Envelope Rossby Research Articles

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

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

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

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

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.

(2 + 1)-Dimensional Coupled Model for Envelope Rossby Solitary Waves and Its Solutions as well as Chirp Effect

Using the method of multiple scales and perturbation method, a set of coupled models describing the envelope Rossby solitary waves in (2+1)-dimensional condition are obtained, also can be called coupled NLS (CNLS) equations. Following this, based on trial function method, the solutions of the NLS equation are deduced. Moreover, the modulation instability of coupled envelope Rossby waves is studied. We can find that the stable feature of coupled envelope Rossby waves is decided by the value ofS. Finally, learning from the concept of chirp in the optical soliton communication field, we study the chirp effect caused by nonlinearity and dispersion in the propagation of Rossby waves.

(2+1)-dimensional dissipation nonlinear Schrödinger equation for envelope Rossby solitary waves and chirp effect**Project supported by the National Natural Science Foundation of China (Grant No. 41406018).

In the past few decades, the (1+1)-dimensional nonlinear Schrödinger (NLS) equation had been derived for envelope Rossby solitary waves in a line by employing the perturbation expansion method. But, with the development of theory, we note that the (1+1)-dimensional model cannot reflect the evolution of envelope Rossby solitary waves in a plane. In this paper, by constructing a new (2+1)-dimensional multiscale transform, we derive the (2+1)-dimensional dissipation nonlinear Schrödinger equation (DNLS) to describe envelope Rossby solitary waves under the influence of dissipation which propagate in a plane. Especially, the previous researches about envelope Rossby solitary waves were established in the zonal area and could not be applied directly to the spherical earth, while we adopt the plane polar coordinate and overcome the problem. By theoretical analyses, the conservation laws of (2+1)-dimensional envelope Rossby solitary waves as well as their variation under the influence of dissipation are studied. Finally, the one-soliton and two-soliton solutions of the (2+1)-dimensional NLS equation are obtained with the Hirota method. Based on these solutions, by virtue of the chirp concept from fiber soliton communication, the chirp effect of envelope Rossby solitary waves is discussed, and the related impact factors of the chirp effect are given.

Nonlinear atmospheric and climate dynamics in China (2003–2006): A review

Recent advances in the study of nonlinear atmospheric and climate dynamics in China (2003–2006) are briefly reviewed. Major achievements in the following eight areas are covered: nonlinear error dynamics and predictability; nonlinear analysis of observational data; eddy-forced envelope Rossby soliton theory; sensitivity and stability of the ocean’s thermohaline circulation; nonlinear wave dynamics; nonlinear analysis on fluctuations in the atmospheric boundary layer; the basic structures of atmospheric motions; some applications of variational methods.

A Barotropic Envelope Rossby Soliton Model for Block–Eddy Interaction. Part III: Wavenumber Conservation Theorems for Isolated Blocks and Deformed Eddies

Abstract In a series of previous papers, an envelope Rossby soliton theory was formulated to investigate the interaction between a preexisting planetary wave and synoptic-scale eddies leading to a typical blocking flow. In this paper, numerical and analytical studies are presented in order to examine the interactive relationship between an isolated vortex pair block and deformed synoptic-scale eddies during their interaction. The deformed blocked flow and eddies are found to satisfy the wavenumber conservation theorem. It is shown that the feedback by a blocked flow on the preexisting synoptic eddies gives rise to two types of eddies: one is the Z-type eddies with a meridional monopole structure that appears at the middle of the channel and the other is the M-type eddies with a meridional tripole structure that have long wavelength and large amplitude. Both the total wavenumber of the blocked flow and M-type eddies and the total wavenumber of the Z- and M-type eddies are conserved. The M- and Z-type eddies are compressed and elongated, respectively, as the blocked flow is elongated zonally during its onset phase, but the reverse is observed during the decay phase. The zonally elongated Z-type eddies are found to counteract the compressed M-type eddies in the blocking region, but strengthen the M-type eddies upstream, causing the split of eddies around the blocking region. In addition, it is also verified theoretically that the blocked flow and synoptic-eddy activity are symbiotically dependent upon one another. The deformed (Z and M type) eddies also display a low-frequency oscillation in amplitude, wavenumber, group velocity, and phase speed, consistent with the blocked flow by the eddy forcing. Thus, it appears that the low-frequency eddy forcing is responsible for the low-frequency variability of the blocked flow and synoptic-eddy activity.

