Nonlinear Shallow-water Theory Research Articles

Global propagation of air pressure waves and consequent ocean waves due to the January 2022 Hunga Tonga-Hunga Ha'apai eruption

On January 15, 2022, the Hunga Tonga-Hunga Ha'apai volcano's huge eruption generated tsunami waves globally due to an unusual source in this unique event: fast-traveling air pressure disturbances originated from the explosion of the volcano. Here we investigate the global propagation of the pressure waves (first cycle) and the consequent ocean waves. The analysis of the measured air pressure waves shows that the speed of air pressure linearly increases with time. The peak and trough amplitudes of the pressure wave exponentially decrease with the traveled distance until half the overall distance; then, it restarts to increase, reaching a peak towards the apogee. Two different modeling approaches are followed to solve the air pressure waves and the resulting ocean waves: i) a synthetic pressure forcing model based on barometric measurements and ii) a numerical model based on nonlinear shallow water theory using an initial disturbance at the volcano. We forced the hydrodynamic model by the produced pressure fields to compute the resulting tsunami waves globally. We present the modeling results, discussing the possible sea level amplification mechanisms and affecting factors. Fairly well agreement between the computed and measured air pressure waves and sea levels around the globe is promising.

The Use of the NAMI-DANCE Computational Complex on the Problem of Tsunami Waves

Mathematical models that are applicable to the simulation of the generation and propagation of tsunami waves from different sources, that is, underwater earthquakes, submarine landslides, and meteotsunamis, are described. The models are based on the well-known nonlinear shallow-water theory and its dispersion generalizations in two horizontal dimensions. Long wave dispersion related to the finiteness of water depth increases the order of the initial equations and, as a consequence, the time of computation. For this reason, physical dispersion in the investigations presented in this paper is replaced by numerical dispersion owing to a special choice of spatial and temporal steps. The numerical scheme for solving the shallow water equations is based on the leapfrog method. The equations are solved in spherical coordinates fixed to the rotating Earth with allowance for dissipative effects in the near-bottom layer with the use of the developed NAMI-DANCE code. For waves of seismic origin, the initial conditions for solving the hydrodynamic equations are taken from the solution of an elasticity theory problem to describe earthquake evolution (the Okada solution). In the case of a meteotsunami, atmospheric factors are taken into account as external forces and included in the right-hand sides of shallow water equations. Finally, generation of tsunami waves by submarine landslides is considered within the framework of a two-layer model with a lower viscous layer modeling the motion of the landslide. Two kinds of boundary conditions are used: free wave passage through open boundaries (in straits, etc.) using linear shallow-water equations and full reflection on the coast or in the near-shore area. Some test problems (benchmarks) on which the computational complex was verified are listed. The difficulties in the representation of tsunami characteristics due to bad data on the bottom bathymetry and topography of the land are mentioned. The possibility of using the developed code for the analysis of tsunami action on coasts and constructions is also discussed.

Comparison of Dispersive and Nondispersive Models for Wave Run-Up on a Beach

The applicability of dispersive and nondispersive wave models for describing long-wave propagation and run-up on a beach in the case of composite bottom topography is investigated: a plane beach transforms into a zone of constant depth. Numerical simulations are performed in the framework of two models: (1) nonlinear shallow-water theory and (2) the dispersive model in the Boussinesq approximation based on modified Peregrine equations. Simulations are compared with the data of a laboratory experiment for different types of waves: regular waves, biharmonic signals, and “vessel-like” wave trains strongly modulated by frequency and amplitude. Conclusions about the applicability of the corresponding theories for describing considered types of waves are drawn based on this comparison.

Tsunami Run-Up on a Plane Beach in a Tidal Environment

We analytically solve the nonlinear shallow water theory for the tsunami run-up on a plane beach in the presence of tide and show that over a plane beach the tide in the nearshore zone can be considered static (uniform in space and frozen in time). With this assumption, we find an exact analytical solution for the tsunami run-up height as a function of the amplitude of the incident wave and analyse the influence of the tide on the tsunami run-up characteristics. We confirm these results by calculation of the total tsunami and tidal run-up field using analytical solutions of the initial nonlinear shallow water equations.

