New Numerical Scheme for Simulation of Hyperbolic Mild-Slope Equation

The original hyperbolic mild-slope equation can effectively take into account the combined effects of wave shoaling, refraction, diffraction and reflection, but does not consider the nonlinear effect of waves, and the existing numerical schemes for it show some deficiencies. Based on the original hyperbolic mild-slope equation, a nonlinear dispersion relation is introduced in present paper to effectively take the nonlinear effect of waves into account and a new numerical scheme is proposed. The weakly nonlinear dispersion relation and the improved numerical scheme are applied to the simulation of wave transformation over an elliptic shoal. Numerical tests show that the improvement of the numerical scheme makes efficient the solution to the hyperbolic mild-slope equation. A comparison of numerical results with experimental data indicates that the results obtained by use of the new scheme are satisfactory.     

一个两时间层分裂显格式海洋环流模式(MASNUM)及其检验

A two-time-level, three-dimensional numerical ocean circulation model（named MASNUM） was established with a two-level, single-step Eulerian forward-backward time-differencing scheme. A mathematical model of large-scale oceanic motions was based on the terrain-following coordinated, Boussinesq, Reynolds-averaged primitive equations of ocean dynamics. A simple but very practical Eulerian forward-backward method was adopted to replace the most preferred leapfrog scheme as the time-differencing method for both barotropic and baroclinic modes. The forward-backward method is of second-order of accuracy, computationally efficient by requiring only one function evaluation per time step, and free of the computational mode inherent in the three-level schemes. This method is superior to the leapfrog scheme in that the maximum time step of stability is twice as large as that of the leapfrog scheme in staggered meshes thus the computational efficiency could be doubled. A spatial smoothing method was introduced to control the nonlinear instability in the numerical integration. An ideal numerical experiment simulating the propagation of the equatorial Rossby soliton was performed to test the amplitude and phase error of this new model. The performance of this circulation model was further verified with a regional（northwest Pacific） and a quasi-global（global ocean simulation with the Arctic Ocean excluded） simulation experiments. These two numerical experiments show fairly good agreement with the observations. The maximum time step of stability in these two experiments were also investigated and compared between this model and that model which adopts the leapfrog scheme.     

MASNUM-WAM海浪模式集合Kalman滤波同化研究——Ⅱ.集合样本对同化效果的影响

背景误差相关结构的确定是影响海浪同化效果的关键因素之一。集合Kalman滤波是一种较为成熟的同化方法,其可以对背景误差进行实时更新和动态估计,现已广泛应用于海洋和大气领域的研究。本文基于MASNUM-WAM海浪模式,分别采用静态样本集合Kalman滤波和EAKF方法,针对2014年全球海域开展海浪数据同化实验,同化资料为Jason-2卫星高度计数据,利用Saral卫星高度计资料对同化实验结果进行检验。结果表明,两组同化方案均有效提高了海浪模式的模拟水平,EAKF方案在风场变化较大的西风带区域表现显著优于静态样本集合Kalman滤波方案,但总体上两者相差不大。综合考虑计算成本和同化效果,静态样本集合Kalman滤波方案更适用于海浪业务化预报。     

Different approaches to model error formulation in 4D-Var: a study with high-resolution advection schemes

All numerical models are imperfect. Weak constraint variational data assimilation ( VDA ), which provides a treatment of the modelling errors, is studied; building on the approach of Vidard et al. (Tellus, 56 A, pp. 177–188, 2004). The evolution of model error ( ME ) is modelled using ordinary differential equations, which involve a scalar parameter. These approaches were tested using different high-resolution advection schemes. The first set of experiments were constructed to see if it is possible to account for (numerical) discretization error within such a framework. In other set of experiments, a systematic source of modelling error was introduced by deliberately specifying an incorrect value for the Coriolis parameter in the model. Results with observational state at half of the model state resolution, are also presented. We also discuss a method of estimating the scalar parameter in the ME through VDA . In all cases, the inclusion of ME provides reduction in forecasting errors. Also, our experiments indicate that different settings of the model (e.g. using different high-resolution advection schemes) would need different ME formulation. Results presented in this paper could be used to formulate sophisticated ME forms to account for systematic errors in higher dimensional models with complex advection schemes.     

