Shallow Water Theory Research Articles

Linear waves in shallow water over an uneven bottom, slowing down near the shore

The exact solutions of the system of equations of the linear theory of shallow water are discussed, representing traveling waves with specific properties for the time propagation, which is infinite when approaching the shore and finite when leaving for deep water. These solutions are obtained by reducing one-dimensional shallow water equations to the Euler–Poisson–Darboux equation with a negative integer coefficient before the lower derivative. The analysis of the wave field dynamics is carried out. It is shown that the shape of a wave approaching the shore will be differentiated a certain number of times, which is illustrated by a number of examples. When a wave moves away from the shore, its profile is integrated. The solutions obtained in the framework of linear theory are valid only for a finite interval of depth variation.

Novel Atmospheric Dynamics Shape the Inner Edge of the Habitable Zone around White Dwarfs

White dwarfs offer a unique opportunity to search nearby stellar systems for signs of life, but the habitable zone around these stars is still poorly understood. Since white dwarfs are compact stars with low luminosity, any planets in their habitable zone should be tidally locked, like planets around M dwarfs. Unlike planets around M dwarfs, however, habitable white dwarf planets have to rotate very rapidly, with orbital periods ranging from hours to several days. Here we use the ExoCAM global climate model to investigate the inner edge of the habitable zone around white dwarfs. Our simulations show habitable planets with ultrashort orbital periods (P ≲ 1 day) enter a “bat rotation” regime, which differs from typical atmospheric circulation regimes around M dwarfs. Bat rotators feature mean equatorial subrotation and a displacement of the surface’s hottest regions from the equator toward the midlatitudes. We qualitatively explain the onset of bat rotation using shallow water theory. The resulting circulation shifts increase the dayside cloud cover and decrease the stratospheric water vapor, expanding the white dwarf habitable zone by ∼50% compared to estimates based on 1D models. The James Webb Space Telescope should be able to quickly characterize bat rotators around nearby white dwarfs thanks to their distinct thermal phase curves. Our work underlines that tidally locked planets on ultrashort orbits may exhibit unique atmospheric dynamics, and guides future habitability studies of white dwarf systems.

Wave propagation over a non-reflective profile of limited depth

Non-reflective wave propagation is of great importance for applications because it allows energy to be transmitted over long distances. The paper discusses the method of reducing the equations of the linear theory of shallow water to a wave equation with a variable coefficient in the form of an inverse hyperbolic sine, the solution of which is represented as a composition of traveling waves. Thanks to this, a new non-reflective bottom profile has been obtained, which reaches a constant at infinity. Wave behavior on the shore is discussed, as well as the conditions under which the wave field remains finite on it. A detailed analysis of the obtained exact solution to the shallow water equations is given in the paper.

Wave Propagation and Bottom Deformation in the Coastal Zone of the Sea Within the Framework of the Nonlinear Dispersion Theory of Shallow Water

Wave Propagation and Bottom Deformation in the Coastal Zone of the Sea Within the Framework of the Nonlinear Dispersion Theory of Shallow Water

