The fluctuating (or disturbance) fields are expressed in terms of Fourier components (i.e., in the form of plane waves with a time-dependent wave vector). Complete analytical solutions for the linear response function are found. In the presence of shear, the solutions are expressed in terms of the Legendre functions of imaginary argument.

By using these solutions, turbulence statistics (spectra, single-point correlations) are determined for an initially isotropic turbulence in inviscid and non diffusive fluid, and in viscous fluid with molecular Prandtl number equals to unity.

For the pure rotating case, the three-dimensional spectra are integrated analytically over wave space to obtain single-point correlations (i.e., scalar intensities and fluxes), and similarities between the turbulent transport of the passive contaminant and the turbulent diffusion are drawn.

In the presence of shear, the three-dimensional spectra are integrated numerically over wave space to obtain scalar intensities and fluxes. To elucidate the role of the mode, the three-dimensional spectra are integrated analytically over the waveplane The product of the integral length scales in the direction by associated single-point correlations (or equivalently, ‘two-dimensional’ energy components) in terms of Bessel functions is then determined. When the flow is unstable (i.e., the ratio of the Coriolis parameter f to the shear rate S ranges between –1 and 0) the long-time behavior of some ratios such as the normalized fluxes or the turbulent Prandtl number is well reproduced by that of their counterparts formulated from the ‘two-dimensional’ energy components.