Nonlinear Wave-wave Interactions Research Articles

Laboratory simulation to study the nonlinear evolution of right-hand circularly polarised wave in laser-produced plasmas

Abstract To simulate the astrophysical scenarios at the scale of laboratory astrophysics, the interaction of high-power lasers with plasmas is increasingly being used. So, in this perspective, a nonlinear wave-wave interaction between right-hand circularly polarised wave (R-mode wave) and electron plasma wave (EPW) is proposed to understand the localization of the pump R-mode wave and associated magnetic turbulence at electron scale in magnetized plasma relevant to laboratory astrophysics. The ponderomotive force and relativistic variation of electron mass are accountable for this nonlinear wave-wave interaction model. For this nonlinear dynamics, coupled model equations for EPW and R-mode wave are developed and solved by laboratory simulations utilizing finite difference and pseudo-spectral methods. The observed simulation findings, like the nonlinear evolution of the pump wave, typical scale size of the formed localized structures, and electron scale turbulent spectra associated with the coherent structures resemble various laboratory and space plasma scenarios. A semi-analytical model in the paraxial limits is also developed to understand the physics behind the localization process of the pump wave inside the plasma medium.

Performance Analysis Of An Innovative Field Measurement Setup For Wave Overtopping At A Dike On A Shallow Foreshore

Low-lying countries typically have mildly-sloping beaches as part of their coastal defense system. Many countries in north-western Europe have coastal urban areas that rely on this type of defense system, which consists of a low-crested impermeable sea dike with a relatively short promenade, and a long (nourished) beach in front that acts as a very/extremely shallow foreshore as defined by Hofland et al. (2017). Along the cross-section of this hybrid beach-dike coastal defense system, storm waves are forced to undergo many transformation processes before they finally overtop the dike. These hydrodynamic processes include shoaling, sea-swell (SS) wave energy transfer to sub- (also infragravity or IG waves) and superharmonics via nonlinear wave-wave interactions, wave dissipation by breaking and bottom friction, reflection against the dike, wave run-up and overtopping on the dike, bore impact on a wall or building, and finally reflection back towards the sea interacting with incoming bores on the promenade. Due to breaking of the SS waves and growth of the IG waves on the shallow foreshore, the IG waves can become as important or even dominant at the toe of the dike (Hofland et al., 2017; Lashley et al., 2020), which influences the overtopping process (van Gent, 1999). Field measurements of all these processes at the same time are very challenging but necessary since field observations do not suffer from scale nor model effects. Field data are therefore crucial to evaluate design methodologies, which rely on physical and numerical modelling. This paper presents the field setup and the design features of the innovative research dike, unique in the world, including a performance analysis.

Field Observations Of The Influence Of Infragravity Waves On Wave Overtopping At A Dike On A Shallow Foreshore

