The quasi-geostrohpic (QG) equation has been used to capture the asymptotic dynamics of the rotating stratified Boussinesq flows in the regime of strong stratification and rapid rotation. In this paper, we establish the invalidity of such approximation when the rotation-stratification ratio is either fixed to be unity or tends to unity sufficiently slowly in the asymptotic regime: the difference between the rotating stratified Boussinesq flow and the corresponding QG flow remains strictly away from zero, independently of the intensities of rotation and stratification. In contrast, we also show that the convergence occurs when the rotation-stratification ratio is fixed to be a number other than unity or converges to unity sufficiently fast. As a corollary, we compute a lower bound of the convergence rate, which blows up as the rotation-stratification ratio goes to unity.
The work of the second author was supported by a KIAS Individual Grant MG086501 at the Korea Institute for Advanced Study. The work of the third author was supported by National Research Foundations of Korea (NRF) grant NRF-2021R1A2C1092830. The authors greatly thank the anonymous referees for their significant suggestions and corrections. The authors' gratitude also goes to Tai-Peng Tsai for his keen comments on important changes in the statements.\\ast Received by the editors March 15, 2023; accepted for publication (in revised form) January 22, 2024; published electronically May 3, 2024. https://doi.org/10.1137/23M1559130 Funding: The work of the second author was supported by a KIAS Individual Grant MG086501 at the Korea Institute for Advanced Study. The work of the third author was supported by National Research Foundations of Korea (NRF) grant NRF-2021R1A2C1092830. \\dagger Mathematics Department, Duke University, Durham, NC 27708 USA (minjun.jo@duke.edu). \\ddagger Department of Mathematics, Ajou University, Suwon, 16499, Republic of Korea (junha02@ kias.re.kr). \\S Department of Mathematics, Chung-Ang University, Seoul, 06974, Republic of Korea (jhleepde@ cau.ac.kr).
