Gravity Wave Turbulence Research Articles

On the kinetics of internal gravity waves beyond the hydrostatic regime

We present a new derivation of the kinetic equation for weak, non-hydrostatic internal gravity wave turbulence. The equation is equivalent to the one obtained by Caillol & Zeitlin (Dyn. Atmos. Oceans, vol. 32, issue 2, 2000, pp. 81–112), but it takes a canonical form. We show that it conserves the energy without involving the resonance condition in frequency, and look for the isotropic part of the steady, scale-invariant solutions. We provide a parametrization of the resonant manifold of non-hydrostatic internal gravity wave triadic interactions. This allows us to simplify the collision integral, and to evaluate the transfer coefficients of all triadic interactions. In the hydrostatic limit, our equation is equivalent to the Hamiltonian description of Lvov & Tabak (Phys. Rev. Lett., vol. 87, issue 16, 2001, 168501).

Turbulence of internal gravity waves in the laboratory

Turbulence of internal gravity waves in the laboratory

Non-local gravity wave turbulence in presence of condensate

The $k^{-23/6}$ wave action spectrum with an inverse cascade is one of the fundamental Kolmogorov–Zakharov solutions for gravity wave turbulence, which is part of the citation for the Dirac Medal in 2003. Instead of confirming this solution, however, several existing simulations and experiments suggest a spectrum of $k^{-3}$ in set-ups corresponding to the inverse cascade. We provide a theoretical explanation for the latter, considering the condensate that naturally forms in finite domains of experiments/simulations. Our new theory hinges on: (1) derivation of a spectral diffusion equation when non-local interactions with the condensate become dominant, for the first time systematically formulated for quartet-interaction systems; and (2) careful analysis of the asymptotics of interaction coefficient with a remarkable cancellation of all leading-order terms.

Conformal Invariance in Water-Wave Turbulence.

We experimentally investigate the statistics of zero-height isolines in gravity wave turbulence as physical candidates for conformal invariant curves. We present direct evidence that they can be described by the family of conformal invariant curves called stochastic Schramm-Löwner evolution (or SLE_{κ}), with diffusivity κ=2.88(8). A higher nonlinearity in the height fields is shown destroy this symmetry, though scale invariance is retained.

Energy Spectra of Nonlocal Internal Gravity Wave Turbulence.

Starting from the classical formulation of the weak turbulence theory in a density stratified fluid, we derive a simplified version of the kinetic equation of internal gravity wave turbulence. This equation allows us to uncover scaling laws for the spatial and temporal energy spectra of internal wave turbulence which are consistent with typical scaling exponents observed in the oceans. The keystone of our description is the assumption that the energy transfers are dominated by a class of nonlocal resonant interactions, known as the "induced diffusion" triads, which conserve the ratio between the wave frequency and vertical wave number. Our analysis remarkably shows that the internal wave turbulence cascade is associated to an apparent constant flux of wave action.

Inverse Cascade Spectrum of Gravity Waves in the Presence of a Condensate: A Direct Numerical Simulation.

During the set of direct numerical simulations of the forced isotropic turbulence of surface gravity waves in the framework of primordial dynamical equations, the universal inverse cascade spectrum was observed. The slope of the spectrum is the same (in the margin of error) for different levels of pumping and nonlinearity as well as dissipation present in the system. In all simulation runs formation of the inverse cascade spectrum was accompanied by the appearance of a strong long wave background (condensate). The observed slope of the spectrum ∼k^{-3.07} is different from the constant wave action flux solution predicted by the wave turbulence theory ∼k^{-23/6}.

