**REVIEWS
ON STUDY OF NONLINEAR ATMOSPHERIC DYNAMICS IN CHINA(1999-2002)**

*DIAO Yina* ^{1,2}, FENG Guolin^{2 }, LIU Shida^{3},*

*LIU Shikuo ^{3}, LUO
Dehai^{1}, HUANG Sixun^{4},*

*LU Weisong ^{5 }* and

1. Department of Atmospheric and Oceanic Sciences, Ocean University of China, Qingdao 266003, China2. LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China3. Department of Atmospheric Sciences, Peking University, Beijing 100871, China4. Air Force Meteorological Institute, Nanjing 211101, China5. Nanjing Institute of Meteorology, Nanjing 210044, China6. Beijing Training Center, China Meteorological Administration, Beijing 100081, China

ABSTRACTResearches on nonlinear atmospheric dynamics in China (1999-2002) are briefly surveyed.

This review includes the major achievements in the branches of nonlinear dynamics as follows: nonlinear stability theory, nonlinear blocking dynamics, 3-D spiral structure in the atmosphere, traveling wave solution of nonlinear evolution equation, numerical predictability in chaos system, and global analysis of climate dynamics. Some applications of nonlinear method such as hierarchy structure of climate and scaling invariance, spatial-temporal series predictive method, nonlinear inverse problem and a new difference scheme with multi-time levels are also introduced.

Key words:nonlinear dynamics, stability, blocking, spiral structure, traveling wave, global analysis

I. INTRODUCTIONNonlinear atmospheric dynamics is an important branch of atmospheric sciences. This field is very active in China during last three decades, and some important advances have been made (see the review paper by Li and Chou, 2003). Recently there are some new studies which have been carried out in many aspects of this area, and some new books on nonlinear atmospheric dynamics have been published, e.g., Nonlinearity and Complexity in Atmospheric Sciences (Chou 2002), Mathematical and Physical Problems in Atmospheric Sciences (Huang and Wu 2001a), Envelope Soliton Theory and Blocking Pattern in the Atmosphere (Luo 1999a), Nonlinear Dynamics of Blocking (Luo 2000a), and External Forcing and the Dynamic Principle of Wave-Flow Interaction (Xu and Gao 2002), etc. These investigations push forward the development of nonlinear atmospheric sciences in China. The aim of this paper is just to briefly review recent advances in nonlinear atmospheric sciences in China during last four years.

II. NONLINEAR STABILITY THEORYProgresses in the problems of nonlinear stability and instability of atmospheric motions, particularly attained by the research group of Mu, are surveyed in Mu and Wu (2001a). In this paper we only present some main results obtained in the past four years.

Symmetric stability is related to mesoscale dynamics with application to the formation of rain-bands. Mu et al. (1999a) studied the symmetric stability of the moist atmosphere and investigated the influence of the northward components of the Coriolis force, which extends previous results from infinitesimal amplitude to the finite amplitude perturbations. In the context of quasigeostrophic motions, Liu and Mu (2001) studied the modified Eady model and established an optimal nonlinear stability in the sense that if it is destroyed, then there always exists a finite periodic zonal channel, in which there is an exponentially growing normal mode. Li et al. (2000) investigated the nonlinear stability of fronts in the ocean on a sloping bottom. By employing the frontal geostrophic model, the nonlinear stability criteria for the fronts in the ocean are obtained by using Arnold's variational principle and a prior estimate method.

For the evolution of initial unstable disturbance in Hamiltonian system, an effect of nonlinearity is to change the basic flow in such a way as to stop the growth of the disturbance after they have reached significant amplitude. The determination of such bounds on the growth of the disturbances to the unstable basic flow is called the saturation problem of instability. By using nonlinear stability criteria for the Phillips model, Xiang et al (2002) and Xiang et al. (2003) established some upper bound on the energy and potential enstrophy of wavy disturbances respectively.

