In a viscous fluid, the energy dissipation is the signature of the breaking of the time-reversal symmetry (hereafter TSB) t->-t, u-> -u, where u is the velocity. This symmetry of the Navier-Stokes equations is explicitly broken by viscosity. Yet, in the limit of large Reynolds numbers, when flow becomes turbulent, the non-dimensional energy dissipation per unit mass becomes independent of the viscosity, meaning that the time-reversal symmetry is spontaneously broken. Natural open questions related to such observation are: what is the mechanism of this spontaneous symmetry breaking? Can we associate the resulting time irreversibility to dynamical processes occurring in the flow? Can we devise tools to locally measure this time irreversibility? In this talk, I first show that the TSB is indeed akin to a spontaneous phase transition in the Reversible Navier-Stokes equations, a modification of the Navier-Stokes equation suggested by G. Gallavotti to ensure energy conservation and relevance of statistical physics interpretation. I then discuss a mechanism of the TSB in Navier-Stokes was first suggested by L. Onsager in 1949, in which quasi-singularities or singularities create a non-viscous dissipation. I exhibit the tools to track these quasi-singularities. I show how the application of these tools to velocity measurements in a turbulent swirling flow allows to detect Eulerian and Lagrangian signatures of irreversibility. This enables me to evidence the structures that are responsible for irreversibility and associate them with peculiar properties of the local velocity field or trajectories.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

Stellar evolution models provide the foundation of several methods applied to study the evolutionary properties of stars and stellar populations, both Galactic and extragalactic. The accuracy of the results obtained with these techniques is tied to the accuracy of the stellar models, and in this context the correct treatment of turbulent convection is crucial. Unfortunately, the modelling of turbulent convection in stellar evolution computations is still affected by sizable uncertainties. The aim of this talk is to highlight the effect of turbulent convection on the most important stellar model predictions in the context of the study of stellar systems like star clusters and galaxies, and the (simple) prescriptions we currently use (out of necessity).

• Thematic program: Mathematics – Magnetism - Materials (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

The three principal dynamical regimes of the atmosphere and the ocean are: i) small-scale turbulence down to the smallest space and time scales ii) internal gravity waves over a wide range of spatial scales iii) geostrophically balanced eddying motion at the largest space and time scales. All regimes are of turbulent character and need parameterisations in ocean components of climate models because of the lack of coarse grid resolution. A few aspects of closures for gravity wave turbulence are presented and closures for eddies in the ocean are discussed.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

Turbulence closure models (parameterization schemes) currently used in numerical models of the atmosphere are discussed. The focus is on truncated one-equation turbulence kinetic energy (TKE) closure schemes that are arguably the present-day draft horses of operational meteorology, e.g., numerical weather prediction. Advantages and shortcomings of one-equation TKE schemes are outlined in the context of various operational constraints. A TKE scalar variance (TKESV) closure scheme is considered in some detail. The TKESV scheme carries transport equations (with due regard for the time-rate-of-change and third-order transport terms) for both the TKE and the variances and covariance of scalar quantities (e.g., temperature and humidity) that characterize turbulence potential energy. It is argued that the TKESV scheme has considerable advantages over the TKE scheme in terms of the essential physics but it can still meet severe operational requirements. Careful consideration is given to a number of tricky parameterization issues, including the pressure-scrambling effects in the Reynolds-stress and scalar-flux equations and the influence of clouds on turbulent mixing. An assumed PDF (probability distribution function) closure approach is briefly outlined. Finally, realizability of turbulence closures is considered within a more general framework of the problem of moments of the probability theory.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

Three examples from geophysical fluid dynamics will showcase mathematical modelling as the "art of judicious simplification": The computational prediction of two seasonal to decadal phenomona, the "quasi-biennial oscillation" (QBO) and the "El Niño Southern Oscillation" (ENSO) became possible only after theoreticians had captured their essential causal structures in convincing reduced mathematical models. With our own research, we aim to similarly untangle the mechanisms behind the "rapid intensification" (RI) of tropical storms during their transition to hurricane strength.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)