Modulation Of Short Waves Research Articles

Optical pattern formation of laser fields in the Rydberg atomic gases

We study the phenomena of long-wave and short-wave modulation instabilities (LMI and SMI) in Rydberg atomic gases within the framework of the nonlocal nonlinear Schrödinger equations in the local and nonlocal regions (characterized by the ratio of beam radius to radius of Rydberg blockade sphere), respectively. We further reveal the rich soliton dynamics arising from interactions of nonlinear waves triggered by LMI. We also show the variety of spatial self-organization pattern formations and their active manipulation due to SMI. Through a synthesis of theoretical analysis and numerical simulations, the self-organized optical structures reported here provide a way for realizing what we believe to be novel optical patterns and solitons based on Rydberg atomic gases.

On the Bimodality of the Wind-Wave Spectrum: Mean Square Slopes and Azimuthal Overlap Integral

Abstract We present an investigation of the azimuthal bimodality of the wind-wave spectrum for waves shorter than the dominant scale comparing numerical model solutions of developing waves from idealized experiments using WAVEWATCH III (WW3). The wave solutions were forced with the “exact” Webb–Resio–Tracy (WRT) nonlinear energy fluxes and the direct interaction approximation (DIA) with three different combinations of wind input and breaking dissipation parameterizations. The WRT gives larger azimuthal bimodal amplitudes compared to the DIA regardless of wind input/dissipation. The widely used wind input/dissipation parameterizations (i.e., ST4 and ST6) generally give narrow directional distributions with relatively small bimodal amplitudes and lobe separations compared to field measurements. These biases are significantly improved by the breaking dissipation of Romero (R2019). Moreover, the ratio of the resolved cross- to downwind mean square slope is significantly lower for ST4 and ST6 compared to R2019. The overlap integral relevant for the prediction of microseisms is several orders of magnitude smaller for ST4 and ST6 compared to R2019, which nearly agrees with a semiempirical model. Significance Statement Spectral gravity wave models generally cannot accurately predict the directional distribution which impacts their ability to predict the resolved down- and crosswind mean square slopes and the generation of microseisms. Our analysis shows that a directionally narrow spectral energy dissipation, accounting for long-wave–short-wave modulation, can significantly improve the directional distribution of the wind-wave spectrum by coupling to the nonlinear energy fluxes due to wave–wave interactions, which has important implications for improved predictions of the mean square slopes and the generation of microseisms.

Discrete and continuum composite solitons in Bose-Einstein condensates with the Rashba spin-orbit coupling in one and two dimensions.

We introduce one- and two-dimensional (1D and 2D) continuum and discrete models for the two-component BEC, with the spin-orbit (SO) coupling of the Rashba type between the components, and attractive cubic interactions, assuming that the condensate is fragmented into a quasidiscrete state by a deep optical-lattice potential. In 1D, it is demonstrated, in analytical and numerical forms, that the ground states of both the discrete system and its continuum counterpart switch from striped bright solitons, featuring deep short-wave modulations of its profile, to smooth solitons, as the strength ratio of the inter- and intracomponent attraction, γ, changes from γ<1 to γ>1. At the borderline, γ=1, there is a continuous branch of stable solitons, which share a common value of the energy and interpolate between the striped and smooth ones. Unlike the 2D system, the 1D solitons, which do not represent the ground state at given γ, are nevertheless stable against small perturbations, and they remain stable too in collisions with other solitons. In 2D, a transition between two different types of discrete solitons, which represent the ground state, viz., semivortices and mixed modes, also takes place at γ=1. A specific property of 2D discrete solitons of both types is their discontinuous transition into a delocalized state at a critical value of the SO-coupling strength. We also address the continuum 2D model in the borderline case of γ=1, which was not studied previously, and demonstrate the existence of an energy-degenerate branch of dynamically stable solitons connecting the semivortex and the mixed mode. Last, it is demonstrated that 1D and 2D discrete solitons are mobile, in a limited interval of velocities.

