Nonlinear Shallow Water Model Research Articles

Numerical study on a possible cause of the ‘strange tide’ in the coastal area of Jiangsu Province, China

There are many eyewitness reports on unexpected water level change called ‘strange tide’ around the area of the radial sand ridges off the coast of Jiangsu Province, China, which has caused several fatal accidents in history. This study aims to prove whether a fast-moving atmospheric pressure disturbance can be a possible cause of the ‘strange tide.’ Based on the nonlinear shallow water model, a large number of scenarios for waves induced by a fast-moving pressure disturbance are numerically simulated and the maximum of the wave amplification ratio in each scenario is discussed. In each scenario, we assume an idealized pressure disturbance passing through the concerned area. The heading angle and translational speed of atmospheric disturbances are considered as variables. The results show that a fast-moving pressure disturbance can induce large wave level fluctuation in this area. In some special situations, the water level rise can be amplified more than 40 times than the water level change caused by a pressure disturbance with equivalent intensity under static equilibrium state. Both the heading angle and translational speed of atmospheric disturbances play a very important role in the occurrence of dangerous situation. The most dangerous combination of the two parameters is discussed, and the disaster prone locations in this area are identified considering the historical accidents and the numerical experiments.

Exact Traveling and Nano-Solitons Wave Solitons of the Ionic Waves Propagating along Microtubules in Living Cells

In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary wave solutions of this model by using the modified Riccati-expansion method then using these solutions to determine the boundary and initial conditions. These conditions are employed to evaluate the semi-analytical wave solutions and calculate the absolute value of error. The values of absolute error show the accuracy of the obtained solutions. Some solutions are sketched to show the perspective view of the solution of this model. Moreover, the novelty of the obtained solutions is illustrated by showing the similarity and differences between our and previous solutions of the model.

Performance assessment of the Tsunami-HySEA model for NTHMP tsunami currents benchmarking. Laboratory data

The Tsunami-HySEA model is used to perform the five numerical benchmarks proposed in the “NTHMP/MMS Benchmarking Workshop: Tsunami currents”. Many different model configurations and sensitivity studies were performed for the different benchmarks, combining numerical methods of first, second, and third order, different friction laws, varying friction coefficients and numerical resolutions, or even different implementations of the boundary conditions. In this work, we include the three benchmark problems dealing with laboratory data, which are BP1 (truncated submerged circular cone), BP4 (Seaside, Oregon, city building model), and BP5 (solitary wave over a complex shelf). Benchmark 1 appeared to be the most difficult one of all for non-linear shallow water (NLSW) models due to its high sensitivity to parameters and the very complex wake structure occurring behind the submerged island. For the rest of the benchmark problems, a good fit was obtained with measured data. An initial version of the Tsunami-HySEA model implementing dispersion was used to reproduce BP4 and BP5. In the related paper Macías et al. (2020), the results corresponding to the two benchmarks dealing with field data that were proposed in the Benchmarking Workshop are described in detail.

Lattice Boltzmann simulation of wind-driven rainfall on a slope surface

In overland flow modeling, the influences of the slope degree, rain intensity and wind condition can be significant. In this study, the effects of the slope gradient and wind-driven rain were integrated in a nonlinear shallow water model based on the lattice Boltzmann method (LBM). Since slope gradient and wind condition can not only affect the flow rate, but also its momentum, we enhanced the mathematical model and derived two LBM schemes that include their hydrodynamic influence on the surface flow. Then, this model was verified by several typical cases. The results indicated that the momentum component of rainfall should be considered during the overland flow simulation influenced by wind, otherwise it would result in an intrinsic error of approximately [Formula: see text] in simulated results.

Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations

This paper considers the dynamical behavior of solutions near explicit self-similar singularity for a class of nonlinear shallow water models including the Dullin–Gottwald–Holm equation, the Korteweg–de Vires equation, the Camassa–Holm equation, the dispersive rod equation and the Benjamin–Bona–Mahony equation. We first show there are explicit self-similar solutions for those five nonlinear shallow water equations, then we find that those explicit self-similar solutions for the Dullin–Gottwald–Holm equation, the Camassa–Holm equation and the dispersive rod equation are asymptotic stable, but for the Korteweg–de Vries equation and the Benjamin–Bona–Mahony equation being unstable.

