Nonlinear Wave-wave Interactions Research Articles

New perspectives on thermosphere tides: 1. Lower thermosphere spectra and seasonal-latitudinal structures

Abstract Thermosphere-Ionosphere-Mesosphere Energetics and Dynamics/Sounding of the Atmosphere using Broadband Emission Radiometry (TIMED/SABER) temperature measurements at 110 km and between ±50° latitude extending from 2002 through 2010 are analyzed to reveal the tidal spectrum entering the ionosphere-thermosphere (IT) system. Seasonal-latitudinal structures are presented for the most prominent spectral components which include DE3, DE2, D0, DW1, DW2, SE1 to SE4, SW1 to SW4, SW6, TE1, TW4, TW5, and TW7. Referring to recent calculations of lower atmosphere heat sources as well as vertical structure characteristics of these waves anticipated from classical tidal theory, we analyze the likely origins of these waves and the nature of their seasonal-latitudinal structures. Several waves are likely to arise through nonlinear wave-wave interactions, and in some cases, this appears to be the sole viable mechanism leading to their existence. The tidal spectrum quantified here is especially relevant to the dynamo generation of electric fields which then impose the tidal variability on the overlying F-region ionosphere. Part 2 of this 2-part study examines penetration of the tidal spectrum to the upper thermosphere. Please see related article: http://www.earth-planets-space.com/content/66/1/122

AN EXPERIMENTAL STUDY ON NONLINEAR WAVE DYNAMICS: ROUGE WAVE GENERATION AND WAVE BLOCKING

Wave modulation due to Bengamin Feir Instability (BFI) in the presence of long wave (swell) is investigated in the elaborate experiments. The mechanically generated waves are composed of three discrete waves; while the newly generated harmonics due to BFI still combining into discrete spectra with the same frequency difference step. These stable discrete waves allow us to further examine the basic properties of nonlinear wave-wave interactions, such as the variation of phase speeds and wave numbers along the wave tank together with speed of crests, especial for the large transient waves. The measured values have a reasonable correspondence to the available nonlinear model estimations. The technique proposed in this study is verified to accurately measure the phase speed and wave number of each harmonic in the wave group. Phase speeds of short waves have a significant amplification in the vicinity of big transient waves, while the speeds of longer waves are close to linear values. The relative long wave (storm wave) travelling with short waves (wind waves) dramatically change the local kurtosis and skewness and may have an important role for the generation of large transient wave and provide an opportunity for triggering the freak waves.

COLLISIONLESS DAMPING OF CIRCULARLY POLARIZED NONLINEAR ALFVÉN WAVES IN SOLAR WIND PLASMAS WITH AND WITHOUT BEAM PROTONS

The damping process of field-aligned, low-frequency right-handed polarized nonlinear Alfven waves (NAWs) in solar wind plasmas with and without proton beams is studied by using a two-dimensional ion hybrid code. The numerical results show that the obliquely propagating kinetic Alfven waves (KAWs) excited by beam protons affect the damping of the low-frequency NAW in low beta plasmas, while the nonlinear wave-wave interaction between parallel propagating waves and nonlinear Landau damping due to the envelope modulation are the dominant damping process in high beta plasmas. The nonlinear interaction between the NAWs and KAWs does not cause effective energy transfer to the perpendicular direction. Numerical results suggest that while the collisionless damping due to the compressibility of the envelope-modulated NAW plays an important role in the damping of the field-aligned NAW, the effect of the beam instabilities may not be negligible in low beta solar wind plasmas.

Simulation of one-dimensional evolution of wind waves in a deep water

A direct wave model based on the one-dimensional nonlinear equations for potential waves is used for simulation of wave field development under the action of energy input, dissipation, and nonlinear wave-wave interaction. The equations are written in conformal surface-fitted nonstationary coordinate system. New schemes for calculating the input and dissipation of wave energy are implemented. The wind input is calculated on the basis of the parameterization developed through the coupled modeling of waves and turbulent boundary layer. The wave dissipation algorithm, introduced to prevent wave breaking instability, is based on highly selective smoothing of the wave surface and surface potential. The integration is performed in Fourier domain with the number of modes M = 2048, broad enough to reproduce the energy downshifting. As the initial conditions, the wave field is assigned as train of Stokes waves with steepness ak = 0.15 at nondimensional wavenumber k = 512. Under the action of nonlinearity and energy input the spectrum starts to grow. This growth is followed by the downshifting. The total time of integration is equal to 7203 initial wave periods. During this time the energy increased by 1111 times. Peak of the spectrum gradually shifts from wavenumber nondimensional k = 512 down to k = 10. Significant wave height increases 33 times, while the peak period increases 51 times. Rates of the peak downshift and wave energy evolution are in good agreement with the JONSWAP formulation.

