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

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.

For the barotropic fluid, based on the quasi-geostrophic potential vorticity equation, an inhomogeneous Boussinesq equation including topographic forcing and an external source is derived by employing the perturbation method and stretching transform of time and space. Through inspection of the evolution of the amplitude of the Rossby wave, it is found that Coridis effect, topography effect and an external source are the key inducing factors of the solitary Rossby wave if the basic stream function has a shear flow. Assuming that there is a balance between nonlinear and topography effects, an inhomogeneous Boussinesq equation is derived. The results show that the topography and the Rossby wave interact in the barotropic flow. The inhomogeneous Boussinesq equation describing the evolution of the amplitude of solitary Rossby waves with the change of Rossby parameter β(y) with latitude β, topographic forcing and the external source is obtained.

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

This study concerns the propagation of three types of Rossby waves—planetary, compressional, and vortex. All these Rossby waves are characterized by flow disturbances with alternating signs of relative vorticity, and their propagation requires a potential vorticity (PV) gradient in the background flow. When fluid parcels are displaced, according to the conservation of PV, the relative vorticity changes, which drives the propagation of Rossby waves. However, these three Rossby waves differ not only by conserving distinct forms of PV but also by propagating in seemingly opposite directions. Relative to the background flow, the propagation is westward for the planetary Rossby waves but eastward for the compressional ones, and the vortex Rossby waves propagate against the cyclonic circulation of a tropical cyclone. Here we illustrate a physical picture for the eastward propagation of the compressional Rossby waves for pedagogical purposes. Furthermore, we use our right hand to aid in explaining the propagation direction of all the Rossby waves. This right-hand rule signifies the disturbance positive vorticity, background PV gradient, and Rossby wave propagation with the thumb, fingers, and palm, respectively. This paper shows that this right-hand rule is critical for explaining the continuous transition between westward- and eastward-propagating disturbances of mixed planetary-compressional Rossby waves (MPCRWs) given the same background flow condition in the equatorial atmosphere. Moreover, this right-hand rule can serve as an aid for teaching about the planetary, compressional, and vortex Rossby waves by providing a unified explanation for the distinct types of Rossby wave propagation.

This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.

Nonlinear Rossby waves in a Boussinesq fluid model which includes both the vertical and horizontal components of Coriolis force are studied by using the semi-geostrophic approximation and the method of travelling-wave solution. Taylor series expansion has been employed to isolate the characteristics of the linear Rossby waves and identify the Rossby cnoidal and solitary waves. Qualitative analysis indicates that if the disturbances are independent of latitude, the effect of horizontal components of Coriolis force disappears.

The authors investigate the relationship between hydrology in the Great Lakes basin—namely, overlake precipitation and transient Rossby waves—using the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data and historical output from phase 3 of the Coupled Model Intercomparison Project (CMIP3). The preferred path of observed Rossby wave trains associated with overlake precipitation on Lake Superior depends strongly on season and appears to be related to the time-mean, upper-level flow. During summer and fall, the Northern Hemisphere extratropical jet is relatively narrow and acts as a waveguide, such that Rossby wave trains traversing the Great Lakes region travel along the extratropical Pacific and Atlantic jets. During other months, the Pacific jet is relatively broad, which allows more wave activity originating in the tropics to penetrate into the midlatitudes and influence Lake Superior precipitation. Analysis is extended to CMIP3 models and is intended to 1) further understanding of how variations in the mean state influence transient Rossby waves and 2) assess models’ ability to capture observed features, such as wave origin and track. Results indicate that Rossby wave train propagation in twentieth-century simulations can significantly differ by model. Unlike observations, some models do not produce a well-defined jet across the Pacific Ocean during summer and autumn. In these models, some Rossby waves affecting the Great Lakes region originate in the tropics. Collectively, observations and model results show the importance of the time-mean upper-level flow on Rossby wave propagation and therefore on the relative influence of the tropics versus the extratropics on the hydroclimate of the Great Lakes region.