Correction of atmospheric effect on ADEOS/OCTS ocean color data: Algorithm description and evaluation of its performance

This paper first describes the atmospheric correction algorithm for OCTS visible band data used at NASDA/EOC. Sharing a basic structure with Gordon and Wang’s Sea WiFS algorithm, it uses 10 candidate aerosol models including the “Asian dust model” introduced in consideration of the unique feature of aerosols over the east Asian waters. Based on the observations at 670 and 865 nm bands, the algorithm selects a pair of aerosol models that account best for the observed spectral reflectances, and synthesizes the aerosol reflectance used for the atmospheric correction. Two different schemes for determining the value of the parameter for the aerosol model selection are presented and their anticipated estimation error is analyzed in terms of retrieved water reflectance at 443 nm. The results of our numerical simulation show that the standard deviation of the estimation error of the “weighted average” scheme is mostly within the permissible level of ±0.002, reducing the error by 18% on average compared to the “simple average” scheme. The paper further discusses the expected error under the old CZCS-type atmospheric correction, which assumes constant aerosol optical properties throughout the given image. Although our algorithm has a better performance than the CZCS algorithm, further analysis shows that the error induced by the assumption taken in the algorithm that the water-leaving radiance at 670 nm band is negligibly small may be large in high pigment concentration waters, indicating the necessity for future improvements.     

A numerically efficient data analysis method with error map generation

The variational inverse model (VIM) for data analysis was already shown to be statistically equivalent to objective analysis (OA) provided the covariance function for OA and the VIM reproducing kernel are identical. The VIM, however does not allow a direct derivation of the error field associated with the analysis. The purpose of the paper is to extend the one-to-one correspondence between the two analysis schemes by proposing a heuristic statistical error expression for the VIM. The numerical efficiency on analysis and error map generation of both methods is compared on quasi-synoptic and climatological data sets. It is shown that the VIM analysis and error map generation offers interesting numerical skills in both case studies.     

MASNUM-WAM海浪模式集合Kalman滤波同化研究-I.风场扰动对海浪模拟影响

模式集合样本的代表性和观测信息的可靠性是制约数据同化效果的重要因素,而前者对海浪模式同化的影响尤为显著。由于海浪模式对初始场的敏感性较弱,来自大气的风输入源函数是海浪的重要能量输入,如何合理地对风输入进行扰动,构造海浪的集合模式运行,是实现和改进海浪模式集合Kalman滤波同化的关键问题。为了实现海浪模式集合运行,本文提出了风场的三种集合扰动方案,分别为：纯随机数、随机场和时间滞后的风场扰动方法。本研究利用2014年1月ECMWF全球风场,基于这三种风场扰动方法开展了集合海浪模式的集合运行实验,并统计分析了海浪特征要素（有效波高）和二维波数谱对风场扰动的响应。结果表明,随机场集合扰动方案所构造的风场集合效果最佳,所得海浪模拟结果的集合样本发散度适中,能够较为合理地反映背景误差的统计特征,可用于进一步的集合Kalman滤波海浪数据同化实验。     

The improvement of the one-dimensional Mellor-Yamada and K-profile parameterization turbulence schemes with the non-breaking surface wave-induced verticalmixing

Both the level 2.5 Mellor-Yamada turbulence closure scheme(MY) and K-profile parameterization(KPP) are popularly used by the ocean modeling community.The MY and the KPP are improved through including the non-breaking surface wave-induced vertical mixing(Bv),and the improved schemes were tested by using continuous data at the Papa ocean weather station(OWS) during 1961–1965.The numerical results showed that the Bv can make the temperature simulations fit much better with the continuous data from Papa Station.The two improved schemes overcame the shortcomings of predicting too shallow upper mixed layer depth and consequently overheated sea surface temperature during summertime,which are in fact common problems for all turbulence closure models.Statistical analysis showed that the Bv effectively reduced the mean absolute error and root mean square error of the upper layer temperature and increased the correlation coefficient between simulation and the observation.Furthermore,the performance of vertical mixing induced by shear instability and the Bv is also compared.Both the temperature structure and its seasonal cycle significantly improved by including the Bv,regardless of whether shear instability was included or not,especially for the KPP mixing scheme,which suggested that Bv played a dominant role in the upper ocean where the mean current was relatively weak,such as at Papa Station.These results may provide a clue to improve ocean circulation models.     