Infragravity (IG) waves are low-frequency waves (0.0033 Hz < f < 0.05 Hz) that exist: (i) being bound to the wave groups of high-frequency sea-swell (SS) waves (0.05 Hz < f < 1 Hz), (ii) generated by the moving breakpoint due to the groupiness of the SS waves (Bertin et al., 2018), and (iii) as free IG waves released from the SS wave groups and reflected from neighbouring coasts (Rijnsdorp et al., 2021). Offshore in deeper water (h/Hm0,o > 4, with h the water depth and Hm0,o the offshore significant wave height), IG waves become most apparent during storm conditions, but are still rather limited in wave height (order of a couple of cm). However, at the toe of coastal structures with a shallow foreshore depth (1 < ht/Hm0,o < 4, with ht the water depth at the toe), the energy of IG waves becomes more significant relative to the SS wave frequency band of the wave spectrum (Hofland et al., 2017), becomes equally important for very shallow foreshores (0.3 < ht/Hm0,o < 1), and even dominates SS wave energy for extremely shallow foreshore depths (ht/Hm0,o < 0.3). IG wave growth on mildly sloped shallow foreshores is a result of IG wave shoaling and SS wave energy transfer to IG waves via nonlinear wave-wave interactions (Bertin et al., 2018), and the relative growth of IG to SS wave energy is further aided by SS wave breaking. Coastal urban areas along the Belgian coast (and in other low-lying countries worldwide) rely on a hybrid beach-dike coastal defense system for protection against flooding. During design storm conditions, the nourished beach erodes to a very to extremely shallow foreshore in front of a low-crested impermeable sea dike. Wave overtopping at this type of structure has been shown to be affected by IG waves, based on physical (Altomare et al., 2016) and numerical modelling (Lashley et al., 2021) for shallow to extremely shallow foreshores. However, field observations of the influence of IG waves on wave overtopping have been scarce (van Gent and Giarusso, 2003) and the influence of free IG waves has not been investigated yet. Indeed, the amount of IG wave energy that is expected to arrive at the coastline, not only depends on IG waves naturally bound to the SS wave groups, but also on free IG waves generated offshore or locally reflected from the coast (edge waves). The Belgian continental shelf is characterized by the prevalence of offshore sandbanks (see Figure 1a), which might be an additional source of free IG waves, due to ss wave breaking on the shallow crest of those banks. These free IG waves are location specific and their existence can only be determined by field observations. Knowledge of the offshore boundary conditions with detailed IG wave information is indispensable for the correct prediction and modelling of surf zone hydrodynamics and beach morphodynamics (Fiedler et al., 2019), and the consequent wave overtopping on the dike. Simultaneous measurements of IG waves, offshore and on the beach-dike system during storm conditions together with wave overtopping has not been done before, as it is complex and needs a specific setup with high precision measurement equipment. In-situ measurements provide an invaluable dataset (i.e., no model nor scale effects) and are essential for validating safety assessment and design methodologies of coastal defense systems.

A numerical study on energy transfer between near-inertial internal waves and super-inertial internal waves in the South China sea under the influence of a typhoon

The energy transfer between near-inertial internal waves (NIWs) and super-inertial internal waves (SIWs) in the South China Sea (SCS) after the passage of Typhoon Son-tinh was investigated on the basis of a three-dimensional numerical model. Our model nicely reproduces the spectral peak (6.6 × 10−2 m2 s−2 cpd−1) within the near-inertial frequency band, which agrees well with that (6.5 × 10−2 m2 s−2 cpd−1) in the in-situ observations at the Xisha mooring. Model results demonstrate that the energy transfer rate from NIWs (0.8f∼1.8f) to SIWs (>1.8f) within the mixed layer in the wake of typhoon Son-tinh is an order of magnitude larger than that during the typhoon-free period. Analogously, the super-inertial shear variance increases by nearly an order of magnitude as well. The increase in the energy of SIWs is mainly due to the energy cascade of NIWs through the nonlinear wave-wave interaction. The interaction between NIWs and SIWs was also revealed by the bicoherence spectrum. Compared with that on the left side of the typhoon track, the NIW kinetic energy on the right side is stronger, where the interaction between NIWs and SIWs is more intense. Several sensitivity experiments were designed to further investigate the effects of three typhoon parameters, namely the radius of maximum typhoon wind speed (Rmax), the maximum typhoon wind speed (Vmax) and the moving speed of typhoon (Um), on the energy transfer rate (ETR) from NIWs to SIWs. It is shown that, the ETR increases linearly with Rmax. Among three parameters, Vmax has the strongest effects on the ETR which grows in power with increasing Vmax. The ETR increases and then decreases with the enhancement of Um, which gets maximum when Um is about 6∼6.5 m/s. Overall, our results highlight the energy transfer between near-inertial internal waves and super-inertial internal waves under the influence of a typhoon.

Numerical simulation of bichromatic wave propagation based on the paddle- and modified mass source wave-maker methods

A second-order numerical wave-maker is realized by combining the paddle wave-maker theory proposed by Schäffer for physical experiments with the Fluent software. The numerical results from the paddle wave-maker method are compared with the results from the modified mass source wave-maker method, the theoretical solutions, and the physical experimental data. The numerical model based on the paddle wave-maker method is verified, and the applicable scopes of the two wave-maker methods are discussed. The paddle wave-maker method is not suitable for bichromatic wave combinations that include shallow-water waves. However, within their common applicable range, the numerical results from the paddle wave-maker method are better than those from the modified mass source wave-maker method, at least for the grid divisions adopted in this study. The effects of the incident wave parameters on the nonlinear wave-wave interaction are also analyzed.

