Amplitude Of Rossby Waves Research Articles

Assessment of climate biases in OpenIFS version 43r3 across model horizontal resolutions and time steps

Abstract. We examine the impact of horizontal resolution and model time step on the climate of the OpenIFS version 43r3 atmospheric general circulation model. A series of simulations for the period 1979–2019 are conducted with various horizontal resolutions (i.e. ∼100, ∼50, and ∼25 km) while maintaining the same time step (i.e. 15 min) and using different time steps (i.e. 60, 30, and 15 min) at 100 km horizontal resolution. We find that the surface zonal wind bias is significantly reduced over certain regions such as the Southern Ocean and the Northern Hemisphere mid-latitudes and in tropical and subtropical regions at a high horizontal resolution (i.e. ∼25 km). Similar improvement is evident too when using a coarse-resolution model (∼100 km) with a smaller time step (i.e. 30 and 15 min). We also find improvements in Rossby wave amplitude and phase speed, as well as in weather regime patterns, when a smaller time step or higher horizontal resolution is used. The improvement in the wind bias when using the shorter time step is mostly due to an increase in shallow and mid-level convection that enhances vertical mixing in the lower troposphere. The enhanced mixing allows frictional effects to influence a deeper layer and reduces wind and wind speed throughout the troposphere. However, precipitation biases generally increase with higher horizontal resolutions or smaller time steps, whereas the surface air temperature bias exhibits a small improvement over North America and the eastern Eurasian continent. We argue that the bias improvement in the highest-horizontal-resolution (i.e. ∼25 km) configuration benefits from a combination of both the enhanced horizontal resolution and the shorter time step. In summary, we demonstrate that, by reducing the time step in the coarse-resolution (∼100 km) OpenIFS model, one can alleviate some climate biases at a lower cost than by increasing the horizontal resolution.

Baroclinic Instability of a Time-Dependent Zonal Shear Flow

In the real atmosphere, the development of large-scale motion is often related to the baroclinic properties of the atmosphere. So, it is necessary to discuss the stability condition of baroclinic flow. It is advantageous to use a layered model to discuss baroclinic instability, not only to apply the potential vortex equation directly, but also to deal with shear of basic flow. The stability and oscillatory shear ability of Rossby waves are studied based on the two-layer Phillips model in the β plane; then, we summarize the baroclinic instability of time-dependent zonal shear flows. The multiscale method is used to eliminate some terms of natural frequency oscillations of nonlinear operators in the third-order expansion, thus generating an equation about the amplitude of the lowest-order Rossby wave in the long-time variable. The large amplitude perturbation begins to decrease, which produces the desired behavior. After the amplitude decreases for some time, the amplitude of Rossby waves can still be found to oscillate periodically with the time variable.

How well is Rossby wave activity represented in the PRIMAVERA coupled simulations?

Abstract. This work aims to assess the performance of state-of-the-art global climate models in representing the upper-tropospheric Rossby wave pattern in the Northern Hemisphere and over the European–Atlantic sector. A diagnostic based on finite-amplitude local wave activity is used as an objective metric to quantify the amplitude of Rossby waves in terms of Rossby wave activity. This diagnostic framework is applied to a set of coupled historical climate simulations at different horizontal resolutions, performed in the framework of the PRIMAVERA project and compared with observations (ERA5 reanalysis). At first, the spatio-temporal characteristics of Rossby wave activity in the Northern Hemisphere are examined in the multimodel mean of the whole PRIMAVERA set. When examining the spatial distribution of transient wave activity, only a minimal improvement is found in the high-resolution ensemble. On the other hand, when examining the temporal variability of wave activity, a higher resolution is beneficial in all models apart from one. In addition, when examining the Rossby wave activity time series, no evident trends are found in the historical simulations (at both standard and high resolutions) and in the observations. Finally, the spatial distribution of Rossby wave activity is investigated in more detail focusing on the European–Atlantic sector, examining the wave activity pattern associated with weather regimes for each model. Results show a marked inter-model variability in representing the correct spatial distribution of Rossby wave activity associated with each regime pattern, and an increased horizontal resolution improves the models' performance only for some of the models and for some of the regimes. A positive impact of an increased horizontal resolution is found only for the models in which both the atmospheric and oceanic resolution is changed, whereas in the models in which only the atmospheric resolution is increased, a worsening model performance is detected.

