Shallow Water System Research Articles

Low dispersion finite volume/element discretization of the enhanced Green–Naghdi equations for wave propagation, breaking and runup on unstructured meshes

We study a hybrid approach combining a finite volume (FV) and a finite element (FE) method to solve a fully-nonlinear and weakly-dispersive depth averaged wave propagation model. The FV method is used to solve the underlying hyperbolic shallow water system, while a standard P1 finite element method is used to solve the elliptic system associated to the dispersive correction. We study the impact of several numerical aspects: the impact of the reconstruction used in the hyperbolic phase; the representation of the FV data in the FE method used in the elliptic phase and their impact on the theoretical accuracy of the method; the well-posedness of the overall method. For the first element we proposed a systematic implementation of an iterative reconstruction providing on arbitrary meshes up to third order solutions, full second order first derivatives, as well as a consistent approximation of the second derivatives. These properties are exploited to improve the assembly of the elliptic solver, showing dramatic improvement of the finale accuracy, if the FV representation is correctly accounted for. Concerning the elliptic step, the original problem is usually better suited for an approximation in H(div) spaces. However, it has been shown that perturbed problems involving similar operators with a small Laplace perturbation are well behaved in H1. We show, based on both heuristic and strong numerical evidence, that numerical dissipation plays a major role in stabilizing the coupled method, and not only providing convergent results, but also providing the expected convergence rates. Finally, the full mode, coupling a wave breaking closure previously developed by the authors, is thoroughly tested on standard benchmarks using unstructured grids with sizes comparable or coarser than those usually proposed in literature.

Two-way wave–vortex interactions in a Lagrangian-mean shallow water model

We derive and investigate numerically a reduced model for wave–vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere–ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of ‘greedy’ waves, i.e. in the long run the wave field always extracts energy from the mean flow.

Simulation of high-frequency dissolved oxygen dynamics in a shallow estuary, the Corsica River, Chesapeake Bay

Understanding shallow water biogeochemical dynamics is a challenge in coastal regions, due to the presence of highly variable land-water interface fluxes, tight coupling with sediment processes, tidal dynamics, and diurnal variability in biogeochemical processes. While the deployment of continuous monitoring devices has improved our understanding of high-frequency (12 - 24 hours) variability and spatial heterogeneity in shallow regions, mechanistic modeling of these dynamics has lagged behind conceptual and empirical models. The inherent complexity of shallow water systems is represented in the Corsica River estuary, a small basin within the Chesapeake Bay ecosystem, where abundant monitoring data have been collected from long-term monitoring stations, continuous monitoring sensors, synoptic sensor surveys, and measurements of sediment-water fluxes. A state-of-the-art modeling system, the Semi-implicit Cross-scale Hydroscience Integrated System Model (SCHISM), was applied to the Corsica domain with a high-resolution grid and nutrient loads from the most recent version of the Chesapeake Bay watershed model. The Corsica SCHISM model reproduced observed high-frequency variability in dissolved oxygen, as well as seasonal variability in chlorophyll-a and sediment-water fluxes. Time-series signal analyses using Empirical Model Decomposition and spectral analysis revealed that the diurnal and M2 tide frequencies are the dominant high-frequency modes and physical transport contributes a larger share to dissolved oxygen budgets than biogeochemical processes on an hourly time scale. Heterogeneity and patchiness in dissolved oxygen resulting from phytoplankton distributions and geometry-driven eddies amplify the physical transport effect, and on longer time scales oxygen is controlled more by photosynthesis and respiration. Our simulation demonstrates that interactions among physical and biological dynamics generate complex high-frequency variability in water quality and non-linear reposes to nutrient loading and environmental forcing in shallow water systems.

A shallow-water dunefield in a microtidal, wind-dominated strait (Stintino, NW Sardinia, Italy)

