A non-hydrostatic algorithm for free-surface ocean modelling

An original implementation of a non-hydrostatic, free-surface algorithm based on a pressure correction method is proposed for ocean modelling. The free surface is implemented through an explicit scheme combined with a mode-spitting method but the depth-averaged velocity and the position of the free surface are updated at each non-hydrostatic iteration. The vertical momentum equation is also integrated up to the surface enabling a natural and accurate treatment of the surface layer. The consistent specification of the numerical schemes provides balanced transfers of potential and kinetic energy. This algorithm is well-suited for implementation as a non-hydrostatic kernel on originally hydrostatic free-surface ocean models such as Symphonie (http://poc.obs-mip.fr/pages/research_topics/modelling/symphonie/symphonie.htm) for which it has originally been developed.Energy balances associated with the propagation of short surface waves and solitary waves are presented for two dedicated well-documented configurations over closed domains. The buoyancy flux, the work rate of the pressure force together with the power of the advective terms are evaluated and discussed for the generation and the propagation of these two types of waves. The dissipation rate is in particular shown to be several orders of magnitude smaller than the work rates of the hydrostatic and non-hydrostatic pressure forces confirming the necessity for the exchanges of energy to be numerically balanced. The algorithm is subsequently applied to the complex generation of non-linear solitary internal waves by surface tides over Georges Bank, in the Gulf of Maine. The generation and the propagation of the observed non-linear and non-hydrostatic features in this region are correctly reproduced.     

Wind-wave variability in a shallow tidal sea—Spectral modelling combined with neural network methods

In this paper the wind-wave variability in the tidal basins of the German Wadden Sea is modelled with combined numerical and neural-network (NN) methods. First, the wave propagation and transformation in the study area are modelled with the state-of-the-art third-generation spectral wave model SWAN. The ability of SWAN to accurately reproduce the phenomena of interest in nonstationary conditions governed by highly variable winds, water levels and currents is shown by comparisons of the modelled and measured mean wave parameters at four stations. The principal component analysis of the SWAN results is then used to reveal the dominating spatial patterns in the data and to reduce their dimensionality, thus enabling an efficient and relatively straightforward NN modelling of mean wave parameters in the whole study area. It is shown that the data produced with the approach developed in this work have statistical properties (discrete probability distributions of the mean wave parameters etc.) very close to the properties of the data obtained with SWAN, thus proving that this approach can be used as a reliable tool for wind wave simulation in coastal areas, complementary to (often computationally demanding) spectral wave models.     

Numerical modelling of tsunami propagation in the open Black Sea

This paper focuses on the numerical modelling of tsunami propagation in the open Black Sea. Two types of numerical models are discussed: a model for the radial propagation of long waves and an evolutionary finite-difference prognostic model. Experimentally derived numerical data on the model source of tsunamis are reported. Translated by Vladimir A. Puchkin.     

Non-hydrostatic pressure in σ-coordinate ocean models

Numerical ocean modelling is computationally very demanding. Traditionally, the hydrostatic approximation has been applied to reduce the computational burden. This approximation is valid in large scale studies with coarse grid resolution. With faster computers and gradually smaller grid sizes, we may expect that more studies will be performed with non-hydrostatic ocean models. In recent papers several methods for including non-hydrostatic pressure in σ-coordinate models have been suggested. In this paper the sensitivity of the non-hydrostatic pressure field, the velocity fields, and the density fields to changes in the method for computing non-hydrostatic pressure in σ-coordinate ocean models is addressed.The first test case used involves the propagation and breaking of an internal wave at an incline in a tank. The other test case concerns tidally driven flow over a sill in a stratified fjord. The results from our numerical exercises suggest that the velocity and density fields are very robust to the model choices investigated here. The differences between the model results are of the same order as the uncertainty due to the internal pressure gradient error, and they are smaller than an estimate of the uncertainty due to subgrid scale closure.     

