We develop a magnetohydrodynamical model of Alfven wave-driven wind in open magnetic flux tubes piercing the stellar surface of Red Giant Branch (RGB) and Asymptotic Giant Branch (AGB) stars, and investigate the physical properties of the winds. The model simulations are carried out along the evolutionary tracks of stars with initial mass in the range of 1.5 to 3.0 $M_{odot}$ and initial metallicity $Z_{rm ini}$=0.02. The surface magnetic field strength being set to be 1G, we find that the wind during the evolution of star can be classified into the following four types; the first is the wind with the velocity higher than 80 km s$^{-1}$ in the RGB and early AGB (E-AGB) phases; the second is the wind with outflow velocity less than 10 km s$^{-1}$ seen around the tip of RGB or in the E-AGB phase; the third is the unstable wind in the E-AGB and thermally pulsing AGB (TP-AGB) phases; the fourth is the stable massive and slow wind with the mass-loss rate higher than 10$^{-7} M_{odot}$ yr$^{-1}$ and the outflow velocity lower than 20 km s$^{-1}$ in the TP-AGB phase. The mass-loss rates in the first and second types of wind are two or three orders of magnitude lower than the values evaluated by an empirical formula. The presence of massive and slow wind of the fourth type suggests the possibility that the massive outflow observed in TP-AGB stars could be attributed to the Alfven wave-driven wind.
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