On the Clark-α model of turbulence:: global regularity and long-time dynamics
On the Clark-α model of turbulence:: global regularity and long-time dynamics
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DOI：
10.1080/14685240500183756
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发表时间：
2005-01-01
影响因子：
1.9
通讯作者：
Titi, ES
中科院分区：
文献类型：
--
作者：
Cao, C;Holm, DD;Titi, ES
在本文中，我们研究了一个著名的三维湍流模型，过滤克拉克模型，或克拉克- a模型。这是一个湍流的大涡模拟（LES）张量扩散率模型，具有宽度为a的显式空间滤波器。我们证明了这个模型的整体适定性与常数的Navier-Stokes粘度。此外，我们建立了这个耗散演化系统的有限维全局吸引子的存在性，我们提供了它的分形维数和Hausdorff维数的分析估计。我们的估计与（L/l（d））（3）成比例，其中L是积分空间尺度，l（d）是粘性耗散长度尺度。这个明确的界限是一致的物理估计的自由度的数量的启发式参数的基础上。使用半严格的物理参数，我们表明，惯性范围的能量谱的克拉克α模型具有通常的k（-5/3）Kolmogorov幂律波数k α> 1。这证明了克拉克-α模型有效地参数化了惯性范围内的大波数，k α>> 1，使得它们包含的平移动能比纳维-斯托克斯方程中的对应物少得多。
In this paper we study a well-known three-dimensional turbulence model, the filtered Clark model, or Clark - a model. This is a large eddy simulation (LES) tensor-diffusivity model of turbulent flows with an explicit spatial filter of width a. We show the global well-posedness of this model with constant Navier - Stokes viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an analytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to (L/l(d))(3), where L is the integral spatial scale and l(d) is the viscous dissipation length scale. This explicit bound is consistent with the physical estimate for the number of degrees of freedom based on heuristic arguments. Using semi-rigorous physical arguments we show that the inertial range of the energy spectrum for the Clark - alpha model has the usual k(-5/3) Kolmogorov power law for wave numbers k alpha > 1. This is an evidence that the Clark - alpha model parameterizes efficiently the large wave numbers within the inertial range, k alpha >> 1, so that they contain much less translational kinetic energy than their counterparts in the Navier - Stokes equations.