A theory of nonlinear AC response in coated composites

A method for determining effective dielectric responses of Kerr-like coated nonlinear composites under the alternating current (AC) electric field is proposed by using perturbation approach. As an example, we have investigated the composite with coated cylindrical inclusions randomly embedded in a host under an external sinusoidal field with finite frequency omega. The local field and potential of composites in general consists of components with all harmonic frequencies. The effective nonlinear AC responses at all harmonics are induced by the coated nonlinear composites because of the nonlinear constitutive relation. Moreover, we have derived the formulae of effective nonlinear AC responses at the fundamental frequency and the third harmonic in the dilute limit.

Effective ac response in weakly nonlinear composites

The perturbation method is developed to deal with the problem of determining the effective nonlinear conductivity of Kerr-like nonlinear media under an external ac electric field. As an example, we have considered the cylindrical inclusion embedded in a host under the sinusoidal external field E-1 sin(omegat) + E-3 sin(3omegat) with frequencies omega and 3omega. The potentials of composites at higher harmonics are derived in both local inclusion particle and host regions. The effective responses of bulk nonlinear composites at basic frequency and harmonics are given for cylindrical composites in the dilute limit. Moreover, the relationships between the nonlinear effective responses at the basic frequency and the third harmonics are derived.

Effective dielectric response of nonlinear composites with external ac and dc electric field

The perturbation method is developed to investigate the effective nonlinear dielectric response of Kerr composites when the external ac and dc electric field is applied. Under the external ac and dc electric field E-app=E-a(1+sin omegat), the effective coupling nonlinear response can be induced by the cubic nonlinearity of Kerr nonlinear materials at the zero frequency, the finite basic frequency omega, the second and the third harmonics, 2omega and 3omega, and so on. As an example, we have investigated the cylindrical inclusions randomly embedded in a host and derived the formulas of the effective nonlinear dielectric response at harmonics in dilute limit. For a higher concentration of inclusions, we have proposed a nonlinear effective-medium approximation by introducing the general effective nonlinear response. With the relationships between the effective nonlinear response at harmonics and the general effective nonlinear response, we have derived a set of formulas of the effective nonlinear dielectric responses at harmonics for a larger volume fraction. (C) 2004 American Institute of Physics.

The nonlinear effective dielectric response of graded composites

The perturbation method is developed to deal with the effective nonlinear dielectric responses of weakly nonlinear graded composites, which consist of the graded inclusion with a linear dielectric function of spatial variables of inclusion material. For Kerr-like nonlinear graded composites, as an example in two dimensions, we have used the perturbation method to solve the boundary value problems of potentials, and studied the effective responses of nonlinear graded composites, where a cylindrical inclusion with linear dielectric function and nonlinear dielectric constant is randomly embedded in a homogeneous host with linear and nonlinear dielectric constants. For the exponential function and the power-law dielectric profiles of cylindrical inclusions, in the dilute limit, we have derived the formulae of effective nonlinear responses of both graded nonlinear composites.

On the nonlinear integral transform of an ocean wave spectrum into an along-track interferometric synthetic aperture radar image spectrum

We present a new nonlinear integral transform relating the ocean wave spectrum to the along-track interferometric synthetic aperture radar (AT-INSAR) image spectrum. The AT-INSAR, which is a synthetic aperture radar (SAR) employing two antennas displaced along the platform's flight direction, is considered to be a better instrument for imaging ocean waves than the SAR. This is because the AT-INSAR yields the phase spectrum and not only the amplitude spectrum as with the conventional SAR. While the SAR and AT-INSAR amplitude spectra depend strongly on the modulation of the normalized radar cross section (NRCS) by the long ocean waves, which is poorly known, the phase spectrum depends only weakly on this modulation. By measuring the phase difference between the signals received by both antennas, AT-INSAR measures the radial component of the orbital velocity associated with the ocean waves, which is related to the ocean wave height field by a well-known transfer function. The nonlinear integral transform derived in this paper differs from the one previously derived by Bao et al. [1999] by an additional term containing the derivative of the radial component of the orbital velocity associated with the long ocean waves. By carrying out numerical simulations, we show that, in general, this additional term cannot be neglected. Furthermore, we present two new quasi-linear approximations to the nonlinear integral transform relating the ocean wave spectrum to the AT-INSAR phase spectrum.

Higher order response of nonlinear composite in external AC electric field

The fifth-order effective nonlinear responses at fundament frequency and higher-order harmonics are given for nonlinear composites, which obey a current-field relation of the form J = sigmaE + x\E\(2) E, if a sinusoidal alternating current (AC) external field with finite frequency omega is applied. As two examples, we have investigated the cylinder and spherical inclusion embedded in a host and, for larger volume fraction, also derived the formulae of effective nonlinear responses at higher-order harmonics by the aid of the general effective response definition. Furthermore, the relationships between effective nonlinear responses at harmonics are given. (C) 2003 Elsevier Science B.V. All rights reserved.

The nonlinear evolution equation on the shallow water wave

Based on the variation principle, the nonlinear evolution model for the shallow water waves is established. The research shows the Duffing equation can be introduced to the evolution model of water wave with time.

