It is known that rapidly rotating turbulent flows are characterized by the emergence of simultaneous upscale and downscale energy transfer. Indeed, both numerics and experiments show the formation of large-scale anisotropic vortices together with the development of small-scale dissipative structures. However the organization of interactions leading to this complex dynamics remains unclear. Two different mechanisms are known to be able to transfer energy upscale in a turbulent flow. The first is characterized by two-dimensional interactions among triads lying on the twodimensional, three-component (2D3C)/slow manifold, namely on the Fourier plane perpendicular to the rotation axis. The second mechanism is three-dimensional and consists of interactions between triads with the same sign of helicity (homochiral). Here, we present a detailed numerical study of rotating flows using a suite of high Reynolds number direct numerical simulations (DNS) within different parameter regimes to analyze both upscale and downscale cascade ranges. We find that the upscale cascade at wave numbers close to the forcing scale is generated by increasingly dominant homochiral interactions which couple the three-dimensional bulk and the 2D3C plane. This coupling produces an accumulation of energy in the 2D3C plane, which then transfers energy to smaller wave numbers thanks to the two-dimensional mechanism. In the forward cascade range, we find that the energy transfer is dominated by heterochiral triads and is dominated primarily by interaction within the fast manifold where kz = 0. We further analyze the energy transfer in different regions in the real-space domain. In particular, we distinguish high-strain from high-vorticity regions and we uncover that while the mean transfer is produced inside regions of strain, the rare but extreme events of energy transfer occur primarily inside the large-scale column vortices. PACS numbers: * postprint version of the manuscript published in Phys. Rev. Fluids 3, 034802, (2018) † Electronic address: michele.buzzicotti@roma2.infn.it ‡ Electronic address: hussein@rochester.edu § Electronic address: biferale@roma2.infn.it ¶ Electronic address: linkmann@roma2.infn.it arXiv:1711.07054v2 [physics.flu-dyn]

To find the path that minimizes the time to navigate between two given points in a fluid flow is known as Zermelo's problem. Here, we investigate it by using a Reinforcement Learning (RL) approach for the case of a vessel which has a slip velocity with fixed intensity, Vs, but variable direction and navigating in a 2D turbulent sea. We show that an Actor-Critic RL algorithm is able to find quasi-optimal solutions for both time-independent and chaotically evolving flow configurations. For the frozen case, we also compared the results with strategies obtained analytically from continuous Optimal Navigation (ON) protocols. We show that for our application, ON solutions are unstable for the typical duration of the navigation process, and are therefore not useful in practice. On the other hand, RL solutions are much more robust with respect to small changes in the initial conditions and to external noise, even when Vs is much smaller than the maximum flow velocity. Furthermore, we show how the RL approach is able to take advantage of the flow properties in order to reach the target, especially when the steering speed is small.Zermelo's point-to-point optimal navigation problem in the presence of a complex flow is key for a variety of geophysical and applied instances. In this work, we apply Reinforcement Learning to solve Zermelo's problem in a multi-scale 2d turbulent snapshot for both frozen-in-time velocity configurations and fully time-dependent flows. We show that our approach is able to find the quasi-optimal path to navigate from two distant points with high efficiency, even comparing with policies obtained from optimal control theory. Furthermore, we connect the learned policy with the topological flow structures that must be harnessed by the vessel to navigate fast. Our result can be seen as a first step towards more complicated applications to surface oceanographic problems and/or 3d chaotic and turbulent flows.

Advent of satellite altimetry brought into focus the pervasiveness of mesoscale eddies $${{{{{{{\bf{{{{{{{{\mathcal{O}}}}}}}}}}}}}}}}({100})$$ O ( 100 ) km in size, which are the ocean’s analogue of weather systems and are often regarded as the spectral peak of kinetic energy (KE). Yet, understanding of the ocean’s spatial scales has been derived mostly from Fourier analysis in small "representative” regions that cannot capture the vast dynamic range at planetary scales. Here, we use a coarse-graining method to analyze scales much larger than what had been possible before. Spectra spanning over three decades of length-scales reveal the Antarctic Circumpolar Current as the spectral peak of the global extra-tropical circulation, at ≈ 104 km, and a previously unobserved power-law scaling over scales larger than 103 km. A smaller spectral peak exists at ≈ 300 km associated with mesoscales, which, due to their wider spread in wavenumber space, account for more than 50% of resolved surface KE globally. Seasonal cycles of length-scales exhibit a characteristic lag-time of ≈ 40 days per octave of length-scales such that in both hemispheres, KE at 102 km peaks in spring while KE at 103 km peaks in late summer. These results provide a new window for understanding the multiscale oceanic circulation within Earth’s climate system, including the largest planetary scales.

The effects of different filtering strategies on the statistical properties of the resolvedto-subfilter scale (SFS) energy transfer are analysed in forced homogeneous and isotropic turbulence. We carry out a priori analyses of the statistical characteristics of SFS energy transfer by filtering data obtained from direct numerical simulations with up to 2048 3 grid points as a function of the filter cut-off scale. In order to quantify the dependence of extreme events and anomalous scaling on the filter, we compare a sharp Fourier Galerkin projector, a Gaussian filter and a novel class of Galerkin projectors with non-sharp spectral filter profiles. Of interest is the importance of Galilean invariance and we confirm that local SFS energy transfer displays intermittency scaling in both skewness and flatness as a function of the cut-off scale. Furthermore, we quantify the robustness of scaling as a function of the filtering type.

The relevance of two-dimensional three-components (2D3C) flows goes well beyond their occurrence in nature, and a deeper understanding of their dynamics might be also helpful in order to shed further light on the dynamics of pure two-dimenional (2D) or three-dimensional (3D) flows and vice versa. The purpose of the present paper is to make a step in this direction through a combination of numerical and analytical work. The analytical part is mainly concerned with the behavior of 2D3C flows in isolation and the connection between the geometry of the nonlinear interactions and the resulting energy transfer directions. Special emphasis is given to the role of helicity. We show that a generic 2D3C flow can be described by two stream functions corresponding to the two helical sectors of the velocity field. The projection onto one helical sector (homochiral flow) leads to a full 3D constraint and to the inviscid conservation of the total (three dimensional) enstrophy and hence to an inverse cascade of the kinetic energy of the third component also. The coupling between several 2D3C flows is studied through a set of suitably designed direct numerical simulations (DNS), where we also explore the transition between 2D and fully 3D turbulence. In particular, we find that the coupling of three 2D3C flows on mutually orthogonal planes subject to smallscale forcing leads to stationary 3D out-of-equilibrium dynamics at the energy containing scales. The transition between 2D and 3D turbulence is then explored through adding a percentage of fully 3D Fourier modes in the volume.