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.

Effective thermal conductivity of graded nonlinear composites with heat contact resistance

The perturbation expansion method is used to find the effective thermal conductivity of graded nonlinear composites having thermal contact resistance on the inclusion surface. As an example, we have studied the graded composites with cylindrical inclusions immersed in a homogeneous matrix. The thermal conductivity of the cylindrical inclusion is assumed to have a power-law profile of the radial distance r measured from its origin. For weakly nonlinear constitutive relations between the heat flow density q and the temperature field T, namely, q = -mu del T - chi vertical bar del T vertical bar(2) del T, in both the inclusion and the matrix regions, we have derived the temperature distributions using the perturbation expansion method. A nonlinear effective medium approximation of graded composites is proposed to estimate the effective linear and nonlinear thermal conductivities. by considering the temperature singularity on the inclusion surface due to the heat contact resistance. (c) 2006 Elsevier B.V. All rights reserved.

Effective AC and DC responses of nonlinear composites at higher concentration

For higher concentration of inclusions, an effective medium approximation (EMA) method is used to investigate the bulk effective response of weakly nonlinear composites, which are subject to the constitutive relation of electric displacement and electric field, D-alpha = epsilon E-alpha + chi(alpha)|E|(2) E. As an example of three dimensions, under the external AC and DC electric fields E-app = E-a (1 + sin omega t), we have derived the general effective nonlinear response of composites by the EMA method for higher concentration of spherical inclusions. Furthermore, the effective nonlinear responses at harmonics are predicted.

Systematic Nonlinear Planning

This paper presents a simple, sound, complete, and systematic algorithm for domain independent STRIPS planning. Simplicity is achieved by starting with a ground procedure and then applying a general and independently verifiable, lifting transformation. Previous planners have been designed directly as lifted procedures. Our ground procedure is a ground version of Tate's NONLIN procedure. In Tate's procedure one is not required to determine whether a prerequisite of a step in an unfinished plan is guarnateed to hold in all linearizations. This allows Tate"s procedure to avoid the use of Chapman"s modal truth criterion. Systematicity is the property that the same plan, or partial plan, is never examined more than once. Systematicity is achieved through a simple modification of Tate's procedure.

Using Special-Purpose Computing to Examine Chaotic Behavior in Nonlinear Mappings

Studying chaotic behavior in nonlinear systems requires numerous computations in order to simulate the behavior of such systems. The Standard Map Machine was designed and implemented as a special computer for performing these intensive computations with high-speed and high-precision. Its impressive performance is due to its simple architecture specialized to the numerical computations required of nonlinear systems. This report discusses the design and implementation of the Standard Map Machine and its use in the study of nonlinear mappings; in particular, the study of the standard map.

Evaluation of nonlinear chromatographic performance by frontal analysis using a simple multi-plate mathematical model

A multi-plate (NIP) mathematical model was proposed by frontal analysis to evaluate nonlinear chromatographic performance. One of its advantages is that the parameters may be easily calculated from experimental data. Moreover, there is a good correlation between it and the equilibrium-dispersive (E-D) or Thomas models. This shows that it can well accommodate both types of band broadening that is comprised of either diffusion-dominated processes or kinetic sorption processes. The MP model can well describe experimental breakthrough curves that were obtained from membrane affinity chromatography and column reversed-phase liquid chromatography. Furthermore, the coefficients of mass transfer may be calculated according to the relationship between the MP model and the E-D or Thomas models. (C) 2004 Elsevier B.V. All rights reserved.