In the theory of internal waves in the coastal ocean, linear stratification plays an exceptional role. This is because the nonlinearity coefficient in KdV theory vanishes, and in the case of large amplitude waves, the DJL theory linearizes and fails to give solitary wave solutions. We consider small, physically consistent perturbations of a linearly stratified fluid that would result from a localized mixing near a particular depth. We demonstrate that the DJL equation does yield exact internal solitary waves in this case. These waves are long due to the weak nonlinearity, and we explore how this weak nonlinearity manifests during shoaling.
Internal waves, DJL theory, shoaling, nearly linear stratification
1. Bell, J. B., D. L. Marcus (1992) , A second-order projection method for variable-density flows, J. Comp. Phys., 101, no. 2, p. 334-348, https://doi.org/10.1016/0021-9991(92)90011-M
2. Bell, J. B., P. Colella, H. M. Glaz (1989) , A second-order projection method for the incompressible navier-stokes equations, J. Comp. Phys., 283, no. 85, p. 257-283, https://doi.org/10.1016/0021-9991(89)90151-4
3. Grimshaw, R., E. Pelinovsky, T. Talipova (1997) , The modified Korteweg-de Vries equation in the theory of large-amplitude internal waves, Nonlinear Processes in Geophysics, 4, p. 237-250, https://doi.org/10.5194/npg-4-237-1997
4. Grimshaw, R., E. Pelinovsky, T. Talipova (2007) , Modelling Internal Solitary Waves in the Coastal Ocean, Surv. Geophys., 28, no. 4, p. 273-2981, https://doi.org/10.1007/s10712-007-9020-0
5. Grimshaw, R., E. Pelinovsky, et al. (2004) , Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves, J. Phys. Oceanogr., 34, no. 12, p. 2774-2791, https://doi.org/10.1175/JPO2652.1
6. Grimshaw, R., E. Pelinovsky, et al. (2010) , Internal solitary waves: propagation, deformation and disintegration, Nonlin. Processes Geophys., 17, p. 633-649, https://doi.org/10.5194/npg-17-633-2010
7. Holloway, P. E., E. N. Pelinovsky, T. G. Talipova (1999) , A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone, J. Geophys. Res., 104, p. 18,333-18,350, https://doi.org/10.1029/1999JC900144
8. Jackson, C. R., J. C. B. Da Silva, G. Jeans (2012) , The generation of nonlinear internal waves, Oceanography, 25, no. 2, p. 108-123, https://doi.org/10.5670/oceanog.2012.46
9. Lamb, K. G. (2007) , Energy and pseudoenergy flux in the internal wave field generated by tidal flow over topography, Cont. Shelf Res., 27, p. 1208-1232, https://doi.org/10.1016/j.csr.2007.01.020
10. Lamb, K. G., B. Wan (1998) , Conjugate flows and flat solitary waves for a continuously stratified fluid, Phys. Fluids, 10, p. 2061-2079, https://doi.org/10.1063/1.869721
11. Lamb, K. G., A. Warn-Varnas (2015) , Two-dimensional numerical simulations of shoaling internal solitary waves at the ASIAEX site in the South China Sea, Nonlin. Processes Geophys., 22, p. 289-312, https://doi.org/10.5194/npg-22-289-2015
12. Lamb, K. G., W. Xiao (2014) , Internal solitary waves shoaling onto a shelf: Comparisons of weakly-nonlinear and fully nonlinear models for hyperbolic-tangent stratifications, Ocean Modelling, 78, p. 17-34, https://doi.org/10.1016/j.ocemod.2014.02.005
13. Liang, J., X.-M. Li (2019) , Generation of second-mode internal solitary waves during winter in the northern South China Sea, Ocean Dyn., 69, p. 313-321, https://doi.org/10.1007/s10236-019-01246-6
14. Pelinovsky, E. N., O. E. Polukhina, K. Lamb (2000) , Nonlinear Internal Waves in the Ocean Stratified in Density and Current, Oceanol, 40, no. 6, p. 757-766
15. Shroyer, E. L., J. N. Moum, J. D. Nash (2010a) , Mode 2 waves on the continental shelf: Ephemeral components of the nonlinear internal wavefield, J. Geophys. Res., 115, p. C07001, https://doi.org/10.1029/2009JC005605
16. Shroyer, E. L., J. N. Moum, J. D. Nash (2010b) , Energy transformations and dissipation of nonlinear internal waves over New Jersey's continental shelf, Nonlin. Processes Geophys., 17, p. 345-360, https://doi.org/10.5194/npg-17-345-2010
