Modulational Instability in JONSWAP Sea States Using the Alber Equation

2017 ◽  
Author(s):  
Odin Gramstad
Keyword(s):  
Stability Analysis ◽  
Nonlinear Evolution ◽  
Modulational Instability ◽  
Wave Spectrum ◽  
Rogue Waves ◽  
Systems Of Nonlinear Equations ◽  
Optimal Parameters ◽  
Weakly Nonlinear ◽  
Marquardt Algorithm ◽  
The Stability

An investigation of the instability of JONSWAP wave fields is carried out within the framework of the Alber equation [1]. The Alber equation describes the weakly nonlinear evolution of an inhomogeneous wave spectrum, and by linear stability analysis of this equation [1–3] the instability of an arbitrary wave spectrum subject to inhomogeneous perturbation is investigated. We are solving the equations for instability using a numerical method based on the Levenberg-Marquardt algorithm for solving systems of nonlinear equations, as implemented in the FORTRAN library MINPACK. Results from previous works addressing related topics [4, 5] are verified and refined, providing new results for the stability of JONSWAP wave spectra. Based on the results of the instability analysis we propose more optimal parameters for parameterizing the effects of modulational instability and probability of rogue waves in JONSWAP sea states. The results from the stability analysis of the Alber equation as well as the proposed parameters for parameterizing the effect of modulational instability are verified and tested by performing phase-resolving numerical simulations with the Higher Order Spectral Method [6, 7].

Long-time behaviour of a random inhomogeneous field of weakly nonlinear surface gravity waves

1983 ◽  
Vol 133 ◽  
pp. 113-132 ◽  
Author(s):  
Peter A. E. M. Janssen
Keyword(s):  
Wave Spectrum ◽  
Inhomogeneous Field ◽  
Time Behaviour ◽  
Phase Mixing ◽  
Weakly Nonlinear ◽  
Large Bandwidth ◽  
Long Time Behaviour ◽  
Homogeneous Wave ◽  
Long Time ◽  
The Stability

In this paper we investigate nonlinear interactions of narrowband, Gaussian-random, inhomogeneous wavetrains. Alber studied the stability of a homogeneous wave spectrum as a function of the width σ of the spectrum. For vanishing bandwidth the deterministic results of Benjamin & Feir on the instability of a uniform wavetrain were rediscovered whereas a homogeneous wave spectrum was found to be stable if the bandwidth is sufficiently large. Clearly, a threshold for instability is present, and in this paper we intend to study the long-time behaviour of a slightly unstable modulation by means of a multiple-timescale technique. Two interesting cases are found. For small but finite bandwidth – the amplitude of the unstable modulation shows initially an overshoot, followed by an oscillation around the time-asymptotic value of the amplitude. This oscillation damps owing to phase mixing except for vanishing bandwidth because then the well-known Fermi–Pasta–Ulam recurrence is found. For large bandwidth, however, no overshoot is found since the damping is overwhelming. In both cases the instability is quenched because of a broadening of the spectrum.

A Study on the Optimal Double Parameters for Steepest Descent with Momentum

2015 ◽  
Vol 27 (4) ◽  
pp. 982-1004 ◽  
Author(s):  
Naimin Zhang
Keyword(s):  
Stability Analysis ◽  
Steepest Descent ◽  
Quadratic Functions ◽  
Optimal Parameters ◽  
Optimal Learning ◽  
Descent Algorithms ◽  
Learning Rates ◽  
Global Optimal ◽  
The Stability

This letter presents the stability analysis for two steepest descent algorithms with momentum for quadratic functions. The corresponding local optimal parameters in Torii and Hagan ( 2002 ) and Zhang ( 2013 ) are extended to the global optimal parameters, that is, both the optimal learning rates and the optimal momentum factors are obtained simultaneously which make for the fastest convergence.

The effects of randomness on the stability of two-dimensional surface wavetrains

1978 ◽  
Vol 363 (1715) ◽  
pp. 525-546 ◽  
Keyword(s):  
Water Waves ◽  
Modulational Instability ◽  
Wave Spectrum ◽  
Random Wave ◽  
Length Scale ◽  
Wave System ◽  
Random Surface ◽  
Homogeneous Wave ◽  
Amplification Rate ◽  
The Stability

A simplified nonlinear spectral transport equation, for narrowband Gaussian random surface wavetrains, slowly varying in space and time, is derived fron the weakly nonlinear equations of Davey & Stewartson. The stability of an initially homogeneous wave spectrum, to small oblique wave perturbations is studied for a range of spectral bandwidths, resulting in an integral equation for the amplification rate of the disturbance. It is shown for random deep water waves that instability of the wavetrain can exist, as in the corresponding deterministic Benjamin-Feir (B-F) problem, provided that the normalized spectral bandwidth σ / k 0 is less than twice the root mean square wave slope, multiplied by a function of the perturbation wave angle ϕ = arctan ( m/l ). A further condition for instability is that the angle ϕ be less than 35.26°. It is demonstrated that the amplification rate, associated with the B-F type instability, diminishes and then vanishes as the correlation length scale of the random wave field ( ca . 1/ σ )is reduced to the order of the characteristic length scale for modulational instability of the wave system.

scholarly journals The influence of front strength on the development and equilibration of symmetric instability. Part 1. Growth and saturation

