CN101308519B

CN101308519B - Simulation method of ocean wave evolution scene with multiple short waves modulated nonlinearly by long waves

Info

Publication number: CN101308519B
Application number: CN2008100479843A
Authority: CN
Inventors: 谢涛; 陈伟; 旷海兰
Original assignee: Wuhan University of Technology WUT
Current assignee: Wuhan University of Technology WUT
Priority date: 2008-06-12
Filing date: 2008-06-12
Publication date: 2010-06-09
Anticipated expiration: 2028-06-12
Also published as: CN101308519A

Abstract

The invention relates to a method for simulating the evolution scene of ocean waves in which multiple short waves are nonlinearly modulated by long waves. Analytical solution of the space-time evolution of the sea surface; the obtained analytical solution is regarded as a long wave 1, and the second short wave 1 is nonlinearly modulated by the long wave; the wave after the nonlinear modulation of the long and short wave is regarded as variable amplitude, variable frequency, and variable wavelength, and the short wave is modulated by the long wave. Long-wave modulation, in turn, the long-wave parameters are affected by the short-wave into "non-linear long-wave", and its amplitude, frequency and other parameters change with time and space. This solves the difficulty that the boundary conditions cannot be used to obtain the analytical solution of nonlinear modulation of one long wave and more than two short waves.

Description

Simulation method of ocean wave evolution scene with multiple short waves modulated nonlinearly by long waves

技术领域technical field

本发明属于海洋动力学工程技术领域，具体涉及一种多个短波受长波非线性调制的海浪演化场景模拟方法。The invention belongs to the technical field of marine dynamics engineering, and in particular relates to a method for simulating a sea wave evolution scene in which multiple short waves are nonlinearly modulated by long waves.

背景技术Background technique

深海海浪时空演化数值模拟是海洋动力学工程技术领域中的一项重要的应用研究工作，是深海石油平台、造船技术领域不可或缺的一部分。目前为止，深海海浪模拟技术局限于线性海浪模型。Numerical simulation of space-time evolution of deep-sea waves is an important applied research work in the field of marine dynamics engineering technology, and an indispensable part of deep-sea oil platform and shipbuilding technology. So far, deep-sea wave simulation techniques have been limited to linear wave models.

Pierson和Moskowitz[1]首先通过对发育完全的风浪及其参数的观察，提出著名的PM海浪谱。Hasselman等对北洋海浪及参数观察提出了JONSWAP谱[2]。最近，J.H.G.M.Alves等[3]对PM谱进行了修正。这些海浪谱模型成为海洋动力学工程技术领域中进行深海海浪模拟基础。Pierson and Moskowitz[1] first proposed the famous PM wave spectrum by observing fully developed wind waves and their parameters. Hasselman et al. proposed the JONSWAP spectrum for the observation of North Ocean waves and their parameters [2]. Recently, J.H.G.M.Alves et al. [3] corrected the PM spectrum. These wave spectrum models become the basis for deep-sea wave simulation in the field of ocean dynamics engineering technology.

目前，人们对深海海浪时空演化模拟方法是：基于上述海浪谱(PM谱、JONSWAP谱)，采用伪谱傅立叶方法对海浪谱进行傅立叶变换，得到由无数个不同波幅、波长、频率单色海浪线性叠加的海面演化函数。例如，F.Berizzi等[4]于1999提出分形海面模型，采用的方法是将大量不同分形维的单色波线性叠加，表征介于随机海面与确定海面的海浪模型。由于深海海浪并不是线性的，长波对短波波长、波幅等参数有调制作用(Longuet-Higgins等[5])，线性波叠加的模拟方法不能准确模拟真实海况，虽然E.Lo，C.C.Mei等[6]提出一种伪谱傅立叶方法对非线性薛定谔演化方程进行数值模拟，但薛定谔演化方程主要是考虑完全非线性孤立波演化的，实际上，深海海浪是由一个长波调制多个短波的线性与非线性叠加的模型。现有的理论不能得到多个短波受长波调制的解析解，从而导致现有海浪模型不能表征长波对多个短波的非线性调制作用。At present, people's simulation method for the spatiotemporal evolution of deep-sea waves is: based on the above-mentioned wave spectrum (PM spectrum, JONSWAP spectrum), the pseudo-spectral Fourier method is used to perform Fourier transform on the wave spectrum, and the linear wave spectrum obtained by countless different amplitudes, wavelengths, and frequencies. Superimposed sea surface evolution functions. For example, F.Berizzi et al. [4] proposed the fractal sea surface model in 1999. The method used is to linearly superimpose a large number of monochromatic waves with different fractal dimensions to represent the wave model between random sea surface and definite sea surface. Since deep sea waves are not linear, long waves have a modulating effect on parameters such as wavelength and amplitude of short waves (Longuet-Higgins et al. [5]), the simulation method of linear wave superposition cannot accurately simulate real sea conditions, although E.Lo, C.C.Mei et al.[ 6] A pseudo-spectral Fourier method is proposed to numerically simulate the nonlinear Schrödinger evolution equation, but the Schrödinger evolution equation mainly considers the evolution of completely nonlinear solitary waves. In fact, deep-sea waves are linear and Models of nonlinear superposition. Existing theories cannot obtain the analytical solution of multiple short waves modulated by long waves, which leads to the fact that existing ocean wave models cannot characterize the nonlinear modulation effect of long waves on multiple short waves.

