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.     

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.     

推广的β平面近似下带有外源和耗散强迫的非线性Boussinesq方程及其孤立波解

在推广的β平面近似下,从包含耗散和外源的准地转位涡方程出发,利用Gardner-Morikawa变换和弱非线性摄动展开法,推导出带有外源和耗散强迫的非线性Boussinesq方程去刻画非线性Rossby波振幅的演变和发展.利用修正的Jacobi椭圆函数展开法,得到Boussinesq方程的周期波解和孤立波解,从解的结构分析了推广的β效应、切变基本流、外源和耗散是影响非线性Rossby波的重要因素.     

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.     

两层流体中具有β变化和地形影响的Rossby波

本文研究了两层流体中具有变化的Rossby参数和地形Rossby波的问题.利用行波法和摄动的方法,获得了Rossby波振幅满足齐次KdV方程和齐次mKdV方程,推广了Rossby参数和地形对Rossby孤立波的影响.     

Wave–Mean-Flow Interactions in a Forced Rossby Wave Packet Critical Layer

This paper describes the nonlinear critical layer evolution of a zonally localized Rossby wave packet forced in mid-latitudes and propagating horizontally on a beta plane in a zonal shear flow. The wave packet has an amplitude that varies slowly in the zonal direction. Numerical solutions of the governing nonlinear equations show that the wave–mean-flow interactions differ from those that would result with a monochromatic forcing. With the localized forcing, the net absorption of the disturbance at the critical layer continues for large time, because there is an outward flux of momentum in the zonal direction. Further insight into the mechanism for this and other aspects of the evolution of the critical layer is obtained through an approximate asymptotic analysis which is valid for large time.     

Rossby Waves on a Shear Flow with Recirculation Cores

Large-amplitude Rossby waves riding on a background flow with a weak shear can be calculated up to a critical amplitude for which the meridional velocity, in a frame traveling with the wave, approaches zero at some point. Here we consider waves with an amplitude slightly greater than the critical amplitude by incorporating a region of recirculating fluid (vortex core) near this critical point. The effect of the vortex core is to introduce an extra nonlinear term into the equation for the wave amplitude proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude. The main effect due to the vortex core is a broadening of the wave profile. Furthermore, we show that the vortex core family has a limiting amplitude, with the limiting amplitude corresponding to a semi-infinite bore.     

Periodic structures of oceanic Rossby wave under the influence of wind stress

A simple oceanic barotropic potential vorticity equation on β-plane with the influence of wind stress is applied to investigate the nonlinear Rossby wave in a shear flow. By the reductive perturbation method, we derived the rotational modified KdV (rmKdV for short) equation. And then with the help of Jacobi elliptic functions, we obtain various periodic structures for these equatorial Rossby waves. It is shown that the wind stress is very important for these periodic structures of rational form.     

Periodic structures of oceanic Rossby wave under the influence of wind stress

A simple oceanic barotropic potential vorticity equation on β-plane with the influence of wind stress is applied to investigate the nonlinear Rossby wave in a shear flow. By the reductive perturbation method, we derived the rotational modified KdV (rmKdV for short) equation. And then with the help of Jacobi elliptic functions, we obtain various periodic structures for these equatorial Rossby waves. It is shown that the wind stress is very important for these periodic structures of rational form.     

Nonlinear Waves in a Shear Flow with a Vorticity Discontinuity

We consider nonlinear finite-amplitude progressive shear-flow waves on a basic velocity profile consisting of two coflowing layers of inviscid equal-density fluid, each of uniform but different vorticity. The problem is formulated as a nonlinear integral equation describing the shape of the vorticity discontinuity in a frame of reference in which the flow is steady. Numerical solutions to this equation are presented for a range of values of the vorticity ratio Ω. For 1 > © ≥ ? 1 the theoretical maximum wave amplitude occurs when the wave crest forms a 90° corner which just touches the appropriate critical-layer stagnation point. The linearized stability of the progressive wave states to arbitrary subharmonic isovortical disturbances is studied numerically. The results indicate stability at moderate values of the wave amplitude.     

Generalized solitary and periodic wave solutions to a (2 + 1)-dimensional Zakharov-Kuznetsov equation

In this paper, the Exp-function method is employed to the Zakharov-Kuznetsov equation as a (2 + 1)-dimensional model for nonlinear Rossby waves. The observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient. The obtained results may be useful for better understanding the properties of two-dimensional coherent structures such as atmospheric blocking events.     

