Group Velocity of Anisotropic Waves-Part Ⅱ: Conservative Properties

It has been argued in Part I that traditional expression of multidimensional group velocity used in meteorology is only applicable for isotropic waves. While for anisotropic waves, it cannot manifest propagation of waves group along the trajectory of a reference wave point, and varies with rotation of coordinates. The general mathematical expression of group velocity which may be used also for anisotropic waves has been derived in Part I. It will be proved that the mean wave energy, momentum and wave action density are all conserved as a wave group propagates at the general group velocity. Since general group velocity represents the movement of a reference point in either isotropic or anisotropic wave trains, it may be used to define wave rays. The variations of wave parameters along the rays in a slowly varying environment are represented by ray-tracing equations. Using the general group velocity, we may derive the anisotropic ray-tracing equations, which give the traditional ray-tracing equations for     

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     

A NOVEL METHOD FOR CALCULATING VERTICAL VELOCITY: A RELATIONSHIP BETWEEN HORIZONTAL VORTICITY AND VERTICAL MOVEMENT

The present work provides a novel method for calculating vertical velocity based on continuity equations in a pressure coordinate system.The method overcomes the disadvantage of accumulation of calculating errors of horizontal divergence in current kinematics methods during the integration for calculating vertical velocity,and consequently avoids its subsequent correction.In addition,through modifications of the continuity equations,it shows that the vorticity of the vertical shear vector(VVSV) is proportional to-ω,the vertical velocity in p coordinates.Furthermore,if the change of ω in the horizontal direction is neglected,the vorticity of the horizontal vorticity vector is proportional to-ω.When ω is under a fluctuating state in the vertical direction,the updraft occurs when the vector of horizontal vorticity rotates counterclockwise;the downdraft occurs when rotating clockwise.The validation result indicates that the present method is generally better than the vertical velocity calculated by the ω equation using the wet Q-vector divergence as a forcing term,and the vertical velocity calculated by utilizing the kinematics method is followed by the O'Brien method for correction.The plus-minus sign of the vertical velocity obtained with this method is not correlated with the intensity of d BZ,but the absolute error increases when d BZ is =40.This method demonstrates that it is a good reflection of the direction of the vertical velocity.     

RADIATIVE COOLING AND BROADBAND PHENOMENON IN LOW-FREQUENCY WAVES

In this paper, we analyze the effects of radiative cooling on the pure baroclinic low-frequency waves under the approximation of equatorial β-plane and semi-geostrophic condition. The results show that radiative cooling does not, exclusively, provide the damping effects on the development of low-frequency waves.Under the delicate radiative-convective equilibrium, radiative effects will alter the phase speed and wave period,and bring about the broadband of phase velocity and wave period by adjusting the vertical profiles of diabaticheating. When the intensity of diabatic heating is moderate and appropriate, it is conductive to the development and sustaining of the low-frequency waves and their broadband phenomena, not the larger, the better. The radiative cooling cannot be neglected in order to reach the moderate and appropriate intensity of diabatic heating.     

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.     

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.     

PROPAGATION OF INTERNAL INERTIO-GRAVITY WAVE SPECTRA IN VERTICAL WIND SHEAR ENVIRONMENTS*

The variation of the spectral structure of the internal inertio-gravity waves (ⅡGWs) propagating in the atmospheric wind shear environments is discussed in this paper.From the hydrodynamic equation set in Boussinesq approximation,a spectral propagation equation ⅡGWs satisfy is derived,then the spectral correspondence in the upper atmosphere is numerically calculated,after a forced spectrum is given as a Van-Zandt one at the lower boundary.The results show that if ⅡGWs do not encounter the critical-layer absorption,then their spectral structure may be not changed significantly; otherwise it may be changed greatly,and a few of spectral components are filtered.Also the isotropy of the assumed VanZandt spectrum is distorted in upward-propagating process.That is the directional filtering effect of the atmospheric wind on the gravity wave spectrum.     

POSSIBILITY TO USE SCH(?)RDINGER EQUATION TO DESCRIBE LARGE-SCALE PROBABILITY WAVES AND ITS APPLICATION IN SEASONAL PREDICTION

Under the influence of a one-dimensional stationary outfield with the equilibrium between kinetic andpotential energy produced by it,a modified Sch(?)rdinger equation in the form i((?)ψ/(?)t)t=a (?)~2ψ/ax~2-ib(?),where b=b_o(?)T/(?)x,is used to describe the behavior of the probability wave on the six-monthdeparture charts at the 500 hPa level.It is found that C=2πa/L-b_o(?)T/ax and when L→∞,then C=-b_o(?)T/(?)x,where C is wave velocity,a and b are constants,and L is wavelength.The motion direction ofprobability waves is against the outfield temperature gradient,and their velocity is related to the absolute valueof temperature gradient.The motion of waves shrinks in heat sinks and expands in heat sources,which have beenverified in practice.Finally the six-month departure probability wave and the modified Sch(?)rdinger equationare used in the MOS predictions of temperature and rainfall in spring-summer 1981-1985 in Jilin Province andthe accuracy for trend predictions is equal to 80%.     

