Advances in Atmospheric Sciences

, Volume 36, Issue 7, pp 733–752 | Cite as

Charney’s Model—the Renowned Prototype of Baroclinic Instability—Is Barotropically Unstable As Well

  • Yuan-Bing Zhao
  • X. San LiangEmail author
Original Paper


The Charney model is reexamined using a new mathematical tool, the multiscale window transform (MWT), and the MWT-based localized multiscale energetics analysis developed by Liang and Robinson to deal with realistic geophysical fluid flow processes. Traditionally, though this model has been taken as a prototype of baroclinic instability, it actually undergoes a mixed one. While baroclinic instability explains the bottom-trapped feature of the perturbation, the second extreme center in the perturbation field can only be explained by a new barotropic instability when the Charney–Green number γ ≪ 1, which takes place throughout the fluid column, and is maximized at a height where its baroclinic counterpart stops functioning. The giving way of the baroclinic instability to a barotropic one at this height corresponds well to the rectification of the tilting found on the maps of perturbation velocity and pressure. Also established in this study is the relative importance of barotropic instability to baroclinic instability in terms of γ. When γ ≫ 1, barotropic instability is negligible and hence the system can be viewed as purely baroclinic; when γ ≪ 1, however, barotropic and baroclinic instabilities are of the same order; in fact, barotropic instability can be even stronger. The implication of these results has been discussed in linking them to real atmospheric processes.

Key words

Charney’s model multiscale window transform canonical transfer baroclinic instability barotropic instability 

摘 要

本研究使用一个新的泛函分析工具——多尺度子空间变换 (MWT) 和基于 MWT 的局地多尺度能量分析法研究了经典Charney模式中的局地稳定性结构和多尺度能量过程。传统上, Charney模式一直被认为是一个纯斜压的模式, 但我们研究发现该模式中实际上存在混合不稳定, 其中斜压不稳定主要位于模式的底层, 而正压不稳定存在于所有层次, 并且在斜压不稳定趋于消失的地方达到最强, 这正好对应着扰动气压场中等位相线由西倾转为垂直的高度。虽然传统的斜压不稳定解释了扰动的底部强化特征, 但扰动场中位于高层的次级中心只能用新发现的正压不稳定来解释 (尤其是当 Charney-Green参数 γ ≪ 1时)。此外, 我们还发现 Charney 模式中正压不稳定和斜压不稳定的相对大小随着参数γ的变化而变化:当 γ ≪ 1时, 正压不稳定与斜压不稳定具有相同量级、甚至超过斜压不稳定;而当 γ ≫ 1时, 正压不稳定可以忽略不计, 此时系统的不稳定性才可以视为是纯斜压的。


Charney模式 多尺度子空间变换 正则传输 斜压不稳定 正压不稳定 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The suggestions of the two anonymous reviewers are appreciated. Yuan-Bing ZHAO thanks Yang YANG, Yi-Neng RONG, and Brandon J. BETHEL for their generous help. This research was supported by the National Science Foundation of China (Grant Nos. 41276032 and 41705024), the National Program on Global Change and Air-Sea Interaction (Grant No. GASIIPOVAI-06), the Jiangsu Provincial Government through the 2015 Jiangsu Program of Entrepreneurship and Innovation Group and the Jiangsu Chair Professorship, and Shandong Meteorological Bureau (Contract No. QXPG20174023).


