Charney-Obukhov Equation Research Articles

New exact solutions of the (3+1) dimensional Charney - Obukhov equation for the ocean in the case of “separation” of variables

Exact solutions describing Rossby waves and vortices in ocean propagating along the zonal direction at a constant velocity are considered for the (3 + 1) -dimensional nonlinear Charney – Obukhov equation. We give examples of solutions of the Charney-Obukhov equation that satisfy the nonlinear boundary conditions for the ocean, for a special case that corresponds to the “separation” of variables. The following example is considered: a lattice of vortices with 3D flow in lattice cells, the inclination of vortex cells in the meridional plane, and the dependence of the inclination angle on depth.

Modeling ocean eddies using exact solutions of the Charney–Obukhov equation

New exact solutions of the Charney–Obukhov equation for the ocean are obtained in the form of a partial superposition of elementary solutions with different wave numbers. The boundary conditions for the ocean are satisfied due to the presence of a carrier zonal flow in the solution. The existing arbitrariness in the choice of wave numbers and other solution parameters makes it possible to simulate an arbitrary stream function profile at a fixed ocean depth on an interval of a fixed length using a Fourier series or in a circle of a fixed radius using a Fourier–Bessel series. An example of modeling a Gaussian stream function profile on the ocean surface in the presence of circular symmetry is considered.

A Lax Representation of the Charney–Obukhov Equation for the Ocean

A Lax Representation of the Charney–Obukhov Equation for the Ocean

On the superposition of solutions of the (3 + 1) dimensional Charney–Obukhov equation for the ocean

New exact solutions describing Rossby waves and vortices in ocean propagating along the zonal direction at a constant velocity are found for the (3 + 1)-dimensional nonlinear Charney–Obukhov equation. These solutions are a partial superposition of previously discovered exact solutions of the Charney–Obukhov equation. Partial superposition is found for that part of the solution of the Charney–Obukhov equation, which complements the zonal stream. The presence of such a superposition in the solutions of a nonlinear equation with two nonlinear boundary conditions is a remarkable property of the Charney–Obukhov equation for the ocean and allows one to simulate a wide class of fluid flows based on exact solutions. As an example, we discuss solutions that include superposition of trigonometric functions and the functions of Bessel in a horizontal plane, and superposition of spherically symmetrical solutions in the vertical coordinate. Visualization of the solutions found shows that, depending on the values of the parameters included in the solutions, they can describe both a flow with a large number of vortices and a periodic structure with alternating high and low pressure fronts.

New exact spatially localized solutions of the (3 + 1)-dimensional Charney–Obukhov equation for the ocean

New exact spatially localized solutions on the background of a zonal flow, propagating along the zonal direction at a constant velocity, are found for the (3 + 1)-dimensional nonlinear Charney–Obukhov equation describing Rossby waves and vortices in ocean. In total, five solutions are presented—two solutions with spherical symmetry and three solutions with cylindrical symmetry. One of the solutions with spherical symmetry is constructed using the Darboux transformation. Visualization of the solutions found shows that, depending on the values of the parameters included in the solutions, they can describe both an irrotational flow and a vortex flow with 1, 2, or more localized vortices. To analytically estimate the vertical localization of vortices, the necessary condition for the instability of the zonal flow and the condition for the maximum total vorticity are used. These estimates are in good agreement with the results of visualization of the vortex flow for the solutions found.

On exact solutions of the Charney Obukhov equation for the ocean

Exact solutions for the (3+1)-dimensional nonlinear Charney–Obukhov equation are obtained and their application to modeling waves and vortices in ocean is discussed. The considered solutions for the ocean, when changing the values of the parameters included in the solutions, demonstrate the transition of a laminar flow to a flow containing vortices.

New solutions for the (3 + 1)-dimensional Charney–Obukhov equation

New solutions are found for the (3+1)-dimensional nonlinear Charney–Obukhov equation describing Rossby waves and vortices. The solutions have the form of stationary structures moving with a constant velocity along the parallel. Some of the solutions found have a functional arbitrariness, which is probably due to the similarity of the Charney–Obukhov equation with an integrable nonlinear system. Functional arbitrariness can be useful when applying the solutions for modeling a wide range of problems.

