Two-layer Shallow Water Model Research Articles

A fully Eulerian two-layer model for the simulation of oil spills spreading over coastal flows

Nowadays, the vast majority of coastal oil spill simulation models are based on Lagrangian methods focused on particle tracking algorithms to represent the oil slick fate. In this work, a fully Eulerian numerical model for the simulation of such environmentally significant disaster is implemented by means of a two-dimensional two-layer shallow water model. A very thin oil layer over a thicker water layer is considered in order to neglect the pressure term that the oil layer exerts over the water. Friction terms between layers are responsible for the layers coupling so that the oil layer flows over a moving water volume. To complete this dynamic model, the temperature transport and evolution under heat exchange for the oil upper layer is considered and the weathering process of evaporation is included. The numerical solution adopted is based on a finite volume upwind scheme with a Roe solver for both oil and water layers. Special care has been taken on the numerical treatment of the two-layer wet-dry boundaries (oil–water–land) and friction terms, since the objective of the model is to compute the oil slick front advancing near the coast.

Flux globalization based well-balanced path-conservative central-upwind scheme for two-layer thermal rotating shallow water equations

We develop a flux globalization based well-balanced path-conservative central-upwind scheme for the two-layer thermal rotating shallow water (TRSW) equations, which arise in both oceanography and atmospheric sciences. As it occurs in all of the two-layer shallow water models, the studied two-layer TRSW system loses hyperbolicity due to the well-known Kelvin-Helmholtz type instabilities arising when the vertical velocity shear between the layers becomes sufficiently large. Therefore, prior to developing the numerical method, we discuss the hyperbolicity criterion. Besides the hyperbolicity loss, there are additional challenges in the development of numerical methods for the studied system. They are related to the presence of nonconservative terms modeling layer interaction and also to a quite complicated structure of steady-states solutions a good well-balanced numerical method should be able to exactly preserve. In order to treat the nonconservative product terms, we use the path-conservative technique, which is implemented within the flux globalization framework: the source and nonconservative terms are incorporated into the fluxes, which results in a quasi-conservative system, which is then numerically solved using the Riemann-problem-solver-free central-upwind scheme. A well-balanced property of the resulting scheme is ensured by performing piecewise linear reconstruction for the equilibrium rather than the conservative variables, by development of special quadratures required for the flux globalization procedure, and by switching off a part of the numerical diffusion when the computed solution is near or at thermo-geostrophic equilibria. The advantages and excellent performance of the proposed scheme are demonstrated on a number of numerical examples.

An Idealized 1½-Layer Isentropic Model with Convection and Precipitation for Satellite Data Assimilation Research. Part II: Model Derivation

AbstractIn this Part II paper we present a fully consistent analytical derivation of the “dry” isentropic 1½-layer shallow-water model described and used in Part I of this study, with no convection and precipitation. The mathematical derivation presented here is based on a combined asymptotic and slaved Hamiltonian analysis, which is used to resolve an apparent inconsistency arising from the application of a rigid-lid approximation to an isentropic two-layer shallow-water model. Real observations based on radiosonde data are used to justify the scaling assumptions used throughout the paper, as well as in Part I. Eventually, a fully consistent isentropic 1½-layer model emerges from imposing fluid at rest (v1= 0) and zero Montgomery potential (M1= 0) in the upper layer of an isentropic two-layer model.

The Momentum Conserving Scheme for Two-Layer Shallow Flows

This paper confronts the numerical simulation of steady flows of fluid layers through channels of varying bed and width. The fluid consists of two immiscible fluid layers with constant density, and it is assumed to be of a one-dimensional shallow flow. The governing equation is a coupled system of two-layer shallow water models. In this paper, we apply a direct extension of the momentum conserving scheme previously used for solving the one layer shallow water equations. Computations of various steady-state solutions are used to demonstrate the performance of the proposed numerical scheme. Under the influence of a given flow rate, the numerical steady interface is generated in a channel topography with a hump. The results obtained confirm the analytic steady interface of the two-layer rigid-lid model. Furthermore, the same scheme was used with an additional artificial damping to simulate the maximal exchange flow in channels of varying width. The numerical steady interface agreed well with the analytical steady solutions.

