Abstract
We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter D. In the limits of large and small D, the 2d-ILW equation respectively tends to the 2d Benjamin-Ono and 2d Zakharov-Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude a and width σ, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane (a, σ) for various values of D. The amplitude threshold a increases as D and σ decrease and tends to infinity at D → 0. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all D except D = 0. But the equation itself has not been proved for small D. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
P. J. Schmidt, D. S. Henningson, and D. F. Jankowski, Stability and Transition in Shear Flows (Appl. Math. Sci., Vol. 142), Springer, New York (2002).
V. V. Voronovich, V. I. Shrira, and Yu. A. Stepanyants, “Two-dimensional models for nonlinear vorticity waves in shear flows,” Stud. Appl. Math.100, 1–32 (1998).
R. I. Joseph, “Solitary waves in a finite depth fluid,” J. Phys. A: Math. Gen.10, L225–L228 (1977).
T. Kubota, D. R. S. Ko, and L. D. Dobbs, “Weakly-nonlinear, long internal gravity waves in stratified fluids of finite depth,” J. Hydronautics12, 157–165 (1978).
R. I. Joseph and R. Egri, “Multi-soliton solutions in a finite depth fluid,” J. Phys. A: Math. Gen.11, L97–L102 (1978).
Y. Matsuno, “Exact multi-soliton solution for nonlinear waves in a stratified fluid of finite depth,” Phys. Lett. A74, 233–235 (1979).
H. H. Chen and Y. C. Lee, “Internal-wave solitons of fluids with finite depth,” Phys. Rev. Lett.43, 264–266 (1979).
V. I. Shrira, “On the ‘sub-surface’ waves of the mixed layer of the upper ocean,” Trans. USSR Acad. Sci., Earth Sci. Sec308, 276–279 (1989).
V. E. Zakharov and E. A. Kuznetsov, “Three-dimensional solitons,” JETP39, 285–286 (1974).
A. I. D’yachenko and E. A. Kuznetsov, “Two-dimensional wave collapse in the boundary layer,” Phys. D87, 301–313 (1995).
D. E. Pelinovsky and V. I. Shrira, “Collapse transformation for self-focusing solitary waves in boundary-layer type shear flows,” Phys. Lett. A206, 195–202 (1995).
V. E. Zakharov and E. A. Kuznetsov, “Solitons and collapses: Two evolution scenarios of nonlinear wave systems,” Phys. Usp.55, 535–556 (2012).
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM Stud. Appl. Math., Vol. 4), SIAM, Philadelphia (1981).
S. Melkonian and S. A. Maslowe, “Two-dimensional amplitude evolution equations for nonlinear dispersive waves on thin films,” Phys. D, 34, 255–269 (1989).
K. Nozaki, “Vortex solitons of drift waves and anomalous diffusion,” Phys. Rev. Lett.46, 184–187 (1981).
V. I. Petviashvili, “Red spot of Jupiter and the drift soliton in a plasma,” JETP Lett.32, 619–622 (1980).
S. Toh, H. Iwasaki, and T. Kawahara, “Two-dimensionally localized pulses of a nonlinear equation with dissipation and dispersion,” Phys. Rev. A40, 5472–5475 (1989).
G. A. Gottwald, “The Zakharov-Kuznetsov equation as a two-dimensional model for nonlinear Rossby waves,” arXiv:nlin/0312009v1 (2003).
D. E. Pelinovsky and Yu. A. Stepanyants, “Self-focusing instability of nonlinear plane waves in shear flows,” JETP78, 883–891 (1994).
D. G. Gaidashev and S. K. Zhdanov, “On the transverse instability of the two-dimensional Benjamin-Ono solitons,” Phys. Fluids16, 1915–1921 (2004).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Natl. Bur. Stds. Appl. Math. Ser., Vol. 55), Dover, New York (1972).
E. A. Kuznetsov, “Stability criterion for solitons of the Zakharov-Kuznetsov-type equations,” Phys. Lett. A382, 2049–2051 (2018).
M. C. Jorge, G. Cruz-Pacheco, L. Mier-y-Teran-Romero, and N. F. Smyth, “Evolution of two-dimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations,” Chaos15, 037104 (2005).
S. A. Orszag, “Numerical methods for the simulation of turbulence,” Phys. Fluids12, II–250–II–258 (1969).
D. A. Kopriva, Implementing Spectral Methods for Partial Differential Equations, Springer, Dordrecht (2009).
Y. S. Kachanov, “Physical mechanisms of laminar-boundary-layer transition,” Ann. Rev. Fluid Mech.26, 411–482 (1994).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare no conflicts of interest.
Additional information
This research was supported in part by the UK NERC (Grant No. NE/M016269/1) and the EU (Grant No. FP7 612610).
The research of J. Oloo was supported by the Commonwealth Scholarship Commission (Grant No. KECA-2016-30), without which this work would not have happened.
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 1, pp. 91–105, April, 2020.
Rights and permissions
About this article
Cite this article
Oloo, J.O., Shrira, V.I. Boundary layer collapses described by the two-dimensional intermediate long-wave equation. Theor Math Phys 203, 512–523 (2020). https://doi.org/10.1134/S0040577920040078
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577920040078