Solitary Rossby Waves Research Articles

Algebraic rossby solitary wave and blocking in the atmosphere

In this paper, the atmosphere is divided into two regions which are governed by different equations. Furthermore, multiple-scale method is used to obtain the Benjamin-Ono equation satisfied by the nonlinear Rossby wave in weak shear zonal flow. The equation has algebraic solitary wave solution, and the stationary streamfunction fields in-ithe atmosphere are calculated by using numerical method, and the results demonstrate that the stationary solution is antisymmetric dipoles with the anticyclone north of the cyclone, and the structure is similar to the equivalent modon obtained by Mcwilliams (1980). Because the modon obtained here is an algebraic solitary wave, the modon may be called “algebraic modon”, and the stationary algebraic modon is consistent with observations of blocking pattern in the atmosphere.

The north-south elongated solitary Rossby wave

As far as the author knows, the previous models of solitary Rossby wave have been restricted to the case of the east-west elongated one. However, now, it is shown by Yano and Tsujimura that the north-south elongated KdV-type solitary Rossby wave is also possible. In this note, a typical example of the north-south elongated elongated KdV-type solitary Rossby wave in the shallow water β-plane model is examined. The conventional east-west elongated solitary Rossby wave is governed by the KdV equation in the longitudinal direction at each latitude. The same is true for the case of the north-south elongated solitary Rossby wave. The main difference is that, however, the KdV-soliton defined at each latitude has drifted by the local phase velocity, which is different for each latitude. Hence, the wave pattern is deformed continuosly with time in the latitudinal direction, and the separable solution is not possible as is in the case of the east-west elongated solitary wave.

The domain of validity of the KdV-type solitary rossby waves in the shallow water β-plane model

Abstract The Domain, where the necessary and sufficient conditions for the existence of the KdV-type solitary Rossby waves are satisfied is defined in the shallow water β-plane model. The KdV-type solitary Rossby waves are the Rossby waves whose time-longitude dependence is determined by the KdV equation. As far as an appropriate amplitude and an appropriate ratio of the scales of the east-west and north-south directions are given, the KdV-type solitary Rossby waves can exist for every basic zonal flow. This result suggests the large validity of the soliton model in geophysical fluid dynamics. The KdV-type solitary Rossby waves are classified into four categories: (1) shear solitons studied by Long, Larsen, Benny, Redekop, and Hukuda, (2) β-divergent solitons studied by Clarke, Yamagata, and Nogami, (3) β-solitons found in the case of the strong stratification, and (4) divergent solitons which exist in the planetary-geostrophic-scale zonal flow. A remarkable result is that, in addition to the conventional east-west elongated solitons, the north-south elongated solitons can also exist for the case of the divergent solitons.

Equatorial solitary waves of tropical atmospheric motion in shear flow

Starting from the primary equations, the author derives the KdV equation which describes solitary Rossby waves in the tropical atmosphere, and indicates that, because these waves are ageostrophic, they differ from the quasigeostrophic solitary Rossby waves studied by Redekopp et al. Owing to nonlinear action, these waves are also different from traditional linear waves of the tropical atmosphere. The author believes that the stationary tropical atmospheric waves reflect the characteristics of solitary waves in that the energy does not disperse.

On the uniqueness of solitary Rossby waves

AbstractThe explicit solitary Rossby wave solutions found by Larichev, Reznik and Berestov are shown to be unique for the model equations considered, in the sense that there are no other antisymmetric wave solutions which are not of these forms. This is done by adapting arguments used by Amick and Fraenkel to show the uniqueness of the Hill's vortex solution. It is based on the maximum principle and the domain folding method of Gidas, Ni and Nirenberg, and involves showing that the function ψ/y is radially symmetric, where ψ is the streamfunction of a solitary wave and y the horizontal cartesian coordinate perpendicular to the x-axis, along which the waves move at steady positive speed. This argument is also used to show the uniqueness of the well-known explicit solutions for cylindrical vortices. The result does not apply directly to rider solutions of Flierl et al., which are not antisymmetric, but it does restrict the possible rider solutions that can form because of their association with particular antisymmetric solutions.