Aircraft observations of gravity wave activity and turbulence in the tropical tropopause layer: prevalence, influence on cirrus clouds, and comparison with global storm-resolving models

Abstract. The tropical tropopause layer (TTL) is a sea of vertical motions. Convectively generated gravity waves create vertical winds on scales of a few to thousands of kilometers as they propagate in a stable atmosphere. Turbulence from gravity wave breaking, radiatively driven convection, and Kelvin–Helmholtz instabilities stirs up the TTL on the kilometer scale. TTL cirrus clouds, which moderate the water vapor concentration in the TTL and stratosphere, form in the cold phases of large-scale (> 100 km) wave activity. It has been proposed in several modeling studies that small-scale (< 100 km) vertical motions control the ice crystal number concentration and the dehydration efficiency of TTL cirrus clouds. Here, we present the first observational evidence for this. High-rate vertical winds measured by aircraft are a valuable and underutilized tool for constraining small-scale TTL vertical wind variability, examining its impacts on TTL cirrus clouds, and evaluating atmospheric models. We use 20 Hz data from five National Aeronautics and Space Administration (NASA) campaigns to quantify small-scale vertical wind variability in the TTL and to see how it varies with ice water content, distance from deep convective cores, and height in the TTL. We find that 1 Hz vertical winds are well represented by a normal distribution, with a standard deviation of 0.2–0.4 m s−1. Consistent with a previous observational study that analyzed two out of the five aircraft campaigns that we analyze here, we find that turbulence is enhanced over the tropical west Pacific and within 100 km of convection and is most common in the lower TTL (14–15.5 km), closer to deep convection, and in the upper TTL (15.5–17 km), further from deep convection. An algorithm to classify turbulence and long-wavelength (5 km < λ < 100 km) and short-wavelength (λ < 5 km) gravity wave activity during level flight legs is applied to data from the Airborne Tropical TRopopause EXperiment (ATTREX). The most commonly sampled conditions are (1) a quiescent atmosphere with negligible small-scale vertical wind variability, (2) long-wavelength gravity wave activity (LW GWA), and (3) LW GWA with turbulence. Turbulence rarely occurs in the absence of gravity wave activity. Cirrus clouds with ice crystal number concentrations exceeding 20 L−1 and ice water content exceeding 1 mg m−3 are rare in a quiescent atmosphere but about 20 times more likely when there is gravity wave activity and 50 times more likely when there is also turbulence, confirming the results of the aforementioned modeling studies. Our observational analysis shows that small-scale gravity waves strongly influence the ice crystal number concentration and ice water content within TTL cirrus clouds. Global storm-resolving models have recently been run with horizontal grid spacing between 1 and 10 km, which is sufficient to resolve some small-scale gravity wave activity. We evaluate simulated vertical wind spectra (10–100 km) from four global storm-resolving simulations that have horizontal grid spacing of 3–5 km with aircraft observations from ATTREX. We find that all four models have too little resolved vertical wind at horizontal wavelengths between 10 and 100 km and thus too little small-scale gravity wave activity, although the bias is much less pronounced in global SAM than in the other models. We expect that deficient small-scale gravity wave activity significantly limits the realism of simulated ice microphysics in these models and that improved representation requires moving to finer horizontal and vertical grid spacing.

Intermittency of gravity wave turbulence on the surface of an infinitely deep fluid: Numerical experiment

The intermittency of gravity wave turbulence on the surface of an infinitely deep fluid has been studied numerically. We estimated the scaling exponent of surface elevation structure-functions directly from the equations of motion, e.g., by numerical solution of the underlying equations of motion, without any additional closure assumptions (e.g., phase randomisation) and reduction to the kinetic equations as in the conventional framework of wave turbulence theory. With our high-resolution numerical methods for accurately modelling the nonlinear water surface, we were able to evaluate the scaling exponents of the structure-function up to relatively high orders (p=15), which has not been previously reported. We investigated the effect of wave steepness and forcing on intermittency and compared our results with other analytical and numerical studies, including results on the intermittency of wave turbulence of a different nature (turbulence of bending waves on a plate and magnetohydrodynamic turbulence). Our results support the conjecture of universality for intermittency phenomena in wave turbulence.

