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

doi:10.1016/j.amc.2018.10.084

Applied Mathematics and Computation

Volume 346, 1 April 2019, Pages 666-679

https://doi.org/10.1016/j.amc.2018.10.084 Get rights and content

Abstract

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.

Introduction

One of the basic models of large scale atmospheric and oceanic circulations in geophysical fluids is the Rossby waves motion model, which is also known as planetary waves. The Rossby solitary waves exist in many real cases, such as the eddy motion of ocean current in the Gulf of Mexico, the atmospheric blocking phenomenon, and the North Atlantic Oscillation (NAO) [2]. Many researchers have made progress in understanding the dynamics of nonlinear Rossby waves over the past years. It was disclosed that the evolution of nonlinear Rossby waves is controlled by multiple physical factors, such as the meridional sheared background flow, topography and dissipation etc. [1]. Historically, several kinds of Nonlinear Partial Differential equations (PDEs) were derived to simulate the evolution of Rossby wave amplitude based on the quasi-geostrophic theory under the beta-plane approximation, in which the Coriolis parameter f = f₀ + βy is assumed. Long firstly used the classical Korteweg–de Vries (KdV) model equation to investigate the dynamics of Rossby waves under beta-plane approximation in [3]. It denoted the existence of Rossby solitary waves with a balance between dispersion and nonlinearity. Benny then considered the velocity and amplitude of Rossby solitons by KdV model equation in [4] and many other following works [5], [6]. Furthermore, the modified KdV (mKdV) equation was derived as another model in characterizing the dynamics of nonlinear Rossby waves by Wadati [7]. Compared to the KdV equation, the mKdV has a stronger nonlinearity implying its priority for the motions with stronger perturbations. Redekopp and Weidman studied the formation of Rossby solitons in a sheared basic flow and derived necessary conditions for the existence of Rossby solitons [8]. Li used mKdV equation to describe the equatorial Rossby solitary waves from the primitive equations approach in [9]. Moreover, Charney and Straus used a two-layer baroclinic model, including topography, heating and friction, to investigate the dynamics of nonlinear atmospheric dynamical systems [10]. Ono put forward a new integral-differential equation (BO equation) in [11], which revealed the existence of algebraic Rossby solitary waves. Recently, Yang et al. developed the integral-differential equation while considering topography [12], [13], [14]. Meanwhile, the Boussinesq equation is also an important model in simulating the evolution of Rossby waves, especially for the cases with dissipation [15]. Furthermore, it is necessary to note that Luo in [16] derived a nonlinear Schrödinger (NLS) equation to characterize the dynamics of atmospheric block. It revealed that the NLS is also an appropriate model equation in simulating the evolution of atmospheres and oceans, and they have achieved great progress in investigating the blocking phenomenon according to the NLS model equation they proposed, see [17], [18], [19], [20], [21], [22]. On the other hand, the higher dimensional model equations were also discussed, such as the Zakharov–Kuznetsov equation [23] and its modified forms [24], [25]. Higher dimensional model equations are more suitable for the real wave motions. As a result, most results indicated that the effects of meridional sheared basic flow and topography are essential for the nonlinear Rossby solitary waves under beta-plane approximation. Unfortunately, the above-mentioned papers did not consider the variation of topography along with time.

However, we wondered about two facts from the previous studies. On the one hand, the traditional beta-plane approximation believes the Rossby parameter to be a constant. Is it the truth? Liu and Tan discussed the variation of beta along with latitude under a specified approximation [26], which described the influence of beta on the nonlinear Rossby waves. Luo further used such an approximation to study the dipole blocking phenomenon [27], and Song et al. promoted it to a general form [28], [29]. All the above-mentioned works reveal that the variation of beta is important in the evolutionary process of nonlinear Rossby solitary waves. On the other hand, does the basic flow change exist only in meridional direction as in most traditional investigations? Since the nonlinearity is easily affected by some disturbances, it is necessary to consider the dynamics of Rossby solitary waves in zonally shear basic flow. Interestingly, Hodyss and Nathanthe [30], [31] used the KdV to investigate nonlinear Rossby waves in zonally varying basic flow. It was revealed that the variation of basic flow in zonal directions determined the connection between coherent structures and low-frequency wave packets.

