The classical equations for ideal vortex dynamics are: (1) This is a conservative system and generic initial positions result in chaotic dynamics. For example, consider 25 vortices initially located on a ring, with γ_{j}=1/25. The movie on the right shows the resulting dynamics. Note that the ring eventually breaks up due to an instability and the result is chaos. |
Chaos in (1) |
However there are special configurations, called stable relative equilibria,
which are rigidly rotating solutions that persist indefinitely.
Here is an example of such a relative equilibria: |
Stable relative equilibrium of (1) |
How did we obtain this configuration? It turns out that the vortex model
(1) is intimately connected to the aggregation model of biological
swarms
which is (2) Here, ω is the angular velocity. There is a one-to-one correspondence between the steady statates x_{j}(t) = ξ_{j} of (2) and the relative equilibrium z_{j}(t) = exp(iω t)ξ_{j} of (1). More importantly, the relative equilibrium for the vortex model (1) is stable if and only if the corresponding steady state for the swarm model (2) is asymptotically stable. The movie on the right shows the evolution of a ring towards the stable lattice equilibrium using the swarming model. |
Swarming model (2) |
In real physical experiments involving vortices, there is usually some
source of ``dissipation'' which breaks the Hamiltonian structure of pure
vortex dynamics, and as a result, random initial conditions that
converge to an asymptotically stable lattice-like relative equilibrium.
This process has been dubbed ``crystallization'' by
[Durkin and Fajans (2000), Phys.Fluids 12 289-293]. One
approach to induce crystallization mathematically, is to add an
artificial damping to vortex dynamics: (3) Here, the μ term models damping. The advantage of this formulation is that any relative equilibrium of (3) is automatically a relative equilibrium of (1). Moreover the relative equilibrium of (3) is stable for any mu>0 if and only if it is stable (neutrally, in Hamiltonian sense) for (1). For example, the movie to the right shows the simulation with μ=0.1, with initial conditions as before. The ring still breaks up, but solution converges to a lattice-like relative equilibrium, rather than chaos. The dashed circle is the asymptotic prediction, see our preprint. |
Damped dynamics (3) leading to stable lattice |
Here are two more examples for unequal weights: N+1 configuration and a bunch of different weights. The slight dashed circles have radius sqrt(γ_{j}) whereas the thick dashed circle is centered at the origin and whose radius is sqrt(Σ γ_{j}). |
Damped dynamics, γ_{1}=0.3, γ_{j}=1/25, j=2..25 |
Damped dynamics, γ_{j}=1/j, j=1..25 |