A numerical method for solving the evolution equation of solitary Rossby waves on a weak shear

In this paper an evoluion equation in integral-differential form for finite amplitude Rossby waves on a weak shear is presented and an efficient method for its numerical solution is set up. It is shown that a propagation of solitary wave is possible whenever a proper weak shear in basic flows acts with the nonlinear effects and dispersion of the media, both in the atmosphere and in the ocean. To test the numerical method for solving the evolution equation, a series of experiments are carried out. The results indicate that the solitary solutions do exist and interact with each other in quite a succinct, manner. Therefore the method is successful and efficient for solving initial value problems of the above equation. The time decoupling problem arising in the numerical scheme and the related filtering technique are discussed. A variety of interesting phenomena such as the interaction of solitary Rossby waves, damping, dispersion and the development of nonlinear wave train are numerically studied.

Kelvin wave fronts, Rossby solitary waves and the nonlinear spin-up of the equatorial oceans : Greatbatch, R.J., 1985. J. geophys. Res., 90(C5):9097–9107

Kelvin wave fronts, Rossby solitary waves and the nonlinear spin-up of the equatorial oceans : Greatbatch, R.J., 1985. J. geophys. Res., 90(C5):9097–9107

Point vortices at the β-plane and Rossby solitary waves: Reznik, G.M., 1986. Okeanologiia, 26(2):165–173. (In Russian, English abstract.)

Point vortices at the β-plane and Rossby solitary waves: Reznik, G.M., 1986. Okeanologiia, 26(2):165–173. (In Russian, English abstract.)

Solitary axisymmetric Rossby waves of finite amplitude

Low-frequency Rossby waves, similar to drift waves in plasma, can propagate in a thin layer of fluid on a rotating sphere. The theory of Rossby waves was developed primarily for quasigeostrophic motions of low amplitude (the relative vortlcity is much less than the angular velocity of rotation of the sphere Ω and the layer thickness H differs little from the unperturbed thickness H0) [1]. Of particular interest are nearly axisymmetric solitary waves describing isolated vortex formations moving uniformly opposite to the direction of rotation of the sphere (to the west). Quasigeostrophic waves of low amplitude, inside which the fluid rotates in the opposite direction to the rotation of the sphere (anticyclonically), were considered in [2, 3]. Anticyclones of finite amplitude were investigated in [4], where it was shown that the structure of a solitary wave depends on the previous history of its formation. In this paper, equations are derived for the description of the slow evolution of flows of finite amplitude in a thin spherical layer of homogeneous rotating fluid and the structure of a smooth axisymmetric solitary Rossby wave is analyzed.

Kelvin wave fronts, Rossby solitary waves and the nonlinear spin‐up of the equatorial oceans

This paper examines the spin‐up from rest of an equatorial ocean with meridional boundaries following the sudden onset of spatially uniform winds by using a nonlinear, one‐active‐layer, reduced‐gravity shallow‐water model. It is shown that the results obtained can differ considerably from corresponding integrations with a linear model and that this occurs within realistic parameter ranges. In the westward wind case a Kelvin wave front can develop, across which there is a discontinuity in the zonal current and in the depth of the model pycnocline. A vortex pair resembling a Rossby solitary wave as described theoretically by Boyd (1980b) is generated by the impact of this front on the eastern boundary. In the eastward wind case a series of equatorial Rossby solitary waves is generated at the eastern boundary. An extension of Boyd's theory is given that allows for a background zonal current and is used to account for the westward propagation speed of these waves. An important effect of nonlinearity in this case is to prevent equatorial waves from being able to bring to rest the equatorial (Yoshida) jet that is established during the first stage of the integration. The model response is shown to depend on the values of only two parameters, one of which (X) measures the nonlinearity and the other (δ) the ratio of the equatorial Rossby radius to the longitudinal basin width L. The former parameter varies as the product of L and the surface wind forcing and as the inverse square of the linear gravity wave speed. It is known that there is a threshold value of X (Xcrit) for the sharp Kelvin wave front to occur. It is shown that the results presented here for a nonlinear, one‐active‐layer, reduced‐gravity shallow‐water model are of relevance to a continuously stratified ocean model, provided there is a thermocline‐trapped mode that is dominantly excited by wind forcing, such as the second baroclinic mode in the equatorial oceans.

Measurements and interpretation of a Jovian, near-equatorial feature

We present evidence to suggest that the large, rapidly moving features in Jupiter's equatorial currents are manifestations of Rossby solitary waves. Their morphological and dynamical similarity to the great red spot and the white ovals is revealed and discussed in the context of the available theory.

