Rossby Wave Equation Research Articles

Variable coefficient (2+1)D KP equation for Rossby waves and its dynamical analysis

Variable coefficient (2+1)D KP equation for Rossby waves and its dynamical analysis

Variable coefficient extended cKP equation for Rossby waves and its exact solution with dissipation

In this article, the variable coefficient (2 + 1)-dimensional extended cylindrical Kadomtsev–Petviashvili (cKP) equation describing Rossby waves was derived from the quasi-geostrophic potential vorticity equation. It is difficult for the variable coefficient cKP equation with dissipation to calculate the exact solution. For obtaining the exact solution, a new transformation was constructed for the first time to reduce the extended cKP equation to the extended KP equation. We emphasize that the exact solution, and not just approximate solution, in Rossby waves flow field can be obtained when dissipation is included. The exact lump and interaction solutions with dissipative effect are given according to the modified Hirota bilinear method, and physics for the evolution of Rossby waves is analyzed based on the obtained solutions. When the dissipative parameter μ0 increases, the structure of the amplitude A changes in the spatial scale y. And when the dissipative parameter increases to a certain value, the structure of Rossby waves tends to be stable. It is pointed out that the dissipative parameter μ0 determines not only the amplitude A of Rossby waves but also structures of Rossby waves flow field, with μ0 acting on the spatial scale y and the timescale t.

Abundance of Exact Solutions of a Nonlinear Forced (2 + 1)-Dimensional Zakharov–Kuznetsov Equation for Rossby Waves

In this paper, an improved tan (φ/2) expansion method is used to solve the exact solution of the nonlinear forced (2 + 1)-dimensional Zakharov–Kuznetsov equation. Firstly, we analyse the research status of the improved tan (φ/2) expansion method. Then, exact solutions of the nonlinear forced (2 + 1)-dimensional Zakharov–Kuznetsov equation are obtained by the perturbation expansion method and the multi-spatiotemporal scale method. It is shown that the improved tan (φ/2) expansion method can obtain more exact solutions, including exact periodic travelling wave solutions, exact solitary wave solutions, and singular kink travelling wave solutions. Finally, the three-dimensional figure and the corresponding plane figure of the corresponding solution are given by using MATLAB to illustrate the influence of external source, dimension variable y, and dispersion coefficient on the propagation of the Rossby wave.

Symmetries and solutions for the inviscid oceanic Rossby wave equation

The -and -dimensional inviscid Rossby wave equations are analysed using Lie symmetry techniques. The travelling-wave reductions for the equation lead to a third-order ordinary differential equation from which the propagation properties are derived. It is observed that the wave has easterly phase velocity and westerly group velocity. Also, the wave propagates slightly faster in the f-plane than the β-plane. The dispersion relation derived from the third-order equation shows that the Rossby equation transports energy both eastward and westward and its speed is reduced successively and the propagation remains eastward. As for the two-dimensional Rossby wave equation, certain solutions which behave like solitary waves after a certain time are plotted. Interestingly, for certain symmetries the reductions lead to the Riccati's, Abel's, Euler's type and to some linearized equations. Moreover, certain reductions lead to highly nonlinear equations which are analysed by the singularity analysis method.

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.

Midlatitude unstable air-sea interaction with atmospheric transient eddy dynamical forcing in an analytical coupled model

While recent observational studies have shown the critical role of atmospheric transient eddy (TE) activities in midlatitude unstable air-sea interaction, there is still a lack of a theoretical framework characterizing such an interaction. In this study, an analytical coupled air-sea model with inclusion of the TE dynamical forcing is developed to investigate the role of such a forcing in midlatitude unstable air-sea interaction. In this model, the atmosphere is governed by a barotropic quasi-geostrophic potential vorticity equation forced by surface diabatic heating and TE vorticity forcing. The ocean is governed by a baroclinic Rossby wave equation driven by wind stress. Sea surface temperature (SST) is determined by mixing layer physics. Based on detailed observational analyses, a parameterized linear relationship between TE vorticity forcing and meridional second-order derivative of SST is proposed to close the equations. Analytical solutions of the coupled model show that the midlatitude air-sea interaction with atmospheric TE dynamical forcing can destabilize the oceanic Rossby wave within a wide range of wavelengths. For the most unstable growing mode, characteristic atmospheric streamfunction anomalies are nearly in phase with their oceanic counterparts and both have a northeastward phase shift relative to SST anomalies, as the observed. Although both surface diabatic heating and TE vorticity forcing can lead to unstable air-sea interaction, the latter has a dominant contribution to the unstable growth. Sensitivity analyses further show that the growth rate of the unstable coupled mode is also influenced by the background zonal wind and the air–sea coupling strength. Such an unstable air-sea interaction provides a key positive feedback mechanism for midlatitude coupled climate variabilities.

