Analytical solution to the 3D tide-induced Lagrangian residual current in a narrow bay with vertically varying eddy viscosity coefficient

Ocean Dynamics - The 3D Lagrangian residual velocity (LRV) is solved analytically in a narrow bay employing a vertically varying eddy viscosity coefficient. The nondimensional vertical profile of...     

Analytical solution for the tidally induced Lagrangian residual current in a narrow bay

In a weakly nonlinear tidal system, the depth-averaged equations for the first-order Lagrangian residual velocity (LRV) are deduced systematically. For the case of a narrow bay, the equations are solved analytically and the results for a specific bottom profile are discussed in detail. According to the pattern of the first-order LRV, the bay can be divided into three parts, namely an inner part, a transitional zone, and an outer part. For the given depth profile, the streamline of the first-order LRV for a shorter bay is a part of that for a longer bay. The first-order LRV depends on a nondimensional parameter that combines the influences of the bottom friction coefficient, the tidal period and the averaged water depth. The form of the bottom friction also has a significant influence on the first-order LRV. The second-order LRV, i.e., the Lagrangian drift, is analytically solved and shows dependence on the initial tidal phase. The LRV differs from the Eulerian residual transport velocity both quantitatively and qualitatively. It is demonstrated that the residual currents obtained according to other definitions may cause misunderstanding of the mass transport in water exchange applications.     

Initial time dependence of wind- and density-driven Lagrangian residual velocity in a tide-dominated bay

Ocean Dynamics - The nonlinear effect of the summer southeast wind and density on the 3D structures of the full Lagrangian residual velocity (LRV) was quantified for a generally nonlinear system,...     

3D analytical solution to the tidally induced Lagrangian residual current equations in a narrow bay

The 3D first-order Lagrangian residual velocity (LRV) equation is established, and its analytical solution is obtained in a narrow bay. The results show clearly the 3D structure of the first-order LRV. When the exponential bottom profile is assumed, the upper half layer of the water flows in through the deep channel from the open boundary directly to the head of the bay. Then the water will return to the area surrounding the lower half of the inflow area. The downwelling area is located mainly at the deep channel, while the upwelling area occupies both sides of the bay. The inter-tidal water transport, obtained by integrating the 3D first-order LRV through the water column, has a pattern similar to the previous study in which the 2D depth-averaged Lagrangian residual current equations were solved. The inter-tidal water transport is used to analyze the water exchange, and it is found that the water exchange at different cross sections increases smoothly with the distance between the cross sections and the head of the bay until about one wavelength. It is also found that the pattern of the breadth-averaged Lagrangian residual current varies with the length of the bay if a non-flat bottom profile is used. The depth-integrated LRV and the breadth-averaged LRV are mainly determined by the different terms of the tidal body force, with the former determined by the bottom friction related term and the latter by the eddy viscosity related term. When the bay is longer than one wavelength, different results in the outer bay can be observed.     

Numerical computation of the tidally induced Lagrangian residual current in a model bay

With a depth-averaged numerical model, the tidally induced Lagrangian residual current in a model bay was studied. To correctly reflect the long-term mass transport, it is appropriate to use the Lagrangian residual velocity (LRV) rather than the Eulerian residual velocity (ERV) or the Eulerian residual transport velocity (ETV) to describe the residual current. The parameter κ, which is defined as the ratio of the typical tidal amplitude at the open boundary to the mean water depth, is considered to be the indicator of the nonlinear effect in the system. It is found that the feasibility of making the mass transport velocity (MTV) approximate the LRV is strongly dependent on κ. The error between the MTV and the LRV tends to increase with a growing κ. An additional error will come from the various initial tidal phases due to the Lagrangian drift velocity (LDV) when κ is no longer small. According to the residual vorticity equation based on the MTV, the Coriolis effect is found to influence the residual vorticity mainly through the curl of the tidal stress. A significant difference in the flow pattern indicates that the LRV is sensitive to the bottom friction in different forms.     

Correction to: Initial time dependence of wind- and density-driven Lagrangian residual velocity in a tide-dominated bay

Ocean Dynamics - A Correction to this paper has been published: https://doi.org/10.1007/s10236-021-01459-8     

3D numerical computation of the tidally induced Lagrangian residual current in an idealized bay

Ocean Dynamics - 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,...     