Nonlinear deformation and run-up of single tsunami waves of positive polarity: numerical simulations and analytical predictions

Abstract. The estimate of an individual wave run-up is especially important for tsunami warning and risk assessment, as it allows for evaluating the inundation area. Here, as a model of tsunamis, we use the long single wave of positive polarity. The period of such a wave is rather long, which makes it different from the famous Korteweg–de Vries soliton. This wave nonlinearly deforms during its propagation in the ocean, which results in a steep wave front formation. Situations in which waves approach the coast with a steep front are often observed during large tsunamis, e.g. the 2004 Indian Ocean and 2011 Tohoku tsunamis. Here we study the nonlinear deformation and run-up of long single waves of positive polarity in the conjoined water basin, which consists of the constant depth section and a plane beach. The work is performed numerically and analytically in the framework of the nonlinear shallow-water theory. Analytically, wave propagation along the constant depth section and its run up on a beach are considered independently without taking into account wave interaction with the toe of the bottom slope. The propagation along the bottom of constant depth is described by the Riemann wave, while the wave run-up on a plane beach is calculated using rigorous analytical solutions of the nonlinear shallow-water theory following the Carrier–Greenspan approach. Numerically, we use the finite-volume method with the second-order UNO2 reconstruction in space and the third-order Runge–Kutta scheme with locally adaptive time steps. During wave propagation along the constant depth section, the wave becomes asymmetric with a steep wave front. It is shown that the maximum run-up height depends on the front steepness of the incoming wave approaching the toe of the bottom slope. The corresponding formula for maximum run-up height, which takes into account the wave front steepness, is proposed.

Probabilistic characteristics of narrow-band long-wave run-up onshore

Abstract. The run-up of random long-wave ensemble (swell, storm surge, and tsunami) on the constant-slope beach is studied in the framework of the nonlinear shallow-water theory in the approximation of non-breaking waves. If the incident wave approaches the shore from the deepest water, run-up characteristics can be found in two stages: in the first stage, linear equations are solved and the wave characteristics at the fixed (undisturbed) shoreline are found, and in the second stage the nonlinear dynamics of the moving shoreline is studied by means of the Riemann (nonlinear) transformation of linear solutions. In this paper, detailed results are obtained for quasi-harmonic (narrow-band) waves with random amplitude and phase. It is shown that the probabilistic characteristics of the run-up extremes can be found from the linear theory, while the same ones of the moving shoreline are from the nonlinear theory. The role of wave-breaking due to large-amplitude outliers is discussed, so that it becomes necessary to consider wave ensembles with non-Gaussian statistics within the framework of the analytical theory of non-breaking waves. The basic formulas for calculating the probabilistic characteristics of the moving shoreline and its velocity through the incident wave characteristics are given. They can be used for estimates of the flooding zone characteristics in marine natural hazards.

Evolution of wave pulses in fully nonlinear shallow-water theory

We consider evolution of wave pulses with formation of dispersive shock waves in framework of fully nonlinear shallow-water equations. Situations of initial elevations or initial dips on the water surface are treated, and motion of the dispersive shock edges is studied within the Whitham theory of modulations. Simple analytical formulas are obtained for asymptotic stage of evolution of initially localized pulses. Analytical results are confirmed by exact numerical solutions of the fully nonlinear shallow-water equations.

Computational NAMI-DANCE complex in the problem of tsunami waves

Mathematical models applicable to the simulation of generation and propagation of tsunamis of different origin - underwater earthquakes, submarine landslides, meteotsunami - are described. The basic model is based on the well-known nonlinear shallow-water theory and its dispersion generalizations in two horizontal dimensions. Long wave dispersion related to the finiteness of water depth increases the order of PDE and leads to serious computational problems. We replace the physical dispersion with the numerical one using specific conditions for spatial and temporal steps. The numerical scheme is based on a Leap-Frog method. The equations are solved in spherical coordinates fixed to the rotating Earth by taking into account dissipative effects in the near-bottom layer with the use of NAMI-DANCE code. For tsunami waves of seismic origin, the initial conditions for hydrodynamic equations are found from the Okada solution of elastic equations describing the development of an earthquake. In the case of meteotsunami, atmospheric factors are modeled by the external forces. The process of generation of tsunami waves of landslide origin is analyzed in the framework of a two-layer model with lower viscous layer modeling the motion of a submarine landslide. There are two kinds of boundary conditions: free wave passage through open boundaries (in straits, etc.) using linear shallow-water equations, and a full reflection on the coast or in near-shore area. The NAMI-DANCE code has been verified with several benchmarks. The difficulties of tsunami modelling associated with the lack of accuracy in the bottom bathymetry and land topography are mentioned. The use of the developed code for the analysis of tsunami action on the coasts and constructions is discussed.

