THE ONSET SPECTRUM OF TURBULENCE

Abstract

A general theoryfor thefluctuation spectrumof the onset ofturbulence is developed, applying to systems that approach turbulence through acascadeof subharmonic bifurcations. Applied to Rayleigh—Bénard flow, we find excellent quantitative agreement with the recentexperimental data of Libchaber and Maurer. Until recent times, the onset of turbulence was viewed as a cascade of anharmonic modes leading to a quasiperiodic motion of the fluid (Landau’s theory) [1]. More recently modern genericityarguments have supplanted this infinite-cascade picture with one in which two or three such modes initially appear followed by an abrupt transition to turbulent behavior(Ruelle and Takens [2,3]). The newer theory received a measure of experimental support several years ago [4,5]. However, a newly completed experiment of higher resolution [61in the case of Rayleigh—Bénard flow has now discovered that after two such Landau bifurcations, a frequency-locking phenomenon occurs producing subharmonics of successive half orders of the original frequency. Thissubharmonic production appears to issue directly into the turbulent regime. Accordingly, neither of the previous pictures correctly describes this kind of onset phenomenon. In this letter we explore a new picture directly rooted in the subbarmonic production — thus, we do not consider how the first few bifurcations have led into this regime. However, from this point onwards the theory holds throughout the onset regime. Moreover,unlike the previous theories, the fluctuation spectrum here is quantitatively determined.