Abstract
We consider slow, compared to the speed of sound, motions of an ideal compressible fluid (gas) in a gravitational field in the presence of two isentropic layers with a small specific-entropy difference between them. Assuming the flow to be potential in each of the layers (v 1, 2 = ▿ϕ1, 2) and neglecting the acoustic degrees of freedom (div(\( \bar \rho \)(z)▿ϕ1, 2) ≈ 0, where \( \bar \rho \)(z) is the average equilibrium density), we derive the equations of motion for the boundary in terms of the shape of the surface z = η(x, y, t) itself and the difference between the boundary values of the two velocity field potentials: ψ(x, y, t) = ψ1 − ψ2. We prove the Hamilto nian structure of the derived equations specified by a Lagrangian of the form ℒ = ∫\( \bar \rho \)(η)η t ψdxdy − ℋ{η, ψ}. The system under consideration is the simplest theoretical model for studying internal waves in a sharply stratified atmosphere in which the decrease in equilibrium gas density due to gas compressibility with increasing height is essentially taken into account. For plane flows, we make a generalization to the case where each of the layers has its own constant potential vorticity. We investigate a system with a model dependence \( \bar \rho \)(z) ∝ e −2αz with which the Hamiltonian ℋ{η, ψ} can be represented explicitly. We consider a long-wavelength dynamic regime with dispersion corrections and derive an approximate nonlinear equation of the form u t + auu x − b[−\( \hat \partial _x^2 \) + α2]1/2 u x = 0 (Smith’s equation) for the slow evolution of a traveling wave.
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References
D. R. Christie, J. Atmos. Sci. 46, 1462 (1989).
J. W. Rottman and F. Einaudi, J. Atmos. Sci. 50, 2116 (1993).
M. G. Wurtele, R. D. Sharman, and A. Datta, Annu. Rev. Fluid Mech. 28, 429 (1996).
T. Kataoka, M. Tsutahara, and T. Akuzawa, Phys. Rev. Lett. 84, 1447 (2000).
Y. V. Lvov and E. G. Tabak, Phys. Rev. Lett. 87, 168501 (2001).
R. Grimshaw, E. Pelinovsky, and O. Poloukhina, Nonlinear Processes Geophys. 9, 221 (2002).
V. Vlasenko, P. Brandt, and A. Rubino, J. Phys. Oceanogr. 30, 2172 (2000).
V. Vlasenko and K. Hutter, J. Phys. Oceanogr. 32, 1779 (2002).
V. Vlasenko and N. Stashchuk, J. Phys. Oceanogr. 36, 1959 (2006).
R. Grimshaw, E. Pelinovsky, and T. Talipova, Surv. Geophys. 27, 273 (2007).
W. Choi and R. Camassa, J. Fluid Mech. 396, 1 (1999).
W. Craig, P. Guyenne, and H. Kalisch, Commum. Pure Appl. Math. 58, 1587 (2005).
A. R. de Zarate and A. Nachbin, Commun. Math. Sci. 6, 385 (2008).
J. L. Bona, D. Lannes, and J.-C. Saut, J. Math. Pures Appl. 89, 538 (2008).
N. N. Romanova and I. G. Yakushkin, Izv. Akad. Nauk, Fiz. Atmos. Okeana 43, 579 (2007).
N. N. Romanova, Izv. Akad. Nauk, Fiz. Atmos. Okeana 44, 56 (2008).
V. P. Goncharov, Izv. Akad. Nauk, Fiz. Atmos. Okeana 22, 468 (1986).
Y. Ogura and N. A. Phillips, J. Atmos. Sci. 19, 173 (1962).
D. R. Durran, J. Atmos. Sci. 46, 1453 (1989).
P. R. Bannon, J. Atmos. Sci. 53, 3618 (1996).
V. P. Ruban, Phys. Rev. D: Part. Fields 62, 127504 (2000).
V. P. Ruban, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 64, 036 305 (2001).
V. E. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968).
V. E. Zakharov, Eur. J. Mech. B/Fluids 18, 327 (1999).
V. E. Zakharov and E. A. Kuznetsov, Usp. Fiz. Nauk 167(11), 1137 (1997) [Phys.-Usp. 40 (11), 1087 (1997)].
T. B. Benjamin, J. Fluid Mech. 29, 559 (1967).
H. Ono, J. Phys. Soc. Jpn. 39, 1082 (1975).
R. Smith, J. Fluid Mech. 52, 379 (1972).
R. J. Joseph, J. Phys. A: Math. Gen. 10, L225 (1977).
H. H. Chen and Y. C. Lee, Phys. Rev. Lett. 43, 264 (1979).
L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut, Physica D (Amsterdam) 40, 360 (1989).
E. Wahlen, Lett. Math. Phys. 79, 303 (2007).
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Original Russian Text © V.P. Ruban, 2010, published in Zhurnal Éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2010, Vol. 138, No. 5, pp. 881–891.
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Ruban, V.P. Internal waves in a compressible two-layer model atmosphere: Hamiltonian description. J. Exp. Theor. Phys. 111, 776–785 (2010). https://doi.org/10.1134/S1063776110110099
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DOI: https://doi.org/10.1134/S1063776110110099