On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Change of Polarity for Periodic Waves in the Variable‐Coefficient Korteweg‐de Vries Equation

We examine the variable‐coefficient Kortweg‐de Vries equation for the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here, we examine the same case but for a modulated periodic wave train. Using an asymptotic analysis, we show that in contrast a periodic wave is preserved with a finite amplitude as it passes through the critical point, but a phase change is generated causing the wave to reverse its polarity.     

Dynamics of Vortex Rossby Waves in Tropical Cyclones,Part 1: Linear Time‐Dependent Evolution on an f‐Plane

Analyses of observational data on hurricanes in the tropical atmosphere indicate the existence of spiral rainbands which propagate outward from the eye and affect the structure and intensity of the hurricane. These disturbances may be described as vortex Rossby waves. This paper describes the evolution of barotropic vortex Rossby waves in a cyclonic vortex in a two‐dimensional configuration where the variation of the Coriolis force with latitude is ignored. The waves are forced by a constant‐amplitude boundary condition at a fixed radius from the center of the vortex and propagate outward. The mean flow angular velocity profile is taken to be a quadratic function of the radial distance from the center of the vortex and there is a critical radius at which it is equal to the phase speed of the waves. For the case of waves with steady amplitude, an exact solution is derived for the steady linearized equations in terms of hypergeometric functions; this solution is valid in the outer region away from the critical radius. For the case of waves with time‐dependent amplitude, asymptotic solutions of the linearized equations, valid for late time, are obtained in the outer and inner regions. It is found that there are strong qualitative similarities between the conclusions on the evolution of the vortex waves in this configuration and those obtained in the case of Rossby waves in a rectangular configuration where the latitudinal gradient of the Coriolis parameter is taken into account. In particular, the amplitude of the steady‐state outer solution is greatly attenuated and there is a phase change of across the critical radius, and in the linear time‐dependent configuration, the outer solution approaches a steady state in the limit of infinite time, while the amplitude of the inner solution grows on a logarithmic time scale and the width of the critical layer approaches zero.     

On the breakdown of acceleration waves in dissociating gas flows

The present paper is devoted to the study of characteristic solution in the neighbourhood of the leading frozen characteristics in dissociating gas flows. It is found that at the cusp of the envelope of intersecting forward characteristics there occurs a breakdown of the wave after a finite critical timet _o. It is observed that there exists a critical value of the initial amplitude of the wave such that all compressive waves with an initial amplitude greater than the critical one will terminate into a shock wave due to non-linear steepening while an initial amplitude less than the critical one will result in a continuous decay. It is also concluded that the breakdown point moves forward along the leading characteristics due to dissociation effects.     

Large-Amplitude Long-Wave Instability of a Supersonic Shear Layer

For sufficiently high Mach numbers, small disturbances on a supersonic vortex sheet are known to grow in amplitude because of slow nonlinear wave steepening. Under the same external conditions, linear theory predicts slow growth of long-wave disturbances to a thin supersonic shear layer. An asymptotic formulation that adds nonzero shear-layer thickness to the weakly nonlinear formulation for a vortex sheet is given here. Spatial evolution is considered for a spatially periodic disturbance having amplitude of the same order, in Reynolds number, as the shear-layer thickness. A quasi-equilibrium inviscid nonlinear critical layer is found, with effects of diffusion and slow growth appearing through a nonsecularity condition. Other limiting cases are also considered, in an attempt to determine a relationship between the vortex-sheet limit and the long-wave limit for a thin shear layer; there appear to be three special limits, corresponding to disturbances of different amplitudes at different locations along the shear layer.     