Focused wave group propagation over fringing reef and its impact on the vertical wall mounted on a reef flat: Experiment and RANS modeling

This paper presents flume experiments and numerical simulations of focused wave group propagation over an idealized fringing reef profile and its impact on the vertical wall mounted on reef flat. The large transient wave group is produced over the steep fore-reef slope by focusing JONSWAP wave trains. The numerical simulations are performed using the compressible two-phase OpenFOAM model. Wavelet and FFT analysis of the surface elevation time histories clearly show wave focusing process and the generation of higher harmonics due to the nonlinear wave-wave interactions. For the high peak frequency cases, the low frequency wave motion increases gradually after wave breaking, but the peak frequency motion still dominates energy spectrum partially due to wave defocusing and reformation of broken waves. However, the magnitude of low frequency wave motion is higher than those around peak frequency and high harmonics for the lower peak frequency cases. It is found that the contribution of low frequency wave motion to the total force exerted on the wall is comparable to or higher than peak frequency wave motion. The effect of geometry characters of fringing reef and focusing position on the maximum horizontal force on the vertical wall is investigated utilizing the validated OpenFOAM model.

Field observations and long short-term memory modeling of spectral wave evolution at living shorelines in Chesapeake Bay, USA

Living shorelines as a nature-based solution for climate change adaptation were constructed in many places around the world. The success of this type of projects requires long-term monitoring for adaptive management. The paper presents a novel framework leveraging scientific machine learning methods for accurate and rapid prediction of long-term hydrodynamic forcing impacting living shorelines using short-term measurements of water levels and wind waves in the largest estuary in the U.S. Different from existing data-driven wave prediction models focusing on significant wave heights, this study is focused on the prediction of wave energy spectra in shallow water using winds and tides as the input feature and short-term measurements of wave spectra and water depths as the label. Long Short-Term Memory (LSTM) models were developed using four-month wave measurements in the stormy seasons to predict integral wave parameters and energy spectra for multiple years. The developed models accurately predicted wave heights, peak periods, and energy spectra around the living shorelines, capturing complex wave dynamics, such as wave generation by wind, nonlinear wave-wave interactions, and depth-limited wave breaking in the shallow water of a large estuary. The validated models were then used to determine the long-term wave forcing impacting the living shorelines based on the modeled wave characteristics and spectra. Model results show that the surrogate models utilizing LSTM to predict wave spectra in the frequency domain enable long-term predictions of spectral wave evolution with a minimal computational cost. Our findings provide valuable insights into the efficacy of living shorelines in attenuating wave energy and demonstrate the utility of this approach in assessing the effectiveness of such living shoreline structures.

Observed oceanic response to Tropical Cyclone Amphan (2020) from a subsurface mooring in the Bay of Bengal

Upper ocean responses to the super Tropical Cyclone (TC) Amphan and modulation of energy dissipation by internal tides were investigated using in situ observations from a subsurface mooring in Bay of Bengal (BoB). The observations showed that the entire observed water column (150–800 m) cooled significantly, up to a maximum of 2 ℃, and salinity increased significantly at 150–200 m within one week after the TC, mainly due to the TC-induced upwelling. Dynamic response was characterized by strong near-inertial oscillation above 120 m. The near inertial waves (NIWs) had a maximum velocity and kinetic energy at 112.6 m, about 0.34 m/s and 59.5 J/m3, respectively. The energy spectrum and bicoherence analysis indicated that, the nonlinear wave-wave interaction between NIWs (f) and semi-diurnal tide (D2), as well as the self-interaction of NIWs were enhanced after the passage of TC. The nonlinear wave-wave interaction processes were able to transfer energy to higher mode internal waves, thus accelerating energy attenuation. Hence, the observed e-folding time for near-inertial kinetic energy (6.2 days) was only 41 % of the theoretical e-folding time. This study provided a new evidence for the mechanism of energy dissipation of NIWs and semi-diurnal tide during a TC condition in the southern BoB.