EFFECT OF MACRO-ROUGHNESS ELEMENTS ON TSUNAMI INUNDATION: SINK TERM FORMULAE FOR DEPTH-AVERAGED NUMERICAL MODELS

Tsunami propagation and inundation are commonly simulated using large-scale depth-averaged models. In such models, the quadratic friction law with a selected Manning’s coefficient is generally applied to account for the effect of bottom surface roughness in each computational element. Buildings and tree vegetation in coastal areas are usually smaller than the computational element size. Using empirical Manning’s coefficients to account for such large objects is not physically sound and, particularly in tsunami inundation modelling, this may result in large uncertainties. Therefore, an improved understanding of the processes associated with the hydraulic resistance of the so-called macro-roughness elements (MRE) is required. Relevant parameters such as shape, height and arrangement of the MRE should be investigated through laboratory experiments or numerical tests using a well-validated three-dimensional CFD model. Given the correlation of such parameters to the MRE-induced hydraulic resistance, empirical formulae were developed and directly implemented as sink terms in depth-averaged numerical solvers such as non-linear shallow-water (NLSW) models.

DEBRIS AND VESSEL TRANSPORT DUE TO TSUNAMI CURRENTS IN PORTS AND HARBORS

The focus of the discussion will be on the debris and vessel transport capacity of the tsunami induced currents in ports and harbors. The tsunami events in the past 15 years proved that understanding these processes within the port/harbor basin has paramount importance in safety, recovery and the long-term resilience planning of the facilities; as all of these depend on the ability of structures or infrastructure to resist damage and the capability of harbors to become functional after the event.
 
 This endeavor requires an accurate representation of the flow field around the floating objects. Particularly, when the size of an average container ship is considered among with its’ fairly high draft to depth ratio, the interaction between flow and the vessels gets stronger. Therefore, in this study, the developed numerical tool, which is coupled with a 2HD nonlinear shallow water model, takes the interaction between the flow and the objects into account, and provides accurate results in a computationally efficient way.
 
 We will also present example simulation results of a numerical modelling study aimed at providing the quantitative guidance on maritime tsunami hazards in ports and harbors. This information can be used in pre-disaster recovery planning with the identification of the safe mooring spots or where the debris will likely accumulate after future tsunamis. Whilst the harbor’s ability to resist damage is a function of reducing the exposure to hazardous conditions as well as the maintaining/upgrading the structures/infrastructure within the harbors.

Efficient numerical scheme based on the method of lines for the shallow water equations

In this paper, a nonlinear shallow-water model of tsunami wave propagation at different points along a coastline of an ocean has been numerically simulated using method of lines. The simulation is carried out for various coastal slopes and the ocean depths. The effects of the coast slope and sea depth on the tsunami wave run-up height and velocity are illustrated. The accuracy of the mathematical model is verified by solving a classical test problem with known analytic solution. The computed run-up height and velocity show satisfactory agreement with the tsunami wave physics.

A Two-Dimensional Dynamical Model for the Subseasonal Variability of the Asian Monsoon Anticyclone

The Asian monsoon anticyclone, which develops in the upper troposphere and lower stratosphere during boreal summer, exhibits significant subseasonal variability with a characteristic spatial structure. The dynamics of this variability is investigated using a nonlinear β-plane shallow-water model. The equivalent depth is estimated using reanalysis data to relate the three-dimensional dynamics in isentropic coordinates to the shallow-water model. Composite analysis reveals the resemblance of the horizontal structures between the Montgomery streamfunction and thickness on the 360-K level. However, the coefficients of the linear regressions between those two variables are strongly dependent on latitude. The estimated equivalent depths of the northern region are more than 2 times greater than those of the southern region. This is attributable to the background thermal structure around the tropopause. Based on this, a latitude-dependent mean depth is incorporated into the shallow-water model to numerically investigate responses to a steady localized forcing in the subtropics. With the inclusion of the latitudinal dependence of the mean depth, the vortex shedding state is able to have a longitudinally confined structure, which differs from the conventional case of constant mean depth. The spatial structure of this numerical solution corresponds to the observed structure, in which low-PV air is largely confined to finite longitudes within the Asian monsoon anticyclone. This suggests the possible role of dynamical instability and the interaction with the subtropical jet in determining the characteristic structure of the Asian monsoon anticyclone.