In this work, we investigated the modulated, nonlinear, ion-acoustic wave (IAW) propagating in an unmagnetized, collisionless, homogeneous plasma where a charged density source of varying velocity due to the debris particles is present. The space-time dynamics of the one dimensional nonlinear ion-acoustic wave is modelled by a forced Nonlinear Schrodinger equation (fNLSE) where the forcing function is generated because of the space debris objects. We obtained : the exact 1 solitary wave, 2 solitary wave followed by N solitary wave solutions and the 1st order rogue wave, the 2nd order rogue wave and then the Nth order rogue wave solutions of the fNLSE for some specific forms of pinned debris functions. The combined system of the nonlinear ion-acoustic wave and the debris function propagates with acceleration. These debris induced, accelerated, exact, modulated nonlinear wave solutions are new in this field of plasma physics as per our study. The condition of modulation instability of such forced system is also discussed with plot. Such exact, nonlinear wave solutions with acceleration may be useful in modelling the experimental data of real astrophysical plasma system.

The generation of Rossby rogue waves (Rossby rogons), as well as the excitation of bright and dark Rossby envelpe solitons are demonstrated on the basis of the modulational instability (MI) of a coherent Rossby wave packet. The evolution of an amplitude-modulated Rossby wave packet is governed by a one-dimensional (1D) nonlinear Schrödinger equation (NLSE). The latter is used to study the amplitude modulation of Rossby wave packets for fluids in Earth's atmosphere and in the solar photosphere. It is found that an ampitude-modulated Rossby wave packet becomes stable (unstable) against quasi-stationary, long-wavelength (in comparision with the Rossby wavelength) perturbations, when the carrier Rossby wave number satisfies k2 < 1/2 or (k2 > 3 or ). It is also shown that a Rossby rogon or a bright Rossby envelope soliton may be excited in the shallow-water approximation for the Rossby waves in solar photosphere. However, the excitation of small- or large-scale perturbations may be possible for magnetized plasmas in the ionosphereic E-layer.

The study of atmospheric Rossby waves could provide a theoretical basis for weather forecasting systems. In this paper, the barotropic quasi-geostrophic model is used to discuss the nonlinear Rossby waves induced by large-scale topography under the β-plane approximation. Starting from the potential vorticity equation, the atmospheric area is divided into internal region and external region, and the perturbation expansions of the two equations are carried out by using the separate multi-scale perturbation method. The Benjamin–Ono equation describing the wave amplitude is obtained by matching solutions of the internal and external equations at the boundaries. The distributions of the perturbation stream function, the total stream function, and the vorticity are obtained. It is found that the algebraic solitary waves induced by large-scale topography do exist. Moreover, the amplitude of the nonlinear Rossby waves is affected along with the changes of topography.

In this article, we investigated new traveling wave solutions for the loaded modified Korteweg-de Vries-Kadomtsev-Petviashvili equation via the functional variable method. The performance of this method is reliable and effective and gives the exact solitary and periodic wave solutions. All solutions of this equation have been examined and three-dimensional graphics of the obtained solutions have been drawn by using the Matlab program. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions and solitary wave solutions of kink type. This method presents a wider applicability for handling nonlinear wave equations.

The horizontal gradient of potential vorticity (PV) across the tropopause typically declines with lead time in global numerical weather forecasts and tends towards a steady value dependent on model resolution. This paper examines how spreading the tropopause PV contrast over a broader frontal zone affects the propagation of Rossby waves. The approach taken is to analyse Rossby waves on a PV front of finite width in a simple single-layer model. The dispersion relation for linear Rossby waves on a PV front of infinitesimal width is well known; here, an approximate correction is derived for the case of a finite-width front, valid in the limit that the front is narrow compared to the zonal wavelength. Broadening the front causes a decrease in both the jet speed and the ability of waves to propagate upstream. The contribution of these changes to Rossby wave phase speeds cancel at leading order. At second order the decrease in jet speed dominates, meaning phase speeds are slower on broader PV fronts. This asymptotic phase speed result is shown to hold for a wide class of single-layer dynamics with a varying range of PV inversion operators. The phase speed dependence on frontal width is verified by numerical simulations and also shown to be robust at finite wave amplitude, and estimates are made for the error in Rossby wave propagation speeds due to the PV gradient error present in numerical weather forecast models.

