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1.
Using Boussinesq scaling for water waves while imposing no constraints on rotationality, we derive and test model equations for nonlinear water wave transformation over varying depth. These use polynomial basis functions to create velocity profiles which are inserted into the basic equations of motion keeping terms up to the desired Boussinesq scaling order, and solved in a weighted residual sense. The models show rapid convergence to exact solutions for linear dispersion, shoaling, and orbital velocities; however, properties may be substantially improved for a given order of approximation using asymptotic rearrangements. This improvement is accomplished using the large numbers of degrees of freedom inherent in the definitions of the polynomial basis functions either to match additional terms in a Taylor series, or to minimize errors over a range. Explicit coefficients are given at O(μ2) and O(μ4), while more generalized basis functions are given at higher order. Nonlinear performance is somewhat more limited as, for reasons of complexity, we only provide explicitly lower order nonlinear terms. Still, second order harmonics may remain good to kh ≈ 10 for O(μ4) equations. Numerical tests for wave transformation over a shoal show good agreement with experiments. Future work will harness the full rotational performance of these systems by incorporating turbulent and viscous stresses into the equations, making them into surf zone models. 相似文献
2.
《Coastal Engineering》1999,37(1):57-96
A Boussinesq model accurate to O(μ)4, μ=k0h0 in dispersion and retaining all nonlinear effects is derived for the case of variable water depth. A numerical implementation of the model in one horizontal direction is described. An algorithm for wave generation using a grid-interior source function is derived. The model is tested in its complete form, in a weakly nonlinear form corresponding to the approximation δ=O(μ2), δ=a/h0, and in a fully nonlinear form accurate to O(μ2) in dispersion [Wei, G., Kirby, J.T., Grilli, S.T., Subramanya R. (1995). A fully nonlinear Boussinesq model for surface waves: Part 1. Highly nonlinear unsteady waves. J. Fluid Mech., 294, 71–92]. Test cases are taken from the experiments described by Dingemans [Dingemans, M.W. (1994). Comparison of computations with Boussinesq-like models and laboratory measurements. Report H-1684.12, Delft Hydraulics, 32 pp.] and Ohyama et al. [Ohyama, T., Kiota, W., Tada, A. (1994). Applicability of numerical models to nonlinear dispersive waves. Coastal Engineering, 24, 297–313.] and consider the shoaling and disintegration of monochromatic wave trains propagating over an elevated bar feature in an otherwise constant depth tank. Results clearly demonstrate the importance of the retention of fully-nonlinear effects in correct prediction of the evolved wave fields. 相似文献
3.
Based on the classical Boussinesq model by Peregrine [Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech. 27 (4), 815–827], two parameters are introduced to improve dispersion and linear shoaling characteristics. The higher order non-linear terms are added to the modified Boussinesq equations. The non-linearity of the Boussinesq model is analyzed. A parameter related to h/L0 is used to improve the quadratic transfer function in relatively deep water. Since the dispersion characteristic of the modified Boussinesq equations with two parameters is only equal to the second-order Padé expansion of the linear dispersion relation, further improvement is done by introducing a new velocity vector to replace the depth-averaged one in the modified Boussinesq equations. The dispersion characteristic of the further modified Boussinesq equations is accurate to the fourth-order Padé approximation of the linear dispersion relation. Compared to the modified Boussinesq equations, the accuracy of quadratic transfer functions is improved and the shoaling characteristic of the equations has higher accuracy from shallow water to deep water. 相似文献
4.