参考文献references

[1] W.J.Pierson and L.Moskowitz，A proposed spectrum formfor fully developed wind seas based on the similarity theory ofS.A，Kitaigorodskii，J.Geophys.Res.，Vol.69，p：5181-5190，1964.[1] W.J.Pierson and L.Moskowitz, A proposed spectrum form for fully developed wind seas based on the similarity theory of S.A, Kitaigorodskii, J.Geophys.Res., Vol.69, p:5181-5190, 1964.

[2]G.J.Komen，L.Cavaleri，M.Donelan，K.Hasselman，S.Hasselman and P.A.E.M.Janssen Dynamics and modelling of oceanwaves，Cambridge University Press，1994.[2] G.J.Komen, L.Cavaleri, M.Donelan, K.Hasselman, S.Hasselman and P.A.E.M. Janssen Dynamics and modeling of oceanwaves, Cambridge University Press, 1994.

[3]J.H.G.M.Alves，M.L Banner and Ian R Young，Revisitingthe Pierson-Moskowitz asymptotic limits for fully developed windwaves.Journal of Physical Oceanography.Vol.33，Iss.7；p：1301-1323，2003。[3] J.H.G.M.Alves, M.L Banner and Ian R Young, Revisiting the Pierson-Moskowitz asymptotic limits for fully developed windwaves. Journal of Physical Oceanography. Vol.33, Iss.7; p:1301-1323, 2003.

[4]F.Berizzi，E.D.Mese and G.Pinelli，One-dimensionalfractal model of the sea surface，IEE Proc.Radar，Sonar Navig.，Vol.146，No.1，pp：55-64，1999.[4] F.Berizzi, E.D.Mese and G.Pinelli, One-dimensionalfractal model of the sea surface, IEE Proc.Radar, Sonar Navig., Vol.146, No.1, pp:55-64, 1999.

[5]M.S.Longuet-Higgins and R.W.Stewart，Changes in theform of short gravity waves on long waves and tidal currents.Journal of fluid Mechanics，Vol.8，p：565-583，1960.[5] M.S.Longuet-Higgins and R.W.Stewart, Changes in the form of short gravity waves on long waves and tidal currents. Journal of fluid Mechanics, Vol.8, p: 565-583, 1960.

[6]E.Lo，C.C.Mei，A numerical study of water-wavemodulation based on a higher-order nonlinear

equation，J.Fluid Mech.Vol.150，p：395 416，1985.[6] E.Lo, CCMei, A numerical study of water-wavemodulation based on a higher-order nonlinear

equation, J. Fluid Mech. Vol. 150, p: 395 416, 1985.

发明内容Contents of the invention

本发明目的在于提供一种多个短波受长波非线性调制的海浪演化场景模拟方法。The purpose of the present invention is to provide a method for simulating the evolution scene of ocean waves in which multiple short waves are nonlinearly modulated by long waves.

本发明提出的非线性海浪演化场景模拟方法，是基于短波受长波非线性调制的海浪演化场景的模拟方法。Longuet-Higgins等根据无限深度海洋动力学边界条件和微扰法，推导出一个短波受一个长波调制的海浪时空演化非线性解析表达式，从解析数学的角度来说，没有办法推到出两个及以上短波受长波调制情形的海浪时空演化解析表达式。本发明针对该问题，提出了一种模拟多个短波受长波非线性调制的海浪演化场景的方法。The simulation method for the nonlinear ocean wave evolution scene proposed by the present invention is based on the simulation method for the ocean wave evolution scene where the short wave is nonlinearly modulated by the long wave. Longuet-Higgins et al. derived a nonlinear analytical expression for the space-time evolution of a short wave modulated by a long wave based on the infinite depth ocean dynamics boundary conditions and the perturbation method. From the perspective of analytic mathematics, there is no way to derive two Analytical expressions of the spatiotemporal evolution of ocean waves in the case of short waves modulated by long waves. Aiming at this problem, the present invention proposes a method for simulating the evolution scene of ocean waves in which multiple short waves are nonlinearly modulated by long waves.