Long Wave Run-up on Plane and “Non-Reflecting” Slopes

The long wave run-up on two types of slopes is investigated numerically within the framework of nonlinear shallow water theory using the CLAWPACK software. One of the slopes represents a plane slope widely used in the laboratory and numerical experiments; the second is the so-called “non-reflecting” slope (h ~ x4/3, where h is the basin depth and x is the distance from the shoreline). In the case of very low wave amplitudes when there is no wave breaking, the run-up height is greater on the non-reflecting beach than that on the plane slope. As the wave amplitude increases, the breaking effects have the stronger impact in the case of non-reflecting beach and the run-up height becomes smaller.

Numerical modelling of flow in Little Pigeon Bay due to the 2016 Kaikoura tsunami

The magnitude 7.8 Kaikoura earthquake on the north-eastern coast of South Island, New Zealand induced a tsunami, which affected many north-facing bays in Banks Peninsula. The V-shaped Little Pigeon Bay experienced pronounced inundation compared to adjacent bays, causing heavy damage to a cottage at the head of the bay. In order to investigate the characteristics of the tsunami flow in the bay, a numerical model based on nonlinear shallow water theory was established, using tidal levels from the neighbouring Sumner sea level gauge. Three incident wave angles were tested, ranging from due north to 30° east of north. The simulated maximum run-up heights were validated with surveyed results. For all the wave directions, a resonance with a period of 6.3 min was found from 50 to 100 min in the simulation. Based on the physical dimensions of Little Pigeon Bay, idealised models were set-up to analyse the influence of V-shaped embayment on maximum water level, maximum velocity and maximum momentum flux. Bays with sand bars of different crest heights were also investigated. A sand bar with a relatively low crest height (dw/d≥0.5) did not significantly reduce the maximum velocity and the maximum momentum flux.

The run-up of nonlinearly deformed sea waves on the coast of a bay with a parabolic cross-section

The run-up of long waves on the coast of a bay with a parabolic cross-section, where the region of constant depth along the principal axis of the bay is connected with the linearly inclined segment, is considered. The study is carried out analytically in the framework of the nonlinear shallow-water theory under the approximation that the height of the initial wave is small compared to the basin depth, and the reflection from the inflection point of the bottom is negligibly small. Three types of incident waves, viz., a sinusoidal wave and solitary waves of positive and negative polarities, are considered in detail. It is shown that a sinusoidal wave undergoes nonlinear deformation at a segment of constant depth faster than solitary waves of positive and negative polarities. Solitary waves of negative polarity steepen somewhat faster than solitary waves of positive polarity. Waves of positive polarity steepen at wave front, while waves of negative polarity steepen at wave rear. These differences in steepness may become crucial at the wave run-up stage, since the wave run-up height on the coast of a bay with a parabolic cross-section is directly proportional to the steepness of a wave that arrives at the slope and can lead to the anomalous run-up of waves on the coast.

Run-up of surface waves on a sea wall built on a convex bottom profile

Wave run-up on a sea wall built on a convex bottom profile is studied in the framework of linear shallow water theory. When the wall is located in “deeper water,” a wave is reflected from the wall without changing its shape and phase, which is fully consistent with classical considerations. If the wall is shifted towards the shore, the shape of the wave changes in a complex way. Note that the wave phase changes to the opposite in the limiting case when the wall is located right on the shore. The role of nonlinear effects is studied by means of numerical simulations using nonlinear shallow water theory. It is shown that the contribution of nonlinear effects and breaking is high on a convex-shaped beach, which makes the structure of the wave field rather complicated.

