CN102508946B - Method for simulating spilled oil sea surface under finite water depth - Google Patents

Abstract

The invention discloses a method for simulating a spilled oil sea surface under a finite water depth. The method comprises the following steps of: acquiring a JONSWAP frequency domain sea wave spectrum of a clean sea surface according to a sea environment parameter; calculating a frequency domain sea wave spectrum of the clean sea surface under the finite water depth according to a water depth parameter; calculating a frequency domain sea wave spectrum of the spilled oil sea surface under the finite water depth according to a spilled oil parameter and a Marangoni theory; performing wavenumber domain conversion on the frequency domain sea wave spectrum of the spilled oil sea surface under the finite water depth according to a dispersion relation of sea waves, thus obtaining a wavenumber domain sea wave spectrum of the spilled oil sea surface under the finite water depth; and acquiring the vertical displacement of the spilled oil sea surface under the finite water depth according to the wavenumber domain sea wave spectrum of the spilled oil sea surface under the finite water depth and a Longuet-Higgins sea wave model. The method is suitable for simulating the spilled oil sea surface in offshore microwave sea remote sensing.

Description

Simulation method for spilled oil sea surface under limited water depth

Technical Field

The invention belongs to the technical field of microwave ocean remote sensing, and particularly relates to a simulation method of an oil spill sea surface under limited water depth.

Background

Synthetic Aperture Radar (SAR) is rapidly developed as a full-time, all-weather and high-resolution microwave remote sensing technology, and plays an active role in monitoring sea surface oil spill. The problem of high false alarm rate has been a difficulty of SAR sea surface oil spill monitoring systems, and although many SAR oil spill identification methods have been proposed, the problem has not been completely solved. The simulation of the oil spill sea surface is an important link of the SAR oil spill monitoring system, and plays a certain promotion role in researching the SAR oil spill identification method, so that the accuracy of SAR oil spill identification can be improved.

The simulation of oil spill sea surface mainly involves two processes: firstly, modeling of sea surface can be roughly divided into two methods, namely a physical model and a geometric model; the physical model is based on a fluid mechanics equation, and a deep water surface wave theory, a small amplitude wave theory and a limited amplitude wave theory are established; usually, a numerical method is adopted for solving, stability analysis is needed, the fidelity of the simulated sea surface is good, the simulated sea surface conforms to the motion of the actual sea surface, but the method has the defects of large calculation amount, low efficiency and the like. The geometric model represents the sea surface by fitting a common geometric curve (e.g., trigonometric function, spline function), wherein Fourier method represents the sea surface by a linear combination of trigonometric functions of different frequencies, amplitudes and phases. Because a definite physical relationship exists between the Fourier coefficient and the frequency domain wave spectrum, the simulation sea surface based on the empirical frequency domain wave spectrum method is widely adopted in the microwave ocean remote sensing, and empirical frequency domain wave spectrum models comprise a Neumann frequency domain wave spectrum, a Bretschneider frequency domain wave spectrum, a Phillips frequency domain wave spectrum, a Pierson-Moskowitz frequency domain wave spectrum, a Fung & Lee frequency domain wave spectrum, a JONSWAP frequency domain wave spectrum and the like. Secondly, the oil stain interacts with the sea surface, and the oil stain can reduce the friction force between wind and the sea surface, so that the short gravity wave and the capillary wave of the spilled oil sea surface are inhibited and attenuated. Research shows that the wave spectrum of the oil spilling sea surface is obviously different from that of the clean sea surface in a high wave number area, and on the basis, a Marangoni theory is established.

Once sea surface oil spill occurs, oil stains can be spread and diffused towards the near coast along with sea waves, wind, ocean currents and the like, and at the moment, the influence of shallow sea topography or water depth on the sea surface needs to be considered. France schetti et al, in simulating the original signals of SAR Oil spilled sea surface, used a method based on frequency domain wave spectrum and Marangoni theory to calculate the Oil spilled sea surface, where the influence of sea bottom topography or water depth was not considered, and thus it is not suitable for the near-coast Oil spill (G.France schetti, A.Iodie, D.Riccio.SAR Raw Signal Simulation of Oil Slicks in Ocean environments. IEEE trans.Geosci.Remote sensing,2002,40(9): 1935-.

