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By Howard C. Rodean

ISBN-10: 1935704117

ISBN-13: 9781935704119

This monograph is meant to offer atmospheric scientists a uncomplicated figuring out of the actual and mathematical foundations of stochastic Lagrangian versions of turbulent diffusion.

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This monograph is meant to provide atmospheric scientists a uncomplicated realizing of the actual and mathematical foundations of stochastic Lagrangian versions of turbulent diffusion.

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Sample text

8c) is drastically changed because the quantities U1 - U 1 and 713 associated with longitudinal diffusion and the mean velocity gradient oUI/ox3 vanish. We use Eq. 9c) with 713 = 0 to get dU3 = [_Coc2 (~) + ~2 (1 + U5) 07OX333 ] dt 733 733 + (Coc)1/ 2dW3(t). 10b) is a companion in two dimensions to Eq. 10a). In addition, it is equivalent to the combination of Eqs. 18) for turbulent diffusion in one dimension. It is an alternate form of the one-dimensional model originally proposed by Wilson et al.

The proper interpretation of Eq. 2) is illustrated by Doob's Eq. 1') on p. 3) 25 26 METEOROLOGICAL MONOGRAPHS where the range of t is in the finite interval [a, bJ: a < t < b. That is, Eq. 2) should be regarded as mathematical shorthand for Eq. 3), with the small increment x(t) - x(a) replacing dx(t). There is no problem in the evaluation of the first term on the right; the classical Riemann-Stieltjes calculus is applicable. But how is the second integral to be interpreted? What value of O'[s,x(s)J in the interval a < s < t should be used?

1987) presented essentially the same mathematics in their proposal that the random displacement model be applied to turbulent diffusion in the atmosphere. These cases considered the equivalent of Eqs. lOa,b), but without the drift correction for inhomogeneous turbulence [the middle term on the right in Eq. lOa)]. Transformations have also been published for stationary inhomogeneous turbulence. Durbin (1984) transformed Eqs. 10a) and (6. 11) These equations are used in two steps to calculate the random displacements of marked particles in turbulent flow.

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Stochastic Lagrangian Models of Turbulent Diffusion by Howard C. Rodean


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