论文标题
Batchelor-Howells-城镇频谱：三维情况
The Batchelor--Howells--Townsend spectrum: three-dimensional case
论文作者
论文摘要
给定速度字段$ u（x，t）$，我们考虑由$ \partial_tθ+ u \ cdot \ cdot \nablaθ=δθ+ g $与时间无关源$ g（x）$支配的被动跟踪$θ$的演变。从某种意义上说，$ \ | u \ | $很小时，Batchelor，Howells and Townsend（1959，J。\ fluid Mech。在我们最近针对二维情况下的工作之后，在本文中，我们证明了BHT缩放率确实可以概率地，对于大型波数，而对于足够小的随机合成三维不可压缩的速度字段$ u（x，t）$。我们还放宽了关于速度和示踪剂源的一些假设，允许两者的有限差异和后者的全功率谱。
Given a velocity field $u(x,t)$, we consider the evolution of a passive tracer $θ$ governed by $\partial_tθ+ u\cdot\nablaθ= Δθ+ g$ with time-independent source $g(x)$. When $\|u\|$ is small in some sense, Batchelor, Howells and Townsend (1959, J.\ Fluid Mech.\ 5:134; henceforth BHT) predicted that the tracer spectrum scales as $|θ_k|^2\propto|k|^{-4}|u_k|^2$. Following our recent work for the two-dimensional case, in this paper we prove that the BHT scaling does hold probabilistically, asymptotically for large wavenumbers and for small enough random synthetic three-dimensional incompressible velocity fields $u(x,t)$. We also relaxed some assumptions on the velocity and tracer source, allowing finite variances for both and full power spectrum for the latter.
