Mathematics
This thesis addresses the challenge of accurately representing oceanic dynamics characterized by a multitude of interacting processes in numerical models. Specifically, it focuses on the simulation of oceanic circular patterns ranging from 10 to 100 km in diameter, known as mesoscale eddies. These eddies play a critical role in transporting energy, water properties, and nutrients across the ocean. This research uses grid resolutions that directly capture some larger mesoscale eddies (resolved) while employing advanced mathematical techniques to represent the effects of smaller, unresolved eddies. The primary aim of the thesis is to develop and incorporate novel mathematical and numerical approaches into the Finite Volume Sea Ice-Ocean Model (FESOM2) to improve the representation of mesoscale eddies while maintaining manageable computational costs.
To bridge the gap between low-resolution and high-resolution simulations, the study enhances the mesoscale eddy modeling framework formulated by Juricke et al. (2019) through the implementation of new components that address unresolved dynamics. This includes the addition of an advection-based component to capture nonlinear interactions between resolved and unresolved eddies, which demonstrates positive performance. Furthermore, stochastic elements are introduced into the governing equations to better represent small-scale variability missing from deterministic formulations. In parallel, the thesis explores alternative and complementary parameterization strategies, offering fresh perspectives on modeling at partially resolved scales. Each enhancement is rigorously evaluated using a suite of diagnostic tools — many developed as part of this work — with a particular focus on spectral analysis and energy pathways. Overall, the thesis proposes an integrated approach to mesoscale eddy modeling, advancing the accuracy and consistency of ocean simulations across eddy-permitting resolutions.
Several oceanic processes depend delicately on mixing of fluid parcels, particularly across density surfaces because of its extremely small magnitude. Even a fractional deviation in its representation can therefore cause large errors in various other ocean modelling aspects like circulation or tracer distribution. Moreover, since this mixing is also vital in maintaining the global energy balance, its accurate representation is highly desirable. This thesis thus deals with the issue of spurious mixing (artificial mixing of numerical or non-physical origin) across density surfaces in general circulation ocean models. It explores ways to properly identify it and also to potentially mitigate it.
The thesis predominantly evolves around Finite volumE Sea Ice-Ocean Model (FESOM2). It develops a split-explicit external model solver together with an asynchronous time-stepping procedure that supports Arbitrary Largangian Eulerian (ALE) coordinates. It also implements a few such ALE coordinates known to reduce spurious mixing across density surfaces. The thesis then further develops a diagnostic technique that provides semi-local in space and time estimates for such spurious mixing on any grid without operator splitting. The work shows the novel solver to be less dissipative and scale better at any given workload without the need for additional temporal-filtering subcycles. It also shows the novel diagnostic technique to provide a local decomposition of various spurious mixing components. It reports levels of spurious mixing across density surfaces for different cases and how it can be much larger than the physical mixing. Finally, it provides discussion on the future possibilities and objectives.
This thesis investigates the use and behavior of balance relations for studying the nearly-geostrophic flow in a bounded domain. The goal is to derive simple, fully nonlinear models for the large scale flow in the vicinity of basin boundaries and investigate their asymptotic behavior. This will help improve our conceptual understanding and provide benchmarks for the calibration of larger numerical ocean models.
The balance models are those in which the Coriolis force balances the pressure gradient force in the limit of small Rossby number. These models are derived using a Lagrangian-based variational approach. The idea was first proposed by Salmon (1983), in which the author derived the approximate model for nearly geostrophic flow for the rotating shallow water equations. He applied the approximations on the Lagrangian of the parent fluid model and then took variations to get the Euler-Lagrangian equations, which he named as L1 balance model. Oliver (2006) generalized this idea and started with the arbitrary change in coordinates to the canonical coordinates, and then consistently truncated the transformation and the Lagrangian to a desired order. This approach gives the one-parameter generalized family of large-scale models (GLSG), among which Salmon’s L1 model is observed to be numerically well-behaved, as noted by Dritschel et al. (2017).
In the current study, we employ the approach detailed in Oliver (2006) and derived the variational L1 balance model for the shallow water equations with constant Coriolis force in the vicinity of the boundaries. At the boundary, zero-flux is assumed in the normal direction and the variational derivation of the model suggests the geostrophic balance up to O(ε) in the tangential direction. We numerically investigated how well the balance dynamics capture the shallow water equations under specified boundary conditions. For this, we initialized the full shallow water equations with the balanced state and allowed it to advect until time T. We then compared the fields using their root mean square (r.m.s.) differences and observed their asymptotic behaviour. Furthermore, Eulerian time scales are also determined at which both the models can be compared. Notably, we observed that the physical boundary interactions result in a slowdown of the time scales when compared to the time scales in the case of periodic boundaries.
