Onset Of Wave Breaking Research Articles

Numerical modelling of wind-modified focused waves in a numerical wave tank

In this paper, we extend the non-breaking second-order corrected focused wave experimental results of Sriram et al. (2015) in a numerical wave tank based on HOS in the presence of wind. The wind is modelled using the modified Jeffrey's Sheltering Mechanism (Kharif et al., 2008). We take care to preserve the consistency of application of pressure terms in the hybrid Stokesian-HOS formulations when applying wind. We confirm the results of Kharif et al. (2008) i.e. the focusing points in presence of wind show an amplification of the measure wave heights although no discernible shifting of focusing location is observed. However, compared to Kharif et al. (2008), the sheltering mechanism is found to be very weak since it depends on exceedance of a threshold slope based on constant steepness or constant amplitude spectrum. The energy flux to local phase celerity (Bx) both in absence and presence of the aforementioned wind model for the detection of onset of wave breaking (Barthelemy et al., 2015b; Saket et al., 2017b) are reported. Moreover, in the absence of wind, if Bx is used as parameter the onset of wave breaking, then for intermediate water wave groups its threshold should be close to 0.9 (computations based on Kurnia and van Groesen (2014)).

Orbitally shaken shallow fluid layers. I. Regime classification

Orbital shakers are simple devices that provide mixing, aeration, and shear stress at multiple scales and high throughput. For this reason, they are extensively used in a wide range of applications from protein production to bacterial biofilms and endothelial cell experiments. This study focuses on the behaviour of orbitally shaken shallow fluid layers in cylindrical containers. In order to investigate the behaviour over a wide range of different conditions, a significant number of numerical simulations are carried out under different configuration parameters. We demonstrate that potential theory—despite the relatively low Reynolds number of the system—describes the free-surface amplitude well and the velocity field reasonably well, except when the forcing frequency is close to a natural frequency and resonance occurs. By classifying the simulations into non-breaking, breaking, and breaking with part of the bottom uncovered, it is shown that the onset of wave breaking is well described by Δh/(2R) = 0.7Γ, ...

Turbulent Flow over Steep Steady and Unsteady Waves under Strong Wind Forcing

AbstractTurbulent flow over strongly forced steep steady and unsteady waves is simulated using large-eddy simulation (LES) with timetand spacexvarying wave heighth(x,t) imposed as a lower boundary condition. With steady waves,h(x,t) is based on measurements of incipient and active breaking waves collected in a wind-wave flume, while a numerical wave code is used to generate an unsteady evolving wave packet (group). Highly intermittent airflow separation is found in the simulations, and the results suggest separation near a wave crest occurs prior to the onset of wave breaking. The form (pressure) drag is most sensitive to the wave slope, and the form drag can contribute as much as 74% to the total stress. Wind and scalar profiles from the LES display log-linear variations above the wave surface; the LES wind profiles are in good agreement with the measurements. The momentum roughness increases as the water surface changes from wind ripples to incipient breaking to active breaking. However, the scalar roughness decreases as the wave surface becomes rougher. This highlights major differences in momentum and scalar transport over a rough wavy surface. For a rapidly evolving, strongly forced wave group, the form drag is highly correlated with the wave slope, and intermittent separation is found early in the packet evolution when the local wave slope −∂h/∂x(x,t) ≥ 0.22. The packet root-mean-square wave slope is 0.084, but the form drag fraction is 2.4 times larger than a comparably forced steady wave. Thus, a passing wave group can induce unsteadiness in the wind stress.

