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.
Summary
We provide a six-component (6-C) polarization model for P-, SV-, SH-, Rayleigh-, and Love-waves both inside an elastic medium as well as at the free surface. It is shown that single-station ...6-C data comprised of three components of rotational motion and three components of translational motion provide the opportunity to unambiguously identify the wave type, propagation direction, and local P- and S-wave velocities at the receiver location by use of polarization analysis. To extract such information by conventional processing of three-component (3-C) translational data would require large and dense receiver arrays. The additional rotational components allow the extension of the rank of the coherency matrix used for polarization analysis. This enables us to accurately determine the wave type and wave parameters (propagation direction and velocity) of seismic phases, even if more than one wave is present in the analysis time window. This is not possible with standard, pure-translational 3-C recordings. In order to identify modes of vibration and to extract the accompanying wave parameters, we adapt the multiple signal classification algorithm (MUSIC). Due to the strong nonlinearity of the MUSIC estimator function, it can be used to detect the presence of specific wave types within the analysis time window at very high resolution. We show how the extracted wavefield properties can be used, in a fully automated way, to separate the wavefield into its different wave modes using only a single 6-C recording station. As an example, we apply the method to remove surface wave energy while preserving the underlying reflection signal and to suppress energy originating from undesired directions, such as side-scattered waves.
Currently, an international network of operating high-resolution microbarographs was established to record wave-induced pressure variations at the Earth’s surface. Based on these measurements, ...simulations are performed to analyze the characteristics of waves corresponding to the observed variations of atmospheric pressure. Such a mathematical problem involves a set of primitive nonlinear hydrodynamic equations considering lower boundary conditions in the form of pressure variations at the Earth’s surface. Selection of upward propagating acoustic-gravity waves (AGWs) generated or reflected at the Earth’s surface requires the Neumann boundary conditions involving the vertical gradients of vertical velocity at the lower boundary. To analyze the correctness of the mathematical problem, linearized equations are used for small-surface wave amplitudes excited near the ground. Using the relation for wave energy, it is proven that the solution of the boundary problem based on the nondissipative approximation is uniquely determined by the variable pressure field at the Earth’s surface. The respective dissipative problem has also a unique solution with the appropriate choice of lower boundary conditions for temperature and velocity components. To test the numerical algorithm, solutions of the linearized equations for AGW modes are used. Developed boundary conditions are implemented into the model describing acoustic-gravity wave propagation from the surface atmospheric pressure source. Atmospheric waves propagating from the observed surface pressure variations to the upper atmosphere are simulated using the obtained algorithms and the computer codes.
A fully consistent 2D parametric model of wave development under spatially and/or time varying winds is developed. Derived coupled equations are written in their characteristic form to provide ...practical means to rapidly assess how the energy, frequency and direction of dominant surface waves are developing and distributed under varying wind forcing conditions. For young waves, nonlinear interactions drive the peak frequency downshift, and the wind energy input and wave breaking dissipation are governing the wave energy evolution. With a prescribed wind wave growth rate, proportional to (u*/c) squared, wave breaking dissipation must follow a power‐function of the dominant wave slope. For uniform wind conditions, this choice for the growth rate imposes solutions to follow fetch laws, with exponents q = −1/4, p = 3/4 correspondingly. This set of exponents recovers the Toba's laws, and imposes the wave breaking exponent equal to 3. A varying wind direction can then drive spectral peak direction changes, leading to the occurrence of focusing/defocusing wave groups over localized areas where wave‐rays merge and cross. Significant (but finite) local variations of the energy are then expected under varying wind forcing. Propagating away from a stormy area, wave rays generally diverge, leading to dispersive swell systems. Examples of practical applications of this model are provided in (Kudryavtsev et al., 2021, companion paper).
