Three-dimensional turbulent flows enhance velocity gradients via strong nonlinear interactions of the rate-of-strain tensor with the vorticity vector, and with itself. For statistically homogeneous flows, their total contributions to gradient production are related to each other by conservation of mass, and so are the total enstrophy and total dissipation. However, locally, they do not obey this relation and have different (often extreme) values, and for this reason both production mechanisms have been subject to numerous studies, often decomposed into multi-scale interactions. In general lines, their dynamics and contributions to the cascade processes and turbulent kinetic dissipation are different, which poses a difficulty for turbulence modelling. In this paper, we explore the consequence of the ‘Betchov’ relations locally, and show that they implicitly define a length scale. This length scale is found to be approximately three times the size of the turbulent structures and their interactions. It is also found that, while the non-locality of the dissipation and enstrophy at a given scale comes mostly from larger scales that do not cancel, the non-local production of strain and vorticity comes from multi-scale interactions. An important consequence of this work is that isotropic cascade models need not distinguish between vortex stretching and strain self-amplification, but can instead consider both entities as part of a more complex transfer mechanism, provided that their detailed point value is not required and a local average of reasonable size is sufficient.

We investigate the nonlinear evolution of pairs of three-dimensional, equal-sized and opposite-signed vortices at finite Froude and Rossby numbers. The two vortices may be offset in the vertical direction. The initial conditions stem from relative equilibria obtained numerically in the quasi-geostrophic regime, for vanishing Froude and Rossby numbers. We first address the linear stability of the quasi-geostrophic opposite-signed pairs of vortices, and show that for all vertical offsets, the vortices are sensitive to an instability when close enough together. In the nonlinear regime, the instability may lead to the partial destruction of the vortices. We then address the nonlinear interaction of the vortices for various values of the Rossby number. We show that as the Rossby number increases, destructive interactions, where the vortices break into pieces, may occur for a larger separation between the vortices, compared to the quasi-geostrophic case. We also show that for well-separated vortices, the interaction is non-destructive, and ageostrophic effects lead to the deviation of the trajectory of the pair of vortices, as the anticyclonic vortex dominates the interaction. Finally, we show that the flow remains remarkably close to a balanced state, emitting only waves containing negligible energy, even when the interaction leads to the destruction of the vortices.

In the context of linear stability analysis, considering unsteady base flows is notoriously difficult. A generalisation of modal linear stability analysis, allowing for arbitrarily unsteady base flows over a finite time, is therefore required. The recently developed optimally time-dependent (OTD) modes form a projection basis for the tangent space. They capture the leading amplification directions in state space under the constraint that they form an orthonormal basis at all times. The present numerical study illustrates the possibility to describe a complex flow case using the leading OTD modes. The flow under investigation is an unsteady case of the Blasius boundary layer, featuring streamwise streaks of finite length and relevant to bypass transition. It corresponds to the state space trajectory initiated by the minimal seed; such a trajectory is unsteady, free from any spatial symmetry and shadows the laminar–turbulent separatrix for a finite time only. The finite-time instability of this unsteady base flow is investigated using the 8 leading OTD modes. The analysis includes the computation of finite-time Lyapunov exponents as well as instantaneous eigenvalues, and of the associated flow structures. The reconstructed instantaneous eigenmodes are all of outer type. They map unambiguously the spatial regions of largest instantaneous growth. Other flow structures, previously reported as secondary, are identified with this method as relevant to streak switching and to streamwise vortical ejections. The dynamics inside the tangent space features both modal and non-modal amplification. Non-normality within the reduced tangent subspace, quantified by the instantaneous numerical abscissa, emerges only as the unsteadiness of the base flow is reduced.