Barotropic Potential Vorticity Equation Research Articles

A new three dimensional dissipative Boussinesq equation for Rossby waves and its multiple soliton solutions

A new three dimensional nonlinear dynamic theoretical model is derived from fluid mechanics system. In this paper, From the quasi-geostrophic barotropic potential vorticity equation, we obtain a three dimensional dissipative Boussinesq equation by the reduced perturbation method, i.e.utt+e1uxx+e2(u2)xx+e3utxy+e4uxxxx+e5uxxyy=0. It is emphasized that the new equation is different from the existing Boussinesq equations, which describe the three dimensional nonlinear Rossby waves in the atmosphere. Moreover, we explore the dispersion relation of the linear wave through the new equation. Using the travelling wave method and simplest equation method, the general solution and soliton solutions of the equation are obtained successfully respectively. Finally, the formation mechanism of Rossby waves is discussed by multiple soliton solutions.

New model and dynamics of higher-dimensional nonlinear Rossby waves

In this work, the propagation of higher-dimensional nonlinear Rossby waves under the generalized beta effect is considered. Using the methods of weak nonlinear perturbation expansions and the multiple scales, we obtain a new (2 + 1)-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation for the first time. Furthermore, a new dispersion relation for the linear Rossby waves is given corresponding to the linearized Boussinesq equation. More importantly, based on the methods of the traveling wave setting and the Jacobi elliptic function expansions, several kinds of exact traveling wave solutions for the higher-dimensional nonlinear Rossby waves, including the periodic solutions, solitary solutions and others are obtained. Finally, we simulate the solitary solutions obtained by using the method of the Jacobi elliptic function. The numerical results show that the amplitude of the Rossby solitary waves is decreasing with the increase of generalized beta effect.

Dynamical Analysis and Exact Solutions of a New (2+1)-Dimensional Generalized Boussinesq Model Equation for Nonlinear Rossby Waves**Supported by the National Natural Science Foundation of China under Grant Nos. 11562014, 11762011, 11671101, 71471020, 51839002; the Natural Science Foundation of Inner Mongolia under Grant No. 2017MS0108; Hunan Provincial Natural Science Foundation of China under Grant No. 2016JJ2061; the Scientific Research Fund of

In this paper, we study the higher dimensional nonlinear Rossby waves under the generalized beta effect. Using methods of the multiple scales and weak nonlinear perturbation expansions [Q. S. Liu, et al., Phys. Lett. A 383 (2019) 514], we derive a new (2+1)-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation. Based on bifurcation theory of planar dynamical systems and the qualitative theory of ordinary differential equations, the dynamical analysis and exact traveling wave solutions of the new generalized Boussinesq equation are obtained. Moreover, we provide the numerical simulations of these exact solutions under some conditions of all parameters. The numerical results show that these traveling wave solutions are all the Rossby solitary waves.

A new model equation for nonlinear Rossby waves and some of its solutions

In this paper, we investigate the (2+1) dimensional nonlinear Rossby waves under non-traditional approximation. Using the asymptotic methods of multiple scales and weak nonlinear perturbation expansions, we derive a new modified Zakharov–Kuznetsov equation from the barotropic potential vorticity equation with the complete Coriolis parameter, the topography and the dissipation. Based on the new auxiliary equation method, new exact solutions of the new mZK equation are obtained when the dissipation is absent. However, the new auxiliary equation method fails to solve the new mZK equation with the dissipative term. Therefore, the weak nonlinear method and the homotopy perturbation method are developed to solve the obtained new mZK equation. Through numerical simulations, the results show the effects of different parameters on Rossby waves.

The disturbance energy of Rossby wave in barotropic atmosphere

Fourier-transformed method is used to investigate the problem of the rapid growth and decay of the disturbance energy of Rossby wave in barotropic atmosphere on the assumption that this kind of atmosphere is adiabatic, non-frictional, unforced, non-dissipative and quasigeostrophic on a beta plane. The analytical solution of linear barotropic potential vorticity equation is obtained for the first time and we go a step further to discuss the relations between the disturbance energy and the initial perturbation flow wave numbers, shear of basic flow and viscid coefficient by using the above analytical solution.