The Predictability of the Extratropical Flow Response to Recurving Atlantic Tropical Cyclones

AbstractGiven the ability of recurving Atlantic tropical cyclones (TCs) to disturb the amplitude of Rossby waves on the extratropical jet, this study investigates whether the predictability of the synoptic-scale flow is significantly modified from climatology downstream from and after TC recurvature events. Predictability is evaluated as the standard deviation of isentropic potential vorticity among a 50-member ensemble and is compared to a model climatology. It is shown that forecast uncertainty is dependent upon the relative location of the nearest trough at the time of recurvature and the relative zonal speed between the aforementioned trough and the TC in the 72 hours after recurvature. Predictability is significantly degraded when recurvature occurs downstream of a trough; the elevated uncertainty subsequently propagates downstream along with the trough axis. Furthermore, this study evaluates predictability in spectral space in order to distinguish between uncertainty tied to the exact location of troughs and ridges and uncertainty in Rossby wave amplitude. The wavelet analysis demonstrates that the increase in uncertainty is not solely limited to the trough location, as there is also significantly elevated uncertainty in the Rossby wave amplitude that originates from the upstream trough and spans across downstream troughs and ridges. Uncertainty is also increased near the recurvature longitude in the subset of cases in which the Rossby wave train propagates zonally slower than the TC after recurvature, which is hypothesized to be linked to baroclinic growth processes.

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.

Space Weather Challenge and Forecasting Implications of Rossby Waves

AbstractRossby waves arise in thin layers within fluid regions of stars and planets. These global wave‐like patterns occur due to the variation in Coriolis forces with latitude. In the past several years observational evidence has indicated that there are also Rossby waves in the Sun. Although Rossby waves have been detected in the Sun's photosphere and corona, they most likely originate in the solar tachocline, the sharp shear layer at the base of the solar convection zone, where the differential rotation driven by convection transitions to the solidly rotating radiative interior. These waves differ from their Earth's counterparts by being strongly modified by toroidal magnetic fields in the solar tachocline. Recent simulations of magnetohydrodynamics of tachocline Rossby waves and magnetic fields are demonstrated to produce strong “tachocline nonlinear oscillations,” which have periods similar to those observed in the solar atmosphere—enhanced periods of solar activity, or “seasons”—occurring at intervals between six months and two years. These seasonal/subseasonal bursts produce the strongest eruptive space weather events. Thus, a key to forecasting the timing, amplitude, and location of future activity bursts, and hence space weather events, could lie in our ability to forecast the phase and amplitude of Rossby waves and associated tachocline nonlinear oscillations. Accurately forecasting the properties of solar Rossby waves and their impact on space weather will require linking surface activity observations to the magnetohydrodynamics of tachocline Rossby waves, using modern data assimilation techniques. Both short‐term (hours to days) and long‐term (decadal to millennial) forecasts of space weather and climate are now being made. We highlight in this article the potential of solar Rossby waves for forecasting space weather on intermediate time scales, of several weeks to months up to a few years ahead.

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.

