Shallow Water System Research Articles

Critical-layer instability of shallow-water magnetohydrodynamic shear flows

In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow,$c-U(y)$, matches the Alfvén wave velocities$\pm B(y)/\sqrt {\mu \rho }$, based on the local magnetic field$B(y)$, the magnetic permeability$\mu$, and the mass density of the fluid$\rho$. It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient$J' = - B''/\mu$in the critical layer, is found to play a complementary role in the instability.

Predicting waves in fluids with deep neural network

In this paper, we present a deep learning technique for data-driven predictions of wave propagation in a fluid medium. The technique relies on an attention-based convolutional recurrent autoencoder network (AB-CRAN). To construct a low-dimensional representation of wave propagation data, we employ a denoising-based convolutional autoencoder. The AB-CRAN architecture with attention-based long short-term memory cells forms our deep neural network model for the time marching of the low-dimensional features. We assess the proposed AB-CRAN framework against the standard recurrent neural network for the low-dimensional learning of wave propagation. To demonstrate the effectiveness of the AB-CRAN model, we consider three benchmark problems, namely, one-dimensional linear convection, the nonlinear viscous Burgers equation, and the two-dimensional Saint-Venant shallow water system. Using the spatial-temporal datasets from the benchmark problems, our novel AB-CRAN architecture accurately captures the wave amplitude and preserves the wave characteristics of the solution for long time horizons. The attention-based sequence-to-sequence network increases the time-horizon of prediction compared to the standard recurrent neural network with long short-term memory cells. The denoising autoencoder further reduces the mean squared error of prediction and improves the generalization capability in the parameter space.

Global Weak Solutions of a Hamiltonian Regularised Burgers Equation

A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter–Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an \(H^1\) energy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parameter \(\ell \) tends to 0 or \(\infty \), limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter–Saxton equation respectively, forced by an unknown remaining term.

Controls of sandstone architecture on hydrocarbon accumulation in a shallow-water delta from the Jurassic Shaximiao Formation of the western Sichuan Basin in China

The thin but widely distributed sandstones in the shallow-water deltas have the potential to hold large deposits of natural gas. The fluvial sand bodies formed from river channels with different sedimentary facies and sandstone architectures have remarkable heterogeneities. The results of this investigation suggest that the sandstone architectures of river channels have a significant impact on the hydrocarbon accumulation and production of sandstone reservoirs. Information regarding the sandstone origin and architecture of shallow-water deltas, especially of the fingered bar type, is still very limited. This investigation focused on the channel sandstones in the shallow-water delta of the Jurassic Shaximiao Formation on the eastern slope of the western Sichuan Basin in China. Based on data from cores and logs of sandstones, seven types of lithofacies and four sedimentary facies were identified. Sand-rich sedimentary facies, including distributary channels, mouth bars, and overbanks, have merged to form four types of compound sandbodies corresponding to the fourth-order architectural unit in the architectural classification. Four types of compound sand bodies were classified, including CS1 (variable distributary channel sand body), CS2 (diverting channel and residual overbank sand body), CS3 (diverting channel and residual mouth bar sand body), and CS4 (terminal channel and residual mouth bar sand body). Relatively good physical properties occur in the sandstone architectures of types CS3 and CS4, with an average porosity of 11.06% and an average permeability of 0.97 mD. Sandstone architectures from the Jurassic Shaximiao Formation of the study area were controlled by water depth and climate, tectonic subsidence, distance from provenance, erosion effect, and channel sinuosity and discharge. With strong lateral erosion, a shallow-water delta would have more sandstone architectures of types CS3 and CS4 in the high sinuosity of channels far from the provenance. Furthermore, sandstone architectures of types CS3 and CS4 are prone to form in connected channel sandbodies with relatively high contents of natural gas of approximately 4.48 × 104 m3/d. The results of this investigation are beneficial to improve the understanding of the characteristics of sandstone distributions in shallow-water deltas, especially in fingered bar types, and to guide oil and gas explorations in shallow-water delta systems.

