Rossby waves in a rotating fluid of spheroidal configuration

The ray method is used to study quasi-geostrophic waves in a thin layer of incompressible, inviscid fluid of constant density both in a rotating spheroidal shell and on a rotating spheroid bounded above by a free surface. Asymptotic approximations to solutions of both the time-dependent and time-reduced problems are found; the dispersion relation obtained has the form of the Rossby-Haurwitz formula when the shell is spherical, and is asymptotically equivalent to that found by Longuet-Higgins (1965) for the free surface problem on a sphere. The approximation is first applied to free oscillations in a rotating spherical shell and Longuet-Higgins' (1964) results are rederived. Spheroidal shells in which the shell thickness depends only upon the latitude are studied next and a necessary and sufficient condition for oscillations to occur is obtained.     

Beta-dispersion of low-frequency Rossby waves

An investigation is made into the dispersion of oceanic internal Rossby waves at annual and semi-annual frequencies. Turning of the group velocity vector due to latitudinal variations in the radius of deformation cannot be neglected, particularly in basins as large as the Pacific. This turning allows disturbances to propagate from high lattitudes into the equatorial zone and distorts the solutions in the western part of the basin. For no mean flow, and a coastline aligned north-south, an almost exact focus of wave energy is found very close to the equator at a distance of just under $\text{πc/4ω}$ from the eastern boundary, where $\text{c}$ is the eigenspeed of a high-frequency internal wave mode, and ω is the angular frequency of the low-frequency wave being studied. The focus depends on a long meridional wavelength excited at the coast, and a frequency small compared with $\text{c/a}$, where $\text{a}$ is the radius of the Earth. For the lowest baroclinic mode and waves of annual period, this distance is about 12 000 km. Equivalence of the ray theory and the theory of equatorial meridional modes is demonstrated for the simple cases where the latter applies.The effects of mean currents and irregular coastlines are examined. Barotropic mean currents may change the turning latitude and ray shapes, inducing critical layers and enhancing reflection. Baroclinic mean currents are seen to affect the rays by simply changing the speed in proportion to the depth of the thermocline. As long as the mean currents are geostrophically balanced, no “effective beta” term from variations in the thermocline depth appears, in contrast to the topographic Rossby wave problem.     

Oscillatory Rossby Solitary Waves in the Atmosphere

The linear Rossby wave frequency expression is expanded at higher accuracy based on the scale difference char?acteristics of atmospheric long waves in the and directions. That the nature of the waves represented by the expan?sion is identical to that of the original ones is demonstrated both in phase velocity and wave energy dispersion speed , followed by the derivation of the nonlinear expression describing atmospheric long wave behaviors with the associated approximate analytic solution obtained. Then, for the first time atmospheric’ oscillatory Rossby solitary wave’ with its dispersion relation is obtained by numerical calculation with the aid of physical parameters of the real atmosphere. The solitary wave is found to be very close to such longwave systems as blocking highs and cut-off de?pressions in the actual atmosphere.     

The algebraic decay of equatorial Rossby waves in a shear flow

Through numerical integration, we show that equatorial Rossby waves, like their midlatitude counterparts, decay algebraically in the limit t → ∞ in a linear shear flow. For small times, the growth expected for some components does not translate into any growth of the wave disturbance as a whole when the initial conditions has a broad Fourier spectrum. The conclusion is that Rossby waves will amplify with time only when the mean flow has an inflection point or when the initial eddy field is strongly concentrated in long waves tilted against the shear.     

大气Rossby波动力学的研究进展

Rossby波是旋转大气和海洋中的一类重要波动,在天气和气候演变中起有重要的作用。该文介绍了北半球中纬度地区的两类重要Rossby波——斜压不稳定波和准定常行星波,在最近20余年来取得的一些重要研究进展。文中涉及的主要研究领域有：斜压不稳定波包动力学、湿斜压过程对斜压不稳定波的作用、北半球风暴轴动力学、斜压波包与高影响天气的预报、准定常行星波的形成机理、准定常行星波的水平传播与能量频散、准定常行星波的垂直传播与能量频散、三维准定常行星波的传播与能量频散。     

N层模式大气中的斜压孤立Rossby波

文中导得了N层模式的KdV方程,作为例子计算了三层模式的孤立Rossby波流型。基本气流的垂直切变、水平切变和Froude数对孤波流型都有重要影响;不同模态的孤波具有不同流型,不同高度位面上的孤波流型一般是不同的。     

弱耗散作用下非线性Rossby波的调制不稳定

本文使用WKB方法对弱耗散作用下基本气流具有弱切变的行星大气中非线性Rossby波进行了研究,得到了弱耗散的非线性Rossby波振幅所满足的复系数非线性Schrdinger方程,并分别就无耗散和弱耗散的Rossby波的边带稳定性问题进行了讨论,得到了一些新结果.     

