Abstract
The mechanisms of instability, whose development leads to the occurrence of the collapse (blow up), have been studied in the scope of the rotating shallow water flows with horizontal density gradient. Analysis shows that collapses in such models are initiated by the Rayleigh-Taylor instability and two scenarios are possible. Both the scenarios evolve according to a power law (t 0 − t)γ, where t 0 is the collapse time, with γ = −1, −2, and γ = −2/3, −1 for the isotropic and anisotropic collapses, respectively. The rigorous criterion of collapse is found on the base of integrals of motion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Foglizzo, F. Masset, J. Guilet, and G. Durand, Phys. Rev. Lett. 108, 051103(4) (2012).
P. A. Gilman, Astrophys. J. 544, L79 (2000).
D. A. Schecter, J. F. Boyd, and P. A. Gilman, Astrophys. J. 551, L185 (2001).
E. A. Kuznetsov and V. E. Zakharov, Lect. Notes Phys. 542, 3 (2000).
E. A. Kuznetsov, Izv. Vyssh. Uchebn. Zaved., Ser. Radiofiz. 56, 342 (2003).
J. Eggers and M. A. Fontelos, Nonlinearity 22, R1 (2009).
V. P. Goncharov and V. I. Pavlov, JETP Lett. 84, 384 (2006).
V. P. Goncharov and V. I. Pavlov, Phys. Rev. E 76, 066314 (2007).
V. P. Goncharov and V. I. Pavlov, Hamiltonian Vortex and Wave Dynamics (Geos, Moscow, 2008) [in Russian].
V. P. Goncharov, JETP Lett. 89, 393 (2009).
V. P. Goncharov and V. I. Pavlov, J. Exp. Theor. Phys. 111, 124 (2010).
V. P. Goncharov, J. Exp. Theor. Phys. 113, 714 (2011).
P. Ripa, Geophys. Astrophys. Fluid Dynam. 70, 85 (1993).
D. L. T. Anderson, Tellus A 36, 278 (1984).
P. S. Schopf and M. A. Cane, J. Phys. Oceanogr. 13, 917 (1983).
P. Ripa, Dynam. Atmos. Oceans 29, 1 (1999).
G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge Univ. Press, Cambridge, 1996).
V. E. Zakharov and E. A. Kuznetsov, Sov. Phys. JETP 64, 773 (1986).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Goncharov, V.P., Pavlov, V.I. Blow-up instability in shallow water flows with horizontally-nonuniform density. Jetp Lett. 96, 427–431 (2012). https://doi.org/10.1134/S0021364012190095
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0021364012190095