Numerical simulations of Rossby–Haurwitz waves

Simulations of Rossby–Haurwitz waves have been carried out using four different high‐resolution numerical shallow water models: a spectral model, two semi‐Langrangian models predicting wind components and potential vorticity respectively, and a finite‐volume model on a hexagonal–icosahedral grid. The simulations show that (i) unlike the nondivergent case, the shallow water Rossby–Haurwitz wave locally generates small‐scale features and so has a potential enstrophy cascade, and (ii) contrary to common belief, the zonal wavenumber 4 Rossby–Haurwitz wave is dynamically unstable and will eventually break down if initially perturbed. Implications of these results for the use of the Rossby–Haurwitz wave as a numerical model test case are discussed. The four models tested give very similar results, giving confidence in the accuracy and robustness of the results. The most noticeable difference between the models is that truncation errors in the hexagonal–icosahedral grid model excite the Rossby–Haurwitz wave instability, causing the wave to break down quickly, whereas for the other models in the configurations tested the instability is excited only by roundoff error at worst, and the Rossby–Haurwitz wave breaks down much more slowly or not at all.     

On resonant Rossby–Haurwitz triads

The dynamics of non-divergent flow on a rotating sphere are described by the conservation of absolute vorticity. The analytical study of the non-linear barotropic vorticity equation is greatly facilitated by the expansion of the solution in spherical harmonics and truncation at low order. The normal modes are the well-known Rossby–Haurwitz (RH) waves, which represent the natural oscillations of the system. Triads of RH waves, which satisfy conditions for resonance, are of critical importance for the distribution of energy in the atmosphere.
We show how non-linear interactions of resonant RH triads may result in dynamic instability of large-scale components. We also demonstrate a mathematical equivalence between the equations for an orographically forced triad and a simple mechanical system, the forced-damped swinging spring. This equivalence yields insight concerning the bounded response to a constant forcing in the absence of damping. An examination of triad interactions in atmospheric reanalysis data would be of great interest.     

Generalized boussinesq model for periodic non-linear shallow-water waves

This paper presents the development of a generalized Boussinesq (gB) model for the periodic non-linear shallow-water waves. An incident cnoidal wave solution for the gB model is derived and applied to the wave simulation. A set of radiation boundary conditions is also established to transmit effectively the cnoidal waves out of the computational domain. The classical solutions of the second-order cnoidal waves are discussed within the content of the KdV equation and the generalized Boussinesq equations. An Euler's predictor-corrector finite-difference algorithm is used for numerical computation. The propagation of normally incident cnoidal waves in a channel is studied. The simulated wave profiles agree well with the analytical results. The temporal and spatial evolution of an obliquely incident cnoidal wave is also modelled. The phenomenon of Mach reflection is discussed.     

斜向入射波与反射波的分离

提出了一种分离斜向入射波和反射波的方法,波浪可以是规则波、不规则波,波向可以任意.在一定的限制条件下,采用两点浪高仪的波浪信号就可将斜向入射波和反射波分离.     

Statistical characteristics of individual waves in laboratory wind waves

On the basis of data on the statistical characteristics of individual waves in laboratory wind waves reported in part I of this series, a self-consistent similarity regime is found to exist among properties of the individual waves, such as the nondimensional frequency, the wave number, the phase speed, and the steepness. Also, it is shown that forms of past empirical formulas for the development of the peak wave can be derived starting from the 3/2-power law, as an extension of the persent laboratory experimental data. In the derivation, only values of the coefficient of the 3/2-power law, and the fraction of momentum transferred from the wind retained by the wind waves, remain on an empirical basis.     

Abnormally high waves due to spectral instability of surface waves

The outcomes of our laboratory experiments corroborate hypotheses advanced earlier, dedicated to the mechanism of origination of abnormally high waves under the development of spectral instability. We have clarified the characteristics of spectral instability development including the dependences of the value of the downshift of the spectral maximum and distances at which it occurs on the steepness of waves and width of the initial spectrum. In addition, we have revealed the dependence of the number of abnormal waves on the fraction of spectral energy transferred to the low-frequency range and on the stage of spectral instability development. Our results offer the basis for creating a statistical model of the origination of abnormally high waves.     

