Spontaneous imbalance in the non-hydrostatic Boussinesq equations

Abstract

Whereas high-frequency waves are valid solutions to the Boussinesq equations in certain limits, their amplitudes are generally observed to be small in large-scale atmospheric and oceanic data. Traditionally, this has led to the development of balance models, reducing the dynamics to only the slow subset. Their solutions, however, can spontaneously generate imbalance in the context of the full equations. To quantify this, we calculate how much energy is transferred from the balanced to the unbalanced part of a turbulent rotating stratified flow that has been initialised to remove high frequencies. We lay out an approach to derive the time evolution of the balanced modes in which their interactions with unbalanced modes are taken into account. This enables us to calculate the budget of balanced (and unbalanced) energy. Our results show that imbalance generation occurs at scales where the Froude and Rossby numbers are still small and the energy spectrum is steep. We find that the scale at which maximum imbalance is generated depends on the peak of the energy spectrum and is invariant to the strength of rotation over the range examined. The unbalanced energy, after being transferred from the balanced component of the flow at larger scales, is cascaded forward and forms a shallow energy spectrum. The steep balanced subrange of the energy spectrum and the shallow subrange cross and form a kink in the total energy spectrum consistent with observed atmospheric and oceanic data. A frequency analysis at different wavenumbers shows that the separation of time scales breaks down at wavenumbers larger than those of maximum imbalance generation, but smaller than the kink of the energy spectrum. Below these scales, there is a single turbulent distribution of frequencies.