Evolution of a line vortex in stratified flow

A distributed vortex in a stratified fluid is treated using numerical simulations in two dimensions. The distributed vortex allows the density field and velocity field to evolve together, as happens in real flows behind lifting surfaces. The numerical simulations treat the anelastic equations using a spectral method in space with periodic horizontal directions and a projection method in time using the third-order Adams-Bashforth method. The primary parameter is the Froude number. For large Froude number, the distributed vorticity quickly rolls up and forms a vortex pair, approximately matching cases initiated as a fully developed vortex pair. For small Froude number, the vortices disintegrate into internal waves. The results indicate that the transition Froude number with a distributed vortex is larger than cases initiated with a vortex pair.

Figure 1

Sketch of the initial conditions, both the vortex pair (left) and the line vortex (right). The downward velocity vector indicates the velocity of the right vortex induced by the left. The left vortex has the same downward velocity (not shown).

Figure 2

Contours of vorticity for the vortex pair (left) and the line vortex (right) with constant density. Background gray is zero vorticity, while dark means positive and light means negative vorticity. There are 200 contour levels evenly distributed over the vorticity interval of $-10\ge \eta \ge 10$.

Figure 3

Vertical position of the centroid of vorticity for the line vortex (solid line) and the vortex pair (dashed line) with constant density.

Figure 4

Contours of vorticity for the vortex pair (left) and the line vortex (right) with $F_r=4$ (same contour increments as Fig. 2).

Figure 5

Vertical position of the centroid of vorticity for the line vortex (solid line) and the vortex pair (dashed line) with $F_r=4$.

Figure 6

Contours of vorticity for the vortex pair (left) and the line vortex (right) with $F_r=0.5$ for early times (same contour increments as Fig. 2).

Figure 7

Contours of vorticity for the vortex pair (left) and the line vortex (right) with $F_r=0.5$ for later times (same contour increments as Fig. 2).

Figure 8

Vertical position of the centroid of vorticity for the line vortex (solid line) and the vortex pair (dashed line) with $F_r=0.5$.

Figure 9

Horizontal position of the centroid of vorticity for the line vortex (solid line) and the vortex pair (dashed line) for the right side of the domain with $F_r=0.5$.

Figure 10

Contours of vorticity for the vortex pair (left) and the line vortex (right) with $F_r=0.1$ (same contour increments as Fig. 2).

Figure 11

Contours of vorticity for the vortex pair (left) and the line vortex (right) with $F_r=1.0$ (same contour increments as Fig. 2).

Figure 12

Contours of vorticity for the line vortex with $F_r=1.0$ for later times (same contour increments as Fig. 2).

Figure 13

Vertical position of the centroid of vorticity for the line vortex (solid line) and the vortex pair (dashed line) with $F_r=1$.

Figure 14

Horizontal position of the centroid of vorticity for the line vortex (solid line) and the vortex pair (dashed line) for the right side of the domain with $F_r=1$.

Figure 15

Contours of vorticity for the vortex pair with $F_r=0.8$ (same contour increments as Fig. 2).

Figure 16

Vertical position of the centroid of vorticity for the line vortex (solid lines) for cases with $1\le F_r\le 2$, as well as the case with $F_r=4$. The dashed line is the vortex pair with $F_r=1$.