Difference between the Lagrangian trajectories and Eulerian residual velocity fields in the southwestern Yellow Sea

The responses to tidal and/or wind forces of Lagrangian trajectories and Eulerian residual velocity in the southwestern Yellow Sea are investigated using a high-resolution circulation model. The simulated tidal harmonic constants agree well with observations and existing studies. The numerical experiment reproduces the long-range southeastward Eulerian residual current over the sloping bottom around the Yangtze Bank also shown in previous studies. However, the modeled drifters deployed at the northeastern flank of the Yangtze Bank in the simulation move northeastward, crossing over this strong southeastward Eulerian residual current rather than following it. Additional sensitivity experiments reveal that the influence of the Eulerian tidal residual currents on Lagrangian trajectories is relatively weaker than that of the wind driven currents. This result is consistent with the northeastward movement of ARGOS surface drifters actually released in the southwestern Yellow Sea. Further experiments suggest that the quadratic nature of the bottom friction is the crucial factor, in the southwestern Yellow Sea, for the weaker influence of the Eulerian tidal residual currents on the Lagrangian trajectories. This study demonstrates that the Lagrangian trajectories do not follow the Eulerian residual velocity fields in the shallow coastal regions of the southwestern Yellow Sea.     

Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay,China

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.

     

Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay, China

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.     

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 three-dimensional numerical sea model with eddy viscosity varying piecewise linearly in the vertical

The solution of the three-dimensional linear hydrodynamic equations which describe wind-driven flow in a homogeneous sea are solved using the eigenfunction method. The eddy viscosity is taken to vary piecewise linearly in the vertical over an arbitrary number of layers. Using this formulation the eigenfunctions are given in terms of Bessel functions. The coefficients of integration as well as the eigenvalues are determined accurately such that the boundary conditions are satisfied. Values of the eigenfunctions at any depth can then be determined very fast and to a high degree of accuracy.Current profiles at any position can hence be computed accurately. The expansion of the horizontal component of current converges very fast at all depths.     

An eddifying Stommel model: fast eddy effects in a two-box ocean

ABSTRACT

A system of stochastic differential equations is formulated describing the heat and salt content of a two-box ocean. Variability in the heat and salt content and in the thermohaline circulation between the boxes is driven by fast Gaussian atmospheric forcing and by ocean-intrinsic, eddy-driven variability. The eddy forcing of the slow dynamics takes the form of a colored, non-Gaussian noise. The qualitative effects of this non-Gaussianity are investigated by comparing to two approximate models: one that includes only the mean eddy effects (the “averaged model”), and one that includes an additional Gaussian white-noise approximation of the eddy effects (the “Gaussian model”). Both of these approximate models are derived using the methods of fast averaging and homogenisation. In the parameter regime where the dynamics has a single stable equilibrium the averaged model has too little variability. The Gaussian model has accurate second-order statistics, but incorrect skew and rare-event probabilities. In the parameter regime where the dynamics has two stable equilibria the eddy noise is much smaller than the atmospheric noise. The averaged, Gaussian, and non-Gaussian models all have similar stationary distributions, but the jump rates between equilibria are too small for the averaged and Gaussian models.     

The computation of the free barotropic oscillations of a global ocean model including friction and loading effects

A set of 61 normal modes with periods between 7.8 and 133.1 h has been calculated, using a 1° model of the global ocean, including the Arctic Ocean. The model explicitly considers frictional forces and ocean self-attraction and loading effects. The latter effects have generally been taken into account by parameterization, but for some modes the effects have also been considered fully. Due to friction, the computed eigenfrequencies are complex, exhibiting also the varying dissipative properties of the modes and their dependence on the distribution of potential and kinetic energies over the oceanic regions. In detail, gravity modes having periods less then 80 h and dominating the semi-diurnal and the diurnal tides, topographically controlled vorticity modes with periods longer than diurnal, and two planetary vorticity modes with periods of 96.8 and 119.4 h have been identified. These planetary vorticity modes have their energies distributed over Pacific, Atlantic, and Indian Oceans, while the other modes with periods longer than 80 h, as vorticity modes, have their energies concentrated on topographic structures of restricted extension. The modes are discussed with respect to their wave properties, e.g., concerning quasi-standing-wave resonances and to the appearance of Kelvin waves of different orders and trapped by different coastlines. In particular, the relevance of specific modes for the development of the fields of the most important semi-diurnal and diurnal tidal constituents is investigated.     