Impacts of wave age on turbulent flow and drag of steep waves

Turbulent flow over steep steady and unsteady wave trains with varying height h(x, t) and propagation speed c is simulated using large-eddy simulation (LES) in a wind-wave channel [17]. The imposed waveshape with steady wave trains is based on measurements of incipient and active breaking waves collected in a wind-wave tank, while a numerical wave code is used to generate an unsteady evolving wave train (or group) [3]. For the adopted waveshapes, process studies are carried out varying the wave age c/u* from ~ 1 to 10: the airflow friction velocity is u*. Under strong wind forcing or low wave age c/u* ~ 1, highly intermittent airflow separation is found in all simulations and the results suggest separation near a wave crest occurs prior to the onset of wave breaking. As wave age increases flow separation is delayed or erased for both steady and unsteady wave trains. Flow visualization shows that near the wave surface vertical velocity w and waveslope ∂h/∂x are positively correlated at c/u* ~ 1 but are negatively correlated at c/u* = 10. The vertical speed of the underlying wave oscillations depends on the local waveslope, increases with phase speed, and is a maximum on the leeward side of the wave. Vigorous boundary movement [8] appears to alter the unsteady flow separation patterns which leads to a reduction in form (pressure) drag as wave age increases. For example, the pressure contribution to the total drag of the active breaker wave train decreases from 74% at c/u* = 1.23 to less than 20% at c/u* = 10. Critical layer dynamics appears to play a secondary role in the air-wave coupling over steep waves, but requires further investigation. For all simulations, the form drag is found to be strongly dependent on both waveslope ∂h/∂x and wave age c/w*. The simulations are in good agreement with experimental results for turbulent flow over steep waves under strong wind forcing.

Simulation of breaking waves using the high-order spectral method with laboratory experiments: wave-breaking energy dissipation

We examine the implementation of a wave-breaking mechanism into a nonlinear potential flow solver. The success of the mechanism will be studied by implementing it into the numerical model HOS-NWT, which is a computationally efficient, open source code that solves for the free surface in a numerical wave tank using the high-order spectral (HOS) method. Once the breaking mechanism is validated, it can be implemented into other nonlinear potential flow models. To solve for wave-breaking, first a wave-breaking onset parameter is identified, and then a method for computing wave-breaking associated energy loss is determined. Wave-breaking onset is calculated using a breaking criteria introduced by Barthelemy et al. (J Fluid Mech https://arxiv.org/pdf/1508.06002.pdf , submitted) and validated with the experiments of Saket et al. (J Fluid Mech 811:642–658, 2017). Wave-breaking energy dissipation is calculated by adding a viscous diffusion term computed using an eddy viscosity parameter introduced by Tian et al. (Phys Fluids 20(6): 066,604, 2008, Phys Fluids 24(3), 2012), which is estimated based on the pre-breaking wave geometry. A set of two-dimensional experiments is conducted to validate the implemented wave breaking mechanism at a large scale. Breaking waves are generated by using traditional methods of evolution of focused waves and modulational instability, as well as irregular breaking waves with a range of primary frequencies, providing a wide range of breaking conditions to validate the solver. Furthermore, adjustments are made to the method of application and coefficient of the viscous diffusion term with negligible difference, supporting the robustness of the eddy viscosity parameter. The model is able to accurately predict surface elevation and corresponding frequency/amplitude spectrum, as well as energy dissipation when compared with the experimental measurements. This suggests the model is capable of calculating wave-breaking onset and energy dissipation successfully for a wide range of breaking conditions. The model is also able to successfully calculate the transfer of energy between frequencies due to wave focusing and wave breaking. This study is limited to unidirectional waves but provides a valuable basis for future application of the wave-breaking model to a multidirectional wave field. By including parameters for removing energy due to wave-breaking into a nonlinear potential flow solver, the risk of developing numerical instabilities due to an overturning wave is decreased, thereby increasing the application range of the model, including calculating more extreme sea states. A computationally efficient and accurate model for the generation of a nonlinear random wave field is useful for predicting the dynamic response of offshore vessels and marine renewable energy devices, predicting loads on marine structures, and in the study of open ocean wave generation and propagation in a realistic environment.

