Negatively buoyant vortices in the Boussinesq-Euler equations

A previous model of a two-dimensional, stratified dipole is applied to a cylindrical vortex of dense fluid traveling across the lower boundary of a semi-infinite, unstratified ambient. The vortex is semicircular in profile. The model is derived from the Boussinesq-Euler equations and hence is an idealization neglecting the effects of viscosity, density diffusion, and friction. This is similar to other idealized models of dense fluid flow. The flow maintains a steady state through a balance of positive and negative vorticity generation as the dense fluid travels around the vortex. Numerical experiments are performed to investigate the stability and behavior of isolated and interacting vortices of this type.

Figure 1

Froude number (27) based on the mean reduced gravity. The horizontal dashed line indicates the $m=0$ value, and the solid line shows the variation of the $m>0$ value with $h$. The vertical dotted line marks the first root of $J_1\left(h\right)$ after $h=0$.

Figure 2

Normalized distribution (the distribution divided by its peak value) of the radial functions for $g^{\prime }$ (black) and $\omega$ (red) at different values of $h$: 0.5 (solid), 1.5 (dashed), 2.5 (dash-dot), 3.5 (dotted).

Figure 3

Dimensionless stream function (a), reduced gravity (b), and vorticity (c) for the $m>0$ solution with $h=2$ and $\sigma =1$; see Eqs. (19) and (20).

Figure 4

Absolute, dimensionless values of vorticity and stream function, plotted against each other for different values of $h$ as shown. These are colocated values taken from a grid of points covering the vortex.

Figure 5

Plots of the single vortex at intervals $\mathrm{\Delta }\overline{t}=10$ for $0\le \overline{t}\le 40$. Panels (a)–(e) show the dimensionless reduced gravity, and (f)–(j) show the dimensionless vorticity. The vortex is initially centered at $\overline{x}=0$ in the top row of panels, and time increases downwards.

Figure 6

Hovmöller diagram of the single vortex, plotting time against cross sections of dimensionless reduced gravity taken at height $\overline{z}=0.25$. The red dotted lines indicate the theoretical vortex speed in an unbounded domain.

Figure 7

Hovmöller diagram of the single vortex with doubled domain resolution compared to that in Fig. 6.

Figure 8

Hovmöller diagram of the single vortex with doubled domain size compared to that in Fig. 6.

Figure 9

Plots of the single, unbalanced vortex at intervals $\mathrm{\Delta }\overline{t}=10$ for $0\le \overline{t}\le 40$. Panels (a)–(e) show the dimensionless reduced gravity, and (f)–(j) show the dimensionless vorticity. The vortex is initially centered at $\overline{x}=0$ in the top row of panels, and time increases downwards.

Figure 10

Plots of the colliding vortices at intervals $\mathrm{\Delta }\overline{t}=10$ for $0\le \overline{t}\le 60$. Panels (a)–(g) show the dimensionless reduced gravity, and (h)–(n) show the dimensionless vorticity. Time increases downwards. The Supplemental Material [28] includes an animation of this experiment.

Figure 11

Hovmöller diagram of the colliding vortices, plotting time against cross sections of dimensionless reduced gravity taken at height $\overline{z}=0.25$. The red dotted lines indicate the theoretical speed of either vortex individually in an unbounded domain.

Figure 12

Plots of the chasing vortices at intervals $\mathrm{\Delta }\overline{t}=20$ for $0\le \overline{t}\le 120$. Panels (a)–(g) show the dimensionless reduced gravity, and (h)–(n) show the dimensionless vorticity. Time increases downwards. The Supplemental Material [28] includes an animation of this experiment.

Figure 13

Hovmöller diagram of the chasing vortices, plotting time against cross sections of dimensionless reduced gravity taken at height $\overline{z}=0.25$. The red dotted lines indicate the theoretical speed of either vortex individually in an unbounded domain.