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

Abstract

Abstract. 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 break up behavior observed in experiments and in simulations.In particular, we demonstrate that beam break up 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.Cyclotrons are efficient and reliable tools for the acceleration of high intensity hadron beams [1]. They are considered a promising option for new applications, including neutrino physics experiments [2,3] and accelerator driven systems [4,5]. For these high intensity applications, a detailed understanding of the beam dynamics is required to avoid uncontrolled beam loss and activation of the structures. Accordingly, a large part of the theoretical effort has focused on characterizing the influence of space charge on beam quality [6,7,8,9,10]. Particle-in-cell (PIC) codes [11] have been predominantly used for that purpose. When combined with supercomputers, they provide quantitative answers that guide the design of machines or help interpret measurements [12]. Even if so, these high performance solvers have large run times, which limits the ability to explore the full range of parameters and configurations, and to identify scaling laws. A complementary approach to study space charge effects is to develop reduced models that retain only the key physical mechanisms to more readily yield scaling laws and physical intuition.In this letter, we derive a simple fluid model by considering a simplified description of cyclotrons. A self-contained mathematical derivation of the model was given in [13]. Here, we present a more physically intuitive derivation. The assumptions are as follows. First, the confining magnetic field is B = B 0 e z , where B 0 is a constant. Second, we focus on coasting beams, that are not accelerated. The motivation here is to describe a regime that has been extensively studied, precisely to characterize space charge effects [8,10]. Our third simplification is to only consider the dynamics in the plane perpendicular to the magnetic field, with physical quantities that only depend on the two coordinates describing that plane. This

FFT-based free space Poisson solvers: why Vico-Greengard-Ferrando should replace Hockney-Eastwood

“…with free-space boundary conditions, such that the electric field E = −∇φ vanishes infinitely far from the charge distribution [1][2][3][4][5][6] . This is the case when the physical system is such that the charge distribution is indeed in free space.…”

Fast and spectrally accurate evaluation of gyroaverages in non-periodic gyrokinetic-Poisson simulations

“…At this point, we have turned (2.1) and (2.2) into the following new system of equations: Unsurprisingly, this system shares many similarities with the two-dimensional inviscid Euler equations in vorticity–streamfunction form (Plunk et al. 2010; Cerfon 2016). The major difference with the Euler equations is that the term couples the dynamics at different values of .…”

“…However, the method we present here is applicable to the more general gyrokinetic systems commonly used to study astrophysical and fusion plasmas. We should mention that (2.1) and (2.2) are a surprisingly accurate description of the dynamics of a beam of charged particles in the plane perpendicular to the magnetic field in high intensity cyclotrons (Cerfon et al 2013;Guadagni 2015;Cerfon 2016 (Peterson & Hammett 2013;Guadagni 2015). Clearly, the challenge that is specific to gyrokinetics is the numerical evaluation of the gyroaverage for the charge density in (2.2) and of the gyroaveraged potential (2.4) when these quantities are not periodic.…”

Crystallization of Bosonic Quantum Hall States

“…By increasing the condensate density such that many Landau levels become populated, we observe a crossover from LLL behaviour to a hydrodynamic instability driven by the sheared internal velocity profile. Analogous phenomena are ubiquitous throughout hydrodynamics, from the diocotron instability in charged plasmas [63] and fragmentation of electron beams [64], to the Kelvin-Helmholtz instability in atmospheric and astrophysical systems [65,66]. In the context of superfluids, for which the circulation is quantized, the Kelvin-Helmholtz instability has been detected in liquid helium [67], and theoretically predicted at the boundary between counterflowing condensates [68,69].…”

“…The Coriolis force 2mv × Ω on each fluid element resulting from the shear flow v = (0, −ω c x) perfectly balances the local gradient of meanfield energy, resulting in an inhomogeneous equilibrium density despite the absence of any scalar potential. Our hydrodynamic stability analysis about this equilibrium state reveals a dynamical snaking instability of the cloud [46] the Kelvin-Helmholtz instability of counterflow in fluid layers [65,66], and the diocotron instability of charged plasmas and electron beams [63,64]. The absence of quantum pressure means that the Thomas-Fermi radius and cyclotron frequency provide the only lengthscale and rate.…”