Simulation of breaking waves using the high-order spectral method with laboratory experiments: Wave-breaking onset

This study investigates a wave-breaking onset criteria to be implemented in the non-linear potential flow solver HOS-NWT. The model is a computationally efficient, open source code, which solves for the free surface in a numerical wave tank using the High-Order Spectral (HOS) method. The goal of this study is to determine the best method to identify the onset of random single and multiple breaking waves over a large domain at the exact time they occur. To identify breaking waves, a breaking onset criteria based on the ratio of local energy flux velocity to the local crest velocity, introduced by Barthelemy et al. (2017) is selected. The breaking parameter is uniquely applied in the numerical model in that calculations of the breaking onset criteria ratio are not made only at the location of the wave crest, but at every point in the domain and at every time step. This allows the model to calculate the onset of a breaking wave the moment it happens, and without knowing anything about the wave a priori. The application of the breaking criteria at every point in the domain and at every time step requires the phase velocity to be calculated instantaneously everywhere in the domain and at every time step. This is achieved by calculating the instantaneous phase velocity using the Hilbert transform and dispersion relation. A comparison between more traditional crest-tracking techniques shows the calculation of phase velocity using Hilbert transform at the location of the breaking wave crest provides a good approximation of crest velocity.The ability of the selected wave breaking criteria to predict single and multiple breaking events in two dimensions is validated by a series of large-scale experiments. Breaking waves are generated by energy focusing and modulational instability methods, with a wide range of primary frequencies. Steep irregular waves which lead to breaking waves, and irregular waves with an energy focusing wave superimposed are also generated. This set of waves provides a wide range of breaking-wave strengths, types and scales for validation of the model. A comparison of calculations made using HOS-NWT with experimental measurements show that the model is successful at predicting the occurrence of wave breaking, as well as accurately calculating breaking onset time and location.Although the current study is limited to a unidirectional wave field, the success of the wave-breaking model presented provides the basis for application of the model in a multidirectional wave field. By including wave breaking onset with the addition of an appropriate energy dissipation model into HOS-NWT, we can increase the application range of the model, as well as decrease the occurrence of numerical instabilities that are associated with breaking waves in a potential flow solver. An accurate description of the wave field is useful for predicting the dynamic response of offshore vessels and marine renewable energy devices, predicting loads on marine structures and the general physics of ocean waves, for example.

Diagnosing the Horizontal Propagation of Rossby Wave Packets along the Midlatitude Waveguide

AbstractIt has been suggested that upper-tropospheric Rossby wave packets propagating along the midlatitude waveguide may play a role for triggering severe weather. This motivates the search for robust methods to detect and track Rossby wave packets and to diagnose their properties. In the framework of several observed cases, this paper compares different methods that have been proposed for these tasks, with an emphasis on horizontal propagation and on a particular formulation of a wave activity flux previously suggested by Takaya and Nakamura. The utility of this flux is compromised by the semigeostrophic nature of upper-tropospheric Rossby waves, but this problem can partly be overcome by a semigeostrophic coordinate transformation. The wave activity flux allows one to obtain information from a single snapshot about the meridional propagation, in particular propagation from or into polar and subtropical latitudes, as well as about the onset of wave breaking. This helps to clarify the dynamics of individual wave packets in cases where other, more conventional methods provide ambiguous or even misleading information. In some cases, the “true dynamics” of the Rossby wave packet turns out to be more complex than apparent from the more conventional diagnostics, and this may have important implications for the predictability of the wave packet.

A Large-Eddy Simulation Study of Atmospheric Boundary Layer Influence on Stratified Flows over Terrain

Abstract The impact of atmospheric boundary layer (ABL) interactions with large-scale stably stratified flow over an isolated, two-dimensional hill is investigated using turbulence-resolving large-eddy simulations. The onset of internal gravity wave breaking and leeside flow response regimes of trapped lee waves and nonlinear breakdown (or hydraulic-jump-like state) as they depend on the classical inverse Froude number, Fr−1 = Nh/Ug, is explored in detail. Here, N is the Brunt–Väisälä frequency, h is the hill height, and Ug is the geostrophic wind. The results here demonstrate that the presence of a turbulent ABL influences mountain wave (MW) development in critical aspects, such as dissipation of trapped lee waves and amplified stagnation zone turbulence through Kelvin–Helmholtz instability. It is shown that the nature of interactions between the large-scale flow and the ABL is better characterized by a proposed inverse compensated Froude number, = N(h − zi)/Ug, where zi is the ABL height. In addition, it is found that the onset of the nonlinear-breakdown regime, ≈ 1.0, is initiated when the vertical wavelength becomes comparable to the sufficiently energetic scales of turbulence in the stagnation zone and ABL, yielding an abrupt change in leeside flow response. Finally, energy spectra are presented in the context of MW flows, supporting the existence of a clear transition in leeside flow response, and illustrating two distinct energy distribution states for the trapped-lee-wave and the nonlinear-breakdown regimes.

