Nonlinear Rossby Research Articles

On the dynamics of nonlinear barotropic–baroclinic interactions through a coupled Gardner hierarchies approach

The aim of this paper is on the propagations of barotropic–baroclinic coherent structures based on the two-layer quasi-geostrophic model (2LQG) through a Fourier spectrum compliant approach. First, by introducing the barotropic and baroclinic stream functions starting from the 2LQG model, a new coupled Gardner-type evolution equations, representing the interaction processes between the barotropic flow and baroclinic one, are obtained by combining the multi-scale method and the perturbation expansion method. Second, based on the obtained coupled model equations, the physical mechanisms of the nonlinear barotropic–baroclinic interaction are analyzed qualitatively. Within the range of parameters chosen in this paper, quantitative results show that the basic flow, the β effect, and the bottom topography are necessary factors to excite the nonlinear Rossby isolated waves. The results also declare that the dipole-like blockings are readily excited in the flow field and move slowly eastward in both barotropic and baroclinic flow fields.

Nonlinear Rossby Wave–Wave and Wave–Mean Flow Theory for Long-term Solar Cycle Modulations

Abstract The Schwabe cycle of solar activity exhibits modulations and frequency fluctuations on slow timescales of centuries and millennia. Plausible physical explanations for the cause of these long-term variations of the solar cycle are still elusive, with possible theories including stochasticity of the alpha effect and fluctuations of the differential rotation. It has been suggested recently in the literature that there exists a possible relation between the spatiotemporal structure of the solar cycle and the nonlinear dynamics of magnetohydrodynamic (MHD) Rossby waves at the solar tachocline, including both wave–wave and wave–mean flow interactions. Here we extend the nonlinear theory of MHD Rossby waves presented in a previous article to take into account long-term modulation effects due to a recently discovered mechanism that allows significant energy transfers throughout different wave triads: the precession resonance mechanism. We have found a large number of Rossby–Hauwirtz wave triads whose frequency mismatches are compatible with the solar cycle frequency. Consequently, by analyzing the reduced dynamics of two triads coupled with a single mode (five-wave system), we have demonstrated that in the amplitude regime in which precession resonance occurs, the energy transfer throughout the system yields significant long-term modulations on the main ∼11 yr period associated with intratriad energy exchanges. We further show that such modulations display an inverse relationship between the characteristic wave amplitude and the period of intratriad energy exchanges, which is consistent with the Waldmeier law for the solar cycle. In the presence of a constant forcing and dissipation, the five-wave system in the precession resonance regime exhibits irregular amplitude fluctuations, with some periods resembling the grand minimum states.

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.

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.

$$\varvec{(2+1)}$$ ( 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.

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.

A steady nonlinear and singular vortex Rossby wave within a rapidly rotating vortex. Part I: Theory

The goal of this paper is to predict the evolution of a rapidly rotating vortex when neutral waves are propagating therein. We consider a singular and nonlinear vortex Rossby helical wave within a linearly stable, barotropic, axisymmetric, and dry vortex in the -plane. The wave enters resonance with the vortex at a certain radius, where the wave phase angular speed is equal to the rotation frequency. The singularity in the modal equation at this radius strongly modifies the flow in the critical layer (CL), the region where the wave/vortical flow interaction occurs. We resolve the singularity by reintroducing nonlinearity in the inner-flow equations in the steady-régime assumption. In this régime, the neutral mode is weakly singular. We indeed assume that the CL forms at the outer edge of the vortex where the basic axial vorticity has become small compared to the inertial frequency, owing to vortex erosion. Through matched asymptotic expansions, we determine the critical-layer induced distorted flow characterized by a secondary mean flow, whose amplitude is larger than the wave amplitude, which diffuses in an asymmetric way at either side of the CL that winds up around the vortex axis. The analysis shows that, after intensifying in the transition stage, the vortex is weakening in the steady stage, but in a negligible way with the slow viscous homogenization time scale. The CL extent is nevertheless evolving with this same time all the more so since the ratio vertical wavenumber over azimuthal one is large.

Equatorial Rossby envelope solitary waves with β effect in a shear flow

With a simple model of shallow-water on an equatorial beta plane, the nonlinear equatorial Rossby waves in a shear flow with beta effect are investigated by the asymptotic method of multiple scales. The nonlinear Schrdinger equation, satisfied by large amplitude Rossby envelope solitary waves in shear basic flow with beta effect, is derived. The effects of basic flow shear and beta effect on the nonlinear equatorial Rossby waves are also analyzed.

Equatorial envelope Rossby solitons in a shear flow

A simple shallow-water model on an equatorial β-plane is employed to investigate the nonlinear equatorial Rossby solitons in a mean zonal flow with meridional shear by the asymptotic method of multiple scales. The cubic nonlinear Schrodinger (NLS, for short) equation, satisfied for large amplitude equatorial envelope Rossby solitons in shear basic flow, is derived. The effects of basic flow shear on the nonlinear equatorial Rossby solitons are also analyzed.