A Barotropic Envelope Rossby Soliton Model for Block–Eddy Interaction. Part IV: Block Activity and Its Linkage with a Sheared Environment

Abstract In this paper, a basic-state flow that has a linear, weak meridional shear is used as a mean flow condition to see if the blocking activity is related to the state of the zonal mean flow based upon the envelope soliton theory proposed in Part I. It is found that an isolated coherent structure similar to a dipole block can arise from the resonant interaction between preexisting planetary- and synoptic-scale waves, but its asymmetry, intensity, and persistence depend strongly upon the horizontal shear of the basic-state flow prior to block onset. The cyclonic shear of the basic-state flow not only plays an important role in preventing the eastward spread of block energy, but also provides a strong diffluent flow as a precursor of block flow. In such an environment, a high amplitude dipole anomaly is maintained and the deformed synoptic-scale eddies tend to deflect northward. But the anticyclonic shear of a basic-state flow plays a reverse role, which causes deformed eddies to deflect southward. It is also found that positive westerly wind anomalies at both high and low latitudes and negative anomalies at middle latitudes are maintained by synoptic-scale eddies through the self-interaction of block. In this case, the zonal mean westerly wind is accelerated at high and low latitudes, and decelerated at middle latitudes. Only for a basic state with cyclonic shear is the meridional profile of the zonal mean wind during the onset of a dipole block in agreement with observations. In addition, it can be shown by computing scatter diagrams of potential vorticity against streamfunction in different sheared environments that an eddy-induced isolated block can exhibit a local free-mode characteristic. Two approximately linear functional relationships between potential vorticity and streamfunction are found to be probably attributed to synoptic eddies.

A Barotropic Envelope Rossby Soliton Model for Block–Eddy Interaction. Part I: Effect of Topography

AbstractA new forced envelope Rossby soliton model in an equivalent barotropic beta-plane channel is proposed to describe the interaction between an incipient block (planetary scale) and short synoptic-scale eddies. This model is based on two assumptions, motivated by observations that (i) there exists a zonal scale separation between the planetary-scale and synoptic-scale waves and (ii) that the range of synoptic-scale zonal wavenumber is comparable to the planetary-scale zonal wavenumber. These assumptions allow an analytical treatment. The evolution of the planetary-scale block under the influence of synoptic-scale eddies is described by a forced nonlinear Schrödinger equation that is solved numerically, while the feedback of block development on the preexisting synoptic-scale eddies is derived analytically. It is shown that the planetary-scale projection of the nonlinear interaction between synoptic-scale eddies is the most important contributor to the amplification and decay of the planetary-scale blocking dipole or anticyclone, while the synoptic–planetary-scale interaction contributes significantly to the downstream development of preexisting synoptic-scale eddies. Large-scale topography plays a secondary role compared to the synoptic-scale eddies in exciting the block. However, it plays a role in inducing a standing planetary-scale ridge prior to block onset, which fixes the geographical location of the block and induces meridional asymmetry in the flow. In particular, the topographically induced planetary-scale ridge that is almost in phase with a dipole component of blocking flow is found to be a controlling factor for the northward deflection of storm tracks associated with blocking anticyclones.

A Barotropic Envelope Rossby Soliton Model for Block–Eddy Interaction. Part II: Role of Westward-Traveling Planetary Waves

Abstract The role of westward-traveling planetary waves in the block onset and the deformation of eddies during the interaction between synoptic-scale eddies and an incipient block is first examined by constructing an incipient block that consists of a stationary dipole wave for zonal wavenumber 2 and a westward-traveling monopole wave with constant amplitude (C wave) for zonal wavenumber 1 or 2. It is shown that the C-wave can affect the onset and strength of blocking through influencing the preblock (diffluent) flow even though it does not affect the amplification of the dipole wave associated with the synoptic-scale eddies. Whether the storm tracks organized by the deformed eddies deflect northward depends upon the zonal wavenumber, amplitude, and phase of the C wave relative to the stationary dipole wave. A typical retrograde blocking anticyclone can arise through the interaction of an incipient block with synoptic-scale perturbations when the C-wave ridge with zonal wavenumber 1 shifts westward from the east of the dipole wave in an incipient block. In this process, a slight northward deflection of organized storm tracks is also observed, particularly under the condition of a large-amplitude C wave. In addition, the interaction between a diffluent flow, consisting of a coupled dipole and monopole waves, and upstream synoptic-scale eddies is investigated. It is found that the eddy forcing tends to not only periodically amplify the dipole soliton and to retard its eastward movement, but to make the monopole wave break up. The breaking of the traveling monopole wave will suppress the eddy-induced blocking ridge that exhibits a surf zone structure where the negative meridional gradient of planetary-scale potential vorticity exists and cause the planetary-scale blocking field to lose its closed circulation compared to that without coupling.