Interaction of Nonlinear Ekman Pumping, Near-Inertial Oscillations, and Geostrophic Turbulence in an Idealized Coupled Model

AbstractA new coupled model is developed to investigate interactions among geostrophic, Ekman, and near-inertial (NI) flows. The model couples a time-dependent nonlinear slab Ekman layer with a two-layer shallow water model. Wind stress forces the slab layer and horizontal divergence of slab-layer transport appears as a forcing in the continuity equation of the shallow water model. In one version of the slab model, self-advection of slab-layer momentum is retained and in another it is not. The most obvious impact of this explicit representation of the surface-layer dynamics is in the high-frequency part of the flow. For example, near-inertial oscillations are significantly stronger when self-advection of slab-layer momentum is retained, this being true both for the slab-layer flow itself and for the interior flow that it excites. In addition, retaining the self-advection terms leads to a new instability, which causes growth of slab-layer near-inertial oscillations in regions of anticyclonic forcing and decay in regions of cyclonic forcing. In contrast to inertial instability, it is the sign of the forcing, not that of the underlying vorticity, that determines stability. High-passed surface pressure fields are also examined and show the surface signature of unbalanced flow to differ substantially depending on whether a slab-layer model is used and, if so, whether self-advection of slab-layer momentum is retained.

Modeling Heavy-Gas Dispersion in Air with Two-Layer Shallow Water Equations

Computation of gas dispersal in urban places or hilly grounds requires a large amount of computer time when addressed with conventional multidimensional models. Those are usually based on two-phase flow or Navier-Stokes equations. Different classes of simplified models exist. Among them, two-layer shallow water models are interesting to address large-scale dispersion. Indeed, compared to conventional multidimensional approaches, 2D simulations are carried out to mimic 3D effects. The computational gain in CPU time is consequently expected to be tremendous. However, such models involve at least three fundamental difficulties. The first one is related to the lack of hyperbolicity of most existing formulations, yielding serious consequences regarding wave propagation. The second is related to the non-conservative terms in the momentum equations. Those terms account for interactions between fluid layers. Recently, these two difficulties have been addressed in Chiapolino and Saurel (2018) and an unconditional hyperbolic model has been proposed along with a Harten-Lax-van Leer (HLL) type Riemann solver dealing with the non-conservative terms. In the same reference, numerical experiments showed robustness and accuracy of the formulation. In the present paper, a third difficulty is addressed. It consists of the determination of appropriate drag effect formulation. Such effects also account for interactions between fluid layers and become of particular importance when dealing with heavy-gas dispersion. With this aim, the model is compared to laboratory experiments in the context of heavy gas dispersal in quiescent air. It is shown that the model accurately reproduces experimental results thanks to an appropriate drag force correlation. This function expresses drag effects between the heavy and light gas layers. It is determined thanks to various experimental configurations of dam-break test problems.

The instability of vertically curved fronts in a two-layer shallow water model

This paper investigates frontal instabilities in the context of a two-layer shallow water model with both linear and curved interfaces. First, we generalize the model previously established to allow for curved interfaces. Second, we revisit the linear interface case but with higher spatial resolution and study the impact of no-normal flow or radiation boundary conditions. The results show that the former prevents a baroclinic mode from occurring. Third, we investigate the novel problem of curved interfaces and deduce that it does not have a significant impact on a baroclinic instability; however, it does alter the regions of Kelvin–Helmholtz and Rossby–Kelvin instabilities. Furthermore, it introduces a new type of baroclinic instability that we refer to as the vertically curved-interface baroclinic instability.

Determination of the Boundaries of Flooded Zones Based on Hydrodynamic Modeling

The paper discusses the problem of determining the boundaries of flooding zones, flooding of territories. The paper describes technique for constructing cadastral maps of flooded zones for various values of the estimated availability of water levels, which is based on hydrodynamic and geoinformation modeling. For creating digital elevation models (DEM) of the conformation, we use spatial data obtained from a large number of sources. We applied the described technique to construct cadastral maps for a number of settlements in the Volgograd region of the Russian Federation. As a result, we get a set of xml-files prepared for registration with the Federal Agency for State Registration, Cadastre and Cartography. For the Republic of Belarus, the problem of flooding the territories is also urgent. The floodplain areas of the Pripyat River are most prone to flooding. We built a DEM for the area in the vicinity of Petrikov city and carried out hydrodynamic modeling. As a result, the inundation maps were obtained. The method proposed in this work is based on a mathematical model of shallow water in a single-layer approximation. This model is successfully applied in simulation of flooding for territories by flood waters. Solving the problem of determining the boundaries of flooding zones requires a transition to a two-layer shallow water model that takes into account the interaction of surface and ground waters.