《Coastal Engineering》2006,53(4):311-318
The extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coastal Eng., 32, 91–117] and Lee et al. [Lee, C., Kim, G., Suh, K.D., 2003. Extended mild-slope equation for random waves. Coastal Eng., 48, 277–287] are compared analytically and numerically to determine their applicability to random wave transformation. The geometric optics approach is used to compare the two models analytically. In the model of Suh et al., the wave number of the component wave with a local angular frequency ω is approximated with an accuracy of O(ω − ω¯) at a constant water depth, where ω¯ is the carrier frequency of random waves. In the model of Suh et al., however, the diffraction effects and higher-order bottom effects are considered only for monochromatic waves, and the shoaling coefficient of random waves is not accurately approximated. This inaccuracy arises because the model of Suh et al. was derived for regular waves. In the model of Lee et al., all the parameters of random waves such as wave number, shoaling coefficient, diffraction effects, and higher-order bottom effects are approximated with an accuracy of O(ω − ω¯). This approximation is because the model of Lee et al. was developed using the Taylor series expansion technique for random waves. The result of dispersion relation analysis suggests the use of the peak and weighted-average frequencies as a carrier frequency for Suh et al. and Lee et al. models, respectively. All the analytical results are verified by numerical experiments of shoaling of random waves over a slightly inclined bed and diffraction of random waves through a breakwater gap on a flat bottom. 相似文献
5.
含强水流高阶Boussinesq水波方程 总被引：13，自引：3，他引：10
采用摄动法并利用已建立的纯波情况下高阶Boussinesq方程,建立了可以考虑强水流与波浪相互作用的高阶Boussinesq方程.水流速度与波浪群速具有相同量级,且随时间和空间的变化尺度远大于波浪周期和波长.方程色散性近似到[4/4]阶Pade展开,对浅水情况方程可以是完全非线性的,可适用于波流相互作用的强非线性问题.通过将水流存在时波长和波幅的结果与一阶斯托克斯波结果对比,讨论了具有不同近似程度的3种含波流相互作用的Boussinesq方程的适用性. 相似文献
6.
Turbulent mixing in the upper ocean(30-200 m) of the northwestern Weddell Sea is investigated based on profiles of temperature,salinity and microstructure data obtained during February 2014.Vertical thermohaline structures are distinct due to geographic features and sea ice distribution,resulting in that turbulent dissipation rates(ε) and turbulent diffusivity(K) are vertically and spatially non-uniform.On the shelf north of Antarctic Peninsula and Philip Ridge,with a relatively homogeneous vertical structure of temperature and salinity through the entire water column in the upper 200 m,both ε and K show significantly enhanced values in the order of O(10~(-7))-O(10~(-6)) W/kg and O(10~(-3))-O(10~(-2)) m~2/s respectively,about two or three orders of magnitude higher than those in the open ocean.Mixing intensities tend to be mild due to strong stratification in the Powell Basin and South Orkney Plateau,where s decreases with depth from O(10~(-8)) to O(10~(-9)) W/kg,while K changes vertically in an inverse direction relative to s from O(10~(-6)) to O(10~(-5)) m~2/s.In the marginal ice zone,K is vertically stable with the order of10~(-4) m~2/s although both intense dissipation and strong stratification occur at depth of 50-100 m below a cold freshened mixed layer.Though previous studies indentify wind work and tides as the primary energy sources for turbulent mixing in coastal regions,our results indicate weak relationship between K and wind stress or tidal kinetic energy.Instead,intensified mixing occurs with large bottom roughness,demonstrating that only when internal waves generated by wind and tide impinge on steep topography can the energy dissipate to support mixing.In addition,geostrophic current flowing out of the Weddell Sea through the gap west of Philip Passage is another energy source contributing to the local intense mixing. 相似文献
7.
Extended Boussinesq equations for rapidly varying topography 总被引：1，自引：0，他引：1
We developed a new Boussinesq-type model which extends the equations of Madsen and Sørensen [1992. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly varying bathymetry. Coastal Engineering 18, 183-204.] by including both bottom curvature and squared bottom slope terms. Numerical experiments were conducted for wave reflection from the Booij's [1983. A note on the accuracy of the mild-slope equation. Coastal Engineering 7, 191-203] planar slope with different wave frequencies using several types of Boussinesq equations. Madsen and Sørensen's model results are accurate in the whole slopes in shallow waters, but inaccurate in intermediate water depths. Nwogu's [1993. Alternative form of Boussinesq equation for nearshore wave propagation. Journal of Waterway, Port, Coastal and Ocean Engineering 119, 618-638] model results are accurate up to 1:1 (V:H) slope, but significantly inaccurate for steep slopes. The present model results are accurate up to the slope of 1:1, but somewhat inaccurate for very steep slopes. Further, numerical experiments were conducted for wave reflections from a ripple patch and also a Gaussian-shaped trench. For the two cases, the results of Nwogu's model and the present model are accurate, because these models include the bottom curvature term which is important for the cases. However, Madsen and Sørensen's model results are inaccurate, because this model neglects the bottom curvature term. 相似文献
8.