Comparing Eulerian and Lagrangian eddy census for a tide-less,semi-enclosed basin,the Baltic Sea

Ocean Dynamics - Eddies in the global and coastal ocean play a key role in mixing and transport processes. Here, we present an eddy census for the Baltic Sea covering the years 2008–2010. The...     

Impact of horizontal eddy diffusivity on Lagrangian statistics for coastal pollution from a major marine fairway

Lagrangian trajectory methods are often applied as deterministic transport models, where transport is due strictly to advection without taking into account stochastic elements of particle dispersion, which raises questions about validity of the model results. The present work investigates the impact of horizontal eddy diffusivity for a case study of coastal pollution in the Gulf of Finland, where the pollutants are assumed to originate from a major fairway and are transported to the coast by surface currents. Lagrangian trajectories are calculated using the TRACMASS model from velocity fields calculated by the Rossby Centre circulation model for 1982 to 2001. Three cases are investigated: (1) trajectory calculation without eddy diffusivity, (2) stochastic modelling of eddy diffusivity with a constant diffusion coefficient and (3) stochastic modelling of eddy diffusivity with a time- and space-variable diffusion coefficient. It is found that the eddy diffusivity effect increases the spreading rate of initially closely packed trajectories and the number of trajectories that eventually reach the coast. The pattern of most frequently hit coastal sections, the probability of hit to each such section and the time the pollution spends offshore are virtually invariant with respect to inclusion of eddy diffusivity.     

Importance of cross-channel bathymetry and eddy viscosity parameterisation in modelling estuarine flow

For a proper understanding of flow patterns in curved tidal channels, quantification of contributions from individual physical mechanisms is essential. We study quantitatively how such contributions are affected by cross-channel bathymetry and three alternative eddy viscosity parameterisations. Two models are presented for this purpose, both describing flow in curved but otherwise prismatic channels with an (almost) arbitrary transverse bathymetry. One is a numerical model based on the full three-dimensional shallow water equations. Special feature of this diagnostic model is that assumptions regarding the relative importance of particular physical mechanisms can be incorporated in the computations by switching corresponding terms in the model equations on or off. We also present an idealized model that provides semi-analytical approximate solutions of the shallow water equations for all three considered alternative eddy viscosity parameterisations. It forms an aid in explaining and theorising about results obtained with the numerical model. Observations regarding Chesapeake Bay serve as a reference case for the present study. We find that the relative importance of both along-channel advective forcing and transverse diffusive forcing depends on local characteristics of the cross-sectional bottom profile rather than global ones. In our reference case, tide-residual along-channel flow induced by these forcings is not small compared to the total tidal residual. Building on this observation, we present an indicative test to judge whether advective processes should be included in leading order in modelling tide-dominated estuarine flow. Furthermore, depending on the applied eddy viscosity parameterisation (uniformly or parabolically distributed over the vertical), we find qualitatively different spatial patterns for the along-channel advective forcing.     

角度域剩余深度对剩余速度的敏感性分析

本文针对地震勘探深度域偏移速度建模研究, 利用角度域共成像点道集(ADCIGS)建立了以剩余速度为自变量, 剩余深度为目标函数的关系式, 及目标函数的梯度公式.利用导出的两个公式分别对剩余深度与剩余速度的关系进行了定量分析.通过理论分析和模型试算证明初始速度模型的误差具有方向敏感性, 即正误差较负误差对速度建模迭代收敛更敏感.利用此结论进行深度域速度建模既可以提高计算效率也可以提高建模精度.     

Effect of bottom slope on the nonlinear triad interactions in shallow water

This paper aims at investigating the effect of bottom slope to the nonlinear triad interactions for irregular waves propagating in shallow water. The physical experiments are conducted in a wave flume with respect to the transformation of waves propagating on three bottom slopes (β?=?1/15, 1/30, and 1/45). Irregular waves with different type of breaking that are mechanically generated based on JONSWAP spectra are used for the test. The obviously different variations of spectra measured on each bottom reveal a crucial role of slope effect in the energy transfer between harmonics. The wavelet-based bispectrum were used to examine the bottom slope effect on the nonlinear triad interactions. Results show that the different bottom slopes which waves are propagated on will cause a significant discrepancy of triad interactions. Then, the discussions on the summed bicoherence which denote the distribution of phase coupling on each frequency further clarify the effect of bottom slope. Furthermore, the summed of the real and imaginary parts of bispectrum which could reflect the intensity of frequency components participating in the wave skewness and asymmetry were also investigated. Results indicate that the value of these parameters will increase as the bottom slope gets steeper.