At the same time, solving PDEs is always an interesting and difficult task. For example, there are the periodic, solitary wave solutions and many other kinds of solutions. Many kinds of analytical methods have been proposed in the last decades, such as the Hirota bilinear method [32], [33], [34], Jacobi elliptic function expansion method [35], Baklund transformations [36], Adomian decomposition method [37], homogeneous balance method [38], Homotopy perturbation technique [39] and so on [40], [41]. There are also some numerical methods, which are powerful treatments in solving nonlinear equations. For example, the Riemann solver, Godunov-type method, Lie group classification are well proposed and developed by Zeidan et al. to two-phase flow equations [42], [43], [44], [45], [46], which gives perfect approaches for the study of wave structure and our future studies. Unfortunately, there is not a unified method for any kind of nonlinear partial differential equation, so it is rather important for us to choose a suitable one to solve the following obtained equation.

In the present work, we use a new model equation to investigate the dynamics of nonlinear Rossby waves in zonally varying flow under generalized beta approximation. It is organized as follows. In Section 2, we derive a new mKdV model equation with variable coefficients for the spatial-temporal evolution of nonlinear Rossby wave amplitude with effects of zonally varying basic flow, generalized beta and topography. We give the qualitative analysis and conservation laws of nonlinear Rossby waves based on the obtained mKdV equation. In Section 3, we analytically discuss the effect of generalized beta on exciting the evolution of nonlinear Rossby solitary waves and obtain asymptotic solutions for the obtained mKdV model equation by both homotopy perturbation method and Adomian Decomposition method. In Section 4, we present results and discussions on the quantitative dynamics of nonlinear Rossby waves by the numerical calculations according to the obtained solution by homotopy perturbation method. Brief conclusions are given at last.

Section snippets

Mathematical model

Starting from a dimensionless barotropic potential vorticity equation with topography and turbulent dissipation in a generalized β(y) channel, the equation is written as [1] $(\frac{\partial}{\partial t} + \frac{\partial Ψ}{\partial x} \frac{\partial}{\partial y} - \frac{\partial Ψ}{\partial y} \frac{\partial}{\partial x}) [\nabla^{2} Ψ + f + h (x, y, t)] = - μ \nabla^{2} Ψ + Q,$ where Ψ is the total stream function, f = f₀ + β(y)y is the vertical component of Coriolis parameter with f₀ = 2Ωsin φ₀, Ω is the angular velocity of the earth rotation, φ₀ is the local latitude. β(y) is the generalized beta approximation proposed by Song et al. [29], [30]. h(x,

Analysis of the meridional structure for the nonlinear Rossby waves

To prove the validity of our previous statements about the excitation of nonlinear Rossby solitary waves by the generalized β(y), we set ${\bar{u}}_{0} (y) = u_{0}$ , h′₀(y) = h₀, [β(y)y]′ = β₀ − δy, where u₀, h₀, β₀ are all constants, which represents uniform basic background flow, linear large scale basic topography and δ is a small parameter indicating weak variation of β(y). Weakly nonlinear method is adopted to solve Eqs. (13) and (17). Let ${\begin{matrix} ϕ_{0} (y) = ϕ_{00} (y) + δ ϕ_{01} (y) + \dots; \\ c_{0} = c_{00} + δ c_{01} + \dots; \\ ϕ_{1} (y) = δ ϕ_{10} (y) + δ^{2} ϕ_{11} (y) + \dots . \end{matrix}$

The

Results and discussions

In this paper, m = 2 and λ = 0.6 are adopted. Firstly, we will make comparisons between the two above-mentioned methods to illustrate the appropriateness of the method we chose below. In Figs. 1 and 2, the evolutionary solutions of wave amplitude are depicted according to (42) by homotopy perturbation method and (45) by Adomian Decomposition method with the same initial condition and parameters. It is easy to conclude that the two methods used are well consistent with each other in such

Conclusions

In this study, we have derived a new mKdV model equation with variable coefficients to simulate the evolution of nonlinear Rossby waves under generalized beta approximation with the effects of topography and dissipation. The physical mechanisms of all kinds of parameters on the Rossby waves were discussed through theoretical analysis and numerical calculations. Moreover, we found that both the generalized beta and the basic sheared topography are important in inducing the evolution of the

Acknowledgments

The authors are deeply appreciated for the very valuable comments from reviewers and constructive suggestions from managing editor and assistant editor, which greatly improve the quality of the paper. Ruigang Zhang, Liangui Yang and Xiaojun Yin are highly thankful to the project of National Natural Science Foundation of China (Grant no. 11762011), Quansheng Liu acknowledges the support by the National Natural Science Foundation of China (Grant no. 11562014).

