Mathematics Ph.D. Dissertations


Exact Calculations for the Lagrangian Velocity

Date of Award


Document Type


Degree Name

Doctor of Philosophy (Ph.D.)



First Advisor

Craig Zirbel (Committee Chair)

Second Advisor

Robert Green (Other)

Third Advisor

Richard Vasques

Fourth Advisor

So-Hsiang Chou


We consider a homogeneous, stationary, and divergence free random velocity field U in R2 to get a statistical description of some of its Lagrangian properties, for example, the Lagrangian auto-covariance, which is closely related to the mean-square displacement of one single particle in a turbulent flow. Velocity field U is written as a sum of finitely many Fourier modes, where each Fourier mode is characterized by an amplitude, a two-dimensional wave number, and a phase; all three can be random. We assume that random phases are independent and identically distributed and independent of other variables to get a general formula for Taylor coefficients of the Lagrangian auto-correlation. This formula is a sum over many terms, the number of which depends on the number of Fourier modes and the degree of the derivative. We prove that odd order derivatives of the Lagrangian auto-covariance vanish at t = 0 and the second order derivative is negative definite with negative main-diagonal entries, so main components of the Lagrangian auto-covariance decay quadratically for small values of t > 0. Assuming that amplitudes and wave numbers are identically distributed and letting the number of Fourier modes go to infinity dramatically reduces the number of terms for Taylor coefficients. For remaining terms, we give an interpretation as interactions among wave numbers. Finally, by assuming isotropy, we prove theoretical results and provide more detailed expressions for Taylor coefficients in terms of wave number magnitudes. We also analyze convergence of the Taylor series for terms having the highest moments of such wave number magnitudes.