ◽  
Fourth Power ◽  
Two Dimensional ◽  
Short Waves ◽  
Surface Gradient

Short water surface waves generated by wind in a water tunnel have been measured by an optical technique that provides a synoptic picture of the water surface gradient over an area of water surface (Zhang & Cox 1994). These images of the surface gradient can be integrated to recover the shape of the water surface and find the two-dimensional wavenumber spectrum. Waveforms and two-dimensional structures of short wind waves have many interesting features: short and steep waves featuring sharp troughs and flat crests are very commonly seen and most of the short waves are far less steep than the limiting wave forms; waveforms that resemble capillary–gravity solitons are observed with a close match to the form theoretically predicted for potential flows (Longuet-Higgins 1989); capillaries are mainly found as parasitics on the downwind faces of gravity waves, and the longest wavelengths of those parasitic capillaries found are less than 1 cm; the phenomenon of capillary blockage (Phillips 1981) on dispersive freely travelling short waves is also observed. The spectra of short waves generated by low winds show a characteristic dip at the transition wavenumber between the gravity and capillary regimes, and the dip becomes filled in as the wind increases. The spectral cut-off at high wavenumbers shows a power law behaviour with an exponent of about minus four. The wavenumber of the transition from the dip to the cut-off is not sensitive to the change of wind speed. The minus fourth power law of the extreme capillary wind wave spectrum can be explained through a model of energy balances. The concept of an equilibrium spectrum is still useful. It is shown that the dip in the spectrum of capillary–gravity waves is a result of blockage of both capillary–gravity wind waves and parasitic capillary waves.

Vertical Wavenumber Spectra of Gravity Waves in the Venus Atmosphere Obtained from Venus Express Radio Occultation Data: Evidence for Saturation

2015 ◽  
Vol 72 (6) ◽  
pp. 2318-2329 ◽  
Author(s):  
Hiroki Ando ◽  
Takeshi Imamura ◽  
Toshitaka Tsuda ◽  
Silvia Tellmann ◽  
Martin Pätzold ◽  
...  
Keyword(s):  
Diffusion Coefficient ◽  
Power Law ◽  
Gravity Waves ◽  
Radio Occultation ◽  
Spectral Power ◽  
European Space Agency ◽  
Small Scale ◽  
Turbulent Diffusion Coefficient ◽  
Venus Express ◽  
Venusian Atmosphere

Abstract By using the vertical temperature profiles obtained by the radio occultation measurements on the European Space Agency (ESA)’s Venus Express, the vertical wavenumber spectra of small-scale temperature fluctuations that are thought to be manifestations of gravity waves are studied. Wavenumber spectra covering wavelengths of 1.4–7.5 km were obtained for two altitude regions (65–80 and 75–90 km) and seven latitude bands. The spectra show a power-law dependence on the high-wavenumber side with the logarithmic spectral slope ranging from −3 to −4, which is similar to the features seen in Earth’s and Martian atmospheres. The power-law portion of the spectrum tends to follow the semiempirical spectrum of saturated gravity waves, suggesting that the gravity waves are dissipated by saturation as well as radiative damping. The spectral power is larger at 75–90 km than at 65–80 km at low wavenumbers, suggesting amplitude growth with height of unsaturated waves. It was also found that the wave amplitude is larger at higher latitudes and that the amplitude is maximized in the northern high latitudes. On the assumption that gravity waves are saturated in the Venusian atmosphere, the turbulent diffusion coefficient was estimated. The diffusion coefficient in the Venusian atmosphere is larger than those in Earth’s atmosphere because of the longer characteristic vertical wavelength of the saturated waves.

Numerical calculations of steady gravity-capillary waves using an integro-differential formulation

1979 ◽  
Vol 94 (4) ◽  
pp. 777-793 ◽  
Author(s):  
James W. Rottman ◽  
D. B. Olfe
Keyword(s):  
Gravity Waves ◽  
Small Amplitude ◽  
Numerical Solutions ◽  
Capillary Wave ◽  
Capillary Waves ◽  
The Other ◽  
Wave Number ◽  
Infinite Depth ◽  
Free Surface Waves ◽  
Differential Formulation

A new integro-differential equation is derived for steady free-surface waves. Numerical solutions of this equation for periodic gravity-capillary waves on a fluid of infinite depth are presented. For the two limiting cases of gravity waves and capillary waves, our results are in excellent agreement with previous calculations. For gravity-capillary waves, detailed calculations are performed near the wave-number at which the classical second-order perturbation solution breaks down. Our calculations yield two solutions in this region, which in the limit of small amplitudes agree with the results obtained by Wilton in 1915; one solution has the small amplitude behaviour of a gravity wave and the other that of a capillary wave, but the numerical results show that at large amplitudes both waves have the characteristics of capillary waves. The calculations also show that the wavenumber range in which two solutions exist increases with increasing wave height.

