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The analytical four-dimensional ensemble variational (A-4DEnVar) data assimilation scheme inherits the advantages of the conventional four-dimensional variational (4D-Var) data assimilation scheme and removes the adjoint model. However, compatible operational improvements such as the reduction of the computational costs and the localization method should be considered when it is used in realistic systems. In this paper, the computational complexity of calculating the inverse of background error covariance (the B matrix) is decreased by a precondition transform method, i.e., introducing a new state variable whose product with the B matrix is the original state variable to be optimized in the cost function. Furthermore, an independent point (IP) scheme is combined to construct an implicit localization method and further decreases the computational cost. Based on the Princeton Ocean Model with the generalized coordinate system (POMgcs), the operational improved A-4DEnVar is applied to optimize the spatially varying bottom friction coefficients (BFCs) of the M 2 constituent in the Bohai and Yellow seas. A twin experiment with idealized observations is designed to validate the effectiveness of the proposed method. In practical experiments, with no more than 10 IPs, the algorithm can assimilate observations from the National Astronomical Observatory (NAO) dataset and obtain a good simulation. The experimental performances increase with the increase of either the IPs or observations, which indicates the efficacy of the proposed algorithm.

Tides are of great importance for ocean mixing and nearshore ocean engineering. Bottom friction is a major factor in tidal dissipation and is usually parameterized by the bottom friction coefficient (BFC). BFC is a critical parameter in numerical tidal models and is known to vary with time and space, as calculated with measured data. However, it is difficult to accurately adjust the spatially-temporally varying BFC in numerical tidal models. Based on the relationship between the spatially-temporally varying BFC estimated by adjoint data assimilation and the simultaneously simulated current speed, an empirical formula of BFC with a dependence on the current speed is proposed. This new empirical formula of BFC is compared with several traditional empirical formulas, including the constant BFC, the Chezy-Manning BFC, and two depth-dependent BFCs. When the four principal tidal constituents ( M 2 , S 2 , K 1 , and O 1 ) in the Bohai, Yellow and East China Seas (BYECS) are simulated, the mean vector error between the simulated results obtained using the current speed-dependent BFC and the TOPEX/Poseidon satellite altimetry data (the tidal gauge data) is 8.81 cm (10.62 cm), which is decreased by up to 8.1% (18.2%) compared with those using the several commonly used empirical formulas of BFC. Furthermore, in the sensitivity experiments where only the M 2 tide in the BYECS, the M 2 , S 2 , K 1 , and O 1 tides in the Bohai and Yellow Sea (BYS), and the M 2 , S 2 , K 1 , and O 1 tides in the South China Sea (SCS) are simulated, the errors between the simulated results obtained by using current speed-dependent BFC and the tidal gauge data are less than those using the other empirical formulas of BFC, further demonstrating the superiority of the current speed-dependent BFC proposed in this study. From numerical model experiments, the current speed-dependent BFC can adequately reflect the spatial and temporal variations of BFC and improve the simulation accuracy of tides, thus having a broad application scope.

A numerical model that solves 3D first-order Lagrangian residual velocity (uL) equations is established by modifying the HAMSOM model. With this model, uL is studied in a wide, idealized bay. The results show that the vertical eddy viscosity term of Stokes’ drift (π1) in the tidal body force determines the overall flow state of uL, and the contribution of the advection term (π2) is responsible for the small correction. In addition, two types of Coriolis effects introduced into the residual current system not only enhance the lateral flow and break the symmetry of the flow regime in the bay but also slightly correct the flow state driven by the entire tidal body force. It is also found by numerical sensitivity experiments that the increase in the aspect ratio δ, implying a decrease in the topographic gradient, can simplify the residual flow state. The increase in tidal amplitude at the open boundary significantly enhances the intensity of uL and causes the residual flow regime to be more complicated in the bay. This can be ascribed to the disproportionate increase in the tidal body force. The proportion of the vertical eddy viscosity term of Stokes’ drift in the tidal body force also varies with the vertical eddy viscosity coefficient, which leads to different residual current states. Compared with the influence of incoming tidal strength on the residual current, the effect of the bottom friction coefficient on the residual current is relatively mild. An increase in the quadratic bottom friction coefficient induces an unbalanced decrease in the tidal body force. Therefore, uL decreases, but the flow regime is more complex. The influence of the nonlinear effect of the bottom friction decreases from the bay head towards the bay mouth. The residual current only changes in magnitude near the bay mouth but changes in pattern near the bay head for different bottom friction coefficients. By keeping the bottom friction coefficient in the zeroth-order tidal equations constant, the sensitivity experiment shows that uL is insensitive to the change in bottom friction coefficient in the governing equations of uL.