Our unintentional large-scale geoengineering project, characterized
by a rapid in- crease in greenhouse gas concentrations, poses significant challenges in predicting and mitigating global and regional consequences. Climate researchers worldwide are constructing and refining climate models to understand and navigate the complex Earth system state and evolution. This thesis focuses on my contributions to this endeavor, specifically the construction, evaluation, and application of the AWI-CM3 coupled climate model.
Additionally, I address the importance of improving the energetic consistency across the critical interface between the atmosphere and ocean. This research was conducted as part of the DFG collaborative Research Center Transregio (TRR) 181 ”Energy Transfers in Atmosphere and Ocean”, which aims to develop mathematically rigorous tools for climate analysis and modelling. By focusing on the interactions between the atmosphere and ocean, I strive to enhance our under- standing of the exchange of heat, momentum, and mass, while incorporating model components for sea ice and river runoff.
I review the selection and method of computation for physical interface fluxes, introduce of stochastic remapping to conserve information across the coupling interface, and adapt vertical ocean mixing parameterizations to enhance the realism of the AWI-CM3 model. Through these efforts, I aim to contribute to the development of a comprehensive Earth System Model and advance our understanding of climate change and its societal implications.
Variational Model Reduction for Non-hydrostatic Stratified Flows in the Mid-latitude and the Equator
(2022)
This thesis studies balance models for a rotating stratified three-dimensional fluid on a tangent plane with full Coriolis force. Derivations are done for two different regions, namely mid-latitude and equator, which we considered separately. Each model is studied via a variational approach which is based on Lagrangian dynamics assuming smallness of the Rossby number and allowing for anisotropy in the horizontal length scales. We assume semigeostrophic scaling, akin to the derivation of the L 1 model by Salmon (1985) for the rotating shallow water equations. Contrary to Salmon’s derivation, we start with an arbitrary change of coordinates and then choose the transformation to fix the degeneracy on the first order of the Lagrangian, L 1, as suggested by Oliver (2006). In our setting, the full projection of the rotation vector of the Earth is considered, so that the horizontal component of the Coriolis vector is taken into account. For each model, conservation laws for the energy and the potential vorticity are valid because of the Hamiltonian structure. Our first model on f-plane is the most general model obtained so far in semi-geostrophic scaling. The other model concerns balance model on the equatorial β-plane. Under the additional assumption of construction of zero-meridional velocity as suggested by the leading order dynamics, an equatorial balance model is obtained.
Optimal balance is a numerical decomposition method of geophysical flows into a balanced and unbalanced components without any asymptotic analysis. It was introduced under optimal potential vorticity (PV) balance by Viúdez and Dritschel (2004) in a special semi-Lagrangian PV-based scheme. The method adiabatically deforms the nonlinear model into its linear form where mode decomposition is exact. It leads to a boundary value problem in time where gravity waves are removed at the linear end and a base-point coordinate is restored at the nonlinear end. This problem is solved by an iterative backward-forward nudging scheme. As global geophysical ocean models use primitive variables, we study optimal balance on an existing f-plane shallow water model in the primitive velocity-height variables. Our model, nevertheless, includes kinematic PV-inversion formulas if the PV is base point.
We, here, systematically investigate our numerical model for several design parameters. We found that optimal balance works with PV-based projectors which are the most robust choice with primitive variable-based projectors which are useful for general domains and global models. The PV-based projectors are the linear oblique projector and the base point PV. The linear oblique projector can be reformulated as a PDE-based projector preserving linear PV. Besides, the height field as a prominent candidate of base point and a linear PDE-based projector support more general cases.
The method returns high-quality balance with rapid convergence of the nudging scheme, but its convergence is, still, an open question. We proved the ''quasi-converge'' of the nudging iterates up to a small termination residual, and this residual is as small as the balance error which is of algebraic order in the time-separation parameter for a lower-dimensional system. Hence, optimal balance is an accurate diagnostic tool in primitive variables and can be implemented on complicated models without fundamental obstacles.
The main goal of this project is to investigate whether the concept of a Hubbard Tree, well established and widely used in polynomial dynamics, is also meaningful for transcendental entire functions.
For a post-critically finite polynomial, its Hubbard Tree is the unique minimal embedded tree that contains all critical points and is forward invariant under the dynamics of the polynomial (and, in a certain sense, normalized on Fatou components). It is not difficult to adapt this definition to post-singularly finite (psf) transcendental entire maps. We show, however, that there are psf entire maps that do not admit a Hubbard Tree. The reason for this is the existence of asymptotic values.