A Climatology of the Extratropical Flow Response to Recurving Atlantic Tropical Cyclones

Abstract Previous case studies have noted a significant extratropical flow response to recurving Atlantic tropical cyclones (TCs), which is often linked to extreme weather events downstream. This study examines the modification of Rossby waves on the extratropical jet in response to recurving Atlantic TCs from a climatological perspective. Changes in amplitude and location of Rossby waves are identified using a wavelet decomposition technique on isentropic potential vorticity. The climatology demonstrates that recurving Atlantic TC events are capable of modifying the amplitude of the extratropical flow. Though the majority of TCs did not produce a significant, systematic modification of the extratropical flow amplitude, a subset of events were associated with a period of significant Rossby wave deamplification occurring from the time of recurvature to 48 h after recurvature, followed by a return of the Rossby wave power beginning around 96 h after recurvature. The characteristics of the TCs were not significantly associated with the resulting extratropical flow modification—a result consistent with previous western North Pacific climatologies. The nature of the extratropical flow response is most strongly tied to the average translation speed of the TC relative to the Rossby wave over the 72 h following recurvature. This study highlights the importance of investigating the extratropical flow response to recurving Atlantic TCs with regards to predictability.

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 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.

Dynamics of nonlinear Rossby waves in zonally varying flow with spatial-temporal varying topography

In the present work, we investigate the dynamics of nonlinear Rossby waves in zonally varying background current under generalized beta approximation. The effects of the zonally varying background current, the spatial-temporal varying topography, the potential forcing and the dissipation on nonlinear Rossby waves are all taken into consideration. We derive a new modified Korteweg–deVries equation with variable coefficients for the Rossby wave amplitude with the help of multiple scales method and perturbation expansions. Based on the obtained model equation, the physical mechanisms of nonlinear Rossby waves are analyzed. Within the present selected parameter ranges, the qualitative results demonstrate that the generalized beta and basic topography are essential factors in exciting the nonlinear Rossby solitary waves. In addition, the zonally varying flow affects the linear phase speed and the linear growth or decay characteristics of the waves. The results also show that the spatial-temporal slowly varying topography, which represents an unstable mechanism for the evolution of Rossby solitary waves, is a factor in linear growth or decay. Furthermore, to validate the efficiency of the obtained model equation, a weakly nonlinear method and numerical simulation are adopted to solve the obtained equation and the results indicate the consistency between the qualitative analysis and the quantitative solutions in explaining the present equation.

([formula omitted])-dimensional ZK–Burgers equation with the generalized beta effect and its exact solitary solution

The (1 ＋1)-dimensional mathematical model had been extensively derived to describe Rossby solitary waves in a line in the past few decades. But as is well known, the (1 ＋1)-dimensional model cannot reflect the generation and evolution of Rossby solitary waves in a plane. In this paper, a (2 ＋1)-dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived to describe the evolution of Rossby wave amplitude by using methods of multiple scales and perturbation expansions from the quasi-geostrophic potential vorticity equations with the generalized beta effect. The effects of the generalized beta and dissipation are presented by the Zakharov–Kuznetsov–Burgers equation. We also obtain the new solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent with the help of the Bernoulli equation, which is different from the common classical solitary solution. Based on the solution, the features of the variable coefficient are discussed by geometric figures Meanwhile, the approximate solitary solution of Zakharov–Kuznetsov–Burgers equation is given by using the homotopy perturbation method. And the amplitude of solitary waves changing with time is depicted by figures. Undoubtedly, these solitary solutions will extend previous results and better help to explain the feature of Rossby solitary waves.

Resolution Dependence and Rossby Wave Modulation of Atmospheric Rivers in an Aquaplanet Model

AbstractAtmospheric rivers (ARs) are examined in a set of aquaplanet simulations using the Model for Prediction Across Scales dynamical core run at multiple horizontal resolutions, namely, 240, 120, and 60 km. As the resolution is increased, there is an increase in the occurrence of long‐lasting ARs. At the same time there is also an increase in the local finite‐amplitude wave activity (LWA) of upper‐tropospheric absolute vorticity, a measure for Rossby wave phase and amplitude that is closely linked with wave breaking. Consistent with the notion that changes in ARs are driven by midlatitude dynamics, a strong relationship is identified between ARs and the equatorward component of LWA. A logistic regression model is used to quantify the probability of AR occurrence based solely on LWA and explains most of the change in AR frequency with resolution. LWA is a diagnostic that may be easily applied to the broadly available output of phase 6 of the Coupled Model Intercomparison project and other model simulations, thus enabling scientists to infer AR and Rossby wave characteristics. AR characteristics, in particular, require higher‐resolution moisture and winds at multiple levels that are not always easily available.

