scholarly journals On Bragg resonances and wave triad interactions in two-layered shear flows

2019 ◽  
Vol 867 ◽  
pp. 482-515 ◽  
Author(s):  
Raunak Raj ◽  
Anirban Guha
Keyword(s):  
Shear Flows ◽  
Resonance Condition ◽  
Piecewise Linear ◽  
Multiple Scale ◽  
Linear Velocity ◽  
Velocity Shear ◽  
Fredholm Alternative ◽  
Bragg Resonance ◽  
Weakly Nonlinear ◽  
Doppler Shifts

The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways: (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the higher-order spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both on the theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of the study of these kinds of problems, to incorporate piecewise constant background shear.

scholarly journals Directional solidification of a binary alloy into a cellular convective flow: localized morphologies

1999 ◽  
Vol 395 ◽  
pp. 253-270 ◽  
Author(s):  
Y.-J. CHEN ◽  
S. H. DAVIS
Keyword(s):  
Directional Solidification ◽  
Binary Alloy ◽  
Temporal Dynamics ◽  
Three Dimensional ◽  
Multiple Scale ◽  
Solute Diffusion ◽  
Two Dimensional ◽  
Morphological Instability ◽  
Flow Strength ◽  
Weakly Nonlinear

A steady, two-dimensional cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behaviour of the moving front is described by a slow, spatial–temporal dynamics obtained through a multiple-scale analysis. The resulting system has a parametric-excitation structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection in general stabilizes two-dimensional disturbances, but destabilizes three-dimensional disturbances. When the flow is weak, the morphological instability is incommensurate with the flow wavelength, but as the flow gets stronger, the instability becomes quantized and forced to fit into the flow box. At large flow strength the instability is localized, confined in narrow envelopes. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundaries and asymptotics are obtained by a WKB analysis. The weakly nonlinear interaction is delivered through the Lyapunov–Schmidt method.

Three Dimensional Fully Localized Waves on Ice-Covered Ocean

2013 ◽  
Author(s):  
Yong Liang ◽  
M.-Reza Alam
Keyword(s):  
Numerical Scheme ◽  
Perturbation Technique ◽  
Three Dimensional ◽  
Multiple Scale ◽  
Governing Equations ◽  
Weakly Nonlinear ◽  
Scale Perturbation ◽  
Speed Of Propagation ◽  
Localized Waves ◽  
Mean Current

We have recently shown [1] that fully-localized three-dimensional wave envelopes (so-called dromions) can exist and propagate on the surface of ice-covered waters. Here we show that the inertia of the ice can play an important role in the size, direction and speed of propagation of these structures. We use multiple-scale perturbation technique to derive governing equations for the weakly nonlinear envelope of monochromatic waves propagating over the ice-covered seas. We show that the governing equations simplify to a coupled set of one equation for the envelope amplitude and one equation for the underlying mean current. This set of nonlinear equations can be further simplified to fall in the category of Davey-Stewartson equations [2]. We then use a numerical scheme initialized with the analytical dromion solution of DSI (i.e. shallow-water and surface-tension dominated regimes of Davey-Stewartson equation) to look for dromion solution of our equations. Dromions can travel over long distances and can transport mass, momentum and energy from the ice-edge deep into the solid ice-cover that can result in the ice cracking/breaking and also in posing dangers to icebreaker ships.

A nonlinear critical layer generated by the interaction of free Rossby waves

1998 ◽  
Vol 371 ◽  
pp. 319-344 ◽  
Author(s):  
JACQUES VANNESTE
Keyword(s):  
Continuous Spectrum ◽  
Rossby Waves ◽  
Resonance Condition ◽  
Inviscid Flow ◽  
Critical Layer ◽  
Weakly Nonlinear ◽  
Critical Layers ◽  
Flow Domain ◽  
Parallel Shear Flow ◽  
Stream Direction

