Shallow water theory for structured bathymetry

A shallow water theory is developed which applies to surface wave propagation over structured bathymetry comprising rapid abrupt fluctuations in depth between two smoothly varying levels. Using a homogenization approach coupled to the depth-averaging process which underpins shallow water modelling, governing equations for the wave elevation are derived which explicitly relate local spatially varying anisotropy of wave speeds to properties of the microstructured bed. The model is applied to two water wave scattering problems both to demonstrate the complex wave propagation characteristics exhibited by structured beds and to provide examples of how to use structured beds to engineer bespoke wave propagation. This includes propagating waves with practically zero reflection and loss of form through circular bends in channels.

2-Day Westward-Propagating Inertio-Gravity Waves during GoAmazon

The dynamical and thermodynamical features of Amazonian 2-day westward-propagating inertia-gravity waves (WIG) are examined. On the basis of a linear regression analysis of satellite brightness temperature and data from the 2014-15 Observations and Modeling of the Green Ocean Amazon (GoAmazon) field campaign, it is shown that Amazonian WIG waves exhibit structure and propagation characteristics consistent with the n=1 WIG waves from shallow water theory. These WIG waves exhibit a pronounced seasonality, with peak activity occurring from March to May and a minimum occurring from June to September. Evidence is shown that mesoscale convective systems over the Amazon are frequently organized in 2-day WIG waves. Results suggest that many of the Amazonian WIG waves come from pre-existing 2-day waves over the Atlantic, which slow down when coupled with the deeper, more intense convection over tropical South America. In contrast to WIG waves that occur over the ocean, Amazonian 2-day WIG waves exhibit a pronounced signature in surface temperature, moisture, and heat fluxes.

Admissibility conditions for Riemann data in shallow water theory

Consideration is given to the shallow-water equations, a hyperbolic system modeling the propagation of long waves at the surface of an incompressible inviscible fluid of constant depth. It is well known that the solution of the Riemann problem associated to this system may feature dry states for some configurations of the Riemann data. This article will discuss various scenarios in which the Riemann problem for the shallow water system arises in a physically reasonable sense. In particular, it will be shown that if certain physical assumptions on the disposition of the Riemann data are made, then dry states can be avoided in the solution of the Riemann problem.

Runup of long waves on composite coastal slopes: numerical simulations and experiment

<p>The goal of this study is to investigate the effect of the bottom shape on wave runup. The obtained results have been confronted with available analytical predictions and a dedicated numerical simulation campaign has been carried out by the team. We study long wave runup on composite coastal profiles. Two types of beach profiles are considered. The Coastal Slope 1 consists of two merged plane beaches with lengths 1.2 m and 5 m and beach slopes tan α = 1:10 and tan β = 1:15 respectively. The Coastal Slope 2 also consists of two sections: plane beach with length 1.2 m and a beach slope α, which is merged with a convex (non-reflecting) beach. The latter is constructed in the way, that its total height and length remain the same as for the Coastal Slope 1.</p><p>The study is conducted with numerical (in silico) and experimental approaches.</p><p>Experiments have been conducted in the hydrodynamic flume of the Nizhny Novgorod State Technical University n.a. R.E. Alekseev. Both composite beach profiles were constructed in 2019. The Coastal Slope 1 consists of three parts made of aluminum. The plain beach part of the Coastal Slope 2 is also made of aluminum, and the convex profile consists of two parts made of curved PLEXIGLAS organic glass. The water surface oscillations are measured using capacitive and resistive wave gauges with recording frequencies of up to 80 Hz and 100 Hz respectively. Wave runup is measured by a capacitive string sensor installed along the slope.</p><p>A series of experiments on the generation and runup of regular wave trains with a period of 1s, 2s, 3s and 4s were carried out. The water level was kept constant for all experiments and was equal to 0.3 meters. Up to now, 21 experiments have been carried out (10 and 11 experiments for each Coastal Slope respectively).</p><p>A comparative numerical study is carried out in the framework of the nonlinear shallow water theory and the dispersive theory in the Boussinesq approximation.</p><p>As a result, we compare the long wave dynamics on these two bottom profiles and discuss the influence of nonlinearity and dispersion on the characteristics of wave runup. It is shown numerically that, in the framework of the nonlinear shallow water theory, the runup height on the Coastal Slope 2 tends to exceed the corresponding runup height on the Coastal Slope 1, that also agrees with our previous results (Didenkulova et al. 2009; Didenkulova et al. 2018). Taking dispersion into account leads to an increase in the spread in values of the wave runup height. As a consequence, individual cases when the runup height on the Coastal Slope 1 is higher than on the Coastal Slope 2 have been observed. In experimental data, such cases occur more often, so that the advantage of one slope over another is no longer obvious. Note also that the most nonlinear breaking waves with a period of 1s have a greater runup height on Coastal Slope 2 for both models and most experimental data.</p>

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

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.

Run-up of nonlinear monochromatic wave on a plane beach in presence of a tide

The nonlinear problem of long wave run-up on a plane beach in a presence of a tide is solved within the shallow water theory using the Carrier-Greenspan approach. The exact solution of the nonlinear problem for wave run-up height is found as a function of the incident wave amplitude. Influence of tide on characteristics of wave run-up on a beach is studied.