Resonant excitation of rossby waves in the equatorial waveguide and their nonlinear evolution

Nonlinear interactions between the baroclinic Rossby waves trapped in the equatorial waveguide and the barotropic Rossby waves freely propagating across the equator are studied within the two-layer model of the atmosphere, or the ocean. It is shown that a barotropic wave can resonantly excite a pair of baroclinic waves with amplitudes much greater than its proper amplitude. The envelopes of the baroclinic waves obey Ginzburg-Landau-type equations and exhibit nonlinear saturation and formation of characteristic "domain-wall" and "dark-soliton" defects.     

Solitary waves in the gulf of quarnaro revealed by LANDSAT thematic mapper

Summary The propagation of solitary wave trains in the Adriatic Sea is studied by means of LANDSAT Thematic-Mapper images. The baroclinic Rossby radius of deformation has been estimated from the width of a baroclinic current observed along the coast of Croatia using AVHRR data. The phase speed of the linear internal waves has been consequently estimated and the compatibility of the Kortweg-de Vries equations with experimental data has been tested. Satellite-derived wave parameters are in good agreement with a plane-bottom, shallow-water solitary-wave model.     

切变基本纬向流中β效应的赤道Rossby孤立波包

从描写赤道Rossby 波的正压大气位涡方程出发,采用多重尺度摄动方法推导出在切变基本纬向流中具有β效应的非线性赤道Rossby波包演变满足非线性Schrdinger方程,并得到单个包络孤立子波解,分析了基本切变流,β效应对非线性赤道Rossby波的影响. 关键词： 赤道Rossby波 β效应 非线性Schrö dinger方程 包络孤立子     

强迫耗散与β效应地形效应作用下的非线性Rossby波包

正压流体中,从有外源的准地转位涡方程出发采用摄动方法和时空伸长变换推导了具有β效应、地形效应和外源的强迫Rossby孤立波包方程,得到孤立Rossby波振幅的演变满足带有地形与外源强迫的非齐次非线性Schrödinger方程的结论. 通过分析孤立Rossby波包振幅的演变,指出了β效应、地形效应以及外源都是诱导Rossby孤立波产生的重要因素,说明了在地形强迫效应和非线性作用相平衡的假定下,Rossby孤立波包振幅的演变满足非齐次非线性Schrödinger 关键词： Rossby波包 β效应')" href="#">β效应 地形 Schrödinger方程     

(2+1)-dimensional coupled Boussinesq equations for Rossby waves in two-layer cylindrical fluid*

In this paper, the existence and propagation characteristics of Rossby waves in a two-layer cylindrical fluid are studied. Firstly, based on the dimensionless baroclinic quasi-geostrophic vortex equations including exogenous and dissipative, we derive new (2+1)-dimensional coupled Boussinesq equations describing wave propagation in polar coordinates by employing a multiscale analysis and perturbation method. Then, the Lie symmetries and conservation laws of the coupled Boussinesq equations are analyzed. Subsequently, by using the $(G^{\prime} /G)$-expansion method, the exact solutions of the (2+1)-dimensional coupled Boussinesq equations are obtained. Finally, the effects of coupling term coefficients on the propagation characteristics of Rossby waves are analyzed.     

Equatorial Rossby Solitary Wave Under the External Forcing

A simple shallow-water model with influence of external forcing on a β-plane is applied to investigate the nonlinear equatorial Rossby waves in a shear flow. By the perturbation method, the extended variable-coefficient KdV equation under an external forcing is derived for large amplitude equatorial Rossby wave in a shear flow. And then various periodic-like structures for these equatorial Rossby waves are obtained with the help of Jacobi elliptic functions. It is shown that the external forcing plays an important role in various periodic-like structures.     

Periodic Structure of Equatorial Envelope Rossby Wave Under Influence of Diabatic Heating

A simple shallow-water model with influence of diabatic heating on a β-plane is applied to investigate the nonlinear equatorial Rossby waves in a shear flow. By the asymptotic method of multiple scales, the cubic nonlinear Schro^edinger (NLS for short) equation with an external heating source is derived for large amplitude equatorial envelope Rossby wave in a shear flow. And then various periodic structures for these equatorial envelope Rossby waves are obtained with the help of Jacob/elliptic functions and elliptic equation. It is shown that phase-locked diabatic heating plays an important role in periodic structures of rational form.     