Structure of equatorial envelope Rossby solitary waves with complete Coriolis force and the external source

In this article, the behavior of the equatorial envelope Rossby solitary waves with complete Coriolis force and the external source are investigated analytically. By the asymptotic method of multiple scales and perturbation expansions, a new cubic nonlinear Schrödinger equation with complete Coriolis force and the external source is derived to describe the evolution of equatorial envelope Rossby solitary waves. The equation is different from the common Schrödinger equation, it is more suitable for describing envelope Rossby solitary waves when the horizontal component Coriolis force is stronger near the equator. Using the equation, the features of generalized beta, the horizontal component of Coriolis force and the external source are presented. And then various periodic structures for equatorial envelope Rossby solitary waves are obtained with the help of Jacobi elliptic functions and elliptic equation. Particularly, the solution of envelope Rossby solitary waves is obtained and graphical presentations are shown for Rossby solitary waves amplitude with the different Coriolis parameters. It is pointed out that with decreasing of Coriolis parameter λ, the amplitude of Rossby solitary waves decreases, whereas the propagating frequency is unchanged. And we find that the periodic solution of the nonlinear Schrödinger equation have different structures with a phase-locked diabatic heating source and without a source.

$$\varvec{(2+1)}$$ ( 2 + 1 ) -Dimensional nonlinear Rossby solitary waves under the effects of generalized beta and slowly varying topography

In this paper, \((2+1)\)-dimensional nonlinear Rossby waves are considered with the generalized beta, the dissipation and the topography which includes both basic part and slowly varying part with time. Starting with a barotropic quasi-geostrophic potential vorticity equation, by using methods of multiscales and perturbation expansions, a generalized forced Zakharov–Kuznetsov equation is obtained in describing the evolution of Rossby wave amplitude. The effects of generalized beta, topography along with latitude and slowly variation with time are all included, indicating that the generalized beta is an essential factor in inducing the nonlinear Rossby solitary waves and the other two are both important factors for the evolution of Rossby wave amplitude. Periodic and solitary wave solutions of Zakharov–Kuznetsov equation are obtained by the elliptic function expansion method; meanwhile, solitary wave solution of generalized forced Zakharov–Kuznetsov equation is obtained by reduced differential transform method. At last, graphical presentations for solitary wave amplitude with different dissipations and slowly varying topographies with time are shown by the Mathematica.

(2[formula omitted]1) dimensional Rossby waves with complete Coriolis force and its solution by homotopy perturbation method

In this paper, the effects of both complete Coriolis force and dissipation on equatorial nonlinear Rossby wave are investigated analytically. From the quasi-geostrophic potential vorticity equation, by using methods of multiple scales and perturbation expansions, a (2+1) dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived in describing the evolution of Rossby wave amplitude. The effects of generalized beta, the horizontal component of Coriolis parameter and the dissipation are presented from the Zakharov–Kuznetsov–Burgers equation. We also obtain the classical solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent by elliptic function expansion method, and the complete Coriolis force effect can be seen by the solution. But the method is failed to Zakharov–Kuznetsov-Burgers equation, therefore, we use the efficient homotopy perturbation method to solve the Zakharov–Kuznetsov–Burgers equation.

