Balance, breakdown of balance, and generation of inertia–gravity waves in vortical flows are fundamental topics of prime importance in geophysical fluid dynamics. This research aims to find the optimum balance relation as well as the Rossby-number variation of the spontaneous generation of inertia–gravity waves in a baroclinic, two-layer, Boussinesq model on a doubly-periodic f-plane geometry for a range of Rossby numbers between 0.07 and 0.79.
The setup of the experiments is an initially axisymmetric vortex in each layer formed by a circular contour of Rossby–Ertel potential vorticity (PV) surrounded by a background, uniform PV of the same magnitude in the two layers. The strength of the uniform PV within the upper-layer (lower-layer) vortex is stronger (weaker) with respect to the background PV. Alternatively, the PV anomaly is positive (negative) in the upper (lower) layer. This setup is amenable to baroclinic instability, which is triggered by inserting random noise of very small magnitude on the contour representation of the vortices in the two layers. The initial distribution of PV is inverted by means of the first three members of the , , and hierarchies of balance relations, comprising nine PV inversion procedures and thus nine ways of determining imbalance through wave–vortex decomposition. For each regime of flow, nine experiments are carried out using one of the nine balance relations to construct the initial conditions. The onset of instability leads to breakdown of symmetry, generation of Rossby waves around the vortices, complex vortex –vortex, wave–vortex, and wave–wave interactions. The focus is, however, on the spontaneous generation of imbalance. For each of the nine experiments, PV inversion by means of the same balance relation employed to construct the initial conditions is used to determine the amount of imbalance. The minimum imbalance is then sought over the nine balance relations.
The energy spectra of imbalance as obtained using various balance relations are also investigated in order to gain insight into the nature of imbalance. Fluctuations on the spectra appear to be related to wave–vortex and wave–wave interactions. The relation between the spectra and the evolution of linear available energy of imbalance is also studied.
As the most important outcome of the present research, the relation between the amplitude of inertia–gravity waves generated during the evolution of unstable, vortical flows and the Rossby number is obtained by spanning the parameter space by changing the strength of the PV anomaly and the nondimensional acceleration due to gravity. The results obtained agree with an exponentially-small functional relation between the amplitude of the inertia–gravity waves and the Rossby number. While this exponentially-small relation agrees with the asymptotic theories available, it is in sharp contrast with the linear relation obtained by Williams et. al, (2008) using their laboratory two-layer experiments.