Analytical investigation of Rossby waves in atmospheric dynamics

The (2+1)-dimensional nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane is analyzed by using the classical Lie group approach. Using the group theory, some types of general exact Rossby wave solutions can be obtained whence a special Rossby wave solution is known. Especially, it is found that the only effect of the time-dependent background basic wind on the Rossby waves is the accumulate motion in the zonal direction. Some types of exact explicit similarity Rossby wave solutions with both nonconstant linear and nonlinear shears are also given.     

Periodic Structures of Rossby Wave under Influence of Dissipation

A simple barotropic potential vorticity equation with the influence of dissipation is applied to investigate the nonlinear Rossby wave in a shear flow in the tropical atmophere. By the reduetive perturbation method, we derive the rotational KdV （rKdV for short） equation. And then, with the help of Jaeobi elliptie functions, we obtain various periodic structures for these Rossby waves. It is shown that dissipation is very important for these periodic structures of rational form.     

Hamiltonian description of barotropic Rossby waves

A transformation into the normal canonical variables is found in the beta-plane approximation for barotropic Rossby waves of an arbitrary amplitude. This transformation is used to derive a matrix of three-wave interaction and to find an expression for the fourth-order term in the interaction Hamiltonian, which describes the modulation instability of Rossby waves. An increment of this instability has been calculated and estimated numerically.     

正压流体中具有β效应与地形效应的强迫Rossby孤立波

正压流体中,从有外源的准地转位涡方程出发采用摄动方法和时空伸长变换推导了具有β效应、地形效应和外源的强迫Rossby孤立波方程,得到孤立Rossby波振幅的演变满足带有地形与外源强迫的非齐次 Boussinesq方程的结论. 通过分析孤立Rossby波振幅的演变,指出β效应、地形效应以及外源都是诱导Rossby孤立波产生的重要因素,说明在地形强迫效应和非线性作用相平衡的假定下,Rossby孤立波振幅的演变满足非齐次Boussinesq方程,给出在切变基本气流下地形和正压流体中R     

Surface gravity waves in deep fluid at vertical shear flows

Special features of surface gravity waves in a deep fluid flow with a constant vertical shear of velocity is studied. It is found that the mean flow velocity shear leads to a nontrivial modification of the dispersive characteristics of surface gravity wave modes. Moreover, the shear induces generation of surface gravity waves by internal vortex mode perturbations. The performed analytical and numerical study show that surface gravity waves are effectively generated by the internal perturbations at high shear rates. The generation is different for the waves propagating in the different directions. The generation of surface gravity waves propagating along the main flow considerably exceeds the generation of surface gravity waves in the opposite direction for relatively small shear rates, whereas the latter wave is generated more effectively for high shear rates. From the mathematical standpoint, the wave generation is caused by non-self-adjointness of the linear operators that describe the shear flow.     

Revisit to frozen-in property of vorticity

Considering some simple topological properties of vorticity vector, the frozen-in property of vorticity herein is revisited. A vortex line, as is analogous to velocity vector along a streamline, is defined as such a coincident material(curve) line that connects many material fluid elements, on which the local vorticity vector for each fluid element is also tangent to the vortex line. The vortex line evolves in the same manner as the material line that it is initially associated with. The vortex line and the material line are both oriented to the same directions, and evolve with the proportional magnitude, just like being ‘frozen' or ‘glued' to the material elements of the fluid under the barotropic assumption. To relax the limits of incompressible and barotropic atmosphere, the frozen-in property is further derived and proved in the baroclinic case. Then two effective usages are given as examples. One is the derivation of potential vorticity conservation from the frozen-in property in both barotropic and baroclinic atmospheres, as a theory application, and the other is used to illuminate the vorticity generation and growth in ideal cases and real severe weather process, e.g., in squall line, tornado, and other severe convection weather with vortex. There is no necessity to derive vorticity equation, and this method is very intuitive to explain vorticity development qualitatively, especially for fast analysis for forecasters. Certainly, by investigating the evolution of vortex line, it is possible to locate the associated line element vector and its development on the basis of the frozen-in property of vorticity. Because it is simple and visualized, it manifests broad application prospects.     

