We consider a two-layer fluid with a depth-dependent upper-layer current (e.g. a river inflow, an exchange flow in a strait, or a wind-generated current). In the rigid-lid approximation, we find the necessary singular solution of the nonlinear first-order ordinary differential equation responsible for the adjustment of the speed of the long interfacial ring wave in different directions in terms of the hypergeometric function. This allows us to obtain an analytical description of the wavefronts and vertical structure of the ring waves for a large family of the current profiles and to illustrate their dependence on the density jump and the type and the strength of the current. In the limiting case of a constant upper-layer current we obtain a 2D ring waves' analogue of the long-wave instability criterion for plane interfacial waves. On physical level, the presence of instability for a sufficiently strong current manifests itself already in the stable regime in the squeezing of the wavefront of the interfacial ring wave in the direction of the current. We show that similar phenomenon can also take place for other, depth-dependent currents in the family.
Stratified shear flows, internal waves, long-wave instability
1. Ablowitz, M. J., H. Segur (1979) , On the evolution of packets of water waves, J. Fluid Mech., 92, p. 691-715, https://doi.org/10.1017/S0022112079000835
2. Bontozoglou, V. (1991) , Weakly-nonlinear Kelvin-Helmholtz waves between fluids of finite depth, Intl. J. Multiphase Flow, 17, p. 509-518, https://doi.org/10.1016/0301-9322(91)90046-6
3. Boonkasame, A., P. A. Milewski (2014) , The stability of large-amplitude shallow interfacial non-Boussinesq flows, Stud. Appl. Math., 133, p. 182-213, https://doi.org/10.1017/jfm.2014.28
4. Ellingsen, S. A. (2014a) , Ship waves in the presence of uniform vorticity, J. Fluid Mech., 742, p. R2, https://doi.org/10.1063/1.4891640
5. Ellingsen, S. A. (2014b) , Initial surface disturbance on a shear current: the Cauchy-Poisson problem with a twist, Phys. Fluids, 26, p. 082104, https://doi.org/10.1063/1.4891640
6. Li, Y., S. A. Ellingsen (2019) , A framework for modelling linear surface waves on shear currents in slowly varying waves, J. Geophys. Res.: Oceans, 124, p. 2527-2545, https://doi.org/10.1029/2018JC014390
7. Grimshaw, R. H. J., L. A. Ostrovsky, V.I. Shrira, et al. (1998) , Long nonlinear surface and internal gravity waves in a rotating ocean, Surv. Geophys., 19, p. 289-338, https://doi.org/10.1023/A:1006587919935
8. Grimshaw, R., E. Pelinovsky, T. Talipova (2007) , Modelling internal solitary waves in the coastal ocean, Surv. Geophys., 28, p. 273-298, https://doi.org/10.1007/s10712-007-9020-0
9. Grimshaw, R., E. Pelinovsky, T. Talipova, et al. (2010) , Internal solitary waves: propagation, deformation and disintegration, Nonlin. Proc. Geophys., 17, p. 633-649, https://doi.org/10.5194/npg-17-633-2010
10. Grimshaw, R. (2015) , Effect of a background shear current on models for nonlinear long internal waves, Fund. Prikl. Gidrofiz., 8, p. 20-23
11. Grimshaw, R. (2019) , Initial conditions for the cylindrical Korteweg-de Vries equation, Stud. Appl. Math., 143, p. 176-191, https://doi.org/10.1111/sapm.12272
12. Helfrich, K. R., W. K. Melville (2006) , Long nonlinear internal waves, Annu. Rev. Fluid Mech., 38, p. 395-425, https://doi.org/10.1146/annurev.fluid.38.050304.092129
13. Johnson, R. S. (1980) , Water wave and Korteweg - de Vries equations, J. Fluid Mech., 97, p. 701-719, https://doi.org/10.1017/S0022112080002765
14. Johnson, R. S. (1990) , Ring waves on the surface of shear flows: a linear and nonlinear theory, J. Fluid Mech., 215, p. 145==160, https://doi.org/10.1017/S0022112090002592
15. Johnson, R. S. (1997) , A Modern Introduction to the Mathematical Theory of Water Waves, 445 pp., Cambridge University Press, Cambridge
16. Khusnutdinova, K. R., C. Klein, V. B. Matveev, et al. (2013) , On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation, Chaos, 23, p. 013126, https://doi.org/10.1063/1.4792268
17. Khusnutdinova, K. R., X. Zhang (2016a) , Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech., 794, p. 17-44, https://doi.org/10.1017/jfm.2016.147
18. Khusnutdinova, K. R., X. Zhang (2016b) , Nonlinear ring waves in a two-layer fluid, Physica D, 333, p. 208-221, https://doi.org/10.1016/j.physd.2016.02.013
19. Lannes, D., M. Ming (2015) , The Kelvin-Helmholtz instabilities in two-fluids shallow water models, Hamiltonian Partial Differential Equations and Applications, Fields Institute Communications, p. 46, Springer, New York, https://doi.org/10.1007/978-1-4939-2950-4_7
20. Lipovskii, V. D. (1985) , On the nonlinear internal wave theory in fluid of finite depth, Izv. Akad. Nauk SSSR, Ser. Fiz., 21, p. 864-871
21. McMilan, J. M., B. R. Sutherland (2010) , The lifecycle of axisymmetric internal solitary waves, Nonlin. Proc. Geophys., 17, p. 443-453, https://doi.org/10.5194/npg-17-443-2010
22. Miles, J. W. (1978) , An axisymmetric Boussinesq wave, J. Fluid Mech., 84, p. 181-191, https://doi.org/10.1017/S0022112078000105
23. Nash, J. D., J. N. Moum (2005) , River plums as a source of large amplitude internal waves in the coastal ocean, Nature, 437, p. 400-403, https://doi.org/10.1038/nature03936
24. Ovsyannikov, L. V. (1979) , Two-layer `shallow water' model, J. Appl. Math. Tech. Phys., 20, p. 127-135, https://doi.org/10.1007/BF00910010
25. Ovsyannikov, L. V., et al. (1985) , Nonlinear Problems in the Theory of Surface and Internal Waves, Nauka, Novosibirsk
26. Ramirez, C., D. Renouard, Yu. A. Stepanyants (2002) , Propagation of cylindrical waves in a rotating fluid, Fluid Dynam. Res., 30, p. 169-196, https://doi.org/10.1016/S0169-5983(02)00040-0
27. Smeltzer, B. K., E. Esoy, S. A. Ellingsen (2019) , Observation of surface wave patterns modified by sub-surface shear currents, J. Fluid Mech., 873, p. 508-530, https://doi.org/10.1017/jfm.2019.424
28. Vlasenko, V., J. C. Sanchez Garrido, N. Staschuk, et al. (2009) , Three-dimensional evolution of large-amplitude internal waves in the Strait of Gibraltar, J. Phys. Oceanogr., 39, p. 2230-2246, https://doi.org/10.1175/2009JPO4007.1
29. Vlasenko, V., N. Staschuk, et al. (2013) , Generation of baroclinic tides over an isolated underwater bank, J. Geophys. Res., 118, p. 4395-4408, https://doi.org/10.1002/jgrc.20304
30. Weidman, P. D., R. Zakhem, et al. (1988) , Cylindrical solitary waves, J. Fluid Mech., 191, p. 557-573, https://doi.org/10.1017/S0022112088001703