Potential Vorticity Dynamics of the Arctic Halocline

AbstractAn idealized two-layer shallow water model is applied to the study of the dynamics of the Arctic Ocean halocline. The model is forced by a surface stress distribution reflective of the observed wind stress pattern and ice motion and by an inflow representing the flow of Pacific Water through Bering Strait. The model reproduces the main elements of the halocline circulation: an anticyclonic Beaufort Gyre in the western basin (representing the Canada Basin), a cyclonic circulation in the eastern basin (representing the Eurasian Basin), and a Transpolar Drift between the two gyres directed from the upwind side of the basin to the downwind side of the basin. Analysis of the potential vorticity budget shows a basin-averaged balance primarily between potential vorticity input at the surface and dissipation at the lateral boundaries. However, advection is a leading-order term not only within the anticyclonic and cyclonic gyres but also between the gyres. This means that the eastern and western basins are dynamically connected through the advection of potential vorticity. Both eddy and mean fluxes play a role in connecting the regions of potential vorticity input at the surface with the opposite gyre and with the viscous boundary layers. These conclusions are based on a series of model runs in which forcing, topography, straits, and the Coriolis parameter were varied.

Efficient modeling of shallow water equations using method of lines and artificial viscosity

In this work, two numerical schemes were developed to overcome the problem of shock waves that appear in the solutions of one/two-layer shallow water models. The proposed numerical schemes were based on the method of lines and artificial viscosity concept. The robustness and efficiency of the proposed schemes are validated on many applications such as dam-break problem and the problem of interface propagation of two-layer shallow water model. The von Neumann stability of proposed schemes is studied and hence, the sufficient condition for stability is deduced. The results were presented graphically. The verification of the obtained results is achieved by comparing them with exact solutions or another numerical solutions founded in literature. The results are satisfactory and in much have a close agreement with existing results.

Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography

We derive a hyperbolic system of equations approximating the two-layer dispersive shallow water model for shear flows recently proposed by Gavrilyuk et al. (2016). The use of this system for modelling the evolution of surface waves makes it possible to avoid the major numerical challenges in solving dispersive shallow water equations, which are connected with the resolution of an elliptic problem at each time instant and realization of non-reflecting conditions at the boundary of the calculation domain. It also allows one to reduce the computation time. The velocities of the characteristics of the obtained model are determined and the linear analysis is performed. Stationary solutions of the model are constructed and studied. Numerical solutions of the hyperbolic system are compared with solutions of the original dispersive model. It is shown that they almost coincide for large time intervals. The system obtained is applied to study non-stationary undular bores produced after interaction of a uniform flow with an immobile wall, non-hydrostatic shear flows over a local obstacle and the evolution of breaking solitary wave on a sloping beach.

Towards a new friction model for shallow water equations through an interactive viscous layer

The derivation of shallow water models from Navier–Stokes equations is revisited yielding a class of two-layer shallow water models. An improved velocity profile is proposed, based on the superposition of an inviscid fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but also on the variations of velocity and thickness of the viscous layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum conservation in which a slight correction on hydrostatic pressure has been made; the third one, known as von Kármán equation, describes the evolution of the viscous layer. This coupled model is shown to be conditionally hyperbolic, and a Godunov-type finite volume scheme is also proposed. Several numerical examples are provided and compared to the Multi-Layer Saint-Venant model. They emphasize the ability of the model to deal with unsteady viscous effects. They illustrate also the phase-lag between friction and topography, and even recover possible reverse flows.

Models and methods for two-layer shallow water flows

Two-layer shallow water models present at least two fundamental difficulties that are addressed in the present contribution. The first one is related to the lack of hyperbolicity of most existing models. By considering weak compressibility of the phases, a strictly hyperbolic formulation with pressure relaxation is obtained. It is shown to tend to the conventional two-layer model in the stiff pressure relaxation limit. The second issue is related to the non-conservative terms in the momentum equations. Analyzing the Riemann problem structure, local constants appear precisely at locations where the non-conservative products need definition. Thanks to these local constants, a locally conservative formulation of the equations is obtained, simplifying the Riemann problem resolution through a HLL-type Riemann solver. The method is compared to literature data, showing accurate and oscillation free solutions. Additional numerical experiments show robustness and accuracy of the method.