The Relationship between Convectively Coupled Waves and the East Pacific ITCZ

Abstract Longstanding climate model biases in tropical precipitation exist over the east Pacific (EP) Ocean, especially during boreal winter and spring when models have excessive Southern Hemisphere (SH) precipitation near the intertropical convergence zone (ITCZ). In this study, we document the impact of convectively coupled waves (CCWs) on EP precipitation and the ITCZ using observations and reanalyses. We focus on the months when SH precipitation peaks in observations: February–April (FMA). CCWs explain 93% of total precipitation variance in the SH, nearly double the percent (48%) of the NH during FMA. However, we note that these percentages are inflated as they inevitably include the background variance. We further investigate the three leading high-frequency wave bands: mixed Rossby–gravity waves and tropical depression–type disturbances (MRG–TD type), Kelvin waves, and n = 0 eastward inertia–gravity waves (IG0). Compared to their warm pool counterparts, these three CCWs have a more zonally elongated and meridionally narrower precipitation structure with circulations that resemble past observational studies and/or shallow water theory. We quantify the contribution of all CCWs to four different daily ITCZ “states”: Northern Hemisphere (NH) (nITCZ), SH (sITCZ), double (dITCZ), and equatorial (eITCZ) using a new precipitation-based ITCZ-state algorithm. We find that the percent of total precipitation variance explained by each of the CCWs is heightened for sITCZs and eITCZs and diminished for nITCZs. Last, we find that nITCZs are most prevalent weeks after strong CCW activity happens in the NH, whereas CCWs and sITCZs peak simultaneously in the SH. Significance Statement Convectively coupled atmospheric waves (CCWs) are a critical feature of tropical weather and are an important source of precipitation near the region of highest precipitation on Earth called the intertropical convergence zone (ITCZ). Given three decades of climate model biases in CCWs and ITCZ precipitation over the east Pacific (EP) Ocean during spring, few studies have examined the relationship between CCWs and the springtime EP ITCZ. We explored the CCWs and EP ITCZ relationship through calculations of the percent of precipitation that comes from CCWs. A significant portion of the tropical precipitation is associated with CCWs during spring. CCWs are even more impactful when the ITCZ is in the SH or on the equator, which are both problematic in climate models.

The Radon and Hilbert transforms and their applications to atmospheric waves

AbstractThe Radon and Hilbert transform and their applications to convectively coupled waves (CCWs) are reviewed. The Hilbert Transform is used to compute the wave envelope, whereas the Radon transform is used to estimate the phase and group velocities of CCWs. Together, they provide an objective method to understand CCW propagation. Results reveal phase speeds and group velocities for fast waves (mixed Rossby‐gravity, westward and eastward inertio‐gravity, and Kelvin) that are consistent with previous studies and with Matsuno's equatorial wave dispersion curves. However, slowly‐propagating tropical depression‐like systems and equatorial Rossby waves exhibit wave envelopes that propagate faster than the individual wave crests, which is not predicted by dry shallow water theory.

Potential vorticity conservation for plasma turbulence in an inhomogeneous magnetic field: Theory and implications

The concept and theory of potential vorticity in drift wave turbulence are extended to the case of an inhomogeneous magnetic field. A one-field magnetic potential vorticity conserving equation is derived via the use of conservative gyrokinetics. The similarity between the corresponding systems for drift wave turbulence and shallow water theory is discussed in detail. Zonal flow physics in an inhomogeneous magnetic field is discussed. In particular, a Charney–Drazin type nonacceleration theorem is derived from the novel system, which conserves magnetic potential vorticity. Extensions of the turbulent equipartition theory to the transport of magnetic potential vorticity are proposed.

Momentum transfer across an open-channel, turbulent flow

The distribution of stress generated by a turbulent flow matters for many natural phenomena, of which rivers are a prime example. Here, we use dimensional analysis to derive a linear, second-order ordinary differential equation for the distribution of stress across a straight, open channel, with an arbitrary cross-sectional shape. We show that this equation is a generic first-order correction to the shallow-water theory in a channel of large aspect ratio. It has two adjustable parameters – the dimensionless diffusion parameter, $\chi$ , and a local-shape parameter, $\alpha$ . By assuming that the momentum is carried across the stream primarily by eddies and recirculation cells with a size comparable to the flow depth, we estimate $\chi$ to be of the order of the inverse square root of the friction coefficient, $\chi \sim C_f^{-1/2}$ , and predict that $\alpha$ vanishes when the flow is highly turbulent. We examine the properties of this equation in detail and confirm its applicability by comparing it with flume experiments and field measurements from the literature. This theory can be a basis for finding the equilibrium shape of turbulent rivers that carry sediment.

Transformation of a plasma boundary curvature into electrical impulses moving along a plasma surface

The self-consistent propagation of electrical impulses and of the accompanying distortions of the electron surface in the framework of a cold plasma model with a sharp boundary has been described with help of a derived system of two equations. The method of ‘shallow water theory’ has been applied for the case of bounded plasma and deriving an equation with which to link the spatial and temporal structures and evolution of the boundary curvature and the surface charge. Under certain conditions, such perturbations can propagate along the boundary without changing their shape for a long distance. An approximate analytical solution has been found, and numerical calculations have been performed. Mutual connections between basic parameters of the considered perturbations (velocity components, electrostatic field, etc.) have been presented.

Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation

Abstract Consideration herein is the stability issue of peaked solitary wave solution for the modified Camassa-Holm-Novikov equation, which is derived from the shallow water theory. This wave configuration accommodates the ordered trains of the modified Camassa-Holm-Novikov-peaked solitary solution. With the application of conservation laws and the monotonicity property of the localized energy functionals, we prove the orbital stability of this wave profile in the H 1 ( R ) {H}^{1}\left({\mathbb{R}}) energy space according to the modulation argument.

Planetary Boundary Layer Flow over Complex Terrain during a Cold Surge Event: A Case Study

Planetary boundary layer (PBL) flow over complex terrain during a cold surge event was investigated using 3-hourly radiosonde measurements in upwind, near ridge, and downwind locations of mountains in the northeastern part of Republic of Korea and using a high-resolution (333-m) numerical simulation. A cold surge occurred on 23 January 2018 and lasted for 4 days. We analyzed onset day of the cold surge when air temperature dropped rapidly. Analysis of the radiosonde data shows that the PBL was characterized by an adiabatic layer with strong capping inversion in early morning and evening as well as during daytime in the upwind and near-ridge sites. The PBL flow at the near-ridge site was strongest among three sites except at 0600 local standard time (LST) when the PBL flow in the lee was strongest. We performed high-resolution (333-m) numerical simulations using the Weather Research and Forecasting (WRF) model. The adiabatic PBL in the upwind site at 0600 LST was simulated, although its depth was underestimated. The model reproduced the strong low-level wind at 0600 LST and large wind shear during the daytime in the lee, but it did not capture the exact timing of the large wind shear. The model showed overall good performance in simulating the vertical profile of the virtual potential temperature and wind below 2 km above ground level at the three sites, with a high index of agreement (IOA) except for the wind at 1200 and 1500 LST in the lee. To examine the cause for the different behavior of PBL flow in the lee of mountains between 0600 LST and the daytime, we calculated the Froude number for PBL flow using radiosonde measurements based on reduced gravity shallow water (RGSW) theory. At 0600 LST, the upwind Froude number F0 was close to 1, while during the daytime, it was much lower than 1. The observed lee flow behavior was consistent with the flow regime change of a single layer over an obstacle with changing F0; the flow with a propagating lee jump changes into that with a stationary lee jump with decreasing F0. Numerical simulation shows that the steepening of streamlines of lee-wave field leads to a jump-like structure in the lee of mountains during the daytime.

Stratified inclined duct: two-layer hydraulics and instabilities

The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears to be linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms by (i) using recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu et al., J. Fluid Mech., vol. 969, 2023, A20), (ii) averaging these DNS into two layers, and (iii) using an inviscid two-layer shallow-water and instability theory to diagnose interfacial wave behaviour and provide physical insight. The laminar flow is subcritical and stable throughout the duct and hydraulically controlled at the ends of the duct. As the tilt is increased, the flow becomes supercritical everywhere and unstable to long waves. An internal jump featuring stationary waves first appears near the centre of the duct, then leads to larger-amplitude travelling waves, and to stronger jumps, wave breaking and intermittent turbulence at the largest tilt angle. Long waves described by the (nonlinear) shallow-water equation are interpreted locally as linear waves on a two-layer parallel base flow described by the Taylor–Goldstein equation. This link helps us to interpret long-wave instability and contrast it with short-wave (e.g. Kelvin–Helmholtz) instability. Our results suggest a transition to turbulence in SID through long-wave instability relying on vertical confinement by the top and bottom walls.