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.     

Effects of terrain-following vertical coordinates on high-resolution NWP simulations

With increasing resolution in numerical weather prediction(NWP)models,the model topography can be described with finer resolution and includes steeper slopes.Consequently,negative effects of the traditional terrain-following vertical coordinate on high-resolution numerical simulations become more distinct due to larger errors in the pressure gradient force(PGF)calculation and associated distortions of the gravity wave along the coordinate surface.A series of numerical experiments have been conducted in this study,including idealized test cases of gravity wave simulation over a complex mountain,error analysis of the PGP estimation over a real topography,and a suite of real-data test cases.The GRAPES-Meso model is utilized with four different coordinates,i.e.,the traditional terrain-following vertical coordinate proposed by Gal-Chen and Somerville(hereinafter referred to as the Gal.C.S coordinate),the one-scale smoothed level(SLEVE1),the two-scale smoothed level(SLEVE2),and the COSINE(COS)coordinates.The results of the gravity wave simulation indicate that the GRAPES-Meso model generally can reproduce the mountain-induced gravity waves,which are consistent with the analytic solution.However,the shapes,vertical structures,and intensities of the waves are better simulated with the SLEVE2 coordinate than with the other three coordinates.The model with the COS coordinate also performs well,except at lower levels where it is not as effective as the SLEVE2 coordinate in suppressing the PGF errors.In contrast,the gravity waves simulated in both the Gal.C.S and SLEVE1 coordinates are relatively distorted.The estimated PGF errors in a rest atmosphere over the real complex topography are much smaller(even disappear at the middle and upper levels)in the GRAPES-Meso model using the SLEVE2 and COS coordinates than those using the Gal.C.S and SLEVE1 coordinates.The results of the real-data test cases conducted over a one-month period suggest that the three modified vertical coordinates(SLEVE1,SLEVE2,and COS coordinates)give better results than the traditional Gal.C.S coordinate in terms of forecasting bias and root mean square error,and forecasting anomaly correlation coefficients.In conclusion,the SLEVE2 coordinate is proved to be the best option for the GRAPES-Meso model.     

Properties and stability of a meso-scale line-form disturbance

By using the 3D dynamic equations for small- and meso-scale disturbances, an investigation is performed on the heterotropic instability (including symmetric instability and traversal-type instability) of a zonal line-like disturbance moving at any angle with respect to basic flow, arriving at the following results: (1) with linear shear available, the heterotropic instability of the disturbance will occur only when flow shearing happens in the direction of the line-like disturbance movement or in the direction perpendicular to the disturbance movement, with the heterotropic instability showing the instability of the internal inertial gravity wave; (2) in the presence of second-order non-linear shear, the disturbance of the heterotropic instability includes internal inertial gravity and vortex Rossby waves. For the zonal line-form disturbance under study, the vortex Rossby wave has its source in the second-order shear of meridional basic wind speed in the flow and propagates unidirectionally with respect to the meridional basic flow. As a mesoscale heterotropic instable disturbance, the vortex Rossby wave has its origin from the second shear of the flow in the direction perpendicular to the line-form disturbance and is independent of the condition in the direction parallel to the flow; (3) for general zonal line-like disturbances, if the second-order shear happens in the meridional wind speed, i.e., the second shear of the flow in the direction perpendicular to the line-form disturbance, then the heterotropic instability of the disturbance is likely to be the instability of a mixed Rossby–internal inertial gravity wave; (4) the symmetric instability is actually the instability of the internal inertial gravity wave. The second-order shear in the flow represents an instable factor for a symmetric-type disturbance; (5) the instability of a traversal-type disturbance is the instability of the internal inertial gravity wave when the basic flow is constant or only linearly sheared. With a second or nonlinear vertical shear of the basic flow taken into account, the instability of a traversal-type disturbance may be the instability of a mixed vortex Rossby – gravity wave.     