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.     

Transverse nonlinear instability for two-dimensional dispersive models

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.     

两层流体界面上的孤立波

本文讨论两水平固壁间两层不可压无粘流体界面上的孤立波,计及界面上的表面张力效应.首先建立了适用于这种模型的基本方程组,并在弱色散近似下应用约化摄动法,导得了一阶界面升高所满足的Korteweg-de Vries方程,指出了按该方程系数α和μ的符号的异同,KdV孤立波可能凸向上或凸向下.然后详细讨论了原有近似下非线性效应与色散效应不能平衡的两种临界情形.在采用了适当的近似之后,对第一种临界情形(α=0)得到了修正的KdV方程,并指出,在所考虑的情形中,当μ>0时孤立波不存在,当μ<0时,孤立波仍可能存在,其形式与KdV孤立波不同;对第二种临界情形(μ=0),导得了推广的KdV方程,这时存在振荡型孤立波.文中还对近临界情形作了讨论.本文结果与一些经典结果完全一致,并把它们作了拓广.     

On the structure and growth rate of unstable modes to the Rossby‐Haurwitz wave

The normal mode instability study of a steady Rossby‐Haurwitz wave is considered both theoretically and numerically. This wave is exact solution of the nonlinear barotropic vorticity equation describing the dynamics of an ideal fluid on a rotating sphere, as well as the large‐scale barotropic dynamics of the atmosphere. In this connection, the stability of the Rossby‐Haurwitz wave is of considerable mathematical and meteorological interest. The structure of the spectrum of the linearized operator in case of an ideal fluid is studied. A conservation law for perturbations to the Rossby‐Haurwitz wave is obtained and used to get a necessary condition for its exponential instability. The maximum growth rate of unstable modes is estimated. The orthogonality of the amplitude of a non‐neutral or non‐stationary mode to the Rossby‐Haurwitz wave is shown in two different inner products. The analytical results obtained are used to test and discuss the accuracy of a numerical spectral method used for the normal mode stability study of arbitrary flow on a sphere. The comparison of the numerical and theoretical results shows that the numerical instability study method works well in case of such smooth solutions as the zonal flows and Rossby‐Haurwitz waves. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

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.     

Solitary wave solutions for high dispersive cubic-quintic nonlinear Schrödinger equation

We consider the higher-order dispersive nonlinear Schrödinger equation including fourth-order dispersion effects and a quintic nonlinearity. This equation describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. By adopting the ansatz solution of Li et al. [Zhonghao Li, Lu Li, Huiping Tian, Guosheng Zhou. New types of solitary wave solutions for the higher-order nonlinear Schrödinger equation. Phys Rev Lett 2000;84:4096], we find two different solitary wave solutions under certain parametric conditions. These solutions are in the form of bright and dark soliton solutions.     

Exact solutions of nonlinear generalizations of the wave equation

Exact solutions of nonlinear generalizations of the wave equation are constructed. In some cases these solutions are solitary waves or solitions. Thus, by explicit construction solitons or solitary waves are shown to exist in dispersionless systems. In contrast to previous solitary wave solutions, these solutions are limiting cases of solutions of nonlinear partial differential equations with dispersion.     

Long-time Solutions of the Ostrovsky Equation

The Ostrovsky equation is a modification of the Korteweg-de Vries equation which takes account of the effects of background rotation. It is well known that the usual Korteweg-de Vries solitary wave decays and is replaced by radiating inertia gravity waves. Here we show through numerical simulations that after a long-time a localized wave packet emerges as a persistent and dominant feature. The wavenumber of the carrier wave is associated with that critical wavenumber where the underlying group velocity is a minimum (in absolute value). Based on this feature, we construct a weakly nonlinear theory leading to a higher-order nonlinear Schrödinger equations in an attempt to describe the numerically found wave packets.     

Generation of Secondary Solitary Waves in the Variable-Coefficient Korteweg–de Vries Equation

We consider the solitary wave solutions of a Korteweg–de Vries equation, where the coefficients in the equation vary with time over a certain region. When these coefficients vary rapidly compared with the solitary wave, then it is well known that the solitary wave may fission into two or more solitary waves. On the other hand, when these coefficients vary slowly, the solitary wave deforms adiabatically with the production of a trailing shelf. In this paper we re-examine this latter case, and show that the trailing shelf, on a very long time-scale, can lead to the generation of small secondary solitary waves. This result thus provides a connection between the adiabatic deformation regime and the fission regime.