Mu et al. (1999b) attempt to apply Arnold type nonlinear stability criteria to the diagnostic study of the dynamical persistence (stability) or breakdown (instability) of the zonal flows in the middle and high latitudes. In the cases of the blocking high, the cut-off low and the zonal flow, the relationships of the geostrophic streamfunction versus the potential vorticity of the observed atmosphere are analyzed by real observation data, which indicates that Arnold's second type nonlinear stability theorem is more relevant to the observed atmosphere than the first one. For both the stable and unstable flows, Arnold's second type nonlinear stability criteria are applied to the diagnosis. The primary results show that the analyses correspond well to the evolution of the atmospheric motions. The synoptically stable atmospheric zonal flows satisfy Arnold's second type nonlinear stability criteria while the synoptically unstable ones violate the nonlinear stability criteria.

The study of instability is closely related to the uncertainty of numerical prediction, and hence to the data assimilation and predictability. Initial error will certainly be amplified by the dynamical instability and finally results in forecast uncertainty. Mu et al (2001c) investigated the effects of four-dimensional variational data assimilation in case of nonlinear instability. On the other hand, singular value and singular vector have been utilized to study the instability of atmospheric motions, ensemble numerical prediction and predictability problems in numerical weather and climate prediction (c.f. Molteni and Palmer 1993; Molteni et al. 1996; Thompson, 1998). In this approach tangent linear model is adopted. It is well-known that the motions of the atmosphere is governed by nonlinear system, and hence the nonlinearity rises the question on the validity of tangent linear model. Mu et al. (2000) and Mu et al. (2001b) using nonlinear stability criteria investigate this problem and found that nonlinear instability effects considerably the length of the valid period of TLM. Generally speaking the length of the valid period of TLM can only be determined a posterior. To avoid answering the validity problem of TLM in the application of singular value and singular vector, Mu (2000) proposed a new concept of nonlinear singular vector and nonlinear singular value. In Mu et al. (2001d), they applied nonlinear singular vector and nonlinear singular value to the study of predictability.

Lu (2001, 2003) presented three new generalized energies as Lyapunov function, and obtained the subcritical criterions of nonlinear barotropic and baroclinic stability and the subcritical criterion of nonlinear mesoscale symmetric stability. His results indicated that the subcritical instability will possibly occur if only the initial disturbance amplitude is lager than another critical value even if the frictional coefficient is greater than a critical value. The theory overcomes the shortcoming of the usual stability criterion that cannot explicitly express nonlinear effect, and may give a new physical mechanism of formations of quasi-stationary system and strong rainstorm, and may also be used to explain supercritical symmetric instability.

Feng et al. (2001a) studied the instability evolution of air-sea oscillator. They expressed a stochastic dynamic model of air-sea interaction as a Fokker-Planck equation (FPE), which is from statistical mechanics used to obtain a time evolution equation about the probability density function and the climatic potential function. Using matrix-continue fraction method (MCFM), they obtained its solution with which effects of greenhouse gases were calculated. As a result, sea surface temperature rises by 1.2°C, and the basic period is about 3-4 years and becomes longer than that under the condition of doubling CO

_{2}. In view of non-equilibrium dynamics, Feng et al. (2002a) wrote the equations of nonlinear stochastic air-sea oscillator as a Fokker-Planck equation (FPE), which gives the evolution from instability to stability state on ENSO event and reveals ENSO existing multi-equilibrium states. Their numerical calculation is coincided with observational data. Using the theory of nonlinear dynamics in combination with both data analysis and numerical experiments, Huang et al. (2003) and Han and Huang (2002) also investigated the nonlinear oscillation of air-sea coupling and ENSO cycle, and revealed the relationship between ENSO and the limit cycle of the ENSO model they discussed.In addition, Xu and Gao (2002) gave a detailed summary on the study of nonlinear stability of wave-flow interaction.

