Study on linear and nonlinear bottom friction parameterizations for regional tidal models using data assimilation

Data assimilation technique (adjoint method) is applied to study the similarities and the differences between the Ekman (linear) and the Quadratic (nonlinear) bottom friction parameterizations for a two-dimensional tidal model. Two methods are used to treat the bottom friction coefficient (BFC). The first method assumes that the BFC is a constant in the entire computation domain, while the second applies the spatially varying BFCs. The adjoint expressions for the linear and the nonlinear parameterizations and the optimization formulae for the two BFC methods are derived based on the typical Largrangian multiplier method. By assimilating the model-generated ‘observations’, identical twin experiments are performed to test and validate the inversion ability of the presented methodology. Four experiments, which employ the linear parameterization, the nonlinear parameterizations, the constant BFC and the spatially varying BFC, are carried out to simulate the M₂ tide in the Bohai Sea and the Yellow Sea by assimilating the TOPEX/Poseidon altimetry and tidal gauge data. After the assimilation, the misfit between model-produced and observed data is significantly decreased in the four experiments. The simulation results indicate that the nonlinear Quadratic parameterization is more accurate than the linear Ekman parameterization if the traditional constant BFC is used. However, when the spatially varying BFCs are used, the differences between the Ekman and the Quadratic approaches diminished, the reason of which is analyzed from the viewpoint of dissipation rate caused by bottom friction. Generally speaking, linear bottom friction parameterizations are often used in global tidal models. This study indicates that they are also applicable in regional ocean tidal models with the combination of spatially varying parameters and the adjoint method.     

A model study of tidal distributions in the Celtic and Irish Sea regions determined with finite volume and finite element models

An unstructured mesh tidal model of the west coast of Britain, covering the Celtic Sea and Irish Sea is used to compare tidal distributions computed with finite element (FE) and finite volume (FV) models. Both models cover an identical region, use the same mesh, and have topography and tidal boundary forcing from a finite difference model that can reproduce the tides in the region. By this means, solutions from both models can be compared without any bias towards one model or another. Two-dimensional calculations show that for a given friction coefficient, there is more damping in the FV model than the FE model. As bottom friction coefficient is reduced, the two models show comparable changes in tidal distributions. In terms of mesh resolution, calculations show that for the M₂ tide, the mesh is sufficiently fine to yield an accurate solution over the whole domain. However, in terms of higher harmonics of the tide, in particular the M₆ component, its small-scale variability in near-shore regions which is comparable to the mesh of the model, suggests that the mesh resolution is insufficient in the near-coastal regions. Even with a finer mesh in these areas, without detailed bottom topography and a spatial varying friction depending on bed types and bed forms, which is not available, model skill would probably not be improved. In addition in the near-shore region, as shown in the literature, the solution is sensitive to the form of the wetting/drying algorithm used in the model. Calculations with a 3D version of the FV model show that for a given value of k, damping is reduced compared to the 2D version due to the differences in bed stress formulation, with the 3D model yielding an accurate tidal distribution over the region.     

Regional ocean tide loading modelling around the Iberian Peninsula

We developed a new 1/12° resolution oceanic tide model in the complex region that surrounds the Iberian Peninsula. The model, named IBER01, allows us to obtain more accurate tidal loading computations for precise geodetic and gravimetric observations in this area. The modelling follows the scheme of data assimilation (coastal tide gauge, bottom pressure sensors and TOPEX/Poseidon altimetry) into a hydrodynamical model, which is based on two-dimensional barotropic depth averaged shallow-water equations. Detailed bathymetry data and quadratic bottom friction with a specific drag coefficient for the region have been considered. Improved ocean load maps for the Iberian Peninsula are obtained for eight harmonic constituents (Q1, P1, O1, K1, N2, M2, S2 and K2), after computing the load effect (Newtonian attraction and elastic contribution) using IBER01 and six present-day global oceanic tide models for comparison. The results achieved verify the quality of the new model. Our ocean loading computations reduce considerably the discrepancies between the theoretical Earth tide parameters and those from observations at the level of 0.3%.     

