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.     

Long Nonlinear Surface Waves in the Presence of a Variable Current

We investigate the eigenvalue problem governing the propagation of long nonlinear surface waves when there is a current beneath the surface, y being the vertical coordinate. The amplitude of such waves evolves according to the KdV equation and it was proved by Burns [ 1 ] that their speed of propagation c is such that there is no critical layer (i.e., c lies outside the range of ). If, however, the critical layer is nonlinear, the result of Burns does not necessarily apply because the phase change of linear theory then vanishes. In this paper, we consider specific velocity profiles and determine c as a function of Froude number for modes with nonlinear critical layers. Such modes do not always exist, the case of the asymptotic suction profile being a notable example. We find, however, that singular modes can be obtained for boundary layer profiles of the Falkner–Skan similarity type, including the Blasius case. These and other examples are treated and we examine singular solutions of the Rayleigh equation to gain insight about the long wave limit of such solutions.     

Nonlinearly Coupled KdV Equations Describing the Interaction of Equatorial and Midlatitude Rossby Waves

Amplitude equations governing the nonlinear resonant interaction of equatorial baroclinic and barotropic Rossby waves were derived by Majda and Biello and used as a model for long range interactions （teleconnections） between the tropical and midlatitude troposphere. An overview of that derivation is nonlinear wave theory, but not in atmospheric presented and geared to readers versed in sciences. In the course of the derivation, two other sets of asymptotic equations are presented： the long equatorial wave equations and the weakly nonlinear, long equatorial wave equations. A linear transformation recasts the amplitude equations as nonlinear and linearly coupled KdV equations governing the amplitude of two types of modes, each of which consists of a coupled tropical/midlatitude flow. In the limit of Rossby waves with equal dispersion, the transformed amplitude equations become two KdV equations coupled only through nonlinear fluxes. Four numerical integrations are presented which show （i） the interaction of two solitons, one from either mode, （ii） and （iii） the interaction of a soliton in the presence of different mean wind shears, and （iv） the interaction of two solitons mediated by the presence of a mean wind shear.     

Nonlinear Evolution of Singular Disturbances to the Bickley Jet

We consider the nonlinear evolution of a disturbance to the Bickley jet, and the critical layer for the disturbance is located very close to the nose of the jet rather than at the inflection points. Using a nonlinear critical layer analysis, equations governing the evolution of the disturbance are derived and discussed. When the critical layer is located exactly at the nose of the jet, we find that the disturbance cannot exist on a linear basis, even with weak viscosity present, but that nonlinear effects inside the critical layer do permit the disturbance to exist if both modes are present. However, when the phase velocity of the disturbance is perturbed sufficiently away from unity, so that we have a pair of critical layers slightly above and below the nose rather than a single critical layer, we find that the waves can exist on a linear basis, and again we derive equations governing the nonlinear evolution of the disturbance.     

The Evolution of a Nonlinear Critical Level

The evolution of inviscid forced Rossby waves on a parallel flow in the presence of a critical layer is discussed. It is shown that the transient critical layer becomes nonlinear after sufficient time has elapsed and that the “? π” logarithmic phase shift is modified accordingly. All harmonics are found to be important in the nonlinear range, even outside the critical layer.     

地形作用下的非线性Rossby波

本文利用一个受地形强迫作用的半地转正压模式讨论了非线性Rossby波的稳定度和解.结果发现,东西向地形和南北向地形对非线性Rossby波的稳定度和相速的影响很不相同.同时也发现,地形强迫下的非线性Rossby波可用著名的KdV方程描述.     

考虑β变化下的Rossby波

本文考虑了Rossby参数β随纬度的变化并引进了γ参数γ≡-dβ/dy=2Ωsin(ф)/a2.同时把β平面近似扩展为含γ参数的近似:f=f₀+β₀y-γ₀y2/2.这就更接近实际,特特是在较高纬度地区.本文着重研究了γ参数对Rossby波的作用.研究指出:γ参数在较高纬地区有较强的作用.它可以形成纯γ参数所产生的Rossby波,并给出了在一般情况下的包含β变化的Rossby波相速公式,它在γ₀=0时退化为著名的Rossby公式.研究还指出:考虑了β的变化,即便基本气流u是y的线性函数也可以出现不稳定,但γ参数通常对Rossby波起稳定的作用.而且,它影响Rossby波的经向尺度和等位相线的结构,但都减缓Rossby波的增长或衰减.     

