Chaotic and Resonant Streamlines in the ABC Flow

Zhao, Xiaohua; Kwek, Keng-Huat; Li, Jibin; Huang, Kelei

doi:10.1137/0153005

Cited by 34 publications

(22 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Upon perturbing C to a small nonzero value, this connection may become transverse. Indeed, a Melnikov perturbation analysis reveals precisely this fact [HZD98,ZKLH93,Gau85]. It thus follows from the Birkhoff-Smale Homoclinic Theorem that there exist parameters for which Equation (2.2) possesses a nontrivial 1-d hyperbolic invariant set as a solution: a suspended 2-shift.…”

Section: A Reeb Field With All Knotsmentioning

confidence: 98%

Contact topology and hydrodynamics III: knotted orbits

Etnyre

2000

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct a steady nonsingular solution to the Euler equations on a Riemannian S 3 whose flowlines trace out closed curves of all possible knot and link types. Using careful contacttopological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on S 3 . Sufficient review of concepts is included to make this paper independent of the previous works in this series.

show abstract

Section: A Reeb Field With All Knotsmentioning

confidence: 98%

Contact topology and hydrodynamics III: knotted orbits

Etnyre

2000

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…When (x, y) stay within a cell, the sign of z does not change, so z can be treated as a time variable and the system can be reduced to a 2D system. This is the approach taken in [17,42]. Then (1.2) can be written as…”

Section: Kam Regions and Edge Orbitsmentioning

confidence: 99%

Ballistic Orbits and Front Speed Enhancement for ABC Flows

McMillen¹,

Xin²,

Yu³

et al. 2016

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the non-integrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular, the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.

show abstract

“…In addition, the total force integrated over the full domain is zero. This vector field is not only an exact 3D steady-state solution to the Euler equations (NSE without the viscous term) but also the only stable solution for the NSE when the viscosity is large [46,45]. In this work we consider an equal magnitude, A = B = C ≡ N, for all terms of the vector field F. This in turn means that the curl of F is equal to F, and thus = 1 in Eq.…”

Section: Weakly-turbulent Flow Resultsmentioning

confidence: 93%