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The stability of unbounded two- and three-dimensional flows subject to body forces: some exact solutions

Published online by Cambridge University Press:  21 April 2006

A. D. D. Craik
A. D. D. Craik
Affiliation:
Department of Mathematical Sciences, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS Scotland
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Abstract

A formulation, previously employed to find exact Navier-Stokes solutions for planar disturbances in two- and three-dimensional flows with spatially uniform rates of strain, is here adapted to incorporate the contribution of various types of body force. In the absence of body forces, it is known that unbounded flows with constant vorticity and elliptical streamlines are unstable to certain planar disturbances, which are amplified by a Floquet mechanism. The influence of a Coriolis force upon this instability mechanism is here described in detail, as an illustration of the general formulation. The results are likely to be of geophysical interest and may also have relevance to the breakdown of closed-eddy structures in turbulence. The final section of the paper reviews other systems for which analogous exact solutions may be obtained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 198 , January 1989 , pp. 275 - 292
Copyright
1989 Cambridge University Press

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References

Babiano, A., Basdevant, C., Legras, B. & Sadourny, R., 1987 Vorticity and passive-scalar dynamics in two-dimensional turbulence. J. Fluid Mech. 183, 379–397.Google Scholar
Bayly, B. J.: 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2171.Google Scholar
Boyd, J. P.: 1983 The continuous spectrum of linear Couette flow with the beta effect. J. Atmos. Sci. 40, 2304–2308.Google Scholar
Caprino, S. & Salusti, E., 1986 Stability of oceanic eddies as vortex patches. Unpublished manuscript (Ocean Modelling 72).
Craik, A. D. D.: 1988 A class of exact solutions in viscous incompressible magnetohydrodynamics. Proc. R. Soc. Lond. A 417, 235–244.Google Scholar
Craik, A. D. D. & Criminale, W. O.1986 Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier–Stokes equations. Proc. R. Soc. Lond. A 406, 13–26.Google Scholar
Criminale, W. O.: 1985 Instabilities in a rotating fluid with constant shear. J. Geophys. Res. 90, 862–868.Google Scholar
Criminale, W. O. & Cordova, J. Q., 1986 Effects of diffusion in the asymptotics of perturbations in stratified shear flow. Phys. Fluids 29, 2054–2060.Google Scholar
Criminale, W. O. & Pinet-Plasencia, R.1985 Unpublished manuscript ‘Wave interaction with mean shear.’ [See also Univ. of Washington M. S. thesis.]
Cushman-Roisin, B.: 1986 Linear stability of large, elliptical warm-core rings. J. Phys. Oceanogr. 16, 1158–1164.Google Scholar
Farrell, B. F.: 1982 The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci. 39, 1663–1686.Google Scholar
Hartman, R. J.: 1975 Wave propagation in s stratified shear flow. J. Fluid Mech. 71, 89–104.Google Scholar
Haynes, P. H.: 1987 On the instability of sheared disturbances. J. Fluid Mech. 175, 463–478.Google Scholar
Kelvin, Lord1887 Stability of fluid motion: rectilinear motion of viscous fluid between two parallel plates. Phil. Mag. 24 (5), 188–196.Google Scholar
Knobloch, E.: 1984 On the stability of stratified plane Couette flow. Geophys. Astrophys, Fluid Dynamics 29, 105–116.Google Scholar
Knobloch, E.: 1985 The stability of non-separable barotropic and baroclinic shear flows. Astrophy. and Space Sci. 116, 149–163.Google Scholar
Lagnado, R. R., Phan-Thien, N. & Leal, L. G.1984 The stability of two-dimensional linear flows. Phys. Fluids 27, 1094–1101.Google Scholar
Landman, M. J. & Saffman, P. G., 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30, 2339–2342.Google Scholar
Love, A. E. H.: 1893 On the stability of certain vortex motions. Proc. Lond. Math. Soc. 25, 18–42.Google Scholar
McWilliams, J. C.: 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
Metcalfe, R. W., Orszag, S. A., Brachet, M. E., Menon, S. & Riley, J. J., 1987 Secondary instability of a temporary growing mixed layer. J. Fluid. Mech. 184, 207–243.Google Scholar
Phillips, O. M.: 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Pierrehumbert, R. T.: 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.Google Scholar
Ripa, P.: 1985 On the stability of Cushman-Roisin's warm eddy solution. Unpublished manuscript (Ocean Modelling, 63).
Ripa, P.: 1987 On the stability of elliptical vortex solutions of the shallow-water equations. J. Fluid Mech. 183, 343–363.Google Scholar
Shepherd, T.: 1985 On the time development of small disturbances to plane Couette flow. J. Atmos. Sci. 42, 1868–1871.Google Scholar
Tung, K. K.: 1983 Initial-value problems for Rossby waves in a shear flow with critical level. J. Fluid Mech. 133, 443–469.Google Scholar
Vladimirov, V. A. & Tarasov, V. F., 1985 Resonance instability of the flows with closed streamlines. In ‘Laminar-Turbulent Transition: Proc. IUTAM Symp. Novosibirsk 1984’ (ed. V. V. Kozlov), pp. 717–722.
Yamagata, T.: 1976 On the propagation of Rossby waves in a weak shear flow. J. Met. Soc. Japan 54, 126–128.Google Scholar