Spectral Properties of One-Dimensional Diffusive Systems Subject to Stochastic Forcing

H-L. Liu High Altitude Observatory, National Center for Atmospheric Research,* Boulder, Colorado

Search for other papers by H-L. Liu in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The vertical wavenumber and frequency spectra of horizontal wind and temperature in stochastically driven systems with diffusion, either due to uniform background eddy and molecular transport, or due to adjustment processes associated with shear or convective instability, are studied. Because of the dominating role of vertical transport in a stratified fluid, one-dimensional Langevin-type equations could be ascribed to such systems in the vertical direction. The linear equation with uniform diffusion is solved explicitly, and the spectra follow power-law distributions if the stochastic force is Gaussian. The nonlinear equations with gradient (either shear or lapse rate) dependent diffusion coefficients are shown to support scale invariance, and the power-law indices of the spectra are determined from dynamic renormalization group (DRG) analysis under rather general conditions. The exact power-law indices vary with the spectrum of the stochastic force and the nonlinearity of the systems. If the wavenumber spectrum of the force is moderately red (between k0 and k−2), the spectral indices of horizontal wind and temperature and the range of their variability are in general agreement with those inferred from wind and temperature measurements. The indices in both linear and nonlinear cases are confirmed by numerical simulations. This theory may suggest an alternative explanation to the universal vertical wavenumber and frequency spectra and their variability. By relating the universal spectra to systems characterized by stochastic forcing and background diffusion or diffusive adjustment due to shear or convective instability, which are ubiquitous in a stratified fluid, the difficulty to associate the time- and location-independent spectral features directly with the highly time- and location-dependent gravity waves or wave-breaking events is avoided. If such systems are suggestive of the real atmosphere, there is a need to be cautious in making assumptions regarding gravity waves solely based on the universal spectra when analyzing and interpreting wind and temperature observations.

* The National Center for Atmospheric Research is sponsored by the National Science Foundation

Corresponding author address: Han-Li Liu, High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000. Email: liuh@ucar.edu

Abstract

The vertical wavenumber and frequency spectra of horizontal wind and temperature in stochastically driven systems with diffusion, either due to uniform background eddy and molecular transport, or due to adjustment processes associated with shear or convective instability, are studied. Because of the dominating role of vertical transport in a stratified fluid, one-dimensional Langevin-type equations could be ascribed to such systems in the vertical direction. The linear equation with uniform diffusion is solved explicitly, and the spectra follow power-law distributions if the stochastic force is Gaussian. The nonlinear equations with gradient (either shear or lapse rate) dependent diffusion coefficients are shown to support scale invariance, and the power-law indices of the spectra are determined from dynamic renormalization group (DRG) analysis under rather general conditions. The exact power-law indices vary with the spectrum of the stochastic force and the nonlinearity of the systems. If the wavenumber spectrum of the force is moderately red (between k0 and k−2), the spectral indices of horizontal wind and temperature and the range of their variability are in general agreement with those inferred from wind and temperature measurements. The indices in both linear and nonlinear cases are confirmed by numerical simulations. This theory may suggest an alternative explanation to the universal vertical wavenumber and frequency spectra and their variability. By relating the universal spectra to systems characterized by stochastic forcing and background diffusion or diffusive adjustment due to shear or convective instability, which are ubiquitous in a stratified fluid, the difficulty to associate the time- and location-independent spectral features directly with the highly time- and location-dependent gravity waves or wave-breaking events is avoided. If such systems are suggestive of the real atmosphere, there is a need to be cautious in making assumptions regarding gravity waves solely based on the universal spectra when analyzing and interpreting wind and temperature observations.

* The National Center for Atmospheric Research is sponsored by the National Science Foundation

Corresponding author address: Han-Li Liu, High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000. Email: liuh@ucar.edu

Save
  • Bak, P., C. Tang, and K. Wiesenfeld, 1987: Self-organized criticality—An explanation of 1/f noise. Phys. Rev. Lett., 59 , 381384.

  • Balsley, B. B., and D. A. Carter, 1982: The spectrum of atmospheric velocity fluctuations at 8 km and 86 km. Geophys. Res. Lett., 9 , 465468.

    • Search Google Scholar
    • Export Citation
  • Broutman, D., C. Macaskill, M. E. McIntyre, and J. W. Rottman, 1997: On Doppler-spreading models of internal waves. Geophys. Res. Lett., 24 , 28132816.

    • Search Google Scholar
    • Export Citation
  • Collins, R. L., A. Nomura, and C. S. Gardner, 1994: Gravity waves in the upper mesosphere over Antarctica: Lidar observations at the South Pole and syowa. J. Geophys. Res., 99 , 54755485.

    • Search Google Scholar
    • Export Citation
  • Dewan, E. M., and R. E. Good, 1986: Saturation and the “universal” spectrum for vertical profiles of horizontal scalar winds in the atmosphere. J. Geophys. Res., 91 , 27422748.

    • Search Google Scholar
    • Export Citation
  • Dewan, E. M., N. Groosbard, A. F. Quesada, and R. E. Good, 1984: Spectral analysis of 10 m resolution scalar velocity profiles in the stratosphere. Geophys. Res. Lett., 11 , 8083.

