  It is known that there must be some weak form of transport (called cool
bottom processing, or CBP) acting in low mass RGB and AGB stars, adding nuclei,
newly produced near the hydrogen-burning shell, to the convective envelope. We
assume that this extra-mixing originates in a stellar dynamo operated by the
differential rotation below the envelope, maintaining toroidal magnetic fields
near the hydrogen-burning shell. We use a phenomenological approach to the
buoyancy of magnetic flux tubes, assuming that they induce matter circulation
as needed by CBP models. This establishes requirements on the fields necessary
to transport material from zones where some nuclear burning takes place,
through the radiative layer, and into the convective envelope. Magnetic field
strengths are determined by the transport rates needed by CBP for the model
stellar structure of a star of initially 1.5 solar mass, in both the AGB and
RGB phases. The field required for the AGB star in the processing zone is B_0 ~
5x10^6 G; at the base of the convective envelope this yields an intensity B_E <
10^4 G (approximately). For the RGB case, B_0 ~ 5x10^4 to 4x10^5 G, and the
corresponding B_E are ~ 450 to 3500 G. These results are consistent with
existing observations on AGB stars. They also hint at the basis for high field
sources in some planetary nebulae and the very large fields found in some white
dwarfs. It is concluded that transport by magnetic buoyancy should be
considered as a possible mechanism for extra mixing through the radiative zone,
as is required by both stellar observations and the extensive isotopic data on
circumstellar condensates found in meteorites.