  1. Badin, G., 2014: On the role of non-uniform stratification and short-wave instabilities in three-layer quasi-geostrophic turbulence. Physics of Fluids, 26, 096603, Scholar
  2. Badin, G., and F. Crisciani, 2018: Variational Formulation of Fluid and Geophysical Fluid Dynamics: Mechanics, Symmetries and Conservation Laws. Advances in Geophysical and Environmental Mechanics and Mathematics, Springer, 182 pp, Scholar
  3. Blackmon, M. L., J. M. Wallace, N.-C. Lau, and S. L. Mullen, 1977: An observational study of the northern hemisphere wintertime circulation. J. Atmos. Sci., 34, 1040–1053,<1040:AOSOTN>2.0. CO;2.Google Scholar
  4. Boyd, J. P., 1976: The noninteraction of waves with the zonally averaged flow on a spherical earth and the interrelationships on eddy fluxes of energy, heat and momentum. J. Atmos. Sci., 33, 2285–2291,<2285:TNOWWT>2.0.CO;2.Google Scholar
  5. Branscome, L. E., 1983: The Charney Baroclinic stability problem: Approximate solutions and modal structures. J. Atmos. Sci., 40, 1393–1409,<1393:TCBSPA>2.0.CO;2.Google Scholar
  6. Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325–334, Scholar
  7. Brown, Jr. J. A., 1969: A numerical investigation of hydrodynamic instability and energy conversions in the quasi-geostrophic atmosphere: Part I. J. Atmos. Sci., 26, 352–365,<0352:ANI0HI>2.0.C0;2.Google Scholar
  8. Burger, A. P., 1962: On the non-existence of critical wavelengths in a continuous baroclinic stability problem. J. Atmos. Sci., 19, 31–38, <0031:OTNEOC’>2.0.CO;2.Google Scholar
  9. Cai, M., and M. Mak, 1990: On the basic dynamics of regional cyclogenesis. J. Atmos. Sci., 47, 1417–1442,<1417:OTBDOR>2.0.CO;2.Google Scholar
  10. Chai, J. Y., and G. K. Vallis, 2014: The role of criticality on the horizontal and vertical scales of extratropical eddies in a dry GCM. J. Atmos. Sci., 71, 2300–2318, Scholar
  11. Chang, E. K. M., 1993: Downstream development of baroclinic waves as inferred from regression analysis. J. Atmos. Sci., 50, 2038–2053,<2038:DDOBWA’>2.0.CO;2.Google Scholar
  12. Chang, E. K. M., and I. Orlanski, 1993: On the dynamics of a storm track. J. Atmos. Sci., 50, 999–1015,<0999:OTDOAS’>2.0.CO;2. Google Scholar
  13. Chapman, C. C., A. M. Hogg, A. E. Kiss, and S. R. Rintoul, 2015: The dynamics of southern ocean storm tracks. J. Phys. Oceanogr., 45, 884–903, Scholar
  14. Charney, J. G., 1947: The dynamics of long waves in a baroclinic westerly current. J. Meteor., 4, 136–162,<0136:TDOLWI>2.0.CO;2.Google Scholar
  15. Charney, J. G., and P. G. Drazin, 1961: Propagation of planetaryscale disturbances from the lower into the upper atmosphere. J. Geophys. Res., 66, 83–109, Scholar
  16. Deng, Y., and M. Mak, 2006: Nature of the differences in the intraseasonal variability of the pacific and Atlantic storm tracks: A diagnostic study. J. Atmos. Sci., 63, 2602–2615, Scholar
  17. Dickinson, R. E., 1969: Theory of planetary wave-zonal flow interaction. J. Atmos. Sci., 26, 73–81,<0073:TOPWZF>2.0.CO;2.Google Scholar
  18. Edmon, H. J., Jr., B. J. Hoskins, and M. E. McIntyre, 1980: Eliassen-palm cross sections for the troposphere. J. Atmos. Sci., 37, 2600–2616,<2600:EPCSFT>2.0.CO;2.Google Scholar
  19. Farrell, B. F., 1982: Pulse asymptotics of the Charney Baroclinic instability problem. J. Atmos. Sci., 39, 507–517,<0507:PAOTCB>2.0.CO;2.Google Scholar
  20. Fels, S. B., and R. S. Lindzen, 1973: The interaction of thermally excited gravity waves with mean flows. Geophys. Fluid Dyn., 5, 211–212. Scholar
  21. Fournier, A., 2002: Atmospheric energetics in the wavelet domain. Part I: Governing equations and interpretation for idealized flows. J. Atmos. Sci., 59, 1182–1197,<1182:AEITWD>2.0.CO;2.Google Scholar