Symmetry Analysis of Barotropic Potential Vorticity Equation

Recently F. Huang [Commun. Theor. Phys. 42 (2004) 903] and X. Tang and P.K. Shukla [Commun. Theor. Phys. 49 (2008) 229] investigated symmetry properties of the barotropic potential vorticity equation without forcing and dissipation on the beta-plane. This equation is governed by two dimensionless parameters, F and β, representing the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth' angular rotation, respectively. In the present paper it is shown that in the case F ≠ 0 there exists a well-defined point transformation to set β = 0. The classification of one- and two-dimensional Lie subalgebras of the Lie symmetry algebra of the potential vorticity equation is given for the parameter combination F ≠ 0 and β = 0. Based upon this classification, distinct classes of group-invariant solutions are obtained and extended to the case β ≠ 0.

A Form of Potential Vorticity Equation for Depth-Integrated Flow with a Free Surface

Abstract A form of linear, barotropic potential vorticity equation is derived for an ocean with a free surface, in which only one scalar variable appears (ocean bottom pressure, or subsurface pressure). Unlike quasigeostrophic or rigid-lid derivations, the only approximation made (apart from linearization) is that changes in the circulation must be slow compared with the inertial frequency. Effects of stratification are included, but only parametrically in the sense that density is treated as a given quantity or forcing term rather than a variable.

Periodic Structures of Rossby Wave under Influenceof Dissipation

A simple barotropic potential vorticity equation with theinfluence of dissipation is applied to investigate the nonlinearRossby wave in a shear flow in the tropical atmophere. By thereductive perturbation method, we derive the rotational KdV (rKdVfor short) equation. And then, with the help of Jacobi ellipticfunctions, we obtain various periodic structures for these Rossby waves. It is shown that dissipationis very important for these periodic structures of rational form.

Periodic structures of oceanic Rossby wave under the influence of wind stress

Abstract A simple oceanic barotropic potential vorticity equation on β-plane with the influence of wind stress is applied to investigate the nonlinear Rossby wave in a shear flow. By the reductive perturbation method, we derived the rotational modified KdV (rmKdV for short) equation. And then with the help of Jacobi elliptic functions, we obtain various periodic structures for these equatorial Rossby waves. It is shown that the wind stress is very important for these periodic structures of rational form.

Numerical modeling of non-linear characteristics for barotropic atmosphere with external forcing

Some characteristics of barotropic atmosphere are illustrated by numerical modeling, basing on a potential vorticity equation with forcing, dissipation, and non -linear interaction. For the barotropic potential vorticity equation (named the barotropic model) on a β-plane channel its solution is a stationary, wave-like pattern under a certain thermal forcing, while a quasi-periodic oscillation betw een two essential flow patterns appears if north-south thermal heating and wave- like topography are added as the model forcing. The oscillation has a period abo ut 15d, very similar to atmospheric index cycle in middle and high latitudes. Th e low-index pattern corresponds to a blocking flow, whereas the high index one i s an intensive westerly flow. More oscillation solutions are expected when the e quation is under periodic boundary condition with a strong thermal forcing, and a similar quasi-periodic solution can be obtained, which also results from a wav e topography forcing. However, its low index pattern is symmetrical, and differe nt from the blocking pattern produced by the β-plane model. When the heating fo rcing becomes weak, there also exists a stationary solution. Therefore, the char acteristics of atmospheric index cycle may not be soundly manifested by an analy tical solution or numerical solution under periodic boundary condition using low -order spectrum method that approximates the barotropic model in the β-plane ch annel.

Coastal currents generated by outflow and vorticity and their interaction with topography