On the Three Dimensional Mass-Weighted Isentropic Time Mean Equation for Rossby Waves

On the Three Dimensional Mass-Weighted Isentropic Time Mean Equation for Rossby Waves

Horizontal advection of temperature and salinity by Rossby waves in the North Pacific

ABSTRACTThe Aquarius satellite has been used for the first time to characterize Rossby waves in sea surface salinity (SSS) measurements for the North Pacific Ocean. Westward propagating wave signals are delineated by the SSS zonal salinity gradients. The phase velocities and spectral properties obtained from zonal salinity gradients are closely correlated with corresponding values obtained from the sea surface temperature (SST) zonal gradient and the altimetry-derived meridional velocity. The westward propagating SSS signals are consistent with Rossby wave advection across the strong meridional gradients of water characteristics. Following Killworth, we attempted to provide satellite-based estimates of the contribution of horizontal Rossby wave advection to the surface transfer of temperature and salinity in the North Pacific Ocean. Westward propagating signals in the SST and SSS zonal gradient fields show that the observed intensity of meridional advection by the ambient gradients of SST and SSS is less than the intensity predicted by an analytical solution of the transfer equation for Rossby waves. Our results extend the previous studies of physical mechanisms of Rossby wave manifestation at the sea surface and we demonstrate that Rossby waves are responsible for low-frequency oscillations in SST and SSS concentration in the North Pacific.

A Theoretical Model of Long Rossby Waves in the Southern Ocean and Their Interaction with Bottom Topography

An analytical model of long Rossby waves is developed for a continuously-stratified, planetary geostrophic ocean in the presence of arbitrary bottom topography under the assumption that the potential vorticity is a linear function of buoyancy. The remaining dynamics are controlled by equations for material conservation of buoyancy along the sea surface and the sea floor. The mean, steady-state surface circulation follows characteristics that are intermediate to f and f / H contours, where f is the Coriolis parameter and H is the ocean depth; for realistic stratification and weak bottom currents, these characteristics are mostly zonal with weak deflections over the major topographic features. Equations are derived for linear long Rossby waves about this mean state. These are qualitatively similar to the long Rossby wave equations for a two-layer ocean, linearised about a state of rest, except that the surface characteristics in the wave equation, which dominate the propagation, follow precisely the same path as the mean surface flow. In addition to this topographic steering, it is shown that a weighted integral of the Rossby propagation term vanishes over any area enclosed by an f / H contour, which has been shown in the two-layer model to lead to Rossby waves “jumping” across the f / H contour. Finally, a nonlinear Rossby wave equation is derived as a specialisation of the result previously obtained by Rick Salmon. This consists of intrinsic westward propagation at the classical long Rossby speed, modified to account for the finite ocean depth, and a Doppler shift by the depth-mean flow. The latter dominates within the Antarctic Circumpolar Current, consistent with observed eastward propagation of sea surface height anomalies.

Rossby waves in an azimuthal wind

Rossby waves in an azimuthal wind are analyzed using an eigen-function expansion. Solutions of the wave equation for the stream-function for Rossby waves are obtained in which depends on where r is the cylindrical radius, is the azimuthal angle measured in the plane relative to the Easterly direction, (the -plane is locally horizontal to the Earth’s surface in which the x-axis points East, and the y-axis points North). The radial eigenfunctions in the -plane are Bessel functions of order and argument kr, where k is a characteristic wave number and have the form in which the satisfy recurrence relations involving , , and . The recurrence relations for the have solutions in terms of Bessel functions of order where is the frequency of the wave and is the angular velocity of the wind and argument . By summing the Bessel function series, the complete solution for reduces to a single Bessel function of the first kind of order . The argument of the Bessel function is a complicated expression depending on r, , a, and kr. These solutions of the Rossby wave equation can be interpreted as being due to wave-wave interactions in a locally rotating wind about the local vertical direction. The physical characteristics of the rotating wind Rossby waves are investigated in the long and short wavelength limits; in the limit as the azimuthal wind velocity ; and in the zero frequency limit in which one obtains a stationary spatial pattern for the waves. The vorticity structure of the waves are investigated. Time dependent solutions with are also investigated.