Disclosure of Invention

The invention aims to provide a simulation method of an oil spill sea surface under a limited water depth aiming at the defects in the prior art. The method combines different sea wave parameters, oil spill parameters and water depth to obtain the vertical displacement suitable for the oil spill sea surface under the limited water depth. The method is suitable for simulating the oil spill sea surface in the near-coast microwave ocean remote sensing.

In order to achieve the purpose, the invention adopts the following technical scheme:

the invention discloses a simulation method of an oil spill sea surface under limited water depth, which is characterized by comprising the following steps:

step one, calculating a JONSWAP frequency domain sea wave spectrum S (omega) of a clean sea surface according to marine environment parameters:

the expression of the JONSWAP frequency domain ocean wave spectrum is

Wherein, $\mathrm{\β}=\exp \[-\frac{{(\mathrm{\ω}-{\mathrm{\ω}}_p)}^2}{{2\mathrm{\δ}}^2{\mathrm{\ω}}_p^2}\],\mathrm{\δ}=\left\{\begin{array}{cc}0.07& {\mathrm{\ω}}_p\≤\mathrm{\ω}\\ 0.09& {\mathrm{\ω}}_p>\mathrm{\ω}\end{array}\right.,{\mathrm{\ω}}_p=7\mathrm{\π}{\left(\frac{g^{2}F}{U^3}\right)}^{-0.33},$ ω_pthe method is characterized in that the method is a spectrum peak angular frequency, omega is the angular frequency of sea waves, gamma is a peak lift factor, delta is a peak shape parameter, g is an attraction constant, alpha is a scale coefficient, U is the wind speed at 10m above the sea surface, and F is the wind area length;

step two, calculating a frequency domain sea wave spectrum S of a clean sea surface under limited water depth according to the water depth parameters_clean(ω)：

Frequency domain wave spectrum of clean sea surface under limited water depth

tanh²(kd), wherein d is water depth and k is wave number of waves;

step three, calculating the frequency domain sea wave spectrum S of the oil spilling sea surface under the finite water depth according to the oil spilling parameters and the Marangoni theory_oil(ω)；

Frequency domain wave spectrum of oil spill sea surface

Wherein q isIs a normalization factor of the sea surface oil spill area, $P\left(\mathrm{\ω}\right)=\frac{1\±2\mathrm{\τ}+{2\mathrm{\τ}}^2-X+Y(X+\mathrm{\τ})}{1\±2\mathrm{\τ}+{2\mathrm{\τ}}^2-2X+{2X}^2},\mathrm{\τ}=\sqrt{\frac{{\mathrm{\ω}}_D}{2\mathrm{\ω}}},X=\frac{E_0{k}^2}{p\sqrt{{2\mathrm{v\ω}}^3}},Y=\frac{E_{0}k}{4\mathrm{vp\ω}},$ E₀is the elastic modulus, v is the kinematic viscosity coefficient, omega_DThe characteristic angular frequency of oil spill is shown, and rho is the density of the oil spill;

step four, converting the frequency domain wave spectrum of the finite water depth underflow oil sea surface into a wave spectrum S of a wavenumber domain according to the dispersion relation of the waves_oil(k_m,k_n)；

Mesh discretization processing (k) is carried out on sea surface wave numbers_m,k_n)，k_mThe wave number of the sea surface in the x direction,

m-1, i.e. M points discretized by sea wave numbers in the x-direction; k is a radical of_nThe wave number in the y-direction of the sea surface,

n-1, i.e. discretizing the sea surface wavenumber in the y-direction by N points, where L is 0,1,2₁Length of sea surface in x direction, L₂The length of the sea surface in the y direction;

calculating the wave number of waves at each grid point $k_{\mathrm{mn}}=\sqrt{k_m^2+k_n^2}$ And angular frequency ${\mathrm{\ω}}_{\mathrm{mn}}=\sqrt{{\mathrm{gk}}_{\mathrm{mn}}\tanh \left(k_{\mathrm{mn}}d\right)+\frac{{\mathrm{\ζk}}_{\mathrm{mn}}^3}{\mathrm{\ρ}}},$ Wave number domain wave spectrum of limited water depth underflow oil sea surface

