  Phase screens above a telescope pupil represent the variation of the phase of
the electromagnetic field induced by atmospheric turbulence. Instances drawn
from such statistics are represented by a vector of random phase amplitudes
which are coefficients of a linear superposition of two-dimensional basis
functions across the pupil. This work shortly reviews Fried's analysis of this
modal decomposition for the case of Kolmogorov statistics of the phase
covariance as a function of separation in the pupil plane.
  We focus on the numerical synthesis of phase screens. The statistically
independent modes are transformed into the eigen-modes of a gradient matrix as
time-dependence is introduced such that on short time scales the instances of
the phase screens are rigidly shifted into a direction imposed by some wind
velocity - known as the Taylor frozen screen approximation. This simple
technique factorizes spatial and temporal variables and aims at binding the
time dependence of the phase screens to the few expansion coefficients of the
basis functions that obey a stochastic time-dependent differential equation.
