## Periodic Reporting for period 1 - BLOC (Mathematical study of Boundary Layers in Oceanic Motions)

**Reporting period:**2015-09-01

**to**2017-02-28

## Summary of the context and overall objectives of the project

"Boundary layer theory is a large component of fluid dynamics. It is ubiquitous in Oceanography, where boundary layer currents, such as the Gulf Stream, play an important role in the global circulation. Comprehending the underlying mechanisms in the formation of boundary layers is therefore crucial for applications. However, the treatment of boundary layers in ocean dynamics remains poorly understood at a theoretical level, due to the variety and complexity of the forces at stake.

The goal of this project is to develop several tools to bridge the gap between the mathematical state of the art and the physical reality of oceanic motion. We will address separately Ekman layers, which are horizontal boundary layers that take place at the top and at the bottom of the ocean, and western boundary layers, which are vertical boundary layers that are localized in the vicinity of the western coast of an oceanic basin. There are four points on which we mainly focus:

1. degeneracy issues: roughly speaking, degeneracy occurs when the expected size of the boundary layer changes drastically at some point of the boundary. In oceanographic models, this might happen in two different contexts. Western boundary layers degenerate at the North and South end of an oceanic basin (this is called the ""geostrophic degeneracy""). And Ekman layers degenerate near the equator (Stewartson layers).

2. rough boundaries, meaning either non-smooth boundaries, or boundaries with small amplitude and high frequency variations;

3. the inclusion of the advection term in the construction of stationary boundary layers;

4. the linear and nonlinear stability of the boundary layers.

This project will allow us to have a better understanding of small scale phenomena in fluid mechanics, and in particular of the inviscid limit of incompressible fluids."

The goal of this project is to develop several tools to bridge the gap between the mathematical state of the art and the physical reality of oceanic motion. We will address separately Ekman layers, which are horizontal boundary layers that take place at the top and at the bottom of the ocean, and western boundary layers, which are vertical boundary layers that are localized in the vicinity of the western coast of an oceanic basin. There are four points on which we mainly focus:

1. degeneracy issues: roughly speaking, degeneracy occurs when the expected size of the boundary layer changes drastically at some point of the boundary. In oceanographic models, this might happen in two different contexts. Western boundary layers degenerate at the North and South end of an oceanic basin (this is called the ""geostrophic degeneracy""). And Ekman layers degenerate near the equator (Stewartson layers).

2. rough boundaries, meaning either non-smooth boundaries, or boundaries with small amplitude and high frequency variations;

3. the inclusion of the advection term in the construction of stationary boundary layers;

4. the linear and nonlinear stability of the boundary layers.

This project will allow us to have a better understanding of small scale phenomena in fluid mechanics, and in particular of the inviscid limit of incompressible fluids."

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Important progress has been made on the construction of nonlinear Ekman layers in general settings. In collaboration with David Gérard-Varet and Yasunori Maekawa, the PI has proved the existence and uniqueness of solutions of nonlinear boundary layer equations for rotating fluids, in general environments. One paper, written with David Gérard-Varet, has been accepted for publication in Analysis and PDE, and another paper with David Gérard-Varet and Yasunori Maekawa is in preparation and should be submitted soon.

The PI has also obtained a breakthrough result with Nader Masmoudi on boundary layer separation. They obtained the first mathematical proof of separation for the stationary Prandtl equation, and showed the relevance of the “Goldstein singularity”. The corresponding paper is almost completed, and should be submitted and posted on the server HAL very soon (in May 2017). The techniques developed by the PI and Nader Masmoudi could likely be extended to western boundary layers in oceanography, and therefore give a mathematical description of separation of the Gulf Stream.

On the other hand, the PI and Matthew Paddick studied the stabilizing effect of rotation on western boundary layers and gave a quantitative condition on the profile of the coastline that prevents recirculation.

Concerning boundary layer denegeracy, the PI and Laure Saint-Raymond have solved the problem of geostrophic degeneracy, which had remained an open problem for almost 20 years. The corresponding paper has been accepted in the Memoirs of the AMS.

Matthew Paddick and Yasunori Maekawa have exhibited some criteria for the convergence of weak solutions of the 2D incompressible Navier-Stokes equations with Navier slip boundary conditions to a strong solution of incompressible Euler.

Other tasks are still in progress. This includes :

- Work on the thermohaline circulation: this task has been addressed by the PI, Marc Briant and Laure Saint-Raymond. A lot of work has been devoted to finding the relevant models and scalings to describe the interaction between the velocity of oceanic currents, the salinity, the density and the temperature. Different sub problems have been identified and are currently under investigation.

- Work on boundary layer denegeracy: a PhD student, Jean Rax, is currently working on Stewartson boundary layers.

- Work on boundary layer separation: the question of what happens after separation is of the utmost mathematical and physical importance, and is currently under investigation by the PI and co-authors (including Frédéric Marbach, who is a post-doc within the project).

The PI has also obtained a breakthrough result with Nader Masmoudi on boundary layer separation. They obtained the first mathematical proof of separation for the stationary Prandtl equation, and showed the relevance of the “Goldstein singularity”. The corresponding paper is almost completed, and should be submitted and posted on the server HAL very soon (in May 2017). The techniques developed by the PI and Nader Masmoudi could likely be extended to western boundary layers in oceanography, and therefore give a mathematical description of separation of the Gulf Stream.

On the other hand, the PI and Matthew Paddick studied the stabilizing effect of rotation on western boundary layers and gave a quantitative condition on the profile of the coastline that prevents recirculation.

Concerning boundary layer denegeracy, the PI and Laure Saint-Raymond have solved the problem of geostrophic degeneracy, which had remained an open problem for almost 20 years. The corresponding paper has been accepted in the Memoirs of the AMS.

Matthew Paddick and Yasunori Maekawa have exhibited some criteria for the convergence of weak solutions of the 2D incompressible Navier-Stokes equations with Navier slip boundary conditions to a strong solution of incompressible Euler.

Other tasks are still in progress. This includes :

- Work on the thermohaline circulation: this task has been addressed by the PI, Marc Briant and Laure Saint-Raymond. A lot of work has been devoted to finding the relevant models and scalings to describe the interaction between the velocity of oceanic currents, the salinity, the density and the temperature. Different sub problems have been identified and are currently under investigation.

- Work on boundary layer denegeracy: a PhD student, Jean Rax, is currently working on Stewartson boundary layers.

- Work on boundary layer separation: the question of what happens after separation is of the utmost mathematical and physical importance, and is currently under investigation by the PI and co-authors (including Frédéric Marbach, who is a post-doc within the project).

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The most important outcome of the project is the result obtained by the PI and by Nader Masmoudi on boundary layer separation. They used techniques that were initially designed for the nonlinear Schrödinger equation, namely modulation of variables techniques, and combined them with maximum principle techniques and with new energy estimates. This result has already attracted a lot of attention from the mathematical community, and the PI has given a large number of talks on this subject (8 invited talks and 2 mini-courses). It is likely that the tools developed for the analysis (e.g. the new energy estimates) will be useful for the study of the Prandtl equation in other contexts (for instance, in order to prove the relevance of the Prandtl Ansatz).

Furthermore, it is expected that the work in progress on boundary layer models describing the solution beyond the separation point will also have a strong impact on the scientific community. The goal is to select boundary layer models that have good stability properties and that accurately describe the boundary layer.

Furthermore, it is expected that the work in progress on boundary layer models describing the solution beyond the separation point will also have a strong impact on the scientific community. The goal is to select boundary layer models that have good stability properties and that accurately describe the boundary layer.