《Coastal Engineering》2002,44(3):205-229
In this paper, we derive and test simplified higher-order Boussinesq equations, i.e., higher-order Boussinesq equations which only show lower-order terms. Simplifications are performed linearly for flat beds and slopes of O(∇h). With proper coefficient choice, dispersion and shoaling properties are found to be good, while interior fluid velocities show relatively greater error at high wavenumbers.The resulting sets of equations are found to be variants of already-existing equations, which may be easily modified to improve performance. The new equations have dispersion identical to previous results but significantly improved shoaling. 相似文献
9.
10.
A new form of higher order Boussinesq equations 总被引：3，自引：0，他引：3
On the basis of the higher order Boussinesq equations derived by the author (1999), a new form of higher order Boussinesq equations is developed through replacing the depth-averaged velocity vector by a new velocity vector in the equations in order to increase the accuracy of the linear dispersion, shoaling property and nonlinear characteristics of the equations. The dispersion of the new equations is accurate to a [4/4] Pade expansion in kh. Compared to the previous higher order Boussinesq equations, the accuracy of quadratic transfer functions is improved and the shoaling property of the equations have higher accuracy from shallow water to deep water. 相似文献
12.
《Coastal Engineering》2006,53(4):319-323
The literature contains empirical knowledge on whether the wave celerity or the group velocity should be used in the line source function for internal wave generation for at given set of Boussinesq or mild-slope equations. Theoretical derivations that confirm and explain these empirical findings are devised. For Boussinesq equations with, e.g. Padé[2,2]-type of dispersion relation some procedures for internal wave generation are affected by their excitation of an evanescent mode. This has some undesirable consequences, but the evanescent-mode excitation can be avoided by the use of an “internal flux boundary”. 相似文献
13.
A note on linear dispersion and shoaling properties in extended Boussinesq equations 总被引：1，自引：0，他引：1
A set of optimum parameter α is obtained to evaluate the linear dispersion and shoaling properties in the extended Boussinesq equations of [Madsen and Sorensen, 1992 and Nwogu, 1993], and [Chen and Liu, 1995]. Optimum α values are determined to produce minimal errors in each wave property of phase velocity, group velocity, or shoaling coefficient relative to the analytical one given by the Stokes wave theory. Comparisons are made of the percent errors in phase velocity, group velocity, and shoaling coefficient produced by the Boussinesq equations with a different set of optimum α values. The case with a fixed value of α = −0.4 is also presented in the comparison. The comparisons reveal that the optimum α value tuned for a particular wave property gives in general poor results for other properties. Considering all the properties simultaneously, the fixed value of α = −0.4 may give overall accuracies in phase velocity and shoaling coefficient for all the types of Boussinesq equations selected in this study. 相似文献
14.
Essential properties of Boussinesq equations for internal and surface waves in a two-fluid system 总被引：1，自引：0，他引：1
Boussinesq equations describing motions of internal waves in a two-fluid system with the presence of free surface are theoretically derived, and the associated essential properties are examined in this study. Eliminating the dependence on the vertical coordinate from all variables, four equations constitute the Boussinesq model with two flexible parameters, zu and zl, which indicate the specific elevations, respectively, in the upper and lower fluids. Similar to the Boussinesq model for a single-layer fluid, zu and zl are determined by matching the linear dispersion relation with Lamb's solution. This determines the optimal model. In the analysis stage, this problem is classified into two cases, the thicker-upper-layer case and the thicker-lower-case case, to avoid the possible divergence of wave properties as the thickness ratio grows. Since there exist two modes of motions that may be excited, cases of both modes are separately analyzed. Linear characteristics including the amplitude ratios and normalized particle velocities are analyzed. Second-order harmonic waves are examined to validate nonlinear behaviors of present model. Results of linear and nonlinear investigations show that the present model indeed extends the applicable range of traditional Boussinesq equations. 相似文献
15.
