Abstract
We described various structures formed during the stabilization of instability in wave and vortex flows of an ideal liquid. The problem of wave structures in an incompressible fluid flow stratified by density and velocity is considered in detail. Instability stabilization occurs as a result of the interaction of an unstable wave with waves forming a resonant triplet with it. In this case, structures of a regular and stochastic nature arise. We analyzed and described the scenario of the transition of the system to the stochastic mode. The formulation corresponds to atmospheric currents under wind shear, but the results can be used in other problems of the theory of nonlinear waves and vortices. In this paper we showed that structures of a similar nature also arise in vortex flows, both ideal and viscous liquids.
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REFERENCES
P. G. Drazin, Introduction to Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, 2002).
G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation (Cambridge Univ. Press, Cambridge, 2017).
G. M. Zaslavskii and R. Z. Sagdeev, Introduction to Nonlinear Physics (Nauka, Moscow, 1988) [in Russian].
E. B. Gledzer, F. V. Dolzhanskii, and A. M. Obukhov, Hydrodynamic-Type Systems and Their Application (Nauka, Moscow, 1987) [in Russian].
J. Jimenez, “Coherent structures in wall-bounded turbulence,” J. Fluid Mech. 842, 1–100 (2018).
P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, 2004).
R. F. Blackwelder, “Coherent structures associated with turbulent transport,” in Proc. 2nd Int. Sump. on Transport Phenomena in Turbulent Flows (Tokyo, 1987), pp. 1–20.
M. Fantini, “Instability of finite-amplitude lower-neutral Eady waves,” Q. J. R. Meteorol. Soc. 132, 2157–2169 (2006).
M. I. Rabinovich and A. L. Fabrikant, “Nonlinear waves in nonequilibrium media,” Radiophys. Quantum Electron. 19 (5), 508–543 (1976).
M. I. Rabinovich, A. L. Fabrikant, and L. Sh. Tsimring, “Finite-dimensional spatial disorder,” Sov. Phys. Usp. 35 (8), 629–649 (1992).
F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, “Stability and vortex structures of quasi-two-dimensional shear flows,” Sov. Phys. Usp. 33 (7), 495–520 (1990).
A. Z. Loesch, “Resonant interactions between unstable and neutral baroclinic waves: Part I,” J. Atmos. Sci. 31, 1177–1201 (1974).
J. Pedlosky, “The amplitude of baroclinic wave triads and mesoscale motion in the ocean,” J. Phys. Oceanogr. 5, 608–614 (1975).
C.-C. P. Caulfield, “Multiple linear instability of layered stratified shear flow,” J. Fluid Mech. 258, 255–285 (1994).
R. Carpenter, E. W. Tetford, E. Heretz, and G. A. Lawrence, “Instability in stratified shear flow: Review of a physical interpretation based on interacting waves,” Appl. Mech. Rev. 64 (6), 060801 (2013).
I. A. Sazonov and I. G. Yakushkin, “Evolution of disturbances in a three-layer model of the atmosphere with shear instability,” Izv., Atmos. Ocean. Phys. 35 (4), 427–434. (1999).
A. A. Guha and G. Lawrence, “A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities,” J. Fluid Mech. 755, 336–364 (2014).
E. Heifetz, J. Mak, J. Nycander, and O. M. Umurhan, “Interacting vorticity waves as an instability mechanism for magnetohydrodynamic shear instabilities,” J. Fluid Mech. 767, 199–225 (2015).
N. N. Romanova and S. Y. Annenkov, “Three-wave resonant interactions in unstable media,” J. Fluid Mech. 539, 57–91 (2005).
S. V. Kostrykin, N. N. Romanova, and I. G. Yakushkin, “On stochastic stabilization of the Kelvin–Helmholtz instability by three-wave resonant interaction,” Chaos 21, 043117 (2011). https://doi.org/10.1063/1.3656800
N. N. Romanova, O. G. Chkhetiani, and I. G. Yakushkin, “Influence of nonlinear interactions on the development of instability in hydrodynamic wave systems,” J. Exp. Theor. Phys. 122 (5) 902–914 (2016).
N. N. Romanova, “Hamiltonian approach to the derivation of evolution equations for wave trains in weakly unstable media,” Nonlinear Proc. Geophys. 5, 241–253 (1998).
N. N. Romanova and I. G. Yakushkin, “Hamiltonian description of motion in a perfect stratified fluid,” Dokl. Phys. 46 (10), 742–746 (2001).
F. V. Dolzhansky, “On the mechanical prototypes of fundamental hydrodynamic invariants and slow manifolds,” Phys.-Usp. 48 (12), 1205–1234 (2005).
F. V. Dolzhanskii, Basics of Geophysical Hydrodynamics (Fizmatlit, Moscow, 2011) [in Russian].
G. M. Zaslavskii, Stochasticity of Dynamical Systems (Nauka, Moscow, 1984) [in Russian].
G. M. Zaslavskii, Physics of Chaos in Hamiltonian Systems (Inst. komp’yuternykh issledovanii, Moscow–Izhevsk, 2004) [in Russian].
M. I. Rabinovich and D. I. Trubetskov, Introduction to the Theory of Oscillations and Waves (Regulyarnaya i khaoticheskaya dinamika, 2000) [in Russian].
E. Van Groesen, “Deformation of coherent structures,” Rep. Prog. Phys. 59, 511–600 (1996).
F. V. Dolzhanskii and V. M. Ponomarev, “Simplest slow manifolds of barotropic and baroclinic motions of a rotating fluid,” Izv., Atmos. Ocean. Phys. 38 (3), 277–290 (2002).
S. N. Gurbatov, A. I. Saichev, and I. G. Yakushkin, “Nonlinear waves and one-dimensional turbulence in nondispersive media,” Sov. Phys. Usp. 26 (10), 857–876 (1983).
ACKNOWLEDGMENTS
The author is grateful to N.N. Romanova and O.G. Chkhetiani for their interest in this paper and our constructive discussions.
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Yakushkin, I.G. On Coherent and Stochastic Structures in Hydrodynamic Flows with a Velocity Shift. Izv. Atmos. Ocean. Phys. 58, 7–17 (2022). https://doi.org/10.1134/S0001433821060104
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DOI: https://doi.org/10.1134/S0001433821060104