Solitary Rossby Waves Research Articles

Challenges and Progress in Computational Geophysical Fluid Dynamics in Recent Decades

Here we present the numerical methods, applications, and comparisons with observations and previous studies. It includes numerical analyses of shallow water equations, Sun’s scheme, and nonlinear model simulations of a dam break, solitary Rossby wave, and hydraulic jump without smoothing. We reproduce the longitude and transverse cloud bands in the Equator; two-day mesoscale waves in Brazil; Ekman spirals in the atmosphere and oceans, and a resonance instability at 30° from the linearized equations. The Purdue Regional Climate Model (PRCM) reproduces the explosive severe winter storms in the Western USA; lee-vortices in Taiwan; deformation of the cold front by mountains in Taiwan; flooding and drought in the USA; flooding in Asia; and the Southeast Asia monsoons. The model can correct the small-scale errors if the synoptic systems are correct. Usually, large-scale systems are more important than small-scale disturbances, and the predictability of NWP is better than the simplified dynamics models. We discuss the difference between Boussinesq fluid and the compressible fluid. The Bernoulli function in compressible atmosphere conserving the total energy, is better than the convective available potential energy (CAPE) or the Froude number, because storms can develop without CAPE, and downslope wind can form against a positive buoyancy. We also present a new terrain following coordinate, a turbulence-diffusion model in the convective boundary layer (CBL), and a new backward-integration model including turbulence mixing in the atmosphere.

A new combined ZK-mZK dynamic model for Rossby solitary wave

In this article, using multi-scale and perturbation methods, a partial differential model of Rossby solitary waves with complete Coriolis force effect is obtained, which is called the combined ZK-mZK equation. This equation can reflect the prop?agation characteristics of Rossby waves in space, and is more suitable for real oceans and atmosphere than the (1+1)-dimensional model. According to the pa?rameter composition of the new model, it can be seen that the effect of the complete Coriolis force affects not only the longitudinal structure of the model, but also the amplitude structure of the non-linear long wave.

Dynamics of Rossby solitary waves with time-dependent mean flow via Euler eigenvalue model

Dynamics of Rossby solitary waves with time-dependent mean flow via Euler eigenvalue model

A new nonlinear integral-differential equation describing Rossby waves and its related properties

This paper focuses on algebraic Rossby solitary waves models in stratified fluids. Starting with the quasi-geostrophic potential vorticity equation, we present the Boussinesq-intermediate long-wave (Boussinesq-ILW) model for the first time using the multi-scale analysis and the perturbation expansion method. Compared with previous models describing algebraic Rossby solitary waves, the Boussinesq-ILW model is more general: the equation transforms into the Boussinesq-BO equation when h1→∞, and when [Ξ(B)]XXX changes to BXXXX, the model reduces to the Boussinesq equation. The conservation law is important for exploring the properties of the model, therefore we propose several common conservation laws: mass, momentum, and energy conservation. Finally, based on the trial function method, we obtain the solution of the Boussinesq-ILW equation and investigate the wave-wave interaction. The results show that the peak value of the Mach stem increases as the parameter γ increases, but the length becomes shorter and changes more rapidly.

Variable coefficient KdV system for the non-linear Rossby waves

The paper deals with a non-linear barotropic vorticity equation in a shear flow, the model is applied to get the non-linear Rossby solitary wave with the effects of topography and other external forcing. Basing on the perturbation method, the variable coefficient KdV equation is derived for Rossby waves. The periodic-like solution for the equation is obtained with the help of Jacobi elliptic functions, the solitary solutions can also be obtained in the limit case. The analysis indicates that the wave amplitude and velocity will be related to the topography effect. It is also shown that the external forcing plays an important role in evolution of the waves.

The barotropic Rossby waves with topography on the earth’sδ-surface

AbstractThe Rossby solitary waves in the barotropic vorticity model which contains the topography on the earth’sδ-surface is investigated. First, applying scale analysis method, obtained the generalized quasi-geostrophic potential vorticity equation (QGPVE). Using The Wentzel–Kramers–Brillouin (WKB) theory, the evolution equation of Rossby waves is the variable-coefficient Korteweg–de Vries (KdV) equation for the barotropic atmospheric model. In order to study the Rossby waves structural change to exist in some basic flow and topography on theδ-surface approximation, the variable coefficient of KdV equation must be explicitly, Chebyshev polynomials is used to solve a Sturm-Liouville-type eigenvalue problem and the eigenvalue Rossby waves, these solutions show that the parameterδusually plays the stable part in Rossby waves and slow down the growing or decaying of Rossby waves with the parameterβ.

