Simple model of the Rayleigh-Taylor instability, collapse, and structural elements

Abstract

The mechanisms and structural elements of the Rayleigh-Taylor instability whose evolution results in the occurrence of the collapse have been studied in the scope of the rotating shallow water model with horizontal density gradient. Analysis of the instability mechanism shows that two collapse scenarios are possible. One scenario implies anisotropic collapse during which the contact area of a collapsing fragment with the bottom contracts into a spinning segment. The other implies isotropic contracting of the area into a point. The rigorous integral criteria and power laws of collapses are found.

Effect of nonhydrostaticity on the final stage of instability in shallow water with a horizontally nonuniform density

“…A power law increase in h max with time indicates that the explosive instability regime, which was pre dicted for hydrostatic models [5][6][7], at the final stage slows down and is transformed to the algebraic insta bility regime. In particular, according to Eq.…”

“…If the hydrostatic balance is slightly violated and the Green-Naghdi correction (the last term in Eq. (1)) can be neglected, we obtain the equations for the active layer model in the hydrostatic approximation [5][6][7]. Nevertheless, the model given by Eqs.…”

“…In the hydrostatic approximation, where the last term on the right hand side of Eq. (1) is omitted, these equations were formulated in [5][6][7] within a two layer model (Fig. 1).…”

“…As in the hydrostatic approximation [6,7], equa tions of motion (1) and (2) can be derived ab initio [9]. They follow from the Hamiltonian formulation with the Hamiltonian (3) and Poisson brackets…”

“…For shal low water models with horizontally nonuniform buoy ancy, which in essence follow from two layer models [5][6][7], the situation is a more complex because a com peting effect appears owing to the effect of the upper layer. Indeed, a tendency to explosive instability char acteristic of the model in the hydrostatic balance approximation can change because allowance for ver tical motions caused by nonhydrostatic effect leads to additional nonlinear and nonlocal dispersion.…”