Partly in order to deal with that issue, we introduce the concept of a Homotopy Hubbard Tree. The essential difference to a Hubbard Tree is that a Homotopy Hubbard Tree is only required to be forward invariant up to homotopy relative to the post-singular set. Our main accomplishment in this work is to show that every psf transcendental entire map admits a Homotopy Hubbard Tree and that this tree is unique up to homotopy relative to the post-singular set.
As a first step towards a classification of psf entire functions in terms of Homotopy Hubbard Trees, we show that a map is uniquely determined by its tree.
The purpose of this Ph.D. thesis is to study and classify primitive ideals of the enveloping algebras U(o(\infty)) and U(sp(\infty)). Let g(\infty) denote any of the Lie algebras o(\infty) or sp(\infty). Then g(\infty)=\Bigcap_{n\geq 2}g(2n) for g(2n) = o(2n) or g(2n) = sp(2n), respectively. We show that each primitive ideal I of U(g(1)) is weakly bounded, i.e., I \ U(g(2n)) equals the intersection of annihilators of bounded weight g(2n)-modules. To every primitive ideal I of g(\infty) we attach a unique irreducible coherent local system of bounded ideals, which is an analog of a coherent local system of finite-dimensional modules, as introduced earlier by A. Zhilinskii. As a result, primitive ideals of U(g(\infty)) are parametrized by triples (x;y;Z) where x is a nonnegative integer, y is a nonnegative integer or half-integer, and Z is a Young diagram. In the case of o(1), each primitive ideal is integrable, and our classification reduces to a classification of integrable ideals going back to A. Zhilinskii, A. Penkov and I. Petukhov. In the case of sp(\infty), only 'half' of the primitive ideals are integrable, and nonintegrable primitive ideals correspond to triples (x;y;Z) where y is a half-integer.
The goal of this research is to study, understand, and robustly model and simulate microwave ablation of liver tumors. This is one of the possible treatments for cancerous tumors that appear in the liver, and there is room for a better understanding and prediction of how such interventions develop, from a mathematical point of view. The areas that could benefit from a more rigorous quantification range from planning of the invasive aspect of the procedure, influence of the vascular system, all the way to predicting the resulting damage in biological tissue. This would ultimately also benefit clinicians, that have much to gain from such results, as it would allow them to plan microwave ablation procedures on a patient-to-patient basis, thus obtaining a better outcome.
This research has been done under the supervision of Prof. Dr. Tobias Preußer, Head of Modeling and Simulation at Fraunhofer MEVIS, and Professor of Mathematics at Jacobs University Bremen, as well as Prof. Dr. Marcel Oliver, Professor of Mathematics at Jacobs University Bremen. I have also been supervised by Dr. Hanne Ballhausen and Dr. Torben Pätz, from Fraunhofer MEVIS, throughout various phases of my research.
Existence, uniqueness, and breakdown of solutions for models of chemical reactions with hysteresis
(2018)
We consider the fast reaction limit of the Keller--Rubinow model for Liesegang rings which was rigorously formulated by D. Hilhorst, R. van der Hout, M. Mimura, and I. Ohnishi in 2009; we shall refer to it as the HHMO-model in the following.
Using a combination of analytical and numerical methods, we demonstrate a mechanism which suggests that the solution of this model possesses an infinite number of precipitation regions, but these regions accumulate in a finite region of space-time. This is done by introducing modifications which simplify the HHMO-model. This simplified model is shown to be equivalent to solving a fixed point functional equation of integral type. Provided that the precipitation region is followed by non-precipitation region and vice versa, we prove that all those regions accumulate at a finite point.
Beyond the accumulation point, the solution can only be continued in a weak sense in which the precipitation indicator function takes fractional values and may be interpreted as a precipitation density function. We demonstrate the existence of that extended solution.
Its uniqueness is shown under the assumption that the precipitation attains fractional values only. The technique involves the theory of Volterra integral equation for weakly degenerate cordial kernel functions.
In separate chapter we present new results for those kernel functions.
Furthermore, numerical evidence suggests that the concentration function converges, in a well-defined sense in the long-time limit, to a self-similar solution for which an explicit expression is derived.
It is achieved when the precipitation functions is interchanged with self-similar scaling profile. That behaviour is proven after modifying the full HHMO-model according to numerical results, to be precise, the precipitation functions is assumed to converge strongly or weakly to the self-similar scaling profile.
Moreover, we establish several uniqueness theorems for the full model.
