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On the Theory of Slope Flows

  • L. Kh. Ingel’
Article
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Paradoxical properties of the classical Prandtl solution for flows occurring in a semibounded liquid (gaseous) medium above an infinite homogeneously cooled/heated inclined plane are analyzed. In particular, the maximum velocity of steady-state slope flow is independent, according to this solution, of the angle of inclination. Consequently, there is no passage to the limit to the case of zero angle where the cooling/heating is unlikely to give rise to homogeneous horizontal flows. It is shown that no paradoxes arise if we do not consider buoyancy sources of infinite spatial scale, which act infinitely long. It follows from the results that, in particular, the solution of the problem for a semibounded medium above a homogeneously cooled surface in the gravity field is unstable to small deviations of this surface from horizontal.

Keywords

slow flows analytical solutions Prandtl model asymptotics at small angles of inclination thermal inhomogeneities 

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References

  1. 1.
    L. Prandtl, Hydroaeromechanics [Russian translation], NITs "Regulyarnaya i Khaoticheskaya Dinamika, Moscow–Izhevsk (2000).Google Scholar
  2. 2.
    L. N. Gutman, Introduction to the Nonlinear Theory of Mesometeorological Processes [in Russian], Gidrometeoizdat, Moscow (1969).Google Scholar
  3. 3.
    R. Mo, On adding thermodynamic damping mechanisms to refine two classical models of katabatic winds, J. Atmos. Sci., 70, 2325−2334 (2013).CrossRefGoogle Scholar
  4. 4.
    D. Zardi and S. Serafin, An analytic solution for daily-periodic thermally-driven slope flow, Quart. J. Roy. Meteorol. Soc., 141, 1968–1974 (2015).CrossRefGoogle Scholar
  5. 5.
    F. K. Chow, S. F. J. DeWekker, and B. Snyder (Eds.,) Mountain Weather Research and Forecasting. Recent Progress and Current Challenges, Springer Atmospheric Sciences, Springer, Berlin (2013).Google Scholar
  6. 6.
    D. E. England and R. T. McNider, Concerning the limiting behavior of time-dependent slope winds, J. Atmos. Sci., 50, No. 11, 1610−1613 (1993).CrossRefGoogle Scholar
  7. 7.
    A. M. J. Davis and R. T. McNider, The development of Antarctic katabatic winds and implications for the coastal ocean, J. Atmos. Sci., 54, 1248–1261 (1997).CrossRefGoogle Scholar
  8. 8.
    L. Kh. Ingel′, On the nonlinear theory of katabatic winds, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 4, 3−12 (2011).Google Scholar
  9. 9.
    B. Grisogono, T. Jurlina, Ž. Večenaj, and I. Güttler, Weakly nonlinear Prandtl model for simple slope flows, Quart. J. Roy. Meteorol. Soc., 141, 883−892 (2015).CrossRefGoogle Scholar
  10. 10.
    A. V. Kistovich and Yu. D. Chashechkin, Structure of nonstationary boundary flow on an inclined plane in a continuously stratified medium, Dokl. Ross. Akad. Nauk, 325, No. 4, 833−837 (1992).Google Scholar
  11. 11.
    A. V. Kistovich and Yu. D. Chashechkin, Structure of Nonstationary Boundary Flow on an Inclined Plane in a Continuously Stratified Liquid, Preprint of the Institute of Applied Mechanics of the Russian Academy of Sciences No. 523, IPM RAN, Moscow (1993).Google Scholar
  12. 12.
    L. Kh. Ingel′, Nonstationary convection in a binary mixture at a plane vertical surface, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 92−97 (2002).Google Scholar
  13. 13.
    L. Kh. Ingel′ and M. V. Kalashnik, Nontrivial features of the hydrothermomechanics of seawater and other stratified solutions, Usp. Fiz. Nauk, 182, No. 4, 379−406 (2012).CrossRefGoogle Scholar
  14. 14.
    A. E. Gill, Atmosphere-Ocean Dynamics [Russian translation], Vol. 1, Nauka, Moscow (1986).Google Scholar
  15. 15.
    B. Gebhart, Y. Jaluria, R. Mahajan, and B. Sammakia, Buoyancy-Induced Flows and Transport [Russian translation], Vol. 1, Mir, Moscow (1991).zbMATHGoogle Scholar
  16. 16.
    M. Abramovitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions [Russian translation], Nauka, Moscow (1979).Google Scholar
  17. 17.
    L. Kh. Ingel′, Convection in a stratified binary mixture at a thermally inhomogeneous vertical surface, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 68−80 (2009).Google Scholar
  18. 18.
    L. Kh. Ingel′, Shear-flow perturbations due to the interaction with convective waves, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 34−37 (2006).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Scientific and Production Association “Typhoon”ObninskRussia
  2. 2.A. M. Obukhov Institute of Atmospheric PhysicsRussian Academy of SciencesMoscowRussia

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