A novel analysis of Cole–Hopf transformations in different dimensions, solitons, and rogue waves for a (2 + 1)-dimensional shallow water wave equation of ion-acoustic waves in plasmas

This work investigates a (2 + 1)-dimensional shallow water wave equation of ion-acoustic waves in plasma physics. It comprehensively analyzes Cole–Hopf transformations concerning dimensions x, y, and t and obtains the dispersion for a phase variable of this equation. We show that the soliton solutions are independent of the different logarithmic transformations for the investigated equation. We also explore the linear equations in the auxiliary function f present in Cole–Hopf transformations. We study this equation's first- and second-order rogue waves using a generalized N-rogue wave expression from the N-soliton Hirota technique. We generate the rogue waves by applying a symbolic technique with β and γ as center parameters. We create rogue wave solutions for first- and second-order using direct computation for appropriate choices of several constants in the equation and center parameters. We obtain a trilinear equation by transforming variables ξ and y via logarithmic transformation for u in the function F. We harness the computational power of the symbolic tool Mathematica to demonstrate the graphics of the soliton and center-controlled rogue wave solutions with suitable choices of parameters. The outcomes of this study transcend the confines of plasma physics, shedding light on the interaction dynamics of ion-acoustic solitons in three-dimensional space. The equation's implications resonate across diverse scientific domains, encompassing classical shallow water theory, fluid dynamics, optical fibers, nonlinear dynamics, and many other nonlinear fields.

Solitary Wave Propagation of the Generalized Rosenau–Kawahara–RLW Equation in Shallow Water Theory with Surface Tension

This paper addresses a numerical approach for computing the solitary wave solutions of the generalized Rosenau–Kawahara–RLW model established by coupling the generalized Rosenau–Kawahara and Rosenau–RLW equations. The solution of this model is accomplished by using the finite difference approach and the upwind local radial basis functions-finite difference. Firstly, the PDE is transformed into a nonlinear ODEs system by means of the radial kernels. Secondly, a high-order ODE solver is implemented for discretizing the system of nonlinear ODEs. The main advantage of this technique is its lack of need for linearization. The global collocation techniques impose a significant computational cost, which arises from calculating the dense system of algebraic equations. The proposed technique estimates differential operators on every stencil. As a result, it produces sparse differentiation matrices and reduces the computational burden. Numerical experiments indicate that the method is precise and efficient for long-time simulation.

Exact solutions for the initial stage of dam-break flow on a plane hillside or beach

Inviscid, incompressible liquid is released from rest by a sudden dam break, accelerating under gravity over a uniformly sloping impermeable plane bed. The liquid flows downhill or up a beach. A linearised model is derived from Euler's equations for the early stage of motion, of duration $2\sqrt {H/g}$ , where H is the depth scale and $g$ is the acceleration due to gravity. Initial pressure and acceleration fields are calculated in closed form, first for an isosceles right-angled triangle on a slope of $45^{\circ }$ . Second, the triangle belongs to a class of finite-domain solutions with a curved front face. Third, an unbounded domain is treated, with a curved face resembling a steep-fronted breaking water wave flowing up a beach. The fluid goes uphill due to a nearshore pressure gradient. In all cases the free-surface-bed contact point is the most accelerated particle, exceeding the acceleration due to gravity. Physical consequences are discussed, and the pressure approximation of shallow water theory is found poor during this early stage, near the steep free surface exposed by a dam break.

A surface-piercing truncated cylindrical meta-structure operating as a wave energy converter

In this paper, we study the interaction of water waves with a surface-piercing truncated cylindrical meta-structure consisting of two overlapping arrays of closely spaced vertical thin plates. The fluid resonance promoted in the narrow vertical channels formed by the cylindrical meta-structure is exploited by a novel design concept of the wave power converter by covering the surface of the cylinder with an array of small cuboid buoys which float in the gaps between the intersecting plate arrays. Each buoy is attached to its own spring and power takeoff damping mechanism, and the vertical displacement of individual buoys is replaced by a continuous two-dimensional function of position which follows from homogenization of the plate/fluid structure of the cylinder. Effective medium equations and boundary conditions are derived under both full depth-dependent theory and shallow-water theory, allowing semi-analytical methods to be developed to investigate the wave scattering and wave energy absorption properties of this cylindrical meta-structure. Results illustrate that the internal resonance of the cylindrical meta-structure can lead to significant wave power capture across a broad range of frequencies.