Structures of Equatorial Envelope Rossby Wave Under a Generalized External Forcing**The project supported by National Natural Science Foundation of China under Grant No. 40305006 and the Ministry of Science and Technology of China Through Special Public Welfare Project under Grant No. 2002DIB20070

The cubic nonlinear Schrödinger (NLS for short) equation with a generalized external heating source is derived for large amplitude equatorial envelope Rossby wave in a shear flow. And then various periodic structures for these equatorial envelope Rossby waves are obtained with the help of a new transformation, Jacobi elliptic functions, and elliptic equation. It is shown that different types of resonant phase-locked diabatic heating play different roles in structures of equatorial envelope Rossby wave.

Periodic Structure of Equatorial Envelope Rossby Wave Under Influence of Diabatic Heating**The project supported by the Ministry of Science and Technology of China through the Special Public Welfare Project under Grant No. 2002DIB20070 and National Natural Science Foundation of China under Grant No. 40305006

A simple shallow-water model with influence of diabatic heating on a -plane is applied to investigate the nonlinear equatorial Rossby waves in a shear flow. By the asymptotic method of multiple scales, the cubic nonlinear Schrodinger (NLS for short) equation with an external heating source is derived for large amplitude equatorial envelope Rossby wave in a shear flow. And then various periodic structures for these equatorial envelope Rossby waves are obtained with the help of Jacobi elliptic functions and elliptic equation. It is shown that phase-locked diabatic heating plays an important role in periodic structures of rational form.

Interaction between a slowly moving planetary—scale dipole envelope rossby soliton and a wavenumber— two topography in a forced higher order nonlinear Schrödinger equation

A parametrically excited higher—order nonlinear Schrodinger (NLS) equation is derived to describe the interaction of a .slowly moving planetary—scale envelope Rossby soliton for zonal wavenumber-two with a wavenumber—two topography under the LG—type dipole near-resonant conditioa The numerical solution of this equation is made. It is found that in a weak background westerly wind satisfying the LG—type dipole near—resonance condition, when an incipient envelope Rossby soliton is located in the topographic trough and propagates slowly, it can be amplified through the near—resonant forcing of wavenumber-two topography and can exhibit an oscillatioa However, this soliton can break up after a long time and excite a train of small amplitude waves that propagate westward. In addition, it is observed that in the soliton—topography interaction the topographically near—resonantly forced planetary—scale soliton has a slowly westward propagation, but a slowly eastward propagation after a certain time. The instantaneous total streamfunction fields of the topographically forced planetary—scale soliton are found to bear remarkable resemblance to the initiation, maintenance and decay of observed omega—type blocking high and dipole blocking. The soliton perturbation theory is used to examine the role of a wavenumber—two topography in near—resonantly forcing omega—type blocking high and dipole blocking. It can be shown that in the amplifying process of forced planetary—scale soliton, due to the inclusion of the higher order terms its group velocity gradually tends to be equal to its phase velocity so that the block envelope and carrier wave can be phase—locked at a certain time. This shows that the initiation of blocking is a transfer of amplified envelope soliton system from dispersion to nondispersion. However, there exists a reverse process during the decay of blocking. It appears that in the higher latitude regions, the planetary—scale envelope soliton—topography interaction could be regarded as a possible mechanism of the establishment of blocking.

Planetary-scale baroclinic envelope Rossby solitons in a two-layer model and their interaction with synoptic-scale eddies

