In this paper we prove that for fixed positive integers α, β, m and m ≥ 2 the equation sum from j=1 ja = (sum form j=1 jβ) holds for n = 2 only when m = 2, α = 3 and β = 1, which is another improvement of LeVequ...In this paper we prove that for fixed positive integers α, β, m and m ≥ 2 the equation sum from j=1 ja = (sum form j=1 jβ) holds for n = 2 only when m = 2, α = 3 and β = 1, which is another improvement of LeVeque’s result, different from Yu’s.展开更多
In this paper, we have introduced a shell-model of Kraichnan＇s passive scalar problem. Different from the original problem, the prescribed random velocity field is non-Gaussian and σ correlated in time, and its intr...In this paper, we have introduced a shell-model of Kraichnan＇s passive scalar problem. Different from the original problem, the prescribed random velocity field is non-Gaussian and σ correlated in time, and its introduction is inspired by She and Levveque （Phys. Rev. Lett. 72, 336 （1994））. For comparison, we also give the passive scalar advected by the Gaussian random velocity field. The anomalous scaling exponents H（p） of passive scalar advected by these two kinds of random velocities above are determined for structure function with values of p up to 15 by Monte Carlo simulations of the random shell model, with Gear methods used to solve the stochastic differential equations. We find that the H（p） advected by the non-Gaussian random velocity is not more anomalous than that advected by the Gaussian random velocity. Whether the advecting velocity is non-Gaussian or Gaussian, similar scaling exponents of passive scalar are obtained with the same molecular diffusivity.展开更多
文摘In this paper we prove that for fixed positive integers α, β, m and m ≥ 2 the equation sum from j=1 ja = (sum form j=1 jβ) holds for n = 2 only when m = 2, α = 3 and β = 1, which is another improvement of LeVeque’s result, different from Yu’s.
基金Project supported by the Major Program of the National Natural Science Foundation （Grant No 10335010） and the National Natural Science Foundation-the Science Foundation of China Academy of Engineering Physics NSAF （Grant No 10576005）.Acknowledgement We are grateful to Professor She Zhen-Su for useful suggestions and Dr Sun Peng and Dr Zhang Xiao- Qiang for extensive discussion.
文摘In this paper, we have introduced a shell-model of Kraichnan＇s passive scalar problem. Different from the original problem, the prescribed random velocity field is non-Gaussian and σ correlated in time, and its introduction is inspired by She and Levveque （Phys. Rev. Lett. 72, 336 （1994））. For comparison, we also give the passive scalar advected by the Gaussian random velocity field. The anomalous scaling exponents H（p） of passive scalar advected by these two kinds of random velocities above are determined for structure function with values of p up to 15 by Monte Carlo simulations of the random shell model, with Gear methods used to solve the stochastic differential equations. We find that the H（p） advected by the non-Gaussian random velocity is not more anomalous than that advected by the Gaussian random velocity. Whether the advecting velocity is non-Gaussian or Gaussian, similar scaling exponents of passive scalar are obtained with the same molecular diffusivity.
