Exploiting Marginal Stability in Slow-Fast Quasilinear Dynamical Systems

The accurate investigation of many geophysical phenomena via direct numerical simulations is computationally not possible nowadays due to the huge range of spatial and temporal scales to be resolved. Therefore advances in this field rely on the development of new theoretical tools and numerical algorithms. In this work we investigate a new mathematical formalism that exploits the property of quasi-linear systems to self-tune towards marginally stable states. The inspiration for this study comes from the asymptotic analysis of strongly stratified flows, performed by Chini et al.. The application of multi-scale analysis to this problem, justified by the presence of scale separation, yields to a simplified quasi-linear model. In this reduced description small-scale instabilities evolve linearly about a large-scale hydrostatic field (whose evolution is fully non-linear) and modify it via a feedback term. From the only assumption of scale separation, two extremely interesting features of this model arise. First the presence of the coupling term between the two dynamics and second the quasi-linearity of the dynamics. The first aspect, generally not present in the hydrostatic approximation, can capture the non-local energy transfer between the small and the large scales, which is key for the quantification of the mixing efficiency in the deep ocean. The second aspect, namely the quasi-linearity, is suggestive of the self-organisation of the dynamics about marginally-stable states. This results in a coupled evolution where the fast dynamics adapts (is slaved) to the mean field, maintaining the marginal stability of the latter. The low-dimensional evolution that arises, enables the integration of the reduced system on temporal scales comparable to the characteristic time scale of the slow dynamics, making this novel approach highly suited to the investigation of the stratified flow problem.Building upon the results obtained by Chini et al. in the present work we extend this methodology addressing three different aspects of the reduced model.As a first case we investigate the twofold nature of the fluctuation feedback, which is not sign-definite and might lead to intense bursting events where the fluctuations exhibit positive growth rates on a fast time scale. In this scenario the scale separation is temporarily lost and the two dynamics have to be co-evolved until a new marginally stable manifold can be approached. Here we propose three different co-evolution techniques and test their efficacy on a one-dimensional model problem.The second aspect we address is the presence of a finite scale separation between the two dynamics. We develop an algorithm that carefully identifies the validity regions of the quasi-linear reduction and determines the relevance of the fluctuation feedback w.r.t. the characteristic time scale of the slow dynamics and the growth rate of the fluctuations.As a third case we derive an efficient extension of the original methodology to two-dimensional model problems, deriving an evolution equation for the wavenumber of the fastest growing mode, which then get slaved to the mean dynamics.Eventually the methodologies derived in the context of the two model problems are applied and discussed for the strongly stratified flow problem.