Incomplete Similarity of Internal Solitary Waves with Trapped Core

Abstract:

The dynamics and internal structure of internal solitary waves with trapped core propagating in a thin pycnocline near the bottom or surface for a wide range of wave amplitude and stratification are studied numerically in the frame of the Navier-Stokes equations for the laboratory and ocean scales. It is shown that the most important characteristics of the dynamics of waves are the local Froude number Fr_m, calculated as the ratio of the maximum local velocity to the phase velocity of the waves, the minimum Richardson number Ri_min and the effective Reynolds number Re_eff , defined as the ratio the product of the phase velocity c of the waves and the wave amplitude a to the kinematic viscosity. Depending on the parameter values Fr_m and Ri_min three main classes of ISW propagating in the pycnocline layer over or under homogeneous deep layer can be identified: (i) the weakly nonlinear waves at Ri_min >1, Fr_m < 1; (ii) the stable strongly nonlinear waves with trapped cores at Ri_min 0.15 and Fr_m ~1.2; and (iii) the unstable strongly nonlinear waves at Ri_min <0.1 and Fr_m ~ 1.25. On the whole, the results of experiments and numerical simulations showed complete similarity in all range of the phase velocity and wavelength at large Re_eff . The experimental and computed dependence of horizontal size of the trapped core on the height of the trapped core also demonstrate self-similarity of the trapped core shape. However, incomplete similarity of Ri_min is found when dependence on viscosity remains at large Re_eff which implies the viscosity effect on the stability of ISW. This dependence is approximated by power law Ri_min~(a / h)^-1.19 Re_eff ^-1.19. With time a wave damping occurs, thus Ri_min is growing following this self-similar dependence.