Statistics of rogue waves in isotropic wave fields

We investigate the statistics of rogue waves occurring in the inverse cascade of surface gravity wave turbulence. In such statistically homogeneous, stationary and isotropic wave fields, low-frequency waves are generated by nonlinear interactions rather than directly forced by a wave maker. This provides a laboratory realization of arguably the simplest nonlinear sea state, in which long-time acquisitions are performed and compared with theoretical models. The analysis of thousands of rogue waves reveals that some of their properties crucially depend on four-wave resonant interactions, large crests being for instance more likely than predicted by second-order models.

Numerical investigation of turbulence of surface gravity waves

In this paper, we numerically study the wave turbulence of surface gravity waves in the framework of Euler equations of the free surface. The purpose is to understand the variation of the scaling of the spectra with wavenumber$k$and energy flux$P$at different nonlinearity levels under different forcing/free-decay conditions. For all conditions (free decay and narrow-band and broad-band forcing) that we consider, we find that the spectral forms approach the wave turbulence theory (WTT) solution$S_\eta \sim k^{-5/2}$and$S_\eta \sim P^{1/3}$at high nonlinearity levels. With a decrease of nonlinearity level, the spectra for all cases become steeper, with the narrow-band forcing case exhibiting the most rapid deviation from WTT. We investigate bound waves and the finite-size effect as possible mechanisms causing the spectral variations. Through a tri-coherence analysis, we find that the finite-size effect is present in all cases, which is responsible for the overall steepening of the spectra and the reduced capacity of energy flux at lower nonlinearity levels. The fraction of bound waves in the domain generally decreases with the decrease of nonlinearity level, except for the narrow-band case, which exhibits a transition at a critical nonlinearity level below which a rapid increase is observed. This increase serves as the main reason for the fastest deviation from WTT with the decrease of nonlinearity in the narrow-band forcing case.

Intermittency of Gravity Wave Turbulence on the Surface of an Infinitely Deep Fluid: Numerical Experiment

Intermittency of Gravity Wave Turbulence on the Surface of an Infinitely Deep Fluid: Numerical Experiment

Gravity wave instability structures and turbulence from more than 1.5 years of OH* airglow imager observations in Slovenia

Abstract. We analysed 286 nights of data from the OH* airglow imager FAIM 3 (Fast Airglow IMager) acquired at Otlica Observatory (45.93∘ N, 13.91∘ E), Slovenia, between 26 October 2017 and 6 June 2019. Measurements have been performed with a spatial resolution of 24 m per pixel and a temporal resolution of 2.8 s. A two-dimensional fast Fourier transform is applied to the image data to derive horizontal wavelengths between 48 m and 4.5 km in the upper mesosphere/lower thermosphere (UMLT) region. In contrast to the statistics of larger-scale gravity waves (horizontal wavelength up to ca. 50 km; Hannawald et al., 2019), we find a more isotropic distribution of directions of propagation, pointing to the presence of wave structures created above the stratospheric wind fields. A weak seasonal tendency of a majority of waves propagating eastward during winter may be due to instability features from breaking secondary gravity waves that were created in the stratosphere. We also observe an increased southward propagation during summer, which we interpret as an enhanced contribution of secondary gravity waves created as a consequence of primary wave filtering by the meridional mesospheric circulation. We present multiple observations of turbulence episodes captured by our high-resolution airglow imager and estimated the energy dissipation rate in the UMLT from image sequences in 25 cases. Values range around 0.08 and 9.03 W kg−1 and are on average higher than those in recent literature. The values found here would lead to an approximated localized maximum heating of 0.03–3.02 K per turbulence event. These are in the same range as the daily chemical heating rates for the entire atmosphere reported by Marsh (2011), which apparently stresses the importance of dynamical energy conversion in the UMLT.

