间歇湍流的分形特征——分数维及分数阶导数的应用  

Fractal behaviors of intermittent turbulence—Applications of fractional dimension and fractional derivatives

作　　者：刘式达[1] 付遵涛[1] 刘式适[1] 

机构地区：[1]北京大学物理学院大气与海洋科学系,气候与海-气实验室,北京100871

出　　处：《地球物理学报》2014年第9期2751-2755,共5页Chinese Journal of Geophysics

基　　金：国家自然科学基金(40975027)资助

摘　　要：分数维由Mandelbrot创立已有30年,分数阶导数在1695年由L'Hospital提出已有400年的历史.本文用物理学中的间歇湍流问题说明分数维及分数阶导数的物理意义.由于间歇湍流涡旋不完全充满空间,所以其维数为2<D<3.由于小涡旋所占的比例缩小,使得小涡旋的功率减小,因此惯性区功率谱的斜率加大(即功率谱指数加大).由于大小涡旋共存,所以湍流游动距离即等待时间差别很大,因此造成涡流的异常扩散.由于湍流涡旋的串级(Cascade)或碰撞(Collision)引起的速度变化并不是暂瞬的,它会影响将来的速度场,这就引起了湍流的记忆性,因此,湍流黏性应该用有记忆的分数阶拉普拉斯算子.正是涡旋并不充满空间,所以涉及到流体力学运算的微点元、微面元和微体积元都要作修改.Fractional dimension was coined by Mandelbrot 30 years ago and the fractional derivatives proposed in 1695 by L ＂Hospital has a history of more than 400 years.In this paper,intermittent turbulence problems in physics have been applied to illustrate the physical significance of fractal dimension and fractional derivatives.Due to incomplete occupation of the intermittent turbulent eddies in the space,so the dimension of intermittent turbulence is 2〈D〈3.Due to the reduced proportion,the power of small vortex is decreased and the slope of power spectrum over the inertial range is increased（i.e.power spectrum index is increased）.The coexistence of vortices of various sizes causes that the walk distance and waiting time of turbulent eddies are very different,which results in that the turbulence is a kind of anomalous diffusion.Since the velocity variations resulted from the cascade or collision of turbulent eddies are not temporary transient,this will affect the velocity field in the future.This will lead to the memory of turbulence motions,so the eddy viscosity of turbulence should be represented by fractional Laplacian operator with memory kernel.Due to the incomplete occupation of the intermittent turbulent eddies in the space,related micro point element,micro area element and micro volume element in fluid mechanics calculation must be modified.

关 键 词：分数阶导数 分数维 间歇湍流 扩散 

分 类 号：P404[天文地球—大气科学及气象学]