Observed upper ocean response to typhoon Megi (2010) in the Northern South China Sea

Typhoon Megi passed between two subsurface moorings in the northern South China Sea in October 2010 and the upper ocean thermal and dynamical response with strong internal tides present was examined in detail. The entire observed water column (60–360 m) was cooled due to strong Ekman-pumped upwelling (up to 50 m in the thermocline) by Megi, with maximum cooling of 4.2°C occurring in thermocline. A relatively weak (maximum amplitude of 0.4 m s−1) and quickly damped (e-folding time scale of 2 inertial periods) near-inertial oscillation (NIO) was observed in the mixed layer. Power spectrum and wavelet analyses both indicated an energy peak appearing at exactly the sum frequency fD1 (with maximum amplitude up to 0.2 m s−1) of NIO (f) and diurnal tide (D1), indicating enhanced nonlinear wave-wave interaction between f and D1 during and after typhoon. Numerical experiments suggested that energy transfer from NIO to fD1 via nonlinear interaction between f and D1 may have limited the growth and accelerated the damping of mixed layer NIO generated by Megi. The occurrence of fD1 had a high correlation with NIO; the vertical nonlinear momentum term, associated with the vertical shear of NIO and vertical velocity of D1 or vertical shear of D1 and vertical velocity of NIO, was more than 10 times larger than the horizontal terms and was responsible for forcing fD1. After Megi, surface-layer diurnal energy was enhanced by up to 100%, attributed to the combined effect of the increased surface-layer stratification and additional Megi-forced diurnal current.

Nonlinear dynamical modelling of chaotic electrostatic ion cyclotron oscillations by jerk equations

Plasma being a nonlinear and complex system, is capable of sustaining a wide spectrum of waves, oscillations and instabilities. These fluctuations interact nonlinearly amongst themselves and also with particles: electrons/ions and thus lead to nonlinear wave-wave or wave-particle interaction. In the presence of coherent waves the particles are accelerated whereas irregular oscillations can give rise to particle heating which is also called stochastic heating. Particle orbits are known to be randomized by the wave fields such that their motion can also become stochastic. For fusion to be sustained one needs a very high temperature plasma for an extended duration. It quite common to deploy external waves like electron cyclotron waves or ion cyclotron waves for plasma heating and current drive. These external waves also work only in certain regimes. Conventional plasma techniques have been able to answer several of the observations of the above processes related to heating transport etc, but nonlinear dynamics as a tool has helped in comprehending the plasma oscillations better. We have for the first time obtained a Third Order nonlinear ordinary differential equation (TONLODE) also known as jerk equation to describe the electrostatic ion cyclotron plasma oscillations in a magnetic field. The interesting feature of this equation is that it does not require an external forcing term to obtain chaotic behaviour.

Characterization of intermittent structures in the solar wind

AbstractThe solar wind (SW) is a suitable natural scenario to study the intermittent nature of magnetohydrodynamic (MHD) turbulence for systems with low dissipation rate. In particular, nonlinear wave-wave interactions can be characterized by the degree of phase correlation and by departures from Gaussianity of the magnetic field. In this work, we study in situ observations of magnetic field intensity from the spacecraft ACE, which is located near one astronomical unit from the Sun, in the SW near Earth. We compute the phase coherence index analyzing two sets of observations, each one consisting of approximately three months during 2008 and 2012, respectively. From these sets of data we characterize intermittent features of the magnetic field intensity corresponding to a solar maximum and a solar minimum.