A Gardner evolution equation for topographic Rossby waves and its mechanical analysis

Topography plays an important role in the excitation and propagation of nonlinear Rossby solitary waves to atmospheres and oceans. In the present study, we investigate the effect of topography from the approach to topographic Rossby waves, not to the geostrophic viewpoint. It is the first time that a new evolution equation, called Gardner equation, is derived to simulate the propagation of nonlinear Rossby waves amplitude by using the methods of multiple scales and weak nonlinearity. In order to investigate the physical mechanisms of topographic Rossby wave, the shooting method is adopted to solve the Sturm-Liouville model equation with fixed boundary conditions and the Fourier spectral method is used to solve the nonlinear Gardner equation. Numerical results reveal that the magnitude of the meridional topography is more important compared to its meridional frequency on the evolution of nonlinear Rossby solitary waves, also, the variation of planetary vorticity is essential for the propagation of Rossby solitary waves.

Solitary waves under curved topography and beta approximation

In this paper, the mechanisms of excitation and propagation of nonlinear Rossby waves are investigated by the approach of topographic balance under the beta approximation for the first time. Using time-space elongation transformation and perturbation expansion method, a Korteweg–de Vries model equation for topographic Rossby wave amplitude is derived. The influences of topography parameters on Rossby solitary waves are discussed through qualitative and quantitative analysis.

A (2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper, we investigate a (2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids. We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasi-geostrophic potential vorticity equation with dissipation and topography under the generalized beta effect, and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method. Through the analysis of this model, it is found that the generalized beta effect and basic topography can induce nonlinear waves, and slowly varying topography is an external impact factor for Rossby waves. Additionally, the conservation laws for the mass and energy of solitary waves are analyzed. Eventually, the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method. Based on the solitary wave solutions obtained, we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.

Topographic solitary waves by the shooting method and Fourier spectral method

The influences of topography on barotropic Rossby solitary waves are discussed in this paper. A Boussinesq model equation of nonlinear Rossby wave amplitude is derived by using the perturbation expansion method and the space–time transformation. A classical problem we treat is the Sturm-Liouville type ordinary differential equation with varied coefficients and fixed boundary conditions. The shooting method is adopted, and the numerical solution of the obtained Boussinesq equation is given by using the Fourier spectral method. The effects of bottom topography and earth’s rotation are both discussed through qualitative and quantitative results, respectively. New conclusions about the topography Rossby waves are declared.

New model and dynamics of higher-dimensional nonlinear Rossby waves

In this work, the propagation of higher-dimensional nonlinear Rossby waves under the generalized beta effect is considered. Using the methods of weak nonlinear perturbation expansions and the multiple scales, we obtain a new (2 + 1)-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation for the first time. Furthermore, a new dispersion relation for the linear Rossby waves is given corresponding to the linearized Boussinesq equation. More importantly, based on the methods of the traveling wave setting and the Jacobi elliptic function expansions, several kinds of exact traveling wave solutions for the higher-dimensional nonlinear Rossby waves, including the periodic solutions, solitary solutions and others are obtained. Finally, we simulate the solitary solutions obtained by using the method of the Jacobi elliptic function. The numerical results show that the amplitude of the Rossby solitary waves is decreasing with the increase of generalized beta effect.

Dynamical Analysis and Exact Solutions of a New (2+1)-Dimensional Generalized Boussinesq Model Equation for Nonlinear Rossby Waves**Supported by the National Natural Science Foundation of China under Grant Nos. 11562014, 11762011, 11671101, 71471020, 51839002; the Natural Science Foundation of Inner Mongolia under Grant No. 2017MS0108; Hunan Provincial Natural Science Foundation of China under Grant No. 2016JJ2061; the Scientific Research Fund of

In this paper, we study the higher dimensional nonlinear Rossby waves under the generalized beta effect. Using methods of the multiple scales and weak nonlinear perturbation expansions [Q. S. Liu, et al., Phys. Lett. A 383 (2019) 514], we derive a new (2+1)-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation. Based on bifurcation theory of planar dynamical systems and the qualitative theory of ordinary differential equations, the dynamical analysis and exact traveling wave solutions of the new generalized Boussinesq equation are obtained. Moreover, we provide the numerical simulations of these exact solutions under some conditions of all parameters. The numerical results show that these traveling wave solutions are all the Rossby solitary waves.

