The Statistical Theory of Stationary Turbulence

This thesis concerns the theory of turbulence as well as that of wavelet transforms. The main contribution to the theory of turbulence is an extension of the von Karman-Howarth equation to turbulence that may be either two- or three-dimensional, and for which external forcing maintains the state of statistical equilibrium. The solution of this equation, for the range of separation where the contribution from viscous forces is negligible, yields the third-order structure function to all orders in the separation. For the case of three-dimensional turbulence, we obtain corrections, in the manner of Yakhot, to Kolmogorov's result from 1941. For the case of two-dimensional turbulence, we obtain novel predictions for the third-order structure function in the ranges dominated, respectively, by a downward enstrophy cascade and an upward energy cascade. Finally, we contribute to the theory of wavelet transforms by demonstrating the existence of a Gibbs phenomenon for the continuous wavelet transform; the overshoot of the reconstructed function, at points of discontinuity, is always smaller than the corresponding overshoot for the Fourier transform.