The classic partitioning between slow-moving, low-wavenumber planetary waves and fast-moving, high-wavenumber synoptic waves is systematically extended by means of a wavenumber/phase speed spectral decomposition to characterize the day-to-day evolution of Rossby wave activity in the upper troposphere. This technique is employed to study the origin and the propagation of circumglobal Rossby wave patterns (CRWPs), amplified Rossby waves stretching across the Northern Hemisphere in the zonal direction and characterized by few, dominant wavenumbers. Principal component analysis of daily anomalies in spectral power allows for two CRWPs to emerge as dominant variability modes in the spectral domain during boreal winter. These modes correspond to the baroclinic propagation of amplified Rossby waves from the Pacific to the Atlantic storm track in a hemispheric flow configuration displaying enhanced meridional gradients of geopotential height over midlatitudes. The first CRWP is forced by tropical convection anomalies over the Indian Ocean and features the propagation of amplified Rossby wave packets over northern midlatitudes, while the second one propagates rapidly over latitudes between 35° N and 55° N and appears to have extratropical origin. Propagation of Rossby waves from the Atlantic eddy-driven jet to the African subtropical jet occurs for both CRWPs following anticyclonic wave breaking.

<strong class="journal-contentHeaderColor">Abstract.</strong> The classic partitioning between slow-moving, low-wavenumber planetary waves and fast-moving, high-wavenumber synoptic waves is systematically extended by means of a space–time spectral decomposition to characterize the day-to-day evolution of Rossby wave activity in the upper troposphere. This technique is employed to study the origin and the propagation of circumglobal Rossby wave patterns (CRWPs), amplified Rossby waves stretching across the Northern Hemisphere in the zonal direction and projecting primarily over few, dominant wavenumber–phase-speed harmonics. Principal component analysis of daily anomalies in spectral power allows for two CRWPs to emerge as leading variability modes in the spectral domain during boreal winter. These modes correspond to the baroclinic propagation of Rossby wave packets (RWPs) from the Pacific to the Atlantic storm track in a hemispheric flow configuration displaying enhanced meridional gradients of geopotential height over midlatitudes. The first CRWP is forced by tropical convection anomalies over the Indian Ocean and features the propagation of amplified RWPs over northern midlatitudes, while the second one propagates rapidly over latitudes between 35 and 55<span class="inline-formula">^∘</span> N and appears to have extratropical origin. An anomalous equatorward propagation of Rossby waves from the Atlantic eddy-driven jet to the North African subtropical jet is observed for both CRWPs. The obtained results highlight the substantial contribution of propagating RWPs to CRWPs, hinting that the two features might have the same nature.

The classic partitioning between slow-moving, low-wavenumber planetary waves and fast-moving, high-wavenumber synoptic waves is systematically extended by means of a wavenumber/phase speed spectral decomposition to characterize the day-to-day evolution of Rossby wave activity in the upper troposphere. This technique is employed to study the origin and the propagation of circumglobal Rossby wave patterns (CRWPs), amplified Rossby waves stretching across the Northern Hemisphere in the zonal direction and characterized by few, dominant wavenumbers. Principal component analysis of daily anomalies in spectral power allows for two CRWPs to emerge as dominant variability modes in the spectral domain during boreal winter. These modes correspond to the baroclinic propagation of amplified Rossby waves from the Pacific to the Atlantic storm track in a hemispheric flow configuration displaying enhanced meridional gradients of geopotential height over midlatitudes. The first CRWP is forced by tropical convection anomalies over the Indian Ocean and features the propagation of amplified Rossby wave packets over northern midlatitudes, while the second one propagates rapidly over latitudes between 35° N and 55° N and appears to have extratropical origin. Propagation of Rossby waves from the Atlantic eddy-driven jet to the African subtropical jet occurs for both CRWPs following anticyclonic wave breaking.