Madden–Julian Oscillations Seen in the Upper-Troposphere Vorticity Field: Interactions with Rossby Wave Trains

AbstractThe vorticity variability associated with the Madden–Julian oscillation (MJO) is examined. The analysis is focused on the 150-hPa pressure level, because a clear dipolar-vortex signal, reminiscent of the theoretically proposed strongly nonlinear solitary Rossby wave solution (albeit with the opposite sign), is seen in raw data at that level. A local empirical orthogonal function (EOF) analysis over the equatorial region of the Eastern Hemisphere (0°–180°E) identifies the two principal components representing an eastward propagation of a dipolar vortex trapped to the equator. Association of this propagation structure with the moist convective variability of the MJO is demonstrated by regressing the outgoing longwave radiation (OLR) against this EOF pair. The obtained evolution of the OLR field is similar to the one obtained by a direct application of the EOF to the OLR. A link of the local vorticity variability associated with the MJO to the global dynamics is further investigated by regressing the global vorticity field against the time series of the identified local EOF pair. The Rossby wave trains tend to propagate toward the Indian Ocean from higher latitudes, just prior to an initiation of the MJO, and in turn, they propagate back toward the higher latitudes from the MJO active region over the Indian Ocean. A three-dimensional regression reveals an equivalent barotropic structure of the MJO vortex pair with the signs opposite to those at 150 hPa underneath. A vertical normal mode analysis finds that this vertical structure is dominated by the equivalent height of about 10 km.

Generalized (2 + 1)-dimensional mKdV-Burgers equation and its solution by modified hyperbolic function expansion method

Abstract We investigate a new higher-dimensional nonlinear dynamics model to describe the generation and evolution of Rossby waves. We derive a generalized (2 + 1)-dimensional modified Korteweg-de Vries (mKdV)-Burgers equation by considering the quasi-geostrophic potential vorticity equation under the generalized β approximation with dissipation and external source in barotropic fluids, and by utilizing multiple scales and the perturbation expansion method. Qualitative analysis yields that the generalized β and shear basic flow can induce the nonlinear Rossby solitary waves, and dissipation has effects on the propagation of Rossby waves as well. Then, we analyze the conservation laws for the mass and energy of Rossby solitary waves. Moreover, the asymptotic kink-shaped solitary wave solution for the generalized (2 + 1)-dimensional mKdV-Burgers equation is explored by modified hyperbolic function expansion method. The solitary wave solutions and figures are analyzed, the results show that the dissipation effect causes the speed and amplitude of Rossby solitary waves to decrease exponentially and the width of Rossby solitary waves to increase exponentially with time.

Nonlinear planetary-synoptic wave interaction under generalized beta effect and its solutions

Abstract The interaction between planetary-scale wave and synoptic-scale wave in atmospheres is important in understanding the physical mechanism of short or long term weather or climate events, such as the blocking phenomena. Kinds of physical factors are disclosed to affect the interaction processes, such as the topography, background current. The effect of beta parameter is investigated in the present paper, it is called the generalized beta effect. By using methods of multiple scales and perturbation expansions, a new nonlinear forced Schrodinger equation is obtained in describing the evolution of planetary-scale envelope Rossby solitary waves, and a modified equation for synoptic-scale waves is derived. By constructing the numerical solution for the nonlinear Schrodinger equation, it reveals that the generalized beta can shift phase and modify the magnitude of planetary-scale envelope solitary waves. An analytical expression for synoptic-scale waves, including the generalized beta effect, is also obtained. It shows that the asymmetry, intensity and persistence of both planetary-scale wave and synoptic-scale wave depend strongly upon the generalized beta. The results provide new theoretical explanations for our understanding of wave-wave interaction.