Spurious diapycnal mixing in terrain-following coordinate models: The problem and a solution

In this paper, we identify a crucial numerical problem in sigma coordinate models, leading to unacceptable spurious diapycnal mixing. This error is a by-product of recent advances in numerical methods, namely the implementation of high-order diffusive advection schemes. In the case of ROMS, spurious mixing is produced by its third-order upwind advection scheme, but our analysis suggests that all diffusive advection schemes would behave similarly in all sigma models. We show that the common idea that spurious mixing decreases with resolution is generally false. In a coarse-resolution regime, spurious mixing increases as resolution is refined, and may reach its peak value when eddy-driven lateral mixing becomes explicitly resolved. At finer resolution, diffusivities are expected to decrease but with values that only become acceptable at resolutions finer than the kilometer. The solution to this problem requires a specifically designed advection scheme. We propose and validate the RSUP3 scheme, where diffusion is split from advection and is represented by a rotated biharmonic diffusion scheme with flow-dependent hyperdiffusivity satisfying the Peclet constraint. The rotated diffusion operator is designed for numerical stability, which includes improvements of linear stability limits and a clipping method adapted to the sigma-coordinate. Realistic model experiments in a southwest Pacific configuration show that RSUP3 is able to preserve low dispersion and diffusion capabilities of the original third-order upwind scheme, while preserving water mass characteristics. There are residual errors from the rotated diffusion operator, but they remain acceptable. The use of a constant diffusivity rather than the Peclet hyperdiffusivity tends to increase these residual errors which become unacceptable with Laplacian diffusion. Finally, we have left some options open concerning the use of time filters as an alternative to spatial diffusion. A temporal discretization approach to the present problem (including implicit discretization) will be reported in a following paper.     

背景误差协方差矩阵不同求逆方案在高度计资料同化试验中的应用比较

针对海表高度计资料的同化,考查了背景误差协方差矩阵的不同求逆方案对同化效果的影响。所使用的求逆方案包括避免求逆的经验正交函数方案(EOF/EOF_var)、递归滤波方案(RF/RF_var)以及采用初等变换法直接求逆的方案(Inv)。基于上述方案开展了热带太平洋地区2002年1-7月的TOPEX/Poseidon高度计资料同化试验,并利用SODA再分析资料和TAO观测资料评估了各方案对温度场的同化效果,主要得到如下结果:与SODA相比较,Inv方案对模式温度场改进甚微,其余四种方案在100~300 m深度之间对温度场改进较多,在其它深度范围内则改进较少;与TAO观测相比较,EOF_var、RF_var方案对模式温度场改进最多,EOF和RF方案次之,Inv方案则对温度场改进甚少。     

Assimilation of surface currents into a regionalmodel over Qingdao coastal waters of China

Surface currents measured by high frequency （HF） radar arrays are assimilated into a regional ocean model over Qingdao coastal waters based on Kalman filter method. A series of numerical experiments are per- formed to evaluate the performance of the data assimilation schemes. In order to optimize the analysis pro- cedure in the traditional ensemble Kalman filter （ENKF）, a different analysis scheme called quasiensemble Kaman filter （QENKF） is proposed. The comparisons between the ENKF and the QENKF suggest that both them can improve the simulated error and the spatial structure. The estimations of the background error covariance （BEC） are also assessed by comparing three different methods： Monte Carlo method; Canadian quick covariance （CQC） method and data uncertainty engine （DUE） method. A significant reduction of the root-mean-square （RMS） errors between model results and the observations shows that the CQC method is able to better reproduce the error statistics for this coastal ocean model and the corresponding external forcing. In addition, the sensibility of the data assimilation system to the ensemble size is also analyzed by means of different scales of the ensemble size used in the experiments. It is found that given the balance of the computational cost and the forecasting accuracy, the ensemble size of 50 will be an appropriate choice in the Qingdao coastal waters.     

Numerical simulation of tidal bores and hydraulic jumps

An implicit finite difference formulation of the nonlinear shallow water equations is developed to allow for the treatment of tidal bores and hydraulic jumps. Five different schemes are investigated involving upwind treatment of convective terms, central differences combined with dissipative interface, forward time-centering and various combinations of these techniques. The schemes are analyzed with respect to their effective amplification portraits, and they are tested on periodic bores, uniform bores and steady hydraulic jumps. In this connection the model results are verified against analytical solutions and a numerical solution obtained with a Godunov Riemann solver. Scheme 4, which combines forward time centering and dissipative interface, is found to be superior to the others and it is applicable for Courant numbers within the range 0.25 to 1.5. This scheme is applied to a case study of the tidal bore in Huangzhou Bay and Qiantang River. The model results are shown to be in very good agreement with field data.     

