Motion of buoyant point vortices

A general formulation is presented for studying the motion of buoyant vortices in a homogeneous ambient fluid. It extends the well-known Hamiltonian framework for interacting homogeneous point vortices to include buoyancy effects acting on the vortices. This is then used to systematically examine the buoyant one-, two-, and three-vortex problems. In doing so we find that two buoyant vortices may evolve either as a pair in bounded circular orbits or as two independent unbounded vortices that drift apart, and a criterion is found to distinguish these cases. Special attention is given to the buoyant vortex couple, consisting of two vortices of equal and opposite circulation and equal buoyancy anomaly. We show that a theoretical maximum height is generally possible for the rise (or fall) of such couples against buoyancy forces. Finally, the possibility and onset of chaotic motions in the buoyant three-vortex problem are addressed. In contrast to the homogeneous three-vortex problem, the buoyant vortex system shows evidence that chaos is present. We also demonstrate the chaotic advection of tracer parcels arising from the flow field induced by just two buoyant vortices.

Figure 1

Streamlines and notation for two interacting vortices. In this case, a vortex couple with unequal circulation strength (${\mathrm{\Gamma }}_1\ne -{\mathrm{\Gamma }}_2$) is sketched.

Figure 2

(a) Phase portrait for the case of $D={\left(4\pi \right)}^{-1}$. The contours represent values of constant $C^2$, with the thick contour representing the separatrix. It corresponds to a critical value of $C$ below which the orbits are bounded to each other, shown by the gray region. The arrows denote the direction of the trajectories in phase space with the colors representing the speed of the point through phase space with red representing fast trajectories and blue representing slower trajectories. (b) Critical curve in the $CD$ plane separating bounded and unbounded orbits.

Figure 3

Vortex trajectories in a frame of reference moving with the center of vorticity for two buoyant vortices with bounded and unbounded orbits. We have used $C=2e^{-1}\pm \epsilon$ to transition between the two cases, with $\epsilon =0.001$, and the minus (plus) sign for the (un)bounded case. In both plots ${\mathrm{\Gamma }}_1=2{\mathrm{\Gamma }}_2$ and $D={\left(4\pi \right)}^{-1}$. A close-up of the dashed rectangular region in (a) with the bounded orbits is shown in (b). The dark line corresponds to vortex 1, and the green line corresponds to vortex 2, with dots indicating the vortex starting positions.

Figure 4

(a) Trajectories of a vortex couple with negative (downward-directed) buoyancy and a circulation oriented as shown. The horizontal dashed line indicates the elevation change of the couple, $\mathrm{\Delta }\overline{y}=0.42$. Parameters are given by ${\mathrm{\Gamma }}_0=-1$, ${\tilde{g}}_0=-1$, ${\zeta }_0=-2$, and ${\eta }_0=0.01$, with the dark line corresponding to vortex 1, the green line corresponding to vortex 2, and dots indicating the vortex starting positions. (b) Maximum elevation change in the vortex couples (made dimensionless through $\mathrm{\Delta }{\overline{y}}_{*}\equiv \mathrm{\Delta }\overline{y}{\tilde{g}}_0/{\mathrm{\Gamma }}_0^2$) as a function of the vortex orientation angle, $\theta$. The singularity at $\theta =0$ represents the case of purely vertical orientation of the couple, as discussed in the text.

Figure 5

Vortex trajectories for the homogeneous (a, c, e) and buoyant (b, d, f) three-vortex problems. Each row shows vortices with the same starting positions [i.e., (1,1), (1.5,1), and (2,2), respectively, for each row, starting from the top] and same circulations (${\mathrm{\Gamma }}_i=\{1,2,3\}$, starting from the top), with the vortices in (b, d, f) having ${\tilde{g}}_i=\{0.02,0,-0.02\}$, respectively.

Figure 6

Maximum Lyapunov exponents for three homogeneous vortices (black curve) and three buoyant vortices (red curve). Circulation and initial position of the vortices are the same in both cases, corresponding to the systems plotted in Fig. 5.

Figure 7

Illustration of chaotic advection in the restricted buoyant three-vortex problem. (a) Trajectories of the two buoyant vortices with nonzero circulation (black and green curves), along with (b) the passive vortex, i.e., neutrally buoyant with vanishing circulation.

Figure 8

Numerical simulation of the horizontal drift of a buoyant vortex patch: (a) at $t=0$ and (b) at $t=100$. (c) The abscissa of the center of the vortex patch as a function of time.