This paper deals with the formation of eddies in a straight parallel or zonal flow and with the subsequent modification of the flow profile. The fluid is taken to be homogeneous and inviscid. Numerical analogues for the physical equations are developed in detail and are analyzed. The work begins with the linear theory of dynamic stability. Numerical analogues are developed to determine the evolution of perturbations. sinusoidal along the flow, which are initially prescribed with arbitrary wave number, amplitude, and tilt variations, and which are superimposed on arbitrary flows. These flows are straight-parallel and are unbounded, or are half bounded or bounded by plane surfaces. Integrations are carried out for an unbounded flow profile with an inflection point. Unstable perturbations are isolated and the unstable spectrum is determined. A numerical analogue for the finite amplitude problem, by which one can study the transfer of energy from the mean flow to the eddy then developed. The most unstable perturbation linearly determined, is taken as a small but finite disturbance. The integration is carried out and reveals the continued growth of the eddy and the modification of the mean flow. This method of investigation with added lapse rate and compressibility is discussed as an approach to turbulence, and to the modification of wind shear and lapse rate by the developed eddies. The general problem of numerical analogues for integrations requiring finite time-steps is also briefly discussed.