References (50)

K.C. Le et al.
Amplitude modulation of waves governed by Korteweg–de Vries equation
Int. J. Eng. Sci.
(2014)
LuoD.
Low-frequency finite-amplitude oscillations in a near resonant topographically forced barotropic flow
Dyn. Atmos. Oceans
(1997)
ZhangR.G. et al.
(2+1) dimensional Rossby waves with complete Coriolis force and its solution by homotopy perturbation method
Comput. Math. Appl.
(2017)
MaW.X. et al.
Hirota bilinear equations with linear subspaces of solutions
Appl. Math. Comput.
(2012)
ZhangJ.B. et al.
Mixed lump–kink solutions to the BKP equation
Comput. Math. Appl.
(2017)
ZhaoH.Q. et al.
Mixed lump–kink solutions to the KP equation
Comput. Math. Appl.
(2017)
LiuS.K. et al.
Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations
Phys. Lett. A
(2001)
G.A. Adomian
Review of the decomposition method and some recent results for nonlinear Equations
Comput. Math. App.
(1991)
WangM.L.
Application of a homogeneous balance method to exact solutions of nonlinear equation in Mathematical Physics
Phys. Lett. A
(1996)
HeJ.H.
Homotopy perturbation technique
Comput. Methods Appl. Mech. Eng.
(1999)

D. Zeidan

Assessment of mixture two-phase flow equations for volcanic flows using Godunov-type methods

Appl. Math. Comput.

(2016)

J. Pedlosky

Geophysical Fluid Dynamics

(1987)

M. Nezlin. et al.

Rossby vortices, Spiral Structures, Solitons, Springer Series in Non-Linear Dynamics

(1993)

R.R. Long

Solitary waves in the westerlies

J. Atmos. Sci.

(1964)

D.J Benney

Long nonlinear waves in fluid flow

J. Math. Phys.

(1966)

J.P. Boyd

Equatorial solitary waves, part I: Rossby solitons

J. Phys. Ocean

(1980)

M. Wadati

The modified Korteweg–deVries equation

J. Phys. Soc. Jpn.

(1973)

L.G. Redekopp et al.

Solitary Rossby waves in zonal shear flows and interactions

J. Atmos. Sci.

(1978)

LiM.C.

Equatorial solitary waves of tropical atmospheric motion in shear flow

Adv. Atmos. Sci.

(1987)

J.G. Charney et al.

Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, oragraphically forced, planetary wave systems

J. Atmos. Sci.

(1980)

OnoH.

Algebraic Rossby wave soliton

J. Phys. Soc. Jpn.

(1981)

YangH.W. et al.

A new integro-differential equation for Rossby solitary waves with topography effect in deep rotational fluids

Abst. Appl. Anal

(2013)

YangH.W. et al.

Interaction of algebraic Rossby solitary waves with topography and atmospheric blocking

Dyn. Atmos. Oceans

(2015)

YangH.W. et al.

Forced ILW-burgers equation as a model for Rossby solitary waves generated by topography in finite depth fluids

J. Appl. Math.

(2012)

K.C. Le et al.

Amplitude modulation of water waves governed by Boussinesq's equation

Nonlinear Dyn.

(2015)

Cited by (43)