Joint modelling of extreme ocean environments incorporating covariate effects

Characterising the joint distribution of extremes of ocean environmental variables such as significant wave height (H_S) and spectral peak period (T_P) is important for understanding extreme ocean environments and in the design and assessment of marine and coastal structures. Many applications of multivariate extreme value analysis adopt models that assume a particular form of extremal dependence between variables without justification. Models are also typically restricted to joint regions in which all variables are extreme, but regions where only a subset of variables is extreme can be equally important for design. The conditional extremes model of Heffernan and Tawn (2004) provides one approach to overcoming these difficulties.Here, we extend the conditional extremes model to incorporate covariate effects in all of threshold selection, marginal and dependence modelling. Quantile regression is used to select appropriate covariate-dependent extreme value thresholds. Marginal and dependence modelling of extremes is performed within a penalised likelihood framework, using a Fourier parameterisation of marginal and dependence model parameters, with cross-validation to estimate suitable model parameter roughness, and bootstrapping to estimate parameter uncertainty with respect to covariate.We illustrate the approach in application to joint modelling of storm peak H_S and T_P at a Northern North Sea location with storm direction as covariate. We evaluate the impact of incorporating directional effects on estimates for return values, including those of a structure variable, similar to the structural response of a floating structure. We believe the approach offers the ocean engineer a straightforward procedure, based on sound statistics, to incorporate covariate effects in estimation of joint extreme environmental conditions.     

A tree-based solver for adaptive ocean modelling

The development of an adaptive (in space and time) ocean model from an existing adaptive finite-volume Navier–Stokes model is described. A flexible and efficient quadtree spatial discretisation is used which requires collocation of all variables (i.e. an A-grid discretisation). We demonstrate that the use of an approximate projection method allows for implicit damping of instabilities generally associated with the A-grid, at the expense of a relatively small amount of numerical energy dissipation, while accurately preserving dispersive properties and geostrophic balance. The finite-volume formulation also maintains second-order spatial accuracy at all solid boundaries. Test cases demonstrate the efficacy of the adaptive ocean model, and the advantages it has in terms of efficient representation of multi-scale behaviour within a single model. The model is freely available as open-source code.     

Coriolis weighting on unstructured staggered grids

There is an increasing interest to move ocean codes from classical Cartesian staggered mesh schemes to unstructured staggered grids. By using unstructured grid models one may construct meshes that follow the coastlines more accurately, and it is easy to apply a finer resolution in areas of special interest.In this paper we focus on how to approximate the Coriolis terms in such unstructured staggered grid models using equivalents of the Arakawa C-grid for the linear equations governing the propagation of the inertia-gravity waves. We base the analysis on a Delaunay triangulation of the region in question and use the Voronoi points and the midpoints on the triangle edges to define a staggered grid for the sea elevation and the velocity orthogonal to the edges of the triangles. It is shown that a standard method for the Coriolis weighting may create unphysical growth of the numerical solutions. A modified Coriolis weighting that conserves the total energy is suggested.In real applications diffusion is often introduced both for physical reasons, but often also in order to stabilise the numerical experiments. The growing modes associated with the unstructured staggered grids and equal weighting may force us to enhance the diffusion more than we would like from physical considerations. The modified weighting offers a simple solution to this problem.     

Similarity method for modelling hydroelastic offshore platforms

Similarity methods provide a very powerful technique for modelling offshore platforms. Such methods are preferable for situations where the wave-structure interaction process is complex, and its modelling cannot be expressed by rigorous mathematical formulations.In modelling the interaction between waves and an offshore platform, the dynamic properties of the wave and structure must be considered together to determine the similitude parameters. The paper gives details of such an investigation using a frequency response function approach. Three different models, which are termed general, distorted and nondistorted Froudian models, are developed. From the general model, it was found that the velocity scales of the wave and structure could be treated differently. Internal and external force scales could also be different. It is observed then that a restrictive modelling criterion is used when the velocity scale of a structure is obtained from Froudian scale modelling. Furthermore, detailed scales for practical use are developed. The model parameters for an acrylic plastic tripod tower platform with length scales of 1/50 and 1/70 are given, and the accuracy involved among the three models determined for these scales. The sensitivity of the results to slight variations in thickness and Young's modulus are also discussed.     