On interaction between intermediate-depth long waves and deep-water short waves

The interaction between a unidirectional deep-water short-wave train and an intermediate water-depth long wave is studied. The steady solutions are derived up to third order in wave steepness, respectively, using two different approaches: a conventional perturbation method employing linear phase functions to describe both long- and short-wave phases and a phase modulation method using a modulational phase function to model the short-wave phase. The two results are shown to be identical for a parametric range χ₁ coth Kdχ₃ ≤ 0.5, where χ₃ is the short-to-long wavelength ratio, χ₁ and K are, respectively, the long-wave steepness and wavenumber, and d the water depth. When χ₁ coth Kd approaches χ₃, the conventional solution converges slowly and eventually diverges for χ₁ coth Kdχ₁. The slow convergence of the conventional solution results from the approximation of a modulated short-wave phase by a linear phase formulation. In addition to the increasing modulation of the short-wave phase, amplitude and wavenumber as the water depth decreases, it is found that the modulation of the short-wave intrinsic frequency and potential amplitude along the long-wave surface become significant. Previous results about virtually non-modulated short-wave intrinsic frequency and potential amplitude are only limited to the case of unidirectional wave modulation in very deep water.     

广义岭估计的谱分解式和Liu型广义岭估计

首先从多参数岭估计,即广义岭参数人手,给出其谱分解形式。然后就岭估计会使估值偏差增大的问题,在Liu估计的基础上,推导出一种新的有偏估计方法——Liu型广义岭估计,并给出该方法的模型、解式。Liu型广义岭估计可以在尽量不增大偏差的同时改进病态性,获得更好的估值。最后通过算例进行了比较分析。     

3D scour below pipelines under waves and combined waves and currents

This paper presents the results of an experimental investigation on three-dimensional local scour below a rigid pipeline subjected to wave only and combined wave and current conditions. The tests were conducted in a conventional wave flume. The major emphasis of the investigation was on the scour propagation speed (free span expansion rate) along the pipeline after local scour was initiated at a controlled location. The effects of flow ratio (steady current velocity vs. combined waves/current velocity), flow incidence angle and pipeline initial embedment depth on free span expansion rate were investigated. It was observed that the scour along the pipeline propagated at a constant rate under wave only conditions. The scour propagation rate decreased with increasing embedment depth, however, increased with the increasing Keuglegan–Carpenter (KC) number. Under combined wave and current conditions, the effect of velocity ratio on scour propagation velocity along the pipeline was quantified. Empirical relationships between the scour propagation rate (V_h) and key parameters such as the KC number and embedment depth (e/D) were established based on the testing results.     

Unevenly compressed flexural-gravitity waves

Free flexural-gravity waves unevenly compressed in an ice-covered basin have been studied using a linear formulation. The conditions were specified allowing determination of the bounds of the angular area of wave disturbances and of the oscillation frequency intervals due to compression. The paper considers the distribution of the wave characteristics over frequency depending on compressive stresses and the direction of wave propagation.Translated by Vladimir A. Puchkin.     

Simulating offshore sand waves

Sand waves form a prominent regular pattern in the offshore seabeds of sandy shallow seas and pose a threat to a range of offshore activities. A two-dimensional vertical (2DV) flow and morphological simulation model describing the behaviour of these sand waves has been developed. The simulation model contains the 2DV shallow water equations, with a free water surface and a general bed load formula. The water movement is coupled to the sediment transport equation with a seabed evolution equation. The domain is non-periodic in both directions. The spatial discretisation is performed by a spectral method based on Chebyshev polynomials. A fully implicit method is chosen for the discretisation in time. Firstly, we validate the simulation model mathematically by reproducing the results obtained using a linear stability analysis for infinitely small sand waves. Hereby, we investigate a steady current situation induced by a wind stress applied at the sea surface. The bed forms we find have wavelengths in the order of hundreds of metres when the resistance at the seabed is relatively large. The results show that it is possible to model the initial evolution of sand waves with a numerical simulation model. This paper forms the necessary first step to investigate the intermediate term behaviour of sand waves.     