Nonlinear Schrödinger equation for envelope Rossby waves with complete Coriolis force and its solution

The physical features of the equatorial envelope Rossby waves including with complete Coriolis force and dissipation are investigated analytically. Staring with a potential vorticity equation, the wave amplitude evolution of equatorial envelope Rossby waves is described as a nonlinear Schrodinger equation by employing multiple scale analysis and perturbation expansions. The equation is more suitable for describing envelope Rossby solitary waves when the horizontal component of Coriolis force is stronger near the equator. Then, based on the Jacobi elliptic function expansion method and trial function method, the classical Rossby solitary wave solution and the corresponding stream function of the envelope Rossby solitary waves are obtained, respectively. With the help of these solutions, the effect of dissipation and the horizontal component of Coriolis parameter are discussed in detail by graphical presentations. The results reveal the effect of the horizontal component of Coriolis force and dissipation on the classical Rossby solitary waves.

Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids

In this paper, the theoretical model of Rossby waves in two-layer fluids is studied. A single quasi-geostrophic vortex equation is used to derive various models of Rossby waves in a one-layer fluid in previous research. In order to explore the propagation and interaction of Rossby waves in two-layer fluids, from the classical quasi-geodesic vortex equations, by employing the multi-scale analysis and turbulence method, we derived a new (2+1)-dimensional coupled equations set, namely the generalized Zakharov-Kuznetsov(gZK) equations set. The gZK equations set is an extension of a single ZK equation; they can describe two kinds of weakly nonlinear waves interaction by multiple coupling terms. Then, for the first time, based on the semi-inverse method and the variational method, a new fractional-order model which is the time-space fractional coupled gZK equations set is derived successfully, which is greatly different from the single fractional equation. Finally, group solutions of the time-space fractional coupled gZK equations set are obtained with the help of the improved ( G ′ / G ) -expansion method.

A forced 3-D time fractional ZK-Burgers model for Rossby solitary waves with dissipation and thermal forcing

In the paper, beginning from the quasi-geostrophic potential vorticity equation with the dissipation and thermal forcing in stratified fluid, by employing multi-scale analysis and perturbation method, we derive a forced 3-D Zakharov Kuznetsor (ZK)-Burgers equation describe the propagation of the Rossby solitary waves within the fractional derivative. The exact solutions are given by virtue of the (G?/G)-expansion method to analyze the excitation effect of thermal forcing on the Rossby waves.

Nonlinear Rossby waves in zonally varying flow under generalized beta approximation

We investigate the dynamics of nonlinear Rossby waves in zonally varying background jet flow under generalized beta approximation. Potential forcing, topography and dissipation are all included as influential physical factors. We derive a new modified (2 + 1) dimensional Zakharov-Kuznetsov model equation with variable coefficients in describing the spatial-temporal evolution of Rossby wave amplitude. Qualitative analysis declares the complex evolutionary processes of the nonlinear Rossby waves affected by multiple physical factors. First, it indicates that the generalized beta and basic topography are essential factors in exciting the nonlinear solitary Rossby waves, especially, for the case without shear basic flow. Second, the zonal variation of basic flow affects the zonal linear phase speed of the Rossby wave amplitude, it also affects the linear growth or decay characteristics of the three dimensional nonlinear Rossby waves, and it plays as a forcing factor on the nonlinear Rossby waves. Third, the spatial-temporal variation for higher order topography affects the evolution of solitary Rossby wave amplitude in different ways. It reveals that the weakly varying topography represents an unstable physical mechanism for the evolution of solitary Rossby waves. Besides, numerical simulations are also given to analyse the dynamics of nonlinear Rossby waves.

Conservation Laws of Space-Time Fractional mZK Equation for Rossby Solitary Waves with Complete Coriolis Force

Abstract The study of Rossby solitary waves are of great significance in physical oceanography, atmospheric physics, water conservancy project, military and communications engineering, etc. All the time, in the study of Rossby solitary waves, people have been focusing on integer order models. Recently, fractional calculus has become a new research hotspot, and it has opened a new door to research atmospheric and ocean. Thus, the fractional order model has the potential value in the study of Rossby solitary waves. In the present paper, according to the quasi-geostrophic potential vorticity equation with the complete Coriolis force, we get a new integer order mZK equation. Using the semi-inverse method and the fractional variational principle, the space-time fractional mZK(STFmZK) equation is obtained. To better understand the property of Rossby solitary waves, we study Lie symmetry analysis, nonlinear self-adjointness, similarity reduction by applying the STFmZK equation. In the end, the conservation and Caputo fractional derivative are discussed, respectively.