$S_{\mathrm{oil}}(k_m,k_n)=\frac{S_{\mathrm{oil}}\left({\mathrm{\ω}}_{\mathrm{mn}}\right)}{2\sqrt{{\mathrm{\ω}}_{\mathrm{mn}}}}\{g\tanh \left(k_{\mathrm{mn}}d\right)+{\mathrm{gk}}_{\mathrm{mn}}d\[1-{\tanh }^2\left(k_{\mathrm{mn}}d\right)\]+3{\mathrm{\ζk}}_{\mathrm{mn}}^2/\mathrm{\ρ})\},$ Wherein, zeta is the surface tension of the oil spill sea surface;

step five, according to the wave number domain wave spectrum S of the spilled oil sea surface under the limited water depth_oil(k_m,k_n) And a Longuet-Higgins sea wave model, and calculating the vertical displacement zeta of the spilled oil sea surface under the limited water depth_oil(x,y;t)：

Vertical displacement of oil spilled sea surface ${\mathrm{\ζ}}_{\mathrm{oil}}(x,y;t)=\underset{m=1}{\overset{M}{\mathrm{\Σ}}}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\η}}_{\mathrm{mn}}\mathrm{exp\; j}(k_{m}x+k_{n}y-\mathrm{\ωt}+\mathrm{\φ})+c.c.,$ Wherein,

c. represents the complex conjugate operator, t is time, (x, y) represents the position coordinates of the sea surface, phi is [ -pi, pi [ -pi ], [ phi ]]Uniformly distributed phase noise, Δ k_xIs the difference between adjacent wave numbers in the x direction, Δ k_yIs the difference between adjacent wave numbers in the y direction.

According to the method, the frequency domain wave spectrum of the oil spill sea surface under the limited water depth is represented by adopting a corrected empirical frequency domain wave spectrum and a Marangoni theory, then the frequency domain wave spectrum of the oil spill sea surface under the limited water depth is converted into a wavenumber domain wave spectrum according to the dispersion relation of waves, and finally the vertical displacement of the oil spill sea surface under the limited water depth is obtained according to the wavenumber domain wave spectrum of the oil spill sea surface under the limited water depth and a Longuet-Higgins wave model. Compared with the prior art, the method is suitable for simulating the oil spill sea surface in the near-coast microwave ocean remote sensing.

Drawings

FIG. 1 is a flow chart of the operation of the present invention;

FIG. 2a is a clean sea surface vertical displacement;

FIG. 2b is a limited water depth clean sea vertical displacement;

FIG. 2c is a vertical displacement of a finite water depth spill sea surface;

FIG. 3a is a gray scale view of the vertical displacement of the clean sea surface;

FIG. 3b is a gray scale graph of vertical displacement of a limited water depth cleaning surface;

fig. 3c is a gray scale view of the vertical displacement of a finite water depth spill sea surface.

Detailed Description

The technical scheme of the invention is explained in detail in the following with the accompanying drawings:

as shown in FIG. 1, the simulation method of the spilled oil sea surface under the limited water depth of the invention comprises the following steps:

the method comprises the following steps: calculating a JONSWAP frequency domain wave spectrum S (omega) of the clean sea surface according to the marine environment parameters:

