Rossby Soliton Research Articles

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.

Barotropic interaction between planetary- and synoptic-scale waves during the life cycles of blockings

In this paper, in an equivalent barotropic framework a new forced nonlinear Schroedinger equation is proposed to examine the interaction between the planetary-scale waves and the localized synoptic-scale eddies upstream. With the help of the perturbed inverse scattering transform method, nonlinear parameter equations can be derived to describe the evolution of the dipole soliton amplitude, frequency, group velocity and phase under the forcing of localized synoptic-scale eddies. The numerical solutions of these equations predict that in the interaction between the weak dipole soliton (weak incipient dipole anomaly) and the synoptic-scale eddies, only when the high-frequency eddies themselves have a moderate parameter match they can near resonantly enhance a quasi-stationary large-amplitude split flow. The instantaneous total streamfunction field (the sum of background westerly wind, envelope Rossby soliton and synoptic-scale waves) is found to be very similar to the observed Berggren-type blocking on the weather map(Berggren et al. 1949). The role of synoptic-scale eddies is to increase the amplitude of large-scale dipole anomaly flow, and to decrease its group velocity, phase velocity and zonal wavenumber so that the dipole anomaly system can be amplified and transferred from dispersive system to very weak dispersive one. This may explain why and how the synoptic-scale eddies can reinforce and maintain vortex pair block. Furthermore, it is clearly found that during the prevalence of the vortex pair block the synoptic-scale eddies are split into two branches around the vortex pair block due to the feedback of amplified dipole block.

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.

Self-organization of the large-scale planetary and plasma drift vortices.

This paper is a semi-review. A new understanding of the self-organization mechanism of solitary (i.e., long-lived and, in this sense, soliton-like) large-scale vortices in geophysical fluid dynamics, as well as that of drift vortices in the magnetized plasma is discussed. This understanding differs from that described in a review paper by Nezlin [Chaos 4, 187-202 (1994)]. Earlier it was believed that formation of the solitary Rossby (and plasma drift) vortices was a result of equilibrium between wave dispersion and KdV-type nonlinearity. Under the influence of experimental data obtained by our team [M. V. Nezlin and E. N. Snezhkin, Rossby Vortices, Spiral Structures, Solitons (Springer-Verlag, Heidelberg, 1993)], it became obvious that the self-organization of the structures inevitably includes an essential effect of other nonlinearities; first, that presented by the Jacobian in the equations. (We replace the term "Rossby soliton" by the more exact one "the Rossby solitary vortex.") It must be noted from the very beginning that the term "self-organization" is used mainly in context with an explanation which factors (dispersion and nonlinearities of different kind) condition formation of the solitary (stable, long-lived) Rossby structures. Although, the experimental fact (see below) that the size of solitary vortices turns out to be close to the Rossby-Obukhov radius, independently of the size of the vortex local source, calls to mind the formation of an attractor. In essence, the Rossby solitary vortex self-organization process (although, only for the case of anticyclones) was described by Nycander and Sutyrin [Dyn. Atmos. Oceans 16, 473-498 (1992)]. Unfortunately, however, the authors did not use the term "self-organization." Our description, being in accord with Nycander and Sutyrin, relates not only to anticyclones, but also to anticyclones and cyclones. Second, a description of the experimental discovery of "anomalous" cyclonic-anticyclonic asymmetry is given. Unlike "normal" asymmetry which manifests, in particular, in that the big vortices dominating in giant planet atmospheres (e.g., the Great Red Spot of Jupiter, the Brown Spot of Saturn, the Great Dark Spot of Neptune, et al.) are anticyclones, the asymmetry described manifests in that large-scale solitary vortices may be as cyclones only, not anticyclones. This phenomenon is observed in the presence of a rather strong and properly directed gradient in rotating shallow water depth. This type of asymmetry exists with drift vortices in magnetized plasma. Third, the physical difference between the planetary atmosphere and the laboratory model based on the shallow water layer in a rotating paraboloid is discussed (following, in principle, the Nycander-93 work). It is shown that laboratory modeling is adequate. Fourth, it is also shown that the most essential behavior of the vortices studied on the so-called "beta-plane" of planets and in plasmas can be described by means of rather simplified and visual equations. These are the so-called generalized Charney-Obukhov equation in fluid dynamics and its plasma counterpart, the generalized Hasegawa-Mima equation. Finally, nonlinearities are revealed, which condition properties of the geostrophic vortices under study on the "f-plane," i.e., in the polar regions of planets. &c