Experimental study on strong interaction between regular waves and wind waves—I

The interaction between mechanically generated regular waves and wind waves is experimentally investigated in a wind-wave tunnel. It is shown that the growth process of regular waves is divided into the four distinct stages as follows: (1) almost independent coexistence of wind waves and regular waves, (2) attenuation of wind waves with simultaneous growth of regular waves, (3) rapid growth of regular waves after disappearance of wind waves, and (4) transition of regular waves to wind waves after the wave breaking. At the second stage there is an apparent relation between the attenuation of wind waves and the growth of regular waves. This fact suggests that there is some strong nonlinear interactions which transfers energy effectively from wind waves to regular waves.     

Detection of ocean waves using satellite altimetry: Application to equatorial Kelvin waves

A new method for wave motion detection from satellite altimetric measurements of sea surface height is presented. The essence of the approach is to construct a two‐dimensional traveling‐wave Fourier series representation of the amplitude field within a prespecified oceanic region. The method employs an iterative, nonlinear least‐squares technique based on the Marquardt‐Levenberg algorithm to solve for model parameters describing characteristic features of the evolving wave system. The Marquardt‐Levenberg Fourier series (MLFS) algorithm was applied to Kelvin waves active during the 1986–1987 El Nino event in the equatorial Pacific ocean using GEOSAT Exact Repeat Mission altimetry data. Characteristics of the wave system were found to be in essential agreement with earlier field measurements and the observations of Cheney and Miller (1987) obtained using time series developed from GEOSAT data. The advantage of the present detection scheme lies in its speed and ability to determine a wave system's dispersion relation over a finite range of wavenumbers, and hence the group velocity of that system.     

Prediction of ocean waves based on the single-parameter growth equation of wind waves

A method for the prediction of ocean waves was developed on the basis of the single-parameter growth equation of wind waves, proposed byToba (1978) on the basis of similarity in growing wind waves. The applicability of the method to actual problems was tested by hindcasting the wave characteristics with the method, for two cases with differing time and space scales, one in Kii Channel Approach, Japan, and the other in the North Atlantic Ocean. The results showed that the present method can predict waves within an error of 1.3 m in wave heights, which ranged from 3 to 12 m.     

Steepness and asymmetry of extreme waves and the highest waves in deep water

Traditional wave steepness s=H/L does not define steep asymmetric waves in a random sea uniquey. Three additional parameters characterising single zero-downcross waves in a time series are crest front steepness, vertical asymmetry factor and horizontal asymmetry factor. Results for steepness and asymmetry from zero-downcross analysis of wave data obtained from full scale measurements in deep water on the Norwegian continental shelf in 58 time series are presented. The analysis demonstrates clearly the asymmetry of both “extreme waves” and the highest waves. The period and height of the highest waves are also given together with their correlation to spectral parameters. The measured maximum wave heights are also compared with predicted values of maximum wave heights showing good agreement.     

实验室长波引起的风浪低频频移

利用实验室波浪水槽观测规则长波对风浪的影响。谱分析显示,较之纯风浪谱,除已被广泛关注的长波抑制风浪这一现象外,当长波波陡较小,且频率远离风浪峰频时,长波还使得风浪谱向低频移动。利用Longuet-HigginsStewart(1960)理论,并考虑到风浪破碎的约束,计算了规则长波的存在对风浪谱的影响,发现可以较好地解释这一现象。表明当长波波陡小且频率远离风浪峰频时,长波对短波的二阶调制及其引起的破碎加强可能是长波影响风浪的主要机制。