Abstract This study focuses on the recent evolution of a shallow-water dunefield system off La Pelosa Beach, located in the wind-dominated, microtidal, asymmetrical Asinelli Strait (Stintino, NW Sardinia, Italy). The system comprises four zones defined by a gradual decrease in current strength from the strait centre to more distal parts that form different bedforms. The strait centre (Zone A) is a substrate-carved channel passing downcurrent to a vast sandy shoal characterized by a subaqueous dunefield (Zone B) that is surrounded by a sandwave–ripple carpet (Zone C). The strait margin (Zone D) is dominated by a sandy cusp, the so-called La Pelosa Beach system. The extensive seagrass meadow located on the SE side of the strait-end zone feeds the system with bioclastic-rich sands. This carbonate sedimentary load is ensured by natural seagrass meadow retreats that are possibly boosted by human impact. The sediment transport is modulated by the interaction between two opposite currents triggered by the east-incoming Greco/Levante and NW-incoming Mistral winds. These currents experience a great variability due to multi-annual fluctuations (4–6 years) of the prevailing wind activities. Such cyclicity is responsible for periodic dune migration and and transport from La Pelosa Beach out of the system below the active part of the submerged beach (depth of closure). Bedform formations (ripples, dunes and sandwaves) and migrations within the strait are the result of complex physical processes observed in modern environments but rarely described in rock formations. The presented conceptual model can be used to recognize sediment ‘spillover’ processes on other modern, microtidal, wind-dominated straits, as well as proposing the main criteria for recognizing this strait sedimentary succession in the geological record.

Formation of singularity for the classical solutions of the rotating shallow water system

In this paper, we investigate the formation of singularity for two dimensional classical solutions of rotating shallow water system from several aspects. First, the formation of singularity for two dimensional solutions is proved via the analysis on the governing equations for the associated moments. Second, the formation of singularity for the solutions with cylindrical symmetry is established by the weighted energy estimate for the moments when the initial data has compact support. Finally, if the cylindrical symmetric solutions are of the form with separated variables, the global existence or formation of singularity for the classical solutions of the rotating shallow water system is obtained with the aid of the analysis on the invariants of the solutions.

On well-balanced implicit-explicit Lagrange-projection schemes for two-layer shallow water equations

This works concerns the study of well-balanced Lagrange-projection schemes applied to the two-layer shallow water system. In particular, a formulation of the mathematical model in Lagrangian coordinates is proposed. The HLL method is then applied to a simplified version of the resulting Lagrangian system. Furthermore, based on the acoustic-transport splitting interpretation, another approximate Riemann solver for the acoustic-Lagrangian step is described. Both an explicit and an implicit-explicit method are proposed, where the latter can allow very fast simulations in sub-critical regimes. Finally, we show numerical simulations in which the outputs are compared with the IFCP method’s results.

Using Different Classic Turbulence Closure Models to Assess Salt and Temperature Modelling in a Lagunar System: A Sensitivity Study

Turbulence modelling is an important issue when dealing with hydrodynamic and transport models for better simulation of the transport of dissolved or suspended substances in a body-water. It controls processes involving physical balances (salt and water temperature) and, therefore, the ecosystem equilibrium. The study arises from the need to model the turbulence more efficiently when dealing with extreme situations on the Ria de Aveiro (Portugal), a coastal lagoon shallow water system dominated by tidal transport. Because the turbulence model is coupled to the hydrodynamic and transport models, a correct estimation of the eddy viscosity is important in simulating the salt and the heat transports. The aim is to assess the performance of four turbulence schemes/models (k, k-ε, Smagorinsky’s, and k-ε/Smagorinsky’s (k-ε/Sma), where k is turbulent kinetic energy and ε the dissipation rate of the turbulent kinetic energy) associated to a coupled hydrodynamic and transport models to simulate the eddy viscosity, the salinity, and the temperature. Overall, the results point out that among the different models/schemes used, the is the one which provides a more realistic value of the eddy viscosity within the range (1–6) m2 s−1, but most probably (1–3) m2 s−1. The application of the sensitivity analysis to some non-universal k-ε/Sma parameters evidenced significant sensitivity for the eddy viscosity and the salinity and moderate sensitivity for the water temperature. A 100% adjustment of the parameter values relative to the reference, translated into variations within the range of (1, 4) m2 s−1, (0, 13) PSU, and (1, 2.20) °C, for the eddy viscosity, salinity, and water temperature, respectively.

Model order reduction strategies for weakly dispersive waves

We focus on the numerical modeling of water waves by means of depth averaged models. We consider in particular PDE systems which consist in a nonlinear hyperbolic model plus a linear dispersive perturbation involving an elliptic operator. We propose two strategies to construct reduced order models for these problems, with the main focus being the control of the overhead related to the inversion of the elliptic operators, as well as the robustness with respect to variations of the flow parameters. In a first approach, only a linear reduction strategies is applied only to the elliptic component, while the computations of the nonlinear fluxes are still performed explicitly. This hybrid approach, referred to as pdROM, is compared to a hyper-reduction strategy based on the empirical interpolation method to reduce also the nonlinear fluxes. We evaluate the two approaches on a variety of benchmarks involving a generalized variant of the BBM–KdV model with a variable bottom, and a one-dimensional enhanced weakly dispersive shallow water system. The results show the potential of both approaches in terms of cost reduction, with a clear advantage for the pdROM in terms of robustness, and for the EIMROM in terms of cost reduction.