Tsunami run-up estimation based on a hybrid numerical flume and a parameterization of real topobathymetric profiles

Abstract. Tsunami run-up is a key value to determine when calculating and assessing the tsunami hazard in a tsunami-prone area. Run-up can be accurately calculated by means of numerical models, but these models require high-resolution topobathymetric data, which are not always available, and long computational times. These drawbacks restrict the application of these models to the assessment of small areas. As an alternative method, to address large areas empirical formulae are commonly applied to estimate run-up. These formulae are based on numerical or physical experiments on idealized geometries. In this paper, a new methodology is presented to calculate tsunami hazard at large scales. This methodology determines the tsunami flooding by using a coupled model that combines a nonlinear shallow water model (2D-H) and a volume-of-fluid model (RANS 2D-V) and applies the optimal numerical models in each phase of the tsunami generation–propagation–inundation process. The hybrid model has been widely applied to build a tsunami run-up database (TRD). The aim of this database is to form an interpolation domain with which to estimate the tsunami run-up of new scenarios without running a numerical simulation. The TRD was generated by simulating the propagation of parameterized tsunami waves on real non-scaled profiles. A database and hybrid numerical model were validated using real and synthetic scenarios. The new methodology provides feasible estimations of the tsunami run-up; engineers and scientists can use this methodology to address tsunami hazard at large scales.

Hydraulic jump and Bernoulli equation in nonlinear shallow water model

A shallow water model was applied to study the hydraulic jump and Bernoulli equation across the jump. On a flat terrain, when a supercritical flow plunges into a subcritical flow, discontinuity develops on velocity and Bernoulli function across the jump. The shock generated by the obstacle may propagate downstream and upstream. The latter reflected from the inflow boundary, moves downstream and leaves the domain. Before the reflected wave reaching the obstacle, the short-term integration (i.e., quasi-steady) simulations agree with Houghton and Kasahara’s results, which may have unphysical complex solutions. The quasi-steady flow is quickly disturbed by the reflected wave, finally, flow reaches steady and becomes critical without complex solutions. The results also indicate that Bernoulli function is discontinuous but the potential of mass flux remains constant across the jump. The latter can be used to predict velocity/height in a steady flow.

Role of Interaction between Magnetic Rossby Waves and Tachocline Differential Rotation in Producing Solar Seasons

Abstract We present a nonlinear magnetohydrodynamic shallow-water model for the solar tachocline (MHD-SWT) that generates quasi-periodic tachocline nonlinear oscillations (TNOs) that can be identified with the recently discovered solar “seasons.” We discuss the properties of the hydrodynamic and magnetohydrodynamic Rossby waves that interact with the differential rotation and toroidal fields to sustain these oscillations, which occur due to back-and-forth energy exchanges among potential, kinetic, and magnetic energies. We perform model simulations for a few years, for selected example cases, in both hydrodynamic and magnetohydrodynamic regimes and show that the TNOs are robust features of the MHD-SWT model, occurring with periods of 2–20 months. We find that in certain cases multiple unstable shallow-water modes govern the dynamics, and TNO periods vary with time. In hydrodynamically governed TNOs, the energy exchange mechanism is simple, occurring between the Rossby waves and differential rotation. But in MHD cases, energy exchange becomes much more complex, involving energy flow among six energy reservoirs by means of eight different energy conversion processes. For toroidal magnetic bands of 5 and 35 kG peak amplitudes, both placed at 45° latitude and oppositely directed in north and south hemispheres, we show that the energy transfers responsible for TNO, as well as westward phase propagation, are evident in synoptic maps of the flow, magnetic field, and tachocline top-surface deformations. Nonlinear mode–mode interaction is particularly dramatic in the strong-field case. We also find that the TNO period increases with a decrease in rotation rate, implying that the younger Sun had more frequent seasons.

Optimal transport for variational data assimilation

Abstract. Usually data assimilation methods evaluate observation-model misfits using weighted L2 distances. However, it is not well suited when observed features are present in the model with position error. In this context, the Wasserstein distance stemming from optimal transport theory is more relevant.This paper proposes the adaptation of variational data assimilation for the use of such a measure. It provides a short introduction of optimal transport theory and discusses the importance of a proper choice of scalar product to compute the cost function gradient. It also extends the discussion to the way the descent is performed within the minimization process.These algorithmic changes are tested on a nonlinear shallow-water model, leading to the conclusion that optimal transport-based data assimilation seems to be promising to capture position errors in the model trajectory.