Wave power variability over the northwest European shelf seas

Regional assessments of the wave energy resource tend to focus on averaged quantities, and so provide potential developers with no sense of temporal variability beyond seasonal means. In particular, such assessments give no indication of inter-annual variability – something that is critical for determining the potential of a region for wave energy convertor (WEC) technology. Here, we apply the third-generation wave model SWAN (Simulating Waves Nearshore) at high resolution to assess the wave resource of the northwest European shelf seas, an area where many wave energy test sites exist, and where many wave energy projects are under development. The model is applied to 7years of wind forcing (2005–2011), a time period which witnessed considerable extremes in the variability of the wind (and hence wave) climate, as evidenced by the variability of the North Atlantic Oscillation (NAO). Our simulations demonstrate that there is much greater uncertainty in the NW European shelf wave resource during October–March, in contrast to the period April–September. In the more energetic regions of the NW European shelf seas, e.g. to the northwest of Scotland, the uncertainty was considerably greater. The winter NW European shelf wave power resource correlated well with the NAO. Therefore, provided trends in the NAO can be identified over the coming decades, it may be possible to estimate how the European wave resource will similarly vary over this time period. Finally, the magnitude of wave power estimated by this study is around 10% lower than a resource which is used extensively by the wave energy sector – the Atlas of UK Marine Renewable Energy Resources. Although this can partly be explained by different time periods analysed for each study, our application of a third-generation wave model at high spatial and spectral resolution significantly improves the representation of the physical processes, particularly the non-linear wave-wave interactions.

Study on reverse calculation for unidirectional waves from shallow water

Feng, X., Shaw, A.G.P. and Zhang, W., 2013. Study on reverse calculation for unidirectional waves from shallow water In: Conley, D.C., Masselink, G., Russell, P.E. and O’Hare, T.J. (eds.), Proceedings 12 th International Coastal Symposium (Plymouth, England), Journal of Coastal Research, Special Issue No. 65, pp. 219-224, ISSN 0749-0208. In the coastal region, some wave measurements are usually deployed in shallow water. However, the knowledge of deep-water waves is more required in some cases. Targeting the east coast of Taiwan with a steep slope, we experimentally detect the capability of the linear spectrum theory in reversely estimating unidirectional deepwater waves from the shallow-water waves. Tank experiments of irregular waves show that the nonlinearity of wave-wave interactions significantly contributes to the total energy transfer. The energy of incident waves around peak frequency is transferred to lower and higher frequency domains. Even though waves are travelling through a short distance under a steep slope, the nonlinear wave-wave interactions cannot be neglected. In the application of the linear spectrum theory, the reversely calculated deep-water spectrum is found to be overestimated in frequency over 2Hz and in total energy of spectrum, but underestimated around peak frequency. There is evidence to show that the ratio of H1/3/Hs is correlated with the spectral bandwidth and Kurtosis.

PROPERTIES OF FREAK WAVES INDUCED BY TWO KINDS OF NONLINEAR MECHANISMS

We investigate the dynamic and kinematic characteristics of freak waves using a direct phase-resolved nonlinear numerical method. The focus is on the understanding of the effects of different nonlinear wave-wave interactions on freak waves development and characteristics in the evolution process of modulated Stokes wave trains. Long time simulations of modulated Stokes wave trains, with different parameters, are obtained. Based on these simulations, we find that there are different kinds of freak waves in different time scales due to two kinds of different nonlinear mechanisms. One is the modulation instability and another related to the wave group interaction. Both the dynamic and kinematic characteristics of the different kinds of freak waves are distinct. Occurrence of freak waves (especially of large height) is usually correlated with broadband wave spectra.

ON THE PROBABILITY DISTRIBUTION OF FREAK WAVES IN FINITE WATER DEPTH

This paper presents the occurrence probability of freak waves based on the analysis of extensive wave data collected during ARSLOE project. It is suggested to use the probability distribution of extreme waves heights as a possible means of defining the freak wave criteria instead of conventional definition which is the wave height greater than the twice of the significant wave height. Analysis of wave data provided such finding as 1) threshold tolerance of 0.2 m is recommended for the discrimination of the false wave height due to noise, 2) no supportive evidence on the linear relationship between the occurrence probability of freak waves and the kurtosis of surface elevation 3) nonlinear wave-wave interactions is not thh primary cause of the generation of freak waves 4) the occurrence of freak waves does not depend on the wave period 5) probability density function of extreme waves can be used to predict the occurrence probability of freak waves. Three different distribution functions of extreme wave height by Rayleigh, Ahn, and Mori were compared for the analysis of freak waves.