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 ${\it\gamma}_{0}$.

scholarly journals Transition from Geostrophic Flows to Inertia–Gravity Waves in the Spectrum of a Differentially Heated Rotating Annulus Experiment

2020 ◽  
Vol 77 (8) ◽  
pp. 2793-2806
Author(s):  
Costanza Rodda ◽  
Uwe Harlander
Keyword(s):  
Gravity Waves ◽  
Large Scale ◽  
Small Scale ◽  
Turbulent Regime ◽  
Weakly Nonlinear ◽  
Geostrophic Flows ◽  
Balanced Flow ◽  
Energy And Momentum ◽  
Open Question ◽  
Inertia Gravity Waves

Abstract Inertia–gravity waves (IGWs) play an essential role in the terrestrial atmospheric dynamics as they can lead to energy and momentum flux when propagating upward. An open question is to what extent IGWs contribute to the total energy and to the flattening of the energy spectrum observed at the mesoscale. In this work, we present an experimental investigation of the energy distribution between the large-scale balanced flow and the small-scale imbalanced flow. Weakly nonlinear IGWs emitted from baroclinic jets are observed in the differentially heated rotating annulus experiment. Similar to the atmospheric spectra, the experimental kinetic energy spectra reveal the typical subdivision into two distinct regimes with slopes k−3 for the large scales and k−5/3 for the small scales. By separating the spectra into the vortex and wave components, it emerges that at the large-scale end of the mesoscale the gravity waves observed in the experiment cause a flattening of the spectra and provide most of the energy. At smaller scales, our data analysis suggests a transition toward a turbulent regime with a forward energy cascade up to where dissipation by diffusive processes occurs.

scholarly journals A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1

2018 ◽  
Vol 854 ◽  
pp. 146-163 ◽  
Author(s):  
H. C. Hsu ◽  
C. Kharif ◽  
M. Abid ◽  
Y. Y. Chen
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Gravity Waves ◽  
Capillary Wave ◽  
Modulational Instability ◽  
Capillary Waves ◽  
Rate Of Growth ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Wave Trains

A nonlinear Schrödinger equation for the envelope of two-dimensional gravity–capillary waves propagating at the free surface of a vertically sheared current of constant vorticity is derived. In this paper we extend to gravity–capillary wave trains the results of Thomas et al. (Phys. Fluids, 2012, 127102) and complete the stability analysis and stability diagram of Djordjevic & Redekopp (J. Fluid Mech., vol. 79, 1977, pp. 703–714) in the presence of vorticity. The vorticity effect on the modulational instability of weakly nonlinear gravity–capillary wave packets is investigated. It is shown that the vorticity modifies significantly the modulational instability of gravity–capillary wave trains, namely the growth rate and instability bandwidth. It is found that the rate of growth of modulational instability of short gravity waves influenced by surface tension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reduced in the presence of positive vorticity and amplified in the presence of negative vorticity; (ii) in finite depth, it is reduced when the vorticity is positive and amplified and finally reduced when the vorticity is negative. The combined effect of vorticity and surface tension is to increase the rate of growth of modulational instability of short gravity waves influenced by surface tension, namely when the vorticity is negative. The rate of growth of modulational instability of capillary waves is amplified by negative vorticity and attenuated by positive vorticity. Stability diagrams are plotted and it is shown that they are significantly modified by the introduction of the vorticity.

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.

Efficient Simulation of Short and Long-Wave Interactions With Applications to Capillary Waves

1994 ◽  
Vol 116 (1) ◽  
pp. 77-82 ◽  
Author(s):  
D. G. Dommermuth
Keyword(s):  
Gravity Waves ◽  
Capillary Wave ◽  
Perturbation Expansion ◽  
Capillary Waves ◽  
Front Face ◽  
Wave Interactions ◽  
Efficient Simulation ◽  
Ripple Formation ◽  
Long Wave ◽  
Unsteady Interaction

A perturbation expansion and a multigrid technique are developed for simulating the fully-nonlinear unsteady-interaction of short waves riding on long gravity waves. Both numerical techniques are capable of simulating wave slopes near breaking and wavelength ratios greater than thirty, but the multigrid technique converges more rapidly and it is more efficient. The results of numerical simulations agree qualitatively with experimental measurements of ripple formation on the front face of a gravity-capillary wave.

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