On the Nonlinear Initial Value Problem for Vortex–Wave Interactions in Shear Flows

Small-amplitude wave systems interacting nonlinearly can produce 0(1)amplitude streamwise vortex structures through the vortex–wave interaction mechanism described, for example, by [1–3]. The key feature of the interaction is that the spanwise velocity component of a vortex is small as compared to the streamwise component so that a nonlinear wave system driving the spanwise velocity component through Reynolds stresses can provoke a 0(1) response of the vortex. The wave system can correspond to either a Rayleigh or Tollmien–Schlichting wave disturbance, but previous work on the initiation of the process has been confined to Rayleigh waves (see, for example, [5, 6]). Here, we address the nonlinear initial value problem for Tollmien–Schlichting wave–vortex interactions in channel flows. The evolution of the disturbances is accounted for using the phase equation approach of [7]. We determine the circumstances, if any, under which the finite amplitude vortex–wave equilibrium states of [4] are generated. Our discussion of the nonlinear evolution of a wave system points toward a possible mechanism for the experimentally observed breakup of three-dimensional instabilities into shorter streamwise scales.     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

Spectrally Stable Encapsulated Vortices for Nonlinear Schr?dinger Equations

Summary. A large class of multidimensional nonlinear Schrodinger equations admit localized nonradial standing-wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are spectrally stable. We find such waves for equations in two space dimensions with focusing-defocusing nonlinearities, such as cubic-quintic. Spectrally stable waves resemble a vortex (nonlocalized solution with asymptotically constant amplitude) cut off at large radius by a kink layer that exponentially localizes the solution. For the evolution equations linearized about a localized spinning wave, we prove that unstable eigenvalues are zeroes of Evans functions for a finite set of ordinary differential equations. Numerical computations indicate that there exist spectrally stable standing waves having central vortex of any degree.     

On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien-Schlichting Waves in Curved Channel Flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien-Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien-Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien-Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien-Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex-wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien-Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien-Schlichting wave amplitude.     

Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.     

On stratified steady periodic water waves with linear density distribution and stagnation points

We use bifurcation theory to construct small periodic gravity stratified water waves with density which depends linearly upon the pseudostream function. As a special feature the density may also decrease with depth and the waves we obtain may posses two different critical layers with cat?s eye vortices. Within the vortex, the density of the fluid has an extremum at the stagnation point.     

Diffusion d'impulsions non linéaires radiales focalisantes dans R1+3

We study the validity of geometric optics for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is critical, linear geometric optics is valid before the focal point, as well as after. The matching between these two régimes is described by a scattering operator, which enlarges the support of the waves.     

Time‐dependent scattering of generalized plane waves by a wedge

We obtain explicit formulas for the scattering of plane waves with arbitrary profile by a wedge under Dirichlet, Neumann and Dirichlet‐Neumann boundary conditions. The diffracted wave is given by a convolution of the profile function with a suitable kernel corresponding to the boundary conditions. We prove the existence and uniqueness of solutions in appropriate classes of distributions and establish the Sommerfeld type representation for the diffracted wave. As an application, we establish (i) stability of long‐time asymptotic local perturbations of the profile functions and (ii) the limiting amplitude principle in the case of a harmonic incident wave. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamics of Vortex Rossby Waves in Tropical Cyclones,Part 2: Nonlinear Time‐Dependent Asymptotic Analysis on a β‐Plane

Vortex Rossby waves in cyclones in the tropical atmosphere are believed to play a role in the observed eyewall replacement cycle, a phenomenon in which concentric rings of intense rainbands develop outside the wall of the cyclone eye, strengthen and then contract inward to replace the original eyewall. In this paper, we present a two‐dimensional configuration that represents the propagation of forced Rossby waves in a cyclonic vortex and use it to explore mechanisms by which critical layer interactions could contribute to the evolution of the secondary eyewall location. The equations studied include the nonlinear terms that describe wave‐mean‐flow interactions, as well as the terms arising from the latitudinal gradient of the Coriolis parameter. Asymptotic methods based on perturbation theory and weakly nonlinear analysis are used to obtain the solution as an expansion in powers of two small parameters that represent nonlinearity and the Coriolis effects. The asymptotic solutions obtained give us insight into the temporal evolution of the forced waves and their effects on the mean vortex. In particular, there is an inward displacement of the location of the critical radius with time which can be interpreted as part of the secondary eyewall cycle.     