  22. Fullmer, J. W. A., 1982: The baroclinic instability of highly structured one-dimensional basic states. J. Atmos. Sci., 39, 2371–2387,<2371:TBIOHS>2.0.CO;2.Google Scholar
  23. Gall, R., 1976a: Structural changes of growing baroclinic waves. J. Atmos. Sci., 33, 374–390,<0374:SCOGBW>2.0.CO;2.Google Scholar
  24. Gall, R., 1976b: A comparison of linear baroclinic instability theory with the eddy statistics of a general circulation model. J. Atmos. Sci., 33, 349–373,<0349:ACOLBI’>2.0.CO;2.Google Scholar
  25. Geisler, J. E., and R. R. Garcia, 1977: Baroclinic instability at long wavelengths on a β-plane. J. Atmos. Sci., 34, 311–321,<0311:BIALWO>2.0.CO;2.Google Scholar
  26. Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, 662 pp.Google Scholar
  27. Green, J. S. A., 1960: A problem in baroclinic stability. Quart. J. Roy. Meteor. Soc., 86, 237–251, Scholar
  28. Green, J. S. A., 1970: Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Quart. J. Roy. Meteor. Soc., 96, 157–185, Scholar
  29. Harrison, D. E., and A. R. Robinson, 1978: Energy analysis of open regions of turbulent flows—Mean eddy energetics of a numerical ocean circulation experiment. Dyn. Atmos. Oceans, 2, 185–211, Scholar
  30. Held, I. M., 1978: The vertical scale of an unstable baroclinic wave and its importance for eddy heat flux parameterizations. J. Atmos. Sci., 35, 572–576,<0572:TVSOAU>2.0.CO;2.Google Scholar
  31. Holopainen, E. O., 1978: A diagnostic study on the kinetic energy balance of the long-term mean flow and the associated transient fluctuations in the atmosphere. Geophysica, 15, 125145.Google Scholar
  32. Hoskins, B. J., and P. J. Valdes, 1990: On the existence of stormtracks. J. Atmos. Sci., 47, 1854–1864,<1854:OTEOST’>2.0.CO;2.Google Scholar
  33. Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877–946, Scholar
  34. Kao, S.-K., and V. R. Taylor, 1964: Mean kinetic energies of eddy and mean currents in the atmosphere. J. Geophys. Res., 69, 1037–1049, Scholar
  35. Kuo, H.-L., 1949: Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere. J. Meteor., 6, 105–122,<0105:DIOTDN>2.0.CO;2.Google Scholar
  36. Kuo, H.-L., 1952: Three-dimensional disturbances in a baroclinic zonal current. J. Meteor., 9, 260–278,<0260:TDDIAB>2.0.CO;2.Google Scholar
  37. Kuo, H.-L., 1979: Baroclinic instabilities of linear and jet profiles in the atmosphere. J. Atmos. Sci., 36, 2360–2378,<2360:BIOLAJ>2.0.CO;2.Google Scholar
  38. Lapeyre, G., and P. Klein, 2006: Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory. J. Phys. Oceanogr., 36, 165–176, Scholar
  39. Liang, X. S., 2016: Canonical transfer and multiscale energetics for primitive and quasigeostrophic atmospheres. J. Atmos. Sci., 73, 4439–4468, Scholar
  40. Liang, X. S., and A. R. Robinson, 2005: Localized multiscale energy and vorticity analysis: I. Fundamentals. Dyn. Atmos. Oceans, 38, 195–230, Scholar
  41. Liang, X. S., and D. G. M. Anderson, 2007: Multiscale window transform. Multiscale Model. Simul., 6, 437–467.Google Scholar
  42. Liang, X. S., and A. R. Robinson, 2007: Localized multi-scale energy and vorticity analysis: II. Finite-amplitude instability theory and validation. Dyn. Atmos. Oceans, 44, 51–76, Scholar
  43. Liang, X. S., and L. Wang, 2018: The cyclogenesis and decay of typhoon damrey. Coastal Environment, Disaster, and Infrastructure, X. S. Liang and Y.Z. Zhang, Eds., IntechOpen, Scholar
  44. Lindzen, R. S., and B. Farrell, 1980: A simple approximate result for the maximum growth rate of baroclinic instabilities. J. Atmos. Sci., 37, 1648–1654,<1648:ASARFT>2.0.CO;2.Google Scholar
