Numerical simulations of large-amplitude internal solitary waves

Abstract

A numerical model based on the incompressible two-dimensional Navier–Stokes equations in the Boussinesq approximation is used to study mode-2 internal solitary waves propagating on a pycnocline between two deep layers of different densities. Numerical experiments on the collapse of an initially mixed region reveal a train of solitary waves with the largest leading wave enclosing an intrusional ‘bulge’. The waves gradually decay as they propagate along the horizontal direction, with a corresponding reduction in the size of the bulge. When the normalized wave amplitude, a, falls below the critical value ac=1.18, the wave is no longer able to transport mixed fluid as it propagates away from the mixed region, and a sharp-nosed intrusion is left behind. The wave structure is studied using a Lagrangian particle tracking scheme which shows that for small amplitudes the bulges have a well-defined elliptic shape. At larger amplitudes, the bulge entrains and mixes fluid from the outside while instabilities develop in the rear part of the bulge. Results are obtained for different wave amplitudes ranging from small-amplitude ‘regular’ waves with a=0.7 to highly nonlinear unstable waves with a=3.8. The dependence of the wave speed and wavelength on amplitude is measured and compared with available experimental data and theoretical predictions. Consistent with experiments, the wave speed increases almost linearly with amplitude at small values of a. As a becomes large, the wave speed increases with amplitude at a smaller rate, which gradually approaches the asymptotic limit for a two-fluid model. Results show that in the parameter range considered the wave amplitude decreases linearly with time at a rate inversely proportional to the Reynolds number. Numerical experiments are also conducted on the head-on collision of solitary waves. The simulations indicate that the waves experience a negative phase shift during the collision, in accordance with experimental observations. Computations are used to determine the dependence of the phase shift on the wave amplitude.