Incomplete similarity of internal solitary waves with trapped cores

Abstract

The dynamics and internal structure of internal solitary waves (ISWs) with trapped cores propagating in a thin pycnocline near the bottom or surface for a wide range of wave amplitudes and stratifications are studied numerically within the framework of Navier-Stokes equations for laboratory and ocean scales. It is shown that the most important characteristics of wave dynamics 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), the ratio of the basin depth to the pycnocline thickness epsilon and the effective Reynolds number Re-eff, defined as a ratio of the product of the phase velocity of the wave 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 layers 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 1 < Fr-m less than or similar to 1.3; and (iii) the unstable strongly nonlinear waves at Ri(min) < 0.1 and Fr-m approximate to 1.35. On the whole, the results of experiments and numerical simulations showed complete similarity in all ranges of the phase velocity and wavelength at large epsilon and Re-eff. The experimental and computed dependence of the horizontal size of the trapped core on the height of the trapped core also demonstrate the self-similarity of the trapped core shape. However, the incomplete similarity of Ri(min) is found when the dependence on viscosity does not vanish at large Re-eff implying a viscosity effect on the stability of ISW. This dependence is approximated by the power law Ri(min) similar to (a/h)(-1.19) Re-eff(-1.19). With the time wave damping occurs, and thus Ri(min) grows following this self-similar dependence.