Comparison between time-evolution of centroids and local-flux vectors flor localized initial conditions in wavenumber space*

Although energy flux in the wavenumber space (k space) is the most fundamental problem in turbulence analyses, no identification method of it in anisotropic turbulence has been established due to its vectorial property. We have proposed a simple method to determine local-flux vectors of inviscid invariants and applied it to various turbulent fields[1][2]. The results were consistent with previous studies. Nazarenko and Quinn [3] examined triple cascade in Charney-Hasegawa-Mima (CHM) turbulence, where an additional positive quadratic invarinant called zonostrphy exists together with the energy and the enstrophy. They performed direct numerical simulations (DNSs) started from the initial condition localized in the k space and also applied a generalized FjΦrtoft argument to predict k-space paths of the three invariants. They inferred the cascade of the invariants from the time-evolution of their centroids in the k space owing to its analogy to the motions of the masses of the invariants, though its relation to the cascade in turbulence, where their spectra are continuously distributed in the whole k space, is not clear. It should be noted that the difference among the motions of invariants comes from the distribution of the spectra even in their simulations.

Our proposed method can be applied to such initial-value problem and directly compared with their results, which may access the adequacy of the proposed method. We performed DNSs of CHM equation with similar localized initial conditions. The results are consistent with those by Nazarenko and Quinn [3]. Since high-wavenumber modes are excited rapidly and the smoothness of the field is lost in inviscid DNSs, the importance of (hyper-)viscosity will be discussed.

¹ N.Yokoyama and M.Takaoka, J.Fluid Mech.908(2021)A17

² M.Takaoka, N.Yokoyama, and E.Sasaki, submitted to Phys.Rev.Fluid

³ S.Nazarenko and B.Quinn, Phys.Rev.Lett.103(2009)118501