Inertio-gravity Poincaré waves and the quantum relativistic Klein–Gordon equation, near-inertial waves and the non-relativistic Schrödinger equation

Shallow water inertio-gravity Poincaré waves in a rotating frame satisfy the Klein–Gordon equation, originally derived for relativistic, spinless quantum particles. Here, we compare these two superficially unrelated phenomena, suggesting a reason for them sharing the same equation. We discuss their energy conservation laws and the equivalency between the non-relativistic limit of the Klein–Gordon equation, yielding the Schrödinger equation, and the near-inertial wave limit in the shallow water system.

Momentum and pseudomomentum in a shallow water equation

A basic shallow water system with variable topography is analyzed from the point of view of a Lagrangian derivation of momentum, energy, and pseudomomentum balances. A two-dimensional action and associated momentum equation are derived. The latter is further manipulated to derive additional equations for energy and pseudomomentum. This revealed structure emphasizes broken symmetries in space and a reference configuration and preserved symmetry in time.

Assessing Through-Water Structure-from-Motion Photogrammetry in Gravel-Bed Rivers under Controlled Conditions

Structure-from-Motion (SfM) photogrammetry has become a popular solution for three-dimensional topographic data collection in geosciences and can be used for measuring submerged bed surfaces in shallow and clear water systems. However, the performance of through-water SfM photogrammetry has not been fully evaluated for gravel-bed surfaces, which limits its application to the morphodynamics of gravel-bed rivers in both field investigations and flume experiments. In order to evaluate the influence of bed texture, flow rate, ground control point (GCP) layout, and refraction correction (RC) on the measurement quality of through-water SfM photogrammetry, we conducted a series of experiments in a 70 m-long and 7 m-wide flume with a straight artificial channel. Bed surfaces with strongly contrasting textures in two 4 m-long reaches were measured under five constant flow regimes with three GCP layouts, including both dry and underwater GCPs. All the submerged surface models with/without RC were compared with the corresponding dry bed surfaces to quantify their elevation errors. The results illustrated that the poorly sorted gravel-bed led to the better performance of through-water SfM photogrammetry than the bed covered by fine sand. Fine sediment transport caused significant elevation errors, while the static sand dunes and grain clusters did not lead to noticeable errors in the corrected models with dry GCPs. The elevation errors of the submerged models linearly increased with water depth for all the tested conditions of bed textures, GCP layouts, and discharges in the uncorrected models, but the slopes of the increasing relations varied with texture. The use of underwater GCPs made significant improvements to the performance of direct through-water SfM photogrammetry, but counteracted with RC. The corrected models with dry GCPs outperformed the uncorrected ones with underwater GCPs, which could still be used to correct the underestimation in surface elevation caused by RC. Based on the new findings, recommendations for through-water SfM photogrammetry in measuring submerged gravel-bed surfaces were provided.

Quantifying Spatial and Temporal Trends of Microplastic Pollution in Surface Water and in the Eastern Oyster Crassostrea virginica for a Dynamic Florida Estuary

Microplastics (MPs) are a ubiquitous pollutant, emphasizing the need to understand their abundance and the factors that influence these patterns around the globe. In a prior study, high numbers of MPs were found in surface waters and tissues of the oyster Crassostrea virginica collected from one location in the Indian River Lagoon (IRL, FL, USA). To better understand spatial and temporal variability of MPs throughout the IRL, for one year, monthly surface water samples were collected from 35 sites, while oysters were collected quarterly from 12 sites. Microscopy and ATR-FTIR were used to quantify MP. In total, 3755 MPs were found in 44% of water samples (mean density ± CI: 1.47 ± 0.09 MP/L). South IRL water had the most MPs, likely associated with proximity to urbanization, inlets (MP sinks) and tributaries (MP sources). MP (n = 3181) were found in 70% of examined C. virginica (n = 1402). Abundances of MP in oysters were lower in the spring and in north IRL. The overall mean abundance was 2.26 ± 0.16 MP/oyster, and the density was 2.43 ± 0.52 MP/g wet tissue weight. Our results provide a more complete picture of MPs in the IRL, a subtropical, shallow-water estuarine system.

