  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 $\alpha-\Omega$
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
$<B_\phi>\simeq3\times10^3$ G (RGB) and $<B_\phi>\simeq5\times10^3$ 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.