Patterns of thermomagnetic convection caused by long- and short-wave modulation of the magnetic field

Patterns of thermomagnetic convection caused by long- and short-wave modulation of the magnetic field

Electromagnetic bias at off‐nadir incidence angles

Measurements of the electromagnetic (EM) bias in altimeter sea surface ranges at nadir and off‐nadir incidence angles were made during the Brigham Young University (BYU) Off‐Nadir Experiment (Y‐ONE) in the months of March and April, 2003. Tower‐mounted C‐ and Ku‐band radars were deployed along with laser rangefinders and a weather station to measure sea surface profiles and environmental parameters. The incidence angle range was nadir to 17°. An off‐nadir bias model incorporating the effects of hydrodynamic modulation of short waves and tilt modulation of long waves was also developed. Both the experimental data set and the theoretical model led to a bias of decreasing magnitude as the incidence angle moves away from nadir. The average measured EM bias at C‐band was approximately zero at 14° incidence, and a small positive mean bias (1% SWH) was observed at 17°. By extrapolating from the Ku‐band measurements, the mean bias vanished at 18.5° for the higher frequency.

Transformation of Short Waves in a Nonuniform Flow Field on the Ocean Surface. The Effect of Wind Growth Rate Modulation

We develop a model of transformation of the short surface wave spectrum in the presence of a nonuniform flow on a water surface, in which the modulation of wind-wave growth rate is taken into account. The model of a turbulent near-water atmospheric layer is used to calculate the modulated growth rate. In this model, turbulent stresses in the wind are described using a gradient approximation with model eddy viscosity specified with allowance for the known laboratory experiments. The examples of short-wave modulation in the presence of nonuniform flows on a water surface, originating from ripples and intense internal waves, are considered. It is shown that deformations of the wind-velocity profile and its long-wavelength perturbation due to the nonlinear interaction between the wind surface waves and the wind has a significant effects on the short-wave growth rate and its modulation. In the case of ripples, this deformation reduces to an increase in the roughness parameter of the wind-velocity profile and to a velocity-profile modulation with ripple period. The modulated growth rate is calculated within the framework of a quasi-linear model of surface-wave generation by a turbulent wind, in which the hypothesis of random phases of the wind-wave field is used. The amplitude and phase of the hydrodynamical modulation transfer function are calculated within the framework of the relaxation model. The calculation results are in reasonable agreement with the available experimental data. A model described by the combined Korteweg–de Vries equation is used to study a surface flow field generated by intense internal waves. The internal-wave parameters are takes from the results of the COPE experiment. We calculate the wind growth-rate dependences on the wave-train phase for the cases of downwind and upwind propagation of an internal wave. The calculation results agree qualitatively with experimental data.

An investigation of altimeter sea state bias theories

Two theoretical formulations of the sea state bias (SSB) by Srokosz [1986] (hereafter S86) and Elfouhaily et al. [2000] (hereafter E00) are applied to directional ocean wave spectra from WAM and from NDBC moored buoys collocated with Topex altimeter data. S86 SSB computations obtained for WAM with idealized ocean current fields reveal a quasi‐linear relationship between the SSB coefficient and the r.m.s. ocean slope, which remains valid for a wide range of wind/wave/current conditions. The SSB coefficient is shown to be better parameterized in terms of slope, instead of the significant wave height and wind speed presently used in empirical models. The theoretical SSB for the buoys' 2‐D wave spectra are compared with estimates from five recent empirical SSB models calculated from the collocated Topex data. The magnitude of the E00 SSB shows a strong sensitivity to the choice of high frequency tail model used to extend the 2‐D wave spectra. We find that the E00 theory can adequately model the radar frequency dependence of the SSB given a suitable choice of high frequency tail. However, the E00 theory is found to systematically underestimate the magnitude of the SSB coefficient with respect to empirical SSB models. In contrast, the long‐waves‐only S86 theory displays good agreement with the empirical SSB estimates. Thus our calculations suggest that the sea state bias may be primarily governed by the slope of long gravity waves, presumably through its well‐documented influence on the modulation of short waves, as also suggested by recent empirical findings.

Transmission of Long Waves Induced by Short-Wave Groups Through a Composite Breakwater

Theoretical findings and laboratory evidences are presented regarding the transmission of long waves induced by short-wave groups through the rubble base of an infinitely long composite breakwater. Multiple-scale perturbation method is applied in describing the long-wave fields in terms of velocity potential, and an analytical solution is derived for transmitted and reflected long waves induced by wave groups. Dependency of the reflection and transmission of free long waves on several parameters is examined through the numerical calculations. It is found that the linear friction factor, the rubble-base height and the modulation of short waves affect the transmission of free long waves significantly. Theoretical results are then compared with the results from laboratory experiments. It is shown that although the conventional rubble-mound breakwaters can reflect short wave energy efficiently, a significant part of the bound long-wave energy associated with shortwave groups is transmitted to the leeward side of the breakwater as free waves.