Asymptotic stability of a polar vortex perturbed by harmonic waves describing atmospheric gravity waves circulating in an equatorial plane of a spherical planet

We examine stability of the vortex that represents one particular class of exact solution of a a nonlinear shallow water model describing atmospheric gravity waves circulating in an equatorial plane of a spherical planet. The mathematical model is represented by a two-dimensional free boundary Cauchy–Poisson problem on the nonstationary motion of a perfect uid around a solid circle with a sufficiently large radius so that the gravity is directed to the center of the circle. It is shown that the model admits two functionally independent nonlinear systems of shallow water equations. Two essential parameters that control stability of the vortex for both systems are identified. The order of their importance is analyzed and it is shown that one of the systems is more resistant to small perturbations and remains stable for larger range of these two parameters.

The 2014 Lake Askja rockslide‐induced tsunami: Optimization of numerical tsunami model using observed data

AbstractA large rockslide was released from the inner Askja caldera into Lake Askja, Iceland, on 21 July 2014. Upon entering the lake, it caused a large tsunami that traveled about ∼3 km across the lake and inundated the shore with vertical runup measuring up to 60–80 m. Following the event, comprehensive field data were collected, including GPS measurements of the inundation and multibeam echo soundings of the lake bathymetry. Using this exhaustive data set, numerical modeling of the tsunami has been conducted using both a nonlinear shallow water model and a Boussinesq‐type model that includes frequency dispersion. To constrain unknown landslide parameters, a global optimization algorithm, Differential Evolution, was employed, resulting in a parameter set that minimized the deviation from measured inundation. The tsunami model of Lake Askja is the first example where we have been able to utilize field data to show that frequency dispersion is needed to explain the tsunami wave radiation pattern and that shallow water theory falls short. We were able to fit the trend in tsunami runup observations around the entire lake using the Boussinesq model. In contrast, the shallow water model gave a different runup pattern and produced pronounced offsets in certain areas. The well‐documented Lake Askja tsunami thus provided a unique opportunity to explore and capture the essential physics of landslide tsunami generation and propagation through numerical modeling. Moreover, the study of the event is important because this dispersive nature is likely to occur for other subaerial impact tsunamis.

A Boussinesq type extension of the GeoClaw model - a study of wave breaking phenomena applying dispersive long wave models

The nonlinear shallow water model is widely used in the study of tsunami propagation, but an increasing number of studies are dedicated to the dispersion dynamics of tsunamis. If the wave dispersion becomes important, Boussinesq-type models are often used. In this work, a general purpose Boussinesq solver, BoussClaw, is introduced for modeling non-linear dispersive tsunami propagation, taking into account inundation. The BoussClaw model is an extension of the GeoClaw tsunami model. It employs a hybrid of finite volume and finite difference methods to solve Boussinesq equations from the literature, which are based on the depth-averaged velocity and include enhanced dispersion properties. On the other hand, in the selected formulation only some non-linearity is retained in the dispersion term. In order to validate BoussClaw, numerical results are compared to analytic solutions, solutions obtained by pre-existing models, and laboratory experiments. Even though the equations of BoussClaw are not fully nonlinear they perform far better than standard Boussinesq equations with only linear dispersion terms. Furthermore, the wave steepening and breaking are carefully scrutinized, and we demonstrate that the point of wave breaking may be wrongly identified in many of the commonly used Boussinesq models.

Numerical investigation of surge–tide interactions in the Bay of Bengal along the Bangladesh coast