A field study of interfacial friction and entrainment in a microtidal salt-wedge estuary

The interfacial friction and entrainment were investigated in a microtidal salt-wedge estuary. A detailed sampling campaign was conducted in the Rjecina River estuary in Croatia from January 2014 to June 2015. The observed vertical profiles of salinity s and temperature T confirmed the presence of a highly stratified estuary, represented by an upper layer of freshwater separated from a lower salt-wedge by a sharp density interface. The entrainment rate E across the interface was estimated by a two-layer box-model, based on the observed freshwater flow rate Q and layer-averaged salinity. Interfacial friction factor $$\lambda _i$$ was estimated by fitting the results of a numerical model to the observed interface depths. For this purpose we applied a numerical two-layer shallow water model extended to account for irregular non-prismatic cross sections of the channel. We found that in microtidal conditions, the strength of the stratification is reduced with increasing Q. Furthermore, we found that as Q increases, so does the shear velocity, the interfacial friction factor and the vertical mixing across the interface. More detail analysis showed that E may be parametrized by bulk non-dimensional parameters, in particular, a combination of bulk Richardson number Ri and average friction factor $$\lambda$$ , which accounts for the channel bed friction and the interfacial friction. On the other hand, $$\lambda _i$$ can be linked to a combination of Reynolds number Re and bulk Richardson Ri. Contrary to previous studies, we showed that in field conditions, $$\lambda _i$$ may increase with Re.

Simulation of arrested salt wedges with a multi-layer Shallow Water Lattice Boltzmann model

The ability to accurately and efficiently model the intrusion of salt wedges into river beds is crucial to assay its interaction with human activities and the natural environment. We present a 2D multi-layer Shallow Water Lattice Boltzmann (SWLB) model able to predict the salt wedge intrusion in river estuaries. The formulation usually employed for the simulation of gravity currents is here equipped with proper boundary conditions to handle both the downstream seaside outlet and the upstream river inlet. Firstly, the model is validated against highly accurate semi-analytical solutions of the steady state 1D two-layer Shallow Water model. Secondly, the model is applied to a more complex, fully 3D geometry, to assess its capability to handle realistic cases. The simple formulation proposed for the shear interlayer stress is proven to be consistent with the general 3D viscous solution. In addition to the accuracy, the model inherits the efficiency of the Lattice Boltzmann approach to fluid dynamics problems.

Sustained gravity currents in a channel

Gravitationally driven motion arising from a sustained constant source of dense fluid in a horizontal channel is investigated theoretically using shallow-layer models and direct numerical simulations of the Navier–Stokes equations, coupled to an advection–diffusion model of the density field. The influxed dense fluid forms a flowing layer underneath the less dense fluid, which initially filled the channel, and in this study its speed of propagation is calculated; the outflux is at the end of the channel. The motion, under the assumption of hydrostatic balance, is modelled using a two-layer shallow-water model to account for the flow of both the dense and the overlying less dense fluids. When the relative density difference between the fluids is small (the Boussinesq regime), the governing shallow-layer equations are solved using analytical techniques. It is demonstrated that a variety of flow-field patterns are feasible, including those with constant height along the length of the current and those where the height varies continuously and discontinuously. The type of solution realised in any scenario is determined by the magnitude of the dimensionless flux issuing from the source and the source Froude number. Two important phenomena may occur: the flow may be choked, whereby the excess velocity due to the density difference is bounded and the height of the current may not exceed a determined maximum value, and it is also possible for the dense fluid to completely displace all of the less dense fluid originally in the channel in an expanding region close to the source. The onset and subsequent evolution of these types of motions are also calculated using analytical techniques. The same range of phenomena occurs for non-Boussinesq flows; in this scenario, the solutions of the model are calculated numerically. The results of direct numerical simulations of the Navier–Stokes equations are also reported for unsteady two-dimensional flows in which there is an inflow of dense fluid at one end of the channel and an outflow at the other end. These simulations reveal the detailed mechanics of the motion and the bulk properties are compared with the predictions of the shallow-layer model to demonstrate good agreement between the two modelling strategies.

A large active wave trapped in Jupiter’s equator

Context. A peculiar atmospheric feature was observed in the equatorial zone (EZ) of Jupiter between September and December 2012 in ground-based and Hubble Space Telescope (HST) images. This feature consisted of two low albedo Y-shaped cloud structures (Y1 and Y2) oriented along the equator and centred on it (latitude 0.5°−1°N).Aims. We wanted to characterize these features, and also tried to find out their properties and understand their nature.Methods. We tracked these features to obtain their velocity and analyse their cloud morphology and the interaction with their surroundings. We present numerical simulations of the phenomenon based on one- and two-layer shallow water models under a Gaussian pulse excitation.Results. Each Y feature had a characteristic zonal length of ~15° (18 000 km) and a meridional width (distance between the north-south extremes of the Y) of 5° (6000 km), and moved eastward with a speed of around 20−40 m s-1 relative to Jupiter’s mean flow. Their lifetime was 90 and 60 days for Y1 and Y2, respectively. In November, both Y1 and Y2 exhibited outbursts of rapidly evolving bright spots emerging from the Y vertex. The Y features were not visible at wavelengths of 255 or 890 nm, which suggests that they were vertically shallow and placed in altitude between the upper equatorial hazes and the main cloud deck. Numerical simulations of the dynamics of the Jovian equatorial region generate Kelvin and Rossby waves, which are similar to those in the Matsuno-Gill model for Earth’s equatorial dynamics, and reproduce the observed cloud morphology and the main properties the main properties of the Y features.

