scholarly journals Topographically forced long waves on a sheared coastal current. Part 1. The weakly nonlinear response

1997 ◽  
Vol 343 ◽  
pp. 131-151 ◽  
Author(s):  
S. R. CLARKE ◽  
E. R. JOHNSON
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Response ◽  
Critical Flow ◽  
Coastal Current ◽  
Weakly Nonlinear ◽  
Constant Vorticity ◽  
Long Wave ◽  
Non Local ◽  
Steady Solutions ◽  
Local Nature

The flow of a constant-vorticity current past coastal topography is investigated in the long-wave weakly nonlinear limit. In contrast to other near-critical weakly nonlinear systems this problem does not exhibit hydraulically controlled solutions. It is shown that near criticality the evolution of the vorticity interface is governed by a forced BDA (Benjamin–Davis–Acrivos) equation. The solutions of this equation are discussed and two distinct near-critical flow regimes are identified. Owing to the non-local nature of the forcing, the first of these regimes is characterized by quasi-steady solutions controlled at the topography with some blocking of the upstream rotational fluid, while in the second regime steady nonlinear wavetrains form downstream of the obstacle with no upstream influence. In the hydraulic limit the velocity band for both of these critical regimes approaches zero.

scholarly journals Topographically forced long waves on a sheared coastal current. Part 2. Finite amplitude waves

1997 ◽  
Vol 343 ◽  
pp. 153-168 ◽  
Author(s):  
S. R. CLARKE ◽  
E. R. JOHNSON
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Coastal Current ◽  
Constant Vorticity ◽  
Long Wave ◽  
Long Time ◽  
Wave Limit ◽  
Parameter Values ◽  
Upstream Flow

This paper analyses the finite-amplitude flow of a constant-vorticity current past coastal topography in the long-wave limit. A forced finite-amplitude long-wave equation is derived to describe the evolution of the vorticity interface. An analysis of this equation shows that three distinct near-critical regimes occur. In the first the upstream flow is restricted, with overturning of the vorticity interface for sufficiently large topography. In the second quasi-steady nonlinear waves form downstream of the topography with weak upstream influence. In the third regime the upstream rotational fluid is partially blocked. Blocking and overturning are enhanced at headlands with steep rear faces and decreased at headlands with steep forward faces. For certain parameter values both overturning and partially blocked solutions are possible and the long-time evolution is critically dependent on the initial conditions. The reduction of the problem to a one-dimensional nonlinear wave equation allows solutions to be followed to much longer times and parameter space to be explored more finely than in the related pioneering contour-dynamical integrations of Stern (1991).

scholarly journals The role of constant vorticity on weakly nonlinear surface gravity waves

Wave Motion ◽  
2020 ◽  
pp. 102702
Author(s):  
M.A. Manna ◽  
S. Noubissie ◽  
J. Touboul ◽  
B. Simon ◽  
R.A. Kraenkel
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Weakly Nonlinear ◽  
Surface Gravity Waves ◽  
Constant Vorticity ◽  
Nonlinear Surface

scholarly journals Spectral analysis of vibration in weakly non-linear systems

2000 ◽  
Vol 22 (3) ◽  
pp. 181-192
Author(s):  
Nguyen Tien Khiem
Keyword(s):  
Nonlinear Systems ◽  
Spectral Density ◽  
Density Function ◽  
Asymptotic Methods ◽  
Random Excitation ◽  
Spectral Density Function ◽  
Kolmogorov Equation ◽  
Spectral Structure ◽  
Random Load ◽  
Weakly Nonlinear

The weakly nonlinear systems subjected to deterministic excitations have been fully and deeply studied by use of the well developed asymptotic methods [1-4]. The systems excited by a random load have been investigated mostly using the Fokker-Plank-Kolmogorov equation technique combined with the asymptotic methods [5-8]. However, the last approach in most successful cases allows to obtain only a stationary single point probability density function, that contains no information about the correlation nor' consequently, the spectral structure of the response. The linearization technique [9, 10] in general permits the spectral density of the response to be determined, but the spectral function obtained by this method because of the linearization eliminates the effect of the nonlinearity. Thus, spectral structure of response of weakly nonlinear systems to random excitation, to the author's knowledge, has not been studied enough. This paper deals with the above mentioned problem. The main idea of this work is the use of an analytical simulation of random excitation given by its spectral density function and afterward application of the well known procedure of the asymptotic method to obtain an asymptotic expression of the response spectral density function. The obtained spectral relationship covers the linear system case and especially emphasizes the nonlinear effect on the spectral density of response. The theory will be illustrated by an example and at the end of this paper there will be a discussion about the obtained results.  