Rossby Waves

An asymptotic solution of the linear shallow water equations for small Rossby number is constructed to describe Ross by waves. It leads to a dispersion or eiconal equation for the phase of the waves and a transport equation for their amplitude. It is shown how these equations can be solved by means of rays for both planetary and topographic Rossby waves. The method is illustrated by constructing the wave field produced by a time harmonic point source in fluid of uniform depth. This solution is a Green's function for the equations.     

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.     

The Small Vorticity Nonlinear Critical Layer for Kelvin Modes on a Vortex

We consider in this paper the propagation of neutral modes along a vortex with velocity profile being the radial coordinate. In the linear stability theory governing such flows, the boundary in parameter space separating stable and unstable regions is usually comprised of modes that are singular at some value of r denoted r_c , the critical point. The singularity can be dealt with by adding viscous and/or nonlinear effects within a thin critical layer centered on the critical point. At high Reynolds numbers, the case of most interest in applications, nonlinearity is essential, but it develops that viscosity, treated here as a small perturbation, still plays a subtle role. After first presenting the scaling for the general case, we formulate a nonlinear critical layer theory valid when the critical point occurs far enough from the center of the vortex so that the vorticity there is small. Solutions are found having no phase change across the critical layer thus permitting the existence of modes not possible in a linear theory. It is found that both the axial and azimuthal mean vorticity are different on either side of the critical layer as a result of the wave–mean flow interaction. A long wave analysis with O (1) vorticity leads to similar conclusions.     

Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

Exact Traveling-Wave Solutions for Model Geophysical Systems

Exact traveling-wave solutions of time-dependent nonlinear inhomogeneous PDEs, describing several model systems in geophysical fluid dynamics, are found. The reduced nonlinear ODEs are treated as systems of linear algebraic equations in the derivatives. A variety of solutions are found, depending on the rank of the algebraic systems. The geophysical systems include acoustic gravity waves, inertial waves, and Rossby waves. The solutions describe waves which are, in general, either periodic or monoclinic. The present approach is compared with the earlier one due to Grundland (1974) for finding exact solutions of inhomogeneous systems of nonlinear PDEs.     

Rossby Elevation Waves in the Presence of a Critical Layer

In a previous paper, we investigated the solitary-wave-like development of small-amplitude Rossby waves propagating in a zonal shear current, for the particular case when the Rossby wave speed equals the mean-flow velocity at a certain latitude in the β-plane. We presented a general theory for the nonlinear critical-layer theory, and illustrated it by explicitly describing the motion of a depression solitary wave (D-wave). Here, we report a continuation of that study and consider the more complex case of an elevation solitary wave (E-wave). The method involves matched asymptotic expansions between the outer flow away from the critical layer and the inner flow inside the latter, both these flows having different scalings. We showed previously that the critical-layer flow expansion diverged in the case of the E-wave on the separatrices bounding the open and closed streamlines, which led us to defer a detailed E-wave study. Thus, in this paper, we examine the motion in the additional layer located along the separatrices where this singularity is removed by using a third scaling and find that the previous undesirable distortions are discarded. The evolution equation is derived and is a Korteveg-de-Vries type-equation modified by new nonlinear terms generated by the nonlinear interactions occuring in the critical layer. This equation supports a family of E-waves provided that the mean flow obeys certain conditions. The energy exchange that occurs between the mean flow and the D or E-wave during the critical-layer formation is evaluated in the quasi-steady régime assumption.     

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.     

The Effect of Meridional and Vertical Shear on the Interaction of Equatorial Baroclinic and Barotropic Rossby Waves

Simplified asymptotic equations describing the resonant nonlinear interaction of equatorial Rossby waves with barotropic Rossby waves with significant midlatitude projection in the presence of arbitrary vertically and meridionally sheared zonal mean winds are developed. The three mode equations presented here are an extension of the two mode equations derived by Majda and Biello [ 1 ] and arise in the physically relevant regime produced by seasonal heating when the vertical (baroclinic) mean shear has both symmetric and antisymmetric components; the dynamics of the equatorial baroclinic and both symmetric and antisymmetric barotropic waves is developed. The equations described here are novel in several respects and involve a linear dispersive wave system coupled through quadratic nonlinearities. Numerical simulations are used to explore the effect of antisymmetric baroclinic shear on the exchange of energy between equatorial baroclinic and barotropic waves; the main effect of moderate antisymmetric winds is to shift the barotropic waves meridionally. A purely meridionally antisymmetric mean shear yields highly asymmetric waves which often propagate across the equator. The two mode equations appropriate to Ref. [ 1 ] are shown to have analytic solitary wave solutions and some representative examples with their velocity fields are presented.     

