Group Velocity of Anisotropic Waves-Part Ⅰ: Mathematical Expression

The group velocity used in meteorology in the last 30 years was derived in terms of conservation of wave energy or crests in wave propagation. The conservation principle is a necessary but not a sufficient condition for deriving the mathematical form of group velocity, because it cannot specify a unique direction in which wave energy or crests propagate. The derived mathematical expression is available only for isotropic waves. But for anisotropic waves, the traditional group velocity may have no a definite direction, because it varies with rotation of coordinates. For these reasons, it cannot be considered as a general expression of group velocity. A ray defined by using this group velocity may not be the trajectory of a reference point in an anisotropic wave train. The more general and precise expression of group velocity which is applicable for both isotropic and anisotropic waves and is independent of coordinates will be derived following the displacement of not only a wave envelope phase but also a     

On the low-frequency,planetary-scale motion in the tropical atmosphere and oceans

Scale analyses for long wave, zonal ultralong wave (with zonal scale of disturbance L1~104 km and meridional scale L2~103 km) and meridional ultralong wave (L1~103 km, L2~104 km) are carried out and a set of approximate equations suitable for the study of these waves in a dry tropical atmosphere is obtained. Under the condition of sheared basic current, frequency analyses for the equations are carried out. It is found that Rossby waves and gravity waves may be separated for n ≥ l where n is the meridional wave number, whereas for n = 0 and L1~1000 km, the mixed Rossby-gravity wave will appear. Hence it is confirmed that the above results of scale analyses are correct. The consistency be-tween frequency analysis and scale analysis is established.The effect of shear of basic current on the equatorial waves is to change their frequencies and phase velocities and hence their group velocities. It increases the velocity of westward travelling Rossby waves and inertia-gravity and mixed waves, but decelerates the eastward inertia-gravity waves and the Kelvin wave. The recently observed low-frequency equatorial ocean wave may be interpreted as an eastward Kelvin wave in a basic current with shear.     

ON THE LOW-FREQUENCY, PLANETARY-SCALE MOTION IN THE TROPICAL ATMOSPHERE AND OCEANS

Scale analyses for long wave, zonal ultralong wave (with zonal scale of disturbance L,-104 km and meridional scale ?-103 km) and meridional ultralong wave (L,-103 km, L2-104 km) are carried out and a set of approximate equations suitable for the study of these waves in a dry tropical atmosphere is obtained. Under the condition of sheared basic current, frequency analyses for the equations are carried out. It is found that Rossby waves and gravity waves may be separated for n≥1 where n is the meridional wave number, whereas for n=0 and L1-1000 km, the mixed Rossby-gravity wave will appear. Hence it is confirmed that the above results of scale analyses are correct. The consistency between frequency analysis and scale analysis is established.The effect of shear of basic current on the equatorial waves is to change their frequencies and phase velocities and hence their group velocities. It increases the velocity of westward travelling Rossby waves and inertia-gravity and mixed waves, but decelerates the ea     

Variations in Amplitudes and Wave Energy along the Energy Dispersion Paths for Rossby Waves in the Quasigeostrophic Barotropic Model

Variations in wave energy and amplitude for Rossby waves are investigated by solving the wave energy equation for the quasigeostrophic barotropic potential vorticity model.The results suggest that compared with rays in the nondivergent barotropic model,rays in the divergent model can have enhanced meridional and zonal propagation,accompanied by a more dramatic variability in both wave energy and amplitude,which is caused by introducing the divergence effect of the free surface in the quasigeostrophic model.For rays propagating in a region enclosed by a turning latitude and a critical latitude,the wave energy approaches the maximum value inside the region,while the amplitude approaches the maximum at the turning latitude.Waves can develop when both the wave energy and amplitude increase.For rays propagating in a region enclosed by two turning latitudes,the wave energy approaches the minimum value at one turning latitude and the maximum value at the other latitude,while the total wavenumber approaches the maximum value inside the region.The resulting amplitude increases if the total wavenumber decreases or the wave energy increases more significantly and decreases if the total wavenumber increases or the wave energy decreases more significantly.The matched roles of the energy from the basic flow and the divergence of the group velocity contribute to the slightly oscillating wave energy,which causes a slightly oscillating amplitude as well as the slightly oscillating total wavenumber.     