  45. Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. Tellus, 7, 157–167, Scholar
  46. Lorenz, E. N., 1967: The Nature and Theory of the General Circulation of the Atmosphere. World Meteorological Organization, 161 pp.Google Scholar
  47. McWilliams, J. C., and J. M. Restrepo, 1999: The wave-driven ocean circulation. J. Phys. Oceanogr., 29, 2523–2540,<2523:TWDOC>2.0.CO;2.Google Scholar
  48. Miles, J. W., 1964: A note on Charney's model of zonal-wind instability. J. Atmos. Sci., 21, 451–452,<0451:ANOCMO>2.0.CO;2.Google Scholar
  49. Nakamura, H., 1992: Midwinter suppression of baroclinic wave activity in the pacific. J. Atmos. Sci., 49, 1629–1642,<1629:MSOBWA>2.0.CO;2.Google Scholar
  50. Orlanski, I., and J. Katzfey, 1991: The life cycle of a cyclone wave in the southern hemisphere. Part I: Eddy energy budget. J. Atmos. Sci., 48, 1972–1998,<1972:TLCOAC>2.0.CO;2.Google Scholar
  51. Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed., Springer-Verlag, 710 pp.Google Scholar
  52. Pierrehumbert, R. T., and K. L. Swanson, 1995: Baroclinic instability. Annual Review of Fluid Mechanics, 27, 419–467, Scholar
  53. Plumb, R. A., 1983: A new look at the energy cycle. J. Atmos. Sci., 40, 1669–1688,<1669:ANLATE>2.0.CO;2.Google Scholar
  54. Pope, S. B., 2000: Turbulent flows. Cambridge University Press, 806 pp.Google Scholar
  55. Ragone, F., and G. Badin, 2016: A study of surface semigeostrophic turbulence: Freely decaying dynamics. J. Fluid Mech., 792, 740–774, Scholar
  56. Simmons, A. J., and B. J. Hoskins, 1978: The life cycles of some nonlinear baroclinic waves. J. Atmos. Sci., 35, 414–432,<0414:TLCOSN>2.0.CO;2.Google Scholar
  57. Simons, T. J., 1972: The nonlinear dynamics of cyclone waves. J. Atmos. Sci., 29, 38–52,<0038:TNDOCW>2.0.CO;2.Google Scholar
  58. Song, R. T., 1971: A numerical study of the three-dimensional structure and energetics of unstable disturbances in zonal currents: Part II. J. Atmos. Sci., 28, 565–586,<0565:ANSOTT>2.0.CO;2.Google Scholar
  59. Trenberth, K. E., 1986: An assessment of the impact of transient eddies on the zonal flow during a blocking episode using localized Eliassen-Palm flux diagnostics. J. Atmos. Sci., 43, 2070–2087,<2070:AAOTIO>2.0.CO;2.Google Scholar
  60. Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 770 pp.Google Scholar
  61. Yin, J. H., 2002: The peculiar behavior of baroclinic waves during the midwinter suppression of the pacific storm track. PhD dissertation, University of Washington, 137 pp.Google Scholar
  62. Zhao, Y.-B., and X. S. Liang, 2018: On the inverse relationship between the boreal wintertime Pacific jet strength and stormtrack intensity. J. Climate, 31, 9545–9564, Scholar
  63. Zhao, Y.-B., X. S. Liang, and J. P. Gan, 2016: Nonlinear multiscale interactions and internal dynamics underlying a typical eddyshedding event at Luzon Strait. J. Geophys. Res. Oceans, 121, 8208–8229, Scholar
  64. Zhao, Y.-B., X. S. Liang, and W. J. Zhu, 2018: Differences in storm structure and internal dynamics of the two storm source regions over East Asia. Acta Meteorologica Sinica, 76(5), 663–679, (in Chinese)Google Scholar

Copyright information

© Institute of Atmospheric Physics/Chinese Academy of Sciences, and Science Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Atmospheric SciencesNanjing University of Information Science and TechnologyNanjingChina
  2. 2.School of Marine SciencesNanjing University of Information Science and TechnologyNanjingChina
  3. 3.School of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingChina

Personalised recommendations