In this paper, planetary-scale baroclinic envelope Rossby solitons for zonal wavenumber 2 in a two-layer model are first investigated. It is found that when the shear of basic state westerly winds between the upper and lower layers is weak, both the upper- and lower-layer envelope Rossby solitons are almost in phase and exhibit vortex pair block structures which have a weak baroclinicity. The possibility of the application of planetary-scale envelope Rossby soliton to observed dipole blocks is discussed. Second, a highly idealized model having weak vertical shear is proposed to investigate the interaction between baroclinic planetary-scale dipole soliton (weak incipient dipole block) and a train of synoptic-scale waves (eddies) upstream. It is shown that in the interaction process, both the planetary-scale dipole soliton and the synoptic-scale eddies are deformed simultaneously. The amplitude of the dipole soliton block can be amplified through the near-resonant forcing of synoptic-scale eddies. In the intensification process of the dipole soliton block, its zonal scale is prolonged, and its phase velocity gradually tends to be equal to its group velocity so that the block envelope and carrier wave can be phase-locked at a certain time. This process reflects a transfer of dipole blocking system from dispersion to nondispersion. This may explain why the synoptic-scale eddies can reinforce and maintain blocking dipole. Conversely, in the decay process, the zonal scale of the dipole soliton block is shortened, and the discrepancy between its phase velocity and group velocity gradually increases. This process may describe a transfer of dipole blocking system from nondispersion to dispersion, which leads to the break-down of the dipole block. Due to the feedback of the intensified dipole block on the synoptic-scale eddies, the upper-layer eddies are split into two branches, and the lower-layer eddies are split into three branches. Moreover, the instantaneous total streamfunction fields (the streamfunction field of planetary-scale envelope Rossby soliton plus the streamfunction field of the deformed synoptic-scale eddies) that describe the coupling between baroclinic dipole envelope soliton and synoptic-scale eddies are found to bear a remarkable resemblance to the synoptic maps observed during a blocking episode. It appears that the eddy-forced envelope soliton model established here could describe the initiation, maintenance and decay of dipole blocking by the baroclinic synoptic-scale eddies. Therefore, this theoretical model seems to provide a theoretical framework for further studying the interaction between synoptic-scale eddies and dipole block.

Near-resonant topographically forced envelope Rossby solitons in a barotropic flow

In this paper, a parametrically excited nonlinear Schrödinger (NLS) equation is proposed to describe the near-resonant interaction between dipole envelope Rossby soliton and wave number-two topography under the LG-type dipole near resonance condition in a weak background westerly wind. Based on this equation, the existence and stability of steady soliton are discussed. It is found that the stability of the steady forced soliton depends on the latitude and setting of the background westerly wind. The numerical method is applied to solve the forced NLS equation. It is found that under favourable conditions small-amplitude envelope Rossby soliton in the topographic trough can be amplified through the near-resonant forcing of wavenumber-two topography. In a moderate parameter range, the forced amplified envelope soliton is stable. In particular, in the soliton amplification process the maximum soliton amplitude exhibits aperiodic oscillation. However, in some parameter range, the amplified soliton is unstable and developes finally into a cnoidal wave after a long time. In addition, the time sequences of the instantaneous total stream function field are found to bear a remarkable resemblance to the initiation, maintenance and decay of localized dipole blockings in the north Atlantic ocean observed from the daily weather maps. Thus, in the weak nearly resonant background westerly wind, the near-resonant interaction between dipole envelope Rossby soliton and wave number-two topography seems to be a possible mechanism for dipole blockings observed in the two oceans.

Parametrically forced envelope Rossby solitons: Soliton-oscillation

In this paper, the non-dimensional equivalent barotropic vorticity equation with dissipation and an external vorticity source induced by diabatic heating was reduced to the generalized periodically forced nonlinear Schrödinger (NLS) equation similar to the parametrically excited NLS equation derived by Miles (1984) by means of the multiplescale method. This forced NLS equation can be reduced to a low-dimensional ordinary differential equation describing the amplitude and phase of the forced envelope Rossby soliton by the inverse-scattering perturbation method. The numerical calculations indicate that for weaker dissipation the envelope Rossby soliton, under a constant amplitude travelling diabatic heating, exhibits multi-periodic oscillations, and its phase portrait is found to be a limit cycle. However, for the forcing of one-year modulated diabatic heating, the forced envelope Rossby soliton can execute chaotic motion when the controlling parameter reaches a certain extent. In addition, the oscillatory behaviour of the forced envelope Rossby soliton is found to be able to explain the establishment and decay of vortex pair blocking usually observed in two oceans.

A THEORY OF BLOCKING FORMATION IN THE ATMOSPHERE

In this paper, a new theory of blocking formation was proposed. The nonlinear Schrdinger equation satisfied by nonlear barotropic Rossby waves for the weak shear zonal flow was obtained by using the WKB method. It was pointed out that when the Rossby wavenumbers sarisfied the relation: k/3m~24k~2 (where k and m are the zonal and meridional wavenumbers respectively), the periodic Rossby wave in the atmosphere might produce the modulational instability and form the envelope Rossby solitary wave. Furthermore, the stream function field of the envelope Rossby solitary wave was calculated, and the structure derived here was similar to the dipole blocking in the atmosphere. On the other hand, we also pointed out that the dipole blocking might be caused through the Rossby wave modulational instability, moreover, the dipole blocking could persist for 25 days and disappear through energy dispersion of Rossby waves.