Cross-equatorial structures of equatorially trapped nonlinear Rossby waves

Analysis of the Sea Surface Height (SSH) from satellite altimeters has shown that equatorially trapped Rossby waves exhibit asymmetric cross-equatorial structures; their northern extrema are much larger in magnitude than their southern counterparts. Such asymmetry is inconsistent with the classical theory for the first baroclinic, first meridional equatorially trapped Rossby mode, which predicts that SSH and zonal velocity are symmetric in latitude and the meridional velocity is latitudinally antisymmetric (Matsuno, 1966). Chelton et al. (2003) attributed the observed asymmetry to the mean-shear-induced modifications of first meridional mode Rossby waves. The present paper examines nonlinear rectification of cross-equatorial wave structures in the presence of different zonal mean currents. Nonlinear traveling Rossby waves embedded in shears are calculated numerically in a 1.5-layer model. Nonlinearity is shown to increase the cross-equatorial asymmetry substantially making the northern extrema even more pronounced. However, nonlinearity only slightly increases the magnitude of the westward phase speed.     

Mechanism of Kelvin and Rossby waves during ENSO events

Summary ?The fields of sea-level height anomaly (SLHA) and surface zonal wind anomaly (SZWA) have been analyzed to investigate the typical evolution of spatial patterns during El Ni?o-Southern Oscillation (ENSO) events. Sea surface temperature (SST) changes during ENSO events are represented as an irregular interplay of two dominant modes, low-frequency mode and biennial mode. Cyclostationary principal component (PC) time series of the former variables are regressed onto the PC time series of the two dominant SSTA modes to find the spatial patterns of SLHA and SZWA consistent with the two SSTA modes. The two regressed patterns of SLHA explain a large portion of SLHA total variability. The reconstruction of SLHA using only the two components reasonably depicts major ENSO events. Although the low-frequency component of SST variability is much larger than the biennial component, the former does not induce strong Kelvin and Rossby waves. The biennial mode induces much stronger dynamical ocean response than the low-frequency mode. Further decomposition of the SLHA modes into Kelvin and Rossby components shows how these two types of equatorial waves evolve during typical ENSO events. The propagation and reflection of these waves are clearly portrayed in the regressed patterns leading to a better understanding of the wave mechanism in the tropical Pacific associated with ENSO. A close examination suggests that the delayed action oscillator hypothesis is generally consistent with the analysis results reported here. Rossby wave development in the central Pacific in the initiation stage of ENSO and the subsequent reflection of Kelvin waves at the western boundary seems to be an important mechanism for further development of ENSO. The development of Kelvin waves forced by the surface wind in the far-western Pacific cannot be ruled out as a possible mechanism for the growth of ENSO. While Kelvin waves in the far-western Pacific serve as an intiation mechanism of ENSO, they also cause the termination of existing ENSO condition in the central and eastern Pacific, thereby leading to a biennial oscillation over the tropical Pacific. The Kelvin waves from the western Pacific erode the thermocline structure in the central Pacific preventing further devlopment of ENSO and ultimately terminating it. It should be emphasized that this wave mechanism is clear and active only in the biennial mode. Received August 15, 2001; revised March 6, 2002     

The effects of zonal flow on nonlinear Rossby waves

In this paper, we using phase plane method have derived the stability criteria of linear and nonlinear Rossby waves under the conditions of semi-geostrophic approximation and have gotten the solutions and geostrophic vorticity of corresponding solitary Rossby waves. It is pointed out that the wave stability is connected with the distri-bution of zonal flow and when the zonal flow is different the solitary wave trough or ridge is formed.     

Topography and the non-linear Rossby wave in the zonal shear basic flow

Under semi geostropical approximation, by means or phase angle function the non-linear ordinary differential equation is derived involving topography and zonal shear basic flow. Conditions for the existence of limited amplitude periodical and isolated wave solutions are directly obtained based on the qualitative theory of the ordinary differentical equation. Analysis is thus made of the influence of topography and zonal shear flow on the existence of wave solution. Finally, explicit wave solutions are determined by function approaching with the result that topogra-phy and zonal shear flow affect not only the existence but also the form of waves, indicating the non-linear features of waves and the effect of topography and shear basic flow on undulation.     

Refraction of Rossby waves on a multiple β-plane

A multiple β-plane is introduced to explore the relation between plane and spherical Rossby waves. The fundamental problem, the refraction of a plane Rossby wave across a discontinuity in β, is solved. It is shown that refraction on the multiple β-plane agrees in the limit with refraction on the full sphere only if a suitable correction is made for the geometric distortion of the β-plane. The full spherical modes of Rossby waves trapped in a band about the equator (Longuet-Higgins, 1964) have their counterpart in a simple model consisting of an “equatorial” β-plane bounded above and below by “polar” β-planes.     