The tidal amplitude at the Bay of Bengal along the Bangladesh coast varies from 2 to 7 m, whereas the tidal phase differs by more than 3 h from the outer part toward the inner part of the Meghna Estuary. At the time of cyclone landfall, the tidal phase has a significant impact on surge height, arrival time and duration. In this paper, the nature of surge–tide interactions at the Bay of Bengal along the Bangladesh coast has been investigated using a two-dimensional nonlinear shallow water model coupled with a storm surge model. Two previous major storm surge events along the Bangladesh coast have been analyzed for surge–tide interaction: Cyclone 1991 and Cyclone Sidr in 2007. To investigate the impact of the tidal phase on surge characteristics, in addition to the base cases, storm surges have been calculated at different phases of an entire tidal cycle at 1-h intervals. The results indicate that during the ebb tides the calculated total surge height of the surge–tide interaction model was higher compared to the total surge height obtained by superimposing surge height that is separately calculated with mean water level over the local tidal water level; the opposite occurs during high tide. The surge propagation time toward the inner part of the bay significantly varied depending on the tidal phases. When the tidal water level along the coast was low, the speed of surge propagation was much slower compared with when the tidal water level was high. The surge duration increases when the falling surge profile encounters an opposing strong tidal current at the landfall location.

Tsunami Simulation by 3D Model Around a Power Station Due to the 2011 Tohoku Earthquake

In the 2011 Tohoku Earthquake, many structures were destroyed by the tsunami whose magnitudes were much larger than the design level. Tsunami defense structures in coastal areas will need reinforcement in the future, and for that reason it is important to evaluate tsunami behavior and resulting tsunami force. This study analyzes tsunami transformation around a Haramachi thermal power station which suffered serious damage by the tsunami due to the 2011 Tohoku Earthquake and which has complex topography and building layout. A three-dimensional (3D) tsunami simulation is carried out for a surrounding region of the Haramachi thermal power station using boundary conditions of the water surface elevation and flow velocity obtained from a two-dimensional (2D) tsunami simulation by a nonlinear shallow water model. Using these boundary conditions, the computer fluid dynamic model FLUENT is employed to simulate the tsunami behavior around the Haramachi thermal power station. The validity of the predictions is examined by comparison with actual traces of tsunami inundation depths (I.D.).

Seasonality of the Structure and Propagation Characteristics of the MJO

Abstract The seasonality of the Madden–Julian oscillation (MJO) is documented in observational data, and a nonlinear shallow-water model is used to help interpret some of the contrasts in MJO structure between the boreal winter season [November–March (NDJFM)] and the Asian summer monsoon period [June–September (JJAS)]. At upper-tropospheric levels, the flanking Rossby waves remain centered around 28°N/S year-round, but they tend to be stronger in the winter hemisphere, where the climatological-mean jet stream is stronger, rendering the subtropical circulation more sensitive to forcing by a near-equatorial heat source. Amplitudes of the MJO-related deep convection and lower-tropospheric zonal wind are stronger in the summer hemisphere, where the column-integrated water vapor is larger. During NDJFM, the equatorial asymmetry is subtle: as in the annual mean, moisture convergence into swallowtail-shaped regions of enhanced deep convection is an integral part of the equatorial Rossby wave signature, and the eastward propagation is due to moistening of the air to the east of the enhanced convection by poleward moisture advection. During the Asian summer monsoon in JJAS, the convection assumes the form of northward-propagating, west-northwest–east-southeast-oriented rainbands embedded within cyclonic shear lines. These features are maintained by frictional convergence of moisture, and their northward propagation is mainly due to the presence of features in the climatological-mean fields: that is, the west–east moisture gradient over India and the Arabian Sea and the southwesterly low-level monsoon flow over the northwest Pacific.

A numerical investigation of the 1783 landslide-induced catastrophic tsunami in Scilla, Italy

The seismic crisis that hit Calabria (South Italy) since the beginning of 1783, lasted 3 years with strong earthquakes concentrated in the first 2 months, causing many fatalities and great destruction, as reported by several detailed historical reports. Following an earthquake that occurred on February 6, 1783, a mass with volume of about 6 Mm3 detached from the flank of Mount Paci and collapsed into the sea. In 1 min, huge waves attacked the nearby village of Scilla and killed most of the people that had gathered in the wide shore named Marina Grande to escape from seismic danger. More than 1500 people died due to the tsunami, many buildings were destroyed. In the paper, such catastrophic event is studied by means of numerical models. First we reconstruct the initial sliding mass and compute its motion through a Lagrangian model; then, we simulate the tsunami generation, propagation and flooding on the coast of Scilla by means of a nonlinear shallow-water equations model. The results of the simulation are compared against the detailed description of the tsunami effects available from several historical accounts and are found to be in a very satisfactory agreement with the 6–8 m inundation height reported in the Marina Grande beach.