Nonlinear Equatorial Rossby Longwave in a Shear Flow

[1] In order to tackle the questions arising from the observations and theoretical studies, and to gain a better understanding of tropical atmospheric dynamics in general, it is necessary to investigate further the efects of the varying basic states upon the equatorially trapped nonlinear Rossby waves in the atmosphere. The purpose of this study is to examine the efects of basic zonal flows on the equatorial Rossby long waves. A simple shallow–water model on an equatorial β–plane is employed to investigate the nonlinear equatorial Rossby waves in a mean zonal flow with meridional shear by the standard reductive perturbation method. The nonlinear Kortewegde Vries (KdV) equation, satisfied by the nonlinear equatorial Rossby long waves of small amplitude, is derived. The analyses indicate that a necessary condition of the soliton formation is the basic flow with shear, and that the shear is not so strong as to create barotropic instability.

Reflection of Nonlinear Baroclinic Rossby Waves and the Driving of Secondary Mean Flows

Abstract The reflection of weakly nonlinear Rossby waves (RWs) from a vertical wall is examined analytically through perturbation methods, with the β-Rossby number (ϵ) as the small parameter. A uniformly valid solution up to O(ϵ3) is constructed using multiple scales. At O(ϵ), the nonlinear interaction between an incident and the reflected RW leads to 1) an Eulerian steady flow, us(1), parallel to the (nonzonal) wall and 2) a transient flow oscillating with a frequency twice (2ω) that of the RW pair. The steady forcing, whose response is us(1), can never be resonant, which implies, under the weak nonlinear regime, that us(1) is stable to the driving RWs. At the next order, the nonlinear interaction between the incident-reflected RW pair and us(1) plus the transient flow produces, in general, resonant forcing leading to a modification of the RWs’ phases: a shift in their offshore wavenumber. The steady flow that occurs at O(ϵ3) is driven by the modified RWs as well as through interactions of several compon...

Nondispersive periodic solution of a barotropic semi-geostrophic model

The existence and solution of the non-dispersive periodic solution are achieved concerning nonlinear barotropic Rossby waves of a barotropic semi-geostrophic model, demonstrating the likelihood of the Taylor evolution, together with the related dimensionlessa-criterion. Finally, the wave velocity expression is proposed with some diagnostic relations among the wave parameters.

Nonlinear Rossby Waves in a Channel of Finite Width

Nonlinear topographic Rossby waves propagating in a channel of finite width are investigated by the reductive perturbation method. When the channel is sufficiently wide and homogeneous in the outside of a wave guide, the resultant equation takes the form of that obtained by Joseph, and Kubota, Ko and Dobbs. On the other hand, in the presence of weak inhomogeneity, it is shown that a Rossby wave soliton undergoes modification such as change in phase velocity and radiation damping according to the properties of the inhomogeneity.

Stationary Response of Barotropic Weakly Nonlinear Rossby Waves to Quasi-Resonant Orographic Forcing

Stationary solutions and their stability properties of nondissipative barctropic flow in a narrow, longitudinally periodic channel on a βplane with a sheared basic current and bottom topography are investigated. An analytical treatment is applied which has been used in studies of solitary Rossby waves and is allowed by the choice of geometry. This leads to an ordinary nonlinear differential equation for the longitude-dependent part of the solution. The equation which is obtained in the present case is formally that of an anharmonic oscillator with external forcing and weakly variable natural frequency. Approximate analytical and numerical solutions are obtained under quasi-resonant conditions. Three possible states are found in a certain range of the parameters. Of these, only two are found to be stable. The implications of the existence of multiple solutions, also found by other authors in various contexts, for the large-scale atmospheric circulation and the phenomenon of blocking are discussed.

Numerical Study of the Nonlinear Rossby Wave Critical Level Development in a Barotropic Zonal Flow

Abstract The fully nonlinear barotropic vorticity equation is integrated in time to study the development of a Rossby wave critical level. The initial conditions consist of a hyperbolic tangent shear flow and a steady forced wave at the northern boundary; a radiation condition is used at the southern boundary. Linear, quasi-linear and nonlinear integrations are made, and the results are compared with previous studies. For small values of the perturbation amplitude, such that nonlinear interactions are important only in the critical layer, a steady state is obtained in the outer domain, in which the Reynolds stress vanishes both above and below the critical level, and the forced wave is totally reflected as suggested by previous analytical nonlinear steady-state solutions; the approach to that steady state and the structure of the critical layer, however, are quite different from the quasi-linear integrations performed by other authors. Some conclusions are drawn with respect to the forcing of the equatori...