Legendre Pseudospectral Approximation of Boussinesq Systems and Applications to Wave Breaking

In this paper, we propose a spectral projection of a regularized Boussinesq system for wave propagation on the surface of a fluid. The spectral method is based on the use of Legendre polynomials, and is able to handle time-dependent Dirichlet boundary conditions with spectral accuracy. The algorithm is applied to the study of undular bores, and in particular to the onset of wave breaking connected with undular bores. As proposed in (2), an improved version of the breaking criterion recently introduced in (5) is used. This tightened breaking criterion together with a careful choice of the relaxation parameter yields rather accurate predictions of the onset of breaking in the leading wave of an undular bore. AMS subject classifications: 35Q53, 65M70, 76B15, 76B25

Raman amplification in the coherent wave-breaking regime.

In regimes far beyond the wave-breaking threshold of Raman amplification, we show that significant amplification can occur after the onset of wave breaking, before phase mixing destroys the coherent coupling between pump, probe, and plasma wave. Amplification in this regime is therefore a transient effect, with the higher-efficiency "coherent wave-breaking" (CWB) regime accessed by using a short, intense probe. Parameter scans illustrate the marked difference in behavior between below wave breaking, in which the energy-transfer efficiency is high but total energy transfer is low, wave breaking, in which efficiency is low, and CWB, in which moderate efficiencies allow the highest total energy transfer.

Capillary effects on wave breaking

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.

Determining the onset and strength of unforced wave breaking in a numerical wave tank

A numerical wave tank is used to investigate the onset and strength of unforced wave breaking, and the waves have three types of initial spectra: constant amplitude spectrum, constant steepness spectrum and Pierson-Moscowitz spectrum. Numerical tests are performed to validate the model results. Then, the onset of wave breaking is discussed with geometric, kinematic, and dynamic breaking criteria. The strength of wave breaking, which is always characterized by the fractional energy loss and breaking strength coefficient, is studied for different spectra. The results show how the energy growth rate is better than the initial wave steepness on estimating the fractional energy losses as well as breaking strength coefficient.

Initial wave breaking dynamics of Peregrine-type rogue waves: A numerical and experimental study

The Peregrine breather, today widely regarded as a prototype for spatio-temporally localized rogue waves on the ocean caused by nonlinear focusing, is analyzed by direct numerical simulations based on two-phase Navier–Stokes equations. A finite-volume approach with a volume of fluid method is applied to study the Peregrine breather dynamics up to the initial stages of wave breaking. The comparison of the numerical results with laboratory experiments to validate the numerical approach shows very good agreement and suggests that the chosen method is an effective tool to study modulation instability and breather dynamics in water waves with high accuracy even up to the onset of wave breaking. The numerical results also indicate some previously unnoticed characteristics of the flow fields below the water surface of breathers, which might be of significance for short-term prediction of rogue waves. Recurrent wave breaking is also observed.

Bathymetric control of surf zone retention on a rip-channelled beach

Simulations from a numerical model address the impact of nearshore morphology on surf zone retention on, open coast, rip-channelled beaches exposed to shore-normal waves. In the model, rip channels are regularly spaced alongshore with a given spacing λ. For a given reference case bathymetry (λ= 200 m), rip current circulations retain floating material at a hourly rate R of about 80 % which is in line with most existing field and laboratory studies in similar settings. The influence of a surf zone rip-channel morphology on surf zone retention is evaluated by a number of morphologic parameters. Results show that rip spacing is important. The ratio of the surf zone width X s to rip spacing λ controls surf zone retention with R rapidly increasing with increasing X s /λ up to a threshold of about 1 above which R levels off to become asymptotic to 100 %. The impact of the presence of a rip head bar is profound but nonlinear. The onset of wave breaking across the rip head bar drives a weak seaward located circulation providing major pathways for surface water exiting the surf zone compartment. Additional simulations suggest that alongshore variations in the offshore bathymetry are important. Patterns in the wave field enforced by wave refraction and potentially wave breaking across offshore bathymetric anomalies can provide a conduit for transporting floating material out of the surf zone and into the inner shelf region. This has major implications for surf zone flushing by inner-bar rips on multiple-barred beaches and on beaches facing bathymetric anomalies on the inner shelf.