Tropical gravity-atmospheric long wave and the walker circulation

Orders of magnitude of terms related to earth’s rotation in linearized vorticity and divergence equations governing tropical large-scale motion are analysed. It is discovered that βyD and βyξ are smaller by one order than βv and βu respectively and then may be neglected. On this basis, tropical wave motions are discussed. It is found that there exists a kind of gravity-atmospheiic long waves which is non-vorticit atmospheric long wave, whereas the Kelvin wave is essentially the gravity-atmospheric long wave with its velocity being much lower than that of gravity. Computation shows that there also exists a kind of large-scale slow waves whose moving speed is lower by one order of magnitude than that of Kelvin wave. Such slow wave is likely to be the Walker Circulation.     

Seasonal Variation of Stationary and Low-Frequency Rossby Wave Rays

The wave rays and their seasonal variation of stationary and low－frequency Rossby waves are studied by using the Runge-Kutta scheme. The results show that for stationary waves the rays can reach lower latitudes in winter, and are limited in higher latitudes in summer. The main differences between the stationary and low-frequency wave rays are that low-frequency waves can propagate across the equator and the easterlies will not be an obstacle on their propagation. It explained to some extent the interaction of disturbances between the Northern and Southern Hemi-spheres. The lower wave frequencies and the stronger easterly flow are, the more difficult low-frequency waves will be to propagate across the equator. The waves with 20-day period are easier to propagate across the equator than that with 50-day period. The winter is the most favorable season for low-frequency waves to propagate into another hem-isphere.     

TROPICAL GRAVITY-ATMOSPHERIC LONG WAVE AND THE WALKER CIRCULATION

Orders of magnitude of terms related to earth's rotation in linearized vorticity and divergence equations governing tropical large-scale motion are analysed. It is discovered that βyD and βyζ are smaller by one order than βv and βu respectively and then may be neglected. On this basis, tropical wave motions are di s-cussed. It is found that there exists a kind of gravity-atmospheric long waves which is non-vorticit atmospheric long wave, whereas the Kelvin wave is essentially the gravity-atmospheric long wave with its velocity being much lower than that of gravity. Computation shows that there also exists a kind of large-scale slow waves whose moving speed is lower by one order of magnitude than that of Kelvin wave. Such slow wavs is likely to be the Walker Circulation.     

Planetary Stationary Waves in the Atmosphere Part II: Thermal Stationary Waves

The contribution of thermal forcing to the planetary stationary waves will be studied also by assuming that heat balance in stationary waves over zonally asymmetric thermal forcing must be maintained over a long time period. Using the same model of geostrophic waves introduced in Part I, we may explain successfully the observed and simulated responses to the thermal forcing in the atmosphere, such as the wave 1 structure at high levels of middle latitudes, the seasonal changes of the stationary waves in the Northern Hemisphere, the opposite phase distributions of stationary waves at high and low levels of the subtropical regions in both hemispheres and so on.     

The Effect of Spatial Structure Character of Heat Source on the Ray Path and the Evolution of Wave Energy of Meridional Wave Train

This paper studies correlations between the spatial structure character of thermal forcing and deformation and the amplitude of rays of meridional wave train. It is shown that if thermal forcing appears a meridional linear variation the rays of quasi-stationary planetary wave may propagate along oblique lines and if the meridional variability of heat source has second order term the rays show distinct deformation as a great circular route. Additionally, the inhomogeneous distribution may cause lower frequency oscillations in mid- and low-latitudes. The combination of zonal and meridional wave numbers and distributive character of heat source may form an inverse mechanism of variational trend of generized wave energy, reflecting in some degree the physical process of transition between meridional and zonal flow patterns.     

Power and Cross-Spectra for the Turbulent Atmospheric Motion and Transports in the Domain of Wave Number Frequency Space: Theoretical Aspects

The study of large-scale atmospheric turbulence and transport processes is of vital importance in the general circulation of the atmosphere. The governing equations of the power and cross-spectra for the atmospheric motion and transports in the domain of wave number frequency space have been derived. The contributions of the nonlinear interactions of the atmospheric waves in velocity and temperature fields to the conversion of kinetic and potential energies and to the meridional transports of angular momentum and sensible heat in the atmosphere have been discussed.     