Energetics of lateral eddy diffusion/advection：Part Ⅱ. Numerical diffusion/diffusivity and gravitational potential energy change due to isopycnal diffusion

Study of oceanic circulation and climate requires models which can simulate tracer eddy diffusion and ad vection accurately. It is shown that the traditional Eulerian coordinates can introduce large artificial hori zontal diffusivity/viscosity due to the incorrect alignment of the axis. Therefore, such models can smear sharp fronts and introduce other numerical artifacts. For simulation with relatively low resolution, large lateral diffusion was explicitly used in models; therefore, such numerical diffusion may not be a problem. However, with the increase of horizontal resolution, the artificial diffusivity/viscosity associated with hori zontal advection in the commonly used Eulerian coordinates may become one of the most challenging ob stacles for modeling the ocean circulation accurately. Isopycnal eddy diffusion （mixing） has been widely used in numerical models. The common wisdom is that mixing along isopycnal is energy free. However, a careful examination reveals that this is not the case. In fact, eddy diffusion can be conceptually separated into two steps： stirring and subscale diffusion. Due to the thermobaric effect, stirring, or exchanging water masses, along isopycnal surface is associated with the change of GPE in the mean state. This is a new type of instability, called the thermobaric instability. In addition, due to cabbeling subscale diffusion of water parcels always leads to the release of GPE. The release of GPE due to isopycnal stirring and subscale diffusion may lead to the thermobaric instability.     

Joint modelling of wave spectral parameters for extreme sea states

Characterising the dependence between extremes of wave spectral parameters such as significant wave height (H_S) and spectral peak period (T_P) is important in understanding extreme ocean environments and in the design and assessment of marine structures. For example, it is known that mean values of wave periods tend to increase with increasing storm intensity. Here we seek to characterise joint dependence in a straightforward manner, accessible to the ocean engineering community, using a statistically sound approach.Many methods of multivariate extreme value analyses are based on models which assume implicitly that in some joint tail region each parameter is either independent of or asymptotically dependent on other parameters; yet in reality the dependence structure in general is neither of these. The underpinning assumption of multivariate regular variation restricts these methods to estimation of joint regions in which all parameters are extreme; but regions where only a subset of parameters are extreme can be equally important for design. The conditional approach of Heffernan and Tawn (2004), similar in spirit to that of Haver (1985) but with better theoretical foundation, overcomes these difficulties.We use the conditional approach to characterise the dependence structure of H_S and T_P. The key elements of the procedure are: (1) marginal modelling for all parameters, (2) transformation of data to a common standard Gumbel marginal form, (3) modelling dependence between data for extremes of pairs of parameters using a form of regression, (4) simulation of long return periods to estimate joint extremes. We demonstrate the approach in application to measured and hindcast data from the Northern North Sea, the Gulf of Mexico and the North West Shelf of Australia. We also illustrate the use of data re-sampling techniques such as bootstrapping to estimate the uncertainty in marginal and dependence models and accommodate this uncertainty in extreme quantile estimation.We discuss the current approach in the context of other approaches to multivariate extreme value estimation popular in the ocean engineering community.     

Numerical modelling of boom and oil spill with SPH

The protection of coastal areas against oil pollution is often addressed with the use of floating booms. These bodies are subject to an empirical design always based on physical models. Indeed, the numerical modelling of a two-phase flow (oil and water) with complicated free surface in the vicinity of a floating body is a challenging issue. The Smoothed Particle Hydrodynamics (SPH) Lagrangian numerical method is appropriate to such simulations since it allows the modelling of complex motions and fluid–structure interactions. In this paper we first study the mechanism of oil leakage by entrainment due to combined turbulent production and buoyancy. Then, we present the main features of the SPH method in a turbulent formalism and apply this model to predict the motion of a boom and an oil spill in an open-channel and a wave flume, for three types of oil (heavy, light and emulsion). Numerical results are compared to experiments and used to depict criteria for oil leakage. It appears that oil leakage by entrainment occurs when the surface water velocity upstream the boom exceeds a critical value which was estimated around 0.5 m/s for a light oil under steady current. A more accurate criterion is derived from theoretical considerations and successfully compared to numerical experiments. In the case of wave flume, no validation from experiments could be made. However, it appears that leakage occurs from a critical wave height between 0.5 and 1.0 m, for the tested wave period of 4 s. A more extended panel of numerical tests would allow a better knowledge of the involved mechanisms and critical parameters. An extensive use of this model should extend our knowledge regarding the mechanisms of oil leakage under a boom and allow a better and easier design of booms in the near future.     