Rip currents and circulation on a high-energy low-tide-terraced beach (Grand Popo, Benin, West Africa)

Castelle, B., Almar, R., Dorel, M., Lefebvre, J.P., Sénéchal, N., Anthony, E.J., Laibi, R., Chuchla, R., du Penhoat, Y., 2014. Rip currents and circulation on a high-energy low-tide-terraced beach (Grand Popo, Benin, West Africa). In: Green, A.N. and Cooper, J.A.G. (eds.), Proceedings 13th International Coastal Symposium (Durban, South Africa), Journal of Coastal Research, Special Issue No. 70, pp. 633–638, ISSN 0749-0208.Rip currents are wave-driven intense seaward-flowing jets of water that are important to both beach morphodynamics and the overall ecosystem. Rip currents are also the leading deadly hazard to recreational beach users worldwide. More specifically, the African region is reported to have the highest rates of drowning in the world, yet both the occurrence and the type of rips developing along the African beaches are unknown. In February 2013, a 12-day field experiment was performed at the high-energy low-tide-terraced sandy beach of Grand Popo beach (Benin, West Africa). Human drifter data and video imagery are combined to address wave-driven circulation and rip current activity. Results show two prevailing rip current types. (1) Low-energy (~ 0.2–0.4 m/s) swash rips, with short life-spans of about 1 minute, extend about 5–10 m offshore and occur preferably at mid to high tide at fixed locations in the center of beach cusps. (2) Higher-energy (0.2–0.8 m/s) surfzone flash rips become active with the onset of intense wave breaking across the low-tide terrace. They tend to migrate downdrift with a longer time-span of about 2–5 minutes. The relatively weak longshore current (0.2–0.55 m/s) measured during the experiment suggests that flash rips were driven by vortices generated by wave breaking rather than shear instabilities of the longshore current. Swash rips and flash rips are common at Grand Popo and often co-exist. We propose a conceptual model of both flash and swash rip activity on this stretch of the West African coast.

Horizontal circulation and jumps in Hamiltonian wave models

Abstract. We are interested in the modelling of wave-current interactions around surf zones at beaches. Any model that aims to predict the onset of wave breaking at the breaker line needs to capture both the nonlinearity of the wave and its dispersion. We have therefore formulated the Hamiltonian dynamics of a new water wave model, incorporating both the shallow water and pure potential flow water wave models as limiting systems. It is based on a Hamiltonian reformulation of the variational principle derived by Cotter and Bokhove (2010) by using more convenient variables. Our new model has a three-dimensional velocity field consisting of the full three-dimensional potential velocity field plus extra horizontal velocity components. This implies that only the vertical vorticity component is nonzero. Variational Boussinesq models and Green–Naghdi equations, and extensions thereof, follow directly from the new Hamiltonian formulation after using simplifications of the vertical flow profile. Since the full water wave dispersion is retained in the new model, waves can break. We therefore explore a variational approach to derive jump conditions for the new model and its Boussinesq simplifications.

BREAKING WAVE KINEMATICS, LOCAL PRESSURES, AND FORCES ON A TRIPOD STRUCTURE

This paper presents breaking wave loads on a tripod structure from physical model tests and numerical simulations. The large scale model tests (1:12) are described as well as the validation of the three dimensional numerical model by comparison of CFD wave gauge data and pressures with measurements in the large wave flume inside and outside the impact area. Subsequently, the impact areas due to a broken wave, a curled wave front as well as for wave breaking directly at the structure with a partly vertical wave front are compared to each other. Line forces in terms of slamming coefficients with variation in time and space are derived from CFD results and the velocity distribution is presented at the onset of wave breaking. Finally, the results are briefly discussed in comparison to other slamming studies.