The offshore spreading of coastal river outflows over shelf-like topography is inhibited by the demands of potential vorticity conservation. As a result, outflows tend to remain coastally trapped and flow parallel to the coast. Alongshore currents may also form when the outflow fluid has vorticity relative to the surrounding fluid, in which case the ‘image’ vorticity drives the outflow parallel to the coast. In the present work, the combined effects of topography and anomalous vorticity of the outflow in forming coastal currents are studied. Shelf-like topography is considered in the form of a sudden step change in depth, running parallel to a straight coast, some distance offshore. A simple model incorporating these effects is formulated using quasigeostrophic dynamics for a barotropic fluid with a rigid lid. The role of topographic wave radiation in establishing coastal currents is studied using the linearised barotropic potential vorticity equation for the case of a weak outflow of fluid with zero anomalous vorticity. The outflow is shown to be confined to the shelf and flows to the right of the source region. For stronger outflows, with relative vorticity, the method of contour dynamics is used to study the nonlinear evolution of the outflow plume and, in particular, its interaction with topography. The effects of outflow strength, magnitude and sign of the relative vorticity of the outflow fluid are considered in detail. In particular, the offshore penetration distance and rate of alongshore spreading of the outflow plume are studied. When the outflow comprises of fluid with positive (cyclonic) vorticity, the topographic and vorticity-driven image effects reinforce each other and increase the spreading rate of the outflow plume to the right of the source. For negative (anticyclonic) vorticity outflows the effects oppose each other leading to enhanced offshore spreading of the outflow. The numerical results demonstrate that the offshore transport is achieved by an efficient dipolar vortex mechanism.

An analytical study of general hyper-diffusivity and barotropic eddies

An exact solution to the barotropic potential vorticity equation is used to examine the properties of barotropic vortices under arbitrary nth-order hyper-diffusivity. Analytical expressions are derived for an eddy's lifetime, meridional drift, decay in size, and energy, as functions of the Coriolis parameter, order and magnitude of diffusivity, and the eddy's size, shape and strength. These expressions provide a simple explanation for many observed features of oceanic and atmospheric vortices. For example, the competition between the Coriolis effect and eddy strength in giving permitted eddy geometries; the bias towards a zonal anisotropy for large vortices but not for small ones; energetic preference for axisymmetry; poleward meridional drift of cyclonic vortices; and meridional speed variation depending on eddy geometry and strength.

Instability of time-dependent wind-driven ocean gyres

The wind-driven ocean circulation at midlatitudes is susceptible to several types of instabilities. One of the simplest models of these flows is the quasigeostrophic barotropic potential vorticity equation in an idealized ocean basin. In this model, the route to complex spatio/temporal flows is through successive bifurcations. The aim of this study is to describe the physics of the destabilization process of a periodic wind-driven flow associated with a secondary bifurcation. Although bifurcation theory has proven to be a valuable tool to determine the physical mechanisms of destabilization of fluid flows, the analysis of the stability of time-dependent (for example, periodic) flows, using this methodology, is computationally unpractical, due to the large number of degrees-of-freedom involved. The approach followed here is to construct a low-order model using numerical Galerkin projection of the full model equations onto the dynamically active eigenmodes. The resulting reduced model is shown to capture the local dynamics of the full model. The physical mechanism of the destabilization of the periodic wind-driven flow is deduced from the reduced model. While there are several stabilizing processes, notably rectification, the destabilization occurs due to time-dependent increase of the background horizontal shear in the flow.

Rossby Waves in Zonal Barotropic Flows with Pseudocritical Levels

Abstract A pseudocritical level is defined as a region where the wave phase speed c equals the zonal basic flow velocity U and, additionally, the background potential vorticity gradient vanishes. In this study Rossby wave propagation is investigated in zonal mean flows when pseudocritical levels are present. For that purpose the barotropic potential vorticity equation linearized about prescribed zonal basic flows is used. The basic flows are constructed such that (i) the amplitude equation for Rossby waves can be solved analytically, (ii) the flows exhibit several pseudocritical levels, and (iii) the flows do not look entirely unrealistic compared to observed zonally averaged flows in atmospheres and oceans. Two examples of basic flows are studied: an unbounded multiple-jet flow on the f plane and a single-jet flow on the β plane bounded to the south by a reflective wall. It is shown that pseudocritical levels are neither critical levels nor turning points of the flow. Global solutions representing trappe...

Vanishing Viscosity in the Barotropic β-Plane

The initial boundary value problems associated with the inviscid barotropic potential vorticity equation in the β-plane and its viscous analogue are considered. It is shown that the solution velocity to the viscous equation converges to the inviscid solution in aC1sense for finite times and that, under additional smoothness assumptions on the inviscid flow, this convergence can be extended toC3. Moreover, this convergence occurs as O(ε), where ε is the viscous parameter. This particular form of vanishing viscosity is of relevance in analysing viscosity induced advection for barotropic models.