Competition mechanism of multiple four-wave mixing in highly nonlinear fiber: spatial instability and satellite characteristics

Competition mechanism in multiple four-wave mixing (MFWM) processes is demonstrated theoretically. Provided considering only two waves injected into a highly nonlinear fiber (HNLF), there are three modes displaying comprehensive dynamic behaviors, such as fixed points, periodic motion, and chaotic motion. Especially, Mode C of MFWM is emphasized by analyzing its phase-space trajectory to demonstrate nonlinear wave-wave interactions. The study shows that, when the phase-space trajectory approaches or gets through a saddle point, a dramatic power depletion for the injected wave can be realized, with the representative point moving chaotically, but when phase-space trajectories are distributed around a center point, the power for the injected wave is retained almost invariable, with the representative point moving periodically. Finally, the evolvement of satellite wave over an optical fiber is investigated by comparing it with the interference pattern in Young’s double-slit experiment.

First simulations with a whole atmosphere data assimilation and forecast system: The January 2009 major sudden stratospheric warming

[1] A Whole atmosphere Data Assimilation System (WDAS) is used to simulate the January 2009 sudden stratospheric warming (SSW). WDAS consists of the Whole Atmosphere Model (WAM) and the 3-dimensional variational (3DVar) analysis system GSI (Grid point Statistical Interpolation), modified to be compatible with the WAM model. An incremental analysis update (IAU) scheme was implemented in the data assimilation cycle to overcome the problem of excessive damping by digital filter in WAM of the important tidal waves in the upper atmosphere. IAU updates analysis incrementally into the model, thus avoids the initialization procedure (i.e., digital filter) during the WAM forecast stage. The WDAS simulation of the January 2009 SSW shows a significant increase in TW3 (terdiurnal, westward propagating, zonal wave number 3) and a decrease in SW2 (semidiurnal, westward propagating, zonal wave number 2) wave amplitudes in the E region during the warming, which can be attributed likely to the nonlinear wave-wave interactions between SW2, TW3 and DW1 (diurnal, westward propagating, zonal wave number 1). There is a delayed increase in SW2 in the E region after the warming, indicating a modulation by the changing large-scale planetary waves in the loweratmosphere during the SSW. These tidal wave responses during SSW appeared to be global in scale. An extended WAM forecast initialized from WDAS analysis shows remarkably consistent tidal wave responses to SSW, indicating a potential forecasting capability of several days in advance of the effects of the large-scale tropospheric and stratospheric dynamics on the thermospheric and ionospheric variability.

Directional spectra of hurricane-generated waves in the Gulf of Mexico

[1] Hurricane-induced directional wave spectra in the Gulf of Mexico are investigated based on the measurements collected at 12 buoys during 7 hurricane events in recent years. Focusing on hurricane-generated wave spectra, we only consider the wave measurements at the buoys within eight times the radius of the hurricane maximum wind speed (Rmax) from the hurricane center. A series of numerical experiments using a third-generation spectral wave prediction model were carried out to gain insight into the mechanism controlling the directional and frequency distributions of hurricane wave energy. It is found that hurricane wave spectra are almost swell-dominated except for the right-rear quadrant of a hurricane with respect to the forward direction, where the local strong winds control the spectra. Despite the complexity of a hurricane wind field, most of the spectra are mono-modal, similar to those under fetch-limited, unidirectional winds. However, bi-modal spectra were also found in both measurements and model results. Four types of bi-modal spectra have been observed. Type I happens far away (>6 × Rmax) from a hurricane. Type II is bi-modal in frequency with significant differences in direction. It happens in the two left quadrants when the direction of hurricane winds deviates considerably from the swell direction. Type III is bi-modal in frequency in almost the same wave direction with two close peaks. It occurs when the energy of locally-generated wind-sea is only partially transferred to the swell energy by non-linear wave-wave interactions. Type IV was observed in shallow waters owing to coastal effects.

Finite difference scheme of a model for nonlinear wave-wave interaction in ionic media

In the paper, the coupled 1D nonlinear Schrodinger system (CNLS) is considered as the model equation for wave-wave interaction in ionic media. A finite difference scheme is derived for the model equations. A new six-point scheme, which is equivalent to the multi-symplectic integrator, is derived. The numerical simulation is also presented for the model equations.