Plain Language Summary
A practical and rapid evaluation of wave conditions under stormy conditions is often required for navigation safety and coastal hazards. Surface waves and breakers are also essential components of the air‐ocean coupled system to possibly control the dynamical evolution of extreme events. Here, a fully consistent 2D parametric model for wave development is solved in the storm frame of reference. Along wave‐rays, peak energy and frequency evolve. Wave‐rays can also bend under changes of the wind direction. The proposed method thus helps rapidly identify localized areas where wave‐rays can merge or cross, leading to dangerous sea state hot spots. The suggested model will efficiently complement operational wave models, to simulate and map surface wave developments generated by moving tropical and extra‐tropical cyclones.
Key Points
A fully consistent 2D parametric model of wave development under space‐time varying winds is suggested
The 2D model is based on first‐principle conservation equations consistently constrained by self‐similar fetch laws
Coupled equations in characteristic form provide rapid assessments on how wave parameters are distributed under varying space‐time wind forcing
Seismic waves radiated by small crustal earthquakes are prone to multiple mode conversions caused by reflection and transmission at interfaces, and scattering by small-scale heterogeneities in the ...bulk of the medium. The goal of this study is to clarify the complex interplay between volume scattering and interface reflections in crustal waveguides and how it will impact the crustal energy propagation. To carry out this task, we have incorporated a rigorous description of wave polarization in the context of Monte–Carlo simulations of the multiple-scattering process by introducing a five-dimensional Stokes vector. To shed light on the wave content of the regional short-period seismic wavefield, we investigate the asymptotic partitioning of seismic energy onto
P
,
SV
and
SH
polarizations in the coda, as well as the angular distribution of energy flux in the waveguide. In full elastic space, equipartition theory predicts that (1) the energy ratio between
P
- and
S
-wave energies tends to
β
3
/
(
2
α
3
)
, (2) an equal distribution of energy among
SV
- and
SH
-waves and (3) that energy fluxes are isotropic. In the presence of interfaces, we find that the isotropy of the wavefield is systematically broken and that energy ratios are shifted to the detriment of
P
-waves and in favor of
SV
-waves in a non-absorbing medium. This implies that a residual polarization is preserved in the waveguide. Through an extensive parametric study, we illustrate in detail how the anisotropy of the wavefield, the partitioning ratios and the shear wave polarization depend on the crustal attenuation parameters. The role of the initial polarization at the source has also been examined. In the case of an explosion and a shear dislocation with equal magnitude, we find that the energy level in the coda can differ by more than one order of magnitude when the effect of crustal scattering becomes very weak compared to reflections or transmissions at interfaces. When comparing different shear dislocation mechanisms, we find that the energy level in the coda can differ by up to 60%. While equipartition, depolarization and coda normalization remain fundamental guides to our understanding of the coda, their application requires a good a priori knowledge of the attenuation properties of the crust.
The 2D‐parametric model suggested in the companion paper is used to simulate waves under tropical cyclones (TCs). Set of equations describing both wind waves and swell evolution in space and time, is ...solved using the method of characteristics. Wave‐ray patterns efficiently chart on how wave trains develop and travel through the TC varying wind field, to leave the storm area as swell systems. Depending on TC main characteristics—maximal wind speed (um), radius (Rm), and translation velocity (V), wave‐train rays superpose to exhibit particular coherent spatial patterns of significant wave height, peak wavelength and direction. Group velocity resonance leads to the appearance of waves with abnormally high energy, further outrunning as long swell systems through the TC front sector. Yet, when the TC translation velocity exceeds a threshold value, waves cannot reach group velocity resonance, and travel backwards, to form a wake of swell systems trailing the forward moving TC. Importantly, the model solutions for TC 2D‐wavefields can be parameterized using 2D self‐similar universal functions. Comparisons between self‐similar solutions and measurements, demonstrate a reasonable agreement to warrant scientific and practical applications. Self‐similar solutions provide immediate estimates of azimuthal‐radial distributions of wave parameters under TCs, solely characterized by arbitrary sets of um, Rm, and V conditions. Self‐similar solutions clearly divide TCs between slow TCs, fulfilling conditions Rm/Lcr > 1, and fast TCs corresponding to Rm/Lcr < 1, where Lcr is a critical fetch. Around the region Rm/Lc = 1, group velocity resonance occurs, leading to the largest possible waves generated by a TC.
