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arXiv 0912.2003physics.flu-dyn
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一类二维turbulence广义模型的有效非线性度

Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence

Chuong V. Tran, David G. Dritschel, Richard K. Scott

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6 journal pages, to appear in Physical Review E
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摘要：

本文研究了由一类模型方程控制的广义二维湍流的小尺度行为，其中活动标量$θ=(-Δ)^{α/2}ψ$被不可压流$\u=(-ψ_y,ψ_x)$平流输送。这个家族的动力学的特点是$θ$的材料守恒，其方差$<θ^2>$优先转移到高波数。随着这种转移进行到越来越小的规模，梯度$\nablaθ$无限增长。这种增长是由于拉伸项$(\nablaθ\cdot\nabla)\u$，它的"有效非线性度"在族中的每个成员之间是不同的。这种程度取决于平流$\u$和活动标量$θ$之间的关系，范围很广，从近似线性到高度超线性。当$\nabla\u$是一个不小于$θ$的尺度的量时，线性动力学被实现，因此它对$θ$方差的直接传递不敏感，而$θ$方差几乎是被动平流的。这种情况对应于$α\ge2$，其中$\nablaθ$的生长在时间上近似呈指数增长，并且是非加速的。对于$α<2$，由于$<θ^2>$的直接转移需要$\nabla\u$的生长，因此实现了超线性动力学，从而增强了$\nablaθ$的产生。这种超线性达到了$α=1$三维湍流的二次非线性，并超过了$α<1$。通常的涡度方程（$α=2$）是边界线，其中$\nabla\u$和$θ$具有相同的尺度，将小尺度动力学的线性和非线性区域分开。我们详细讨论这些制度，强调的地方直接转让。

英文摘要：

We study the small-scale behavior of generalized two-dimensional turbulence governed by a family of model equations, in which the active scalar $θ=(-Δ)^{α/2}ψ$ is advected by the incompressible flow $\u=(-ψ_y,ψ_x)$. The dynamics of this family are characterized by the material conservation of $θ$, whose variance $<θ^2>$ is preferentially transferred to high wave numbers. As this transfer proceeds to ever-smaller scales, the gradient $\nablaθ$ grows without bound. This growth is due to the stretching term $(\nablaθ\cdot\nabla)\u$ whose ``effective degree of nonlinearity'' differs from one member of the family to another. This degree depends on the relation between the advecting flow $\u$ and the active scalar $θ$ and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when $\nabla\u$ is a quantity of no smaller scales than $θ$, so that it is insensitive to the direct transfer of the variance of $θ$, which is nearly passively advected. This case corresponds to $α\ge2$, for which the growth of $\nablaθ$ is approximately exponential in time and non-accelerated. For $α<2$, superlinear dynamics are realized as the direct transfer of $<θ^2>$ entails a growth in $\nabla\u$, thereby enhancing the production of $\nablaθ$. This superlinearity reaches the familiar quadratic nonlinearity of three-dimensional turbulence at $α=1$ and surpasses that for $α<1$. The usual vorticity equation ($α=2$) is the border line, where $\nabla\u$ and $θ$ are of the same scale, separating the linear and nonlinear regimes of the small-scale dynamics. We discuss these regimes in detail, with an emphasis on the locality of the direct transfer.

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