Vortex Dynamics and Shear-Layer Instability in High-Intensity Cyclotrons

We show that the space-charge dynamics of high-intensity beams in the plane perpendicular to the magnetic field in cyclotrons is described by the two-dimensional Euler equations for an incompressible fluid. This analogy with fluid dynamics gives a unified and intuitive framework to explain the beam spiraling and beam breakup behavior observed in experiments and in simulations. Specifically, we demonstrate that beam breakup is the result of a classical instability occurring in fluids subject to a sheared flow. We give scaling laws for the instability and predict the nonlinear evolution of beams subject to it. Our work suggests that cyclotrons may be uniquely suited for the experimental study of shear layers and vortex distributions that are not achievable in Penning-Malmberg traps.

Figure 1

Spiraling and axisymmetrization of an elongated beam. The dynamics of the beam is given by Eq. (4), and the initial density distribution is $n(x,y)=\exp (-x^2/2{\sigma }_x^2-y^2/2{\sigma }_y^2)$, with $2{\sigma }_x=2.52$, $2{\sigma }_y=13.4$, and ${\delta }^2=0.8$. The beam propagates in the negative $y$ direction.

Figure 2

$\mathbf{E}\times \mathbf{B}$ velocity field for a highly elongated SIR beam. The velocity field is represented by red arrows, and it was computed for the density distribution $n(x)=\exp (-\beta x^2)$ if $-15\le y\le 15$ and $n(x)=0$ otherwise, with $\beta =-4\ln (10^{-3})$, so that $n=10^{-3}$ at $x=\pm 1/2$.

Figure 3

Breakup of the beam shown in Fig. 2. We added a small periodic perturbation to the initial beam profile in order to trigger the instability: for $-15\le y\le 15$, the initial beam distribution is given by $n(x)=\exp (-\beta x^2/h_p^2)$, with $\beta =-4\ln (10^{-3})$, and the perturbed layer width is $h_p=1+\alpha \cos (ky)$, with $\alpha =.025$ and $k=1.68$. The evolution of the beam is determined by solving Eq. (4). Turns 36 and 52 show the development of vortex mergers leading to the formation of a single large round cluster.