Plain Language Summary
A practical and rapid evaluation of wave conditions under tropical cyclone (TC) is often required for navigation safety and coastal hazards. Building on the fully consistent 2D‐parametric model suggested in the companion paper, a method is presented to map the distribution of wave energy, peak frequency and direction along wave‐rays. Wave‐rays help to visualize how wave trains develop and travel through the TC varying wind field, and how they leave the storm area as swell systems. Depending on the main TC characteristics—maximal wind speed (um), radius (Rm), and translation velocity (V)—the most striking feature of wavefields is generally a strong azimuthal asymmetry, resulting from group velocity resonance between traveling waves and moving TC. This effect can lead to extreme waves, further outrunning as swell forerunners in the TC heading direction. Importantly, it is demonstrated that immediate directional characteristics of TC wavefields can be evaluated using 2D self‐similar universal functions. For scientific and practical applications, these solutions provide fast estimates of waves generated by moving TC with arbitrary sets of um, Rm, and V.
Key Points
Superposition of wave‐trains rays provides efficient visualization on how waves develop under tropical cyclone and leave storm area as swell
Parametric model solutions are described using 2D self‐similar universal functions and tested against the measurements
Self‐similar solutions demonstrate good agreement with measurements that warrant their scientific and practical applications
Seismic detection of the martian core Stähler, Simon C.; Khan, Amir; Banerdt, W. Bruce ...
Science,
07/2021, Volume:
373, Issue:
6553
Journal Article
Peer reviewed
Open access
Single seismometer structure
Because of the lack of direct seismic observations, the interior structure of Mars has been a mystery. Khan
et al.
, Knapmeyer-Endrun
et al.
, and Stähler
et al.
used ...recently detected marsquakes from the seismometer deployed during the InSight mission to map the interior of Mars (see the Perspective by Cottaar and Koelemeijer). Mars likely has a 24- to 72-kilometer-thick crust with a very deep lithosphere close to 500 kilometers. Similar to the Earth, a low-velocity layer probably exists beneath the lithosphere. The crust of Mars is likely highly enriched in radioactive elements that help to heat this layer at the expense of the interior. The core of Mars is liquid and large, ∼1830 kilometers, which means that the mantle has only one rocky layer rather than two like the Earth has. These results provide a preliminary structure of Mars that helps to constrain the different theories explaining the chemistry and internal dynamics of the planet.
Science
, abf2966, abf8966, abi7730, this issue p.
434
, p.
438
, p.
443
see also abj8914, p.
388
Data from the InSight mission on Mars help constrain the structure and properties of the martian interior.
Clues to a planet’s geologic history are contained in its interior structure, particularly its core. We detected reflections of seismic waves from the core-mantle boundary of Mars using InSight seismic data and inverted these together with geodetic data to constrain the radius of the liquid metal core to 1830 ± 40 kilometers. The large core implies a martian mantle mineralogically similar to the terrestrial upper mantle and transition zone but differing from Earth by not having a bridgmanite-dominated lower mantle. We inferred a mean core density of 5.7 to 6.3 grams per cubic centimeter, which requires a substantial complement of light elements dissolved in the iron-nickel core. The seismic core shadow as seen from InSight’s location covers half the surface of Mars, including the majority of potentially active regions—e.g., Tharsis—possibly limiting the number of detectable marsquakes.
Wave breaking represents one of the most interesting and challenging problems for fluid mechanics and physical oceanography. Over the last 15 years our understanding has undergone a dramatic leap ...forward, and wave breaking has emerged as a process whose physics is clarified and quantified. Ocean wave breaking plays the primary role in the air-sea exchange of momentum, mass and heat, and it is of significant importance for ocean remote sensing, coastal and ocean engineering, navigation and other practical applications. This book outlines the state of the art in our understanding of wave breaking and presents the main outstanding problems. It is a valuable resource for anyone interested in this topic: researchers, modellers, forecasters, engineers and graduate students in physical oceanography, meteorology and ocean engineering.