Atmospheric turbulence spectrum in high resolution mode detected by a high power-aperture sodium lidar over Yanqing, Beijing (40.47°N, 115.97°E)

With the support of the Meridian Project of China, an integrated two-wavelength and three-channel lidar system was completed for atmospheric detection in Yanqing, Beijing. This lidar system emitted a 589 nm laser beam for detecting the 80–110 km sodium atomic layer. A 532 nm laser system was used to detect the 30–80 km atmospheric temperature and density and also to provide the atmosphere aerosol information from the ground to a height of 30 km. Because all of the technique details were carefully designed, this lidar obtained a very good signal and achieved the full-scale detection of the atmosphere from near-ground to a height of 110 km. With the conventional mode (3 minutes, 100 m), the atmospheric 532 nm Rayleigh detecting channel could reach a height of 80 km, and the 589 nm channel photo counting peak could reach 37000. This specification actually reached the high power-aperture sodium lidar standard in advance, which caused the high-resolution detection mode (15 m, 6.7 s) to be realized. For the high-resolution detecting mode of our Yanqing Lidar, observations of the spectrum of a gravity wave in the turbulence region were given according to the advantages of the high signal/noise ratio. Additionally, the properties of the turbulence spectrum and their variations were first obtained in the northern hemisphere over China. The observed gravity wave turbulence spectra well fitted with the theoretical prediction. These results achieved new field detections, which will lead us to further discover the nature and essence of a gravity wave and deeply study the behavior of upper atmosphere circulation on a large scale.

Saturation of the Inverse Cascade in Surface Gravity-Wave Turbulence

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-crested waves.

Emergence of Peregrine solitons in integrable turbulence of deep water gravity waves

We study experimentally the early stages of integrable turbulence of unidirectional deep water gravity waves. By generating partially coherent waves in a 148 m long wave flume, we observe the emergence of high-amplitude structures formed by nonlinear focusing, commonly referred to as rogue waves. This work confronts the experiment with two recent results obtained in the framework of the nonlinear Schrodinger equation (NLSE), namely that (i) these structures can be locally fitted by a Peregrine soliton and (ii) their emergence leaves a visible trace on the evolution of statistical parameters such as kurtosis. Although Peregrine solitons have been observed for almost ten years in experiments using a deterministic forcing, we report their first systematic study in hydrodynamics with a random forcing. We show that (i) yields accurate results as long as the wave steepness remains moderate, whereas (ii) is very robust and remains valid beyond the assumption of integrability. Numerical simulations of the NLSE and of the fully nonlinear dynamical equations are also performed to support these results.

Generation of weakly nonlinear turbulence of internal gravity waves in the Coriolis facility

We investigate experimentally stratified turbulence forced by waves. Stratified turbulence is present in oceans and it is expected to be dominated by nonlinear interaction of internal gravity waves as described by the Garrett & Munk spectrum. In order to reach turbulent regimes dominated by stratification we use the Coriolis facility in Grenoble (France) which large size enables us to reach regimes with both low Froude number and large Reynolds number. Stratification is obtained by using vertically linearly varying salt concentration and we force large scale waves in a $6\times6\times 1$ m$^3$ domain. We perform time-resolved PIV to probe the space-time structure of the velocity field. We observe a wide band spectrum which is made of waves. Discrete modes are observed due to the square shape of the flow container as well as a continuum part which appears consistent with an axisymmetric superposition of random weakly nonlinear waves. Our observations support the interpretation of turbulence of a strongly stratified fluid as wave turbulence of internal waves although our spectrum is quite different from the Garrett & Munk spectrum. Weak turbulence proceeds down to a small cutoff length scale (the buoyancy wavelength) at which a transition to more strongly nonlinear turbulence is expected.

Energy-based analysis and anisotropic spectral distribution of internal gravity waves in strongly stratified turbulence