On the solution of the carbonate chemistry system in ocean biogeochemistry models

We present a simplified method for solving the local equilibrium carbonate chemistry in numerical ocean biogeochemistry models. Compared to the methods typically used, the scheme is fast, efficient and compact. The accuracy of the solution is dictated by the number of species retained in the expression for alkalinity and there is almost no computational penalty for retaining minor contributions. We demonstrate that this scheme accurately reproduces the results of the commonly used method in the context of a three-dimensional global ocean carbon cycle model. Using this model we also show that neglecting the regional variations in surface dissolved inorganic phosphorus and silicic acid concentrations can lead to significant systematic bias in regional estimates of air–sea carbon fluxes using such models.     

Response of data from numerical modelling of the Equatorial Atlantic circulation to fluctuations of turbulence coefficients

A numerical model for ocean thermohydrodynamics is considered whose difference scheme permits a number of linear and quadratic invariants to be retained. The model is used as the basis for adaptive computations in the Equatorial Atlantic ocean. Model equations were integrated using different values of the coefficients of diffusion and momentum turbulent exchange. It has been shown that variations of these coefficients strongly influence the intensity of jet streams and the structure of the thermocline.Translated by Vladimir A. Puchkin.     

Numerical modelling and analysis of temperature controlled density currents in Tomales Bay, California

The coastal region adjacent to Tomales Bay, California is dominated by wind-driven upwelling during spring and summer and the cold, upwelled water is moved towards Tomales Bay, entering the estuary with the flood tide. If the tidal excursion is ≥6 km and the cold water subducts beneath the warmer, less dense estuarine water, a temperature controlled density current may form and intrude towards the head of the estuary as a thermally stratified bottom layer. The numerical modelling was aimed at determining the capability of the Delft3D-FLOW model to reproduce the cold ocean water intrusion events, the response (development and progression) of these intrusions to differing physical scenarios and the comparative importance of the parameters to the intrusions. The numerical model successfully reproduced the density intrusions and showed that the persistence and break down of the density intrusions were affected by a number of physical parameters, to varying degrees. The sensitivity analysis showed that density intrusion formation is controlled by tidal conditions and ocean water temperature. The strength and persistence of the developed density intrusions are influenced by wind, insolation and estuary depth. Fresh water inflow at the head of the estuary had no impact on the density intrusions. Three-dimensional numerical modelling is thus a valuable tool in understanding the estuary and its functioning.     

Certain results of numerical simulation of processes in the Arctic Ocean

Variations in hydrophysical parameters in the Arctic Ocean and the North Atlantic are studied on the basis of numerical simulation with the use of an ocean circulation model (including ice formation and drift). The main circulation and ice-drift modes have been ascertained depending on atmospheric cycles. The possibilities of the parameterization of intermediate and deep water formation in numerical models of polar ocean dynamics are considered. The effect of the interannual variability of the discharge of Siberian rivers on the distribution and propagation of fresh water in this region are estimated from numerical experiments. The simulation results of the propagation of the dissolved methane from Siberian rivers are presented.     