FIELD MEASUREMENTS OF WAVE INTERACTIONS WITH A DIKE ON A SHALLOW FORESHORE USING AN “ARTIFICIAL DIKE” CONCEPT

Low-lying countries typically have mildly-sloping beaches as part of their coastal defence system. Many countries in north-western Europe have coastal urban areas that rely on this type of defence system, which consists of a lowcrested impermeable sea dike with a relatively short promenade, and a long (nourished) beach in front that acts as a very/extremely shallow foreshore as defined by Hofland (2017). Along the cross-section of this hybrid beach-dike coastal defence system, storm waves are forced to undergo many transformation processes before they finally overtop the dike. These hydrodynamic processes include shoaling, sea and swell wave energy transfer to sub- and superharmonics via nonlinear wavewave interactions, wave dissipation by breaking and bottom friction, reflection against the dike, wave run-up and overtopping on the dike, bore impact on a wall or building, and finally reflection back towards the sea interacting with incoming bores on the promenade. Field measurements of all these processes at the same time are very challenging but necessary since these suffer from neither scale nor model effects. Field data are therefore crucial to evaluate design methodologies, which rely on physical and numerical modelling. This paper presents the field setup and the design features of the innovative artificial dike, unique in the world.

Chaos and complexity in the dynamics of nonlinear Alfvén waves in a magnetoplasma.

The nonlinear dynamics of circularly polarized dispersive Alfvén wave (AW) envelopes coupled to the driven ion-sound waves of plasma slow response is studied in a uniform magnetoplasma. By restricting the wave dynamics to a few number of harmonic modes, a low-dimensional dynamical model is proposed to describe the nonlinear wave-wave interactions. It is found that two subintervals of the wave number of modulation k of AW envelope exist, namely, (3/4)kc<k<kc and 0<k<(3/4)kc, where kc is the critical value of k below which the modulational instability (MI) occurs. In the former, where the MI growth rate is low, the periodic and/or quasi-periodic states are shown to occur, whereas the latter, where the MI growth is high, brings about the chaotic states. The existence of these states is established by the analyses of Lyapunov exponent spectra together with the bifurcation diagram and phase-space portraits of dynamical variables. Furthermore, the complexities of chaotic phase spaces in the nonlinear motion are measured by the estimations of the correlation dimension as well as the approximate entropy and compared with those for the known Hénon map and the Lorenz system in which a good qualitative agreement is noted. The chaotic motion, thus, predicted in a low-dimensional model can be a prerequisite for the onset of Alfvénic wave turbulence to be observed in a higher dimensional model that is relevant in the Earth's ionosphere and magnetosphere.

A WAVEWATCH III® model approach to investigating ocean wave source terms for West Africa: Non-linear wave-wave interaction source terms

• Investigation of ocean wave model parameterization schemes for West Africa. • Selection of appropriate non-linear source terms for ocean wave projections. • DIA term best estimates wave heights for sub-sections of West Africa. • Application of no non-linear term best estimates wave heights for the entire region. Ocean wave forecasting for West Africa has mostly relied on global parameterization schemes and data outputs. Investigating non-linear ocean wave-wave interaction source terms ( S nl ) for the region, that forms part of the wave model parameterization schemes, is a contribution of this study towards developing a regional wave modelling scheme for West Africa. The study evaluates five non-linear source term configurations with the WAVEWATCH III® (WW3) numerical ocean wave model version 5.16; (i) No source term applied - NL0, (ii) Discrete Interaction Approximation (DIA) – NL1, (iii) Exact Interaction Approximation – NL2, (iv) Generalized Multiple DIA – NL3, and (v) Two-Scale Approximation – NL4, in order to determine best performing S nl for projecting significant wave heights and directions for the region. The wave simulations were run on three separate grids comprising a low resolution West Africa grid (latitudes 10 o S – 30 o N; longitudes 35 o W – 15 o E), high resolution south-eastern (latitudes 2 o S – 8 o N; longitudes 10 o W – 10 o E) and north-western (latitudes 10 o N – 25 o N; longitudes 30 o W – 10 o W) sub-grids. Simulations for the entire West Africa grid produced higher accuracies for the wave parameters compared to the higher resolution sub-grids. NL0 best estimates significant wave height whereas NL3 best estimates wave directions for the West Africa grid. For combined stations of the south-eastern sub-grid, while NL1 best estimates wave heights, NL3 best estimates the wave directions with possible alternation with NL1 due to insignificant differences. Similarly, for the north-western sub-grid, while NL1 best estimates significant wave heights, NL0 best estimates wave directions with similar ability to alternate with NL2 in projecting wave directions. Generally, insignificant differences exist between the source terms in projecting wave directions, especially for the north-western sub-region, implying that any of the source terms may be used in projecting wave directions without significantly compromising on accuracy. Thus, NL1 which best estimates significant wave heights for both south-eastern and north-western sub-regions can also be used in estimating the wave directions. Similarly, NL0 would be appropriate for the entire West Africa region.

