The available information on isotopic abundances in the atmospheres of low-mass Red Giant Branch (RGB) and Asymptotic Giant Branch (AGB) stars requires that episodes of extensive mixing occur below the convective envelope, reaching down to layers close to the hydrogen burning shell (Cool Bottom Processing). Recently \cite{Busso:2007jw} suggested that dynamo-produced buoyant magnetic flux tubes could provide the necessary physical mechanisms and also supply sufficient transport rates. Here, we present an $��-��$ dynamo in the envelope of an RGB/AGB star in which shear and rotation drain via turbulent dissipation and Poynting flux. In this context, if the dynamo is to sustain throughout either phase, convection must resupply shear. Under this condition, volume-averaged, peak toroidal field strengths of $\simeq3\times10^3$ G (RGB) and $\simeq5\times10^3$ G (AGB) are possible at the base of the convection zone. If the magnetic fields are concentrated in flux tubes, the corresponding field strengths are comparable to those required by Cool Bottom Processing.

Replaced to correct small error in published version: In \S 2.1, paragraphs 2 and 3 incorrectly refer to the poloidal field when qualitatively discussing magnetic diffusion in the shear zone. The correct physical interpretation is that the toroidal field diffuses through the shear zone consistent with the value of $\beta_\phi$