Impact of tidal mixing with different scales of bottom roughness on the general circulation

The current study deals with a parameterization of diapycnal diffusivity in an ocean model. The parameterization estimates the diapycnal diffusivity depending on the location of tidal-related energy dissipation over rough topography. The scheme requires a bottom roughness map that can be chosen depending on the scales of topographic features. Here, we implement the parameterization on an ocean general circulation model, and we examine the sensitivity of the modeled circulations to different spatial scales of the modeled bottom roughness. We compare three simulations that include the tidal mixing scheme using bottom roughness calculated at three different ranges of spatial scales, with the largest scale varying up to 200?km. Three main results are discussed. First, the dependence of the topographic spectra with depth, characterized by an increase in spectral energy over short length scales in the deep ocean, influences the vertical profile of the diffusivity. Second, the changes in diffusivities lead to different equilibrium solutions in the Atlantic meridional overturning circulation and bottom circulation. In particular, the lower cell of the Atlantic overturning and the bottom water transport in the Pacific Ocean are stronger for stronger diffusivities at the corresponding basins and depths, and the strongest when using the small-scale roughness map. Third, a comparison of the density fields of the three simulations with the density field of World Ocean Atlas dataset, from which the models are initialized, shows that among the simulations with three different roughness maps, the one using small-scale bottom roughness map has the smallest density bias.     

Impact of tidal mixing with different scales of bottom roughness on the general circulation

The current study deals with a parameterization of diapycnal diffusivity in an ocean model. The parameterization estimates the diapycnal diffusivity depending on the location of tidal-related energy dissipation over rough topography. The scheme requires a bottom roughness map that can be chosen depending on the scales of topographic features. Here, we implement the parameterization on an ocean general circulation model, and we examine the sensitivity of the modeled circulations to different spatial scales of the modeled bottom roughness. We compare three simulations that include the tidal mixing scheme using bottom roughness calculated at three different ranges of spatial scales, with the largest scale varying up to 200 km. Three main results are discussed. First, the dependence of the topographic spectra with depth, characterized by an increase in spectral energy over short length scales in the deep ocean, influences the vertical profile of the diffusivity. Second, the changes in diffusivities lead to different equilibrium solutions in the Atlantic meridional overturning circulation and bottom circulation. In particular, the lower cell of the Atlantic overturning and the bottom water transport in the Pacific Ocean are stronger for stronger diffusivities at the corresponding basins and depths, and the strongest when using the small-scale roughness map. Third, a comparison of the density fields of the three simulations with the density field of World Ocean Atlas dataset, from which the models are initialized, shows that among the simulations with three different roughness maps, the one using small-scale bottom roughness map has the smallest density bias.

     

Optimal boundary conditions for ORCA-2 model

A 4D-Var data assimilation technique is applied to ORCA-2 configuration of the NEMO in order to identify the optimal parametrization of boundary conditions on the lateral boundaries as well as on the bottom and on the surface of the ocean. The influence of boundary conditions on the solution is analyzed both within and beyond the assimilation window. It is shown that the optimal bottom and surface boundary conditions allow us to better represent the jet streams, such as Gulf Stream and Kuroshio. Analyzing the reasons of the jets reinforcement, we notice that data assimilation has a major impact on parametrization of the bottom boundary conditions for u and v. Automatic generation of the tangent and adjoint codes is also discussed. Tapenade software is shown to be able to produce the adjoint code that can be used after a memory usage optimization.     

Synthetic tidal parameters for gravity over China and its neighbor area

The synthetic tidal parameters with high spatial resolution for gravity over China and its neighbor area are con- structed with Earth’s tidal model and ocean tide loading calculated using TPXO7 global ocean tide model as well as tidal data over China seas. The comparison between synthetic parameters and ones observed by spring gravime- ters at some seismic network stations and Hong Kong station and one observed by super-conducting gravimeter at Wuhan station shows that the average differences in amplitude factors and phases are smaller than 0.005 and 0.5° respectively; and that the discrepancies between observational and synthetic parameters are dependent on gravim- etric technique in that the synthetic parameters are in well agreement with the superconducting gravimetric obser- vations. This also indicates that the synthetic result is a good estimation for tidal gravity, and the numerical results in the present paper not only can provide ground and space gravimetry such as absolute gravimetry with correction model of tidal gravity, but also provide effective tidal parameters over areas where no observation is carried out.     