III. NEW THEORIES OF NONLINEAR BLOCKING DYNAMICS

During recent four years nonlinear blocking dynamics has been extensively studied and been made a significant progress. Luo (1999a, 2000a) systemically introduced and investigated nonlinear blocking dynamics and the envelope soliton theory of blocking in his two books. Luo (1999b, 2000b) presented the theory of envelope Rossby soliton forced by synoptic-scale eddies, and theoretically proved that under some conditions a blocking embryo described by the envelope Rossby soliton can develop to blocking pattern through the forced effect of antecedent synoptic-scale eddies. The onset process of this kind of blocking is a transfer process from dispersion to non-dispersion or weak dispersion, and its decay is an opposite process. Luo et al. (2001a) and Luo and Li (2002) further studied this problem and confirmed from the observational studies that the life cycle of blocking associated with synoptic-scale eddies is a transfer process between dispersion and non-dispersion. Employing numerical experiments, Luo and Li (2002) also found that the synoptic-scale eddies seem to play a dominant role in the amplification of blocking, while the topography effect appears to play a phase-locking role (Luo 1999). At the same time, the synoptic eddies tend to split into two branches during the onset of blocking. These numerical results are in accordance with the observational results.

Luo (2001b) proposed a new nonlinear Schrödinger equation with higher order nonlinear term, which is an extension of the classical Schrödinger equation and has come to more attention. This new equation and its transformed equation can be used to study the mechanism of blocking. In an equivalent barotropic framework, for example, Luo and Li (2000) applied the new forced nonlinear Schrödinger equation to examine the interaction between the planetary-scale waves and the localized synoptic-scale eddies upstream, and explained why and how the synoptic-scale eddies can reinforce and maintain vortex pair block. Furthermore, Luo and Li (2001b) used a parametrically excited higher-order nonlinear Schrödinger equation to describe the interaction of a slowly moving planetary-scale envelope Rossby soliton for zonal wavenumber-two with a wavenumber-two topography under the LG-type dipole near-resonance condition. Their results showed that the initiation and decay of blocking are the transfer processes of amplified envelope soliton system between dispersion and non-dispersion or weak dispersion. They therefore suggested that in the higher latitude regions, the planetary-scale envelope soliton-topography interaction could be regarded as a possible mechanism of the establishment of blocking.

Tao (2000) analyzed the general nonlinear free mode of the atmosphere by use of observational data. Lu et al. (1998a) employed the free mode as the forced field to carry out some numerical experiments. Their results showed that under the condition of weak dissipation and forcing, the low-frequency oscillation caused by the transform between zonal and meridional patterns occurs, and that under the condition of strong dissipation and forcing, the atmospheric evolution tends to the free mode of the forcing field. Based on the results they proposed new physical mechanism of blocking in the mid-high latitudes and low-frequency oscillation. Besides, Lu et al. (1998b) also used the nonlinear critical layer theory to explain the formation, maintain and oscillation of the subtropical high.

Additionally, Xu and Gao (2002) discussed the wave-flow interaction in the maintaining process of blocking. Wu and Mu (1999) studied the Liapunov stability of Modons solution and blocking in the atmosphere, proved the instability of Modons solution in the sense of Liapunov stability, and introduced a concept of blocking “life span”to study blocking process.

IV. 3-D SPIRAL STRUCTURE IN THE ATMOSPHERERecent investigations imply that the 3-D spiral structure may be a basic feature of the atmosphere. Liu et al. (2000a, b) studied the 3-D spiral structure pattern in the atmosphere. Through the balance relationship among the pressure gradient force, Coriolis force and frictional force, they obtained the nonlinear 3-D velocity field from the equations describing the mesoscale vortex in the geophysical fluid. By analyzing the 3-D wind field, the 3-D spiral structure in physical space for unstable stratification was found. The 3-D spiral vortex is very similar to typically real cyclones (Liu et al. 2000b). They pointed out that the existence of 3-D spiral structure in the mesoscale vortex is a result of the mass conservation and effect of earth's rotation, therefore the rotation of the earth and turbulent viscosity of the air play an important role in the 3-D spiral structure of mesoscale vortex.