Dual-polarized Ku-band backscatter signatures of hurricane ocean winds

The Ku-band dual-polarized backscatter signatures of ocean surfaces are described in this paper with the airborne scatterometer measurements collected in the Hurricane Ocean Wind Experiment in September 1997. The data collected from flights over Hurricane Erika provide a direct evidence that there are wind direction signals in the vertically and horizontally polarized Ku-band backscatter of ocean surfaces under the influence of hurricane force winds. At 46/spl deg/ incidence angle, the vertically polarized backscatter acquired at the upwind direction increases by about 1 dB as the wind speed increases from 22 m/spl middot/s/sup -1/ to 35 m/spl middot/s/sup -1/, while the horizontally polarized backscatter appears to be twice as sensitive with a change of about 2 dB. At 35 m/spl middot/s/sup -1/ wind speeds, the difference between upwind and crosswind observations of vertically polarized backscatter is about 1.5 dB, smaller than the 2 dB difference for the horizontally polarized backscatter. This demonstrates that the horizontal polarization has a greater sensitivity to wind speed and direction than the vertical polarization in the high wind regime. The data also suggest that the upwind and downwind asymmetry of Ku-band backscatter decreases with increasing wind speed and can fall below 0 dB at small incidence angles (<35/spl deg/) for the vertical polarization. A combined interaction of the geometric optics scattering and the short wave modulation by long waves is proposed to interpret this phenomenon and appears to agree with the dependence of the signature on incidence, wind speed, and polarization. The aircraft flight data support the feasibility of dual-polarized Ku-band radar for hurricane ocean wind measurements, although the data do suggest a reduced wind speed and direction sensitivity in the high wind regime. Also, the differing polarization backscatter signatures-suggest the relative contributions of various surface scattering mechanisms. An improved Ku-band GMF is described.

The normalized radar cross section of the sea at 10° incidence

Measurements of the normalized radar cross section of the sea at K/sub u/ band at an incidence angle of 10/spl deg/ were performed from a manned airship off the Oregon coast in September and October of 1993. The cross section at this incidence angle is often assumed to have little dependence on windspeed and direction. Their measurements, however, indicate that at windspeeds below 6-7 m/s, the cross section is in fact dependent on these quantities, and the azimuthal modulation can reach values on the order of 5-8 dB. Comparisons of the measured values with the predictions of the quasispecular scattering model are presented. The theory is shown to be accurate in predicting the azimuthal modulation and the strength of the backscatter if the effects of swell are included or if measured wind directions are ignored and the upwind direction is forced to be near the maximum cross section. Values of mean-square wind-wave slope and effective-reflection coefficient required to obtain these fits are very close to those obtained by previous investigators. In particular, mean-square wind-wave slopes are about 70-80% of those of Cox and Munk (1954) because the radar responds only to facets larger than about 10 cm, with smaller ripples acting to reduce the reflection coefficient. If swell is included, they find that mean-square slopes in the direction of the swell, that are as much as ten times the measured swell slopes, are required to fit the model to the cross-section data at low windspeeds. They suggest that this may be due to high-order effects of the hydrodynamic modulation of short waves by the swell. They believe that this explanation is more likely than assuming that wind directions were incorrectly measured.

Microstructure of Ocean Surface Roughness: A Study of Spatial Measurement and Laboratory Investigation of Modulation Analysis*

Abstract This paper describes a technique to measure the spatial structure of short capillary–gravity waves on the water surface. The method is based on optical refraction of a single laser beam crossing the air–water interface to derive the surface slope information. By quickly scanning the laser beam, the spatial and temporal evolution of the surface undulation can be studied in great detail. The spatial measurement of the surface fluctuation also allows direct computation of the wavenumber spectra in the capillary–gravity wave regime. This approach avoids the difficulty of resolving the Doppler frequency shift encountered in the processing of high-frequency spectra derived from a single-point sensor. The modulation of short waves by the orbital velocity of surface long waves is analyzed from the distribution of the wavenumber spectra of short waves along the phase of long waves. Key parameters determining the modulation magnitude are identified to be the slope of wavenumber spectra in the neighborhood ...