为了实现上述目的，本发明所采用的方法是：In order to achieve the above object, the method adopted in the present invention is:

第一步骤：采用微扰法、根据深海动力学边界条件，求解一个短波(简称：短波0)受一个长波(简称：长波0)非线性调制的海面时空演化解析解；The first step: using the perturbation method and according to the deep-sea dynamic boundary conditions, solve a short wave (referred to as: short wave 0) nonlinearly modulated by a long wave (referred to as: long wave 0) analytical solution of sea surface space-time evolution;

第二步骤：将得到的解析解看成一个长波(简称：长波1)，第二个短波(简称：短波1)受此长波非线性调制；The second step: regard the obtained analytical solution as a long wave (abbreviation: long wave 1), and the second short wave (abbreviation: short wave 1) is nonlinearly modulated by the long wave;

第三步骤：针对不能求解其非线性调制的解析解，一是将长波1看成随时间和空间变幅、变频、变波数的“非线性长波1”，二是不同具体某个时间点某个空间点对应波幅、频率的长波都满足深海动力学边界条件，将长波1看成是这些长波分别非线性调制短波1时在对应时间点和空间点的解析解的叠加(称为离散的解析解)；The third step: for the analytical solution that cannot solve the nonlinear modulation, one is to regard the long-wave 1 as a "non-linear long-wave 1" that varies in amplitude, frequency, and wavenumber with time and space; The long waves corresponding to the amplitude and frequency of each space point satisfy the deep-sea dynamic boundary conditions, and the long wave 1 is regarded as the superposition of the analytical solutions at the corresponding time point and space point when these long waves respectively nonlinearly modulate the short wave 1 (called the discrete analytical solution). untie);

第四步骤：采用同样的方法求解第3、4、5......个短波受前面“非线性长波2、3、4......”调制的“离散的解析解”；The fourth step: adopt the same method to solve the "discrete analytical solution" that the 3rd, 4th, 5th short waves are modulated by the front "non-linear long waves 2, 3, 4...";

第五步骤：根据PM谱，求出不同风速下长、短波波幅及其对应的频率和波长等参数作为输入，通过以“离散的解析解”为算子进行数值模拟，可模拟出不同海况、深海多个短波受长波非线性调制的海浪演化场景。The fifth step: According to the PM spectrum, obtain the parameters such as long-wave and short-wave amplitudes and their corresponding frequencies and wavelengths under different wind speeds as input, and use the "discrete analytical solution" as the operator to carry out numerical simulation to simulate different sea conditions, Ocean wave evolution scenario with multiple short waves modulated nonlinearly by long waves in the deep sea.

本发明中，提出了将长短波非线性调制后的海浪看成变幅、变频、变波长海浪的思想。短波受长波调制，反过来长波参数受短波的影响变成“非线性长波”，其幅度、频率等参数随时空变化而改变，这是本发明核心技术的前提条件。In the present invention, the idea of considering the waves after nonlinear modulation of the long and short waves as waves with variable amplitude, variable frequency and variable wavelength is proposed. The short wave is modulated by the long wave, and the long wave parameters are affected by the short wave to become "non-linear long wave".

本发明提出“非线性长波”离散化的核心技术。深海动力学边界条件也只适用固定不同参数的正弦波相互作用，因此变幅、变频、变波长的“非线性长波”与另外短波的非线性调制情形，不可能利用边界条件求出其解析解，而具体某个时间点某个空间点对应波幅、频率和波长是固定的，将“非线性长波”离散化，即将“非线性长波”看成是在对应时间点和空间点，对应固定波幅、频率等参数的不同长波分别非线性调制短波时的解析解的叠加，这样就解决了不能利用边界条件求出一个长波与两个以上短波非线性调制解析解的困难。The invention proposes the core technology of "non-linear long wave" discretization. The boundary conditions of deep-sea dynamics are only applicable to the interaction of sine waves with different parameters fixed. Therefore, it is impossible to obtain the analytical solution for the "nonlinear long wave" with variable amplitude, variable frequency, and variable wavelength and the nonlinear modulation of other short waves by using boundary conditions. , and the amplitude, frequency and wavelength corresponding to a certain time point and a certain space point are fixed, and the "nonlinear long wave" is discretized, that is, the "nonlinear long wave" is regarded as corresponding to a time point and a space point, corresponding to a fixed amplitude The superposition of the analytical solutions when different long waves nonlinearly modulate the short waves with different parameters such as , frequency, etc., solves the difficulty that the boundary conditions cannot be used to obtain the analytical solutions for nonlinear modulation of one long wave and more than two short waves.