High resolution numerical simulation of typhoon Mujigae (2015) and analysis of vortex Rossby waves

Mesoscale weather research and forecasting model with high resolution is used to investigate the landfall process of typhoon Mujigae (2015). The simulation well reproduces the path, intensity and rainfall of the typhoon, especially before and after the landfall. The fine thermal and dynamical structures of the typhoon Mujigae and its macroscopic characteristics of rain bands are examined with the simulation output. The rain band regions from the eyewall outward are composed of mixing rain band, secondary rain band, principal rain band and distant rain band. The lower-level inflow and upper-level outflow are observed in the eyewall. The maximum tangential wind, strong updraft and positive temperature anomaly are located in the eyewall and tilted outward with height. The convective systems in the eyewall with high radar reflectivity are much deeper than those in the principal rain band, secondary rain band and distant rain band.In order to analyze the vortex Rossby waves, the fast Fourier transform is performed to decompose the model output variables into perturbations with different wavenumbers. The vorticity perturbations in the wavenumbers 1 and 2 have significant features in the azimuthal and radial propagation. The amplitude of wavenumber 1 is larger than that of wavenumber 2, while the wavenumber 2 propagates much faster than the wavenumber 1 both in azimuthal and radial directions. The waves propagate with a speed less than 10 m/s, which are in consistent with the magnitudes of the radial velocities in spiral rain band. The amplitude of vortex Rossby waves decreases quickly beyond the stagnation radius which is about 90 km from the cyclone center. For the perturbations of wavenumbers 1 and 2, there are some intrinsic relations among the vertical vorticity, divergence and vertical velocity. The positive values of vertical vorticity with the two wavenumbers are associated with the strong reflectivity indicating deep convections. When the dipole patterns of positive vorticity in the upper level and negative vorticity in the lower level over the rainfall region are coupled with the pattern of divergence, the upper-level divergence and lower-level convergence are promoted. Then, updrafts are enhanced, which is favorable for the development of convective system and the increase of precipitation. On the other hand, the updrafts can be weakened in two cases: i) the vertical distribution of negative vorticity in the upper level and positive vorticity in the lower level is similar to the divergence distribution; ⅱ) the vertical distribution of vorticity is opposite to that of divergence. Consequently, the convective systems are inhibited and less rainfall is produced. The dynamical structures of vortex Rossby waves with wavenumbers 1 and 2 affect the development of deep convective system and precipitation in the typhoon Mujigae.

On the barotropic instability using wave ray theory

Based on the spherical Rossby wave ray theory, dynamic characteristics of the barotropic disturbances associated with the westerly jet are discussed, and the barotropic instability conditions are reconsidered. The well-known Rayleigh-Kuo instability criterion indicates that the gradient of potential vorticity must change sign in the unstable area. Kuo uses the method of eigenvalues by assuming disturbances to be simple harmonic waves; however, in this work we assume the atmosphere to be a slowly varying medium, and calculate the amplitude of the Rossby wave by ray tracing. We find a band where the quasi-geostrophic potential vorticity gradient ( β M) is negative, and the amplitude of the northward perturbation grows substantially in this area, making it an unstable region for northward perturbations. In addition the perturbations cannot cross the line of β M=0, and therefore, this is referred to as the “trapped line”. Characteristics of the nonstationary Rossby wave are first analyzed using an ideal zonal mean wind profile u = 30 sin 8 ( 2 φ ) + 0.2 , with the “trapped line” located at 58°N and 70°N, and β M l = 8, but with different zonal wave numbers k . The results suggest that the behavior of the disturbances depends on the initial wave numbers k and l . Northward disturbances with smaller k can propagate further, while disturbances with larger k cannot reach high latitudes and turn back towards the south at turning points β M / ( u M − σ k ) − k 2 = 0. As such, only long Rossby waves with small k values can reach the “trapped line”, while short Rossby waves with larger k values cannot reach it. Long Rossby waves with k = 1, l = 8, first propagate northwest, and then turn eastwards at the “trapped line”. Their energy increases while propagating northward, reaching its maximum in the northern part of the jet stream, and then decays rapidly to zero near the “trapped line”. The energy increases during northward propagation, which means the disturbance obtains energy from the base flow in the southern part of the jet stream, and gives it back to the base flow near the “trapped line”, which completes energy transport from south to north. Further, we also calculated the energy evolution with realistic January and June wind profiles using NCEP data. We conclude that long waves with k <3 are easily trapped by the “trapped line” in both winter and summer, while the propagation of short waves differs substantially between winter and summer. The non-stationary disturbances at low latitudes are easily blocked by easterly winds during winter; however, they can cross easterlies and then propagate to the southern hemisphere in summer, forming a westward wave in the easterlies. It is known that stationary Rossby waves cannot propagate in the easterlies, because only non-stationary waves that meet many conditions can propagate in the easterlies; therefore, this scenario will not be detailed in this paper.

