Head-on collision of two Lamb dipoles

In this simulation there is a collion of two identical Lamb dipoles moving in opposite direction along the same line.

(You can click on an image for a somewhat larger version;

The two Lamb dipoles are initially (top-left picture) placed with their axes along the x-axis in such a way that they move towards each other (top-centre). During this motion a small tail is formed in the wake of each dipole. At a certain moment they hit each other and deform. In the top-right picture the four extrema of vorticity are closest together and from one vortex structure consisting of four patches of vorticity (a 'quadrupole'). This configuration is not stable and it falls apart in two dipole moving in opposite direction along the y-axis (bottom-left): there has been an exchange of partners. The newly formed dipoles now move away from each other (bottom-centre) and gradually assume the structure and characteristics of two Lamb dipoles again (bottom-right), leaving behind a little vorticity in their wakes.

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Some further notes concerning dipole collisions:

• A computation of such a collision of two dipoles using a Contour Dynamics Method looks very similar; see the movie at this page.
• The overall behaviour shown here is very similar to a collision of a Lamb dipole with a stress-free wall, when the dipoles sort of collides with its reflection in the wall. The main difference is the clear formation of a small tail by in the wake of each dipole before they collide (cf. the pictures above). In the case of the collision with a stress-free wall a tail is also formed but at a lower vorticity level, unseen in the pictures on that page.
• Since both dipoles start exactly at x-axis (top-left picture above), the motion and collision of the dipoles is symmetric about the axes. A collison of two dipoles not exactly head-on does not show such a symmetry and gives rise e.g. to the formation of larger tails and entrainment of surrounding fluid; see for instance the movie made with the Contour Dynamics Method at this page.
• The two Lamb dipoles used in this example are identical: they have the same strength and the same radius (these two parameters determine the Lamb dipole). Collisions of two Lamb dipoles that are not idential lead to a (much) more complicated behaviour, since asymmetric dipoles can be formed, and these asymmetric dipoles move along curved paths.
• Dipoles behave also differently in the presence of a background rotation; see for examples links on a page on bottom-topography effects under Vortex Dynamics at this site.

The motion of the Lamb dipoles is computed with a Finite Difference Method which solves the two-dimensional vorticity (Navier-Stokes) equation. Time and distances are given in dimensionless units. What you see in the simulation above are lines of tracers: passive 'particles' that advection by the motion. In this particular case, the tracers are placed on streamlines of the Lamb dipole.

===> Some details on the computation presented on this page for those who are interested.