Symmetric and asymmetric inertial instability of zonal jets on the -plane with complete Coriolis force

Symmetric and asymmetric inertial instability of the westerly mid-latitude barotropic Bickley jet is analysed without the traditional approximation which neglects the vertical component of the Coriolis force, as well as the contribution of the vertical velocity to the latter. A detailed linear stability analysis of the jet at large Rossby numbers on the non-traditional $f$-plane is performed for long waves in both the two-layer rotating shallow-water and continuously stratified Boussinesq models. The dependence of the instability on both the Rossby and Burger numbers of the jet is investigated and compared to the traditional case. It is shown that non-traditional effects significantly increase the growth rate of the instability at small enough Burger numbers (weak stratifications) for realistic aspect ratios of the jet. The main results are as follows. (i) Two-layer shallow-water model. In the parameter regimes where the jet is inertially stable on the traditional $f$-plane, the symmetric inertial instability with respect to perturbations with zero along-jet wavenumber arises on the non-traditional $f$-plane. Both non-traditional symmetric and asymmetric (small but non-zero wavenumbers) inertial instabilities have higher growth rates than their traditional counterparts. (ii) Continuously stratified model. It is shown that by a proper change of variables the linear stability problem for the barotropic jet, on the non-traditional $f$-plane, can be rendered separable and analysed along the same lines as in the traditional approximation. Neutral, weak and strong background stratifications are considered. For the neutral stratification the jet is inertially unstable if the traditional approximation is relaxed, while its traditional counterpart is not. For a sufficiently weak stratification, both symmetric and asymmetric inertial instabilities have substantially higher growth rates than in the traditional approximation. The across-jet structure of non-traditional unstable modes is strikingly different, as compared to those under the traditional approximation. No discernible differences between the two approximations are observed for strong enough stratifications. The influence of dissipation and non-hydrostatic effects upon the instability is quantified.

Local wind effect on the Kuroshio path state off the southeastern coast of Kyushu

Small meanders of the Kuroshio southeast of Kyushu, Japan, are conventionally known to form in winter or early spring, but rarely in other seasons. This study examines the seasonal nature of the small meander formation as it relates to the seasonal monsoon winds over the East China Sea, based on observational data analyses, theoretical considerations and numerical experiments. Positional data of the Kuroshio path indicate that the small meander experiences phase locking with the seasonal cycle, as is conventionally thought, although its intensity is modulated on decadal time scales. Monthly mean data of the surface geostrophic current and wind stress fields indicate that the southwestward wind blowing against the Kuroshio in the autumn and early winter causes an inshore-ward shift of the Kuroshio in the northern Okinawa Trough. This effect acts to strengthen the convoluted pattern of the Kuroshio path around Kyushu, and to develop the small meander southeast of the same island. Assuming that the Kuroshio is a surface geostrophic jet with a double-exponential velocity profile, the observed evidence is well explained by a combination of Ekman layer dynamics and quasi-geostrophic dynamics. More specifically, Ekman pumping due to nonlinear Ekman divergence over the jet decreases (increases) the surface velocity of the offshore (inshore) side of the jet, resulting in the inshore-ward shift of the jet. A two-layer shallow water model with idealized topography shows that the response of the modeled Kuroshio to the local wind stress is consistent with observational and theoretical results.

Local Well-Posedness of the Two-Layer Shallow Water Model with Free Surface

In this paper, we address the question of the hyperbolicity and the local well-posedness of the two-layer shallow water model, with free surface, in two dimensions. We first provide a general criterion that proves the symmetrizability of this model, which implies hyperbolicity and local well-posedness in $\mathcal{H}^s(\mathbb{R}^2)$, with $s>2$. Then, we analyze rigorously the eigenstructure associated with this model and prove a more general criterion for hyperbolicity and local well-posedness, under a weak density-stratification assumption. Finally, we consider a new conservative two-layer shallow water model, prove the hyperbolicity and the local well-posedness, and relate it to the basic two-layer shallow water model.