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Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 333--351 | DOI:10.5890/DNC.2015.09.009

S.M. Churilov

Institute of Solar–Terrestrial Physics SB RAS, 126a Lermontov Street, Irkutsk 664033, Russia

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Abstract

Weakly stratified flows of the class under study have a wide 3D spectrum of the most unstable waves with very close growth rates and phase velocities so that their individual critical layers merge into a common one. The analysis of evolution equations for those waves has shown that throughout a weakly nonlinear stage of development their amplitudes grow explosively. During the first (three-wave) phase, the most rapidly growing are low-frequency waves whereas at the next phase, when numerous and diverse higher-order wave interactions come into play, the growth of highfrequency waves is accelerated and they overtake low-frequency waves. The results obtained are illustrated by numerical calculations for some ensembles of waves.

Acknowledgments

The work was supported in part by RFBR Grants No. 10-05-00094 and No. 14-05-00080.

References

1. [1]	Maslowe, S.A. (1986), Critical layers in shear flows, Annual Reviews in Fluid Mechanics, 18, 405-432.

2. [2]	Churilov, S.M. and Shukhman, I.G. (1996), Critical layer and nonlinear evolution of disturbances in weakly supercritical shear layer, Izvestiya, Atmospheric and Oceanic Physics, 31, 534-546.

3. [3]	Timofeev, A.V. (1971), Oscillations of inhomogeneous flows of plasma and liquids, Sov. Phys. Uspekhi, 13, 632-646.

4. [4]	Andronov, A.A. and Fabrikant, A.L. (1979) Landau damping, wind waves and whistle. In: Nonlinear Waves (edited by A.V. Gaponov-Grekhov), 68-104, Nauka, Moscow (in Russian).

5. [5]	Fabrikant, A. (2002), Plasma-hydrodynamic analogy for waves and vortices in shear flows. Sound-Flow Interactions. Lecture Notes in Physics, Vol. 586 (edited by Aurégan Y. et al.), 192-209, Springer-Verlag, New York.

6. [6]	Wu, X. and Stewart, P.A. (1996), Interaction of phase-locked modes: a new mechanism for the rapid growth of threedimensional disturbances, Journal of Fluid Mechanics, 316, 335-372.

7. [7]	Holmboe, J. (1962), On the behaviour of symmetric waves in stratified shear layers. Geofys. Publik., 24, 67-112.

8. [8]	Garaud, P. (2001), Latitudinal shear instability in the solar tachocline, Monthly Notices of the Royal Astronomical Society, 324, 68-76.

9. [9]	Gavrilov, N.M., Fukao, S., Hashiguchi, H., Kita, K., Sato, K., Tomikawa, Y. and Fujiwara, M. (2006), Combined MU radar and ozonesonde measurements of turbulence and ozone fluxes in the tropo-stratosphere over Shigaraki, Japan, Geophysical Research Letters, 33, L09803.

10. [10]	Yoshida, S., Ohtani, M., Nishida, S. and, Linden, P.F. (1998),Mixing processes in a highly stratified river. In: Physical Processes in Lakes and Oceans (edited by J. Imberger), 389-400. AGU, Washington, DC.

11. [11]	Tedford, E. W., Carpenter, J. R., Pawlowicz, R., Pieters, R. and Lawrence G. A. (2009), Observation and analysis of shear instability in the Fraser River estuary, Journal of Geophysical Research, 114, C11006.

12. [12]	Carpenter, J.R, Tedford, E. W., Rahmani, M. and Lawrence, G. A. (2010), Holmboe wave fields in simulation and experiment, Journal of Fluid Mechanics, 648, 205-223.

13. [13]	Churilov, S.M. (2008), Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points. Part 2. Continuous density variation, Journal of Fluid Mechanics, 617, 301-326.

14. [14]	Churilov, S.M. (2010), Three-dimensional instability of shear flows with inflection-free velocity profiles in stratified media with a high Prandtl number, Izvestiya, Atmospheric and Oceanic Physics, 46, 159-168.

15. [15]	Squire, H.B. (1933), On the stability of three-dimensional disturbances of viscous flow between parallel walls, Proceedings of the Royal Society London A, 142, 621-628.

16. [16]	Yih, C.-S. (1955), Stability of two-dimensional parallel flows for three-dimensional disturbances, Quarterly Applied Mathematics, 12, 434-435.

17. [17]	Smyth, W. D. and Peltier, W. R. (1990), Three-dimensional primary instabilities of a stratified, dissipative, parallel flow, Geophysical and Astrophysical Fluid Dynamics, 52, 249-261.

18. [18]	Drazin, P.G. and Reid, W.H. (2004), Hydrodynamic Stability, 2nd edition, Cambridge University Press, Cambridge.

19. [19]	Churilov, S.M. (2011), Resonant three-wave interaction of Holmboe waves in a sharply stratified shear flow with an inflection-free velocity profile. Physics of Fluids, 23, 114101.

20. [20]	Churilov, S.M. (2009), Nonlinear stage of instability development in a stratified shear flow with an inflection-free velocity profile. Physics of Fluids, 21, 074101.

21. [21]	Craik, A.D.D. (1985),Wave Interactions and Fluid Flows, Cambridge University Press, Cambridge.

22. [22]	Bridgman, P.W. (1932), Dimensional Analysis, Yale University Press, New Haven.

23. [23]	Churilov, S.M. and Shukhman I.G. (1994), Nonlinear spatial evolution of helical disturbances to an axial jet, Journal of Fluid Mechanics, 281, 371-402.

24. [24]	Landau, L.D. and Lifshitz, E.M. (1987), Fluid Dynamics, Pergamon Press, Oxford.

25. [25]	Moiseev, S.S., Sagdeev, R.Z., Tur, A.V. and Yanovsky, V.V. (1979), Effect of vortices on acoustic turbulence spectrum. In: Nonlinear Waves (edited by A.V. Gaponov-Grekhov), 105-115, Nauka: Moscow (in Russian).