Figure 9 - uploaded by James Montaldi
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Vortex pairs trapped in mirrors The amplitude of the smooth zigzags is shown in Figure 10, as a function of the angle of departure θ with the horizontal (the abscissa is θ/π). The two curves are for different initial values of Im z; the lower one for Im z = 0.2, the upper for Im z = 0.5 (where z is the mid-point of the initial position of the pair). As can be seen in the previous figure, zigzags take place in the strip −2 < Im z < 2 where S(z) < 0. The maximum of the curves is attained by pairs facing almost vertically upward. (If the pair faces exactly upward, then it tends to Im z = 2 where S(z) = 0 in infinite time.) The minimum occurs where the pair faces horizontally, and the motion has the initial point as an extremum.
Source publication
The dynamics of point vortices is generalized in two ways: first by making
the strengths complex, which allows for sources and sinks in superposition with
the usual vortices, second by making them functions of position. These
generalizations lead to a rich dynamical system, which is nonlinear and yet has
conservation laws coming from a Hamiltonian-...
Context in source publication
Context 1
... released from near z = 0 where S(z) < 0 move in a kind of smoothed zigzag, drifting along the real axis. Their trajectories are confined within a certain strip near the bottom of the trough, as depicted in Figure 9. The width of the confinement strip depends on the direction the pair initially faces. ...
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Citations
... Dynamics of poles with position-dependent strengths and its optical analogues [43] Montaldi and Tokieda (2011) This is a both conceptual and somewhat heuristic paper, which in a way reflects the authors' personalities. Some of the results appeared in chapter 3 of Anik Soulière's Ph.D. dissertation under Tokieda [50]. ...
... An interesting phenomenon is that of leapfrogging, which is usually observed only for two pairs of point vortices. However, also one pair of point vortices can leapfrog by itself (see Figure 11) if the circulation is position dependent [55]. In this case, the function ϑ in the Lagrange is nonlinear, hence this provides an interesting test case for our integrators. ...
We propose and compare several projection methods applied to variational integrators for degenerate Lagrangian systems, whose Lagrangian is of the form $L = \vartheta(\bq) \cdot \bv - H(\bq)$ and thus linear in velocities.
While previous methods for such systems only work reliably in the case of $\vartheta$ being a linear function of $\bq$, our methods are long-time stable also for systems where $\vartheta$ is a nonlinear function of $\bq$.
We analyse the properties of the resulting algorithms, in particular with respect to the conservation of energy, momentum maps and symplecticity.
In numerical experiments, we verify the favourable properties of the projected integrators and demonstrate their excellent long-time fidelity.
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... By contrast the paper [17] uses WKB theory to analyse the evolution of smallscale waves and instabilities to flows of rotating, self-gravitating fluid, Riemann ellipsoids, in a vertical magnetic field. The classical problem of the dynamics of point vortices and other distributional solutions is the subject of [18,19]. The first paper explores the dynamics of classical point vortices generalised to include sources and sinks or to allow the strengths to depend on position so they can be refracted and reflected. ...
