Abstract
Despite much previous effort, the establishment of an accurate model of the western Pacific subtropical high (WPSH) and analysis of its chaotic behavior has proved to be difficult. Based on a phase-space technique, a nonlinear dynamical model of the WPSH ridge line and summer monsoon factors is constructed here from 50 years of data. Using a genetic algorithm, model inversion and parameter optimization are performed. The Lyapunov spectrum, phase portraits, time history, and Poincaré surface of section of the model are analyzed and an initial-value sensitivity test is performed, showing that the model and data have similar phase portraits and that the model is robust. Based on equilibrium stability criteria, four types of equilibria of the model are analyzed. Bifurcations and catastrophes of the equilibria are studied and related to the physical mechanism and actual weather phenomena. The results show that the onset and enhancement of the Somali low-level jet and the latent heat flux of the Indian monsoon are among the most important reasons for the appearance and maintenance of the double-ridge phenomenon. Violent breakout and enhancement of the Mascarene cold high will cause the WPSH to jump northward, resulting in the “empty plum” phenomenon. In the context of bifurcation and catastrophe in the dynamical system, the influence of the factors considered here on the WPSH has theoretical and practical significance. This work also opens the way to new lines of research on the interaction between the WPSH and the summer monsoon system.
Similar content being viewed by others
References
Adamowski JF (2008) Development of a short-term river flood forecasting method for snowmelt driven floods based on wavelet and cross-wavelet analysis. J Hydrol 14:5321–5334
Bergman JW, Hendon HH (2000) The impact of clouds on the seasonal cycle of radiative heating over the Pacific. J Atmos Sci 57:545–566
Blanchard P, Devaney RL, Hall GR (2006) Differential equations. Thompson, London, pp. 96–111
Boudjema G, Cazelles B (2001) Extraction of nonlinear dynamics from short and noisy time series. Chaos Soliton Fractals 12:2051–2069
von Bremen HF, Udwadia FE, Proskurowski W (1997) An efficient QR based method for the computation of Lyapunov exponents. Physica D 101:1–16
Cao J, Huang RH, Xie YQ, Tao Y (2003) Evolution mechanism of the western Pacific subtropical high. Sci China (Series D) 46(3):257–268
Cehelsky P, Tung KK (1987) Theories of multiple equilibria and weather regimes—a critical reexamination (part II): baroclinic two-layer models. J. A. S. 44(21):3282–3303
Chan JCL, Ai W, Xu J (2002) Mechanisms responsible for the maintenance of the 1998 South China Sea summer monsoon. J Meteor Soc Japan 80:1103–1113
Chang CP, Zhang YS, Li T (2000) Interannual and interdecadal variations of the East Asian summer monsoon and tropical Pacific SSTs. Part I: roles of the subtropical ridge. J Clim 13:4310–4340
Charney JG, Devore JG (1979) Multiple flow equilibria in the atmosphere and blocking. J. A. S. 36:1205–1216
Chen H, Lee C (2004) Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals 21:957–965
Deser C, Phillips AS (2006) Simulation of the 1976/77 climate transition over the North Pacific: sensitivity to tropical forcing. J Clim 19:6170–6180
Ding YH (2007) The variability of the Asian summer monsoon. J Meteorol Soc Jpn 85:21–54
Dong B, Jifan C (1988) The observational analysis and the theoretic simulation of the seasonal change of the western Pacific subtropical high ridge line. Acta Meteorologica Sinica (in Chinese) 3:361–363
Fraedrich K (1987) Estimating weather and climate predictability on attractors. J. A tmos. Sci. 44:722–728
Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33(2):1134–1140
Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208
Grinsted JC, SJ M (2004) Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Process Geophys 11:561–566
Hong M, Wang D, Zhang R, et al. (2015) Reconstruction and forecast experiments of a statistical–dynamical model of the Western Pacific subtropical high and East Asian summer monsoon factors. Weather and Forecasting 30:206–216
Hong M, Zhang R, Li JX, Liu KF (2013) Inversion of the Western Pacific subtropical high dynamic model and analysis of dynamic characteristics for its abnormality. Nonlinear Proc Geoph. 20:131–142
Hong, M., R. Zhang, Qian L. X., et al. (2014) Bifurcations in a nonlinear dynamical model between Western Pacific subtropical high ridge line index and its summer monsoon impact factors. Discrete dynamics in nature and society, Volume 2014, Article ID 709589, 10 pages.
