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1.
Weakly nonlinear triad interactions between spherical Rossby harmonics are studied. Each of the harmonics has the form APnm(sin θ)exp[i(mλ − σt)], where A is an amplitude and Pnm is the associated Legendre function. Equations for the amplitudes are derived and resonance conditions are analysed. The resonance conditions differ substantially from the usual resonance conditions on a β-plane and include a Diophantine equation and a few inequalities for the integer wavenumbers n and m of the interacting modes. Particular analytical series of solutions to the resonance conditions are constructed, which show that modes with arbitrary large wavenumbers can participate in the interactions. A numerical study of the resonance conditions reveals that no more than 21% of the Rossby harmonics can participate in the triad interactions and that chains of the interacting triads soon break. Thus precise interactions (for which the resonance conditions hold exactly) do not result in any significant redistribution of energy over the spectrum. On the other hand, approximate interactions (for which the resonance conditions hold approximately) generate an intensive energy redistribution among short Rossby modes with typical scales much smaller than the Earth's radius. Thus the energy cascade is concentrated mainly in the short-wave part of the spectrum and is very weak in the large-scale domain. The efficiency of the triad interaction of Rossby modes with scales much smaller than the Earth's radius depends strongly on the existence of the so-called interaction latitude at which the local wave-vectors and frequencies of the interacting modes satisfy resonance conditions for plane Rossby waves on the β-plane approximating the neighbourhood of the latitude. If the interaction latitude exists, the interaction is intensive; in the opposite case, it is weak. 相似文献
2.
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. 相似文献
3.
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. 相似文献
4.
5.
When the problem of the reflection of spatially localized Rossby waves from a coast is treated using the quasigeostrophic (QG) approximation, the total fluid mass and the along-shore circulation calculated from the geostrophic height field are not conserved. To understand the correct mass balance and the degree to which the QG equations and boundary conditions may be in error, we analyze an initial-value problem for the Laplace tidal equations on a β-plane in the asymptotic limit 1, where is the ratio of the spatial scale of the motion to the Earth's radius.It is shown that there is a coupling between QG and O() fields. Physically, the coupling occurs by a peculiar adjustment process in the O() approximation in which fast gravity waves are permanently generated to build up a quasi-stationary edge Kelvin wave. Different temporal scales (large for O(1) Rossby waves and small for the O() gravity waves make comparable the contributions of the waves to the mass and circulation balance equations. However, QG analysis itself describes the reflection of Rossby waves correctly, but is incomplete, and for satisfactory balances one has to take into account the fields of both orders of the approximation.Applications of the results to closed basins, baroclinicity, and variable bottom topography are discussed. It is conjectured that an interaction of strong oceanic eddies with a coast (continental slope) may give rise to noticeable along-shore jet currents. 相似文献
6.
John P. Boyd 《Dynamics of Atmospheres and Oceans》1984,8(2):173-184
It is shown that a mean flow with shear makes the Kelvin wave dispersive. This in turn modifies its nonlinear behavior and makes it necessary to replace the one-dimensional advection equation derived in an earlier work of the author's by the Korteweg-deVries equation instead. The frontogenesis predicted in the earlier paper will still occur, but the wave breaking will not. Instead, once a steep front has formed, it will disintegrate into a train of solitary waves. These then propagate towards the east at a faster-than-linear rate. It is also shown that Kelvin solitary waves will have much smaller zonal widths than Rossby solitons of the same height; “round” Kelvin solitary waves (equal zonal and latitudinal width) are to be expected, and are fully consistent with the small amplitude, weak dispersion theory. An interesting implication of the Korteweg-deVries model is that the peak signal from a nonlinear Kelvin wave packet may be roughly double that of a linear Kelvin wavetrain. 相似文献
7.
