scholarly journals A method for computing unsteady fully nonlinear interfacial waves

1997 ◽  
Vol 351 ◽  
pp. 223-252 ◽  
Author(s):  
JOHN GRUE ◽  
HELMER ANDRÉ FRIIS ◽  
ENOK PALM ◽  
PER OLAV RUSÅS
Keyword(s):  
Solitary Waves ◽  
Bottom Topography ◽  
The Body ◽  
Interfacial Waves ◽  
Nonlinear Method ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Taylor Series Expansions ◽  
Layer Flow ◽  
Fully Nonlinear Method

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchy's theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

Transcritical two-layer flow over topography

1987 ◽  
Vol 178 ◽  
pp. 31-52 ◽  
Author(s):  
W. K. Melville ◽  
Karl R. Helfrich
Keyword(s):  
Steady Flow ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Single Layer ◽  
Cubic Nonlinearity ◽  
Arbitrary Length ◽  
Weakly Nonlinear ◽  
Flow Over Topography ◽  
Layer Flow

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

scholarly journals Anomalous width variation of rarefactive ion acoustic solitary waves in the context of auroral plasmas

2004 ◽  
Vol 11 (2) ◽  
pp. 219-228 ◽  
Author(s):  
S. S. Ghosh ◽  
G. S. Lakhina
Keyword(s):  
Solitary Waves ◽  
Nonlinear Theory ◽  
Acoustic Waves ◽  
Large Amplitude ◽  
Magnetized Plasma ◽  
Amplitude Variation ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

Abstract. The presence of dynamic, large amplitude solitary waves in the auroral regions of space is well known. Since their velocities are of the order of the ion acoustic speed, they may well be considered as being generated from the nonlinear evolution of ion acoustic waves. However, they do not show the expected width-amplitude correlation for K-dV solitons. Recent POLAR observations have actually revealed that the low altitude rarefactive ion acoustic solitary waves are associated with an increase in the width with increasing amplitude. This indicates that a weakly nonlinear theory is not appropriate to describe the solitary structures in the auroral regions. In the present work, a fully nonlinear analysis based on Sagdeev pseudopotential technique has been adopted for both parallel and oblique propagation of rarefactive solitary waves in a two electron temperature multi-ion plasma. The large amplitude solutions have consistently shown an increase in the width with increasing amplitude. The width-amplitude variation profile of obliquely propagating rarefactive solitary waves in a magnetized plasma have been compared with the recent POLAR observations. The width-amplitude variation pattern is found to fit well with the analytical results. It indicates that a fully nonlinear theory of ion acoustic solitary waves may well explain the observed anomalous width variations of large amplitude structures in the auroral region.

scholarly journals The transformation of an interfacial solitary wave of elevation at a bottom step

2009 ◽  
Vol 16 (1) ◽  
pp. 33-42 ◽  
Author(s):  
V. Maderich ◽  
T. Talipova ◽  
R. Grimshaw ◽  
E. Pelinovsky ◽  
B. H. Choi ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Study ◽  
Nonlinear Effects ◽  
Ocean Model ◽  
Gardner Equation ◽  
Fully Nonlinear ◽  
Layer Flow ◽  
The One ◽  
Bottom Step

Abstract. In this paper we study the transformation of an internal solitary wave at a bottom step in the framework of two-layer flow, for the case when the interface lies close to the bottom, and so the solitary waves are elevation waves. The outcome is the formation of solitary waves and dispersive wave trains in both the reflected and transmitted fields. We use a two-pronged approach, based on numerical simulations of the fully nonlinear equations using a version of the Princeton Ocean Model on the one hand, and a theoretical and numerical study of the Gardner equation on the other hand. In the numerical experiments, the ratio of the initial wave amplitude to the layer thickness is varied up one-half, and nonlinear effects are then essential. In general, the characteristics of the generated solitary waves obtained in the fully nonlinear simulations are in reasonable agreement with the predictions of our theoretical model, which is based on matching linear shallow-water theory in the vicinity of a step with solutions of the Gardner equation for waves far from the step.

THE RISK OF HITTING THE ZERO LOWER BOUND AND THE OPTIMAL INFLATION TARGET

2017 ◽  
Vol 22 (2) ◽  
pp. 402-425 ◽  
Author(s):  
Phuong V. Ngo
Keyword(s):  
Lower Bound ◽  
Nonlinear Function ◽  
Zero Lower Bound ◽  
Nonlinear Method ◽  
Linear Quadratic ◽  
Inflation Target ◽  
Fully Nonlinear ◽  
Optimal Target ◽  
Dynamic Stochastic ◽  
Fully Nonlinear Method

I examine the optimal inflation target in a dynamic stochastic New Keynesian model featuring an occasionally binding zero lower bound on nominal interest rate (ZLB). To this end, I first calibrate the shock needed to generate the risk of hitting the ZLB that matches the U.S. data, based on a fully nonlinear method. I then resolve the model with different inflation targets and find that the optimal target is 3.4%. In addition, the optimal inflation target is a nonlinear function of the risk of hitting the ZLB and inflation indexation. It is always greater than 2% if the risk is greater than 2.5% or if the inflation indexation is higher than 0.5. Finally, the linear–quadratic approach overestimates the true optimal inflation target. In particular, based on the benchmark calibration, it generates an optimal target of 5.5%, compared with 3.4% found by the fully nonlinear method.

