**Dwight Barkley**

Recent years have witnessed a profound change in our understanding of the route to turbulence in wall-bounded shear flows such as pipes, ducts, and channels. These lectures will review our current knowledge of the dynamics of transitional turbulence on a wide range of scales. Considerable focus will be given to quantifying the complex spatiotemporal intermittency observed in experiments and numerical simulations. A theoretical underpinning of the route to turbulence in subcritical shear flows will be presented. Finally, lectures will include a discussion of outstanding open questions.

Turbulence closure modelling in the coastal ocean: the essential effect of stable stratification on vertical mixing

**Hans Burchard**

Small-scale turbulent mixing in the ocean is highly variable, with eddy viscosities and diffusivities ranging over several orders of magnitudes. While turbulence in the nearly unstratified surface and bottom boundary layers is generally high and only bounded geometrically by the thickness of the boundary layers, turbulence in the stratified interior is strongly suppressed. Specifically in the coastal ocean, temporal variability is high and boundary layers may occupy a substantial part of the water column. In this presentation, turbulence closure models are introduced which account for this spatial and temporal variability. Two-equation turbulence closure models are argued to be an optimal compromise between efficiency and accuracy for the purpose of calculating vertical fluxes of momentum, heat and tracers in coastal ocean modelling. They provide enough degrees of freedom to be calibrated to the most prominent properties of coastal ocean mixing, but are still numerically robust and computationally efficient. One essential ingredient for a working turbulence cosure in the ocean is the proper calibration of the suppression of vertical mixing by stratification. Major implementational and numerical aspects are presented. Some focus will be on the inherent problem of numerically-induced mixing which together with the physically-induced mixing gives the effective mixing in ocean models. Vertically adaptive coordinates are presented as one possibility to reduce numerical mixing. Some examples for thermocline mixing in the Northern North Sea, physically and numerically induced mixing in the Western Baltic Sea as well as basinwide mixing in the Central Baltic Sea are presented. All three examples highlight the importance of using well-calibrated turbulence closure models together with vertically adaptive coordinates.

The problem of turbulence: bounding solutions to equations of fluid mechanics & other dynamical systems

**Sergei Chernyshenko**(Assisted by Giovanni Fantuzzi)

Advances in computing technology have enabled the calculation of complex and chaotic solutions to nonlinear dynamical systems, including in some cases turbulent solutions to the fundamental equations of fluid mechanics. However, numerical simulations have two inherent drawbacks. The first is that one is often interested only in a few quantities, such as the lift and drag of an aircraft, but computing them typically requires very high-fidelity simulations, whose computational cost can be prohibitive. The famous problem of turbulence consists in discovering rigorous and computationally efficient methods to calculate only the quantities of interest, without having to compute also the fine details of the flow. The second drawback is that, even when a mathematical model is known to precisely represent a physical system, the approximation error of numerical solutions cannot be calculated exactly. In safety- or performance-critical applications, overcoming the uncertainty in numerical errors necessitates calculations with higher precision than essential, or even possible.

In the last few years a rigorous general approach addressing these drawbacks has been proposed. If X is the quantity of interest, the approach gives lower and upper bounds (A and B, respectively) such that X is mathematically guaranteed to lie between A and B. This bounding framework combines a generalisation of the century-long idea of a Lyapunov function with advances in computational semi-algebraic geometry made at the start of the millennium, and it is related both to the well-known nonlinear energy stability theory and to the "background method" for bounding time averages. The crucial observation is that the bounds A and B can be computed numerically without simulating the underlying system, thus promising a reduction in computational complexity compared to current practice. At the expense of additional computational cost, bounds can also be tightened systematically as much as needed to guarantee that any safety or performance specifications are met.

In these lectures and associated exercise sessions we will introduce the theory behind this new approach, showcase specific examples, and provide a hands-on experience of computing bounds for a few simple nonlinear systems.

**Cancelled for health reasons**

**Charles Doering**

Spatiotemporal chaos

**Paul Manneville**

We will scrutinize the emergence and some properties of spatiotemporal chaos, i.e. mild turbulence, in systems with dimensions large when compared to intrinsic scales generated by instabilities. The key characteristic is the slow dynamics in time and space that results from the proximity of a bifurcation and from the continuous symmetries typical of the unbounded-domain limit. A brief review of relevant elements of stability theory and bifurcation analysis will first be given. A key feature is the continuous or discontinuous nature of the primary bifurcation away from the system's base state of interest. Considered first, the globally supercritical transition scenario takes place when the primary instability is continuous. This case is amenable to analysis via multiple-scale expansions that introduce envelopes and phases as natural tools apt to account for pattern formation and phase turbulence. The subcritical case implies the coexistence of concurrent locally stable states in both phase space and physical space. In extended systems, this leads to a spin-like reduction to be studied within the framework of statistical physics, hence an analysis of, e.g., spatiotemporal intermittency in terms of phase transitions and critical phenomena. To conclude, some general properties of far-from-equilibrium systems will be discussed.