Large Internal Solitary Waves in Shallow Waters
107
shoaling internal waves and solibore formation (Fig. 10). The steady-state solution
of BM describes also the nonsymmetric solitary wave of the second mode shown in
Fig. 6.
The submodels ULM, BLM, SM of the basic model BM allow us to find the
soliton-like waves in two- or three-layer stratified fluids. One-parameter family of
solitary solutions for ULM, BLM or for SM depends on the Froude number Fr or
Fr s , correspondingly. It is worth to note that BM doesn’t have such solutions for
arbitrary initial flow parameters.
Decay of internal waves due to breaking and entrainment processes is very important for wave dynamics. It is shown for the symmetric solitary waves of the second
mode that the decay of internal waves can be simulated by the corresponding friction terms (27) (Fig. 9). Nevertheless, future development of the multi-layer shallow
water approach must include the modelling of mixing and entrainment processes at
the interfaces.
Acknowledgements This work was supported by the Russian Foundation for Basic Research
(Grant No. 15-01-03942).
References
1. Carr, M., Franklin, J., King, S. E., Davies, P. A., Grue, J., & Dritschel, D. G. (2017). The
characteristics of billows generated by internal solitary waves. Journal of Fluid Mechanics,
812, 541–577.
2. Choi, W. (2000). Modeling of strongly nonlinear internal gravity waves. In: Y. Goda, M. Ikehata
and K. Suzuki (Eds.), Proceedings of the Fourth International Conference on Hydrodynamics
(pp. 453–458).
3. Choi, W., & Camassa, R. (1999). Fully nonlinear internal waves in a two-fluid system. Journal
of Fluid Mechanics, 386, 1–36.
4. Derzho, O. G., & Grimshaw, R. (2007). Asymmetric internal solitary waves with a trapped
core in deep fluids. Physics of Fluids, 19, 096601.
5. Ermanyuk, E. V., & Gavrilov, N. V. (2007). A note on the propagation speed of a weakly
dissipative gravity current. Journal of Fluid Mechanics, 574, 393–403.
6. Fructus, D., Carr, M., Grue, J., Jensen, A., & Davies, P. A. (2009). Shear-induced breaking of
large internal solitary waves. Journal of Fluid Mechanics, 620, 1–29.
7. Gavrilov, N. V., & Lyapidevskii, V Yu. (2009). Symmetric solitary waves in a two-layer fluid.
Doklady Physics, 54(11), 508–511.
8. Gavrilov, N. V., & Liapidevskii, V. Yu. (2010). Finite-amplitude solitary waves in a two-layer
fluid. Journal of Applied Mechanics and Technical Physics, 51(4), 471–481.
9. Gavrilov, N., Liapidevskii, V., & Gavrilova, K. (2011). Large amplitude internal solitary waves
over a shelf. Natural Hazards and Earth System Sciences, 11, 17–25.
10. Gavrilov, N., Liapidevskii, V., & Gavrilova, K. (2012). Mass and momentum transfer by solitary internal waves in a shelf zone. Nonlinear Processes in Geophysics, 19, 265–272.
11. Gavrilov, N. V., Liapidevskii, V. Y., & Liapidevskaya, Z. A. (2013). Influence of dispersion
on the propagation of internal waves in a shelf zone. Fundam. prikl. gidrofiz., 6, 25–34 (in
Russian).
12. Gavrilov, N. V., Liapidevskii, V Yu., & Liapidevskaya, Z. A. (2015). Transformation of large
amplitude internal waves over a shelf. Fundam. prikl. gidrofiz., 8(3), 32–43 (in Russian).
107
shoaling internal waves and solibore formation (Fig. 10). The steady-state solution
of BM describes also the nonsymmetric solitary wave of the second mode shown in
Fig. 6.
The submodels ULM, BLM, SM of the basic model BM allow us to find the
soliton-like waves in two- or three-layer stratified fluids. One-parameter family of
solitary solutions for ULM, BLM or for SM depends on the Froude number Fr or
Fr s , correspondingly. It is worth to note that BM doesn’t have such solutions for
arbitrary initial flow parameters.
Decay of internal waves due to breaking and entrainment processes is very important for wave dynamics. It is shown for the symmetric solitary waves of the second
mode that the decay of internal waves can be simulated by the corresponding friction terms (27) (Fig. 9). Nevertheless, future development of the multi-layer shallow
water approach must include the modelling of mixing and entrainment processes at
the interfaces.
Acknowledgements This work was supported by the Russian Foundation for Basic Research
(Grant No. 15-01-03942).
References
1. Carr, M., Franklin, J., King, S. E., Davies, P. A., Grue, J., & Dritschel, D. G. (2017). The
characteristics of billows generated by internal solitary waves. Journal of Fluid Mechanics,
812, 541–577.
2. Choi, W. (2000). Modeling of strongly nonlinear internal gravity waves. In: Y. Goda, M. Ikehata
and K. Suzuki (Eds.), Proceedings of the Fourth International Conference on Hydrodynamics
(pp. 453–458).
3. Choi, W., & Camassa, R. (1999). Fully nonlinear internal waves in a two-fluid system. Journal
of Fluid Mechanics, 386, 1–36.
4. Derzho, O. G., & Grimshaw, R. (2007). Asymmetric internal solitary waves with a trapped
core in deep fluids. Physics of Fluids, 19, 096601.
5. Ermanyuk, E. V., & Gavrilov, N. V. (2007). A note on the propagation speed of a weakly
dissipative gravity current. Journal of Fluid Mechanics, 574, 393–403.
6. Fructus, D., Carr, M., Grue, J., Jensen, A., & Davies, P. A. (2009). Shear-induced breaking of
large internal solitary waves. Journal of Fluid Mechanics, 620, 1–29.
7. Gavrilov, N. V., & Lyapidevskii, V Yu. (2009). Symmetric solitary waves in a two-layer fluid.
Doklady Physics, 54(11), 508–511.
8. Gavrilov, N. V., & Liapidevskii, V. Yu. (2010). Finite-amplitude solitary waves in a two-layer
fluid. Journal of Applied Mechanics and Technical Physics, 51(4), 471–481.
9. Gavrilov, N., Liapidevskii, V., & Gavrilova, K. (2011). Large amplitude internal solitary waves
over a shelf. Natural Hazards and Earth System Sciences, 11, 17–25.
10. Gavrilov, N., Liapidevskii, V., & Gavrilova, K. (2012). Mass and momentum transfer by solitary internal waves in a shelf zone. Nonlinear Processes in Geophysics, 19, 265–272.
11. Gavrilov, N. V., Liapidevskii, V. Y., & Liapidevskaya, Z. A. (2013). Influence of dispersion
on the propagation of internal waves in a shelf zone. Fundam. prikl. gidrofiz., 6, 25–34 (in
Russian).
12. Gavrilov, N. V., Liapidevskii, V Yu., & Liapidevskaya, Z. A. (2015). Transformation of large
amplitude internal waves over a shelf. Fundam. prikl. gidrofiz., 8(3), 32–43 (in Russian).