附图说明Description of drawings

图1为本发明一个长波分别调制2、3、4、5、6个短波的“非线性长波”频率空间演化图。Fig. 1 is a "non-linear long wave" frequency space evolution diagram of the present invention, where one long wave modulates 2, 3, 4, 5, and 6 short waves respectively.

图2为本发明一个长波分别调制2、3、4、5、6个短波的“非线性长波”频率时间演化图。Fig. 2 is a "non-linear long wave" frequency-time evolution diagram of the present invention, in which one long wave modulates 2, 3, 4, 5, and 6 short waves respectively.

图3为本发明一个长波分别调制2、3、4、5、6个短波的“非线性长波”初相位空间演化图。Fig. 3 is the spatial evolution diagram of the initial phase of "non-linear long wave" modulated by one long wave of 2, 3, 4, 5, and 6 short waves respectively in the present invention.

图4为本发明一个长波分别调制2、3、4、5、6个短波的“非线性长波”初相位时间演化图。Fig. 4 is a time evolution diagram of the initial phase of "non-linear long wave" modulated by one long wave of 2, 3, 4, 5, and 6 short waves in the present invention.

图5为本发明一个长波分别调制2、3、4、5、6个短波的“非线性长波”幅度空间演化图。Fig. 5 is a "non-linear long wave" amplitude spatial evolution diagram of one long wave modulating 2, 3, 4, 5, and 6 short waves respectively in the present invention.

图6为本发明一个长波分别调制2、3、4、5、6个短波的“非线性长波”幅度时间演化图。Fig. 6 is a "non-linear long wave" amplitude-time evolution diagram of one long wave modulating 2, 3, 4, 5, and 6 short waves respectively in the present invention.

图7为本发明一个长波分别调制2、3、4、5、6个短波的空间演化图。Fig. 7 is a spatial evolution diagram of one long wave modulating 2, 3, 4, 5, and 6 short waves respectively in the present invention.

图8一个长波分别调制2、3、4、5、6个短波的时间演化图图中：生成7个波对应的19.5米高风速：U＝[20 7.5 7 6.5 65.5 5]；根据PM谱计算得到7个波的幅度：a＝[4.27 0.6 0.52 0.450.38 0.32 0.27]；根据PM谱计算得到7个波的频率：f＝[0.07 0.180.2 0.21 0.23 0.25 0.27]；7个波的波长：Lamda＝[318.31 48.1438.99 35.37 29.48 24.96 21.40]；7个波对应的角频率：w＝[0.441.13 1.26 1.32 1.45 1.57 1.70]；7个波各自的波数k＝[0.020.13 0.16 0.18 0.21 0.25 0.29]；7个波的初始相位设为：thet＝[0 185 124 35 105 20 141]。Figure 8: Time evolution of a long wave modulating 2, 3, 4, 5, and 6 short waves respectively In the figure: generate 7 waves corresponding to 19.5 meters high wind speed: U=[20 7.5 7 6.5 65.5 5]; calculated according to PM spectrum Get the amplitude of the 7 waves: a＝[4.27 0.6 0.52 0.450.38 0.32 0.27]; calculate the frequency of the 7 waves according to the PM spectrum: f＝[0.07 0.180.2 0.21 0.23 0.25 0.27]; the wavelength of the 7 waves: Lamda＝[318.31 48.1438.99 35.37 29.48 24.96 21.40]; the angular frequencies corresponding to the 7 waves: w＝[0.441.13 1.26 1.32 1.45 1.57 1.70]; the respective wave numbers of the 7 waves k＝[0.2 0.2 0.1 0.1 0.29]; the initial phase of the 7 waves is set as: thet=[0 185 124 35 105 20 141].

具体实施方式Detailed ways

下面结合附图对本发明作进一步的详细描述。The present invention will be described in further detail below in conjunction with the accompanying drawings.

本发明提出多个短波受长波非线性调制的海浪演化场景模拟方法，主要包含三个步骤：The present invention proposes a method for simulating the ocean wave evolution scene where multiple short waves are nonlinearly modulated by long waves, which mainly includes three steps:

1.针对具体长短波系统，解出系统的“离散的解析解”，获取演化算子；1. For the specific long-short wave system, solve the "discrete analytical solution" of the system and obtain the evolution operator;

2.根据PM谱模型，通过数值计算，获取不同风速海况下，对应海浪波幅、波长及频率等参数信息；2. According to the PM spectrum model, through numerical calculation, obtain the corresponding wave amplitude, wavelength and frequency and other parameter information under different wind speed and sea conditions;

3.利用步骤1获得的系统演化算子，以步骤2获得的系统参数信息为输入，模拟输出多个短波受长波非线性调制的海浪演化场景。3. Using the system evolution operator obtained in step 1, taking the system parameter information obtained in step 2 as input, simulate and output multiple short wave evolution scenarios in which short waves are nonlinearly modulated by long waves.