Wave turbulence in the two-layer ocean model

AbstractThis paper looks at the two-layer ocean model from a wave-turbulence (WT) perspective. A symmetric form of the two-layer kinetic equation for Rossby waves is derived using canonical variables, allowing the turbulent cascade of energy between the barotropic and baroclinic modes to be studied. It is already well known that in two-layers, energy is transferred via triad interactions from the large-scale baroclinic modes to the baroclinic and barotropic modes at the Rossby deformation scale, where barotropization takes place, and from there to the large-scale barotropic modes via an inverse transfer. However, by applying WT theory, we find that energy is transferred via dominant$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\{+--\}$triads with one barotropic component and two baroclinic components, and that the direct transfer of energy is local and the inverse energy transfer is non-local. We study this non-locality using scale separation and obtain a system of coupled equations for the small-scale baroclinic component and the large-scale barotropic component. Since the total energy of the small-scale component is not conserved, but the total barotropic plus baroclinic energy is conserved, the baroclinic energy loss at small scales will be compensated by the growth of the barotropic energy at large scales. Using the frequency resonance condition, we show that in the presence of the beta-effect this transfer is mostly anisotropic and mostly to the zonal component.

Nonlinear self-adjointness and invariant solutions of a 2D Rossby wave equation

AbstractThe paper investigates the nonlinear self-adjointness of the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane. It is a particular form of Rossby equation which does not possess variational structure and it is studied using a recently method developed by Ibragimov. The conservation laws associated with the infinite-dimensional symmetry Lie algebra models are constructed and analyzed. Based on this Lie algebra, some classes of similarity invariant solutions with nonconstant linear and nonlinear shears are obtained. It is also shown how one of the conservation laws generates a particular wave solution of this equation.

Numerical wave propagation for the triangular P1 DG– P2 finite element pair

The f-plane and β-plane wave propagation properties are examined for discretisations of the linearised rotating shallow-water equations using the P1 DG – P2 finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the f-plane (planar geometry, Coriolis force independent of space) case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the P1 DG – P2 finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency f, and which do not propagate. A restriction of the P1 DG velocity space is proposed in which these modes are not present, leading to a finite element discretisation which is completely free of spurious modes. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised P1 DG – P2 equations in the β-plane (planar geometry, Coriolis force linear in space) case resulting in a Rossby wave equation which is also third-order accurate. This means that the dispersion relation for the wave propagation is very accurate; an illustration of this is provided by a numerical dispersion analysis in the case of a triangulation consisting of equilateral triangles.

Coupling coefficients and kinetic equation for Rossby waves in multi-layer ocean

Abstract. The kinetic description of baroclinic Rossby waves in multi-layer model ocean is analysed. Explicit analytical expressions for the coupling coefficients describing energy exchange intensity between different modes are obtained and their main properties are established for the three-layer model. It is demonstrated that several types of interactions vanish in the case of simple vertical structures of the ocean, e.g. when all layers have equal depth. These cases correspond to a zero component of the eigenvectors of the potential vorticity equations. The kinetic equation always possesses a fully barotropic solution. If energy is concentrated in the baroclinic modes, the barotropic mode will necessarily be generated. Motion systems consisting of a superposition of the barotropic and a baroclinic mode always transfer energy to other baroclinic modes.

