Hamiltonian structure of inviscid rotating horizontal convection

Abstract

A stratified and rotating, ideal (i.e., inviscid and nondiffusive) fluid, contained in a box subject to meridional differential heating (“forcing”), is described by two state vectors. These represent (1) the three-dimensional center-of-mass (COM) location relative to the basin center and (2) the nonlinearly coupled basin-averaged angular momentum vector, associated with overturning circulations in zonal, meridional, and horizontal directions. Without wind, horizontal circulation (associated with the vertical component of the basin-averaged angular momentum) is absent. The COM displacements can be likened to the motion of a "fluid pendulum." In addition to the two angles characterizing the motion of a spherical pendulum relative to the vertical and zonal directions, a stratified fluid’s "pendulum length"—proportional to the fluid’s stratification strength—is changing dynamically too. In this ideal fluid setting, in the absence of forcing or rotation, the remaining equations, though nonlinear, are integrable, and circulation and COM motion are periodic. When rotation and forcing (i.e., differential heating in the meridional plane) are both present, assuming the rotation axis to be antiparallel to gravity, direct integration of the governing equations reveals that the dynamics is inconclusive: both periodic and aperiodic state trajectories are observed. This is attributed to the fact that the resulting two-degrees-of-freedom Hamiltonian system is subject to the Kolmogorov-Arnold-Moser (KAM) theorem. The latter predicts that some of the periodic orbits that occur without rotation or forcing are replaced by chaotic orbits. As this Hamiltonian system, including gyroscopic terms, can be rewritten as a forced, complex Duffing equation, these results demonstrate the presence of Hamiltonian chaos in this type of Duffing equation.