A new nonlinear integral-differential equation describing Rossby waves and its related properties
2022, Physics Letters, Section A: General, Atomic and Solid State Physics
Citation Excerpt :
Rossby wave is a kind of large-scale permanent wave generated by the ocean atmosphere with a long life history [1,2]. Due to its important role in shaping weather and climate, and it is linked to many natural phenomena, such as: North Atlantic Gulf Stream, Kuroshio Current, El Niño phenomenon and so on [3–5]. Therefore, since it was discovered, Rossby wave has been the research hotspot of many scholars.
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 $h_{1} \to \infty$ , and when ${[Ξ (B)]}_{X X X}$ changes to $B_{X X X X}$ , 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.
The multiple kink solutions and interaction mechanism with help of the coupled Burgers' equation
2022, Chinese Journal of Physics
Citation Excerpt :
These models include the Korteweg-de Vries (KdV)equation [4], the Benjamin-Davis-Ono (BDO) equation [5], the fractional BO equation [6], the Boussinesq equation [7], the space-time fractional mZK equation [8], the mBO-mKdV-Burgers equation [9], and so on [10]. Corresponding, the special weather can be better explained by these mathematical models [11, 12, 13, 14, 15]. Recently, the higher dimensional theoretical models are utilized to describe Rossby waves.
This paper presents a study of the traditional approximation potential vorticity equation. A nonlinear forced Zakharov-Kuznetsov equation by using the perturbation expansion method and multiple temporal and spatial scales method is obtained. And the multiple kink solutions of the forced Zakharov-Kuznetsov equation are obtained and the one- and two-soliton solutions are given by making use of the simplest equation method, namely the coupled Burgers’ equation, which is of a low-order than obtained equation and the multiple kink solutions of the equation are already given by utilizing the simplified Hirota's method. The results show that the proposed method is a powerful mathematical method to solve the nonlinear evolution equation arising in various areas of nonlinear science. Based on the one- and two-soliton solutions, graphical figures are portrayed by taking definite values to parameters and the features of the variable parameters are discussed in detail. Some results are given about interaction of two-soliton solutions and we found the wave amplitude of the two-soliton is influenced with the Coriolis parameters. Meanwhile, the soliton of the model with the external source is also obtained.
Coherent structures of nonlinear barotropic-baroclinic interaction in unequal depth two-layer model
2021, Applied Mathematics and Computation
The nonlinear barotropic-baroclinic interaction theory is of great importance for the understanding of the atmosphere and the oceans. In this paper, a mathematical model based on analytical methods is established to discuss the propagation of the coherent structure of the nonlinear barotropic-baroclinic interaction of two-layer fluids in geophysics. It is the first time that the coupled Boussinesq equations are derived by using perturbation and spatiotemporal extensions transformations to simulate the evolutions of nonlinear barotropic and baroclinic waves. The solitary wave solutions are analytically obtained to explain the excitation, propagation and decrease of the coherent structures. The physical mechanism of the nonlinear wave-wave coherent structures is discussed by considering the effects of different physical factors, such as the topography, the Earth’s rotation and the basic shear flow. It is found that both topography and beta parameter have essential effects on the evolutions of the coherent structures. Furthermore, the effect of the unequal-depth parameter on the propagation of coherent structures is discussed.
Riemann-Hilbert problems and soliton solutions of nonlocal real reverse-spacetime mKdV equations
2021, Journal of Mathematical Analysis and Applications
Citation Excerpt :
It is also worth mentioning that it would be particularly interesting to find a certain kind of connections among different solution approaches, including the Hirota direct method [16], the Wronskian technique [9,28] and the Darboux transformation [33]. Moreover, various recent studies have exhibited great richness of other kinds of solutions to nonlinear dispersive wave integrable equations, such as lump and rogue wave solutions and their interaction solutions [31,40,24,29], Rossby wave solutions [42,41], algebro-geometric solutions [13,22] and solitonless solutions [25,27]. Definitely, it would be very interesting to understand how to construct those exact solutions, for example, lump solutions and rogue wave solutions, through the Riemann-Hilbert perspective or a generalized Riemann-Hilbert perspective.
We would like to analyze a kind of nonlocal reverse-spacetime integrable PT-symmetric multicomponent modified Korteweg-de Vires (mKdV) equations by making a group of nonlocal reductions, and establish their associated Riemann-Hilbert problems which determine generalized Jost solutions of higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the associated Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. The Riemann-Hilbert problems in the reflectionless case are solved explicitly, and the resulting formulation of solutions enables us to present solitons to the nonlocal reverse-spacetime integrable PT-symmetric mKdV equations.
A new dynamic model of ocean internal solitary waves and the properties of its solutions
2021, Communications in Nonlinear Science and Numerical Simulation
Citation Excerpt :
When describing complex physical problems, the physical meaning of fractional order model is clearer and more concise. In addition, we found that although the application of fractional order calculus is more and more widespread, in the study of internal solitary waves in the ocean, the study of fractional order model is still very few [15–18]. Therefore, it is necessary to study the fractional order model in depth.
In this paper, considering the frictional dissipation, we use the multi-scale analysis and perturbation methods to obtain a new model: the $(2 + 1)$ -dimensional modified Kadomtsev-Patviashvili-Benjamin-Ono-Burgers (mKP-BO-Burgers) equation, to describe internal solitary waves in the ocean. Compared with the traditional BO model, the mKP-BO-Burgers model takes account of frictional dissipation, and its nonlinearity is stronger. Next, in order to better understand the characteristics of internal solitary waves, the integer-order mKP-BO-Burgers equation is converted into the time-fractional form by using semi-inverse and Agrawal’s methods. And then, conservation laws of the time-fractional mKP-BO-Burgers equation are derived. The results show that when the dissipation disappears, the mass, momentum and energy are conserved. Finally, we obtain the $N$ -soliton solutions of the new model, analyze the asymptotic behaviors of the $N$ -soliton solutions, discuss the interaction between the two solitons, and get some conclusions.
Optical dromions and domain walls in (2+1)-dimensional coupled system
2021, Optik
Citation Excerpt :
To study exact solutions of nonlinear partial differential equations (NLPDEs) is one of the most fascinating and excited area of research for many researchers and mathematicians since several years because they are widely appearing to describe the nonlinear wave phenomena in mathematical studies and applied physics with potential applications in many scientific and natural sciences including plasma physics, solid state physics, optics, nuclear physics, nano-fiber technology, optical communication systems, electromagnetism, bio-mathematics, mathematical physics, chemistry and many other engineering disciplines [1–33].
In this paper we have discussed the analytical treatment of coupled integrable (2+1)-dimensional nonlinear Schrödinger equation with the help of our newly introduced method extended modified auxiliary equation mapping method(EMAEMM). By using this newly proposed method we have found some new and more general variety of exact traveling wave solutions which are collecting some kind of semi half bright, dark, bright, semi half dark, doubly periodic, combined, periodic, half hark and half bright via three parametric values which is the primary key point of difference of our technique. These results are highly applicable to develop new theories of plasma physics, quantum mechanics, biomedical problems, soliton dynamics, nuclear physics, optical physics, fluid dynamics, electromagnetism, industrial studies, mathematical physics, and in many other natural and physical sciences. For detailed physical dynamical representation of our results we have shown them with graphs in different dimensions using Mathematica 10.4 to get complete understanding in a more efficient manner to observe the behavior of different new dynamical shapes of solutions.