A Diagnostic Barotropic Finite-Element Ocean Circulation Model

Abstract The finite-element method possesses many advantages over more traditional numerical techniques used to solve systems of differential equations. These advantages include a number of conservation properties and a natural treatment of boundary conditions. The method's piecewise nature makes it useful when dealing with irregular domains and similarly when using variable horizontal resolution. To take advantage of these properties, a finite-element representation of the linearized, steady-state, barotropic potential vorticity equation is developed. The Stommel problem is used as an initial test for the model. A fourth-order eddy viscosity term is then added, and the resulting problem is solved in both simply and multiply connected domains under both slip and no-slip boundary conditions. The beta-plane assumption is then relaxed, and the model is reformulated in spherical coordinates. A realistic geography and topography version of this model is also used to examine the barotropic circulation in the No...

Resonance of Fofonoff's Mode in a Rotated Basin

Abstract Fofonoff solutions to the inviscid barotropic potential vorticity equation are found for the steady, free flow in a basin rotated at an arbitrary angle to a latitude circle. These solutions am used to study the inertial recirculation of the subtropical gyre, which is forced by anomalously low values of potential vorticity within the separated Gulf Stream. It is found, depending on the sense of rotation of the northern boundary relative to latitude circles, that closed potential vorticity contours can exist that allow a large-amplitude inertial response, a resonance of the Fofonoff gyre. A barotropic ocean model is used to confirm thew ideas, in which flow is forced both by potential vorticity anomalies prescribed at the boundaries, and also a field of Ekman pumping, which spins up an anticyclonic gyre. It is shown that the recirculation is sensitive to both the orientation and extension of the inertial jet along the northern boundary.

On the Linear Stability of Verkley's Modons.

Verkley (1987, 1990) constructed modons on a sphere as models of atmospheric blocking phenomena. These are exact solitary wave solutions to the barotropic potential vorticity equation on a sphere. In this paper, the linear Liapunov stability of Verkley's 1987 modons is proved by means of Arnol'd's invariant, and moreover it is shown that Verkley's 1990 modons are linearly spectrally stable. In spite of the fact that Arnol'd's invariant does not exist for the 1990 modon, it is still possible for the 1990 modon to be proved linearly spectrally stable because of the fact that the 1990 modon is a limiting case of the 1987 modon.

A Relation between Verkley's 1987 Modon and 1990 Modon.

Verkley (1987, 1990) constructed modons as a model of the atmospheric blocking phenomenon. They are exact solutions of the barotropic potential vorticity equation on a sphere. In this note, it is shown that the 1987 modon changes continuously into the 1990 modon as the value of -dQ/dΨ within the modon goes to zero, where Q and Ψ are potential vorticity and streamfunction of the 1987 modon, respectively. Thereby it is revealed that the stability properties of the 1990 modon can be studied by means of Arnol'd's invariant of the 1987 modon though Arnol'd's invariant can not be defined for the 1990 modon.

Topographic Rossby waves above a random array of seamountains

The barotropic potential vorticity equation or topographic wave equation is not linear in topography: the solution structure for topography formed from a sum of obstacles is not the sum of solutions for the obstacles in isolation, even when these individual solutions have identical frequencies. This paper considers the mechanism by which normal modes of oscillation above one mountain are modified by interactions with its neighbours. Exact explicit solutions for the normal modes above a pair of circular seamountains show that the interactions between the mountains rapidly approaches the large-separation approximation obtained by considering solely the first reflection of the disturbance of one mountain at the other. For mountains of one diameter separation at the closest point, the approximation is accurate to within 1%. Perhaps surprisingly, coupling between two identical mountains is weak and resonance occurs between mountains and dales of equal and opposite height.The accurate approximate solutions enable consideration of the effects on a mountain of an infinite set of randomly distributed neighbours. The ensembleaveraged frequency for a mountain of given height is determined in terms of the area fraction of the other mountains. The idea of an effective topography is introduced for the ensemble-averaged stream function: it is that (non-random) topography generating a stream function identical to the ensemble-averaged stream function. This differs markedly from the ensemble-averaged topography. The explicit form of the effective topography is derived for a set of right circular cylinders.