The modulation of short waves riding on solitary waves

The modulation of linear short waves riding on a long finite-amplitude solitary wave has been analyzed numerically. It is found that the maximum modulated wave number, frequency, and amplitude of short waves always occur at the crest of solitary waves. This paper shows that the modulated wave number on the crest of solitary waves increases significantly as the amplitude of the solitary waves increases, and that the modulated short wave frequency and amplitude on the crest increase almost linearly.

Short-wave measurements by a fixed tower-based and a drifting buoy system

The small-scale roughness of the sea surface acts as an important link in air-sea interaction processes. Radar and sonar waves are scattered by short surface waves providing the basis for remote sensing methods of the sea surface. At high wind speeds, breaking waves occur. Bubbles penetrate into the water and drastically increase acoustical reverberation, transmission loss and ambient noise. Thus, the development of short waves and wave breaking have to be known to apply radar remote sensing to the surface and to deduce from radar backscatter which sonar conditions prevail. To measure the wind dependence of short waves an experimental device was constructed for use from stationary platforms. It is nearly all-weather capable and can easily be handled by a crane. On the other hand, frequencies of short waves measured in a fixed position are extremely frequency shifted by currents. This limits the usefulness of tower-based measurements, e.g., the short wave modulation by wind and waves or currents can only be estimated in a rough approximation. Consequently, a buoy was developed to reduce the frequency shifts. The principle of the buoy is to drift in the local surface current and to follow the amplitudes of long waves. Therefore, short waves are measured in facets of long waves and the Doppler shifts are minimized. The wind is measured at a constant height above the long wave profile and relative to the moving facets. The paper describes the conventional measuring device and points out the necessity of the drifting buoy system. Examples of wind and wave spectra are presented and short wave modulations by long waves are depicted, too. From these measurements, new insights in short wave behaviour have to be expected. >

Surface Waves and Coastal Dynamics

This review covers theoretical advances made during the past decade for predicting or understanding wind-induced waves and associated coastal processes. First, the offshore propagation over many wavelengths is discussed. An effective approximation which permits practical calculations of both refraction and diffraction over a mildly sloping bottom is given. For still longer distances nonlinear effects accumulate to become important; recent progress in the parabolic approximation for the refraction and diffraction of short waves is also reviewed. The nearshore region is discussed and the infragravity waves which are caused by nonlinear interactions of short waves within a narrow frequency band are surveyed. The effects of a gradual or sudden change of depth or a steady current on the modulation of short waves and the generation of long infragravity waves is also discussed. A discussion of infragravity waves which resonate in harbours and the strong resonance of edge waves on beaches is made. The final topics are concerned with the causes and effects of bars and ripples on the seabed.

A three-frequency scatterometer technique for the measurement of ocean wave spectra

A microwave technique for the measurement of ocean wave spectra has been compared with wave gauge output during extensive field testing. The method is based on the dual-frequency technique for detecting long ocean waves by matching the modulation of short waves with the beat wavelength between two transmitted microwave frequencies. The new method, however, utilizes three microwave frequencies in order to reduce mean backscatter not related to short-wave modulation. Two prototype scatterometers have been built using three frequencies at L-band and at K/sub u/-band. Wave spectra have been measured by both radar systems which, when properly normalized, agree well with simultaneous in situ measurements taken by conventional wave gauges at the pier site. Thirteen sets of spectra have been computed, five of which correspond to a situation in which a local wind sea was generated and then decayed. The present experiment does not demonstrate the directionality of this new technique.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Measurement of the hydrodynamic modulation of centimeter waves

An investigation of the hydrodynamic modulation of 1‐ to 3.5‐cm wind waves in a wave tank is described. The modulation of short waves at wind speeds from 4 to 10 m s−1 was due to a mechanically generated 2‐s surface wave. Measurement of the surface slope was obtained using a laser‐optical slope sensor. The equilibrium height wave number spectrum was determined using the two components of the slope frequency spectrum. It was found to be consistent with k−3.75 cosp (θ/2), where p is a decreasing function of k and wind speed. The linear modulation transfer function was found to be 2–3 times that predicted by a linearized action‐conservation model based on Hughes' (1978) theory, and to increase with wind speed. Phase agreement is also unsatisfactory, the modulation being in phase with the oscillating current at high wind speeds.