Nonlinear waves in barotropic model

In this paper, from the system of equation describing a barotropic atmosphere using the method of Taylor expansion for the nonlinear terms, the periodic solutions of the nonlinear inertio-surface gravity waves and Rossby waves have been obtained. The finite-amplitude nonlinear inertio-surface gravity waves and Rossby waves with horizontal divergence satisfy all the KdV equation. The solutions are all the cnoidal function, i. e., the cnoidal waves which include the linear waves and form the solitary waves under certain conditions. For the finite-amplitude Rossby waves with horizontal divergence, we find the new dispersive relation including both the wave number and the amplitude parameter. In case of small amplitude it is reduced to the Yeh formula. It is shown that the larger the amplitude and width, the faster the finite-amplitude inertio-surface gravity waves and the slower the finite-amplitude Rossby waves with horizontal divergence propagate. The blocking or cut-off system in which the amplitude and width are large may be considered as Rossby solitary waves.

Equatorial Solitary Waves. Part 3: Westward-Traveling Modons

Abstract Boyd's previous work on equatorial Rossby solitary waves which derived the Korteweg-deVries equation using the method of multiple scales is here extended in several ways. First, the perturbation theory is carried, to the next highest order to (i) assess the accuracy and limitations of the zeroth-order theory and (ii) analytically explore solitons of moderate amplitude. Second, using the refined theory, it is shown that Rossby solitary waves will carry a region of closed recirculating fluid along with the wave as it propagates provided that the amplitude of the wave is greater than some (moderate) threshold. The presence of such closed “streaklines”, i.e., closed streamlines in a coordinate system moving with the wave, is an important property of modons in the theory of Flieri, McWilliams and others. The “closed-streakline” Rossby waves have many other properties in common with modons including (i) phase speed outside the linear range, (ii) two vortex centers of equal magnitude and opposite sign, ...

The Evolution of Finite Amplitude Solitary Rossby Waves on a Weak Shear

The evolution equation urn:x-wiley:00222526:media:sapm1983692127:sapm1983692127-math-0001 is derived for finite amplitude, long Rossby waves on a weak shear generalizing an earlier version given by Benney [1].

Numerical investigation of the collisions of two-dimensional solitary Rossby waves : Larichev, V.D. and G.M. Reznik, 1982. Dokl. Akad. Nauk SSSR, 264(1):229–233. (In Russian)

Numerical investigation of the collisions of two-dimensional solitary Rossby waves : Larichev, V.D. and G.M. Reznik, 1982. Dokl. Akad. Nauk SSSR, 264(1):229–233. (In Russian)

Topographic effects on solitary Rossby waves

The model for baroclinic solitary waves previously described by Flierl is developed to include effects of bottom variations.Topographic disturbances are shown to act, through their first and second derivatives, on both the phase speed correction and the maximum horizontal amplitude.The combined effects of mean shear flow steepening and bottom variations are considered; an equation is derived, showing the relative importance of each effect.

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.

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.

Solitary rossby waves over variable relief and their stability. Part II: numerical experiments

The barotropic, quasi-geostrophic vorticity equation describing large scale, rotating flows over zonal relief supports nonlinear permanent form solutions, namely nonlinear topographic Rossby waves. Through an analytical theory, these solutions have been shown to be neutrally stable to infinitesimal perturbations. Numerical algorithms, which necessarily truncate the infinite number of degrees of freedom of any continuum model to a finite number, are capable of reproducing the numerical equivalent of these form-preserving solutions. Moreover, these numerical solutions are shown to preserve their shape throughout the numerical experiment not only in the limit of small amplitude, but also for high amplitude (Rossby number → O(1)). Through numerical simulation, the stability analysis is carried far beyond the analytical limit of infinitesimal perturbations. The solutions maintain their stability in agreement with the analytical theory, up to perturbations having intensities almost of the same order as the solutions themselves. For higher-amplitude perturbations, the solutions break up and typical turbulent behavior ensues. The passage from wave-like to turbulent behavior, upon surpassing a critical perturbation value, can be observed in the sudden loss of phase locking of the permanent solution Fourier modes.

Solitary Rossby waves over variable relief and their stability. Part I: The analytical theory

Nonlinear permanent form solutions have been found for the barotropic, quasi-geostrophic divergenceless vorticity equation describing large scale, rotating flows over zonal relief. In the linear limit these solutions are topographic Rossby waves. The analytical procedure is an expansion in two small dimensionless parameters, an amplitude parameter (the Rossby number) and the aspect ratio between North-South (cross-relief) and East-West length scales. Permanent form solutions exist when these two parameters, and the related effects of dispersion and nonlinearity, mutually balance. By the same expansion procedure, an analytical linearized stability theory has been formulated which proves the neutral stability of these solutions to infinitesimal, two-dimensional perturbations.