The bottom friction is critical for the dissipation of the global tidal energy. The bottom friction coefficient is traditionally determined using the Manning’s n formulation in tidal models. The Manning’s n coefficient in the Manning’s n formulation is vital for the accurate simulation and prediction of the tide in coastal shallow waters, but it cannot be directly measured and contains large amounts of uncertainties. Based on a two-dimensional multi-constituent tidal model with the adjoint data assimilation, the estimation of the Manning’s n coefficient is investigated by assimilating satellite observations in the Bohai, Yellow and East China Seas with the simulation of four principal tidal constituents M 2 , S 2 , K 1 and O 1 . In the twin experiments, the Manning’s n coefficient is assumed to be constant, and it is estimated by assimilating the synthetic observations at the spatial locations of the satellite tracks. Regardless the inclusion of artificial random observational errors associated with synthetic observations, the model performance is improved as evaluated by the independent synthetic observations. The prescribed ‘real’ Manning’s n coefficient is reasonably estimated, indicating that the adjoint data assimilation is an effective method to estimate the Manning’s n coefficient in multi-constituent tidal models. In the practical experiments, the errors between the independent observations at the tidal gauge stations and the corresponding simulated results of the four principal tidal constituents are substantially decreased under both scenarios of the constant and spatially-temporally varying Manning’s n coefficient estimated by assimilating the satellite observations with the adjoint data assimilation. In addition, the estimated spatial and temporal variation trend is robust and not affected by the model settings. The spatially-temporally varying Manning’s n coefficient is negatively correlated with the current speed and shows significant spatial variation in the shallow water areas. This study demonstrates that the Manning’s n coefficient can be reasonably estimated by the adjoint data assimilation, which allows significant improvement in accurate simulation of the ocean tide.

In this study, the tsunami-induced bottom boundary layer was investigated based on actual waveforms obtained by the GPS buoys along the coast of the Tohoku region during the 2011 Great East Japan Earthquake tsunami. The k-ω model was utilized for the numerical analysis in this study. As a result, the tsunami boundary layer thickness was found to be extremely thin compared to the water depth. The velocity distribution was similar to that of the bottom boundary layer under wind-generated waves. The flow regime is located in the transition from smooth turbulence to rough turbulence. Because of this, the gradient of the flow across the layer is much greater than the gradients in the steady flow direction. Therefore, the bottom friction is underestimated if the steady friction factor, such as in the Manning formula, is used. This study proposes a new simple method for calculating the bottom shear stress due to an irregular tsunami based on the wave friction law, and the k-ω model results are used to validate the proposed methods.

The quasi-geostrophic two-layer model is a widely used tool to study baroclinic instability in the ocean. One instability criterion for the inviscid two-layer model is that the potential vorticity (PV) gradient must change sign between the layers. This has a well-known implication if the model includes a linear bottom slope: for sufficiently steep retrograde slopes, instability is suppressed for a flow parallel to the isobaths. This changes in the presence of bottom friction as well as when the PV gradients in the layers are not aligned. We derive the generalised instability condition for the two-layer model with non-zero friction and arbitrary mean flow orientation. This condition involves neither the friction coefficient nor the bottom slope; even infinitesimally weak bottom friction destabilises the system regardless of the bottom slope. We then examine the instability characteristics as a function of varying slope orientation and magnitude. The system is stable across all wavenumbers only if friction is absent and if the planetary, topographic and stretching PV gradients are aligned. Strong bottom friction decreases the growth rates but also alters the dependence on bottom slope. In conclusion, the often mentioned stabilisation by steep bottom slopes in the two-layer model holds only in very specific circumstances, thus probably plays only a limited role in the ocean.

