Theory of Weak Homogeneous Turbulence II

The expansion in powers of Reynolds number R for weak homogeneous turbulence is applied to one-dimensional Burgers\u27 model fluid. In this part of paper, the second approximation for the case of initially normal fluctuations is of particular interest. The relevant equations are formally identical with those in the approximation of the fourth-cumulant discard. Here, however, the vanishing fourth-cumulant is not an assumption but an outcome of the theory. As illustrating examples, three kinds of initial energy spectrum are introduced: two continuous ones and Dirac\u27s o-spectrum. In each case, the solution is obtained in an analytical form for all values of the wave-number k and the time t. The second approximation is found to be ineffective when R exceeds about 2 owing to the appearance of negative energy spectrum, and such a marginal value of R seems to diminish as the initial spectrum profile becomes sharper and sharper. Another interesting result is that contrary to expectation the second-order nonlinear transfer of energy acts so as to accelerate the decay of total energy. Lastly, a serious doubt is suggested about the convergence of the series in the range of high wave-numbers, but further discussion together with the fourth approximation and other related problems are reserved for the succeeding part