其中第一步为系统的演化算子获取，1个长波调制2个短波的演化算子具体获取方式为：The first step is to obtain the evolution operator of the system. The specific acquisition method of the evolution operator of one long wave modulating two short waves is as follows:

根据文献[5]，深海海浪动力学边界条件满足：According to literature [5], the boundary conditions of deep ocean wave dynamics satisfy:

$u u = = &dtri; &dtri; φ φ - - - - - - ((11 a a))$

${&dtri; &dtri;}^{22} φ φ = = 00 - - - - - - ((11 b b))$

$\frac{p p}{ρ ρ} + + gz gz + + \frac{11}{22} {u u}^{22} + + \frac{&PartialD; &PartialD; φ φ}{&PartialD; &PartialD; t t} = = 00 - - - - - - ((11 c c))$

其中u，φ，p，ρ，ζ分别表示速度、速度势、压强、密度和表面高度。当长波对短波微扰，相应参数为：where u, φ, p, ρ, and ζ represent velocity, velocity potential, pressure, density, and surface height, respectively. When the long wave perturbs the short wave, the corresponding parameters are:

u＝u⁽⁰⁾+εu⁽¹⁾+… (2a)u＝u ⁽⁰⁾ +εu ⁽¹⁾ +...(2a)

φ＝φ⁽⁰⁾+εφ⁽¹⁾+… (2b)φ＝φ ⁽⁰⁾ +εφ ⁽¹⁾ +... (2b)

ζ＝ζ⁽⁰⁾+εζ⁽¹⁾+… (2c)ζ = ζ ⁽⁰⁾ + εζ ⁽¹⁾ + ... (2c)

$\frac{p p}{ρ ρ} + + gz gz = = {p p}^{((00))} + + ϵ ϵ {p p}^{((11))} + + . . . . . . - - - - - - ((22 d d))$

对于一个长波调制二短波系统，用下标1表示长波的各参量，短波2、3分别表示两短波的各参量。For a long-wave modulation two short-wave system, the subscript 1 represents the parameters of the long-wave, and the short-wave 2 and 3 represent the parameters of the two short-waves respectively.

首先考虑1(长波)对2(短波1)的调制作用，可根据边界条件式(1)及微扰法，求出各微扰量，其中波高度ζ为：First consider the modulation effect of 1 (long wave) on 2 (short wave 1), and then calculate the perturbation quantities according to the boundary condition formula (1) and the perturbation method, where the wave height ζ is:

${ζ ζ}_{1212} = = (({a a}_{11} sin sin {ψ ψ}_{11} - - \frac{11}{22} ϵ ϵ {a a}_{11}^{22} {k k}_{11} sin sin 22 {ψ ψ}_{11})) + + (({a a}_{22} sin sin {ψ ψ}_{22} - - \frac{11}{22} ϵ ϵ {a a}_{22}^{22} {k k}_{22} sin sin 22 {ψ ψ}_{22})) - - - - - - ((33))$

$- - ϵ ϵ {a a}_{11} {a a}_{22} (({k k}_{22} cos cos {ψ ψ}_{11} cos cos {ψ ψ}_{22} - - {k k}_{11} sin sin {ψ ψ}_{11} sin sin {ψ ψ}_{22})) + + . . . . . .$

其中k₁x-σ₁t+θ₁＝ψ₁，k₂x-σ₂t+θ₂＝ψ₂，k₂＞＞k₁，a表示波幅，k表示波数，σ表示角频率，ε＝k₁a₁为长波峰度即微扰。由ζ₁₂，可以得到长波1的各个参数受短波2的影响，表示为：Where k ₁ x-σ ₁ t+θ ₁ =ψ ₁ , k ₂ x-σ ₂ t+θ ₂ =ψ ₂ , k ₂ >>k ₁ , a represents wave amplitude, k represents wave number, σ represents angular frequency, ε = k ₁ a ₁ is the long-wave kurtosis or perturbation. From ζ ₁₂ , it can be obtained that each parameter of long wave 1 is affected by short wave 2, expressed as:

a₁₂＝a₁(1+εa₂k₁sinψ₂) (4a)a ₁₂ ＝a ₁ (1+εa ₂ k ₁ sinψ ₂ ) (4a)

k₁₂＝k₁(1+εa₂k₁sinψ₂) (4b)k ₁₂ ＝k ₁ (1+εa ₂ k ₁ sinψ ₂ ) (4b)

