  In this paper, we propse a series expansion of the baroclinic torque in
low-Mach-number flows, so that the accuracy and universality of any buoyancy
term could be examined analytically, and new types of buoyancy terms could be
constructed and validated. We first demonstrate that the purpose of introducing
a buoyancy term is to approximate the baroclinic torque, and straightforwardly
the accuracy of any buoyancy term could be measured by the deviation of its
curl from the baroclinic torque. Then a regular perturbation method is
introduced for the elliptic equation of the hydrodynamic pressure in
low-Mach-number flows, resulting in a sequence of Poisson equations, whose
solutions lead to the series representation of the baroclinic torque and the
new types of buoyancy terms. With the error definition of buoyancy terms and
the series representation of the baroclinic torque, the classical gravitational
and centrifugal buoyancy term, as well as some other previously proposed
buoyancy terms are revisited. Finally, numerical simulations confirm that, with
a decreasing density variation or an increasing order of our newly proposed
buoyancy term, the simplified equations with one of the new types of buoyancy
terms can converge to the original low-Mach-number equations.