View all citing articles on Scopus

View full text

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

Abstract

Introduction

Section snippets

Mathematical model

Analysis of the meridional structure for the nonlinear Rossby waves

Results and discussions

Conclusions

Acknowledgments

Int. J. Eng. Sci.

Dyn. Atmos. Oceans

Comput. Math. Appl.

Appl. Math. Comput.

Comput. Math. Appl.

Comput. Math. Appl.

Phys. Lett. A

Comput. Math. App.

Phys. Lett. A

Comput. Methods Appl. Mech. Eng.

Appl. Math. Comput.

Geophysical Fluid Dynamics

Rossby vortices, Spiral Structures, Solitons, Springer Series in Non-Linear Dynamics

Solitary waves in the westerlies

J. Atmos. Sci.

Long nonlinear waves in fluid flow

J. Math. Phys.

Equatorial solitary waves, part I: Rossby solitons

J. Phys. Ocean

The modified Korteweg–deVries equation

J. Phys. Soc. Jpn.

Solitary Rossby waves in zonal shear flows and interactions

J. Atmos. Sci.

Equatorial solitary waves of tropical atmospheric motion in shear flow

Adv. Atmos. Sci.

Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, oragraphically forced, planetary wave systems

J. Atmos. Sci.

Algebraic Rossby wave soliton

J. Phys. Soc. Jpn.

A new integro-differential equation for Rossby solitary waves with topography effect in deep rotational fluids

Abst. Appl. Anal

Interaction of algebraic Rossby solitary waves with topography and atmospheric blocking

Dyn. Atmos. Oceans

Forced ILW-burgers equation as a model for Rossby solitary waves generated by topography in finite depth fluids

J. Appl. Math.

Amplitude modulation of water waves governed by Boussinesq's equation

Nonlinear Dyn.