The wavenumber-frequency characteristics of the tropical waves in an aqua-planet GCM

Based on the aqua-planet experiments, the wavenumber-frequency characteristics of tropical waves and their influencing factors in SST distribution and the convective parameterization scheme are investigated using the spectral atmospheric general circulation model （SAMIL）. Space-time spectral analysis is used to obtain the variance of convectively coupled tropical waves. In the Control experiment with maximum SST located at the equator the simulated tropical-wave behaviors are in agreement with those in observations and theoretical solutions. When the maximum SST is located at 5°N, the symmetric and antisymmetric waves are much weaker than those in the control experiment, suggesting that tropical wave activities are very sensitive to the SST distributions. Importantly, the variance maximum of Madden-Julian oscillation （MJO） is found to occur around 5°N, which suggests that the development of the MJO depends largely on the latitude of maximum SST. Furthermore, the seasonal variations of MJO may be mainly caused by the seasonal variations of the maximum SST. The experiment results with two different cumulus schemes the Manabe moist convective adjustment and Zhang-McFarlane （ZM） convective scheme, were also compared to examine the impacts of convective parameterization. Weakened variances of each individual tropical wave when the ZM scheme is used suggest that the ZM scheme is not favorable for the tropical wave activities. However, the wave characteristics are different when the ZM scheme is used in different models, which may imply that the simulated basic state is important to the meridional distributions of the waves. The MJO signals suggest that the parameterization scheme may have great influence on the strength, but have less direct impact on the MJO distribution. The frequency of the tropical waves may be associated with the moisture control of convection and the large-scale condensation scheme used in the model.     

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.     

Blocking Distributions in the Atmosphere

The zonal momentum generation in forced stationary waves may exceed the requirement for momentum balance after long, if the waves do not change their patterns. This suggests that the changes in stationary wave patterns would be required by maintenance of momentum balance over the external forcings. It will be found that the low frequency anomalies like blocking regimes may produce reversed zonal momentum variations, if they happen in the observed centre areas. The zonal momentum balance in the stationary waves may be maintained effectively by alternation between the normal and blocking circulation regimes. Thus, from the point of long-term zonal momentum balance, we may explain the geographical distributions of the blocking centres and the seasonal variations in blocking areas and frequencies.     

Wintertime Stratospheric Anomalies-Part II: Sudden Warmings

The process of stratospheric sudden warmings from development of planetary waves to.the sudden cooling after reversal of mean zonal circulation will be studied with the primitive equations of heat and momentum balances. It will be explained that the sudden warmings may occur only in the polar regions of winter stratosphere where zonal mean temperature decreases poleward. The heating rate in the order of major warmings is produced by developed planetary waves in the stratospheric breaking layers. The particular perturbation structure characterized by large amplitude of wave 1 together with minimum of wave 2 discovered by Labitzke (1977) is crucial for initiation of major warmings. The cooling by the same mechanism can be produced in the regions with reversed mean temperature gradient.     

Dynamic and Numerical Study of Waves in the Tibetan Plateau Vortex简

In terms of its dynamics, The Tibetan Plateau Vortex （TPV） is assumed to be a vortex in the botmdary layer forced by diabatic heating and friction. In order to analyze the basic characteristics of waves in the vortex, the governing equations for the vortex were established in column coordinates with the balance of gradient wind. Based on this, the type of mixed waves and their dispersion characteristics were deduced by solving the linear model. Two numerical simulations with triple-nested domains--one idealized large-eddy simulation and one of a TPV that took place on 14 August 2006---were also carried out. The aim of the simulations was to validate the mixed wave deduced from the governing equations. The high-resolution model output data were analyzed and the results showed that the tangential flow field of the TPV in the form of center heating was cyclonic and convergent in the lower levels and anticyclonic and divergent in the upper levels. The simulations also showed that the vorticity of the vortex is uneven and might have shear flow along the radial direction. The changing vorticity causes the formation and spreading of vortex Rossby （VR） waves, and divergence will cause changes to the n~otion of the excitation and evolution of inertial gravity （IG） waves. Therefore, the vortex may contain what we call mixed ：inertial gravity-vortex Rossby （IG-VR） waves. It is suggested that some strongly developed TPVs should be studied in the future, because of their effects on weather in downstream areas.     

The Structure and Propagation of Stationary Planetary Wave Packet in the Barotropic Atmosphere

Monthly or seasonally mean anomalies of large-scale atmospheric circulation are better represented by wave packets or their combination. Both qualitative and quantitative analyses of equations of wave packet dynamics, which are obtained by the use of WKB approximation, are very helpful for the understanding of structure, formation and propagation of stationary and quasi-stationary planetary wave packet patterns in the atmosphere. Indeed, these equations of wave packet dynamics can be directly solved by the method of characteristic lines, and the results can be simply and clearly interpreted by physical laws. In this paper, a quasi-geostrophic barotropic model is taken for simplicity, and the wave packets superimposed on several ideal profiles of the basic current and excited by some ideal forcings are investigated in order to make comparison of the accuracy of calculation with the analytical solution. It is revealed that (a) the rays of stationary planetary wave packet do not coincide with but go away f     