Transport of rarefied gases inside a vortex tube induced by a surface wave

We derive the entrainment of rarefied gases induced by a surface wave (SW) propagating along the wall or interface of the vortex core in a rather long straight vortex-tube (vortex or vorticity filament) by using the perturbation method. The values of the critical reflux pressure-gradient for time-averaged macroscopic flows of rarefied gases decrease as the Knudsen number increases from zero. Nonlinear coupling due to the boundary effect and the streaming flow of the rarefied gases is observed as the Reynolds number (based on the wave speed) increases to 50 and the wave number of the surface wave is larger. We also address the zero-flux states of our results. Received: November 22, 2004; revised: May 17, 2005     

A new stabilizing technique for boundary integral methods for water waves

Boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities. A previous stability analysis by Beale, Hou and Lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability. Such balance can be preserved by applying suitable numerical filtering at certain places of the discretization. While this filtering technique is effective for two-dimensional (2-D) periodic fluid interfaces, it does not apply to nonperiodic fluid interfaces. Moreover, using the filtering technique alone does not seem to be sufficient to stabilize 3-D fluid interfaces.

Here we introduce a new stabilizing technique for boundary integral methods for water waves which applies to nonperiodic and 3-D interfaces. A stabilizing term is added to the boundary integral method which exactly cancels the destabilizing term produced by the point vortex method approximation to the leading order. This modified boundary integral method still has the same order of accuracy as the point vortex method. A detailed stability analysis is presented for the point vortex method for 2-D water waves. The effect of various stabilizing terms is illustrated through careful numerical experiments.

     

Nonlinear Propagation of Wave Packets for Near-Critical Wave Numbers in a Liquid that Is Piecewise Nonuniform with Depth

This is a study of the evolution of finite amplitude wave packets at the contact surface between a semi-infinite liquid medium and a liquid located above it near a solid lid. The limiting case of near-critical wave numbers for propagation of wave packets in the liquid system is discussed. Perturbation expansions are obtained for the deviation of the contact surface from its unperturbed position and for wave numbers near the critical value. This is compared with the solution for the analogous limiting cases of one or two semi-infinite liquids. A version in which the wave numbers are far from critical is examined.     

Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate

ABSTRACT

We study the traveling waves of reaction-diffusion equations for a diffusive SEIR model with a general nonlinear incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Its proof is showed by introducing an auxiliary system, applying Schauder fixed point theorem and then a limiting argument. The non-existence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Finally, we present some examples to support our theoretical results.     

Periodic Travelling Waves and Compactons in Granular Chains

We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz’s contact forces. Each bead periodically undergoes a compression phase followed by free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behavior, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wave number, we observe fast instabilities of the periodic travelling waves, leading to a disordered regime.     

Nonlinear Alfvén/Beltrami waves—An integrable structure built around the Casimir

We formulate a nonlinear wave equations that describe amplitude and pitch modulations of one-dimensional Alfvén waves propagating on a dispersive nonlinear plasma. The well-known fact that the ideal Alfvén wave can propagate on a homogeneous ambient magnetic field with conserving an arbitrary wave shape of any amplitude is explained by invoking the Casimirs stemming from a “topological defect” (or, a kernel) in the Poisson bracket operator of the ideal magnetohydrodynamic (MHD) system. Including the Hall term, however, the Alfvén waves are affected by the dispersive effect, and the aforementioned simplicity of the ideal Alfvén waves is greatly lost; an arbitrary wave can no longer propagate with a constant shape. Yet, we observe an integrable structure in the nonlinear modulation (induced by a compressible motion) of the Alfvén waves, which is described as nonlinear deformation of “Beltrami vortex” pertaining to the Casimirs.