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Starting from hydrodynamic equations, we have established a set ofhydrodynamic equations for average flow and a set of dynamic equations of auto-and cross-correlations of turbulent velocity and temperature fluctuations,following the classic Reynold's treatment of turbulence. The combination of thetwo sets of equations leads to a complete and self-consistent mathematicalexpressions ready for the calculations of stellar structure and oscillations.In this paper, non-locality and anisotropy of turbulent convection areconcisely presented, together with defining and calibrating of the threeconvection parameters ($c_1$, $c_2$ and $c_3$) included in the algorithm. Withthe non-local theory of convection, the structure of the convective envelopeand the major characteristics of non-adiabatic linear oscillations aredemonstrated by numerical solutions. Great effort has been exercised to thechoice of convection parameters and pulsation instabilities of the models, theresults of which show that within large ranges of all three parameters ($c_1$,$c_2$ and $c_3$) the main properties of pulsation stability keep unchanged.

Turbulent convection and pulsation stability of stars - I. Basic equations for calculations of stellar structure and oscillations