Shallow-water hydrothermal vent system as an extreme proxy for discovery of microbiome significance in a crustacean holobiont

The shallow-water hydrothermal vent (HV) system off Kueishan Island lies at the end of the Okinawa Trough to the northeast of Taiwan. Near its submarine vent openings, aperiodic vent discharges generate a dynamic acidic (pH 5.5-8.1) and sulfidic (9-3000 μM) ecosystem. The dominant metazoan in this unique environment is the brachyuran vent crab, Xenograpsus testudinatus, which has developed robust metabolic strategies and highly adaptive acid-base regulatory mechanisms to maintain its physiological homeostasis. X. testudinatus is considered a holobiont, but the symbiotic mechanisms underlying acid and sulfur tolerance in the host-microbe system remain largely unclear. In this study, we used LoopSeq long-read sequencing of the full-length 16S rRNA gene to identify the bacterial communities present in the gills and carapace surface of X. testudinatus. The alpha diversity analysis, Venn diagram, and principal coordinate analysis (PCoA) indicated that the gills and carapace surface exhibit different bacterial constituents. Further measurements of relative abundance, coupled with functional predictions and fluorescence in situ hybridization (FISH), revealed a predominance of Sulfurovum sp. NBC37-1, a key bacterium that can perform sulfur and hydrogen oxidation to support denitrification processes. Consequently, our findings suggest that the symbiotic bacteria may play a critical role in conferring the extraordinary acid and sulfur tolerances of X. testudinatus, allowing the crustacean holobiont to thrive in its ecological niche within one of the most extreme marine habitats on Earth.

A bedload transport in shallow water model applied for sediment transport in open channel flows

This paper presents the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modelled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a non-conservative hyperbolic system. To discretize it, first we consider a Non-Homogeneous Riemann Solver scheme as well as a variant based on the use of flux limiters. In order to develop second-order scheme, we use a MUSCL method incorporating slope limiters in the spatial approximation and a two-step Runge-Kutta method for time integration. The comparison between results based on the proposed scheme and analytical results shows good agreement..

Carbonate contourite drifts in the southwest South China Sea: Sedimentary, paleoceanographic and economic implications

Contourite drifts are significant sedimentary features and provide clues for the reconstruction of paleoceanography and paleoenvironment. Although they have been increasingly identified in the world’s ocean, shallow-water contourite drifts (< 300 m depth) remain poorly understood and the examples are rare. This study documents a Middle Miocene shallow-water contourite depositional system in the southwest South China Sea by interpreting seismic reflection data and calibrating results with the previous chronological framework. The depositional system consisted of six mounded drifts and six moats. The contourite features were generated in seismic unit III (16-10.5 Ma) and distributed adjacent to carbonate reefs. They were formed on the proto-continental shelf (50-200 m depth) and shaped by the wind-driven currents. Changes in the sedimentary stacking patterns suggest three evolutionary stages of the contourite features. Stage I represents the growth of the Middle Miocene contourite depositional system between 16 and 10.5 Ma. Stage II marks the termination of carbonate drifts and the burial of the Late Miocene sedimentation during 10.5-5.3 Ma. Stage III started with the development of modern deep-water sedimentary systems since 5.3 Ma. The contourite features are compared with the examples on other South China Sea margins. Significant changes in the paleoceanography occurred at 10.5 Ma and 6.5-5.3 Ma when the dominated bottom currents shifted from the monsoonal wind-driven currents to the North Pacific waters, and then the modern circulation system. The Middle Miocene mounded drifts were likely sourced by the coarse-grained carbonate sands. Fluid flow escaped from the coarse-grained contourite layers and natural gas leakage occurs on the seafloor. Shallow-water carbonate contourite drifts can be served as a good gas reservoir and have great economic potential.

The effects of acidification on arsenic concentration and speciation in offshore shallow water system

The effects of acidification on speciation and transportation of arsenic in shallow seawater system were investigated based on data from acidification simulation experiments in lab scale tanks, in which enhanced levels of pCO2 corresponding to pHT were processed. The results showed that: (1) the concentration of DIAs (Dissolved inorganic arsenic), As5+ and As3+ in the overlying water increased with experimental CO2 enrichment; (2) while the ratio of As5+/As3+ decreased; (3) acidification could cause more DIAs transport into the overlying water from sediments or suspended particulate matters, and would be favorable to the existence of As3+. Thus, DIAs is available to microorganisms and can be taken in effectively by microorganisms in the shallow water system, resulting in toxic effects of As on microorganisms and thus the inhibition of the growth of microorganisms.

