Abstract Based on the turbulent convection model (TCM) of Li & Yang, we have studied the characteristics of turbulent convection in the envelopes of 2 and 5M⊙ stars at the red giant branch and asymptotic giant branch phases. The TCM has been successfully applied over the entire convective envelopes, including the convective unstable zone and the overshooting regions. We find that the convective motions become progressively stronger when the stellar models are located farther up along the Hayashi line. In the convective unstable zone, we find that the turbulent correlations are proportional to functions of a common factor (∇−∇ad), which explains similar distributions in those correlations. For the TCM we find that if the obtained stellar temperature structure is close to that of the mixing length theory (MLT), the convective motion will have a much larger velocity and thus be more violent. However, if the turbulent velocity is adjusted to be close to that of the MLT, the superadiabatic convection zone would be much more extended inward, which would lead to a lower effective temperature of the stellar model. For the overshooting distance, we find that the e-folding lengths of the turbulent kinetic energy k in both the top and bottom overshooting regions decrease as the stellar model is progressively located farther up along the Hayashi line, but both the extents of the decrease are not obvious. The overshooting distances of the turbulent correlation are almost the same for the different stellar models with the same set of TCM parameters. For the decay modes of the kinetic energy k, we find that they are very similar for different stellar models based on the same set of TCM parameters, and there is a nearly linear relationship between lg k and ln P for different stellar models. When Cs or α increases while the other parameters are fixed, the obtained linearly decaying distance will become longer.

Keywords stars: RGB and AGB — convection of stars — turbulent convection model: convective overshooting

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