$S\left(\mathrm{\ω}\right)=\frac{{\mathrm{\αg}}^2}{{\mathrm{\ω}}^5}\exp \[-\frac{5}{4}{\left(\frac{{\mathrm{\ω}}_p}{\mathrm{\ω}}\right)}^4\]{\mathrm{\γ}}^{\mathrm{\β}}---\left(1\right)$

wherein, $\mathrm{\β}=\exp \[-\frac{{(\mathrm{\ω}-{\mathrm{\ω}}_p)}^2}{{2\mathrm{\δ}}^2{\mathrm{\ω}}_p^2}\],\mathrm{\δ}=\left\{\begin{array}{cc}0.07& {\mathrm{\ω}}_p\≤\mathrm{\ω}\\ 0.09& {\mathrm{\ω}}_p>\mathrm{\ω}\end{array}\right.,{\mathrm{\ω}}_p=7\mathrm{\π}{\left(\frac{g^{2}F}{U^3}\right)}^{-0.33},$ ω_pthe method is characterized in that the method is a spectrum peak angular frequency, omega is the angular frequency of sea waves, gamma is a peak lift factor, delta is a peak shape parameter, g is an attraction constant, alpha is a scale coefficient, U is the wind speed at 10m above the sea surface, and F is the wind area length;

step two: calculating the frequency domain wave spectrum S of the clean sea surface under the limited water depth according to the water depth parameter_clean(ω)：

$S_{\mathrm{clean}}\left(\mathrm{\ω}\right)=S\left(\mathrm{\ω}\right){\[1+\frac{2\mathrm{kd}}{\sinh \left(2\mathrm{kd}\right)}\]}^{-1}{\tanh }^2\left(\mathrm{kd}\right)---\left(2\right)$

Wherein d is water depth, and k is wave number of sea waves;

step three: according to the oil spill parameters and the Marangoni theory, calculating the frequency domain ocean wave spectrum S of the oil spill sea surface under the finite water depth_oil(ω)；

$S_{\mathrm{oil}}\left(\mathrm{\ω}\right)=S_{\mathrm{clean}}\left(\mathrm{\ω}\right)(1-q+\frac{q}{p\left(\mathrm{\ω}\right)})---\left(3\right)$

Wherein q is a normalization factor of the sea surface oil spilling region, $P\left(\mathrm{\ω}\right)=\frac{1\±2\mathrm{\τ}+{2\mathrm{\τ}}^2-X+Y(X+\mathrm{\τ})}{1\±2\mathrm{\τ}+{2\mathrm{\τ}}^2-2X+{2X}^2},\mathrm{\τ}=\sqrt{\frac{{\mathrm{\ω}}_D}{2\mathrm{\ω}}},$ E₀is the elastic modulus, v is the kinematic viscosity coefficient, omega_DThe characteristic angular frequency of oil spill is shown, and rho is the density of the oil spill;

step four: according to the dispersion relation of sea waves, the frequency domain sea wave spectrum of the limited water depth underflow oil sea surface is converted into the sea wave spectrum S of the wavenumber domain_oil(k_m,k_n)；

Mesh discretization processing (k) is carried out on sea surface wave numbers_m,k_n)，k_mThe wave number of the sea surface in the x direction,

m-1, i.e. M points discretized by sea wave numbers in the x-direction; k is a radical of_nThe wave number in the y-direction of the sea surface,

n-1, i.e. discretizing the sea surface wavenumber in the y-direction by N points, where L is 0,1,2₁Length of sea surface in x direction, L₂The length of the sea surface in the y direction;

calculating the wave number of waves at each grid point $k_{\mathrm{mn}}=\sqrt{k_m^2+k_n^2}$ And angular frequency ${\mathrm{\ω}}_{\mathrm{mn}}=\sqrt{{\mathrm{gk}}_{\mathrm{mn}}\tanh \left(k_{\mathrm{mn}}d\right)+\frac{{\mathrm{\ζk}}_{\mathrm{mn}}^3}{\mathrm{\ρ}}},$ Wave number domain wave spectrum S of limited water depth underflow oil sea surface_oil(k_m,k_n)

$S_{\mathrm{oil}}(k_m,k_n)=\frac{S_{\mathrm{oil}}\left({\mathrm{\ω}}_{\mathrm{mn}}\right)}{2\sqrt{{\mathrm{\ω}}_{\mathrm{mn}}}}\{g\tanh \left(k_{\mathrm{mn}}d\right)+{\mathrm{gk}}_{\mathrm{mn}}d\[1-{\tanh }^2\left(k_{\mathrm{mn}}d\right)\]+3{\mathrm{\ζk}}_{\mathrm{mn}}^2/\mathrm{\ρ})\}---\left(4a\right)$