Modulation of short waves by surface currents: A numerical solution

Modulation of short waves by surface currents subjected to wind input and breaking dissipation is investigated. The dissipation is expressed as a power law function of the action density. The solution is presented in the form of an integral equation, and the method of characteristics is applied to obtain numerical results. The numerical solution compares favorably with the relaxation theory when the surface current is weak, and it can be applied to a strong current condition, where the linear assumption used in the derivation of the relaxation theory is no longer valid. The modulation response to various wind and wave parameters, such as short‐wave wavelength, surface wind stress, long‐wave frequency and steepness, different dissipation functions, and interaction length or duration between long and short waves, are discussed.

Short surface waves with growth and dissipation

The modulation of surface waves by larger‐scale flows is central to many forms of remote imaging of the sea surface yet is an incompletely understood process. An excellent example is the modulation of short waves by long ones (e.g., by swell). Observed modulations are of the order of 10 times the long‐wave steepness, decrease with both the long‐wave frequency and windspeed, and have stable phase, with maximum short‐wave amplitudes just forward of the long‐wave crests. Assuming a balance between propagation, growth, and drift‐enhanced dissipation, the short‐wave modulations arise owing to (1) direct straining by the long‐wave orbital motion, (2) straining of the wind drift layer (modulating the dissipation rate), (3) variations of the apparent gravity in the short‐wave frame, and (4) induced variations of the applied wind stress (affecting both the short‐wave growth and the drift‐enhanced dissipation). As modeled here, only the last (stress variation) can reproduce the observed wind speed and frequency dependence. In addition, the drift layer and the short waves are closely matched, suggesting direct coupling. The implied fractional modulation of stress is of the order of 20 times the long‐wave steepness.

Measurements of wind velocity and pressure with a wave follower during Marsen

Near sea‐surface air pressure and wind velocities at several levels above the instantaneous water surface were measured using a wave follower during the Marine Remote Sensing (MARSEN) Experiment. The rate of momentum transfer from wind to waves is determined from the air pressure data. From the wind velocity measurements, the wave‐induced airflow and its coherence with waves are obtained for various wind velocities and phase speeds of the ocean waves. The pressure results indicate that momentum transfer to waves can be specified by the equation γi = 0.12 (μ − 1)2 in the ranges μ = 1 to 7.4, where γi is related to the imaginary part of the wave‐induced pressure, as defined by equation (2), μ = (8/3π)(U/c), U is wind speed at 10 m above the mean water level, and c is wave phase speed. The wind velocity results indicate that the wave‐induced airflow is much smaller than the mean wind speed. An empirical equation is proposed for the modulation of atmospheric transfer to short waves induced by the orbital velocity of long waves. Such modulation explains adequately the excess modulation of short waves by long waves as suggested by Wright et al. [1980] when the short waves are 13 cm long but not when they are 2.3 cm long. The agreement for the 13 cm waves supports their suggestion that the excess modulation is due to a modulation in atmospheric transfer. For the shorter 2.3 cm waves the less adequate agreement and the scatter in the results of Wright et al. suggest that other mechanisms such as short wave blocking by long wave orbital velocity, short wave breaking, and other strong nonlinear transfer among the short wave spectral components also may be contributing to the excess modulation of short waves by long waves.

On the Skewness of Sea-Surface Slopes

Sunlight reflected from a wind-roughened sea surface produces a glitter pattern in which the region of maximum intensity tends to be shifted horizontally by an apparent angle Δ, depending on the wind speed. It is shown that Δ is related directly to the skewness of the distribution of surface slopes. From the observed data of Cox and Munk (1956) it is possible to deduce a simple correlation between Δ and the wind stress τ. The physical mechanism underlying slope skewness is investigated. The skewness which results from damping of individual waves is shown to be negligible. A two-scale model is proposed, in which damped ripples or short gravity waves ride on the surface of longer gravity waves. The model is found to give skewness of the observed magnitude. The sign of the skewness depends on the angle between the wind maintaining the ripples and the direction of the longer waves, in agreement with observation. Certain theoretical relations between Δ and the phase γ of the short-wave modulation may be of interest in interpreting observations of the sea surface by other types of remote sensing.