In the coastal ocean, interactions of waves and currents with large roughness elements, similar in size to wave orbital excursions, generate drag and dissipate energy. These boundary layer dynamics differ significantly from well-studied small-scale roughness. To address this problem, we derived spatially and phase-averaged momentum equations for combined wave–current flows over rough bottoms, including the canopy layer containing obstacles. These equations were decomposed into steady and oscillatory parts to investigate the effects of waves on currents, and currents on waves. We applied this framework to analyse large-eddy simulations of combined oscillatory and steady flows over hemisphere arrays (diameter $D$), in which current ($U_c$), wave velocity ($U_w$) and period ($T$) were varied. In the steady momentum budget, waves increase drag on the current, and this is balanced by the total stress at the canopy top. Dispersive stresses from oscillatory flow around obstacles are increasingly important as $U_w/U_c$ increases. In the oscillatory momentum budget, acceleration in the canopy is balanced by pressure gradient, added-mass and form drag forces; stress gradients are small compared to other terms. Form drag is increasingly important as the Keulegan–Carpenter number $KC=U_wT/D$ and $U_c/U_w$ increase. Decomposing the drag term illustrates that a quadratic relationship predicts the observed dependences of steady and oscillatory drag on $U_c/U_w$ and $KC$. For large roughness elements, bottom friction is well represented by a friction factor ($f_w$) defined using combined wave and current velocities in the canopy layer, which is proportional to drag coefficient and frontal area per unit plan area, and increases with $KC$ and $U_c/U_w$.

Coastal trapped waves (CTWs) carry the ocean’s response to changes in forcing along boundaries and are important mechanisms in the context of coastal sea level and the meridional overturning circulation. Motivated by the western boundary response to high-latitude and open-ocean variability, we use a linear, barotropic model to investigate how the latitude dependence of the Coriolis parameter ( β effect), bottom topography, and bottom friction modify the evolution of western boundary CTWs and sea level. For annual and longer period waves, the boundary response is characterized by modified shelf waves and a new class of leaky slope waves that propagate alongshore, typically at an order slower than shelf waves, and radiate short Rossby waves into the interior. Energy is not only transmitted equatorward along the slope, but also eastward into the interior, leading to the dissipation of energy locally and offshore. The β effect and friction result in shelf and slope waves that decay alongshore in the direction of the equator, decreasing the extent to which high-latitude variability affects lower latitudes and increasing the penetration of open-ocean variability onto the shelf—narrower continental shelves and larger friction coefficients increase this penetration. The theory is compared with observations of sea level along the North American east coast and qualitatively reproduces the southward displacement and amplitude attenuation of coastal sea level relative to the open ocean. The implications are that the β effect, topography, and friction are important in determining where along the coast sea level variability hot spots occur.

In recent years, magnesium-based alloys and composites have great attention in automobile, structural, and biomedical industries due to their desirable characteristics such as lower density, low elastic modulus, high specific strength, better damping properties, and excellent castability. The pure magnesium was used as a matrix material and reinforced with fly ash fillers of different compositions with a weight percentage of 2.5%, 5%, and 7.5% to compare with the pure magnesium. The three different weights (wt%) of fly ash/Mg samples were prepared using the bottom pouring stir casting method. Fabricated samples sliding wear characteristics and mechanical behaviour (ASTM standard) were studied. Wear, tensile, and hardness results portray that 7.5 wt% fly ash composites possess better elongation and hardness and good wear resistance. The tensile strength values were improved by 42% in sample 4 compared with pure Mg. Hardness values were also improved by 21% in sample 4 compared with pure Mg. The wear rate and coefficient of friction are also reduced by the increased weight percentage of fly ash reinforcement. SEM images display casted pure magnesium’s morphology and wear-tested samples’ worn surface characteristics.

Measurement of bed shear stress is always a challenging task for engineers. In river engineering, bed shear is a fundamental variable and is important in estimating flow resistance and sediment transport. In this study, experiments are carried out in diverging compound channel with smooth bed (perspex sheet) and rough bed (Gravel) conditions to determine the effect of roughness. The shear velocity is estimated from universal logarithmic law. The effect of geometry and roughness on Von-Karman constant, eddy viscosity coefficient, friction factor is studied. The mass conservation and momentum conservation equations are used to derive apparent shear forces at interface of main channel and floodplain. A genetic algorithm model is developed to predict percentage of shear force (%Sfp) carried by sub-sections. To perform better with less and unseen data K-Fold cross-validation technique is used. The model is compared with available models in literature and it is observed that developed model gave better predictions with low MAPE.