An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays

By assuming the flow is uniform along the narrow long bays, the 2-D nonlinear shallow-water equations are reduced to a linear semi-axis variable-coefficient 1-D wave equation via the generalized Carrier–Greenspan transformation. The run-up of long waves in constantly sloping U-shaped and V-shaped bays is studied both analytically and numerically within the framework of the 1-D nonlinear shallow-water theory. An analytic solution, in the form of a double integral, to the resulting linear wave equation is obtained by utilizing the Hankel transform, and consequently the solution to the tsunami run-up problem is developed by applying the inverse generalized Carrier–Greenspan transform. The presented solution is a generalization of the solutions found by Carrier et al. (2003) and Didenkulova and Pelinovsky (2011) for the case of a plane beach and a parabolic bay, respectively. The shoreline dynamics in U-shaped and V-shaped bays are computed via a double integral through standard integration techniques.

Run-up of nonlinear long waves in U-shaped bays of finite length: analytical theory and numerical computations

Run-up of long waves in sloping U-shaped bays is studied analytically in the framework of the 1-D nonlinear shallow water theory. By assuming that the wave flow is uniform along the cross section, the 2-D nonlinear shallow water equations are reduced to a linear semi-axis variable-coefficient 1-D wave equation via the generalized Carrier-Greenspan transformation (Rybkin et al. in J Fluid Mech 748:416–432, 2014). A spectral solution is developed by solving the linear semi-axis variable-coefficient 1-D equation via separation of variables and then applying the inverse Carrier-Greenspan transform. To compute the run-up of a given long wave, a numerical method is developed to find the eigenfunction decomposition required for the spectral solution in the linearized system. The run-up of a long wave in a bathymetry characteristic of a narrow canyon is then examined.

Influence of the shape of the bay cross section on wave run-up

This article compares the characteristics of wave run up on a shore in sloped bays of arbitrary section as calculated in terms of strict solutions of nonlinear shallow water theory. The solitary waves of the same amplitude, “significant” duration, and different shapes are considered. For each geometry of a bay, parameterization of formulas is performed in order to eliminate the influence of the shape of waves approaching the shore on these formulas describing extreme wave run-up characteristics. The results obtained earlier for a plane beach and a parabolic beach are reproduced, and the new bay shapes (with a triangular cross section and with a cross-section profile varying proportionately to the coordinate in fourth and tenth degree) are considered. It is shown that in narrower bays the maximal run-up and run-down heights and the respective run-up and run-down velocities are higher than those in wider bays. At a run up of a solitary wave in a parabolic bay, run-up and run-down heights coincide, whereas in wider bays the run-up height exceeds the run-down height and in narrower bays, for example, in triangular ones, the opposite pattern (run-down height exceeds run-up height) is observed.

Runup of Nonlinear Long Waves in Trapezoidal Bays: 1-D Analytical Theory and 2-D Numerical Computations

Long nonlinear wave runup on the coasts of trapezoidal bays is studied analytically in the framework of one-dimensional (1-D) nonlinear shallow-water theory with cross-section averaging, and is also studied numerically within a two-dimensional (2-D) nonlinear shallow water theory. In the 1-D theory, it is assumed that the trapezoidal cross-section channel is inclined linearly to the horizon, and that the wave flow is uniform in the cross-section. As a result, 1-D nonlinear shallow-water equations are reduced to a linear, semi-axis variable-coefficient 1-D wave equation by using the generalized Carrier–Greenspan transformation [Carrier and Greenspan (J Fluid Mech 1:97–109, 1958)] recently developed for arbitrary cross-section channels [Rybkinet al. (Ocean Model 43–44:36–51, 2014)], and all characteristics of the wave field can be expressed by implicit formulas. For detailed computations of the long wave runup process, a robust and effective finite difference scheme is applied. The numerical method is verified on a known analytical solution for wave runup on the coasts of an inclined parabolic bay. The predictions of the 1-D model are compared with results of direct numerical simulations of inundations caused by tsunamis in narrow bays with real bathymetries.