Rossby waves in the magnetic fluid dynamics of a rotating plasma in the shallow-water approximation

We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.     

The coherent structures of shallow-water turbulence: Deformation-radius effects,cyclone/anticyclone asymmetry and gravity-wave generation

Over a large range of Rossby and Froude numbers, we investigate the dynamics of initially balanced decaying turbulence in a shallow rotating fluid layer. As in the case of incompressible two-dimensional decaying turbulence, coherent vortex structures spontaneously emerge from the initially random flow. However, owing to the presence of a free surface, a wealth of new phenomena appear in the shallow-water system. The upscale energy cascade, common to strongly rotating flows, is arrested by the presence of a finite Rossby deformation radius. Moreover, in contrast to near-geostrophic dynamics, a strong asymmetry is observed to develop as the Froude number is increased, leading to a clear dominance of anticyclonic vortices over cyclonic ones, even though no beta effect is present in the system. Finally, we observe gravity waves to be generated around the vortex structures, and, in the strongest cases, they appear in the form of shocks. We briefly discuss the relevance of this study to the vortices observed in Jupiter's atmosphere.     

正压Rossby波扰动能量

利用Fourier变换方法,研究准地转近似下beta平面上绝热、无摩擦、无强迫耗散的正压大气Rossby波扰动能量在有限时段内的快速发展和衰减情形.给出线性正压位势涡度方程扰动流函数的解析解,并进一步分析扰动能量与东西波数、南北波数、基流切变和黏性系数之间的关系.     

Cherenkov interaction of vortices with a free surface

The interaction of vortex filaments in an ideal incompressible fluid with the free surface of the latter is investigated in the canonical formalism. A Hamiltonian formulation of the equations of motion is given in terms of both canonical and noncanonical Poisson brackets. The relationship between these two approaches is analyzed. The Lagrangian of the system and the Poisson brackets are obtained in terms of vortex lines, making it possible to study the dynamics of thin vortex filaments with allowance for finite thickness of the filaments. For two-dimensional flows exact equations of motion describing the interaction of point vortices and surface waves are derived by transformation to conformal variables. Asymptotic steady-state solutions are found for a vortex moving at a velocity lower than the minimum phase velocity of surface waves. It is found that discrete coupled states of surface waves above a vortex are possible by virtue of the inhomogeneous Doppler effect. At velocities higher than the minimum phase velocity the buoyant rise of a vortex as a result of Cherenkov radiation is described in the semiclassical limit. The instability of a vortex filament against three-dimensional kink perturbations due to interaction with the “image” vortex is demonstrated. Zh. éksp. Teor. Fiz. 115, 894–919 (March 1999)     

大尺度浅水波方程中相互调制的非线性波

基于一般的浅水波方程, 根据大尺度正压大气的特点, 得到无量纲的控制大尺度大气的动力学非线性方程组. 利用多尺度法, 由无量纲的动力学方程组导出了扰动位势的非线性控制方程. 采用椭圆方程构造该扰动位势控制方程的解, 获得了扰动位势和速度的多周期波与冲击波(爆炸波) 并存的解析解. 扰动位势的解表明经向和纬向具有不同周期和波长的周期波, 且都受纬向孤波的调制; 速度的解表明大尺度大气流动存在气旋和反气旋周期性分布的现象. 关键词： 浅水波方程 大尺度正压大气 解析解 非线性波     

Helicity generation in uniform helical flows

Helicity generation conditions are derived for helical flows of Joukowski type with allowance for effects due to viscosity, buoyancy, temperature nonuniformity, and solid-body rotation. The upper and lower limits are determined for the rotation-frequency interval in which helicity can be generated by viscous forces. These conditions correspond to the regime of an isolated tornado-like vortex. An exact solution to the time-independent equations of motion for inviscid incompressible flow is obtained. The solution describes a generalized Kelvin-Helmholtz vortex having the form of a localized cylindrical vortex with nontrivial stable topological vortex-core structure determined by a finite value of helicity. For linear traveling inertia waves, which must have uniform helical structure, a general representation is found that characterizes helical structures of different origin.