• Andrews, D. G., 1984: On the stability of forced nonzonal flows. Quart. J. Roy. Meteor. Soc.,110, 657–662.

  • Baines, P. G., 1976: The stability of planetary waves on a sphere. J. Fluid Mech.,73, 193–213.

  • Borges, M. D., and P. D. Sardeshmukh, 1995: Barotropic Rossby wave dynamics of zonally varying upper-level flows during northern winter. J. Atmos. Sci.,52, 3779–3796.

  • Branstator, G. W., and I. M. Held, 1995: Westward propagating normal modes in the presence of stationary background waves. J. Atmos. Sci.,52, 247–262.

  • Brunet, G., and P. H. Haynes, 1996: Low-latitude reflection of Rossby wave trains. J. Atmos. Sci.,53, 482–496.

  • Frederiksen, J. S., and P. J. Webster, 1988: Alternative theories of atmospheric teleconnections and low-frequency fluctuations. Rev. Geophys.,26, 459–494.

  • Gill, A. E., 1974: The stability of planetary waves. Geophys. Fluid Dyn.,6, 29–47.

  • Grose, W. L., and B. J. Hoskins, 1979: On the influence of orography on large-scale atmospheric flow. J. Atmos. Sci.,36, 223–234.

  • Hack, J. J., and R. Jakob, 1992: Description of a shallow water model based on the spectral transform method. NCAR Tech. Note NCAR/TN-343+STR, 39 pp. [Available from NCAR, P.O. Box 3000, Boulder, CO 80307.].

  • Horel, J. D., 1984: Complex principal component analysis: Theory and examples. J. Climate Appl. Meteor.,23, 1660–1673.

  • Huang, H. P., and W. A. Robinson, 1995: Barotropic model simulations of the North Pacific retrograde disturbances. J. Atmos. Sci.,52, 1630–1641.

  • Sardeshmukh, P. D., M. Newman, and M. D. Borges, 1997: Free barotropic Rossby wave dynamics of the wintertime low-frequency flow. J. Atmos. Sci.,54, 5–23.

  • Simmons, A. J., J. M. Wallace, and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci.,40, 1363–1392.

  • Waugh, D. W., L. M. Polvani, and R. A. Plumb, 1994: Nonlinear, barotropic response to a localized topographic forcing: Formation of a “tropical surf zone” and its effect on interhemispheric propagation. J. Atmos. Sci.,51, 1401–1416.

  • Webster, P. J., and J. R. Holton, 1982: Cross-equatorial response to middle-latitude forcing in a zonally varying basic state. J. Atmos. Sci.,39, 722–733.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 138 22 0
PDF Downloads 40 16 0

Time Variability and Simmons–Wallace–Branstator Instability in a Simple Nonlinear One-Layer Model

View More View Less
  • 1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York
  • | 2 Program in Atmospheres, Ocean and Climate, Massachusetts Institute of Technology, Cambridge, Massachusetts
Restricted access

Abstract

Using a global, one-layer shallow water model, the response of a westerly flow to a localized mountain is investigated. A steady, linear response at small mountain heights successively gives way first to a steady flow in which nonlinearities are important and then to unsteady, but periodic, flow at larger mountain heights. At first the unsteady behavior consists of a low-frequency oscillation of the entire Northern Hemisphere zonal flow. As the mountain height is increased further, however, the oscillatory behavior becomes localized in the diffluent jet exit region downstream of the mountain. The oscillation then takes the form of a relatively rapid vortex shedding event, followed by a gradual readjustment of the split jet structure in the diffluent region. Although relatively simple, the model exhibits a surprisingly high sensitivity to slight parameter changes. A linear stability analysis of the time-averaged flow is able to capture the transition from steady to time-dependent behavior, but fails to capture the transition between the two distinct regimes of time-dependent response. Moreover, the most unstable modes of the time-averaged flow are found to be stationary and fail to capture the salient features of the EOFs of the full time-dependent flow. These results therefore suggest that, even in the simplest cases, such as the one studied here, a linear analysis of the time-averaged flow can be highly inadequate in describing the full nonlinear behavior.

Corresponding author address: Dr. Lorenzo M. Polvani, Department of Applied Physics and Applied Mathematics, Columbia University, Seeley W. Mudd Bldg., Rm. 209, New York, NY 10027.

Email: polvani@columbia.edu

Abstract

Using a global, one-layer shallow water model, the response of a westerly flow to a localized mountain is investigated. A steady, linear response at small mountain heights successively gives way first to a steady flow in which nonlinearities are important and then to unsteady, but periodic, flow at larger mountain heights. At first the unsteady behavior consists of a low-frequency oscillation of the entire Northern Hemisphere zonal flow. As the mountain height is increased further, however, the oscillatory behavior becomes localized in the diffluent jet exit region downstream of the mountain. The oscillation then takes the form of a relatively rapid vortex shedding event, followed by a gradual readjustment of the split jet structure in the diffluent region. Although relatively simple, the model exhibits a surprisingly high sensitivity to slight parameter changes. A linear stability analysis of the time-averaged flow is able to capture the transition from steady to time-dependent behavior, but fails to capture the transition between the two distinct regimes of time-dependent response. Moreover, the most unstable modes of the time-averaged flow are found to be stationary and fail to capture the salient features of the EOFs of the full time-dependent flow. These results therefore suggest that, even in the simplest cases, such as the one studied here, a linear analysis of the time-averaged flow can be highly inadequate in describing the full nonlinear behavior.

Corresponding author address: Dr. Lorenzo M. Polvani, Department of Applied Physics and Applied Mathematics, Columbia University, Seeley W. Mudd Bldg., Rm. 209, New York, NY 10027.

Email: polvani@columbia.edu

Save