Hoskins BJ (1996) On the existence and strength of the summer subtropical anticyclones. Bull. Amer. Meteor. Soc. 77:1287–1292
Huang RH, Zhang RH, Yan BL (2001) Dynamical effect of the zonal wind anomalies over tropical western Pacific on ENSO cycles. Science in China, Ser D 44(12):1089–1098
Kang H, Tsuda I (2009) On embedded bifurcation structure in some discretized vector fields. Chaos 19:033132
Kurihara K (1989) A climatological study on the relationship between the Japanese summer weather and the subtropical high in the western northern Pacific. Geophys Mag 43:45–104
Li MC, Luo ZX (1983) The nonlinear mechanism of the abrupt change of the atmosphere general circulation during June and October. Science in China, Ser B 26(7):746–774
Lindzen RS, Hou AY (1988) Hadley circulations for zonally averaged heating centered off the equator. J Atmos Sci 45:2416–2427
Liu CJ, Tao SY (1983) Northward jumping of the subtropical highs and CUSP catastrophe. Scie China, Ser B 26(10):1065–1074
Liu SD, Liu SS (2011) Atmospheric vortex dynamics. Meteorology, Beijing, pp. 75–78
Liu Y, Wu G (2004) Progress in the study on the formation of the summertime subtropical anticyclone. Adv Atmos Sci 21:322–342
Lu KL, Wang BQ (1996) On problems of highly truncated low-spectral model in studies of multiple equilibria in the atmosphere. J Tropical Meteorology (in Chinese) 1:51–59
Manneville P, Pomeau Y (1979) Intermittency and the Lorenz Model. Phys Lett 75(1):l–2
Miu JH, Ding MF (1985) The abrupt change and seasonal change of the atmospheric equilibrium states driven by the thermal forcing, the northward jump of the subtropical high. Sci China (in Chinese) Ser B 1:87–96
Miyasaka T, Nakamura H (2005) Structure and formation mechanisms of the Northern Hemisphere summertime subtropicalhighs. J Climate 18:5046–5065
Pandhiani SM, Shabri AB (2013) Time series forecasting using wavelet-least squares support vector machines and wavelet regression models for monthly stream flow data. Open J Statistics 14:114–126
Pang Y, Xu W, Yu L, Ma J, Lai KK, Wang SY, Xu S (2011) Forecasting the crude oil spot price by wavelet neural networks using OECD petroleum inventory levels. New Mathematics Natural Computation 7:281–297
Poincaré H (1900) La théorie de Lorentz et le principe de réaction. Archives néerlandaises des sciences exactes et naturelles 5:252–278
Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys.Rev. Lett. 170(6):421–428
Qi L, Zhang ZQ, He JH, et al. (2008) A probe into the maintaining mechanism of one type of the double-ridges processes of West Pacific subtropical high. Chinese J. Geophys. 51(3):682–691
Rodwell MJ, Hoskins BJ (2001) Subtropical anticyclones and summer monsoons. J Clim 14:3192–3211
Rulkov NF (2001) Regularization of synchronized chaotic bursts. Phys Rev Lett 86(1):183–186
Sajin H, Yamagata T (2003) Possible impacts of Indian Ocean dipole mode events on global climate. Clim Res 25(2):151–169
Seager R, Murtugudde R, Naik N, Clement A, Gordon N, Miller J (2003) Air–sea interaction and the seasonal cycle of the subtropical anticyclones. J Clim 16:1948–1966
Shigeo Y (1987) Bifurcation properties of a stratospheric vacillation model. J. Atmos Sci 44(14):1723–1733
Su TH, Xue F (2010) The intraseasonal variation of summer monsoon circulation and rainfall in East Asia. Chinese J Atmospheric Sci (in Chinese) 34(3):611–628
Takens F (1981) Determining strange attractors in turbulence. Lecture Notes in Math 898:361–381
Tao SY, Ni YQ, Zhao SX, et al. (2001) The study on formation mechanism and forecast of the 1998 summer rainstorm in China. Meteorological Press 20:19–31
Tao SY, Wei J (2006) The westward and northward advance of the subtropical high over the west Pacific in summer. J ApplMeteor Sci 17:513–524
The Central Meteorological Station Forecast Group (1976) The long-range weather forecast technology experience. The Central Meteorological Station Press, Beijing, pp. 5–6
Tung KK, Rosenthal AJ (1985) Theories of multiple equilibria—a critical reexamination (part I): barotropic models. J A S 42(24):2804–2819
Udwadia FE, von Bremen HF (2001) Computation of Lyapunov characteristic exponents for continuous dynamical systems. Z Angew Math Phys 53:123–128
Wu GX, Liu YM, Liu P, et al. (2002) Relationship between zonal mean subtropical high and Hadley sinking circulation. Acta Meteorologica Sinica 60(5):635–636
Wu GX, Chou JF, Liu YM, et al. (2003) Review and prospect of the study on the subtropical anticyclone. Chinese J Atmospheric Sci 27(4):503–517
