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Numerical experiments of dynamo action designed to understand the generation of Earth's magnetic field produce different regime branches identified within bifurcation diagrams. Notable are distinct branches where the resultant magnetic field is either weak or strong. Weak‐field solutions are identified by the prominent role of viscosity (and/or inertia) on the motion, whereas the magnetic field has a leading‐order effect on the flow in strong‐field solutions. We demonstrate the persistence of the strong‐field branch, preserving the expected force balance of Earth's core, and provide scaling laws governing its onset as parameters move toward values appropriate for the Geodynamo. We introduce a new output parameter, based on dynamically important parts of rotational and magnetic forces, that captures expected values of strong‐field solutions throughout input parameter space. This new measure of the field strength and our bounds on scaling laws can guide future studies in locating strong‐field dynamos in parameter space.

This paper is devoted to the asymptotic analysis of strongly rotating and stratified fluids, under a $\beta$-plane approximation, and within a three-dimensional spatial domain with strong topography. Our purpose is to propose a linear idealized model, which is able to capture one of the key features of western boundary currents, in spite of its simplicity: the separation of the currents from the coast. Our simplified framework allows us to perform explicit computations, and to highlight the intricate links between rotation, stratification and bathymetry. In fact, we are able to construct approximate solutions at any order for our system, and to justify their validity. Each term in the asymptotic expansion is the sum of an interior part and of two boundary layer parts: a ``Munk'' type boundary layer, which is quasi-geostrophic, and an ``Ekman part'', which is not. Even though the Munk part of the approximation bears some similarity with previously studied 2D models, the analysis of the Ekman part is completely new, and several of its properties differ strongly from the ones of classical Ekman layers. Our theoretical analysis is supplemented with numerical illustrations, which exhibit the desired separation behavior.

In this thesis, we study interpretable groups and fields in various theories of enriched fields, using tools from geometric model theory. The work is divided into the following three parts. The group configuration theorem for generically stable types. Following the proof of the group configuration theorem in the usual stable setting, we generalize it to the case of generically stable group configurations in arbitrary theories. More explicitly, we show how one can construct a type-definable group (action) from a generically stable sextuple of points satisfying the usual algebraicity and independence properties of a group configuration. On groups and fields interpretable in NTP2 fields. We show that, in NTP2 theories of enriched fields, under mild model-theoretic and algebraic assumptions, any definably amenable interpretable group admits a definable morphism to an algebraic group with purely imaginary kernel, i.e. that does not admit definable maps to the field sort with infinite image. We deduce a structure theorem for interpretable fields, which we instantiate for henselian valued fields of characteristic 0. We also extend these results to NIP (possibly enriched) differential fields, and prove a full classification of interpretable fields for differentially closed valued fields. In passing, we prove that in arbitrary theories, if K and F are definable fields such that the group of affine transformations F+ ⋊ F × can be definably embedded into an algebraic group over K, then F admits a definable field embedding into a finite extension of the field K. On groups and fields definable in D-henselian fields. Finally, we focus on the theory of D-henselian valued fields with differentially closed residue field and divisible value group, studied by Scanlon and Rideau-Kikuchi. Adapting the proof of Hrushovski’s p-configuration theorem, we prove that groups definable in the valued field sort with generically stable generics orthogonal to all differentially algebraic types, admit definable group homomorphisms to alge- braic groups, with kernels of finite rank. We then show that any definable field, in the valued field sort, with a generating subring admitting such a generic, is definably isomorphic to the valued field itself, assuming its Kolchin closure is of infinite rank.