Wave number domain wave spectrum S of clean sea surface under limited water depth_oil(k_m,k_n)

$S_{\mathrm{clean}}(k_m,k_n)=\frac{S_{\mathrm{clean}}\left({\mathrm{\ω}}_{\mathrm{mn}}\right)}{2\sqrt{{\mathrm{\ω}}_{\mathrm{mn}}}}\{g\tanh \left(k_{\mathrm{mn}}d\right)+{\mathrm{gk}}_{\mathrm{mn}}d\[1-{\tanh }^2\left(k_{\mathrm{mn}}d\right)\]+3{{\mathrm{\ζ}}_{1}k}_{\mathrm{mn}}^2/{\mathrm{\ρ}}_1)\}---\left(4b\right)$

Wave number domain wave spectrum S for cleaning sea surface_oil(k_m,k_n)

$S(k_m,k_n)=\frac{S\left({\mathrm{\ω}}_{\mathrm{mn}}\right)}{2\sqrt{{\mathrm{\ω}}_{\mathrm{mn}}}}\{g\tanh \left(k_{\mathrm{mn}}d\right)+{\mathrm{gk}}_{\mathrm{mn}}d\[1-{\tanh }^2\left(k_{\mathrm{mn}}d\right)\]+3{\mathrm{\ζ}}_1{k}_{\mathrm{mn}}^2/{\mathrm{\ρ}}_1)\}---\left(4c\right)$

Therein, ζ₁For cleaning the surface tension of the sea, ζ is the surface tension of a spilled oil sea, ρ₁Is the density of seawater;

step five: according to the wave number domain wave spectrum S of the spilled oil sea surface under the limited water depth_oil(k_m,k_n) And a Longuet-Higgins sea wave model, and calculating the vertical displacement zeta of the spilled oil sea surface under the limited water depth_oil(x,y;t)：

Vertical displacement of spilled oil sea surface at limited depth of water

${\mathrm{\ζ}}_{\mathrm{oil}}(x,y;t)=\underset{m=1}{\overset{M}{\mathrm{\Σ}}}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\η}}_{\mathrm{mn}}\mathrm{exp\; j}(k_{m}x+k_{n}y-\mathrm{\ωt}+\mathrm{\φ})+c.c.---\left(5a\right)$

Vertical displacement of clean sea surface under limited depth of water

${\mathrm{\ζ}}_{\mathrm{clean}}(x,y;t)=\underset{m=1}{\overset{M}{\mathrm{\Σ}}}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\η}}_{\mathrm{mn},1}\mathrm{exp\; j}(k_{m}x+k_{n}y-\mathrm{\ωt}+\mathrm{\φ})+c.c.---\left(5b\right)$

Cleaning vertical displacements of the sea

$\mathrm{\ζ}(x,y;t)=\underset{m=1}{\overset{M}{\mathrm{\Σ}}}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\η}}_{\mathrm{mn},2}\mathrm{exp\; j}(k_{m}x+k_{n}y-\mathrm{\ωt}+\mathrm{\φ})+c.c.---\left(5c\right)$

Wherein, ${\mathrm{\η}}_{\mathrm{mn}}=\sqrt{{2S}_{\mathrm{oil}}(k_m,k_n)\mathrm{\Δ}{k}_x\mathrm{\Δ}{k}_y},{\mathrm{\η}}_{\mathrm{mn},1}=\sqrt{{2S}_{\mathrm{clean}}(k_m,k_n)\mathrm{\Δ}{k}_x\mathrm{\Δ}{k}_y},{\mathrm{\η}}_{\mathrm{mn},2}=\sqrt{2S(k_m,k_n)\mathrm{\Δ}{k}_x\mathrm{\Δ}{k}_y},$ c. represents the complex conjugate operator, t is time, (x, y) represents the position coordinates of the sea surface, phi is [ -pi, pi [ -pi ], [ phi ]]Uniformly distributed phase noise, Δ k_xIs the difference between adjacent wave numbers in the x direction, Δ k_yIs the difference between adjacent wave numbers in the y direction.

