From AMS Glossary
It is the phase difference for any frequency between two time series that yields the greatest correlation. The phase spectrum is computed as the arctangent of the ratio of the quadrature spectrum to the cospectrum. For example, turbulence usually consists of vertical velocity and potential temperature either in phase (for daytime convection) or 180° out of phase (for turbulence in a statically stable environment). However, gravity waves usually consist of vertical velocity and potential temperature that are 90° out of phase. Thus, if one were analyzing the cross spectra at night in a stable boundary layer, and found 90° phase shift for the longer wavelengths, but 180° for the shorter wavelengths, then one could infer that long-wavelength gravity waves are propagating through a region of small-eddy turbulence.