Proof of Six-Wave Resonance Conditions of Ocean Surface Gravity Waves in Deep Water

A necessary big step up in the modern water wave theories and their widespread application in ocean engineering is how to obtain 6-wave resonance conditions and to prove it. In the light of the existing forms and characteristics of 3-wave, 4-wave and 5-wave resonance conditions, the 6-wave resonance conditions are proposed and proved for currently a maximum wave-wave resonance interactions of the ocean surface gravity waves in deep water, which will be indispensable to both the Kolmogorov spectrum of the corresponding universal wave turbulence and a synthetic 4-5-6-wave resonant model for the ocean surface gravity waves.

Particle Motion in Surface Gravity Waves

Fluid particles in translating surface gravity waves have an orbital motion which decreases in size with increasing mean depth. These wave characteristics came from observations and were not forecast theoretically. The classical potential flow model is incapable of explaining the particle movement due to the irrotational assumption and to a flaw in carrying out the method. When a wave passes by an observer from left to right, the particles move clockwise under a crest and a trough. This correct conclusion is consistent with what the incorrect standard theory implies but should not be considered to have been derived from it.

On Surface Gravity Wave Energies

Kinetic and potential energies of a propagating surface gravity wave are calculated for the orbiting fluid particles. These energies are then compared with the respective standard theoretical quantities. Based on earlier work, it is concluded that the two sets of energies will not agree. To substantiate the prediction observations are needed, or a new theory for the depth decay rate of the particle orbits would work. Considerably less algebraic effort is involved in the present analysis than what was done in the past.

Impact of the Waves on the Sea Surface Roughness under Uniform Wind Conditions:Idealized Cases for Uniform Winds(Part I)

The effect of the surface gravity waves over sea surface roughness length (z0) is investigated from several idealized numerical experiments with the Wave-Watch-III (WW3) model. The WW3 model is combined with a simplified model to estimate z0, CD, u* and U10 as function of the sea state. The impacts related to the presence of the ocean waves over z0 are obtained from conditions of growing (young waves) and mature seas (old waves). The wave spectrum is obtained from WW3 model for each idealized simulation under uniform wind conditions. Uniform wind experiments range from 15 to 45 m/s. The simplified algorithm determines z0, CD, u* and U10 for cases of young waves, old waves and by the Charnock method. The results show that when the ocean is characterized by young waves, both z0 and CD (drag coefficient) increase while U10 is reduced. In Charnock case, the values of z0, CD and U10 have no dependence with the presence of gravity waves. Experiments using winds higher than 30 m/s result in young waves’ CD values higher than the CD value for old waves. Even for young waves CD values are high for cases of strong winds. The results also show that in experiments using winds higher than 30 m/s the dependence between CD and wave age becomes stronger, which is in accordance with other studies.

Impact of the Waves on the Sea Surface Roughness Length under Idealized Like-Hurricane Wind Conditions (Part Ⅱ)

In this study the effect of the surface waves over sea surface roughness (z0) and drag coefficient (CD) is investigated by combining an ocean wave model and a simplified algorithm, which estimates z0 and CD with and without dependence on the sea state. This investigation was possible from several numerical simulations with the Wave-Watch-III (WW3) model for complex wind conditions. The numerical experiments were performed for idealized like-hurricanes with different translation speed (0, 5 and 10 m/s) and maximum wind speed (MWS) at the centre (35, 45 and 55 m/s). It is observed that z0 and CD are strongly dependent on the sea state, via substantial modification in Charnock parameterization (zch). As the hurricane translation speed increases more discrepancies in z0 and CD are observed in opposite quadrants around the region of MWS. As for instance, higher, longer and older (or more developed) waves, located in the front-right quadrant, produce lower values of z0 and CD. In the rear-left quadrant, where the waves are lower, shorter and younger (or less developed), higher values of z0 and CD are observed. In addition the difference between values on opposite quadrants increases as the hurricane intensity increases, showing the hurricane intensification dependence. Interesting aspects are observed in scatter plotting wave age versus Charnock coefficient. It is also observed that zch, which has a constant value of 0.0185, is modified by the sea state, where young waves produce higher values of zch, while old waves are related to lower values of zch when compared with zch without dependence on sea state.

涉及垂直多孔膜的斜向入射波散射问题研究

Oblique surface waves incident on a fixed vertical porous membrane of various geometric configurations is analyzed here.The mixed boundary value problem is modified into easily resolvable problems by using a connection.These problems are reduced to that of solving a couple of integral equations.These integral equations are solved by a one-term or a two-term Galerkin method.The method involves a basis functions consists of simple polynomials multiplied with a suitable weight functions induced by the barrier.Coefficient of reflection and total wave energy are numerically evaluated and analyzed against various wave parameters.Enhanced reflection is found for all the four barrier configurations.

On the Downshift of Wave Frequency for Bragg Resonance

For surface gravity waves propagating over a horizontal bottom that consists of a patch of sinusoidal ripples,strong wave reflection occurs under the Bragg resonance condition.The critical wave frequency,at which the peak reflection coefficient is obtained,has been observed in both physical experiments and direct numerical simulations to be downshifted from the well-known theoretical prediction.It has long been speculated that the downshift may be attributed to higher-order rippled bottom and free-surface boundary effects,but the intrinsic mechanism remains unclear.By a regular perturbation analysis,we derive the theoretical solution of frequency downshift due to third-order nonlinear effects of both bottom and free-surface boundaries.It is found that the bottom nonlinearity plays the dominant role in frequency downshift while the free-surface nonlinearity actually causes frequency upshift.The frequency downshift/upshift has a quadratic dependence in the bottom/free-surface steepness.Polychromatic bottom leads to a larger frequency downshift relative to the monochromatic bottom.In addition,direct numerical simulations based on the high-order spectral method are conducted to validate the present theory.The theoretical solution of frequency downshift compares well with the numerical simulations and available experimental data.

On Rayleigh expansion for nonlinear long water waves

We consider strongly nonlinear long waves on the surface of a homogeneous fluid layer.By modifying the formulation for the high-order spectral(HOS)method for waves in water of finite depth,we present a higher-order nonlinear system for the surface elevation and the velocity potential on the free surface to describe the two-dimensional evolution of large amplitude long waves.It is shown that the resulting system preserves the Hamiltonian structure of the Euler equations and can be transformed to the strongly nonlinear long-wave model for the depth-averaged velocity.Due to truncation of the linear dispersion relation for water waves,both the system for the surface velocity potential and that for the depth-averaged velocity are ill-posed when the order of approximation is odd and even,respectively.To avoid this ill-posedness,fully dispersive models are also proposed.Under the same order approximation,the long-wave model is found more effective for numeral studies of large amplitude long waves than the finite-depth model.