Examples

On the Matlab platform, it was simulated. Firstly, giving JONSWAP frequency domain wave spectrum parameters: the scale coefficient alpha is 0.0081, the peak rising factor gamma is 3.3, the wind area F is 100km, the wind speed U is 10m/s, the gravity constant g is 9.80655m/s²(ii) a Then, sea surface parameters are given: the length of the sea surface in the x direction is 100m, the length of the sea surface in the y direction is 100m, the sea wave direction is 45 degrees, the water depth d is 8m, and the density rho of the sea water₁＝10³kg/m³Surface tension zeta of clean sea surface₁＝74×10^-3N/m; finally, oil spill parameters are given: the normalization factor q of the sea surface oil spill area is 1, and the characteristic angular frequency omega of the oil spill is_D8rad/s, modulus of elasticity E₀＝23×10^-3N/m, kinematic viscosity coefficient v 34 × 10^-6m²(s) oil spill density ρ 870kg/m³Surface tension zeta of oil spill sea surface 28 × 10^-3N/m。

The realization process is as follows: calculating a JONSWAP frequency domain wave spectrum S (omega) by using the formula (1); calculating the frequency domain wave spectrum S of the clean sea surface under the limited water depth by using the formula (2)_clean(ω); calculating the frequency domain wave spectrum S of the spilled oil sea surface under the finite water depth by using the formula (3)_oil(ω); using the formulas (4 a), (4 b) and (4 c), the frequency domain wave spectrum S of the spilled oil sea surface under the limited water depth is obtained_oilFrequency domain wave spectrum S of clean sea surface under limited water depth_clean(omega) and clean sea surface frequency domain wave spectrum S (omega) are subjected to wave number domain conversion to respectively obtain wave number domain wave spectrum S_oil(k_m,k_n)、S_clean(k_m,k_n) And S (k)_m,k_n) (ii) a Using equations (5 a), (5 b) and (5 c), the vertical displacement ζ of the water surface at a finite depth is calculated_oil(x, y; t), vertical displacement ζ of clean sea surface at finite depth_clean(x, y; t) and the vertical displacement of the clean sea surface ζ (x, y; t) as shown in FIGS. 2a, 2b and 2 c; the results are displayed in grayscale for comparison of detail changes in clean, finite depth clean and finite depth spill seas, as shown in figures 3a, 3b and 3 c.

Claims (1)

1. A simulation method of an oil spill sea surface under a limited water depth; the method is characterized by comprising the following steps:

step one, calculating a JONSWAP frequency domain wave spectrum S (omega) of a clean sea surface according to marine environment parameters:

the expression of the JONSWAP frequency domain ocean wave spectrum is

Wherein, $\mathrm{\β}=\exp \[-\frac{{(\mathrm{\ω}-{\mathrm{\ω}}_p)}^2}{{2\mathrm{\δ}}^2{\mathrm{\ω}}_p^2}\],\mathrm{\δ}=\left\{\begin{array}{cc}0.07& {\mathrm{\ω}}_p\≤\mathrm{\ω}\\ 0.09& {\mathrm{\ω}}_p>\mathrm{\ω}\end{array}\right.,{\mathrm{\ω}}_p=7\mathrm{\π}{\left(\frac{g^{2}F}{U^3}\right)}^{-0.33},$ ω_pis the spectral peak angular frequency, omega is the angular frequency of the sea wave, gamma is the peak lift factor, delta is the peak shape parameter, g is the gravityConstant, alpha is a scale coefficient, U is the wind speed at 10m above the sea surface, and F is the wind zone length;

step two, calculating a frequency domain sea wave spectrum S of the clean sea surface under the limited water depth according to the water depth parameter_clean(ω)：