DYNAMICS OF LATERAL BOUNDARY MESO-SCALE JET IN THE OCEAN AND ATMOSPHERE

By using an ageostrophie shallow water model, it is pointed out that a kind of lateral boundary meso-seales jet can be established near the plateau or coast. The charactcristie width of this kind of jet is proportional to the scale of Lc = L(?)C(?)(?),where Lo= Cof is the radius of Rossby deformation, Co= (y*H)the speed of gravity wave and g* the redueced gravity. In general, L(?) is of the order of one hundred kilometers and lens of kilometers in the atmosphere and in the ocean respectively. The large-scale gcostrophic current is an important background condition for forming this kind of jet. From this view point it seems that this kind of atmospheric meso-scale jet only occurs in late spring and summer in the eastern part of Asia, because there is a large-scale south monsoon over there. For the ocean, this kind of meso-scale jct seems to be a semi-persistant syslem and not to show a signilieanl seasonal variation, and it can be established on both sides of the ocean.     

Diagnosis of the forcing of inertial-gravity waves in a severe convection system

The non-hydrostatic wave equation set in Cartesian coordinates is rearranged to gain insight into wave generation in a mesoscale severe convection system. The wave equation is characterized by a wave operator on the lhs, and forcing involving three terms—linear and nonlinear terms, and diabatic heating—on the rhs. The equation was applied to a case of severe convection that occurred in East China. The calculation with simulation data showed that the diabatic forcing and linear and nonlinear forcing presented large magnitude at different altitudes in the severe convection region. Further analysis revealed the diabatic forcing due to condensational latent heating had an important influence on the generation of gravity waves in the middle and lower levels. The linear forcing resulting from the Laplacian of potential-temperature linear forcing was dominant in the middle and upper levels. The nonlinear forcing was determined by the Laplacian of potential-temperature nonlinear forcing. Therefore, the forcing of gravity waves was closely associated with the thermodynamic processes in the severe convection case. The reason may be that, besides the vertical component of pressure gradient force, the vertical oscillation of atmospheric particles was dominated by the buoyancy for inertial gravity waves. The latent heating and potential-temperature linear and nonlinear forcing played an important role in the buoyancy tendency. Consequently, these thermodynamic elements influenced the evolution of inertial-gravity waves.     

DIAGNOSIS OF WAVE ACTIVITY OVER RAINBAND OF LANDFALL TYPHOON

A generalized wave-activity density, which is defined as an absolute value of production of three-dimensional vorticity vector perturbation and gradient of general potential temperature perturbation, is introduced and its wave-activity law is derived in Cartesian coordinates. Constructed in an agoestrophic and nonhydrostatic dynamical framework, the generalized wave-activity law may be applicable to diagnose mesoscale weather systems leading to heavy rainfall. The generalized wave-activity density and wave-activity flux divergence were calculated with the objective analysis data to investigate the character of wave activity over heavy-rainfall regions. The primary dynamical processes responsible for disturbance associated with heavy rainfall were also analyzed. It was shown that the generalized wave-activity density was closely correlated to the observed 6-h accumulative rainfall. This indicated that the wave activity or disturbance was evident over the frontal and landfall-typhoon heavy-rainfall regions in middle and lower troposphere. For the landfall-typhoon rainband, the portion of generalized wave-activity flux divergence, denoting the interaction between the basic-state cyclonic circulation of landfall typhoon and mesoscale waves, was the primary dynamic process responsible for the evolution of generalized wave-activity density.     

THE NONDISPERSION SOLUTION OF NONLINEAR SPIRAL PLANETARY WAVES

In this paper,by using the simplified hydrodynamic equations of barotropic nondivergent atmosphere incylindrial coordinates,the nondispersion cnoidal wave solution of spiral planetary waves is discovered.The formulas of wave speed and the diagnostic relationships between wave parameters are obtained,mostof which are consistent with observed facts.It might mean that the nondispersion cnoidal wave does existin the real atmosphere.At least it might be a first approximation to the actual nonlinear wave.     

THE LYAPUNOV STABILITY OF NONLINEAR WAVE MOTION IN THE BAROTROPIC NONDIVERGENT ATMOSPHERE

Introducing the concept of pseudo-momentum, a generalized Arnold-Dikii functional is established, and then thesufficient condition for stability of nonlinear wave motion in the barotropic nondivergent atmosphere is derived by useof variational principle. It is found that the stability of nonlinear wave motion depends not only on its streamfield distri-bution, but also on its phase speed for the propagating nonlinear wave motion. Moreover, the stability criterion of trav-elling modon is also obtained, and it is shown that the travelling modon is stable if the scale of disturbance superimposedon the travelling modon remains to be less than that of the travelling modon.     

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.     

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.     

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.