Double shock dynamics induced by the saturation of defocusing nonlinearities

We show that the saturation of defocusing nonlinearities leads to qualitative changes in the onset of wave breaking, determining double shock formation whose regularization occurs in terms of antidark solitons. In a given material, the crossover between different regimes can be controlled by changing the input intensity.

An eddy viscosity model for two-dimensional breaking waves and its validation with laboratory experiments

An eddy viscosity model to describe energy dissipation in two-dimensional breaking waves in deep water is implemented in a numerical model for the evolution of nonlinear surface waves and evaluated with experimental results. In the experiments, to develop a reliable eddy viscosity model, breaking waves are generated by both energy focusing and modulated wave groups. Local wave parameters prior to and following breaking are defined and then determined. Significant correlations between the pre-breaking and post-breaking parameters are identified and adopted in the eddy viscosity model. The numerical model detects automatically wave breaking onset based on local surface slope, determines pre-breaking local wave parameters, predicts post-breaking time and length scales, and estimates eddy viscosity to dissipate energy in wave breaking events. Numerical simulations with the model are performed and compared to the experiments. It is found that the model predicts well the total energy dissipation due to breaking waves. In addition, the computed surface elevations after wave breaking agree reasonably well with the measurements for the energy focusing (plunging) wave groups. However, for breaking wave groups due to modulational instability (plunging and spilling), a relatively large discrepancy between the surface elevation predictions and the experimental measurements is observed, in particular, at the downstream wave probe locations. This is possibly due to wave reflection and three-dimensionality in the experiments. To further validate the eddy viscosity model, the evolution of highly nonlinear irregular waves is studied numerically and the numerical solutions are compared with additional independent laboratory experiments for long-crested irregular waves. It is shown that the numerical model is capable of predicting the wave evolution subsequent to wave breaking.

Frequency spectra evolution of two-dimensional focusing wave groups in finite depth water

AbstractAn experimental and numerical study of the evolution of frequency spectra of dispersive focusing wave groups in a two-dimensional wave tank is presented. Investigations of both non-breaking and breaking wave groups are performed. It is found that dispersive focusing is far more than linear superposition, and that it undergoes strongly nonlinear processes. For non-breaking wave groups, as the wave groups propagate spatial evolution of wave frequency spectra, spectral bandwidth, surface elevation skewness, and kurtosis are examined. Nonlinear energy transfer between the above-peak ($f/ {f}_{p} = 1. 2{{\ndash}}1. 5$) and the higher-frequency ($f/ {f}_{p} = 1. 5\text{{\ndash}} 2. 5$) regions, with ${f}_{p} $ being the spectral peak frequency, is demonstrated by tracking the energy level of the components in the focusing and defocusing process. Also shown is the nonlinear energy transfer to the lower-frequency components that cannot be detected easily by direct comparisons of the far upstream and downstream measurements. Energy dissipation in the spectral peak region ($f/ {f}_{p} = 0. 9\text{{\ndash}} 1. 1$) and the energy gain in the higher-frequency region ($f/ {f}_{p} = 1. 5\text{{\ndash}} 2. 5$) are quantified, and exhibit a dependence on the Benjamin–Feir Index (BFI). In the presence of wave breaking, the spectral bandwidth reduces as much as 40 % immediately following breaking and eventually becomes much smaller than its initial level. Energy levels in different frequency regions are examined. It is found that, before wave breaking onset, a large amount of energy is transferred from the above-peak region ($f/ {f}_{p} = 1. 2\text{{\ndash}} 1. 5$) to the higher frequencies ($f/ {f}_{p} = 1. 5\text{{\ndash}} 2. 5$), where energy is dissipated during the breaking events. It is demonstrated that the energy gain in the lower-frequency region is at least partially due to nonlinear energy transfer prior to wave breaking and that wave breaking may not necessarily increase the energy in this region. Complementary numerical studies for breaking waves are conducted using an eddy viscosity model previously developed by the current authors. It is demonstrated that the predicted spectral change after breaking agrees well with the experimental measurements.