V. TRAVELING WAVE SOLUTION OF NONLINEAR EVOLUTION EQUATIONTraveling wave solution in the atmosphere becomes an active field during last ten years. The traveling wave solution of nonlinear evolution equation contains spiral wave (Liu et al. 2002b), solitary wave, shock, periodic wave, etc. Recently, Liu et al. (2001a, b) have proposed a new method known as the Jacobi elliptic function expansion method to study wave solutions of nonlinear wave equations. This method is more general than the other methods such as the hyperbolic tangent function expansion method. Applying this method they obtained new periodic solutions of a kind of nonlinear wave equation and the envelope periodic solution of some nonlinear wave equations with the variable coefficients (Fu et al. 2001; Liu et al. 2002c, d). Liu et al. (2002b) also discussed the Hopf bifurcation and spiral wave solution of the complex Gianzburg-Landau equation. Huang (1998, 1999), Huang and Zhao (2000b), and Huang and Wu (2001a) adopted the topological method, singular perturbation method, etc. to discuss the existence of various traveling wave solutions and their bifurcation structures, and used these theoretical results to explain related weather phenomena.

VI. NUMERICAL PREDICTABILITY IN CHAOS SYSTEMRound-off error can cause numerical uncertainly (Li et al. 2000, 2001a). Through the numerical and theoretical study on round-off error, Li et al. (2000, 2001a) presented a computational uncertainty principle in numerical nonlinear system, which implies an inherent relation between the uncertainty due to the imperfection of numerical method itself and the uncertainty due to the inherent inaccuracy of digital computers. Specifically, if the discretization error and the round-off error are treated as two “adjoint variables”, the computational uncertainty principle reveals that the smaller one of them, the greater will be the other adjoint variable. Owing to the inherent relationship between the two uncertainties due to numerical method and computer respectively, it naturally causes a limitation in the width of interval of effectively numerical solution. This is just the root cause of the inevitable existence of maximally effective computation time (MECT) and optimal stepsize (OS) in numerical nonlinear system. This indicates that, once the precision of calculation machine used is given, the best degree of accuracy which can be achieved for the numerical solution obtained by a numerical method is determined entirely. Thus, the computational uncertainty principle gives a certain limitation to the computational capacity of numerical method under the inherent property of finite machine precision. However, OS is corresponding to MECT. Feng and Chou (2001b) used a climate model, the Rossler system and the super chaos system to investigate MECT and OS in these systems. Their results suggested that by using of OS in solving numerically nonlinear ordinary differential equations the self-memorization theory of chaos systems provides a new approach of numerical weather prediction.

VII. GLOBAL ANALYSIS OF CLIMATE DYNAMICSThe global analysis of climate dynamics is a useful approach for studying long-term behavior of the atmosphere. Li and Chou (2003a, b) briefly summarized the idea and main theoretical results of the global analysis theory of climate system and introduced its main applications, especially, the adjustment and evolution processes of climate, the principle of numerical model design and the optimally numerical integration. For a class of nonlinear evolution equations, Li and Chou (1999a) studied their global attractors and discussed the existence of their inertial manifolds using the truncated method. They proved that the operator equation of the atmospheric motion with dissipation and external forcing belongs to this class of nonlinear evolution equations. Therefore there exists an inertial manifold of the atmospheric equations if the spectral gap condition for the dissipation operator is satisfied. These results furnish a basis for studying further dynamical properties of global attractor of the atmospheric equations and for designing better numerical scheme. Using the properties of integral with parameters, Li and Chou (1999b) analyzed the properties of operators of the full atmospheric operator equation. Based on them, the asymptotic behavior of the solutions of the atmospheric equations was discussed and the existence of the global attractor of the atmospheric equations was proved. Based on the global qualitative theory of atmospheric dynamical equations, Li and Chou (2001b) presented a new method for simplifying equations, i.e. the operator constraint principle, and discussed the general rule of the method and its mathematical strictness. Besides, Fan et al. (1999) applied the global analysis theory to investigate climate predictability.

