Algebraic instability in shallow water flows with horizontally nonuniform density

The regimes and mechanisms of the Rayleigh-Taylor instability have been studied in the scope of the nonhydrostatic shallow water model with horizontally nonuniform density. As analysis shows, the nonhydrostaticity has a crucial influence on the instability. It is for this reason that at the final stage a collapse tendency predicted on the base of the hydrostatic scenario slows down and turns into the regime of algebraic instability. The numerical testing has shown that in spite of its simplicity, the model is quite able to describe realistically a number of effects. For example, the model captures the shallowing effect, which manifests itself as profile concavities on either side of the jet coming out of the boundary layer.

Figure 1

The model of the active (lower) layer with horizontally nonuniform density.

Figure 2

Phase portrait of the nonhydrostatic model in the regime of instability $\kappa <0$.

Figure 3

The solid curve $c$ describes the transition from the collapsing regime to the regime of algebraic instability. The initial asymptotics $a$ corresponds to the collapsing solution (14) while the solution (16) gives the final asymptotics $b$.

Figure 4

Phase portrait of the nonhydrostatic model in the regime of stability $\kappa >0$.

Figure 5

Profiles of the interface over the period $\lambda$ in the model [27]. The real profile is shown by solid curve, the polygonal dashed line corresponds to the schematic profile while the central dashed line is the unperturbed interface location.

Figure 6

Layer's surface elevation $h$ as function of $x$ at time $t=25$.

Figure 7

Dependence of $h_{\max }$ on time. The dashed line corresponds to asymptotics (17), the solid line is calculated numerically.

Figure 8

Dependence of the shallowing level $h_{\min }$ on time.