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Initial-value problems for Rossby waves in a shear flow with critical level

Published online by Cambridge University Press:  20 April 2006

Ka Kit Tung
Ka Kit Tung
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
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Abstract

The time-dependent evolution of sheared Rossby waves starting from an initial disturbance is studied for the simple case in which the shear is uniform. The uniform-shear assumption allows explicit solutions to be obtained which are useful in addressing the issue of the long-time asymptotic approach to normal modes and in assessing the relative importance of viscosity, nonlinearity and time-dependence in the evolution of Rossby waves in the presence of critical layers.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 133 , August 1983, pp. 443 - 469
Copyright
© 1983 Cambridge University Press

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References

Achbson, D. J. 1976 On over-reflection J. Fluid Mech. 77, 433–472.Google Scholar
Béland, M. 1976 Numerical study of the nonlinear Rossby wave critical level development in a barotropic zonal flow J. Atoms. Sci. 33, 2066–2078.Google Scholar
Benney, D. J. & Bergeron, R. G. 1969 A new class of nonlinear waves in parallel flows Stud. Appl. Maths 48, 181–204.Google Scholar
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow J. Fluid Mech. 27, 513–539.Google Scholar
Bretherton, F. P. 1966 The propagation of groups of internal gravity waves in a shear flow Q. J. R. Met. Soc. 92, 466–480.Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1969 Wave trains in inhomogeneous moving media Proc. R. Soc. Lond. A302, 529–554.Google Scholar
Brown, S. N. & Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Part II Geophys. Astrophys. Fluid Dyn. 10, 1–24.Google Scholar
Case, K. M. 1960 Stability of inviscid plane Couette flow Phys. Fluids 3, 143–148.Google Scholar
Dickinson, R. E. 1969 Planetary Rossby waves propagating vertically through weak westerly wind wave guides J. Atmos. Sci. 25, 984–1002.Google Scholar
Dickinson, R. E. 1970 Development of a Rossby wave critical level J. Atmos. Sci. 27, 627–633.Google Scholar
Farrell, B. F. 1981 Baroclinic instability as an initial value problem. Ph.D. thesis, Harvard University.
Farrell, B. F. 1982 The initial growth of disturbances in a baroclinic flow J. Atmos. Sci. 39, 1663–1686.Google Scholar
Fjørtoft, R. 1950 Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex Geofis. Publ. 17, no. 5.Google Scholar
Gradshteyn, I. S. & Ryzhick, I. M. 1975 Table of Integrals, Series and Products. Academic.
Haberman, R. 1972 Critical layers in parallel flows Stud. Appl. Maths 51, 139–160.Google Scholar
Hartman, R. J. 1973 The dynamics of infinite shear flows. Ph.D. thesis, University of California, Santa Barbara.
Hartman, R. J. 1975 Wave propagation in a stratified shear flow J. Fluid Mech. 71, 89–104.Google Scholar
Lin, C. C. 1945 On the stability of two-dimensional parallel flows. Q. Appl. Maths 3, 117–142, 218–234.Google Scholar
Lin, C. C. 1957 On uniformly valid asymptotic solutions of the Orr-Sommerfeld equation. In Proc. 9th Intl Congress Appl. Mech., Brussels, vol. 1, pp. 136–148.Google Scholar
Lindzen, R. S. & Tung, K. K. 1978 Wave overreflection and shear instability J. Atmos. Sci. 35, 1626–1632.Google Scholar
Lu, P.-S. & Zeng, Q.-C. 1981 On the evolution process of disturbances in the barotropic atmosphere Sci. Atmos. Sinica 5, 1–8.Google Scholar
Marcus, P. S. & Press, W. H. 1977 On Green's functions for small disturbances of plane Couette flow J. Fluid Mech. 79, 525–534.Google Scholar
Orr, W. M. 1907 The stability or instability of the steady motions of a liquid Proc. R. Irish Acad. A27, 69–138.Google Scholar
Pedlosky, J. 1964 An initial value problem in the theory of baroclinic instability Tellus 16, 12–17.Google Scholar
Phillips, O. M. 1966 Dynamics of the Upper Ocean, 1st edn. Cambridge University Press.
Rayleigh, J. W. S. 1880 On the stability, or instability, of certain fluid motions Proc. Lond. Math. Soc. 9, 57–70.Google Scholar
Rosen, G. 1971 General solution for perturbed plane Couette flow Phy. Fluids 14, 2767–2769.Google Scholar
Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave Geophys. Astrophys. Fluid Dyn. 9, 185–200.Google Scholar
Tung, K. K. 1979 A theory of stationary long waves. Part III: Quasi-normal modes in a singular waveguide. Mon. Weather Rev. 107, 751–774.Google Scholar
Tung, K. K. 1981 Barotropic instability of zonal flows J. Atmos. Sci., 38, 308–321.Google Scholar
Warn, T. & Warn, H. 1976 Development of a Rossby wave critical level J. Atmos. Sci. 33, 2021–2024.Google Scholar
Warn, T. & Warn, H. 1978 The evolution of a nonlinear critical level Stud. Appl. Maths 59, 37–71.Google Scholar
Yamagata, T. 1976a On the propagation of Rossby waves in a weak shear flow. J. Met. Soc. Japan 54, 126–128.Google Scholar
Yamagata, T. 1976 On trajectories of Rossby wave-packets released in a lateral shear flow J. Oceanogr. Soc. Japan 32, 162–168.Google Scholar