Analysis on observing optimization for the wind-driven circulation by an adjoint approach

The adjoint approach is a variational method which is often applied to data assimilation widely in meteorology and oceanography. It is used for analyses on observing optimization for the wind-driven Sverdrup circulation. The adjoint system developed by Thacker and Long (1992), which is based on the GFDL Byran-Cox model, includes three components, i. e. the forward model, the adjoint model and the optimal algorithm. The GFDL Byran-Cox model was integrated for a long time driven by a batch of ideal wind stresses whose meridional component is set to null and zonal component is a sine function of latitudes in a rectangle box with six vertical levels and 2 by 2 degree horizontal resolution. The results are regarded as a "real" representative of the wind-driven Sverdrup circulation, from which the four dimensional fields are allowed to be sampled in several ways, such as sampling at the different levels or along the different vertical sections. To set the different samples, the fields of temperature, salinity and velocities function as the observational limit in the adjoint system respectively where the same initial condition is chosen for 4D VAR data assimilation. By examining the distance functions which measure the misfit between the circulation field from the control experiment of the adjoint system with a complete observation and those from data assimilation of adjoint approach in these sensitivity experiments respectively, observing optimizations for the wind-driven Sverdrup circulation will be suggested under a fixed observational cost.     

Tide model comparison over the Southwestern Atlantic Shelf

Sea surface height (SSH) as measured by satellites has become a powerful tool for oceanographic and climate related studies. Whereas in the open ocean good accuracy has been achieved, more energetic dynamics and a number of calibration problems have limited applications over continental shelves and near the coast. Tidal ranges in the Southwestern Atlantic (SWA) continental shelf are among the highest in the world ocean, reaching up to 12 m at specific locations. This fact highlights the relevance of the accuracy of the tidal correction that must be applied to the satellite data to be useful in the region. In this work, amplitudes and phases of tidal constituents are extracted from five global tide models and three regional models and compared to the corresponding harmonics estimated from coastal tide gauges (TGs) and satellite altimetry data. The Root Sum Square (RSS) of the misfit of the common set of the five tidal constituents solved by the models (M₂, N₂, S₂, K₁ and O₁) is higher than 18 cm close to the coast for two of the regional models and higher than 24.5 cm for the rest of the models considered. Both values are too high to provide an accurate estimation of geostrophic non-tidal currents from satellite altimetry in the coastal region. On the other hand, the global model with the highest spatial resolution has a RSS lower than 4.5 cm over the continental shelf even when the non-linear M₄ overtide is considered. Comparison with in-situ current measurements suggests that this model can be used to de-tide altimetry data to compute large-scale patterns of SSH and associated geostrophic velocities. It is suggested that a local tide model with very high resolution that assimilates in-situ and satellite data should meet the precision needed to estimate geostrophic velocities at a higher resolution both close to the coast and over the Patagonian shelf.     

Time-domain modeling of global ocean tides generated by the full lunisolar potential

Traditionally, ocean tides have been modeled in frequency domain with a forcing from selected tidal constituents. It is a natural approach; however, it implicitly neglects non-linearities of ocean dynamics. An alternative approach is time-domain modeling with a forcing given by the full lunisolar potential, i.e., all tidal waves are a priori included. This approach has been applied in several ocean tide models; however, some challenging tasks still remain, for example, assimilation of satellite altimetry data. In this paper, we introduce the assimilative scheme applicable in a time-domain model, which is an alternative to existing techniques used in assimilative ocean tide models. We present results from DEBOT, a global barotropic ocean tide model, which has two modes: DEBOT-h, a purely hydrodynamical mode, and DEBOT-a, an assimilative mode. The accuracy of DEBOT in both modes is assessed through a series of tests against tide gauge data which demonstrate that DEBOT is comparable to state-of-the-art global ocean tide models for major tidal constituents. Furthermore, as signals of all tidal frequencies are included in DEBOT, we also discuss modeling of minor tidal constituents and non-linear compound tides. Our modeling approach can be useful for those applications where the frequency domain approach is not suitable.     

Variational Data Assimilation Technique in Mathematical Modeling of Ocean Dynamics

Problems of the variational data assimilation for the primitive equation ocean model constructed at the Institute of Numerical Mathematics, Russian Academy of Sciences are considered. The model has a flexible computational structure and consists of two parts: a forward prognostic model, and its adjoint analog. The numerical algorithm for the forward and adjoint models is constructed based on the method of multicomponent splitting. The method includes splitting with respect to physical processes and space coordinates. Numerical experiments are performed with the use of the Indian Ocean and the World Ocean as examples. These numerical examples support the theoretical conclusions and demonstrate the rationality of the approach using an ocean dynamics model with an observed data assimilation procedure.     