Nonlinear Stability of Nonstationary Cross-Flow Vortices in Compressible Boundary Layers

The nonlinear evolution of long-wavelength non stationary cross-flow vortices in a compressible boundary layer is investigated; the work extends that of Gajjar [1] to flows involving multiple critical layers. The basic flow profile considered in this paper is that appropriate for a fully three-dimensional boundary layer with O(1) Mach number and with wall heating or cooling. The governing equations for the evolution of the cross-flow vortex are obtained, and some special cases are discussed. One special case includes linear theory, where exact analytic expressions for the growth rate of the vortices are obtained. Another special case is a generalization of the Bassom and Gajjar [2] results for neutral waves to compressible flows. The viscous correction to the growth rate is derived, and it is shown how the unsteady nonlinear critical layer structure merges with that for a Haberman type of viscous critical layer.     

A hard ferromagnetic and elastic beam-plate sandwich structure

In this paper, the deformation of a composite hard ferromagnetic-elastic beam-plate structure is investigated. A sandwich structure, composed of two thin hard ferromagnetic layers, with a linear elastic layer in between, is considered. The deformation is due to the self generated magnetic field (magnetostriction). The aim is to assess the interaction forces among the perfectly bonded layers, through a consistent application of the classical nonlinear magneto-elastic theory. Once the general mechanical model is stated, the analysis is specialized to study longitudinal elongation, given its great relevance in technical applications. Owing to the non-local character of the magnetic action, a nonlinear integro-differential equation is derived. Some qualitative properties of the solution are pointed out and the asymptotic behavior near the end sections is examined in detail. A finite differences approach allows writing an approximating nonlinear system of equations in the non asymptotic part of the solution, which is solved through a Newton’s iterative scheme. The numerical results are discussed and it is shown how the asymptotic part of the solution well approximates the full behavior of the structure. Furthermore, the longitudinal interaction force density is found to be singular at the end cross-sections, regardless of the assumed bonding type.     

A hard ferromagnetic and elastic beam-plate sandwich structure

In this paper, the deformation of a composite hard ferromagnetic-elastic beam-plate structure is investigated. A sandwich structure, composed of two thin hard ferromagnetic layers, with a linear elastic layer in between, is considered. The deformation is due to the self generated magnetic field (magnetostriction). The aim is to assess the interaction forces among the perfectly bonded layers, through a consistent application of the classical nonlinear magneto-elastic theory. Once the general mechanical model is stated, the analysis is specialized to study longitudinal elongation, given its great relevance in technical applications. Owing to the non-local character of the magnetic action, a nonlinear integro-differential equation is derived. Some qualitative properties of the solution are pointed out and the asymptotic behavior near the end sections is examined in detail. A finite differences approach allows writing an approximating nonlinear system of equations in the non asymptotic part of the solution, which is solved through a Newton’s iterative scheme. The numerical results are discussed and it is shown how the asymptotic part of the solution well approximates the full behavior of the structure. Furthermore, the longitudinal interaction force density is found to be singular at the end cross-sections, regardless of the assumed bonding type.     

地形对正压大气Rossby波非线性相互作用的影响

本文采用弱非线性近似，推导出地形和Ｅｋｍａｎ摩擦共同作用下连续谱正压Ｒｏｓｓｂｙ波的非线性时空演化方程．根据这组方程，我们研究了窄角谱Ｒｏｓｓｂｙ波包的波波相互作用问题，当一个大振幅Ｒｏｓｓｂｙ波包通过大气传播时，如果它的振幅超过某个阈值，非线性相互作用会使一个尺度比它大的Ｒｏｓｓｂｙ波包和一个尺度比它小的Ｒｏｓｓｂｙ波包的振幅随时间指数增长，这两个次级波的本征频率会发生改变，Ｅｋｍａｎ摩擦、频率不匹配、地形坡度以及波包的空间演变共同决定了主波振幅的阈值及次级波本征频率的改变量．     

The Generation of Clear Air Turbulence by Nonlinear Waves

The structure of the critical layer in a stratified shear flow is investigated for finite-amplitude waves at high Reynolds numbers. Under such conditions, which are characteristic of the Clear Air Turbulence environment, nonlinear effects will dominate over diffusive effects. Nevertheless, it is shown that viscosity and heat-conduction still play a significant role in the evolution of such waves. The reason is that buoyancy leads to the formation of thin diffusive shear layers within the critical layer. The local Richardson number is greatly reduced in these layers and they are, therefore, likely to break down into turbulence. A nonlinear mechanism is thus revealed for producing localized instabilities in flows that are stable on a linear basis. The analysis is developed for arbitrary values of the mean flow Richardson number and results are obtained numerically.