Two free waves propagating in a parallel shear flow generate a critical layer when their nonlinear interaction induces a perturbation whose phase velocity matches the basic-state velocity somewhere in the flow domain. The condition necessary for this to occur may be interpreted as a resonance condition for a triad formed by the two waves and a (singular) mode of the continuous spectrum associated with the shear. The formation of the critical layer is investigated in the case of freely propagating Rossby waves in a two-dimensional inviscid flow in a β-channel.A weakly nonlinear analysis based on a normal-mode expansion in terms of Rossby waves and modes of the continuous spectrum is developed; it leads to a system of amplitude equations describing the evolution of the two Rossby waves and of the modes of the continuous spectrum excited during the interaction. The assumption of weak nonlinearity is not however self-consistent: it breaks down because nonlinearity always becomes strong within the critical layer, however small the initial amplitudes of the Rossby waves. This demonstrates the relevance of nonlinear critical layers to monotonic, stable, unforced shear flows which sustain wave propagation.A nonlinear critical-layer theory is developed that is analogous to the well-known theory for forced critical layers. Differences arise because of the presence of the Rossby waves: the vorticity in the critical layer is advected in the cross-stream direction by the oscillatory velocity field due to the Rossby waves. An equation is derived which governs the modification of the Rossby waves that results from their interaction; it indicates that the two Rossby waves are undisturbed at leading order. An analogue of the Stewartson–Warn–Warn analytical solution is also considered.

T-periodic solutions for some second order differential equations with singularities

1992 ◽  
Vol 120 (3-4) ◽  
pp. 231-243 ◽  
Author(s):  
Manuel del Pino ◽  
Raúl Manásevich ◽  
Alberto Montero
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Positive Solutions ◽  
Resonance Condition ◽  
Second Order ◽  
Fredholm Alternative ◽  
Positive Slope ◽  
Second Order Differential Equations ◽  
Image Position

SynopsisWe study the existence of T-periodic positive solutions of the equationwhere f(t, .) has a singularity of repulsive type near the origin. Under the assumption that f(t, x) lies between two lines of positive slope for large and positive x, we find a non-resonance condition which predicts the existence of one T-periodic solution.Our main result gives a Fredholm alternative-like result for the existence of T-periodic positive solutions for

Stability of solitary-wave solutions to a modified Zakharov–Kuznetsov equation

2000 ◽  
Vol 64 (4) ◽  
pp. 411-426 ◽  
Author(s):  
S. MUNRO ◽  
E. J. PARKES
Keyword(s):  
Acoustic Waves ◽  
Multiple Scale ◽  
Travelling Wave ◽  
Magnetized Plasma ◽  
Initial Growth ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Instability Range ◽  
Scale Perturbation ◽  
Weakly Nonlinear Waves

In the context of ion-acoustic waves in a magnetized plasma comprising cold ions and non-isothermal electrons, small-amplitude, weakly nonlinear waves have been shown previously by Munro and Parkes to be governed by a modified version of the Zakharov–Kuznetsov equation. In this paper, we consider solitary travelling-wave solutions to this equation that propagate along the magnetic field. We investigate the initial growth rate γ(k) of a small transverse sinusoidal perturbation of wavenumber k. The instability range is shown to be 0 < k < 3. We use the multiple-scale perturbation method developed by Allen and Rowlands to determine a consistent expansion of γ about k = 0 and k = 3. By combining these results in the form of a Padé approximant, an analytical expression for γ is found that is valid for 0 < k < 3. γ is also determined by using the variational method developed by Bettinson and Rowlands. The two results for γ are compared with a numerical determination.

scholarly journals Weakly Nonlinear and Numerical Analysis of Auto-Oscillatory Dynamics in a Solid Propellant Combustion Model

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Michael Reardon ◽  
Jun Yu
Keyword(s):  
Hopf Bifurcation ◽  
Solid Propellant ◽  
Numerical Solutions ◽  
Multiple Scale ◽  
Combustion Model ◽  
Small Perturbations ◽  
Weakly Nonlinear ◽  
Propellant Combustion ◽  
Oscillatory Dynamics ◽  
Numerical Predictions

In solid combustion, a planar traveling flame wave may lose stability to oscillatory modes in certain parameter regimes. The loss of stability is commonly achieved through a Hopf bifurcation in which linearly unstable modes grow to fixed amplitudes. This situation has been well studied in models of strictly condensed combustion and has been observed experimentally. In this paper, we describe the onset and evolution of oscillatory modes arising in a free-interface model of solid propellant combustion. After a brief introduction, the structure and stability of small perturbations to a planar flame wave are then investigated. Multiple-scale expansions are utilized to describe changes in the amplitude and the phase of a single, weakly unstable mode, and the results are compared to numerical predictions. It is found the over much of the parameter space that the analytical and numerical solutions predict a supercritical Hopf bifurcation similar to those found in models of strictly condensed combustion.

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