Wu GX, Liu YM, Ren RC, et al. (2004) The relationships between the given normal subtropical high and Hadley vertical movement. Acta Meteorologica Sinica 62(5):587–597
Xue F, Wang HJ, He JH (2003) Interannual variability of Mascarene high and Australian high and their influences on summer rainfall over East Asia. Chin Sci Bull 48(5):492–497
Yu DD, Zhang R, Hong M (2007) A charateristic correlation analysis between the asia summer monsoon memberships and west Pacific subtropical high. J Trop Meteorol 01:58–67
Zhan RF, Li JP, He JH (2004) Influence of the double ridges of West Pacific subtropical high on the second MeiYu over the Yangtze river valley during 1998. Acta Meteorologica Sinica 62:294–307
Zhang R, Yu ZH (2000) Numerical and dynamical analyses of heat source forcing and restricting subtropical high activity. Advances in Atmospheric Sci 17(1):61–71
Zhang R, Hong M, et al. (2007) Non-linear dynamic model retrieval of subtropical high based on empirical orthogonal function and genetic algorithm. Appl Math Mech 27:1645–1654
Zhang Y, Zhang Q, Zhao L, Yang C (2006) Dynamical behaviors and chaos control in a discrete functional response model. Chaos, Solitons Fractals 8(21):132–136
Zhu BZ, Jin FF, Liu ZY (1991) An introduction to the nonlinear dynamics of the atmosphere and ocean (in Chinese). Ocean Press, Beijing
Acknowledgments
This study is supported by the Chinese National Natural Science Fund for young scholars (41005025/D0505), the Chinese National Natural Science Fund (41075045), and the Chinese National Natural Science Fund (BK2011123) of Jiangsu Province.
Author information
Authors and Affiliations
Corresponding authors
Appendix: Reconstruction idea of the dynamical model
Appendix: Reconstruction idea of the dynamical model
Suppose that the physical law of a nonlinear system evolving with time can be expressed as follows:
where f i is the generalized nonlinear function of q 1 , q 2,. . . , q i , . . . , q N and N is the number of state variables. Ncan generally be determined by the complexity of dynamical system and measured by calculating its fractal dimensions. The difference form of Eq. 7 can be written as follows:
where Mis the length of time series of observed data. The model parameters and the system structure can be gained by inversion algorithm based on the observed data.\( {f}_i\left({q}_1^{j\varDelta t},{q}_2^{j\varDelta t},\dots, {q}_i^{j\varDelta t},\dots, {q}_N^{j\varDelta t}\right) \)is an unknown nonlinear function and we assume that \( {f}_i\left({q}_1^{j\varDelta t},{q}_2^{j\varDelta t},\dots, {q}_i^{j\varDelta t},\dots, {q}_N^{j\varDelta t}\right) \) contains two parts: G jk representing the expanding items containing variable q i and P ik just representing corresponding parameters which are real numbers ( i = 1 , 2 , . . . N,j = 1 , 2 , . . . M, k = 1 , 2 , . . . , K). It can be supposed that \( {f}_i\left({q}_1,{q}_2,\dots, {q}_n\right)={\displaystyle \sum_{k=1}^K{G}_{jk}{P}_{ik}} \)(8). The matrix form of Eq. 8 is D = GP, in which
Coefficients of the above-generalized unknown equation can be identified through inverting the observed data. Given a vector D, the vector P can be solved to satisfy the above equation. It is a nonlinear system with respect to q; however, it is a linear system with respect to P(assume P is unknown). So the classical least square method can be introduced to estimate the equation and the regular equation G T GP = G T D can be derived by making the residual sum of squares S = (D − GP)T(D − GP) minimum.
As G T G is usually a singular matrix, its eigenvalues and eigenvectors can be solved easily. After removing those with 0 value, the remaining components are K numbers of λ 1 , λ 2,. . . , λ i which can make up a diagonal matrix Λ k , and the corresponding K numbers eigenvectors which can form the diagnostic matrix U L .
with \( {V}_L=\frac{G{U}_i}{\lambda_i} \), \( H={U}_L{\varLambda}^{-1}{V}_L^T \), equation P = HD can be solved, and the parameter Pcan be obtained.
Based on the above approach, coefficients of the nonlinear dynamical systems can be determined and the nonlinear dynamical equations of observed data can be established.
Rights and permissions
About this article
Cite this article
Hong, M., Zhang, R., Li, M. et al. Bifurcations and catastrophes in a nonlinear dynamical model of the western Pacific subtropical high ridge line index and its evolution mechanism . Theor Appl Climatol 129, 363–384 (2017). https://doi.org/10.1007/s00704-016-1777-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00704-016-1777-y