反射地震数据的逐层波形反演

本文针对层状介质并结合梯度法波形反演,提出逐层波形反演的方法. 首先给出介质扰动响应的概念,并在此基础上分析了梯度法波形反演方法. 波形反演实质上是将实测地震记录和预测地震记录的波形残差信息转化为实际地质模型和预测地质模型的模型残差信息. 波形反演的优点是利用大量振幅相位信息得到高分辨率的反演结果, 其缺点是运行耗时大;当初始模型和实际模型相差较大时,迭代算法容易陷入局部极小点,这是因为目标函数和初始模型同实际模型间的差异是非线性的关系. 逐层波形反演方法是使自上而下每一层的目标函数最小,这样总的目标函数也是最小的. 利用二分法速度扫描确定每一层速度不仅提高了运算速度也避免了迭代算法陷入局部极小点的问题. 结合介质扰动响应和目标函数值变化可以更为准确迅速地确定每一层速度和该层界面位置.     

On the sensitivity of tidal residuals off the west coast of Britain to mesh resolution

An irregular mesh model of the west coast of Britain is used to examine the sensitivity of tidal residuals to mesh resolution in the region. Computed residuals are compared with earlier published results determined with a high resolution (1 km grid) finite difference model of the eastern Irish Sea. Initial calculations show that tidal residuals are largest in nearshore regions particularly in the vicinity of headlands. Local refinement of the mesh in these regions leads to a more detailed picture of the flow field, particularly adjacent to the coast. Although large scale offshore features of the flow can be resolved using the high resolution finite difference model, such an approach leads to a “stair case” representation of the coastal boundary with an adjacent near coastal region of spurious tidal residuals. By using an irregular mesh that follows the coast, this effect is removed. In the Mersey river region the tidal residual is resolved with a mesh resolution of 120 m, although calculations show that its distribution is particularly sensitive to small scale features of the topography. A variable mesh that can accurately represent the lateral variations in river width and details of topography in both the nearshore and estuarine environment appears essential in modelling the coastal spread of freshwater plumes from rivers and pollutants discharged into the near coastal environment.     

Tidal boundary conditions in SEAWAT

SEAWAT, a U.S. Geological Survey groundwater flow and transport code, is increasingly used to model the effects of tidal motion on coastal aquifers. Different options are available to simulate tidal boundaries but no guidelines exist nor have comparisons been made to identify the most effective approach. We test seven methods to simulate a sloping beach and a tidal flat. The ocean is represented in one of the three ways: directly using a high hydraulic conductivity (high-K) zone and indirect simulation via specified head boundaries using either the General Head Boundary (GHB) or the new Periodic Boundary Condition (PBC) package. All beach models simulate similar water fluxes across the upland boundary and across the sediment-water interface although the ratio of intertidal to subtidal flow is different at low tide. Simulating a seepage face results in larger intertidal fluxes and influences near-shore heads and salinity. Major differences in flow occur in the tidal flat simulations. Because SEAWAT does not simulate unsaturated flow the water table only rises via flow through the saturated zone. This results in delayed propagation of the rising tidal signal inland. Inundation of the tidal flat is delayed as is flow into the aquifer across the flat. This is severe in the high-K and PBC models but mild in the GHB models. Results indicate that any of the tidal boundary options are fine if the ocean-aquifer interface is steep. However, as the slope of that interface decreases, the high-K and PBC approaches perform poorly and the GHB boundary is preferable.     

A multiscale adjoint method to compute sensitivity coefficients for flow in heterogeneous porous media

A multiscale adjoint (MSADJ) method is developed to compute high-resolution sensitivity coefficients for subsurface flow in large-scale heterogeneous geologic formations. In this method, the original fine-scale problem is partitioned into a set of coupled subgrid problems, such that the global adjoint problem can be efficiently solved on a coarse grid. Then, the coarse-scale sensitivities are interpolated to the local fine grid by reconstructing the local variability of the model parameters with the aid of solving embedded adjoint subproblems. The approach employs the multiscale finite-volume (MSFV) formulation to accurately and efficiently solve the highly detailed flow problem. The MSFV method couples a global coarse-scale solution with local fine-scale reconstruction operators, hence yielding model responses that are quite accurate at both scales. The MSADJ method is equally efficient in computing the gradient of the objective function with respect to model parameters. Several examples demonstrate that the approach is accurate and computationally efficient. The accuracy of our multiscale method for inverse problems is twofold: the sensitivity coefficients computed by this approach are more accurate than the traditional finite-difference-based numerical method for computing derivatives, and the calibrated models after history matching honor the available dynamic data on the fine scale. In other words, the multiscale based adjoint scheme can be used to history match fine-scale models quite effectively.     

