Some Progress in Turbulence Theory 



reasonably satisfactory first approximations to a wide variety of turbulence 

 phenomena, yielding qualitatively correct behavior, with quantitative errors of 

 the order of 10% in favorable cases (e.g., the eddy diffusivity found in the section 

 on Approximation for Turbulent Diffusion). Two questions arise at this point. 

 Can the errors be bounded, or estimated, analytically, without recourse to ex- 

 perimental comparisons? Can a convergent sequence of higher approximations 

 to desired statistical functions be constructed, with the direct- interaction as a 

 base? Incomplete investigations, which we report briefly below, suggest that 

 the answer to both questions is yes, with the qualification that the estimates and 

 higher corrections may be difficult to extract and evaluate numerically. 



In the section on Dynamical Equations for Statistical Quantities, it was stated 

 that all known techniques for forming dynamical equations for the statistical 

 functions lead to divergent, infinite series or sequences of some kind. In particu- 

 lar, there is a divergent series of integrals on the right-hand side of Eq. (3), the 

 dynamical equation whose truncation yields the direct-interaction approximation 

 for turbulent dispersion by a random velocity field. As an aid to understanding 

 the nature of the divergence, its significance, and how to cope with it, we shall 

 consider a simpler case of a divergent series. Suppose that f(X) is defined by 

 the integral . . y 



f(\) 





If the integrand is expanded in a power series in X. , and the integration is per- 

 formed, the result is 



f(\) = 1 - \2 + 2! \^ - 3! \6 ^ . . . , ■ ' (23) 



a strongly divergent series. .;■; = /. ' ?• 



Although Eq. (23) is divergent, it is not meaningless. A power-series ex- 

 pansion of a function may be considered a kind of encoding. In the case of a 

 convergent series, the decoding process can be simple summation. This no 

 longer is possible for divergent series, but the code may nevertheless be un- 

 ambiguously soluble, given some qualitative information about the function. In 

 the case of Eq. (23), rapidly converging approximants to f (\) are yielded by the 

 Pade table [36]. This is an array of functions, each the ratio of two polynomials, 

 chosen so that, for polynomials of given orders in numerator and denominator, 

 the power- series expansion of the approximating function reproduces as many 

 coefficients of Eq. (23) as possible. 



The Pade technique may be stated in a more intuitive form, which brings 

 out better its broad significance. Suppose we know, or have reason to believe, 

 that an unknown function f(\) has a representation of the form 



Jp (a) da In A\ 







807 