2021 ◽  
Vol 926 ◽  
Author(s):  
A.F. Wienkers ◽  
L.N. Thomas ◽  
J.R. Taylor
Keyword(s):  
Stability Analysis ◽  
Shear Instability ◽  
Nonlinear Stability Analysis ◽  
Coriolis Parameter ◽  
Weakly Nonlinear ◽  
Thermal Wind ◽  
Inertial Instability ◽  
Slantwise Convection ◽  
The Stability ◽  
Symmetric Instability

Submesoscale fronts with large horizontal buoyancy gradients and $O(1)$ Rossby numbers are common in the upper ocean. These fronts are associated with large vertical transport and are hotspots for biological activity. Submesoscale fronts are susceptible to symmetric instability (SI) – a form of stratified inertial instability which can occur when the potential vorticity is of the opposite sign to the Coriolis parameter. Here, we use a weakly nonlinear stability analysis to study SI in an idealised frontal zone with a uniform horizontal buoyancy gradient in thermal wind balance. We find that the structure and energetics of SI strongly depend on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Vertically bounded non-hydrostatic SI modes can grow by extracting potential or kinetic energy from the balanced front and the relative importance of these energy reservoirs depends on the front strength and vertical stratification. We describe two limiting behaviours as ‘slantwise convection’ and ‘slantwise inertial instability’ where the largest energy source is the buoyancy flux and geostrophic shear production, respectively. The growing linear SI modes eventually break down through a secondary shear instability, and in the process transport considerable geostrophic momentum. The resulting breakdown of thermal wind balance generates vertically sheared inertial oscillations and we estimate the amplitude of these oscillations from the stability analysis. We finally discuss broader implications of these results in the context of current parameterisations of SI.

On the nonlinear stability of the oscillatory viscous flow of an incompressible fluid in a curved pipe

1999 ◽  
Vol 379 ◽  
pp. 145-163 ◽  
Author(s):  
TRUDI A. SHORTIS ◽  
PHILIP HALL
Keyword(s):  
Stability Analysis ◽  
Differential Operators ◽  
Circular Cross Section ◽  
Nonlinear Effects ◽  
Taylor Number ◽  
Leading Order ◽  
Cross Sectional ◽  
Weakly Nonlinear ◽  
Curved Pipe ◽  
The Stability

The stability of the flow of an incompressible, viscous fluid through a pipe of circular cross-section, curved about a central axis is investigated in a weakly nonlinear regime. A sinusoidal pressure gradient with zero mean is imposed, acting along the pipe. A WKBJ perturbation solution is constructed, taking into account the need for an inner solution in the vicinity of the outer bend, which is obtained by identifying the saddle point of the Taylor number in the complex plane of the cross-sectional angle coordinate. The equation governing the nonlinear evolution of the leading-order vortex amplitude is thus determined. The stability analysis of this flow to axially periodic disturbances leads to a partial differential system dependent on three variables, and since the differential operators in this system are periodic in time, Floquet theory may be applied to reduce it to a coupled infinite system of ordinary differential equations, together with homogeneous uncoupled boundary conditions. The eigenvalues of this system are calculated numerically to predict a critical Taylor number consistent with the analysis of Papageorgiou (1987). A discussion of how nonlinear effects alter the linear stability analysis is also given. It is found that solutions to the leading-order vortex amplitude equation bifurcate subcritically from the eigenvalues of the linear problem.

Stability Analysis of the Nanjung Water Diversion Twin Tunnels based on Convergence Measurement

2019 ◽  
Vol 1 (1) ◽  
pp. 49-60
Author(s):  
Simon Heru Prassetyo ◽  
Ganda Marihot Simangunsong ◽  
Ridho Kresna Wattimena ◽  
Made Astawa Rai ◽  
Irwandy Arif ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Convergence Rate ◽  
Critical Strain ◽  
Convergence Rates ◽  
Strain Analysis ◽  
Water Diversion ◽  
Tunnel Face ◽  
Tunnel Stability ◽  
The Stability ◽  
Twin Tunnels

This paper focuses on the stability analysis of the Nanjung Water Diversion Twin Tunnels using convergence measurement. The Nanjung Tunnel is horseshoe-shaped in cross-section, 10.2 m x 9.2 m in dimension, and 230 m in length. The location of the tunnel is in Curug Jompong, Margaasih Subdistrict, Bandung. Convergence monitoring was done for 144 days between February 18 and July 11, 2019. The results of the convergence measurement were recorded and plotted into the curves of convergence vs. day and convergence vs. distance from tunnel face. From these plots, the continuity of the convergence and the convergence rate in the tunnel roof and wall were then analyzed. The convergence rates from each tunnel were also compared to empirical values to determine the level of tunnel stability. In general, the trend of convergence rate shows that the Nanjung Tunnel is stable without any indication of instability. Although there was a spike in the convergence rate at several STA in the measured span, that spike was not replicated by the convergence rate in the other measured spans and it was not continuous. The stability of the Nanjung Tunnel is also confirmed from the critical strain analysis, in which most of the STA measured have strain magnitudes located below the critical strain line and are less than 1%.

The Stability of Radiatively Cooling Jets. II. Nonlinear Evolution

1997 ◽  
Vol 483 (1) ◽  
pp. 136-147 ◽  
Author(s):  
James M. Stone ◽  
Jianjun Xu ◽  
Philip E. Hardee
Keyword(s):  
Nonlinear Evolution ◽  
The Stability
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