Axisymmetric critical withdrawal of a rotating fluid

The problem of inertial critical flow out of a rotating cylindrical container is addressed theoretically and experimentally. The inviscid steady-state Navier—Stokes equations with the approximation of no vertical shear and the hydrostatic approximation are solved by closing the problem with a critical condition which is applied at the radius of the exit sink. Although the results are presented in general, attention is focused upon the problem when the exit radius is less than the source radius, in which case rotation can be important even when the apparatus size is much less than the Rossby radius of deformation. The results of the analysis are compared to experiments to see if the assumptions were justified. The experiment leads to results that agree closely with predictions of the theory. However, one result is gleaned from the experiment which the theoretical results did not predict. It is that an instability in the flow occurs and is manifested as an azimuthally travelling wave. The role of axisymmetry in producing intense rotational effects is discussed in conext with the experimental findings, and possible implications to withdrawal of oceanic waters by ocean thermal difference power plants are discussed.     

Propagation of the Rossby waves on two dimensional rectangular grids

Summary A simple two-dimensional quasi-geostrophic linearized model of the atmosphere is used to investigate the behaviour of the quasi-geostrophic modes for five horizontal rectangular grids. Numerical expressions for frequencies of Rossby waves for all grids are evaluated. It was found that the B and C grids produce only negative frequencies as well as the continuous case. The D grid has negative and zero frequencies. Finally, it was found that the A and E grids produce positive frequencies and eastward moving Rossby waves.With 2 Figures     

Vortex–ridge interaction in a rotating fluid

The evolution of barotropic vortices interacting with a topographic ridge on a f-plane is studied by means of laboratory experiments in a rotating tank and numerical simulations. The initial condition in all experiments is a cyclonic vortex created at a certain distance from the ridge. The results are presented in two main scenarios: (a) weak interactions, which occur at early stages of the experiments, when the vortex is far from the ridge, and thus weakly experiences the influence of the topography. In these situations, the vortex slowly drifts towards the ridge with a leftward inclination due to the ascending slope of the topography. Such a behaviour is similar to the “northwestern” motion of cyclones over a weak sloping bottom. The circular shape of the monopolar vortex is preserved. (b) Strong interactions, in which the vortex core reaches the ridge and presents a more complicated evolution. The cyclone “climbs” to the top of the topography and crosses to the other side. Once the vortex experiences the opposite slope, it moves backwards trying to return to the original side of the ridge. For strong enough vortices, this process may be repeated a number of times until the vortex is dissipated by viscous effects. During these interactions the shape of the vortex is strongly deformed and several filaments are produced. In some cases the vortex is cleaved in two parts when crossing the ridge, one at each side of it and moving in opposite directions.Weak and strong interactions are numerically simulated by using a quasi-two-dimensional model. The results confirm that the vortex behaviour is governed by stretching and squeezing effects associated with changes in depth over the ridge and, at latter stages, by Ekman damping due to the solid bottom. The main results observed during strong interactions on a f-plane are also found on preliminar topographic β-plane experiments.     

Measurements in a turbulent patch in a rotating,linearly-stratified fluid

A series of experiments is described in which a turbulent patch is generated locally by an oscillating grid positioned at one end and mid-depth of a rotating channel filled initially with a linearly-stratified fluid. Measurements have been made of vertical density profiles through the patch both during sustained oscillations and following cessation of grid forcing. Temporal variations in patch size and structure, Thorpe scales, mixedness parameter and available potential energy are deduced from these measurements, and the effects thereon of varying the background rotation rate, initial buoyancy frequency and grid action are investigated. For the growth phase of the patch, previous results obtained by other workers are confirmed and extended. Because the rapid turbulent motions implied a large Rossby number, rotation was not important during this phase. During the decay phase, rotational effects are shown to become important, and the presence of rotation is found to retard the decay of both the mixedness and the Thorpe scales of the density overturns within the patch. The work is novel in that measurements of the patch parameters listed above have not previously been carried out in the presence of rotation. The results are relevant to studies of such patches that have been observed in the ocean and atmosphere.     

大地形对切变基流上罗斯贝波稳定性的影响

利用WKB近似讨论大地形对切变基流上Rossby波稳定性的影响.北半球东西走向地形有利导式长波槽辐合能量于地形北坡附近,对曳式长波槽有使能量从地形南坡区域辐散的趋势.南半球地形的影响正好相反.散度顶有利于导式长波槽在东西风带过渡区域的能量辐射,有利于曳式长波槽能量在这里辐合.     

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

In this paper an evolution 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 propa-gation 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.