Full-field drift Hamiltonian particle orbits in axisymmetric tokamak geometry

A Hamiltonian/Lagrangian theory to describe guiding center orbit drift motion that is canonical in Boozer magnetic coordinates is developed to include full electrostatic and electromagnetic perturbed fields in axisymmetric tokamak geometry. Furthermore, the radial component of the equilibrium magnetic field in the covariant representation is retained and the background equilibrium state extends to anisotropic plasma pressure conditions. A gauge transformation on the perturbed vector potential is imposed to guarantee canonical structure in the Boozer frame. Perturbed field nonlinear wave-wave interactions affect only the evolution of the guiding center particle parallel gyroradius. The evolution of the particle coordinate positions retains only linear wave-particle interactions. For particle motion in magnetohydrodynamic (MHD) instability structures, the electrostatic potential is linked mainly to the binormal component of the perturbed displacement vector when finite δA⊥ components are included.

Recent Capabilities of CMS-Wave: A Coastal Wave Model for Inlets and Navigation Projects

Abstract The Coastal Inlets Research Program (CIRP) of the U.S. Army Engineer Research and Development Center (ERDC), Coastal and Hydraulics Laboratory (CHL) has developed a nearshore spectral wave transformation numerical model to address needs of the U.S. Army Corps of Engineers (USACE) navigation projects. The model is called CMS-Wave and is part of Coastal Modeling System (CMS) for wave estimates in the vicinity of coastal and estuarine navigation channels. It can simulate important wave processes at coastal inlets including wave diffraction, refraction, reflection, wave breaking and dissipation mechanisms, wave-current interaction, and wave generation and growth. This paper describes recent improvements in CMS-Wave that include semi-empirical estimates of wave run-up and overtopping, nonlinear wave-wave interactions, and wave dissipation over muddy bottoms. CMS-Wave may be used with nested grids and variable rectangular cells in a rapid mode to assimilate full-plane wave generation for circulation an...

RAPID CALCULATION OF NONLINEAR WAVE-WAVE INTERACTIONS

This paper presents an efficient numerical algorithm for the nonlinear wave-wave interactions that can be important in the evolution of coastal waves. Indeed, ocean waves truly interact with each others. However, because ocean waves can also interact with the atmosphere such as under variable wind and pressure fields, and waves will deform from deep to shallow water, it is generally difficult to differentiate the actual amount of the nonlinear energy transfer among spectral waves mixed with the atmospheric input and wave breaking. The classical derivation of the nonlinear wave energy transfer has involved tedious numerical calculation that appears impractical to the engineering application. The present study proposed a theoretically based formulation to efficiently calculate nonlinear wave-wave interactions in the spectral wave transformation equation. It is approved to perform well in both idealized and real application examples. This rapid calculation algorithm indicates the nonlinear energy transfer is more significant in the intermediate depth than in deep and shallow water conditions.

Pelagic and coastal sources of P‐wave microseisms: Generation under tropical cyclones

Nonlinear wave‐wave interactions generate double‐frequency (DF) microseisms, which include both surface waves (mainly Rayleigh‐type) and compressional (P) waves. Although it is unclear whether DF surface waves generated in deep oceans are observed on land, we show that beamforming of land‐based seismic array data allows detection of DF P waves generated by ocean waves from Super Typhoon Ioke in both pelagic and coastal regions. Two distinct spectral bands associated with different P‐wave source locations are observed. The short‐period DF band (0.16–0.35 Hz) is dominated by P waves generated in the deep ocean by local wind seas under the storm. In contrast, P waves in the long‐period DF band (0.1–0.15 Hz) are weaker and generated closer to the coast of Japan from swell interactions. The accurate identification of DF P‐wave microseism source areas is useful to monitor ocean wave‐wave interactions due to tropical cyclones and to image Earth structure using ambient seismic noise.

Nonlinear regimes of Farley-Buneman instability

The dynamic suppression of the instability of a quasi-monochromatic wave by nonlinear wave-wave interaction is considered. It is shown that, near the threshold of linear instability, the process of decay into two strongly damped waves leads to the onset of a quasi-periodic or a stochastic nonlinear stabilization regime involving a small number of modes. A case study is made of the Farley-Buneman instability in an isothermal magnetized current-carrying plasma in which particle collisions play an important role. Typical characteristic features of different stabilization regimes are analyzed as functions of current and other plasma parameters.