R. H. J. Grimshaw 《Dynamics of Atmospheres and Oceans》1985,9(2):85-120
We consider the three-dimensional reflection and diffraction properties of internal waves in a continuously stratified rotating fluid which are incident on the junction of a vertical slit and a half-space. This geometry is a model for submarine canyons on continental slopes in the ocean, where various physical phenomena embodying reflection and diffraction effects have been observed. Three types of incident wave are considered: (1) Kelvin waves in the slit (canyon); (2) Kelvin waves on the slope; and (3) plane internal waves incident from the half-space (ocean). These are scattered into Kelvin and Poincaré waves in the slit, a Kelvin wave on the slope and Poincaré waves in the half-space. Most of the discussion is centered around case (1). Various properties of the wave field are calculated for ranges of the parameters c/cot θ, γα and ƒ/ω where cot θ is the topographic slope, c is the internal wave ray slope, α is the canyon half-width, γ is the down-slope wave-number, ƒ is the Coriolis parameter and ω is the wave frequency. Analytical results are obtained for small γα and some approximate results for larger values of γα. The results show that significant wave trapping may occur in oceanic situations, and that submarine canyons may act as source regions for internal Kelvin waves on the continental slope. 相似文献
8.
Stefano Pierini Adam M. Fincham Dominique Renouard Maria Rosaria D&#x;Ambrosio Henry Didelle 《Dynamics of Atmospheres and Oceans》2002,35(3)
The physical modeling of topographic Rossby normal modes carried out at the “Coriolis” Rotating Platform (Grenoble), is presented. The basic feature of the bottom topography is a linear slope of 4.3 m×2 m delimited by two lateral walls. Since the studied motions are essentially barotropic, homogeneous water was used. Unsheared currents were generated by a simple movement of a wavemaker located in front of the topographic barrier. The conservation of potential vorticity for the currents flowing onto the channel slope produced Rossby waves: reflections at the lateral boundaries then led to the formation of propagating barotropic Rossby normal modes, whose frequencies and spatial structures were selected by the physical system. The currents were measured through the correlation imaging velocimetry (CIV) method, which allowed an extremely detailed synoptic map of the horizontal velocities in an area (13 m2) including the slope to be obtained every 30 s.A variety of experiments were performed in order to provide a complete process study in which the effect of different channel lengths and rotation periods could be tested. Two different lengths of the linear slope, 4.3 and 3.3 m, and rotation periods ranging from 30 to 50 s were considered. The qualitative analysis of the 2D current patterns, and the good agreement found between the measured eigenperiods and the periods obtained by means of a simple analytical model, show that in all cases the first Rossby normal mode was generated. Moreover, numerical simulations based on the shallow-water equations, for a geometry and paddle movements that match closely the experimental setup, allow to calibrate the analytical model and provide useful information on a discrepancy found between experimental and analytical eigenperiods due to an oscillation of the normal mode trajectory. 相似文献
9.
In this study we investigate which conditions are needed to assure uniqueness of solutions of the barotropic QG model, and, how these conditions are related to the conservation laws of mass, vorticity and energy. Uniqueness and conservation laws are analyzed for a simply connected domain (a closed basin) and for a doubly connected domain (a periodic or re-entrant channel).For the multiply connected domain we find, besides the model proposed by McWilliams in 1977, another consistent model whose solutions satisfy the conservation laws. The additional conditions in our model are: (a) the sum of the circulations around all closed solid walls is time-independent; (b) the value of the QG streamfunction (Φ) is the same at all closed walls. McWilliams' model and ours are not equivalent.As a simple application we study the free Rossby normal modes in a channel. For a non-zonal channel on a β-plane there are solutions (modes) that are independent of the coordinate along the channel. These are used to compare the modes and frequencies obtained from three different, but well posed, models. Solutions that are independent of the along-channel coordinate do not exist for the planetary geostrophic and for the shallow water equations on a β-plane. 相似文献
10.
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. 相似文献
11.
The effect of barotropic shear in the basic flow on baroclinic instability is investigated using a linear multilevel quasi-geostrophic β-plane channel model and a nonlinear spherical primitive equation model. Barotropic shear has a profound effect on baroclinic instability. It reduces the growth rates of normal modes by severely restricting their structure, confirming earlier results with a two-layer model. Dissipation, in the form of Ekman pumping and Newtonian cooling, does not change the main characteristics of the effect of the shear on normal mode instability.Barotropic shear in the basic state, characterized by large shear vorticity with small horizontal curvature, also effects the nonlinear development of baroclinic waves. The shear limits the energy conversion from the zonal available potential energy to eddy energy, reducing the maximum eddy kinetic energy level reached by baroclinic waves. Barotropic shear, which controls the level of eddy activity, is a major factor which should be considered when parameterizing the eddy temperature and momentum fluxes induced by baroclinic waves in a climate model. 相似文献
12.
