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Spatially amplifying modes of the Charney baroclinic-instability problem

Published online by Cambridge University Press:  21 April 2006

R. T. Pierrehumbert
R. T. Pierrehumbert
Affiliation:
Geophysical Fluid Dynamics Laboratory/NOAA, Princeton University, NJ 08542, USA
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Abstract

We determine the circumstances under which baroclinic instability in the Charney model subjected to localized time-periodic forcing manifests itself as a wavetrain that oscillates at the source frequency and amplifies in space with distance from the source; analytical and numerical results describing the salient characteristics of such waves are presented. The spatially amplifying disturbance is a hitherto unsuspected part of the response to a pulsating source, and coexists with the more familiar neutral Rossby wavetrains; it is likely to play a role in a wide range of atmospheric and oceanic phenomena.

The central results rely on a careful application of a causality criterion due to Briggs. These results illustrate a practical means of attacking spatial instability problems, which can be applied to a broad class of systems besides the one at hand. We have found that the Charney problem with positive vertical shear is not absolutely unstable, so long as the wind at the ground is non-negative. This implies that spatial instability and forced stationary-wave problems are well posed in an open domain under typical atmospheric circumstances.

The amplifying waves appear on the downstream side of the source, have eastward (downstream) phase propagation and have wavelengths that increase monotonically with decreasing frequency, becoming infinite at zero frequency. When the surface wind is not too large, the spatial amplification rate has a single maximum near the frequency ωm = (f/N)Uz, where f is the Coriolis parameter, N is the stability frequency and Uz is the vertical shear; the rate approaches zero at zero frequency and asymptotes algebraically to zero at large frequency for any positive surface wind. Distinct Charney and Green modes do not appear until the surface wind is made very large. The amplification rate at ωm becomes infinite as surface wind approaches zero, suggesting a mechanism for the concentration of eddy activity.

We also discuss the relationship of these results to the structure of low- and high-frequency atmospheric variability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 170 , September 1986 , pp. 293 - 317
Copyright
© 1986 Cambridge University Press

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References

Betchov, R. & Criminale W. O.1966 Spatial instability of the inviscid jet and wake. Phys. Fluids 9, 359–362.Google Scholar
Blackmon M. L.1976 A climatological spectral study of the 500 mb geopotential height of the Northern Hemisphere. J. Atmos. Sci. 33, 1607–1623.Google Scholar
Blackmon M. L., Lee, Y.-H. & Wallace J. M.1984a Horizontal structure of 500 mb height fluctuations with long, intermediate and short time scales. J. Atmos. Sci. 41, 961–979.Google Scholar
Blackmon M. L., Lee Y.-H., Wallace, J. M. & Hsu H.-H.1984b Time evolution of 500 mb height fluctuations with long, intermediate and short time scales as deduced from lag-correlation statistics. J. Atmos. Sci. 41, 981–991.Google Scholar
Briggs R. J.1964 Electron-Stream Interaction with Plasmas, chap. 2, 8–46. MIT Press.
Charney J. G.1946 The dynamics of long waves in a baroclinic westerly current J. Meteor, 4, 135–162.Google Scholar
Farrell B. F.1982 Pulse asymptotics of the Charney baroclinic instability problem. J. Atmos. Sci. 39, 507–517.Google Scholar
Farrell B. F.1983 Pulse asymptoties of three-dimensional baroclinic waves. J. Atmos. Sci. 40, 2202–2209.Google Scholar
Gaster M.1962 A note on the relation between temporally increasing and spatially increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google Scholar
Gaster M.1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433–441.Google Scholar
Held I. M., Panetta, R. L. & Pierrehumbert R. T.1985 Stationary external Rossby waves in vertical shear. J. Atmos. Sci. 42, 863–883.Google Scholar
Held I. M., Pierrehumbert, R. T. & Panetta R. L.1986 Dissipative destabilization of external Rossby waves. J. Atmos. Sci. 43, 388–396.Google Scholar
Hogg N. G.1976 On spatially growing baroclinic waves in the ocean. J. Fluid Mech. 78, 217–235.Google Scholar
Hoskins, B. J. & Karoly D. J.1981 The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci. 38, 1117–1318.Google Scholar
Huerre, P. & Monkewitz P. A.1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.Google Scholar
Kok, C. J. & Opsteegh J. P.1985 Possible causes of anomalies in seasonal mean circulation patterns during the 1982–83 El Niño Event. J. Atmos. Sci. 42, 677–694.Google Scholar
Landahl M. T.1972 Wave mechanics of breakdown. J. Fluid Mech. 56, 773–802.Google Scholar
Lindzen, R. S. & Rosenthal A. J.1981 A WKB asymptotic analysis of baroclinic instability. J. Atmos. Sci. 38, 619–629.Google Scholar
Mcintyre M.1972 Baroclinic instability of an idealized model of the polar night jet. Q. J. R. Met. Soc. 98, 165–173.Google Scholar
Merkine L.-O.1977 Convective and absolute instability of baroclinic eddies. Geophys. Astrophys. Fluid Dyn. 9, 129–157.Google Scholar
Merkine L.-O.1982 The stability of quasi-geostrophic fields induced by potential vorticity sources. J. Fluid Mech. 116, 315–342.Google Scholar
Michalke A.1965 On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23, 521–544.Google Scholar
Miles J.1964a A note on Charney's model of zonal-wind instability. J. Atmos. Sci. 21, 451–452.Google Scholar
Miles J.1964b Baroclinic instability of the zonal wind. Rev. Geophys. 2, 155–176.Google Scholar
Pedlosky J.1979 Geophysical Fluid Dynamics. Springer.
Pierrehumbert R. T.1984 Local and global baroclinic instability of zonally varying flow. J. Atmos. Sci. 41, 2141–2162.Google Scholar
Plumb R. A.1986 Three-dimensional propagation of transient quasigeostrophic eddies and its relationship with the eddy forcing of the time-mean flow. J. Atmos. Sci. (in press).Google Scholar
Sardeshmukh, P. D. & Hoskins B. J.1985 Vorticity balances in the tropics during the 1982–83 El Niño—Southern Oscillation event. Q. J. R. Met. Soc. 111, 261–278.Google Scholar
Thacker W. C.1976 Spatial growth of Gulf Stream meanders. Geophys. Astrophys. Fluid Dyn. 7, 271–295.Google Scholar