Fluid flow stimulates chemoautotrophy in hydrothermally influenced coastal sediments

Hydrothermalism in coastal sediments strongly impacts biogeochemical processes and supports chemoautotrophy. Yet, the effect of fluid flow on microbial community composition and rates of chemoautotrophic production is unknown because rate measurements under natural conditions are difficult, impeding an assessment of the importance of these systems. Here, in situ incubations controlling fluid flow along a transect of three geochemically distinct locations at a shallow-water hydrothermal system off Milos (Greece) show that Campylobacteria dominated chemoautotrophy in the presence of fluid flow. Based on injected 13C-labelled dissolved inorganic carbon and its incorporation into fatty acids, we constrained carbon fixation to be as high as 12 µmol C cm−3 d−1, corresponding to areal rates up to 10-times higher than previously reported for coastal sediments, and showed the importance of fluid flow for supplying the necessary substrates to support chemoautotrophy. Without flow, rates were substantially lower and microbial community composition markedly shifted. Our results highlight the importance of fluid flow in shaping the composition and activity of microbial communities of shallow-water hydrothermal vents, identifying them as hotspots of microbial productivity.

Geometrical demonstration for persistence properties of a bi-Hamiltonian shallow water system

We present a geometrical demonstration for the persistence properties for a bi-Hamiltonian system modeling waves in a shallow water regime. Both periodic and non-periodic cases are considered, and a key ingredient in our approach is one of the Hamiltonians of the system.

Nitrogen Fixation Influenced by Phosphorus and Nitrogen Availability in the Benthic Bloom-forming Cyanobacterium Hydrocoleum sp. Identified in a Temperate Marine Lagoon.

The nitrogen-fixing, non-heterocystous cyanobacterium Hydrocoleum sp. (Oscillatoriales) is a common epiphytic and benthic bloom-former in tropical and subtropical shallow water systems but shares high phylogenetic similarity with the planktonic, globally important diazotroph Trichodesmium. Multiphasic observations in this study resulted in unexpected identification of Hydrocoleum sp. in mass accumulations in a coastal lagoon in the Western temperate North Atlantic Ocean. Hydrocoleum physiology was examined in situ through measurements of N2 and CO2 fixation rates and expression of genes involved with N2 fixation, CO2 fixation, and phosphorus (P) stress. Bulk N2 fixation rates and Hydrocoleum nifH expression peaked at night and were strongly suppressed by dissolved inorganic nitrogen (DIN). The expression of high affinity phosphate transporter (pstS) and alkaline phosphatase (phoA) genes of Hydrocoleum was elevated during the night and negatively responded to phosphate amendments, as evidence that these mechanisms contribute to P acquisition during diazotrophic growth of Hydrocoleum in situ. This discovery at the edge of the previously known Hydrocoleum habitat range in the warming oceans raises intriguing questions about diazotrophic cyanobacterial adaptations and transitions on the benthic-pelagic continuum.

In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. An extension to second-order of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic is presented. This extension is based on the combination of the first-order in-cell reconstruction with the standard MUSCL-Hancock reconstruction. The first-order strategy allowed in particular to capture exactly the isolated shocks and this new second-order extension keep this property. Moreover, the well-balanced property of the method is also studied. Several numerical tests are proposed to validate the methods for the Coupled-Burgers system, Gas dynamics equations in Lagrangian coordinates and the modified shallow water system.

An automatically well-balanced formulation of pressure forcing for discontinuous Galerkin methods for the shallow water equations

This paper begins a development of methods for addressing variable bottom topography in discontinuous Galerkin numerical methods for multi-layer, variable-density models of ocean circulation. For numerical models of ocean circulation, it is a widespread practice to split the fast (external) and slow (internal) dynamics into separate subsystems that are solved by different techniques. The fast dynamics are modeled by a vertically-integrated system that is very similar to the shallow water equations for a hydrostatic fluid of constant density. As a first step, the present paper focuses on variable bottom topography for the shallow water equations; extensions to the multi-layer case will be reported elsewhere.A central point of this work, for both the shallow water and the multi-layer cases, is the representation of the pressure forcing in the momentum equations. For the shallow water case, the present work does not use the standard representation of the pressure forcing that is widely used for the shallow water system. Instead, it begins with a more basic form of the momentum equations for a fluid flow, and it proceeds directly to a weak Galerkin form via integration over a suitable fluid region. The resulting formulation of the momentum equations is automatically well-balanced, subject to an assumption that the algorithms used to compute quantities at cell edges reproduce the continuous values in the case where all functions involved are continuous. In addition, this formulation has a structure that is analogous to that of the momentum equations in the individual layers of a multi-layer fluid, and this facilitates consistency between the two subsystems that are used to model such a fluid.The numerical computations described here include tests of well-balancing and tests of wave propagation over variable topography.