Experiments with a weakly damped monopile, either fixed or free to oscillate, exposed to irregular waves in deep water, obtain the wave-exciting moment and motion response. The nonlinearity and peak ...wavenumber cover the ranges:
$\unicodeSTIX{x1D716}_{P}\sim 0.10{-}0.14$
and
$k_{P}r\sim 0.09{-}0.14$
where
$\unicodeSTIX{x1D716}_{P}=0.5H_{S}k_{P}$
is an estimate of the spectral wave slope,
$H_{S}$
the significant wave height,
$k_{P}$
the peak wavenumber and
$r$
the cylinder radius. The response and its statistics, expressed in terms of the exceedance probability, are discussed as a function of the resonance frequency,
$\unicodeSTIX{x1D714}_{0}$
in the range
$\unicodeSTIX{x1D714}_{0}\sim 3{-}5$
times the spectral peak frequency,
$\unicodeSTIX{x1D714}_{P}$
. For small wave slope, long waves and
$\unicodeSTIX{x1D714}_{0}/\unicodeSTIX{x1D714}_{P}=3$
, the nonlinear response deviates only very little from its linear counterpart. However, the nonlinearity becomes important for increasing wave slope, wavenumber and resonance frequency ratio. The extreme response events are found in a region where the Keulegan–Carpenter number exceeds
$KC>5$
, indicating the importance of possible flow separation effects. A similar region is also covered by a Froude number exceeding
$Fr>0.4$
, pointing to surface gravity wave effects at the scale of the cylinder diameter. Regarding contributions to the higher harmonic forces, different wave load mechanisms are identified, including: (i) wave-exciting inertia forces, a function of the fluid acceleration; (ii) wave slamming due to both non-breaking and breaking wave events; (iii) a secondary load cycle; and (iv) possible drag forces, a function of the fluid velocity. Also, history effects due to the inertia of the moving pile, contribute to the large response events. The ensemble means of the third, fourth and fifth harmonic wave-exciting force components extracted from the irregular wave results are compared to the third harmonic FNV (Faltinsen, Newman and Vinje) theory as well as other available experiments and calculations. The present irregular wave measurements generalize results obtained in deep water regular waves.
Capillary effects on wave breaking Deike, Luc; Popinet, Stephane; Melville, W. Kendall
Journal of fluid mechanics,
04/2015, Volume:
769
Journal Article
Peer reviewed
Open access
We investigate the influence of capillary effects on wave breaking through direct numerical simulations of the Navier–Stokes equations for a two-phase air–water flow. A parametric study in terms of ...the Bond number,
$\mathit{Bo}$
, and the initial wave steepness,
${\it\epsilon}$
, is performed at a relatively high Reynolds number. The onset of wave breaking as a function of these two parameters is determined and a phase diagram in terms of
$({\it\epsilon},\mathit{Bo})$
is presented that distinguishes between non-breaking gravity waves, parasitic capillaries on a gravity wave, spilling breakers and plunging breakers. At high Bond number, a critical steepness
${\it\epsilon}_{c}$
defines the onset of wave breaking. At low Bond number, the influence of surface tension is quantified through two boundaries separating, first gravity–capillary waves and breakers, and second spilling and plunging breakers; both boundaries scaling as
${\it\epsilon}\sim (1+\mathit{Bo})^{-1/3}$
. Finally the wave energy dissipation is estimated for each wave regime and the influence of steepness and surface tension effects on the total wave dissipation is discussed. The breaking parameter
$b$
is estimated and is found to be in good agreement with experimental results for breaking waves. Moreover, the enhanced dissipation by parasitic capillaries is consistent with the dissipation due to breaking waves.
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