Wave modelling in the vicinity of submerged breakwaters

This paper describes a simple method for modelling wave breaking over submerged structures, with the view of using such modelling approach in a coastal area morphodynamic modelling system.A dominant mechanism for dissipating wave energy over a submerged breakwater is depth-limited wave breaking. Available models for energy dissipation due to wave breaking are developed for beaches (gentle slopes) and require further modifications to model wave breaking over submerged breakwaters.In this paper, wave breaking is split into two parts, namely: 1) depth-limited breaking modelled using Battjes and Janssen's (1978) theory [Battjes, J.A. and Jannsen, J.P.F.M. (1978). Energy loss and setup due to breaking of random waves. Proceedings of the 16th Int. Conf. Coast. Eng., Hamburg, Germany, pp. 569-587.] and 2) steepness limited breaking modelled using an integrated form of the Hasselmann's whitecapping dissipation term, commonly used in fully spectral wind–wave models. The parameter γ₂, governing the maximum wave height at incipient breaking (H_max = γ₂d) is used as calibration factor to tune numerical model results to selected laboratory measurements. It is found that γ₂ varies mainly with the relative submergence depth (ratio of submergence depth at breakwater crest to significant wave height), and a simple relationship is proposed. It is shown that the transmission coefficients obtained using this approach compare favourably with those calculated using published empirical expressions.     

Kraken海洋声学模型及其声传播与衰减的数值试验

针对射线、简正波、PE、FFP等传播模型的算法原理及其适用的海洋环境,建立了以Kraken声学模型计算软件为基础的海洋声场数值预报系统.应用该预报系统对4组典型的海洋声场进行了数值试验,结果表明:在相同的海面和海底边界条件下,声场分布是由声速剖面和声源位置决定的.在负梯度声场中,所有声线都折向海底,在极限声线外产生阴影区.声源位于声道轴附近的温跃层中会产生波导传播.用射线理论解释了上述现象的成因,指出了其实际应用价值.     

Accurate representation of geostrophic and hydrostatic balance in unstructured mesh finite element ocean modelling

Accurate representation of geostrophic and hydrostatic balance is an essential requirement for numerical modelling of geophysical flows. Potentially, unstructured mesh numerical methods offer significant benefits over conventional structured meshes, including the ability to conform to arbitrary bounding topography in a natural manner and the ability to apply dynamic mesh adaptivity. However, there is a need to develop robust schemes with accurate representation of physical balance on arbitrary unstructured meshes. We discuss the origin of physical balance errors in a finite element discretisation of the Navier–Stokes equations using the fractional timestep pressure projection method. By considering the Helmholtz decomposition of forcing terms in the momentum equation, it is shown that the components of the buoyancy and Coriolis accelerations that project onto the non-divergent velocity tendency are the small residuals between two terms of comparable magnitude. Hence there is a potential for significant injection of imbalance by a numerical method that does not compute these residuals accurately. This observation is used to motivate a balanced pressure decomposition method whereby an additional “balanced pressure” field, associated with buoyancy and Coriolis accelerations, is solved for at increased accuracy and used to precondition the solution for the dynamical pressure. The utility of this approach is quantified in a fully non-linear system in exact geostrophic balance. The approach is further tested via quantitative comparison of unstructured mesh simulations of the thermally driven rotating annulus against laboratory data. Using a piecewise linear discretisation for velocity and pressure (a stabilised P₁P₁ discretisation), it is demonstrated that the balanced pressure decomposition method is required for a physically realistic representation of the system.     

适于模拟不规则水域波浪的缓坡方程两种数值模型比较

本文分析比较了适于不规则水域波浪模拟的椭圆型缓坡方程两种数值模型。两种数值模型均采用有限体积法离散,分别基于四叉树网格和非结构化三角形网格建立。首先结合近岸缓坡地形上波浪传播的经典物理模型实验对两种数值模型分别进行了验证,并结合计算结果对比分析了两种模型的计算精度和效率。计算结果表明,两种数值模型均可有效地模拟近岸波浪的传播变形;相对非结构化三角形网格下的模型,基于四叉树网格建立的数值模型在数值离散和求解过程中无需引入形函数、不产生复杂的交叉项,离散简单,易于程序实现,且节约计算存储空间,计算效率高。