Instabilities in the Shallow-Water System with a Semi-Lagrangian, Time-Centered Discretization

Abstract Conventional wisdom suggests that the combination of semi-Lagrangian advection and an implicit treatment of gravity wave terms should result in a combined scheme for the shallow-water equations stable for high Courant numbers. This wisdom is well justified by linear analysis of the system about a uniform reference state with constant fluid depth and velocity, but it is only assumed to hold true in more complex scenarios. This work finds that this conventional wisdom no longer holds in more complicated flow regimes, in particular when the background state is given by steady-state flow past topography. Instead, this background state admits a wide range of instabilities that can lead to noise in atmospheric forecasts. Significance Statement This work shows that solutions to the shallow-water equations with a semi-Lagrangian treatment of advection and an implicit, time-centered treatment of gravity wave terms can be unstable when there is a background state of flow over topography. This basic algorithm is used by many operational weather-forecasting models to simulate the meteorological equations, and showing an instability in the simplified, shallow-water system suggests that a similar mechanism may be responsible for “noise” in operational weather forecasts under some circumstances. If this problem can be addressed, it could allow numerical weather models to operate with less dissipation, improving forecast quality.

Data Assimilation by Neural Network for Ocean Circulation: Parallel Implementation

Data assimilation (DA) is an essential issue for operational prediction centers, where a computer code is applied to simulate physical phenomena by solving differential equations. The procedure to determine the best initial condition combining data from observation and previous forecasting (background) is carried out by a data assimilation method. The Kalman filter (KF) is a technique for data assimilation, but it is computationally expensive. An approach to reduce the computational effort for DA is to emulate the KF by a neural network. The multi-layer perceptron neural network (MLP-NN) is employed to emulate the Kalman in a 2D ocean circulation model, and algorithmic complexity to KF and NN is presented. A shallow-water system models the ocean dynamics. Synthetic measurements are used for evaluating the MLP-NN for the data assimilation process. Here, a parallel version for the DA procedure by the neural network is described and tested, showing the performance improvement for a parallel version of the NN-DA.

Microbial Biofilms Along a Geochemical Gradient at the Shallow-Water Hydrothermal System of Vulcano Island, Mediterranean Sea.

Shallow water hydrothermal vents represent highly dynamic environments where strong geochemical gradients can shape microbial communities. Recently, these systems are being widely used for investigating the effects of ocean acidification on biota as vent emissions can release high CO2 concentrations causing local pH reduction. However, other gas species, as well as trace elements and metals, are often released in association with CO2 and can potentially act as confounding factors. In this study, we evaluated the composition, diversity and inferred functional profiles of microbial biofilms in Levante Bay (Vulcano Island, Italy, Mediterranean Sea), a well-studied shallow-water hydrothermal vent system. We analyzed 16S rRNA transcripts from biofilms exposed to different intensity of hydrothermal activity, following a redox and pH gradient across the bay. We found that elevated CO2 concentrations causing low pH can affect the response of bacterial groups and taxa by either increasing or decreasing their relative abundance. H2S proved to be a highly selective factor shaping the composition and affecting the diversity of the community by selecting for sulfide-dependent, chemolithoautotrophic bacteria. The analysis of the 16S rRNA transcripts, along with the inferred functional profile of the communities, revealed a strong influence of H2S in the southern portion of the study area, and temporal succession affected the inferred abundance of genes for key metabolic pathways. Our results revealed that the composition of the microbial assemblages vary at very small spatial scales, mirroring the highly variable geochemical signature of vent emissions and cautioning for the use of these environments as models to investigate the effects of ocean acidification on microbial diversity.

Influence of mid-term base level cycle changes on the development characteristics of shallow water delta sand bodies: A case study of the Fuyu oil layer in the Mu 17 block in the southern Songliao Basin

The Fuyu oil layer in the southern Songliao Basin is a shallow-water delta sedimentary system. Frequent changes in the base level lead to obvious differences in the superimposed style, scale and geometry of the sand bodies, which seriously affects the distribution of remaining oil. Taking the Fuyu oil layer in the Mu 17 block in the southern Songliao Basin as an example, using a large number of cores and logging data in the dense well pattern development area, based on the relationship between the accommodated space (A) and the sediment supply (S), it is clear that The Fuyu oil layer develops the superimposed styles of cut-stack, superimposed and isolated sand bodies. In the depositional period of MSC1 and MSC2, the superimposed style of the cut-and-stacked sand body was the main one; in the depositional period of MSC3, the superimposed style of the superimposed sand body was dominant; in the depositional period of MSC4, the sand body was the least developed, and the superimposed style of the isolated sand body was the main one.This study has important guiding significance for clarifying the influence of medium-term base level cycle changes on the superposition law of shallow water delta sand bodies.