Numerical model for zonal flow generation by magnetized Rossby waves in the ionosphere with the background shear flow

A theoretical-numerical description of zonal flow generation in the turbulent ionosphere by controlled inhomogeneous background wind is given. The generalized Charney–Obukhov equation, which describes the nonlinear interaction of five different-scale modes (primary modes, relatively short-wave ultra-low frequency (ULF) magnetized Rossby waves (MRWs) (pumping waves), two satellites of these MRWs, long-wave zonal mode, and large-scale background shear flows (inhomogeneous wind)) is used. New features of energy transfer from relatively small-scale waves and the background shear flow into that of largescale zonal flows and nonlinear self-organization of the five-wave collective activity in the ionospheric medium are identified based on the numerical solution of the corresponding system of equations for perturbation amplitudes (generalized eigenvalue problems). It is shown that if there is the background shear flow with a moderate amplitude growth the modulation instability increment and intensifies the zonal flow generation, while a very strong shear flow significantly reduces the modulation instability increment and can even suppress the generation process.

Symmetry group application for the ([formula omitted])-dimensional Rossby waves

The (3+1)-dimensional nonlinear Charney–Obukhov equation is analyzed by means of the classical Lie group approach. The classical Lie symmetry group calculated allows one to obtain new exact Rossby wave solutions when a particular Rossby wave solution is known. New solutions obtained assist in the understanding how the Rossby waves change when the background wind changes.

Zonal flow interaction with Rossby waves in the Earth's atmosphere: A numerical simulation

A numerical simulation of the Charney–Obukhov equation modified by the presence of a sheared zonal flow is carried out. The zonal flow is assumed to propagate longitudinally and is sheared along the meridians. It is shown that owing to the nonlinear interaction of the sheared zonal flow with the initially given disturbances the energy of the zonal flow is accumulating into the formations which are broken into several pieces. As a result new solitary vortex structures arise to produce the structural turbulence

Synoptic-scale nonlinear stationary magnetized Rossby waves in the ionospheric E-layer

Magnetized Rossby waves are produced by a dynamo electric field and represent the ionospheric generalization of tropospheric Rossby waves in a rotating atmosphere with a spatially inhomogeneous geomagnetic field. They are described by the modified Charney-Obukhov equation with a Poisson-bracket convective nonlinearity. This type of equation has solutions in the form of synoptic-scale nonlinear solitary dipole vortex structures of 1000–3000 km in diameter. With the use of equivalence conditions, various stationary nonlinear solutions are obtained and investigated analytically. The basic characteristics of stationary vortex structures for magnetized Rossby waves are investigated.

Investigation and numerical resolution of initial-boundary value problem for the generalized Charney–Obukhov and Hasagewa–Mima equations

In the present work, first-order accurate implicit difference schemes for the numerical solution of the nonlinear generalized Charney–Obukhov and Hasegawa–Mima equations with scalar nonlinearity are constructed. On the basis of numerical calculations accomplished by means of these schemes, the dynamics of a two-dimensional nonlinear solitary vortical structures is studied. For the considered equations the initial-boundary value problem is set when at the initial moment the solution in the form of solitary dipole structure is taken. For this problem uniqueness of the solution in case of periodic boundary conditions is proved. The dynamic relation between solutions of the generalized Charney–Obukhov and Hasegawa–Mima equations is established. It is shown that, in spite of the existing opinion, the scalar nonlinearity in case of the generalized Hasegawa–Mima equation develops monopolar anticyclone, while in case of the generalized Charney–Obukhov equation develops monopolar cyclone.

Some remarks on a Hasegawa–Mima–Charney–Obukhov equation

This paper deals with the local Cauchy problem and the existence of weak solutions for an Hasegawa–Mima–Charney–Obukhov equation arising in geophysics and plasmas theory. This model can be obtained as an asymptotic model from the Euler equations with free surface under a quasi-geostrophic velocity field assumption. We use a parabolic regularization of the equation and a priori estimates on the solutions. Moreover, in order to pass to the limit we strongly use the Jacobian structure of the nonlinear term.

