UC Irvine

In this paper, we prove that the celebrated Arnold-Beltrami-Childress (ABC) flow with parameters A = B = C = 1, given by x = sin z + cos y, y = sin x + cos z, z = sin y + cos x, has periodic orbits on (2πT)3 with rotation vectors parallel to (1, 0, 0), (0, 1, 0), and (0, 0, 1). Despite ABC flows being studied since the 1960s, this seems to be the first time that existence of nonperturbative periodic orbits has been established for them. The main difficulty here is the lack of a variational structure for these flows, and our proof instead relies on their symmetry properties. As an application of our result, we show that the well-known G-equation model of turbulent combustion with this ABC flow on R3 has a linear (i.e., maximal possible) flame speed enhancement rate as the flow amplitude grows to infinity. To the best of our knowledge, this is the first time an asymptotic flame speed growth law has been established for a natural three-dimensional incompressible flow with such a complex structure.