Implementation of high-order particle-tracking schemes in a water column model

Stochastic differential equations (SDEs) offer an attractively simple solution to complex transport-controlled problems, and have a wide range of physical, chemical, and biological applications, which are dominated by stochastic processes, such as diffusion. As for deterministic ordinary differential equations (ODEs), various numerical scheme exist for solving SDEs. In this paper various particle-tracking schemes are presented and tested for accuracy and efficiency (time vs. accuracy). To test the schemes, the particle tracking algorithms are implemented into a community wide used 1D water column model. Modelling individual particles allows a straightforward physical interpretation of the involved processes. Further, this approach is strictly mass conserving and does not suffer from the numerical diffusion that plagues grid-based methods. Moreover, the Lagrangian framework allows to assign properties to the individual particles, that can vary spatially and temporally. The movement of the particles is described by a stochastic differential equation, which is consistent with the advection–diffusion equation. Here, the concentration profile is represented by a set of independent moving particles, which are advected according to the velocity field, while the diffusive displacements of the particles are sampled from a random distribution, which is related to the eddy diffusivity field.The paper will show that especially the 2nd order schemes are accurate and highly efficient. At the same level of accuracy, the 2nd order scheme can be significantly faster than the 1st order counterpart. This gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost.     

基于POM的物质输运方程数值格式研究

在三维海洋模式POM基础上建立水质模型,采用中心差分格式、迎风格式以及Smolarkiewicz迎风格式离散物质输运方程.以三维理想水槽中连续源排放的浓度场预测为例,分析3种离散格式求解所得的浓度场.结果表明,3种格式的数值解与解析解的偏差均小于20%.中心差分格式会引起解的震荡,导致物质的反向输移,出现浓度负值.迎风格式能够保证浓度的正值,但该格式带来的数值耗散导致数值解与解析解偏离较大.Smolarkiewicz迎风格式在普通迎风格式基础上引入抗扩散流速,经多次叠代,能有效降低计算中的数值耗散,提高了计算精度.     

Hybrid N-order Lagrangian Interpolation Eulerian?Lagrangian Method for Salinity Calculation

The Eulerian?Lagrangian method (ELM) has been used by many ocean models as the solution of the advection equation, but the numerical error caused by interpolation imposes restriction on its accuracy. In the present study, hybrid N-order Lagrangian interpolation ELM (LiELM) is put forward in which the N-order Lagrangian interpolation is used at first, then the lower order Lagrangian interpolation is applied in the points where the interpolation results are abnormally higher or lower. The calculation results of a step-shaped salinity advection model are analyzed, which show that higher order (N=3?8) LiELM can reduce the mean numerical error of salinity calculation, but the numerical oscillation error is still significant. Even number order LiELM makes larger numerical oscillation error than its adjacent odd number order LiELM. Hybrid N-order LiELM can remove numerical oscillation, and it significantly reduces the mean numerical error when N is even and the current is in fixed direction, while it makes less effect on mean numerical error when N is odd or the current direction changes periodically. Hybrid odd number order LiELM makes less mean numerical error than its adjacent even number order LiELM when the current is in the fixed direction, while the mean numerical error decreases as N increases when the current direction changes periodically, so odd number of N may be better for application. Among various types of Hybrid N-order LiELM, the scheme reducing N-order directly to 1st-order may be the optimal for synthetic selection of accuracy and computational efficiency.     

TY01和O02两种海面空气动力粗糙度方案的比较

本文采用美国国家浮标站(44008)2003年1～3月的资料，通过COARE算法(版本2．6b)，比较了O02和TY01这两种海面空气动力粗糙度长度的参数化方案。通过对摩擦速度、拖曳系数、海面粗糙度及风应力等物理因子的计算得出：在粗糙的海面上，TY01和O02两种参数化方案的计算结果是比较一致的。在考虑浪的信息方面，TY01和O02都是很好的参数化方案．它们都可以适用于不同的风速条件，适用于各种尺度的海洋及湖泊。但是这两种方案在处理幼波时存在不连续的缺点。并且，对于风速较小的光滑海面，尽管它们计算的结果较一致，但是仍然存在偏差。据此，本文的结果对于理论分析和数值计算如何正确使用上述两种海面空气动力粗糙度参数化方案，可提供必要的参考价值。     