Dissipative effects on the resonant flow of a stratified fluid over topography

1988 ◽  
Vol 192 ◽  
pp. 287-312 ◽  
Author(s):  
N. F. Smyth
Keyword(s):  
Bottom Topography ◽  
Stratified Fluid ◽  
Flow Speed ◽  
Resonant Condition ◽  
Weakly Nonlinear ◽  
Dissipative Effects ◽  
Long Wave ◽  
Near Resonance ◽  
Wave Limit ◽  
Wave Modes

The effect of dissipation on the flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit for the case when the flow is near resonance, i.e. the basic flow speed is close to a linear long-wave speed for one of the long-wave modes. The two types of dissipation considered are the dissipation due to viscosity acting in boundary layers and/or interfaces and the dissipation due to viscosity acting in the fluid as a whole. The effect of changing bottom topography on the flow produced by a force moving at a resonant velocity is also considered. In this case, the resonant condition is that the force velocity is close to a linear long-wave velocity for one of the long-wave modes. It is found that in most cases, these extra effects result in the formation of a steady state, in contrast to the flow without these effects, which remains unsteady for all time. The flow resulting under the action of boundary-layer dissipation is compared with recent experimental results.

scholarly journals A non-local formulation of rotational water waves

2011 ◽  
Vol 689 ◽  
pp. 129-148 ◽  
Author(s):  
A. C. L. Ashton ◽  
A. S. Fokas
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Velocity Potential ◽  
Vries Equation ◽  
Travelling Waves ◽  
Explicit Dependence ◽  
Local Equation ◽  
Constant Vorticity ◽  
Small Parameters ◽  
Non Local

AbstractThe classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free surface. Here, in the two-dimensional case: (a) we generalize the relevant formulation to the case of constant vorticity, as well as to the case where the free surface is described by a multivalued function; (b) in the case of travelling waves we derive an upper bound for the free surface; (c) in the case of constant vorticity we construct a sequence of nearly Hamiltonian systems which provide an approximation in the asymptotic limit of certain physical small parameters. In particular, the explicit dependence of the vorticity on the coefficients of the Korteweg–de Vries equation is clarified.

On the Equivalence of Normal Form Theory and Multiple Time Scale Method

2007 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Differential Equations ◽  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Perturbation Scheme ◽  
Normal Form Theory ◽  
Weakly Nonlinear ◽  
Form Theory

Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales are introduced that serve as the independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear systems as simple as possible. The “simplest” differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the equivalence of these two methods for constructing periodic solutions is proven, and it is explained why some studies have found the results obtained by the two techniques to be inconsistent.

scholarly journals Collective vibrations of a hydrodynamic active lattice

2020 ◽  
Vol 476 (2239) ◽  
pp. 20200155 ◽  
Author(s):  
S. J. Thomson ◽  
M. Durey ◽  
R. R. Rosales
Keyword(s):  
Nonlinear Dynamical System ◽  
Bath Surface ◽  
Nonlinear Dynamical ◽  
Weakly Nonlinear ◽  
New Class ◽  
Subcritical Hopf Bifurcation ◽  
Geometrical Constraints ◽  
Non Local ◽  
Collective Vibrations ◽  
Spatio Temporal

Recent experiments show that quasi-one-dimensional lattices of self-propelled droplets exhibit collective instabilities in the form of out-of-phase oscillations and solitary-like waves. This hydrodynamic lattice is driven by the external forcing of a vertically vibrating fluid bath, which invokes a field of subcritical Faraday waves on the bath surface, mediating the spatio-temporal droplet coupling. By modelling the droplet lattice as a memory-endowed system with spatially non-local coupling, we herein rationalize the form and onset of instability in this new class of dynamical oscillator. We identify the memory-driven instability of the lattice as a function of the number of droplets, and determine equispaced lattice configurations precluded by geometrical constraints. Each memory-driven instability is then classified as either a super- or subcritical Hopf bifurcation via a systematic weakly nonlinear analysis, rationalizing experimental observations. We further discover a previously unreported symmetry-breaking instability, manifest as an oscillatory–rotary motion of the lattice. Numerical simulations support our findings and prompt further investigations of this nonlinear dynamical system.

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