First- and second-order accurate implicit difference schemes for the Charney–Obukhov equation

In the present work, first- and second-order accurate implicit difference schemes for the numerical solution of the nonlinear Charney–Obukhov equation are constructed. For each constructed scheme, the approximation error is estimated and the convergence of the iterative process is investigated. On the basis of numerical calculations accomplished by means of these schemes, the propagation of a two-dimensional nonlinear solitary Rossby vortex structure is studied. A comparative analysis of the obtained numerical results is carried out.

New solution for nonlinear pancake solitary Rossby vortices

The new solution of the three-dimensional nonlinear Charney–Obukhov equation describing solitary pancake Rosssby vortices is found. The solution is represented in the form of an axially symmetric cylindrical monopole (anticyclonic) vortical structure moving with constant velocity. A close-packed array of non-overlapping monopole vortices with alternating cyclone and anticyclone structures is also a solution.

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

What can asymptotic expansions tell us about large-scale quasi-geostrophic anticyclonic vortices?

Abstract. The problem of the large-scale quasi-geostrophic anticyclonic vortices is studied in the framework of the baratropic rotating shallow- water equations on the β-plane. A systematic approach based on the multiplescale asymptotic expansions is used leading to a hierarchy of governing equations for the large-scale vortices depending on their characteristic size, velocity and a free surface elevation. Among them are the Charney-Obukhov equation, the intermediate geostrophic model equation, the frontal dynamics equation and some new nonlinear quasi-geostrophic equation. We are looking for steady-drifting axisymmetric anticyclonic solutions and find them in a consistent way only in this last equation. These solutions are soliton-like in the sense that the effects of weak non-linearity and dispersion balance each other. The same regimes on the paraboloidal β-plane are studied, all giving a negative result in what concerns the axisymmetric steady solutions, except for a strong elevation case where any circular profile is found to be steadily propagating within the accuracy of the approximation.

Rossby solitary vortices, on giant planets and in the laboratory

This is a review of laboratory experiments with a layer of shallow water having a free surface and rotating together with a vessel of parabolic form. Such a (rather original) setup has allowed one to create Rossby solitary vortex for the first time. The latter is an anticyclonic Rossby vortex not subjected to dispersive spread owing to its compensation by the nonlinearity of KdV type. By its structural, collisional, and other properties, including clear-cut cyclonic-anticyclonic asymmetry, it may be considered as a physical prototype of the large-scale long-lived anticyclonic Rossby vortices like the Great Red Spot of Jupiter or the Great Dark Spot of Neptune (this remarkable vortex was discovered by the spacecraft Voyager-2 during its farewell to the Solar System) and other vortices dominating in the atmospheres of giant planets and created by the unstable zonal flows. It has been shown that the vortex under study is a long-lived entity provided it satisfies "antitwisting condition," i.e., it has rather large amplitude (at which it rotates more quickly than it propagates and thereby carries the trapped fluid). In this case, it is not subjected to the "twisting" deformation and may be ascribed by the generalized Charney-Obukhov equation for Rossby vortices on shallow water with a free surface. The results of creating the vortex under consideration by the different methods have been compared with the results obtained by other authors in the experiments on shear-flow generation of Rossby vortices.

Some remarks on coherent structures out of chaos in planetary atmospheres and oceans.

In the context of planetary atmospheres and oceans, it is natural to define "coherent structures" as "long-lived," or "solitary," Rossby vortices. These can be described by the generalized Charney-Obukhov equation (in fluid dynamics) or the analogous generalized Hasegawa-Mima equation (in plasma physics). These two equations contain KdV-type nonlinearities which (together with the compensating dispersive spreading) determine the formation of the coherent structures and explain the clear-cut cyclonic/anticyclonic asymmetry observed experimentally in long-lived planetary Rossby vortices. Examples are given of natural vortices which are (and which are not) coherent structures.