${σ σ}_{1212} = = \sqrt{g g {k k}_{1212}} - - - - - - ((44 c c))$

其次，在1对2调制作用的基础上，再来考虑对短波3的调制作用。将ζ₁₂看成变幅、变波数和变频的“非线性长波”，其对短波3作用后的波高用ζ₁₂₃表示。由于不可能根据ζ₁₂表达式，利用边界条件和微扰法求出非线性长波12对3非线性调制的解析解，因此本发明提出“非线性长波”离散化思想，即对于某时间点下某空间点的波，对应于相应的固定波幅、波数和频率的正弦波，即空间i点j时刻某波高ζ_(12，i，j)又可表示为：Secondly, on the basis of the 1-to-2 modulation effect, consider the modulation effect on the short-wave 3. Consider ζ ₁₂ as a "nonlinear long wave" with variable amplitude, variable wave number and variable frequency, and the wave height after it acts on the short wave 3 is represented by ζ ₁₂₃ . Since it is impossible to obtain the analytical solution of nonlinear long-wave 12-to-3 nonlinear modulation according to the expression _ζ12 using boundary conditions and perturbation method, the present invention proposes the idea of "non-linear long-wave" discretization, that is, for a certain time point A wave at a point in space corresponds to a sine wave with a corresponding fixed amplitude, wave number, and frequency, that is, a certain wave height ζ _{(12, i, j)} at point j in space can be expressed as:

ζ_(12，i，j)＝a_(12，i，j)sinψ_(l2，i，j) (5)ζ _{(12, i, j)} = a _{(12, i, j)} sinψ _{(l2, i, j)} (5)

其中ψ_(12，i，j)＝k_(12，i，j)x_i-σ_(12，i，j)t_j+θ_(12，i，j)，θ_(12，i，j)可由

求出。Where ψ _{(12, i, j)} = k _{(12, i, j)} x _i -σ _{(12, i, j)} t _j + θ _{(12, i, j)} , θ _{(12, i, j)} can be given by

Find out.

将ζ_(12，i，j)延拓为ζ₁₂＝a₁₂sinψ₁₂，将其看成不同时空点不同波幅、波数和频率的正弦波ζ₁₂分别与短波3调制的结果，ζ₁₂₊₃则为不同对应点处(i，j)的离散化结果叠加而成。这样ζ₁₂对短波3的调制就能通过边界条件和微扰法求出，从而解出系统的“离散的解析解”，获取演化算子。Extend ζ _{(12, i, j)} to ζ ₁₂ =a ₁₂ sinψ ₁₂ , and regard it as the result of modulation of sine wave ζ ₁₂ with different amplitudes, wave numbers and frequencies at different time-space points and short wave 3, ζ ₁₂₊₃ It is formed by superimposing the discretization results of (i, j) at different corresponding points. In this way, the modulation of ζ ₁₂ on the short wave 3 can be obtained through the boundary conditions and the perturbation method, so as to solve the "discrete analytical solution" of the system and obtain the evolution operator.

本系统的算子为，对应点(i，j)处波高为：The operator of this system is, and the wave height at the corresponding point (i, j) is:

${ζ ζ}_{((1212 + + 33,, i i,, j j))} = = {ζ ζ}_{1212 + + 33}^{((00))} {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} + + ϵ ϵ {ζ ζ}_{((1212 + + 33,, i i,, j j))}^{((11))} - - - - - - ((66))$

${ζ ζ}_{1212 + + 33}^{((00))} = = (({a a}_{11} sin sin {ψ ψ}_{11} - - \frac{11}{22} ϵ ϵ {a a}_{11}^{22} {k k}_{11} sin sin 22 {ψ ψ}_{11})) + + (({a a}_{22} sin sin {ψ ψ}_{22} - - \frac{11}{22} ϵ ϵ {a a}_{22}^{22} {k k}_{22} sin sin 22 {ψ ψ}_{22})) - - - - - - ((77))$

$- - ϵ ϵ {a a}_{11} {a a}_{22} (({k k}_{22} cos cos {ψ ψ}_{11} cos cos {ψ ψ}_{22} - - {k k}_{11} sin sin {ψ ψ}_{11} sin sin {ψ ψ}_{22})) + + {ζ ζ}_{33}$