Calculation method of submerged plant biomass in small and medium urban rivers: Based on the investigation and analysis of Shanghai City

沉水植物的盖度和生物量是衡量浅水系统恢复程度的重要指标，但城市河道沉水植物的生境由于人为干扰及修复过程，具有较高的异质性，使得城市河道沉水植物盖度和生物量测定更加困难，目前尚无准确、快速的计算方法.本研究以上海市内中小河道为研究对象，通过实测4条河道330余个代表样方的沉水植物生物，研究了基于采样次数的沉水植物单位面积生物量计算方法、采样布点方法和河道总盖度测算方法.结果显示：（1）至少需30~35次面积为0.04 m²的采样可准确估算上海中小河道内沉水植物单位面积的生物量；（2）样点设置可选用随机或等距设置；（3）利用Rededge-M多光谱相机对河道拍摄5波段影像，并进行5~8次重复处理，可较为准确获得整条河道沉水植物盖度.本研究可为城市中小河道沉水植被恢复、河道的分类管理与保护提供参考及依据.;The coverage and biomass of submerged plants are the key indicators of the restoration level of shallow water systems. But the habitat of submerged plants in urban rivers usually has high heterogeneity because of the human disturbance and restoration process. This makes it more difficult to determine the coverage and biomass of submerged plants in urban rivers and lacks an accurate and fast method. In this study, we measured the biomasses of submerged plants in more than 330 representative plots in 4 rivers to investigate the method of calculating the biomass per unit area of submerged plants using the numbers of samplings, as well as the methods of sampling and calculation of total coverage of submerged plants in rivers. The results showed that:(1) At least 30 to 35 samplings (0.04 m²) are required to accurately estimate the biomass per unit area of submerged plants in small and medium rivers in Shanghai; (2) Random or equidistant setting can be selected for sample setting; (3) Getting the band-5 images of the river channels using the Rededge-M multi-spectral camera image and repeating the process for 5 to 8 times, the coverage of submerged plants in the rivers can be obtained with higher accuracy. This study provides crucial reference and knowledge for the restoration of submerged vegetation in small and medium urban rivers and the management and protection of rivers.

Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations

Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadrature points in parameter space. Based on our theoretical results for the stochastic Galerkin formulation, we develop a corresponding well-balanced hyperbolicity-preserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche’s method. Convergence of the numerical method is analysed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

Mathematical modeling of the pollutant distribution process in the Gelendzhik Bay

Currently, the pollution problem of coastal sea waters in the resort areas of the Black Sea is becoming increasingly urgent. Thousands of chemicals, industrial and household waste enter the water basins every year, which significantly worsens the state of marine waters. Storm drains are saturated with pollutants at precipitation, washing out various chemical compounds, garbage and transporting them to the sea. In addition, a separate problem is poorly self-cleaning of the Gelendzhik Bay. A complete water change occurs in the period from 1 to 6 days. This paper covers the development, research and numerical implementation of a mathematical model of the pollution transport, including petroleum products, in the Gelendzhik Bay taking into account a number of important hydrodynamic and hydrophysical factors, methods of its numerical implementation, which allow predictive modeling of the pollution spread in shallow water systems in a limited time. A hydrobiological model of a coastal system characterized by significant depth differences has been developed. A three-dimensional mathematical model is designed to research the transformation of the phosphorus, nitrogen and silicon forms in the plankton dynamics problem. It takes into account the convective and diffusive transports, absorption and release of nutrients by phytoplankton, salinity, temperature, oxygen regime, etc. Using a spatial-three-dimensional hydrodynamics model, taking into account the physical properties of water environment of the coastal system, calculation results are used as input data for the development of scenarios for the dynamics of transport processes and the transformation of pollution biogenic elements in the water.