Frequency domain wave spectrum of clean sea surface under limited water depth

Wherein d is the water depth, and k is the wave number of the sea waves;

step three, calculating the frequency domain wave spectrum S of the oil spilling sea surface under the finite water depth according to the oil spilling parameters and the Marangoni theory_oil(ω)；

Frequency domain wave spectrum of oil spill sea surface

Wherein q is a normalization factor of the sea surface oil spilling region, $P\left(\mathrm{\ω}\right)=\frac{1\±2\mathrm{\τ}+{2\mathrm{\τ}}^2-X+Y(X+\mathrm{\τ})}{1\±2\mathrm{\τ}+{2\mathrm{\τ}}^2-2X+{2X}^2},\mathrm{\τ}=\sqrt{\frac{{\mathrm{\ω}}_D}{2\mathrm{\ω}}},X=\frac{E_0{k}^2}{\mathrm{\ρ}\sqrt{{2\mathrm{v\ω}}^3}},Y=\frac{E_{0}k}{4\mathrm{v\ρ\ω}},$ E₀is the elastic modulus, v is the kinematic viscosity coefficient, omega_DThe characteristic angular frequency of oil spill is shown, and rho is the density of the oil spill;

step four, according to the dispersion relation of the sea waves, converting the frequency domain sea wave spectrum of the limited water depth underflow oil sea surface into the sea wave spectrum S of the wavenumber domain_oil(k_m,k_n)；

Mesh discretization processing (k) is carried out on sea surface wave numbers_m,k_n)，k_mThe wave number of the sea surface in the x direction,

m-1, i.e. M points discretized by sea wave numbers in the x-direction; k is a radical of_nThe wave number in the y-direction of the sea surface,

n-1, i.e. discretizing the sea surface wavenumber in the y-direction by N points, where L is 0,1,2₁Length of sea surface in x direction, L₂The length of the sea surface in the y direction;

calculating the wave number of waves at each grid point $k_{\mathrm{mn}}=\sqrt{k_m^2+k_n^2}$ And angular frequency ${\mathrm{\ω}}_{\mathrm{mn}}=\sqrt{{\mathrm{gk}}_{\mathrm{mn}}\tanh \left(k_{\mathrm{mn}}d\right)+\frac{{\mathrm{\ζk}}_{\mathrm{mn}}^3}{\mathrm{\ρ}}},$ Wave number domain wave spectrum of limited water depth underflow oil sea surface

$S_{\mathrm{oil}}(k_m,k_n)=\frac{S_{\mathrm{oil}}\left({\mathrm{\ω}}_{\mathrm{mn}}\right)}{2\sqrt{{\mathrm{\ω}}_{\mathrm{mn}}}}\{g\tanh \left(k_{\mathrm{mn}}d\right)+{\mathrm{gk}}_{\mathrm{mn}}d\[1-{\tanh }^2\left(k_{\mathrm{mn}}d\right)\]+3{\mathrm{\ζk}}_{\mathrm{mn}}^2/\mathrm{\ρ})\},$ Wherein, zeta is the surface tension of the oil spill sea surface;

step five, according to the wave number domain wave spectrum S of the limited water depth underflow oil sea surface_oil(k_m,k_n) And a Longuet-Higgins sea wave model, and calculating the vertical displacement zeta of the spilled oil sea surface under the limited water depth_oil(x,y;t)：

Vertical displacement of oil spilled sea surface ${\mathrm{\ζ}}_{\mathrm{oil}}(x,y;t)=\underset{m=1}{\overset{M}{\mathrm{\Σ}}}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\η}}_{\mathrm{mn}}\mathrm{exp\; j}(k_{m}x+k_{n}y-\mathrm{\ωt}+\mathrm{\φ})+c.c.,$ Wherein,

c. represents the complex conjugate operator, t is time, (x, y) represents the position coordinates of the sea surface, phi is [ -pi, pi [ -pi ], [ phi ]]Uniformly distributed phase noise, Δ k_xIs the difference between adjacent wave numbers in the x direction, Δ k_yIs the difference between adjacent wave numbers in the y direction.

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