Michael Spillane 《Dynamics of Atmospheres and Oceans》1978,2(5):427-453
When a broad ocean current encounters a large-scale topographic feature, standing Rossby wave patterns can be generated. Short Rossby waves with a scale Li = √ Q/β (Q is the speed of the approaching flow; β is the meridional gradient of f) are generated east of the topography. If the zonal scale of the topography, L, is planetary, long standing Rossby waves can be generated west of the topography, when the current has a meridional component. The long waves focus the disturbance zonally and produce alternating regions of intensified or reduced zonal flow. The meridional scale that characterizes these zonal bands is the intermediate scales, L = Li2/3L1/3. When the meridional topographic scale is comparable to L, the amplitude of the long-wave disturbance is dominant. Using multiple-scale methods to exploit the scale gap between the planetary, intermediate and Rossby wave scales, the topographically induced pressure and velocity fields due to a zonal ridge are obtained. When the planetary-scale flow field is directed poleward, a westward counterflow can occur along the poleward flank of the ridge. The meridional scales of these topographically induced flows are comparable to those observed along the Indian-Antarctic Ridge by Callahan (1971). 相似文献
13.
基本流场切变对赤道长Rossby波的影响 总被引：9，自引：2，他引：9
文中应用赤道β平面近似 ,建立一个简单的斜压大气半地转模式 ,在热力学方程中引入表征基本位温场 (θ)经向分布特征的无量纲参数 σ,对线性化的扰动方程进行了频率分析 ,研究基本位温场经向非均匀分布以及基本气流垂直切变对赤道纬向超长尺度 Rossby波动的影响 ,并指出仅考虑基本气流垂直切变或者基本位温场变化的作用是不合适的。定性分析结果表明 :基本位温场经向温差必然有基本气流垂直切变与其相匹配 ,而基本气流垂直切变将导致赤道长 Rossby波动不稳定并影响其纬向传播速度 相似文献
14.
It is shown how symmetric dipolar vortices can be formed by the action of an impulsive jet in a homogeneous single layer of fluid in a rotating tank. These dipoles are allowed to interact with a constant topographic slope, which can model a β-plane or a continental shelf. A dipole's trajectory bends toward the right when climbing a slope and to the left when descending, as predicted by numerical simulations and analytical arguments. The maximum penetration of the dipoles over a slope, the adjustment to the slope, and formation of trailing lobes are compared with both numerical simulations and a two-point vortex model. The results suggest that Rossby wave radiation plays an important role in the interaction process. 相似文献
15.
The system of linearized shallow water equations is formulated in this paper on any rotating and smooth surface M in terms of differential geometry. The system decouples into two separate equations: a scalar one for the height deviation and a vector one for the velocity field. For low and high frequencies these equations yield asymptotic equations whose solutions are the generalizations of the Poincare and Rossby waves to smooth surface. The application of these equations to the β-plane yields both new and previously known equations for the height deviation and for the velocity components. The application of the equations to the rotating spherical Earth shows that the meridional amplitudes of Poincare and Rossby waves are both described by the prolate angular spheroidal wave functions. The asymptotic and the power series expansions of the eigenvalues of these functions yield new approximations for the dispersion relations of these waves on a sphere. The new dispersion relations are very accurate in the physically relevant range of the single nondimensional model parameter – the square of the nondimensional gravity waves’ phase speed. The invariant formulation can also be applied to other surfaces that are of geophysical interest such as an oblate ellipsoid of revolution. 相似文献
16.