Collision Interactions of Envelope Rossby Solitons in a Barotropic Atmosphere

Abstract A theory is developed here to describe the propagation of nonlinear Rossby wave packets in a barotropic atmospheric model and their interactions by using the multiple-scale method. It is shown that the propagation of a single Rossby wave packet can be described by the nonlinear Schrodinger equation that has envelope soliton solutions. For two interacting packets with slightly different wavenumbers they satisfy a set of two coupled nonlinear Schrodinger equations. These equations are used to study the collision interactions of two envelope Rossby solitons. It is found that despite the complexity of the interaction, the energy of each soliton is conserved, while the shapes and velocities of the two solitons may be altered significantly by the interaction. The action of one soliton on another is realized by providing a field of force or potential for it through the cross-modulation terms.

Propagation of an envelope Rossby soliton over topography

Abstract By using a simple barotropic model and the multiple-scale perturbation method, a modified nonlinear Schrodinger equation with variable coefficient, which describes the propagation of a nonlinear Rossby wave packet over topography, is derived. With the exception of several specific cases the equation is nonintegrable. The behavior of an envelope soliton traveling toward a topography is studied numerically. It is found that when the initial velocity of the soliton is high, the topography has little effect on it. When the initial velocity is low, the topographic effects on the soliton are significant: The soliton may be reflected by the topography, or may oscillate around and be captured by the topography, depending on the wave-topography interaction coefficient.

Statistical mechanical theory of the great red spot of jupiter

In a previous study a permanent isolated vortex like the Great Red Spot of Jupiter was obtained as a statistical equilibrium for the classical quasigeostrophic model of atmospheric motion on rapidly rotating planets. We provide here a theoretical basis for this work and relate it to a previous model of the spot (Rossby soliton).

Dynamics of singular vortices on a beta-plane

A new singular-vortex theory is presented for geostrophic, beta-plane dynamics. The stream function of each vortex is proportional to the modified Bessel function Ko(pr), where p can be an arbitrary positive constant. If p−1 is equal to the Rossby deformation scale Rd, then the vortex is a point vortex; for p−1 ≠ Rd the relative vorticity of the vortex contains an additional logarithmic singularity. Owing to the β-effect, the redistribution of the background potential vorticity produced by the vortices generates a regular field in addition to the velocity field induced by the vortices themselves. Equations governing the joint evolution of singular vortices and the regular field are derived. A new invariant of the motion is found for this system. If the vortex amplitudes and coordinates are set in a particular way then the regular field is zero, and the vortices form a system moving along latitude circles at a constant speed lying outside the range of the phase velocity of linear Rossby waves. Each of the systems is a discrete two-dimensional Rossby soliton and, vice versa, any distributed Rossby soliton is a superposition of the singular vortices concentrated in the interior region of the soliton. An individual singular vortex is studied for times when Rossby wave radiation can be neglected. Such a vortex produces a complicated spiral-form regular flow which consists of two dipoles with mutually perpendicular axes. The dipoles push the vortex westward and along the meridian (cyclones move northward, and anticyclones move southward). The vortex velocity and trajectory are calculated and applications to oceanic and atmospheric eddies are given.

Cyclonic-anticyclonic asymmetry and a new soliton concept for rossby vortices in the laboratory, oceans and the atmospheres of giant planets

Abstract It is demonstrated in laboratory experiments with rotating shallow water that large scale Rossby vortices, greater than the Rossby-Obukhov radius in size, have dispersive and non-linear properties that are fundamentally different for the two possible polarities. We call this “cyclonic-anticyclonic asymmetry”. This asymmetry manifests itself in the following way: first, anticylones, unlike cyclones, do not undergo the dispersive spreading inherent in a linear wave packet. and therefore, having a considerably longer natural lifetime, are obvious candidates for Rossby solitons; second, dipolar vortices are, because of the comparatively rapid decay of a cyclone, transformed into anticyclonic solitons; third, anticyclones are much more readily generated by zonal flows of the type existing in planetary atmospheres. The evident dominance of anticyclones amongst the long-lived vortices in the atmospheres of giant planets strongly suggests that the cyclonic-anticyclonic symmetry plays a decisive role in t...

Solitons in a Continuously Stratified Equatorial Ocean

The equatorial soliton studies of Boyd are extended to include the effects of continuous vertical stratification. We use vertical profiles of density measured in the equatorial Pacific Ocean and an idealized profile. The wavenumber intervals of existence/nonexistence of wavepacket solitons are similar to Boyd for the Rossby mode, less similar for the eastward propagating gravity mode, and least similar for the westward gravity mode. Competing resonances and vertical diffusion are discussed. The observed stratification for the equatorial Pacific has a significant effect on solitons. Some of the most dramatic effects are for the KdV solitons of Boyd. We propose that the longitudinal variation of stratification may result in the existence of the antisymmetric (with respect to the equator) Rossby solitons in only the western Pacific Ocean. The nonlinear coefficients of the symmetric KdV Rossby soliton decrease monotonically eastward by approximately 30% across the Pacific Ocean.

