A direct interaction approximation treatment of turbulence in a compressible fluid. I. Formalism

The direct interaction approximation (DIA) is used to treat turbulence in a compressible medium with constant mean gradients. The set of coupled nonlinear integrodifferential equations is derived that is satisfied by the transverse and longitudinal energy spectral functions, QT and QL, and by the transverse and longitudinal response functions, GT and GL. Finally, expressions for the average of the product of pairs of physically relevant fluctuating quantities (velocity, temperature, density) are derived in terms of QT and QL. The numerical solution of this set of equations will be presented in part II of this work.