Dissipative Nonlinear Schrödinger Equation with External Forcing in Rotational Stratified Fluids and Its Solution**Supported by Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant No. 41421005, National Natural Science Foundation of China under Grant Nos. 41376030, 41376029, 41476019, NSFC–Shandong Joint Fund for Marine Science Research Centers Grant (U1406401), Special Funds for Theoretical Physics of

The dissipative nonlinear Schrödinger equation with a forcing item is derived by using of multiple scales analysis and perturbation method as a mathematical model of describing envelope solitary Rossby waves with dissipation effect and external forcing in rotational stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt–Vaisala frequency and β effect are important factors in forming the envelope solitary Rossby waves. By employing Jacobi elliptic function expansion method and Hirota's direct method, the analytic solutions of dissipative nonlinear Schrödinger equation and forced nonlinear Schrödinger equation are derived, respectively. With the help of these solutions, the effects of dissipation and external forcing on the evolution of envelope solitary Rossby wave are also discussed in detail. The results show that dissipation causes slowly decrease of amplitude of envelope solitary Rossby waves and slowly increase of width, while it has no effect on the propagation speed and different types of external forcing can excite the same envelope solitary Rossby waves. It is notable that dissipation and different types of external forcing have certain influence on the carrier frequency of envelope solitary Rossby waves.

Nonlinear solitary Rossby waves with external heating source and effect topographic effect in stratified flows described by the inhomogeneous Schrdinger equation

Rossby waves are intrinsic in the large-scale systems of fluids, so they are the most important waves in the atmosphere and ocean. Theory and observation show that their basic characteristic is to satisfy the quasi-geostrophic and quasi-static equilibrium approximations. In stratified fluids, we discuss the long waves in a homogenous atmosphere and obtain the KdV equation, but the analysis is limited to the case that the velocity shear is small compared with a basic uniform zonal motion, and it gives no insight pertaining to the kinds of stream-line-flow patterns accompanying these waves. Here, the -plane approximation f= f0+ 0 y (0 is a constant) is extended into f= f0+ (y) y, which includes a nonlinear function (y) taking the place of in the -plane approximation. Such an approximation can depict more precisely the motion of the atmosphere and ocean, especially in the middle and high latitude regions. It generalizes the theory developed by Helfrich and Pedlosky for time-dependent coherent structures in a marginally stable zonal flow by including forcing. Such forcing could be due to topography or external source. We take the basic flow to be a shear and the Visl-Brunt frequency N a function of variable z. For the stratified fluids, based on the lower boundary with external heating source and topography, as well as the quasi-geostrophic potential vorticity equation with external heating source, an inhomogeneous nonlinear Schrdinger equation (including topographic forcing and an external heating source) is derived by using the perturbation method and stretching transforms of time and space. It is found that the external heating source, effect and topography effect are the important factors that could induce the nonlinear solitary Rossby by inspection of the evolution of the amplitude of Rossby waves. On the assumption that nonlinear topographic effects and the dissipation of external heating source are balanced, an inhomogenous equation in which the coefficients depend on (y), u(y,z) and N(z) is derived. Results show that the topography, external heating source and Rossby waves will interact with a basic stream function that has a shear. In stratified fluids, the inhomogeneous nonlinear Schrdinger equation is obtained for describing the evolution of the amplitude of solitary Rossby envelop solitary waves as the change of Rossby parameter (y) with latitude y, topographic forcing and the external heating source.