Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T21:54:59.731Z Has data issue: false hasContentIssue false
  • English
  • Français

The role of forcing in the local stability of stationary long waves. Part 1. Linear dynamics

Published online by Cambridge University Press:  28 March 2007

DANIEL HODYSS  and
TERRENCE R. NATHAN
DANIEL HODYSS
Affiliation:
Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida, USA
TERRENCE R. NATHAN
Affiliation:
Atmospheric Science Program, Department of Land, Air, and Water Resources, University of California Davis, Davis, California, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The local linear stability of forced, stationary long waves produced by topography or potential vorticity (PV) sources is examined using a quasi-geostrophic barotropic model. A multiple scale analysis yields coupled equations for the background stationary wave and low-frequency (LF) disturbance field. Forcing structures for which the LF dynamics are Hamiltonian are shown to yield conservation laws that provide necessary conditions for instability and a constraint on the LF structures that can develop. Explicit knowledge of the forcings that produce the stationary waves is shown to be crucial to predicting a unique LF field. Various topographies or external PV sources can be chosen to produce stationary waves that differ by asymptotically small amounts, yet the LF instabilities that develop can have fundamentally different structures and growth rates. If the stationary wave field is forced solely by topography, LF oscillatory modes always emerge. In contrast, if the stationary wave field is forced solely by PV, two LF structures are possible: oscillatory modes or non-oscillatory envelope modes. The development of the envelope modes within the context of a linear LF theory is novel.

An analysis of the complex WKB branch points, which yields an analytical expres-sion for the leading-order eigenfrequency, shows that the streamwise distribution of absolute instability and convective growth is central to understanding and predicting the types of LF structures that develop on the forced stationary wave. The location of the absolute instability region with respect to the stationary wave determines whether oscillatory modes or envelope modes develop. In the absence of absolute instability, eastward propagating wavetrains generated in the far field can amplify via local convective growth in the stationary wave region. If the stationary wave region is streamwise symmetric (asymmetric), the local convective growth results in a local change in wave energy that is transient (permanent).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 576 , 10 April 2007 , pp. 349 - 376
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, D. G. 1984 On the stability of forced non-zonal flows. Q. J. R. Met. Soc. 110, 657–662.Google Scholar
Arnol'd, V. I. 1965 Conditions for nonlinear instability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk. SSSR 162, 975–978. (Engl. transl. Sov. Math. 6, 773–777, 1965.)Google Scholar
Bar-Sever, Y. & Merkine, L. O. 1988 Local instabilities of weakly non-parallel large-scale flows: WKB analysis. Geophys. Astrophys. Fluid Dyn. 41, 233–286.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw–Hill.Google Scholar
Boyd, J. P. 1999 The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series. Acta Appl. Maths. 56, 1–98.CrossRefGoogle Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas, pp. 8–46. MIT Press.CrossRefGoogle Scholar
Charney, J. & Eliassen, A. 1949 A numerical method for predicting the perturbations of the middle latitude westerlies. Tellus 1, 38–54.CrossRefGoogle Scholar
Gaster, M. 1962 A note on a relation between temporally increasing and spatially increasing disturbances in hydrodynamics instability. J. Fluid Mech. 14, 222–224.CrossRefGoogle Scholar
Held, I. M., Ting, M. & Wang, H. 2002 Northern winter stationary waves: theory and modeling. J. Climate 15, 2125–2144.2.0.CO;2>CrossRefGoogle Scholar
Helfrich, K. R. & Pedlosky, J. 1995 Large-amplitude coherent anomalies in baroclinic zonal flows. J. Atmos. Sci. 52, 1615–1629.2.0.CO;2>CrossRefGoogle Scholar
Hodyss, D. & Nathan, T. R. 2004a Effects of topography and potential vorticity forcing on solitary Rossby waves in zonally varying flow. Geophys. Astrophys. Fluid Dyn. 98, 175–202.CrossRefGoogle Scholar
Hodyss, D. & Nathan, T. R. 2004b The connection between coherent structures and low-frequency wave packets in large-scale atmospheric flow. J. Atmos. Sci. 61, 2616–2626.Google Scholar
Hodyss, D. & Nathan, T. R. 2004c Long waves in streamwise varying shear flows: new mechanisms for a weakly nonlinear instability. Phys. Rev. Lett. 93, 074502.CrossRefGoogle ScholarPubMed
Hodyss, D. & Nathan, T. R. 2006 Instability of variable media to long waves with odd dispersion relations. Commun. Math. Sci. 4, 669–676.CrossRefGoogle Scholar
Holton, J. R. 2004 An Introduction to Dynamic Meteorology. Academic.Google Scholar
Hoskins, B. J., James, I. N. & White, G. H. 1983 The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci. 40, 1595–1612.Google Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities (ed. Goldréche, C. & Manneville, P.), pp. 81–294. Cambridge University Press.CrossRefGoogle Scholar
Kamenkovich, I. V. & Pedlosky, J. 1994 Instability of baroclinic currents which are locally non-zonal. J. Phys. Oceanogr. 51, 2418–2433.Google Scholar
Kushnir, Y. & Wallace, J. 1989 Low-frequency variability in the northern hemisphere winter: geographical distribution, structure and time-scale dependence. J. Atmos. Sci. 46, 3122–3142.2.0.CO;2>CrossRefGoogle Scholar
Le, D izes, S., Huerre, P., Chomaz, J. M. & Monkewitz, P. A. 1996 Linear global modes in spatially developing media. Phil. Trans. R. Soc. Lond. A 354, 169–212.Google Scholar
Li, L. & Nathan, T. R. 1997 Effects of low-frequency tropical forcing on intraseasonal tropical–extratropical interactions. J. Atmos. Sci. 54, 332–346.2.0.CO;2>CrossRefGoogle Scholar
McIntyre, M. E. & Shepherd, T. G. 1987 An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems. J. Fluid. Mech. 181, 527–565.Google Scholar
Magnusdottir, G. & Haynes, P. H. 1999 Reflection of planetary waves in three-dimensional tropospheric flows. J. Atmos. Sci. 56, 652–670.2.0.CO;2>CrossRefGoogle Scholar
Merkine, L.-O. 1977 Convective and absolute instabilities of baroclinic eddies. Geophys. Astrophys. Fluid Dyn. 9, 129–157.CrossRefGoogle Scholar
Merkine, L.-O. 1982 The stability of quasigeostrophic fields induced by potential vorticity sources. J. Fluid Mech. 116, 315–342.CrossRefGoogle Scholar
Nathan, T. R. 1997 Nonlinear spatial baroclinic instability in slowly varying zonal flow. Dyn. Atmos. Oceans 27, 81–90.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.Google Scholar
Pierrehumbert, R. T. 1983 Bounds on the growth of perturbations to non-parallel steady flow on the barotropic beta plane. J. Atmos. Sci. 40, 1207–1217.Google Scholar
Pierrehumbert, R. T. 1984 Local and global baroclinic instability of zonally varying flow. J. Atmos. Sci. 41, 2141–2162.Google Scholar
Pierrehumbert, R. T. 1986 Spatially amplifying modes of the Charney baroclinic-instability problem. J. Fluid Mech. 170, 293–317.CrossRefGoogle Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Shepherd, T. G. 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287–338.CrossRefGoogle Scholar
Simmons, A. J., Wallace, J. M. & Branstator, G. W. 1983 Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci. 40, 1363–1392.2.0.CO;2>CrossRefGoogle Scholar
Swanson, K. L. 2002 Dynamical aspects of extratropical troposphere low-frequency variability. J. Climate 15, 2145–2162.2.0.CO;2>CrossRefGoogle Scholar