The sedimentary response of mixed lithoclastic-bioclastic Lower- Pleistocene shallow-marine systems to tides and waves in the south Apennine foredeep (Basilicata, southern Italy)Tidalites Field Trips Special Volume - Tidalites, Matera, Field Trip T3

This two-day-long field trip is associated with the 10th International Congress of Tidal Sedimentology (Tidalites), Matera, Italy. The foreland-basin system of southern Italy preserves mixed lithoclastic-bioclastic deposits mostly accumulated in shallow-marine environments, reproducing types of mixing and degrees of segregation (sensu Chiarella and Longhitano, 2012; Chiarella et al., 2017) that are thought to be indicative of a variety of geological processes, including surficial waves and tides, acting at different time scales.During the Late Pliocene and Early Pleistocene, small open piggyback basins reproduced marginal-marine settings with various hydrodynamic conditions in the wedge-top depozone. Siliciclastic-bioclastic sediments accumulated under the influence of geomorphological elements, such as coastal sheltering, promontories, presence of tectonic highs and local inlets, hosting shoreface to offshore-transition zones, whose cross-stratified facies indicate how sensitive shallow-water mixed systems are to recording surficial waves and weak tidal currents.Along the outer-foredeep depozone, carbonate-bioclastic coastal wedges back-stepped over the gently sloping rocky flanks of a structural high, leading to the retrogradational stacking of seismic-scale prograding bodies. Their size and cross-sectional outcrop views apparently resemble those typical of large tidal sand waves developed in wide tide-dominated oceans. In contrast, internal facies features reveal the dominance of surficial waves and gravity-driven avalanches along clinoform depositional slopes, and a very negligible to no tidal influence.

Analogy between electro-magnetic waves in cold unmagnetized plasma and shallow water inertio-gravity waves in geophysical systems

The fundamental dispersion relation of transverse electro-magnetic waves in a cold collisionless plasma is formally equivalent to the two-dimensional dispersion relation of inertio-gravity waves in a rotating shallow water system, where the Coriolis frequency can be identified with the plasma frequency, and the shallow water gravity wave phase speed plays the role of the speed of light. Here we examine this equivalence and compare between the propagation wave mechanisms in these seemingly unrelated physical systems.

Effects of sea level and upwelling on development of a Miocene shallow-water tropical carbonate ramp system, Ponce, Puerto Rico

ABSTRACT A Miocene (Langhian–Tortonian, ca. 15–10 Ma) tropical ramp system exposed in southern Puerto Rico is characterized by shallow-water facies consisting of heterozoans, red algae, large benthic foraminifera (LBF), and corals, which occur as isolated corals, segment- and cluster-type reefs, and reworked accumulations. Photozoan association components are limited to corals (Montastraea, Porites, Goniopora, and Agaricia) and LBF (amphisteginids, soritids, gypsinids, miliolids) that have been documented to tolerate elevated nutrients, turbidity, and cooler water conditions. Similar shallow-water carbonate systems are found throughout the Caribbean, and this regional development is thought to have resulted from the well-documented upwelling in the Caribbean during the Miocene. Sea-level fluctuations also exerted a major control on facies distributions and shifts in the Puerto Rico ramp, including a vertical facies pattern that occurs in each of three sequences. Basal parts of sequences, deposited during sea-level rises, are dominantly composed of mollusks, echinoderms, red algae, LBF, bryozoans, and solitary corals that formed in low-energy seagrass-bed environments with local associated higher-energy shoal environments. Coral facies occur only in upper parts of sequences and formed in shallow-water, low- to high-energy environments closely associated with seagrass beds during late highstands and sea-level falls. A similar vertical facies pattern occurs in time-equivalent sequences elsewhere around the Caribbean. Strontium-isotope age data indicate two sequence boundaries reflecting sea-level falls formed at about 12.3 Ma and 11.1 Ma. Correlation with time-equivalent unconformities in other well-dated areas in the Caribbean and to sea-level lows on eustatic curves suggests a global signature for sequence development. The connection between the Caribbean and the Pacific along the Central American Seaway (CAS), impacted by local tectonic episodes and sea-level fluctuations during the Miocene, affected nutrient influx and upwelling in the Caribbean, which may be reflected in the vertical facies pattern in shallow-water carbonate sequences. Times of restricted connection during sea-level falls and lows resulted in reduced nutrients and upwelling, which may have been more conducive to coral development. Time-equivalent tropical carbonate systems in the Mediterranean and Indo-Pacific show similarities to those in the Caribbean, indicating influence of global processes (cooling, temperature gradients, oceanographic circulation). Differences between areas indicates the importance of local and regional controls, which in the Caribbean was dominantly the opening and closure of the CAS.