Karhunen-Loéve expansion of the derivative of an inhomogeneous process

The properties of the Karhunen-Loéve (KL) expansion of the derivative u_x(x) of an inhomogeneous random process possessing viscous boundary-layer behavior are studied in relation to questions of efficient representation for numerical Galerkan schemes for computational simulation of turbulence. Eigenfunctions and eigenvalue spectra are calculated for the randomly forced one-dimensional Burgers' model of turbulence. Convergence of the expansion of u_x is much slower than convergence of the expansion of u(x), and direct expansion of u_x is not significantly more efficient than differentiating the expansion of u. The ordered eigenvalue spectrum of u_x is proportional to the square of the order parameter times the eigenvalue spectrum of u. The underlying cause of slow convergence is the earlier onset of locally sinusoidal behavior of the KL eigenfunctions when the expansion is performed over the entire domain of the solution.