武汉超导重力仪观测最新结果和海潮模型研究

利用武汉台站GWR_C032超导重力仪观测资料,在对原始数据进行有效预处理的基础上作调和分析,获得反映地球内部介质特征的重力潮汐参数.基于卫星测高技术和有限元方法同时考虑验潮站数据作约束条件获得的多个全球海潮模型,利用负荷理论和数值褶积积分技术计算了重力负荷,对周日和半日频段内的重力潮汐参数实施负荷改正,提出了“负荷改正有效性”概念,研究了全球海潮模型适应性.数值结果说明,海潮改正的有效性高达91%（O1,NAO99）和92%（M2,ORI96）.基于11个海潮模型对主波（O1,K1,M2和S2）的负荷改正说明平均有效性为（86%,70%,73%和84%）,振幅因子与理论模型间的差异分别从（212%,155%,116%和080%）降到（031%,039%,034%和008%）,同时还说明利用NAO99和ORI96全球海潮模型能获得比其他模型更佳的负荷改正效果.文章还利用国际地球动力学计划网络其他7个台站的超导重力仪观测研究了全球海潮模型的适定性问题,结果说明不同模型中不同潮波具有明显的区域特点,早期构制的SCW80全球海潮模型仍可作为大地测量研究中的重要参考模型.     

Equilibration mechanisms in an adjoint ocean general circulation model

We examine the equilibrated and time-evolving adjoint solutions of an ocean general circulation model. Adjoint models calculate the sensitivity of a diagnostic, (here, the strength of the meridional overturning) to all forcing fields in a single integration. The time evolution of the sensitivity patterns demonstrates the validity of the adjoint modeling approach over climatological time scales in coarse-resolution ocean models. Our objective is to identify the principle adjustment mechanisms through which the meridional overturning strength adapts to perturbations in wind and buoyancy forcing. The adjoint approach is shown to be a valuable alternative to traditional perturbation methods in highlighting the processes and time scales important to ocean and climate modeling.     

Offset‐variable density improves acoustic full‐waveform inversion: a shallow marine case study

We have previously applied three‐dimensional acoustic, anisotropic, full‐waveform inversion to a shallow‐water, wide‐angle, ocean‐bottom‐cable dataset to obtain a high‐resolution velocity model. This velocity model produced an improved match between synthetic and field data, better flattening of common‐image gathers, a closer fit to well logs, and an improvement in the pre‐stack depth‐migrated image. Nevertheless, close examination reveals that there is a systematic mismatch between the observed and predicted data from this full‐waveform inversion model, with the predicted data being consistently delayed in time. We demonstrate that this mismatch cannot be produced by systematic errors in the starting model, by errors in the assumed source wavelet, by incomplete convergence, or by the use of an insufficiently fine finite‐difference mesh. Throughout these tests, the mismatch is remarkably robust with the significant exception that we do not see an analogous mismatch when inverting synthetic acoustic data. We suspect therefore that the mismatch arises because of inadequacies in the physics that are used during inversion. For ocean‐bottom‐cable data in shallow water at low frequency, apparent observed arrival times, in wide‐angle turning‐ray data, result from the characteristics of the detailed interference pattern between primary refractions, surface ghosts, and a large suite of wide‐angle multiple reflected and/or multiple refracted arrivals. In these circumstances, the dynamics of individual arrivals can strongly influence the apparent arrival times of the resultant compound waveforms. In acoustic full‐waveform inversion, we do not normally know the density of the seabed, and we do not properly account for finite shear velocity, finite attenuation, and fine‐scale anisotropy variation, all of which can influence the relative amplitudes of different interfering arrivals, which in their turn influence the apparent kinematics. Here, we demonstrate that the introduction of a non‐physical offset‐variable water density during acoustic full‐waveform inversion of this ocean‐bottom‐cable field dataset can compensate efficiently and heuristically for these inaccuracies. This approach improves the travel‐time match and consequently increases both the accuracy and resolution of the final velocity model that is obtained using purely acoustic full‐waveform inversion at minimal additional cost.