In a recent publication “Glory phenomenon informs of presence and phase state of liquid water in cold clouds” Nevzorov [Nevzorov, A., 2006. Glory phenomenon informs of presence and phase state of liquid water in cold clouds. Atmospheric Research 82, 367–378] claims that “the convincing evidence has been provided that this sort of glory forms as a first-order bow from spherical particles with a refractive index of 1.81–1.82 and diameter over 20 μm”. This is a highly unusual finding because the refractive index of liquid water and ice is between 1.30 and 1.35 in the visible spectral range. The author concludes that “once more corroboration is gained […] of droplets of liquid water in specific phase state referred to amorphous water, or A-water”. Here we show that the phenomena described by the author are easily explained assuming liquid water with a refractive index of 1.33 and a realistic droplet size distribution with an effective radius of around 10 μm. We conclude that this type of observations does not corroborate the existence of amorphous water in the atmosphere. In a recent publication we showed how to quantitatively derive cloud optical thickness, effective droplet radius, and even the width of the size distribution from observations of the glory [Mayer, B., Schröder, M., Preusker, R., Schüller, L., 2004. Remote sensing of water cloud droplet size distributions using the backscatter glory: a case study. Atmospheric Chemistry and Physics 4, 1255–1263]. 相似文献
17.
切变基流中赤道Rossby包络孤立波 总被引：1，自引：0，他引：1
A simple shallow-water model on an equatorialβ-plane is employed to investigate the nonlinear equatorial Rossby solitons in a mean zonal flow with meridional shear by the asymptotic method of multiple scales. The cubic nonlinear Schrodinger (NLS, for short) equation, satisfied for large amplitude equatorial envelope Rossby solitons in shear basic flow, is derived. The effects of basic flow shear on the nonlinear equatorial Rossby solitons are also analyzed. 相似文献
18.
Observations of internal tide generation over continental slopes in a laboratory experiment have been carried out, with the objectives of making comparisons with linear generation theory and investigating its limitations. Both continuous and layered stratification have been considered. A measure of the amplitude of the barotropic tidal forcing (and hence of non-linearity) is given by the Froude Number F = usb/cw, where usb is the maximum barotropic tidal velocity at the shelf break, and cw is the long-wave speed of the lowest internal mode.For continuous stratification, good agreement was obtained for “steep” slopes (α/c > 1), where α is the slope at the continental slope and c is the slope of the internal wave rays of tidal frequency), even for quite large amplitude motions (F < 1.6), and the upper limit of its quantitative usefulness was not reached. For “flat” slopes (α/c < 1) reasonable agreement was also obtained, even up to quite amplitudes (F < 3.1), although some departure from linear theory was apparent.For two-layer flows the applicability of linear theory was much more restricted. For F 0.5 there was only qualitative agreement and for larger F (>1) significant differences were observed. The latter were due to the substantial advection and associated hydraulic jumps which occured seaward of the shelf-break during the ebb-phase of the barotropic tide. Shelf-break values of F > 1 are common in the ocean. 相似文献
19.
Frdric O. Vandermeirsch Xavier J. Carton Yves G. Morel 《Dynamics of Atmospheres and Oceans》2003,36(4):271-296
In a two-and-a-half-layer quasi-geostrophic model, a process study is conducted on the interaction between a vortex and a zonal jet, both with constant potential vorticity. The vortex is a stable anticyclone, initially located north of the eastward jet. The potential vorticity of the jet is allowed to have various vertical structures, while the vortex is concentrated in only one layer. The flow parameters are set to values characteristic of the Azores region.First, the jet is stable. Weak vortices steadily drift north of the jet without crossing it while strong vortices can cross the jet and tear off a cyclone with which they pair as a heton (baroclinic dipole). This heton often breaks later in the shear exerted by the jet; the two vortices finally drift apart. When crossed by deep anticyclones, the jet develops meanders with 375 km wavelength. These results exhibit a noticeable similarity with the one-and-a-half-layer case studied in Part I.Secondly, the jet is allowed to be linearly unstable. In the absence of the vortex, it develops meanders with 175 km wavelength and 25-day e-folding time on the β-plane. For various vertical structures of the jet, baroclinic instability is shown to barely affect jet–vortex interaction if the linear growth rate of unstable waves is smaller than 1/(14 days). Further simulations with a linearly unstable, nonlinearly equilibrated jet evidence its strong temporal variability when crossed by a deep vortex on the β-plane. In particular, long waves can dominate the spectrum for a few months after jet crossing by the vortex. Again in this process, the deep vortex couples with a surface cyclone and both drift southwestward. 相似文献