${ζ ζ}_{((1212 + + 33,, i i,, j j))}^{((11))} = = - - \frac{11}{22} {a a}_{33}^{22} {k k}_{33} sin sin 22 {ψ ψ}_{33} - - \frac{11}{22} {a a}_{1212}^{22} {k k}_{1212} sin sin 22 {ψ ψ}_{1212} - - - - - - ((88))$

$- - {a a}_{1212} {a a}_{33} (({k k}_{33} cos cos {ψ ψ}_{1212} cos cos {ψ ψ}_{33} - - {k k}_{1212} sin sin {ψ ψ}_{1212} sin sin {ψ ψ}_{33})) {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}}$

其中ζ₃＝a₃sinψ₃，ψ₃＝k₃x-σ₃t+θ₃。Where ζ ₃ =a ₃ sinψ ₃ , ψ ₃ =k ₃ x−σ ₃ t+θ ₃ .

1个长波调制n个(n＝3，4，5...)短波的演化算子具体获取方式为：The specific acquisition method of the evolution operator for modulating n (n=3, 4, 5...) short waves by one long wave is as follows:

用ζ_12…n-1表示长波调制前n-2个短波的波高，将ζ_12…n-1看成变幅、变波数和变频的“非线性长波”，其对短波n作用后的波高用ζ_12…n-1表示。，可以得到“非线性长波”(21…n-1)的各个参数受短波n的影响，表示为：Use ζ _12...n-1 to represent the wave heights of n-2 short waves before long wave modulation, and regard ζ _12...n-1 as a "non-linear long wave" with variable amplitude, variable wave number and variable frequency, and the wave height after it acts on the short wave n It is represented by ζ _12...n-1 . , it can be obtained that each parameter of the "nonlinear long wave" (21...n-1) is affected by the short wave n, expressed as:

${a a}_{1212 . . . . . . n no} = = {a a}_{1212 . . . . . . n no - - 11} ((11 + + {ϵ ϵ}_{1212 . . . . . . n no - - 11} {a a}_{n no} {k k}_{1212 . . . . . . n no - - 11} sin sin {ψ ψ}_{n no})) {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} - - - - - - ((99 a a))$

${k k}_{1212 . . . . . . n no} = = {k k}_{1212 . . . . . . n no - - 11} ((11 + + {ϵ ϵ}_{1212 . . . . . . n no - - 11} {a a}_{n no} {k k}_{1212 . . . . . . n no - - 11} sin sin {ψ ψ}_{n no})) {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} - - - - - - ((99 b b))$

${σ σ}_{1212 . . . . . . n no} = = \sqrt{g g {k k}_{1212 . . . . . . n no}} {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} - - - - - - ((99 c c))$

对于某时间点下某空间点的波，对应于相应的固定波幅、波数和频率的正弦波，即空间i点j时刻某波高ζ_{(12…n-1，i，j)}又可表示为：For a wave at a certain spatial point at a certain time point, corresponding to a sine wave with a corresponding fixed amplitude, wave number and frequency, that is, a certain wave height ζ _{(12…n-1, i, j) at} a point j in space can be expressed as:

ζ_{(12…n-1，i，j)}＝a_{(12…n-1，i，j)}sinψ_{(12…n-1，i，j)} (10)ζ _{(12...n-1, i, j)} = a _{(12...n-1, i, j)} sinψ _{(12...n-1, i, j)} (10)

其中ψ_{(12…n-1，i，j)}＝k_{(12…n-1，i，j)}x_i-σ_{(12…n-1，i，j)}t_j+θ_{(12…n-1，i，j)}，θ_{(12…n-1，i，j)}可由

求出。where ψ _{(12…n-1, i, j)} = k _{(12…n-1, i, j)} x _i -σ _{(12…n-1, i, j)} t _j + θ _{(12…n- 1, i, j)} , θ _{(12…n-1, i, j)} can be obtained by

Find out.

将ζ_{(12…n-1，i，j)}延拓为

将其看成不同时空点不同波幅、波数和频率的正弦波ζ_12…n-1分别与短波n调制的结果，ζ_12…n则为不同对应点处(i，j)的离散化结果叠加而成。这样ζ_12…n-1对短波n的调制就能通过边界条件和微扰法求出，从而解出系统的“离散的解析解”，获取演化算子。Extend ζ _{(12…n-1, i, j)} to

Think of it as the result of modulation of sine waves ζ _12...n-1 with different amplitudes, wave numbers and frequencies at different time and space points and short wave n respectively, and ζ _12...n is the superposition of discretization results at different corresponding points (i, j) made. In this way, the modulation of ζ _12...n-1 to short-wave n can be obtained through boundary conditions and perturbation method, so as to solve the "discrete analytical solution" of the system and obtain the evolution operator.