VIII. SOME APPLICATIONS OF NONLINEAR METHODS1. Hierarchy Structure of Climate and Scaling Invariance

A climatic time series consists of many time scales. Whether cooling or warming will depend on time scales, Liu et al. (2000c) therefore indicated that the climatic hierarchy could be constructed by analyzing different scales. Their work on the time series of Northern Hemisphere surface atmosphere temperature showed that the numbers of climatic jump point are approximately according to the Fibonacci sequences with scale hierarchy. They therefore concluded that there exists a scaling invariance in nonlinear climate system which may reflects evolutional laws of climatic jump time and hierarchical structure of climatic series. Moreover, Liu et al. (2000c) and She et al. (2002) pointed out that a nonlinear dynamical system which leaves climatic scaling invariance can be uncovered from the climatic jump point location. Liu and Ling (2000) further suggested that turbulence of geophysics also has the scaling invariance in which scaling index varies with atmospheric different temperature stratification. Besides, using the analysis of nonlinear dynamics Liu et al. (2002c) showed that climatic time series contains many complex structures.

2. Spatial-Temporal Series Predictive Method

By using of the idea of nonlinear reconstruction on spatial-temporal series, Yang et al. (2000) built a nonlinear regional predictive model on atmospheric ozone in China. Their numerical experiments indicated that for one-month prediction the correlation coefficient between observational and predicted ozone anomaly fields was over 43%. Chen et al. (2003) also employed the reconstruction of phase-space theory and the spatial-temporal series predictive method to construct a monthly pentad-mean nonlinear dynamical prediction model of the zonal-mean geopotential height. They suggested that the systematic error in zonal mean height could be reduced significantly by the numerical model.

3. Nonlinear Inverse Problem

The significant features of inversion of satellite remote sensing are its ill-posedness and strong nonlinearity. The accuracy of inversion of satellite remote sensing is not good because of the two characteristics so that it cannot satisfy the need of numerical weather prediction. Huang and Li (2000a), Li and Huang (2001), and Zhang and Huang (2003a) studied the problem how to improve the accuracy of inversion of satellite remote sensing. They applied and developed the regularization method to investigate the problem, and designed an iterative scheme of the inversion, which not only considers ill-posedness as well as nonlinear effect, but also optimally choose regular parameters. The results of numerical experiments indicated that their method is apparently better than the other methods.

Huang and Han (2001b, 2002a) and Huang et al. (2003a) systemically discussed applications of the regularization method to variational data assimilation. Through the synthetical modification for initial value condition, boundary value condition and model parameters, under the conditions of global or local data they applied the regularization method to analyze the variational data assimilation of sea temperature. Their method reduces the numbers of iteration and highly improves the accuracy of assimilation. Additionally, for the non-differential system Huang and Han (2002b) proposed the idea of weak form to study its variational data assimilation and derived its adjoint form.

Physical process with “on-off” switch is a nonlinear one, of which the treatment in the inverse problems by adjoint approach, e.g. variational data assimilation, is a key problem (c.f. Zou 1997; Qin 1996). Recently Mu et al. (2003) and Wang et al. (2002) proposed a new approach to deal with this problem. They also demonstrated the advantage of this approach by numerical results.

4. A New Difference Scheme with Multi-Time Levels

To make the best utilization of information from observational data, Feng et al. (2001c) presented a new multi-time level difference scheme and discussed some mathematical characteristics of the scheme. The results of some numerical experiments showed the computational accuracy of numerical linear advection equation and numerical nonlinear advection equation with the difference scheme is higher than that with the leapfrog scheme. The scheme may be applied to numerical analysis in many fields, such as meteorology, engineering physics, astronautics, environment and economy etc. To put more information into a difference scheme of certain differential equation, on the basis of the memorial dynamics Feng et al. (2002b) and Feng and Chou (2002c) proposed a new kind of time integration scheme, known as the retrospective scheme. Stability criteria of the scheme of an advection equation in certain conditions are derived mathematically. It was shown from the numerical results of the advection equation with its retrospective scheme that the accuracy of the scheme is much higher than that of the leapfrog difference scheme.

Acknowledgments:This work was jointly supported by the National Natural Science Foundation of China (Nos. 40221503, 40275025, 40023001), and Chinese Academy of Sciences (No. KZCX2-SW-210).

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