Flows of a viscous fluid confined in a shallow rotating annulus with imposed temperature gradients

Abstract

A numerical study is made of flows of a fluid confined in a shallow rotating container characterized by horizontally-imposed thermal gradients, which have been recognized as the benchmark model for the baroclinic instability. Extensive numerical solutions to the governing Navier-Stokes equations are acquired for the large and small Richardson number, in order to capture the qualitative essentials of three-dimensional and symmetric baroclinic waves of finite-amplitude. The structures and energetics of baroclinic waves are described in detail. In the first, cases for the large Richardson number are considered. Flow and heat transfer characteristics of baroclinic waves using a new rotating annulus, referred to as the Miller-Fowlis (MF) model, are investigated. Special features of this annulus are: (1) the aspect ratio (height to width) is small, (2) the thermal gradients are imposed on the horizontal boundaries. In an effort to expand upon the previous experimental measurements, comprehensive numerical data are acquired for the finite-amplitude wave-present flow regime. A highly-accurate basic state is not significantly affected by the Ekman number, and the Richardson number is a function of the Prandtl number only. Parallel numerical efforts are devoted to simulating fully nonlinear symmetric baroclinic waves for small Richardson number, 0.2$\leq$Ri$\leq$2.0. By spanning a wide range of the thermal Rossby number, both the hydrostatic ($Ro>>1$) and nonhydrostatic (Ro$\sim$O(1)) cases are taken into account. Results of the numerical parameter study are summarized in a stability diagram constructed in Ro-Ri domain for the modified Hadley cell model, which exhibit similar features to those for the Eady and Hadley cell models. Two regimes of symmetric baroclinic waves are observed by exemplifying the time-dependent characteristics of the waves. The steady finite-amplitude baroclinic waves are developed for all the nonhydrostatic cases of Ri less than Ric and the hydro...