Balanced Meshless Method for Numerical Simulation of Pollutant Transport by Shallow Water Flow over Irregular Bed: Application in the Strait of Gibraltar

This paper focuses on the implementation of an accurate meshless scheme for the simulation of the advection–diffusion of non-active pollutant in a two-dimensional depth-averaged flow. The depth-averaged flow model includes the shallow water system and the pollutant propagation described by the advection–diffusion equation with diffusion tensor. The mathematical model was implemented by using a meshless method based on local radial basis functions. This method was used to numerically evaluate the spatial derivatives and complemented with the second-order Runge–Kutta method for the time evolution. To remove the non-physical oscillations, which appear at the discontinuity, a filter based on a hyperviscosity operator was applied. To investigate the effectiveness and accuracy of the proposed scheme, a number of tests are presented, including the dam break problem, and the pure transport of a pollutant in a long channel. Finally, a hypothetical example of a pollutant transport in the Strait of Gibraltar is modeled. The results obtained are compared both with analytical solutions and with simulation results obtained by a finite volume method based on the Roe-MUSCL scheme. The main advantages of the proposed method are: (i) the simplicity of implementation, (ii) the ability to handle calculations of slowly varying flows or concentrations, as well as rapidly varying flows containing shocks or discontinuities, and (iii) the ability to satisfy the C-property and guarantee positive values of both water level and pollutant concentration.

Hg in the hydrothermal fluids and gases in Baia di Levante, Vulcano, Italy

The importance of fluid and gaseous mercury (Hg) emissions from hydrothermal systems in the shallow, coastal ocean is poorly constrained. However, there are indicators that they could be a significant natural Hg source. We evaluated the hydrothermal Hg emissions around Vulcano Island, Aeolian Arc, Italy, which is host to a marine shallow-water hydrothermal system (MSWHS) in Baia di Levante. Fluids were collected with porewater probes, and gases were collected into Tedlar© bags. Total Hg (THg) concentrations in the hydrothermal fluids ranged from 2.9 to 2888 pM. The concentrations of volatile Hg were below 8 pM and trended positively with increasing temperature. Monomethyl Hg (MMHg) was not detected. Total Hg in the gases ranged from 0.03 to 1.82 μmol/m3.High concentrations of THg were associated with low Cl-concentrations, low pH-values, and high K/Cl and Mg/Cl ratios. Concentrations of THg in the hydrothermal fluids resulted from mixing between meteoric water, seawater, condensed fumarolic vapor, and a deep hydrothermal fluid. However, not all low-Cl fluids were high THg samples. Samples taken along the coast of the La Fossa crater contained substantially less THg (2.9 to 29.4 pM), despite similar hydrothermal indicators as those in the Baia di Levante samples. These data support subsurface circulation models, which discuss the downslope flow of condensed La Fossa crater gases to the coast. In the hydrothermally active area of Baia di Levante, concentrations of THg were elevated relative to background surface seawater, ranging from 40 to 5110 pM. In comparison, the remainder of the bay ranged from 0.8 to 2.4 pM.The flux of Hg to the atmosphere from surface waters was calculated through dissolved Hg concentrations. The flux at each sampled site ranged from 0 to 19.6 pmol/m2/hr. The highest flux rates were determined for those areas with visible hydrothermal activity, particularly those with high gaseous emission rates. Surface water concentrations declined rapidly away from point sources, indicating atmospheric emission, dilution through mixing, or scavenging and sedimentation. In hydrothermally active areas, Baia di Levante sediments contained THg concentrations of 2.42 nmol/g and 49.52 nmol/g, which were significantly above background (0.03 nmol/g).The largest gaseous point source, Bambino, released >2 L of gas per second. Although, Hg concentrations in the gas were low (113 to 122 nmol/m3) relative to other measured point sources near the beach (1768 to 1817 nmol/m3), due to the volume of discharge, surface water concentrations were elevated (131 pM).The Hg present in the hydrothermal system of Vulcano contributes Hg to the atmosphere and local seawater as a natural source.

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any ntimes n system of conservation laws has a solution. The solution concept is an extension of the notion of singular delta -shocks which have been used to provide solutions for Riemann problems in various systems, for example in cases where strict hyperbolicity or the genuine-nonlinearity condition are not satisfied, or in cases where initial conditions have large variation. We also introduce admissibility conditions which eliminate a wide range of unreasonable solutions. Finally, we provide an example from the shallow water system which justifies introduction of delta -distributions as a part of solutions to systems of conservation laws.

Efficient GPU Implementation of Multidimensional Incomplete Riemann Solvers for Hyperbolic Nonconservative Systems: Applications to Shallow Water Systems with Topography and Dry Areas

Efficient GPU Implementation of Multidimensional Incomplete Riemann Solvers for Hyperbolic Nonconservative Systems: Applications to Shallow Water Systems with Topography and Dry Areas