Perturbation analysis of baroclinic torque in low-Mach-number flows

TL;DR

This paper develops a series expansion method to analyze and construct buoyancy terms in low-Mach-number flows, enabling precise approximation of baroclinic torque and improving flow modeling accuracy.

Contribution

It introduces a perturbation-based series expansion for baroclinic torque, allowing analytical assessment and creation of new buoyancy terms in low-Mach-number flow equations.

Findings

New series representation of baroclinic torque.

Classical and new buoyancy terms analyzed and compared.

Numerical results show convergence to original equations with higher-order terms.

Abstract

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.