Influence of the Rossby waves on the seasonal cycle in the tropical Atlantic

The importance of the long “midlatitude” Rossby waves to the seasonal cycle in the Tropical Atlantic is investigated. An analytical model of the baroclinic Rossby waves is forced with the annual and semiannual components of the climatological wind stress in the reopica (20°S to 20°N), excluding the Equator. The Rossby waves are fast enough in the tropics to influence the ocean on the seasonal timescale. The seasonal migrations of the sea surface troughs (lows) and ridges (highs) are linked to the forced Rossby waves and to their latitudinal speed dependency. The sea surface height of the analytical Rossby wave solutions are compared to data from the TOPEX/Poseidon satellite, the Levitus et al. [1994] climatology, a high‐resolution ocean general circulation model, a baroclinic primitive equation model, and a numerical solution of the Rossby wave equation. The regions where the Rossby waves seem to play an important role are those with the strongest observed seasonal variability, in particular the region just north of Brazil. The forced Rossby wave is composed of one free Rossby wave part and one part due to the local forcing, both of equal importance in the whole region. The observed phase propagation is westward and decreases with latitude but not as much and as clearly as the theoretical Rossby waves do.

A Simple Model of the Decadal Response of the Ocean to Stochastic Wind Forcing*

Abstract A simple linear model is used to estimate the decadal response of the extratropical ocean to wind stress forcing, assuming a flat bottom, a mean state at rest, and no dissipation. The barotropic fields are governed by a time-dependent Sverdrup balance, the baroclinic ones by the long Rossby wave equation. The ocean is bounded by a coast in the east and a radiation condition is used in the west. At each frequency, the baroclinic response consists of a forced response plus a Rossby wave generated at the eastern boundary. For zonally independent forcing, the response propagates westward at twice the Rossby phase speed. The wind stress is assumed to be stochastic with a white frequency spectrum, so the model represents the continuous excitation of the ocean interior by the weather fluctuations. The model predicts the shape and level of the frequency spectra of the oceanic pressure field and their variation with longitude and latitude. The baroclinic response is spread over a continuum of frequencies,...

Rossby Wave Equation for Long Wavelength

It is well known that the nonlinear Rossby wave equation has a dipole vortex solution, the so-called modon, in the short-wavelength regime. For a steady state in the long-wavelength regime, we find an appropriate variable transformation to express the potential vorticity in conservation form, and obtain a nonlinear Schrodinger equation for the stream function. It is found by numerical analysis that the equation has a monopole vortex solution for small separatrix radius.

Rossby Wave Scattering by a Meridional Line Barrier in an Infinitely Long Zonal Channel

The interaction of a prescribed channel mode Rossby wave with a meridional line barrier situated in an infinitely long, zonally aligned channel is considered. An analytical solution for the scattered field is presented for a barrier that spans half the channel width. The solution is obtained using the Wiener-Hopf technique. For other barrier widths the scattered field is determined by numerically solving the Rossby wave equation. In all cases the scattered field is described in terms of propagating cross-channel eigenmodes, together with an infinite number of evanescent modes. The solutions are interpreted in terms of the energy density and energy flux associated with each scattered field eigenmode. Mode one dominates the energy balance in the scattered field for most barrier widths in response to a mode-one incident wave. However, for a particular range of barrier widths the second cross-channel mode is found to be the most energetic. The energy density of each scattered field mode is larger on the eastern side of the barrier than on the west. The implication of the model results for Rossby wave propagation in the Southern Ocean are examined.

Canonical variables for rossby waves and plasma drift waves

The Poisson bracket defining a hamiltonian formulation of the Rossby wave equation is transformed to the Gardner bracket via a special functional change. The diagonal form of the bracket enables us to introduce the normal canonical variables in the considered hamiltonian system. The first terms of the hamiltonian expansion in powers of the canonical variables are calculated. The proposed method of the Poisson bracket diagonalization is relevant for other physically significant problems: barotropic waves above an uneven bottom, waves in the presence of a scalar nonlinearity and quasigeostrophic flow of a vertically stratified fluid, including the baroclinic effects of topography as dynamical boundary conditions.

On The Solitary Vortex Solutions of Nonlinear Drift Wave Equations

The Hasegawa-Mima equation for a quasi-two-dimensional electrostatic drift wave is generalized to include arbitrary distributions of the equilibrium electrostatic potential and the density and temperature of plasma. It is shown that the spatial dependence of the equilibrium density distribution plays a decisive role in determining the behavior of the steady vortex solution, that is, whether it is a single soliton or pair-solitons, etc. The relation between the drift wave equation and the barotropic equation for Rossby waves is discussed, and, in particular, a remark is made on Petviashvili's equation which has an additional nonliruear term proportional to the temperature gradient.