Nonlinear excitation of zonal flows by turbulent energy flux.

The nonlinear excitation of zonal flows (ZFs) generated by the ion-temperature-gradient turbulence in a tokamak plasma is investigated by using the global gyrokinetic code nlt. It is found that ZFs are initially driven by the nonlinear self-interaction of the eigenmode. In the nonlinear saturation, the modulational instability becomes important, and its contribution to ZFs can finally be comparable to that of the self-interaction mechanism. More importantly, both types of nonlinear wave-wave interactions can be well described by the turbulent energy flux model.

A Generalized Two–Scale Approximation for Ocean Wave Models

The two-scale approximation (hereafter, TSA) was previously presented as a new method to approximate and estimate transfer rates in wind – wave spectra. It was shown to perform well for a variety of idealized and observed sea state conditions and to compare well with respect to the total Boltzmann integral for nonlinear quadruplet wave-wave interactions. Here, we present a generalized formulation of TSA, allowing for multiple peaked spectra, sheared spectra, sea – swell combinations, etc. This formulation is implemented in a modern operational wave model, WAVEWATCHIII™, and shown to provide a significant improvement over the standard approximation used in wave models, the discrete interaction approximation (DIA). Additional tests involve the simulation of waves generated in Hurricane Teddy (2020).

A Characteristics Set Computation Model for Internal Wavenumber Spectra and Its Validation with MODIS Retrieved Parameters in the Sulu Sea and Celebes Sea

The quasi-linear or nonlinear interactions among different ocean motions dominate the system internal structure and appearance feature presented in spatial and temporal evolution. However, deficiency of the characteristics set computation model for internal wavenumber spectra proves to be a serious barrier to derive interaction mechanisms of internal waves with large or small scale ocean motions. In this study, a characteristics set computation model for internal wavenumber spectra is proposed for complicated offshore environments. The refraction of current shear instability, bottom topography and the reflection at surface and bottom are attentively considered in the complicated characteristics inlaid scheme. Model results are validated with MODIS retrieved internal wave parameters in the Sulu Sea and Celebes Sea. This original characteristics set computation model for internal wavenumber spectra can be used widely and can further improve the understandings of generation, dissipation, nonlinear wave-wave interaction and mixing process of internal waves.

Energy transfer of trapped electron turbulence in tokamak fusion plasmas

The first principle gyrokinetic simulations of trapped electron turbulence in tokamak fusion plasmas demonstrate the energy transfers from the most linearly unstable modes at high k_theta rho _isim 1 to intermediate k_theta via parametric decay process in a short period of linear-nonlinear transition phase. Dominant nonlinear wave-wave interactions occur near the mode rational surface msimeq nq. In fully nonlinear turbulence, inverse energy cascade occurs between a cutoff wave number k_c and k_theta rho _isim 1 with a power law scaling |phi (k_theta )|^2propto k^{-3}, while modes with k<k_c are suppressed. The numerical findings show fair agreement with both the weak turbulence theory and realistic experiments on Tore Supra tokamak.