Stratified turbulence shows scale- and direction-dependent anisotropy and the coexistence of weak turbulence of internal gravity waves and strong turbulence of eddies. Straightforward application of standard analyses developed in isotropic turbulence sometimes masks important aspects of the anisotropic turbulence. To capture detailed structures of the energy distribution in the wave-number space, it is indispensable to examine the energy distribution with non-integrated spectra by fixing the codimensional wave-number component or in the two-dimensional domain spanned by both the horizontal and vertical wave numbers. Indices which separate the range of the anisotropic weak-wave turbulence in the wave-number space are proposed based on the decomposed energies. In addition, the dominance of the waves in the range is also verified by the small frequency deviation from the linear dispersion relation. In the wave-dominant range, the linear wave periods given by the linear dispersion relation are smaller than approximately one third of the eddy-turnover time. The linear wave periods reflect the anisotropy of the system, while the isotropic Brunt-V\"ais\"al\"a period is used to evaluate the Ozmidov wave number, which is necessarily isotropic. It is found that the time scales in consideration of the anisotropy of the flow field must be appropriately selected to obtain the critical wave number separating the weak-wave turbulence.

Identifying four-wave-resonant interactions in a surface gravity wave turbulence experiment

The nonlinear dynamics of waves at the sea surface is believed to be ruled by the Weak Turbulence framework. In order to investigate the nonlinear coupling among gravity surface waves, we developed an experiment in the Coriolis facility which is a 13-m diameter circular tank. An isotropic and statistically stationary wave turbulence of average steepness of 10\% is maintained by two wedge wave makers. The space and time resolved wave elevation is measured using a stereoscopic technique. Wave-wave interactions are analyzed through third and fourth order correlations. We investigate specifically the role of bound waves generated by non resonant 3-wave coupling. Specifically, we implement a space-time filter to separate the dynamics of free waves (i.e. following the dispersion relation) from the bound waves. We observe that the free wave dynamics causes weak resonant 4-wave correlations. A weak level of correlation is actually the basis of the Weak Turbulence Theory. Thus our observations support the use of the Weak Turbulence to model gravity wave turbulence as is currently been done in the operational models of wave forecasting. Although in the theory bound waves are not supposed to contribute to the energy cascade, our observation raises the question of the impact of bound waves on dissipation and thus on energy transfers as well.

The transition of gravity wave or mesoscale flow to boundary layer turbulence and implications for infrasound propagation conditions

Knowledge of the atmospheric boundary layer (ABL) is important for weather and climate forecasting as well as infrasound detection. In spite of many advances, the ABL is still not represented realistically in weather models, which leads to uncertainties in weather predictions. At the same time, the influence of the ABL on infrasound detectability is not well understood, which potentially limits the capability of the technique. In this work, a network of anemometers, microbarometers, and ceilometers is used to characterize gravity wave activity and turbulence in the ABL. The sensors are co-located at the Cabauw Experimental Site for Atmospheric Research (CESAR). The sensor network allows for the estimation of velocity and pressure spectra, as well as boundary-layer structure and mixing height. The goal of this work is to study the transition from gravity wave or mesoscale flow to boundary layer turbulence and to consider implications for infrasound propagation conditions. Here, an analysis of spectra under convective and stable conditions is presented with respect to both gravity wave and boundary-layer turbulence theory. Observed distinctions in the velocity and pressure spectral transitions will be discussed, including large-scale turbulence and the mesoscale spectral gap.Knowledge of the atmospheric boundary layer (ABL) is important for weather and climate forecasting as well as infrasound detection. In spite of many advances, the ABL is still not represented realistically in weather models, which leads to uncertainties in weather predictions. At the same time, the influence of the ABL on infrasound detectability is not well understood, which potentially limits the capability of the technique. In this work, a network of anemometers, microbarometers, and ceilometers is used to characterize gravity wave activity and turbulence in the ABL. The sensors are co-located at the Cabauw Experimental Site for Atmospheric Research (CESAR). The sensor network allows for the estimation of velocity and pressure spectra, as well as boundary-layer structure and mixing height. The goal of this work is to study the transition from gravity wave or mesoscale flow to boundary layer turbulence and to consider implications for infrasound propagation conditions. Here, an analysis of spectra under...

Three-wave and four-wave interactions in gravity wave turbulence

Water surface deformation is reconstructed by stereoscopic imaging to develop a high-order statistical analysis of randomly forced turbulent gravity waves. Resonant wave coupling is investigated and laboratory data compared to field measurements in the Black Sea.