1个长波调制n个(n＝3，4，5...)短波的演化算子为，对应点(i，j)处波高为：The evolution operator of one long wave modulating n (n=3, 4, 5...) short waves is, and the wave height at the corresponding point (i, j) is:

${ζ ζ}_{((1212 \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; n no,, i i,, j j))} {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} = = {ζ ζ}_{1212 \cdot \cdot \cdot \cdot \cdot &Center Dot; n no}^{((00))} {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} + + {ϵ ϵ}_{1212 \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; n no} {ζ ζ}_{((1212 \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; n no,, i i,, j j))}^{((11))} {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} - - - - - - ((1111))$

${ζ ζ}_{1212 . . . . . . n no}^{((00))} {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} = = {ζ ζ}_{1212 . . . . . . n no - - 11} {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} + + {ζ ζ}_{n no} {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}} - - - - - - ((1212))$

$ζ_{(12 . . . n, i, j)}^{(1)} |_{x = x_{i}, t = t_{j}} = - \frac{1}{2} a_{n}^{2} k_{n} \sin 2 ψ_{n} |_{x = x_{i}, t = t_{j}} - \frac{1}{2} a_{12 . . . n - 1}^{2} k_{21 . . . n - 1} \sin 2 ψ_{12 . . . n - 1} |_{x = x_{i}, t = t_{j}}$ (13) $ζ_{(12 . . . no, i, j)}^{(1)} |_{x = x_{i}, t = t_{j}} = - \frac{1}{2} a_{no}^{2} k_{no} \sin 2 ψ_{no} |_{x = x_{i}, t = t_{j}} - \frac{1}{2} a_{12 . . . no - 1}^{2} k_{twenty one . . . no - 1} \sin 2 ψ_{12 . . . no - 1} |_{x = x_{i}, t = t_{j}}$ (13)

$- - {a a}_{1212 . . . . . . n no - - 11} {a a}_{n no} (({k k}_{n no} cos cos {ψ ψ}_{1212 . . . . . . n no - - 11} cos cos {ψ ψ}_{n no} - - {k k}_{1212 . . . . . . n no - - 11} sin sin {ψ ψ}_{1212 . . . . . . n no - - 11} sin sin {ψ ψ}_{n no})) {| |}_{x x = = {x x}_{i i},, t t = = {t t}_{j j}}$

其中

ζ_n＝a_nsinψ_n，ψ_n＝k_nx-σ_nt+θ_n。in

ζ _n = a _n sinψ _n , ψ _n = k _n x−σ _n t+θ _n .

获取系统演化算子后，根据PM谱模型，通过数值计算，获取不同风速海况下，对应海浪波幅、波长及频率等参数信息作为系统输入，模拟输出多个短波受长波非线性调制的海浪演化场景。After obtaining the system evolution operator, according to the PM spectrum model, through numerical calculation, obtain the corresponding wave amplitude, wavelength and frequency and other parameter information under different wind speed and sea conditions as the system input, and simulate and output multiple short-wave nonlinearly modulated wave evolution scenarios .

本说明书未作详细描述的内容属于本领域专业技术人员公知的现有技术。The content not described in detail in this specification belongs to the prior art known to those skilled in the art.

Claims

1. A method for simulating the evolution of ocean waves in which multiple short waves are nonlinearly modulated by long waves, the method used is:

The first step: using the perturbation method and according to the deep-sea dynamic boundary conditions, solve the analytical solution of the space-time evolution of the sea surface in which a short-wave wave is nonlinearly modulated by a long-wave wave;

The second step: regard the obtained analytical solution as a long wave 1, and the second short wave 1 is nonlinearly modulated by this long wave;

The third step: for the analytical solution that cannot be solved for its nonlinear modulation, one is to regard the long wave 1 as a nonlinear long wave 1 that varies in amplitude, frequency, and wavenumber with time and space; The long waves corresponding to the amplitude and frequency of the points all satisfy the boundary conditions of deep-sea dynamics, and the long wave 1 is regarded as the superposition of the analytical solutions at the corresponding time points and space points when these long waves nonlinearly modulate the short wave 1 respectively;

The fourth step: use the same method to solve the superposition of the nth short wave modulated by the previous n-1 nonlinear long wave, where n≥3;

Step 5: According to the above steps, the evolution operator is obtained, and according to the PM spectrum, the long and short wave amplitudes and their corresponding frequency and wavelength parameters under different wind speeds are obtained as input for numerical simulation, and different sea conditions and multiple short waves in deep sea are simulated by long waves. Nonlinearly modulated ocean wave evolution scenario.