16.
Turbulent mixing in the central equatorial Pacific has been quantitatively evaluated by analyzing data from microstructure measurements and conductivity temperature depth profiler (CTD) observations in a meridionally and vertically large region. The result that strong turbulent mixing with dissipation rate ε (>O(10?7) W kg?1), continuing from sea-surface mixed layer to low Richardson number region below, in the area within 1° of the equator, shows that turbulent mixing has a close relationship to shear instability. ε > O(10?7) W kg?1 and turbulent diffusivity K ρ > O(10?3) m2 s?1 were obtained from near-surface to 85 db at stations even southwardly beyond 3°S, where it is already far from the southern boundary (~2°S) of the Equatorial Undercurrent. Turbulence-induced heat flux and salinity flux were calculated, and both had their maxima in the equatorial upwelling region, though the former was downward and the latter was upward. Accordingly, vertical velocity in the upwelling region was estimated to be similar to the results derived by other methods. These fluxes and the vertical velocity suggest the critical importance of turbulent mixing in maintaining the well-mixed upper layer. Secondly, in the intermediate region (>500 db), turbulent eddies were investigated by applying Thorpe’s method to the CTD data. A large number of overturns were detected, with spatial-averaged K ρ (700–1,000 db) being 3.3 × 10?6 m2 s?1, and the corresponding K ρ-max reaching to O(10?4) m2 s?1 in the north (3°–13°N). The results suggest that, in the intermediate region, considerable turbulent mixing occurs and moderates the properties of the water masses. 相似文献
17.
This study deals with the general numerical model to simulate the two-dimensional tidal flow, flooding wave (long wave) and shallow water waves (short wave). The foundational model is based on nonlinear Boussinesq equations. Numerical method for modelling the short waves is investigated in detail. The forces, such as Coriolis forces, wind stress, atmosphere and bottom friction, are considered. A two-dimensional implicit difference scheme of Boussinesq equations is proposed. The low-reflection outflow open boundary is suggested. By means of this model,both velocity fields of circulation current in a channel with step expansion and the wave diffraction behind a semi-infinite breakwater are computed, and the results are satisfactory. 相似文献
18.
G. S. Golitsyn 《Izvestiya Atmospheric and Oceanic Physics》2011,47(3):393-398
A relationship between the statistical parameters of horizontal diffusion and the parameters of the energy-containing part of the frequency spectrum of sea-surface elevations is found depending on the wave age Ω and the ratio between the wind speed at 10 m and the phase velocity of the peak of a wave. It has been observed in [1–7] that the diffusion coefficient K(r) of a patch of size r increases as r β, where 1.15 < β < 4/3, and the patch area S(t) increases with time as t γ, where 2 < γ < 3. As was calculated in [15], in the energy-containing part of the elevation frequency spectrum, S(ω) ～ ω?n , where n = 13/3 for young waves with Ω > 2, n = 4 for waves with 1.2 < ψ < 2, and n = 11/3 for developed waves with 0.83 < Ω < 1.2. It is found that β = (n + 1)/4 and γ = 8/(7 ? n). These relations explain the entire set of observed exponents: β = 4/3 and γ = 3 for young waves and β = 1.15 and γ = 2.34 for large sizes (up to 1000 km) and times (up to a month) when it is found here that β = 7/6 and γ = 2.4. 相似文献
19.
通过改进二阶全非线性 Boussinesq 波浪方程中的色散项,得到了一组没有改变原方程的数学形式但适用于更大变化水深的新方程,其色散性能和变浅性能都比原方程有了很大改进,所适用的水深范围更大,能更好地描述从深水到近岸浅水处的波浪传播;并基于新方程建立了波浪数值模型,通过模拟波浪从浅水到深水的传播变形来验证新方程的有效性. 相似文献
20.
建立具有色散性的水平二维非线性波浪方程,方程的非线性近似到了三阶。方程以波面升高和自由表面速度势表达的微分-积分型数学方程,给出方程的数值求解方法和算例,对方程积分项的处理给出了计算方法。计算结果与Boussinesq方程模型和缓坡方程模型的对应计算结果进行了对比。 相似文献