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Spatially growing waves, intermittency, and convective chaos in an open- flow system. (English) Zbl 0611.76061
We study a model open-flow system: the time-dependent generalized Ginzburg-Landau equation under conditions when it is convectively (i.e. spatially) unstable. In the presence of low-level external noise, this system exhibits a selective spatial amplification of noise resulting in spatially growing waves.
Three distinct spatial regions are found:
1) A linear region in which spatially growing waves are formed;
2) a transition region characterized by the transition from the spatially growing wave frequencies of the linear region to the turbulent frequencies of the fully developed region; and
3) a fully developed region in which the nonlinear dynamics dominates.
In the transition region, the microscopic noise plays an important role in the macroscopic dynamics of the system. In particular, the external noise initiates the transition to turbulence and is responsible for the intermittent turbulent behavior observed in the transition region. Power spectra and correlation functions are calculated and analytical expressions are derived for these quantities in the linear region. Similarities between this system and fluid systems are discussed. Also, a few analytical results for coupled-map lattices are derived.

76E15 Absolute and convective instability and stability in hydrodynamic stability
76F99 Turbulence
76M99 Basic methods in fluid mechanics
35Q30 Navier-Stokes equations
Full Text: DOI
[1] Reynolds, O., Phil. trans. R. soc., 174, 935, (1883)
[2] Lorenz, E.N., J. atmos. sci., 20, 130, (1963)
[3] Newell, A.C.; Whitehead, J.A., J. fluid mech., 38, 279, (1969)
[4] Kogelman, S.; DiPrima, R.C., Phys. fluids, 13, 1, (1970)
[5] Stewartson, K.; Stuart, J.T., J. fluid. mech., 48, 529, (1971)
[6] Blennerhasset, P.J., Phil. trans. roy. soc. lond., 298A, 451, (1980)
[7] Sreenivasan, K.R., (), For some discussion on “closed flow“ and “open flow” fluid systems see · Zbl 0968.76576
[8] Deissler, R.J., J. stat. phys., 40, 371, (1985)
[9] Kuramoto, Y., Prog. theor. phys., Suppl. prog. theor. phys., 64, 346, (1978)
[10] Moon, H.T.; Huerre, P.; Redekopp, L.G., Phys. rev. lett., Physica, 7D, 135, (1983)
[11] Nozaki, K.; Bekki, N., Phys. rev. lett., 51, 2171, (1983)
[12] Keefe, L.R., Ph.D. thesis, Studies in app. math., 78, 91, (1985)
[13] Kuramoto, Y.; Koga, S., Phys. lett., 92A, 1, (1982)
[14] Farmer, J.D.; Farmer, J.D., Deterministic noise amplifiers, Los alamos report LA-UR-83-1450, Los alamos report, (1984), J.D. Farmer and R.J. Deissler, in preparation · Zbl 0744.58040
[15] R.J. Deissler and K. Kaneko, “Velocity-Dependent Liapunov Exponents as a Measure of Chaos for Open Flow Systems”, Los Alamos Report LA-UR-85-3249, accepted for publication in Phys. Lett. A;
[16] Deissler, R.J.; Kaneko, K., ()
[17] Deissler, R.J., Physics letters, 100A, 451, (1984)
[18] Briggs, R.J., Electron-stream interaction with plasmas, (1964), M.I.T. Press Cambridge, MA
[19] Bers, A., ()
[20] Dee, G.; Langer, J.S.; Ben-Jacob, E.; Brand, H.; Dee, G.; Kramer, L.; Langer, J.S., Phys. rev. lett., Physica, 14D, 348, (1985), Also see
[21] Lapidus, L.; Pinder, G.F., Numerical solution of partial differential equations in science and engineering, (1982), J. Wiley New York · Zbl 0584.65056
[22] Stuart, J.T.; DiPrima, R.C.; Newell, A.C., (), Lect. appl. math., 15, 157, (1974), also see
[23] Schlichting, H., Boundary-layer theory, (1968), McGraw-Hill New York
[24] Dryden, H.L., ()
[25] White, F.M., Viscous fluid flow, (1974), McGraw-Hill New York · Zbl 0356.76003
[26] Goldstein, M.E., J. fluid mech., 154, 509, (1985)
[27] Orszag, S.A.; Patera, A.T., J. fluid mech., 128, 347, (1983)
[28] S.A. Orszag, R.B. Pelz, and B.J. Bayly, “Secondary Instabilities, Coherent Structures, and Turbulence”, AIAA-85-1486-CP, AIAA 7th Computational Fluid Dynamics Conference.
[29] Klebanoff, P.S.; Tidstrom, K.D.; Sargent, L.M., J. fluid mech., 12, 1, (1962)
[30] Nishioka, M.; Asai, M.; Iida, S., ()
[31] Huerre, P.; Monkewitz, P.A., J. fluid mech., 159, 151, (1985)
[32] Smith, A.M.O.; Stuart, J.T., (), 7, 565, (1960), Also see
[33] Rubin, Y.; Wygnanski, I.; Haritonidis, J.H., ()
[34] Teitgen, R., ()
[35] Wygnanski, I.J.; Champagne, F.H., J. fluid mech., 59, 281, (1973)
[36] Crutchfield, J., Noisy chaos, Ph.D. thesis, (1983)
[37] Kaneko, K., (), 72, 480, (1984), to be published by
[38] Kaneko, K., Prog. theor. phys., 74, 1033, (1985)
[39] J. Crutchfield and K. Kaneko, “Statistical Mechanics of Lattice Dynamical Systems”, in preparation.
[40] Kapral, R., Phys. rev. A, 31, 3868, (1985)
[41] Aizawa, Y., Prog. theor. phys., 72, 662, (1984)
[42] Yamada, T.; Fujisaka, H., Prog. theor. phys., 73, (1985)
[43] J. Keeler and D. Farmer, in preparation.
[44] Kaneko, K., Phys. lett. A, 111, 321, (1985)
[45] R.J. Deissler, “Turbulent Bursts, Spots, and Slugs in a Generalized Ginzburg-Landau Equation”, Los Alamos Report LA-UR-86-2067, submitted to Phys. Lett. A.
[46] Nita, T., J. met. soc. Japan, 40, 13, (1962)
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