The Skill Assessment of Weather and Research Forecasting and WAVEWATCH-III Models During Recent Meteotsunami Event in the Persian Gulf

This study aims to use a fully realistic high-resolution mesoscale atmospheric and wave model to reproduce met-ocean conditions during a meteotsunami in the Persian Gulf. The atmospheric simulations were performed with the Weather and Research Forecasting (WRF) model by varying planetary boundary layer, microphysics, cumulus, and radiations parameterizations. The atmospheric results were compared to the meteorological observations (e.g., air pressure and wind speed) from the coastal and island synoptic and buoy stations of the nearest area to the meteotsunami event. The results show that using Mellor-Yamada-Nakanishi-Niino (MYNN) scheme for planetary boundary and surface layer had the best performance for stations over the water, whereas applying Mellor-Yamada-Janjic scheme for planetary boundary and Eta similarity surface layer had the best performance for stations over the land. For wave simulations, the WAVEWATCH-III model was employed with the well-known WAM-Cycle4 formulation and a more recent ST6 package. Six WRF experiments and ERA5 wind data were used to force the wave models. The new error parameter was introduced to identify the optimum wind data for wave simulation. EXP4 configuration which uses the MYNN scheme for planetary boundary and surface layer was led to minimum error, while ERA5 severely underestimated Hs and Tp parameters. For the first time, the Gaussian Quadrature Method (GQM) was implemented in the WAVEWATCH-III model and combined with a depth scale to be used in the Persian Gulf. This method is more accurate for non-linear wave-wave interaction than the default Discrete Interaction Approximation (DIA) method. Lower coefficients for dissipation term were required for GQM and the resulted bulk wave parameters were improved compared to the DIA method. The calibrated ST6 formulation with GQM resulted in a more realistic prediction of wave spectrum than the default settings of the WAVEWATCH-III.

Performance evaluation of spectral wave model caused by ERA-Interim, ERA5, and CFSR wind fields in the Black Sea

The main objective of the present study is to evaluate the performance of the MIKE 21 SW (Spectral Wave) in a semi-closed basin (Black Sea). Wind data were obtained from ECMWF ERA-Interim, ECMWF ERA5, and NCEP CFSR. The wave model was calibrated and validated with wave measurements recorded at seven different stations along the Black Sea coastlines. In the calibration, several different physical parameters were tested to determine the optimal model settings, and the whitecapping parameter (Cds) was more influential than the bottom friction parameter (kn), wave breaking parameter (γ), and nonlinear wave-wave interactions in the prediction of the Black Sea wave properties. The wave results modeled using ERA-Interim showed less agreement with wave measurements than those obtained with ERA-5 and CFSR wind fields. Although the significant wave height and wave period modeled using ERA5 and CFSR wind fields were reasonably well matched at all measurement stations, ERA5 wind fields provided slightly better performance owing to having the largest correlation coefficient (R) and lowest statistical error measures (bias, RMSE, SI) in the Black Sea.

Impact of anticyclonic eddy on nonlinear wave-wave interaction in the southern South China Sea during late summer 2020

Impact of anticyclonic eddy on nonlinear wave-wave interaction in the southern South China Sea during late summer 2020

Two-fluid approach to weak plasma turbulence

Weakly turbulent processes that take place in plasmas are customarily formulated in terms of kinetic theory. However, owing to an inherent complexity associated with the problem, thus far the theory is fully developed largely for unmagnetized plasmas. In the present paper it is shown that a warm two fluid theory can successfully be employed in order to partially formulate the weak turbulence theory in spatially uniform plasma. Specifically, it is shown that the nonlinear wave-wave interaction, or decay processes, can be reproduced by the two-fluid formalism. The present finding shows that the same approach can in principle be extended to magnetized plasmas, which is a subject of future work.

Nonlinear evolution of localized structures and turbulent magnetic field amplification by extraordinary laser beam in laser produced plasmas

In the present investigation, the nonlinear coupling of waves is evaluated to understand the turbulent magnetic field in the laser-produced plasma. The model equations are set up for the extraordinary (x-mode) laser and upper hybrid oscillations, incorporating the relativistic electron mass variation and the nonlinear ponderomotive force. The upper hybrid oscillations are excited due to the nonlinear ponderomotive force exerted by the x-mode laser. The nonlinear coupled system of x-mode laser and the upper hybrid oscillations were simulated numerically. Simulation results depict the evolution of localized structures with time. The ensemble-averaged turbulent power spectrum is obtained through numerical simulations. Furthermore, the frequency spectra associated with upper hybrid oscillation lie in the terahertz (THz) frequency range. A semi-analytical approach in the paraxial approximation has been studied to get a better perspective of the field localization process. Such studies of nonlinear wave-wave interaction are crucial to understand the turbulent magnetic field generation relating to various laboratory and astrophysical scenarios.