    • Search Google Scholar
    • Export Citation
  • Diamond, P. H., and T. S. Hahm, 1995: On the dynamics of turbulent transport near margmal stability. Phys. Plasmas, 2 , 36403649.

  • Eckermann, S. D., 1997: Influence of wave propagation on the Doppler-spreading of atmospheric gravity waves. J. Atmos. Sci., 54 , 25542573.

    • Search Google Scholar
    • Export Citation
  • Eckermann, S. D., 1999: Isentropic advection by gravity waves: Quasi-universal m−3 vertical wavenumber spectra near the onset of instability. Geophys. Res. Lett., 26 , 201204.

    • Search Google Scholar
    • Export Citation
  • Endlich, R., R. C. Singleton, and J. W. Kaufman, 1969: Spectral analysis of detailed vertical wind speed profiles. J. Atmos. Sci., 26 , 10301041.

    • Search Google Scholar
    • Export Citation
  • Forster, D., D. R. Nelson, and M. J. Stephen, 1977: Large distance and long-time properties of a randomly stirred fluid. Phys. Rev. A, 16 , 732749.

    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., and W. Lu, 1993: Spectral estimates of gravity wave energy and momentum fluxes, II: Parameterization of wave forcing and variability. J. Atmos. Sci., 50 , 36953713.

    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., and M. J. Alexander, 2003: Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys., 41 .1003, doi:10.1029/2001RG000106.

    • Search Google Scholar
    • Export Citation
  • Gage, K. S., and G. D. Nastrom, 1985: On the spectrum of atmospheric velocity fluctuations seen by MST/ST radar and their interpretation. Radio Sci., 20 , 13391348.

    • Search Google Scholar
    • Export Citation
  • Gardner, C. S., 1994: Diffusive filtering theory of gravity-wave spectra in the atmosphere. J. Geophys. Res., 99 , 2060120622.

  • Hertzog, A., and F. Vial, 2001: A study of the dynamics of the equatorial lower stratosphere by use of ultra-long-duration balloons. 2. Gravity waves. J. Geophys. Res., 106 , 2274522761.

    • Search Google Scholar
    • Export Citation
  • Hines, C. O., 1991a: The saturation of gravity waves in the middle atmosphere. Part I: Critique of linear-instability theory. J. Atmos. Sci., 48 , 13481359.

    • Search Google Scholar
    • Export Citation
  • Hines, C. O., 1991b: The saturation of gravity waves in the middle atmosphere. Part II: Development of Doppler-spread theory. J. Atmos. Sci., 48 , 13601379.

    • Search Google Scholar
    • Export Citation
  • Hines, C. O., 1997a: Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere, 1, basic formulation. J. Atmos. Solar Terr. Phys., 59 , 371386.

    • Search Google Scholar
    • Export Citation
  • Hines, C. O., 1997b: Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere, 2, broad and quasi monochromatic spectra, and implementation. J. Atmos. Solar Terr. Phys., 59 , 387400.

    • Search Google Scholar
    • Export Citation
  • Hines, C. O., 2001: Theory of the Eulerian tail in the spectra of atmospheric and oceanic internal gravity waves. J. Fluid Mech., 448 , 289313.

    • Search Google Scholar
    • Export Citation
  • Holloway, G., 1981: Theoretical approaches to interactions among internal waves, turbulence and finestructure. Nonlinear Properties of Internal Waves, B. J. West, Ed., American Institute of Physics, 47–77.

    • Search Google Scholar
    • Export Citation
  • Holloway, G., 1983: A conjecture relating oceanic internal waves and small-scale processes. Atmos.–Ocean, 21 , 107122.

  • Hostetler, C. A., and C. S. Gardner, 1994: Observations of horizontal and vertical wave number spectra of gravity wave motions in the stratosphere and mesosphere over the mid-Pacific. J. Geophys. Res., 99 , 12831302.

    • Search Google Scholar
    • Export Citation
  • Hwa, T., and M. Kadar, 1992: Avalanches, hydrodynamics, and discharge events in models of sandpiles. Phys. Rev. A, 45 , 70027023.

  • Kloeden, P. E., and E. Platen, 1999: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, 636 pp.

  • Kraichnan, R. H., 1987: An interpretation of the Yakhot–Orszag turbulence theory. Phys. Fluids, 30 , 24002405.

  • Liu, H-L., 2000: Temperature changes due to gravity wave saturation. J. Geophys. Res., 105 , 1232912336.

  • Liu, H-L., P. Charbonneau, A. Pouquet, T. J. Bogdan, and S. W. McIntosh, 2002: Continuum analysis of an avalanche model for solar flares. Phys. Rev. E, 66 .doi:10.1103/PhysRevE.66.056111.

    • Search Google Scholar
    • Export Citation
  • Lumley, J. L., 1964: The spectrum of nearly inertial turbulence in a stably stratified fluid. J. Atmos. Sci., 21 , 99102.