NONLINEAR GRAVITY-WAVES IN DEEP-WATER

In consideration of the problem on the boundary condition of nonlinear free water wave, coordinate transform is used to handle the free boundary. Supposing the solution form be the traveling wave, the ordinary differential equations of the one-order autonomous system with two variables are caused, then expanding the nonlinear terms at the equilibrium point with the Taylor expansion, we obtained the solution to traveling wave. The linear approximate equation near the equilibrium point is the small amplitude wave. A new nonlinear periodic traveling wave and nonlinear dispersion relation are shown when expanding to the second-order terms. A conclusion that the expansion of dispersion relation does not contain any odd-power terms of wave steepness and because of the nonlinear effort an oscillate structure is produced in the vertical direction is drawn.

A SINGULAR EXACT SOLUTION FOR NONLINEAR SHALLOW-WATER ON A ROTATING EARTH

In this paper, we present an exact solution for nonlinear shallow water on a rotating planet. It is a kind of solitary waves with always negative wave height and a celerity smaller than linear shallow water propagation speed square-root gh. In fact, it propagates with a speed equal to (1 + a/h) square-root gh(1 + a/h) where a is the negative wave height. The lowest point of the water surface is a singular point where the first order derivative has a discontinuity of the first kind. The horizontal scale of the wave has actually no connection with the water depth.

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation (CNOP), which is a nonlinear generalization of the linear singular vector (LSV), is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming (SQP) algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, and the spectral projected gradients (SPG2) algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.

Characteristics of nonlinear internal waves observed in the northern South China Sea

The South China Sea (SCS) is one of the most active areas of internal waves. We undertook a program of physical oceanography in the northern South China Sea from June to July of 2009, and conducted a 1-day observation from 15:40 of June 24 to 16:40 of June 25 using a chain of instruments, including temperature sensors, pressure sensors and temperature-pressure meters at a site (117.5A degrees E, 21A degrees N) northeast of the Dongsha Islands. We measured fluctuating tidal and subtidal properties with the thermistor-chain and a ship-mounted Acoustic Doppler Current Profiler, and observed a large-amplitude nonlinear internal wave passing the site followed by a number of small ones. To further investigate this phenomenon, we collected the tidal constituents from the TPXO7.1 dataset to evaluate the tidal characteristics at and around the recording site, from which we knew that the amplitude of the nonlinear internal wave was about 120 m and the period about 20 min. The horizontal and vertical velocities induced by the soliton were approximately 2 m/s and 0.5 m/s, respectively. This soliton occurred 2-3 days after a spring tide.

Comparison of nonlinear and linear PCA on surface wind, surface height, and SST in the South China Sea

We compared nonlinear principal component analysis (NLPCA) with linear principal component analysis (LPCA) with the data of sea surface wind anomalies (SWA), surface height anomalies (SSHA), and sea surface temperature anomalies (SSTA), taken in the South China Sea (SCS) between 1993 and 2003. The SCS monthly data for SWA, SSHA and SSTA (i.e., the anomalies with climatological seasonal cycle removed) were pre-filtered by LPCA, with only three leading modes retained. The first three modes of SWA, SSHA, and SSTA of LPCA explained 86%, 71%, and 94% of the total variance in the original data, respectively. Thus, the three associated time coefficient functions (TCFs) were used as the input data for NLPCA network. The NLPCA was made based on feed-forward neural network models. Compared with classical linear PCA, the first NLPCA mode could explain more variance than linear PCA for the above data. The nonlinearity of SWA and SSHA were stronger in most areas of the SCS. The first mode of the NLPCA on the SWA and SSHA accounted for 67.26% of the variance versus 54.7%, and 60.24% versus 50.43%, respectively for the first LPCA mode. Conversely, the nonlinear SSTA, localized in the northern SCS and southern continental shelf region, resulted in little improvement in the explanation of the variance for the first NLPCA.

Highly nonlinear internal solitary waves over the continental shelf of the northwestern South China Sea

Large amplitude internal solitary waves (ISWs) often exhibit highly nonlinear effects and may contribute significantly to mixing and energy transporting in the ocean. We observed highly nonlinear ISWs over the continental shelf of the northwestern South China Sea (19A degrees 35'N, 112A degrees E) in May 2005 during the Wenchang Internal Wave Experiment using in-situ time series data from an array of temperature and salinity sensors, and an acoustic Doppler current profiler (ADCP). We summarized the characteristics of the ISWs and compared them with those of existing internal wave theories. Particular attention has been paid to characterizing solitons in terms of the relationship between shape and amplitude-width. Comparison between theoretical prediction and observation results shows that the high nonlinearity of these waves is better represented by the second-order extended Korteweg-de Vries (KdV) theory than the first-order KdV model. These results indicate that the northwestern South China Sea (SCS) is rich in highly nonlinear ISWs that are an indispensable part of the energy budget of the internal waves in the northern South China Sea.

Statistical distribution of nonlinear random wave height

A statistical model of random wave is developed using Stokes wave theory of water wave dynamics. A new nonlinear probability distribution function of wave height is presented. The results indicate that wave steepness not only could be a parameter of the distribution function of wave height but also could reflect the degree of wave height distribution deviation from the Rayleigh distribution. The new wave height distribution overcomes the problem of Rayleigh distribution that the prediction of big wave is overestimated and the general wave is underestimated. The prediction of small probability wave height value of new distribution is also smaller than that of Rayleigh distribution. Wave height data taken from East China Normal University are used to verify the new distribution. The results indicate that the new distribution fits the measurements much better than the Rayleigh distribution.