Numerical simulation of the propagation of ship-induced Riemann waves of depression into the Venice Lagoon

Large in situ measured ship-induced depression waves (Bernoulli wakes) in the Malamocco-Marghera industrial channel of the Venice Lagoon are interpreted as long-living strongly nonlinear Riemann (simple) waves of depression. The properties of these depressions are numerically replicated using nonlinear shallow water theory and the CLAWPACK software. The further behaviour of measured depressions is analysed by means of replicating the vessel-induced disturbances with the propagation of initially smooth free waves. It is demonstrated that vessel-driven depressions of substantial height (> 0.3 m) often propagate for more than 1 km from the navigation channel into areas of the lagoon of approximately 2 m water depth. As a depression wave propagates into the lagoon, its front slope becomes gradually less steep, but the rear slope preserves an extremely steep bore-like appearance and the amplitude becomes almost independent of the initial properties of the disturbance. Analysis suggests that even modest ships in terms of their size, sailing speed, and blocking coefficient may generate deep depressions that travel as compact and steep entities resembling asymmetric solitary waves over substantial distances into shallow water adjacent to navigation channels. Their impact may substantially increase the environmental impact of ship wakes in this and similar water bodies.

LONG WAVE PROPAGATION, SHOALING AND RUN-UP IN NEARSHORE AREAS

This paper discusses the possibility to study propagation, shoaling and run-up of these waves over a slope in a 300- meter long large wave flume (GWK), Hannover. For this purpose long bell-shaped solitary waves (elongated solitons) of different amplitude and the same period of 30 s are generated. Experimental data of long wave propagation in the flume are compared with numerical simulations performed within the fully nonlinear potential flow theory and KdV equations. Shoaling and run-up of waves on different mild slopes is studied hypothetically using nonlinear shallow water theory. Conclusions about the feasibility of using large scale experimental facility (GWK) to study tsunami wave propagation and run-up are made.

Effective coastal boundary conditions for tsunami wave run-up over sloping bathymetry

Abstract. An effective boundary condition (EBC) is introduced as a novel technique for predicting tsunami wave run-up along the coast, and offshore wave reflections. Numerical modeling of tsunami propagation in the coastal zone has been a daunting task, since high accuracy is needed to capture aspects of wave propagation in the shallower areas. For example, there are complicated interactions between incoming and reflected waves due to the bathymetry and intrinsically nonlinear phenomena of wave propagation. If a fixed wall boundary condition is used at a certain shallow depth contour, the reflection properties can be unrealistic. To alleviate this, we explore a so-called effective boundary condition, developed here in one spatial dimension. From the deep ocean to a seaward boundary, i.e., in the simulation area, we model wave propagation numerically over real bathymetry using either the linear dispersive variational Boussinesq or the shallow water equations. We measure the incoming wave at this seaward boundary, and model the wave dynamics towards the shoreline analytically, based on nonlinear shallow water theory over bathymetry with a constant slope. We calculate the run-up heights at the shore and the reflection caused by the slope. The reflected wave is then influxed back into the simulation area using the EBC. The coupling between the numerical and analytic dynamics in the two areas is handled using variational principles, which leads to (approximate) conservation of the overall energy in both areas. We verify our approach in a series of numerical test cases of increasing complexity, including a case akin to tsunami propagation to the coastline at Aceh, Sumatra, Indonesia.

Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier–Greenspan approach

AbstractWe present an exact analytical solution of the nonlinear shallow water theory for wave run-up in inclined channels of arbitrary cross-section, which generalizes previous studies on wave run-up for a plane beach and channels of parabolic cross-section. The solution is found using a hodograph-type transform, which extends the well-known Carrier–Greenspan transform for wave run-up on a plane beach. As a result, the nonlinear shallow water equations are reduced to a single one-dimensional linear wave equation for an auxiliary function and all physical variables can be expressed in terms of this function by purely algebraic formulas. In the special case of a U-shaped channel this equation coincides with a spherically symmetric wave equation in space, whose dimension is defined by the channel cross-section and can be fractional. As an example, the run-up of a sinusoidal wave on a beach is considered for channels of several different cross-sections and the influence of the cross-section on wave run-up characteristics is studied.