THE EFFECTS OF THE PLATEAU''S TOPOGRAPHIC GRADIENT ON ROSSBY WAVES AND ITS NUMERICAL SIMULATION

By using barotropic model equations, this article analyzed the characteristics of Rossby waves, the propagation features of wave energy and the influence of dynamic and thermal effects of the Tibetan Plateau on Rossby waves, and the focus is on discussing the plateau's topographic gradient effects on atmospheric Rossby waves. Then based on the WRF3.2 and the NCEP/NCAR FNL reanalysis data, we devised comparative tests of changing the plateau's topographic gradient and simulated a process of persistent heavy rain that happened in May 2010 in South China. The results are shown as follows. The Tibetan Plateau’s topography is conducive to the formation of atmospheric Rossby waves. while the plateau's terrain, its friction and heating effects can all make the atmospheric Rossby waves develop into the planetary waves; The effects of plateau's north and south slopes on the Rossby wave’ phase velocity is opposite, and when the slope reached a certain value can the quasi-steady normal fluctuations be generated; Simultaneously, due to the plateau's topographic gradient, descending motion appears at the west side of the plateau while ascending motion appears at the east side, and the vertical movement increased with the amplification of topographic gradients. The plateau's topographic gradient also obviously amplified the precipitation in South China, and the rainfall area increased with the amplification of topographic gradients and gradually moved from south to north and from west to east, which is conducive to the occurrence and development of convective activities in the downstream areas of the Tibetan Plateau; Moreover, for the plateau’s dynamic and thermal effects, the Rossby wave’ propagation shows upstream effects of energy dispersion, so the plateau can then affect the weather in downstream areas. Moreover, the wave group velocity increased with the degree of topographic slope.     

ON THE INFLUENCES OF LARGE-SCALE INHOMOGENEITY OF SEA TEMPERATURE UPON THE OCEANIC WAVES IN THE TROPICAL REGIONS——PART I : LINEAR THEORETICAL ANALYSIS

By using a linear oceanic mixed layer model, the influences of the horizontal gradients of sea surface temperature (SST) and the depth variations of the mixed layer upon tropical oceanic waves are investigated. The equatorial Rossby wave will be modified and a kind of slower thermal wave has been revealed under the influences of inhomogeneities of large-scale sea temperature field. An interesting result is that the propagating direction of the thermal wave is opposite to that of the classical Rossby wave. The result also shows that the thermal wave becomes dominant when the meridional gradient of sea temperature in the mixed layer exceeds a critical value. As a first approximation, it seems that both waves obtained by this study may be used to explain the observational facts that the SST anomalies can usually propagate in both directions, that is, eastward and westward, during the El Nino events.     

THE NONLINEAR INTERACTION BETWEEN DIFFERENT WAVE COMPONENTS AND THE PROCESS OF CYCLE OF GENERAL CIRCULATION

By using a two-level quasi-geostrophic truncated spectral model taking account of the nonlinear interaction between different wave components (i. e. basic current, ultra-long waves and long waves), the index cycle of general circulation is investigated. The calculated results show that the circulation index has a, quasi-periodic vacillation with a period of 8 to 16 days, which can be created by the nonlinear interaction and that the nonlinear interaction between different wave components may cause the tilted-trough vacillation, amplitude vacillation of wave pattern and quasi-periodic change of wave number of flow pattern.     

Variations in Wave Energy and Amplitudes along the Energy Dispersion Paths of Nonstationary Barotropic Rossby Waves

The variations in the wave energy and the amplitude along the energy dispersion paths of the barotropic Rossby waves in zonally symmetric basic flow are studied by solving the wave energy equation,which expresses that the wave energy variability is determined by the divergence of the group velocity and the energy budget from the basic flow.The results suggest that both the wave energy and the amplitude of a leading wave increase significantly in the propagating region that is located south of the jet axis and enclosed by a southern critical line and a northern turning latitude.The leading wave gains the barotropic energy from the basic flow by eddy activities.The amplitude continuously climbs up a peak at the turning latitude due to increasing wave energy and enlarging horizontal scale(shrinking total wavenumber).Both the wave energy and the amplitude eventually decrease when the trailing wave continuously approaches southward to the critical line.The trailing wave decays and its energy is continuously absorbed by the basic flow.Furthermore,both the wave energy and the amplitude oscillate with a limited range in the propagating region that is located near the jet axis and enclosed by two turning latitudes.Both the leading and trailing waves neither develop nor decay significantly.The jet works as a waveguide to allow the waves to propagate a long distance.