A note on waveless subcritical flow past symmetric bottom topography

2016 ◽  
Vol 28 (4) ◽  
pp. 562-575
Author(s):  
R. J. HOLMES ◽  
G. C. HOCKING
Keyword(s):  
Nonlinear Analysis ◽  
Nonlinear Problem ◽  
Vries Equation ◽  
Bottom Topography ◽  
Finite Depth ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Subcritical Flow ◽  
Weakly Nonlinear Analysis ◽  
Subcritical Solutions

This paper re-examines the problem of the flow of a fluid of finite depth over two Gaussian-shaped obstructions on the stream bed. A weakly nonlinear analysis in the form of the Korteweg–de Vries equation is used to compare with the results of the fully nonlinear problem. The main focus is to find waveless subcritical solutions, and contours showing the obstruction height and separation values that result in waveless solutions are found for different Froude numbers and different obstruction widths.

scholarly journals Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves

2011 ◽  
Vol 18 (3) ◽  
pp. 351-358 ◽  
Author(s):  
M. Dunphy ◽  
C. Subich ◽  
M. Stastna
Keyword(s):  
Euler Equations ◽  
Solitary Waves ◽  
Dynamic Simulations ◽  
Large Lakes ◽  
Full Spectrum ◽  
Density Profiles ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
First Order ◽  
Spectral And Pseudospectral Methods

Abstract. Internal solitary waves are widely observed in both the oceans and large lakes. They can be described by a variety of mathematical theories, covering the full spectrum from first order asymptotic theory (i.e. Korteweg-de Vries, or KdV, theory), through higher order extensions of weakly nonlinear-weakly nonhydrostatic theory, to fully nonlinear-weakly nonhydrostatic theories and finally exact theory based on the Dubreil-Jacotin-Long (DJL) equation that is formally equivalent to the full set of Euler equations. We discuss how spectral and pseudospectral methods allow for the computation of novel phenomena in both approximate and exact theories. In particular we construct markedly different density profiles for which the coefficients in the KdV theory are very nearly identical. These two density profiles yield qualitatively different behaviour for both exact, or fully nonlinear, waves computed using the DJL equation and in dynamic simulations of the time dependent Euler equations. For exact, DJL, theory we compute exact solitary waves with two-scales, or so-called double-humped waves.

scholarly journals Nonlinear hydroelastic waves on a linear shear current at finite depth

2019 ◽  
Vol 876 ◽  
pp. 55-86 ◽  
Author(s):  
T. Gao ◽  
Z. Wang ◽  
P. A. Milewski
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Modulational Instability ◽  
Finite Depth ◽  
Time Dependent ◽  
Mapping Technique ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Constant Vorticity

This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

The inverse water wave problem of bathymetry detection

2013 ◽  
Vol 714 ◽  
pp. 562-590 ◽  
Author(s):  
Vishal Vasan ◽  
Bernard Deconinck
Keyword(s):  
Numerical Solution ◽  
Water Wave ◽  
Bottom Topography ◽  
Wave Surface ◽  
Nonlinear Method ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Ill Posed ◽  
Non Local ◽  
Fully Nonlinear Method

AbstractThe inverse water wave problem of bathymetry detection is the problem of deducing the bottom topography of the seabed from measurements of the water wave surface. In this paper, we present a fully nonlinear method to address this problem in the context of the Euler equations for inviscid irrotational fluid flow with no further approximation. Given the water wave height and its first two time derivatives, we demonstrate that the bottom topography may be reconstructed from the numerical solution of a set of two coupled non-local equations. Owing to the presence of growing hyperbolic functions in these equations, their numerical solution is increasingly difficult if the length scales involved are such that the water is sufficiently deep. This reflects the ill-posed nature of the inverse problem. A new method for the solution of the forward problem of determining the water wave surface at any time, given the bathymetry, is also presented.

scholarly journals Properties of large-amplitude internal waves

1999 ◽  
Vol 380 ◽  
pp. 257-278 ◽  
Author(s):  
JOHN GRUE ◽  
ATLE JENSEN ◽  
PER-OLAV RUSÅS ◽  
J. KRISTIAN SVEEN
Keyword(s):  
Fresh Water ◽  
Solitary Waves ◽  
Particle Tracking Velocimetry ◽  
Wave Speed ◽  
Propagation Speed ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Nonlinear Interface ◽  
The Waves ◽  
Theoretical Results

Properties of solitary waves propagating in a two-layer fluid are investigated comparing experiments and theory. In the experiments the velocity field induced by the waves, the propagation speed and the wave shape are quite accurately measured using particle tracking velocimetry (PTV) and image analysis. The experiments are calibrated with a layer of fresh water above a layer of brine. The depth of the brine is 4.13 times the depth of the fresh water. Theoretical results are given for this depth ratio, and, in addition, in a few examples for larger ratios, up to 100[ratio ]1. The wave amplitudes in the experiments range from a small value up to almost maximal amplitude. The thickness of the pycnocline is in the range of approximately 0.13–0.26 times the depth of the thinner layer. Solitary waves are generated by releasing a volume of fresh water trapped behind a gate. By careful adjustment of the length and depth of the initial volume we always generate a single solitary wave, even for very large volumes. The experiments are very repeatable and the recording technique is very accurate. The error in the measured velocities non-dimensionalized by the linear long wave speed is less than about 7–8% in all cases. The experiments are compared with a fully nonlinear interface model and weakly nonlinear Korteweg–de Vries (KdV) theory. The fully nonlinear model compares excellently with the experiments for all quantities measured. This is true for the whole amplitude range, even for a pycnocline which is not very sharp. The KdV theory is relevant for small wave amplitude but exhibit a systematic deviation from the experiments and the fully nonlinear theory for wave amplitudes exceeding about 0.4 times the depth of the thinner layer. In the experiments with the largest waves, rolls develop behind the maximal displacement of the wave due to the Kelvin–Helmholtz instability. The recordings enable evaluation of the local Richardson number due to the flow in the pycnocline. We find that stability or instability of the flow occurs in approximate agreement with the theorem of Miles and Howard.

Sign in / Sign up

Export Citation Format

Share Document