Diffusion reduction in an arbitrary scale third generation wind wave model

The numerical schemes for the geographic propagation of random, short-crested, wind-generated waves in third-generation wave models are either unconditionally stable or only conditionally stable. Having an unconditionally stable scheme gives greater freedom in choosing the time step (for given space steps). The third-generation wave model SWAN (“Simulated WAves Nearshore”, Booij et al., 1999) has been implemented with this type of scheme. This model uses a first order, upwind, implicit numerical scheme for geographic propagation. The scheme can be employed for both stationary (typically small scale) and nonstationary (i.e. time-stepping) computations. Though robust, this first order scheme is very diffusive. This degrades the accuracy of the model in a number of situations, including most model applications at larger scales. The authors reduce the diffusiveness of the model by replacing the existing numerical scheme with two alternative higher order schemes, a scheme that is intended for stationary, small-scale computations, and a scheme that is most appropriate for nonstationary computations. Examples representative of both large-scale and small-scale applications are presented. The alternative schemes are shown to be much less diffusive than the original scheme while retaining the implicit character of the particular SWAN set-up. The additional computational burden of the stationary alternative scheme is negligible, and the expense of the nonstationary alternative scheme is comparable to those used by other third generation wave models. To further accommodate large-scale applications of SWAN, the model is reformulated in terms of spherical coordinates rather than the original Cartesian coordinates. Thus the modified model can calculate wave energy propagation accurately and efficiently at any scale varying from laboratory dimensions (spatial scale O(10 m) with resolution O(0.1 m)), to near-shore coastal dimension (spatial scale O(10 km) with resolution O(100 m)) to oceanic dimensions (spatial scale O(10 000 km) with resolution O(100 km).     

黄海近岸海雾的初始场择优集合预报试验:个例研究

选取2014年4月发生的一次黄海近岸海雾个例,利用WRF(Weather Research and Forecasting)模式开展了集合预报试验研究。依据每个集合成员初始场中海平面气压、2 m温度、2 m水汽混合比与2 m相对湿度(relative humidity, RH)4个变量的均方根误差(root mean square error, RMSE)与RMSE集合平均值的相对大小,以剔除高于者而保留低于者的原则,设计了4种不同的初始场集合体择优方案,实施了一系列数值预报试验,比较了不同择优方案的集合预报效果。研究结果表明:(1)蒙特卡罗方法所生成的集合体中存在不少海雾预报效果较差的成员,这会降低集合预报效果,因此初始场择优十分必要;(2)以RH作为择优变量的择优方案(记为RH-RMSE方案),集合预报效果明显优于其他3种方案;(3)对比不择优集合预报,采用RH-RMSE方案的择优集合预报效果不仅节省了50%左右的计算时间,并且公正预兆评分(equitable threat score,ETS)改进率高达36%左右。本研究提出的RH-RMSE方案具有业务化应用前景。     

Data assimilation of partitioned HF radar wave data into Wavewatch III

In this study the assimilation of HF radar data into a high resolution, coastal Wavewatch III model is investigated. An optimal interpolation scheme is used to assimilate the data and the design of a background error covariance matrix which reflects the local conditions and difficulties associated with a coastal domain is discussed. Two assimilation schemes are trialled; a scheme which assimilates mean parameters from the HF radar data and a scheme which assimilates partitioned spectral HF radar data. This study demonstrates the feasibility of assimilating partitioned wave data into a coastal domain. The results show that the assimilation schemes provide satisfactory improvements to significant wave heights but more mixed results for mean periods. The best improvements are seen during a stormy period with turning winds. During this period the model is deficient at capturing the change in wave directions and the peak in the waveheights, while the high sea state ensures good quality HF radar data for assimilation. The study also suggests that there are both physical and practical advantages to assimilating partitioned wave data compared to assimilating mean parameters for the whole spectrum.     

资料同化中的伴随方法及其在海洋中的应用

较系统地综述了资料同化中各种方法的分类,并着重说明了伴随方法的基本原理和具体实现步骤。并总结了伴随方法在海洋中的应用现状,并指出由于观测资料特别是遥感资料的日益丰富,伴随方法将有广泛地应用前景。