  A model for the motion of slender vortex filaments is extended to include the
effect of gravity. The model, initially introduced by Callegari and Ting (SIAM,
J. of App. Math., (1978), vol. 35, pp. 148-175), is based on a matched
asymptotic expansion in which the outer solution, given by the Biot-Savart law,
is matched with the inner solution derived from the Navier-Stokes equations.
Building on recent work by Harikrishnan et al (Phys. of Fluids, (2023), vol.
35) the Boussinesq approximation is applied such that the density variations
only enter in the gravity term. However, unlike Harikrishnan et al. (2023) the
density variation enters at a lower order in the asymptotic expansion, and thus
has a more significant impact on the self-induced velocity of the vortex
filament. In this regime, which corresponds to the regime studied by Chang and
Smith (J. of Fl. Mech., (2018), vol. 857), the effect of gravity is given by an
alteration of the core constant, which couples the motion of the filament to
the motion within the vortical core, in addition to a change in the
compatability conditions (evolution equations) which determine the leading
order azimuthal and tangential velocity fields in the vortex core. The results
are used to explain certain properties of bouyant vortex rings, as well as
qualitatively explore the impact of gravity on tornado type atmospheric
vorticies.