Rossby solitons (Experimental investigations and laboratory model of natural vortices of the Jovian Great Red Spot type)

This review is devoted to a new object in nonlinear physics—Rossby solitons. The following questions are examined in greatest detail: 1) experimental observation of Rossby solitons in the laboratory; the geometric and kinematic properties, stability, collisions, and methods of generation of Rossby solitons; 2) self-organization of a Rossby autosoliton in unstable oppositely directed (zonal) flows; 3) a stationary laboratory model of natural vortices of the Jovian Great Red Spot (JGRS) type based on the Rossby autosoliton; 4) the question of the uniqueness of the JGRS along the perimeter of the planet; and 5) paired Rossby vortices. The theory and experiment are compared. The analogy between Rossby vortices and drift vortices in a plasma is pointed out.

Rossby autosoliton and stationary model of the jovian Great Red Spot

A theory proposed about 10 years ago claimed that the jovian Great Red Spot (GRS) was a solitary wave vortex (Rossby soliton) kept stationary by counter–streaming zonal winds. We have attempted to verify this soliton theory experimentally. The jovian atmosphere is modelled by a rotating thin parabolic layer of fluid (shallow water) with a free surface in which counter-streaming (zonal) flows are excited mechanically. We have found that instability of these flows can generate a Rossby autosoliton, that is, an undamped stationary solitary vortex which is alone on the perimeter of the system. This vortex rotates around its axis in the anticyclonic sense and drifts in the opposite direction to the global rotation of the system. As the observed Rossby soliton can be considered as a physical analogue (or rather as a stationary physical model) of a natural vortex such as the GRS, the results of our experiments can be considered to support the soliton theory of the GRS. We have compared the soliton model with another physical model of the GRS based on thermoconvection in rotating deep water under the action of a transverse non-monotonic temperature gradient. A new model based on a synthesis of these ideas is urgently required.

Solitony Rossbi (Eksperimental'nye issledovaniya i laboratornaya model' prirodnykh vikhrei tipa Bol'shogo Krasnogo Pyatna Yupitera)

Solitony Rossbi (Eksperimental'nye issledovaniya i laboratornaya model' prirodnykh vikhrei tipa Bol'shogo Krasnogo Pyatna Yupitera)

Electron drift solitons in an inhomogeneous magnetized plasma

It is shown that electron drift solitons can exist in an inhomogeneous magnetized plasma. Two such types of solitons are discussed, one which is similar to the solitons described by the one-dimensional Korteweg-de Vries equation, and a second which is an analog of the two-dimensional Rossby solitons.

Equatorial solitary waves Part 4. Kelvin solitons in a shear flow

It is shown that a mean flow with shear makes the Kelvin wave dispersive. This in turn modifies its nonlinear behavior and makes it necessary to replace the one-dimensional advection equation derived in an earlier work of the author's by the Korteweg-deVries equation instead. The frontogenesis predicted in the earlier paper will still occur, but the wave breaking will not. Instead, once a steep front has formed, it will disintegrate into a train of solitary waves. These then propagate towards the east at a faster-than-linear rate. It is also shown that Kelvin solitary waves will have much smaller zonal widths than Rossby solitons of the same height; “round” Kelvin solitary waves (equal zonal and latitudinal width) are to be expected, and are fully consistent with the small amplitude, weak dispersion theory. An interesting implication of the Korteweg-deVries model is that the peak signal from a nonlinear Kelvin wave packet may be roughly double that of a linear Kelvin wavetrain.

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.

Solitary Rossby Waves in the Presence of Vertical Shear

The effects of vertical shear on regular neutral mode Rossby solitons driven by a horizontal shear are investigated in light of the proposition that certain features in the Jupiter atmosphere may be explained by solitary Rossby waves. Consideration is given to a two-layer quasi-geostrophic model in which the motion in each layer consists of a different zonal shear flow, with vertical shear concentrated at the interface between the layers. In the case of a strong vertical shear, it is found that only a very restricted set of flows will admit Rossby neutral model solitons. For the more realistic case of a weak vertical shear, results indicate similar, but latitudinally shifted, wave patterns in each layer. It is noted that no such slant has yet been detected in the Great Red Spot.