  • Ma, S. K., and G. F. Mazenko, 1975: Critical dynamics of ferromagnets in 6 − ϵ dimensions: General discussion and detailed calculation. Phys. Rev. B, 11 , 40774100.

    • Search Google Scholar
    • Export Citation
  • Morton, J. B., and S. Corrsin, 1970: Consolidated expansions for estimating the response of a randomly driven nonlinear oscillator. J. Stat. Phys., 2 , 153194.

    • Search Google Scholar
    • Export Citation
  • Nakamura, T., T. Tsuda, H. Miyagawa, Y. Matsushita, H. Fukunishi, Y. Takahashi, and Y. Yamada, 1998: Propagation directions of gravity wave patterns observed in OH CCD images during the SEEK campaign. Geophys. Res. Lett., 25 , 17931796.

    • Search Google Scholar
    • Export Citation
  • Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1996: Numerical Recipes in Fortran 77: The Art of Scientific Computing. 2d ed. Cambridge University Press, 919 pp.

    • Search Google Scholar
    • Export Citation
  • Sica, R. J., and A. T. Russell, 1999: How many waves are in the gravity wave spectrum? Geophys. Res. Lett., 26 , 36173620.

  • Smith, L. M., and S. L. Woodruff, 1998: Renormalization-group analysis of turbulence. Annu. Rev. Fluid Mech., 30 , 275310.

  • Smith, S. A., D. C. Fritts, and T. E. VanZandt, 1987: Evidence for a saturated spectrum of atmospheric gravity waves. J. Atmos. Sci., 44 , 14041410.

    • Search Google Scholar
    • Export Citation
  • Staquet, C., and J. Sommeria, 2002: Internal gravity waves: From instabilities to turbulence. Annu. Rev. Fluid Mech., 34 , 559593.

  • Taylor, M. J., P. J. Espy, D. J. Baker, R. J. Sica, P. C. Neal, and W. R. Pendleton Jr., 1991: Simultaneous intensity, temperature and imaging measurements of short period wave structure in the OH nightglow emission. Planet. Space Sci., 39 , 11711188.

    • Search Google Scholar
    • Export Citation
  • Taylor, M. J., Y. Y. Gu, X. Tao, and C. S. Gardner, 1995: An investigation of intrinsic gravity wave signatures using coordinated lidar and nightglow image measurements. Geophys. Res. Lett., 22 , 28532856.

    • Search Google Scholar
    • Export Citation
  • Tsuda, T., T. Inoue, D. C. Fritts, T. E. VanZandt, S. Kato, T. Sato, and S. Fukao, 1989: MST radar observations of a saturated gravity wave spectrum. J. Atmos. Sci., 46 , 24402447.

    • Search Google Scholar
    • Export Citation
  • Tsuda, T., T. E. VanZandt, M. Mizumoto, S. Kato, and S. Fukao, 1991: Spectral analysis of temperature and Brunt-Väisälä frequency fluctuations observed by radiosondes. J. Geophys. Res., 96 , 1726517278.

    • Search Google Scholar
    • Export Citation
  • VanZandt, T. E., 1982: A universal spectrum of buoyancy waves in the atmosphere. Geophys. Res. Lett., 9 , 575578.

  • Vincent, R. A., 1984: Gravity wave motions in the mesosphere. J. Atmos. Terr. Phys., 46 , 119128.

  • Vinnichenko, N. K., 1970: The kinetic energy spectrum in the free atmosphere-1 second to 5 years. Tellus, 12 , 158166.

  • Walterscheid, R. L., 1981: Dynamical cooling induced by dissipating internal gravity-waves. Geophys. Res. Lett., 8 , 12351238.

  • Weinstock, J., 1978: On the theory of turbulence in the buoyancy subrange of stably stratified flows. J. Atmos. Sci., 35 , 634649.

  • Weinstock, J., 1985: Theoretical gravity wave spectrum in the atmosphere: Strong and weak wave interactions. Radio Sci., 20 , 12951300.

    • Search Google Scholar
    • Export Citation
  • Weinstock, J., 1990: Saturated and unsaturated spectra of gravity waves and scale-dependent diffusion. J. Atmos. Sci., 47 , 22112225.

  • Wilson, K. G., 1970: Model of coupling-constant renormalization. Phys. Rev. D, 2 , 14381472.

  • Wu, Q., and T. L. Killeen, 1996: Seasonal dependence of mesospheric gravity waves (<100 km) at Peach Mountain Observatory, Michigan. Geophys. Res. Lett., 23 , 22112214.

    • Search Google Scholar
    • Export Citation
  • Wu, Y-F., J. Xu, H-U. Widdel, and F-J. Lübken, 2001: Mean characteristics of the spectrum of horizontal velocity in the polar summer mesosphere and lower thermosphere observed by foil chaff. J. Atmos. Solar Terr. Phys., 63 , 18311839.

    • Search Google Scholar
    • Export Citation
  • Yamada, Y., H. Fukunishi, T. Nakamuara, and T. Tsuda, 2001: Breakdown of small-scale quasi-stationary gravity wave and transition to turbulence observed in OH airglow. Geophys. Res. Lett., 28 , 21532156.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 198 78 4
PDF Downloads 105 78 7