Nonlinear Effects and Run-up of Coastal Waves Generated by Billiards with Semi-rigid Walls in the Framework of Shallow Water Theory

Nonlinear Effects and Run-up of Coastal Waves Generated by Billiards with Semi-rigid Walls in the Framework of Shallow Water Theory

Modulation of the reflected signal due to a nonlinear surface charge excited on a plasma boundary by an electromagnetic beam

The description of the nonlinear surface charge was supplemented by an equation linking the distortion of the surface shape of the plasma boundary with the characteristics of the motion of electrons in this region for the selected type of external influence. As a result, it became possible to study the interrelated appearance and development of these two effects in conditions of commonality of processes that cause their occurrence. On a simple example when a flat boundary of a cold plasma interacts with intense electromagnetic radiation, it was shown that there is development of non linear surface charge, which results in change of the surface curvature and causes fluctuations in the amplitude of the reflected wave. The “shallow water theory” has been used for the derivation of the equation which to link the spatial and temporal structures and evolution of surface charge and curvature, which affect the reflected wave. Both qualitative and numerical analyses have been performed.

Directing Shallow-Water Waves Using Fixed Varying Bathymetry Designed by Recurrent Neural Networks

Directing shallow-water waves and their energy is highly desired in many ocean engineering applications. Coastal infrastructures can be protected by reflecting shallow-water waves to deep water. Wave energy harvesting efficiency can be improved by focusing shallow-water waves on wave energy converters. Changing water depth can effectively affect wave celerity and therefore the propagation of shallow-water waves. However, determining spatially varying bathymetry that can direct shallow-water waves to a designed location is not trivial. In this paper, we propose a novel machine learning method to design and optimize spatially varying bathymetry for directing shallow-water waves, in which the bathymetry is assumed fixed in time without considering morphodynamics. Shallow-water wave theory was applied to establish the mapping between water wave mechanics and recurrent neural networks (RNNs). Two wave-equivalent RNNs were developed to model shallow-water waves over fixed varying bathymetry. The resulting RNNs were trained to optimize bathymetry for wave energy focusing. We demonstrate that the bathymetry optimized by the wave-equivalent RNNs can effectively reflect and refract wave energy to various designed locations. We also foresee the potential that new engineering tools can be similarly developed based on the mathematical equivalence between wave mechanics and recurrent neural networks.

Interpretation of Signals Recorded by Ocean-Bottom Pressure Gauges during the Passage of Atmospheric Lamb Wave on 15 January 2022

The eruption of the Hunga Tonga–Hunga Ha’apai volcano on 15 January 2022 was the first powerful explosive eruption in history to be recorded with high quality by a wide range of geophysical equipment. The atmospheric Lamb wave caused by the explosion repeatedly circled the Earth and served as one of the reasons for the formation of tsunami waves. In this paper, the Lamb wave manifestations are analyzed in the recordings of tsunamimeters, i.e., in data from DONET and DART pressure sensors located in the area of the Japanese Islands. The work is aimed at studying the physics of the formation of pressure variations at the ocean floor in order to develop a method for isolating free gravity waves in records obtained by bottom pressure sensors. Within the framework of shallow water theory, an analysis of the response of the water layer to the atmospheric Lamb wave was performed. This response combines a forced perturbation, the amplitude of which depends on the depth of the ocean, and free gravity waves arising as a result of the restructuring of the forced perturbation on the submarine slopes. Analytical formulas are given for the amplitude and energy of the forced perturbation and free waves arising at the depth jump. With the aid of numerical simulation, the finite length of a slope was revealed to significantly affect the parameters of free waves when exceeding 50 km. The analysis of in situ data (DONET, DART) confirms the validity of theoretical concepts presented in the work. In particular, it is shown that variations of bottom pressure in the deep ocean exceed the amplitude of atmospheric pressure fluctuations in the Lamb wave.