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.

Convex and concave types of second baroclinic mode internal solitary waves

“…Deeper in the water column, the opposite displacement occurs, creating what has been described as a bulge-shaped, double-humped, varicose, or sausage-type wave (Davis and Acrivos, 1967;Stamp and Jacka, 1995;Ostrovsky and Stepanyants, 2005;Moum et al, 2008). Pioneering investigations have described this type of wave, including theoretical investigations (Benjamin, 1967;Davis and Acrivos, 1967;Akylas and Grimshaw, 1992;Vlasenko, 1994;Grimshaw, 1997), laboratory experiments (Davis and Acrivos, 1967;Kao and Pao, 1980;Maxworthy, 1980;Honji et al, 1995;Stamp and Jacka, 1995;Vlasenko and Hutter, 2001;Mehta et al, 2002;Sutherland, 2002), numerical analyses (Tung et al, 1982;Terez and Knio, 1998;Rubino et al, 2001;Vlasenko and Hutter, 2001;Rusås and Grue, 2002;Stastna and Peltier, 2005;Vlasenko and Alpers, 2005), and field observations (Farmer and Smith, 1980;Konyaev et al, 1995;Imberger, 1998, 2001;Antenucci et al, 2000;Boegman et al, 2003;Duda et al, 2004;Yang et al, 2004;Bougucki et al, 2005;Sabinin and Serebryany, 2005;Moum et al, 2008;Shroyer et al, 2010).…”

Head-on collision of internal waves with trapped cores

“…This kind of head-on collision occurs in the range of approximately 1 ≤ α ≤ 1.6. Note that in the numerical study (Terez and Knio, 1998), the wave lost trapped fluid in the process of interaction even at α = 2.1. As shown in Fig.…”

“…The waves carry out trapped fluid, but the cores gradually lose trapped fluid to the wake through KH billows shifting to the wave rear and through recirculation in the trapped core (Terez and Knio, 1998;Maderich et al, 2001;Lamb, 2002). Figure 6b shows the collision of waves with equal amplitude α = α L = α R = 6.4 (case (A13; A13), with the parameters Fr max = 1.31 and Ri min = 0.06).…”

“…Head-on collision as a special case of oblique interaction was considered by Matsuno (1998) using a second-order analysis of small-amplitude interfacial waves in deep fluid. A mostly qualitative description of the head-on collision of mode-2 waves with trapped cores, obtained by conducting several laboratory experiments (Kao and Pao, 1980;Honji and Matsunaga, 1995;Stamp and Jacka, 1995;Gavrilov et al, 2012) and via a numerical simulation (Terez and Knio, 1998), has been given. These experiments showed that (i) waves experience a phase shift during collision, (ii) largeamplitude waves transport trapped masses of fluid after collision, (iii) waves of unequal amplitude exchange masses of trapped fluid in the interaction process, and (iv) some trapped fluid is ejected upon collision.…”

Damping of quasi-stationary waves between two miscible liquids

“…1 The velocity field can also be expressed by the same functional relationship. Bifurcation of the wavelength of the interface is obtained computationally usingfinite difference techniques, witha 90x90 gridsize, tosolve forthesetof governing equations (6,11,12). A flux-corrected transport algorithm is usedto resolve thesharp gradient atthe interface.…”