Ian Roulstone

Quaternions and particle dynamics in the Euler fluid equations

Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet-Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.

Holomorphic structures in hydrodynamical models of nearly geostrophic flow

We study complex structures arising in Hamiltonian models of nearly geostrophic flows in hydrodynamics. In many of these models an elliptic Monge–Ampère equation defines the relationship between a ‘balanced’ velocity field, defined by a constraint in the Hamiltonian formalism, and the materially conserved potential vorticity. Elliptic Monge–Ampère operators define an almost–complex structure, and in this paper we show that a natural extension of the so–called geostrophic momentum transformation of semi–geostrophic theory, which has a special importance in theoretical meteorology, defines Kahler and special Kähler structures on phase space. Furthermore, analogues of the ‘geostrophic momentum coordinates’ are shown to be special Lagrangian coordinates under conditions which depend upon the physical approximations under consideration. Certain duality properties of the operators are studied within the framework of the Kähler geometry.

Examples of quaternionic and Kähler structures in Hamiltonian models of nearly geostrophic flow

We study 2-forms on phase spaces of Hamiltonian models of nearly geostrophic flows. A quaternionic structure is identified, and the complex part of a symplectic representation of this structure corresponds to an elliptic Monge - Ampere equation. The real part is an invariant Kahler structure.

The mathematical structure of theories of semigeostrophic type

In the past twenty years, semigeostrophic equations have become a prominent model for describing certain atmospheric motions on a synoptic scale, including the presence of fronts. Theoretical studies of them have revealed Hamiltonian features, and novel numerical methods, motivated by the need to improve weather forecasts, have been explored. A shallow–water theory analogue has been used as a paradigm for some aspects. This paper sets out to uncover the mathematical structure of the semigeostrophic equations that has been essential to finding solutions and developing numerical techniques. We study the shallow–water and atmospheric theories side–by–side, and we introduce a generalized form which encapsulates the differences between them. When the Coriolis parameter, f, is a constant, it is found that a lift transformation is at the heart of the theory, and the consequences of this are developed. When f is not a constant, the role of the lift transformation is, in some respects, looser; we explore the extent to which it still offers a worthwhile guide. In particular, it can be viewed as motivating a generalization of the geostrophic momentum transformation for planetary semigeostrophic equations. The paper is broadly self–contained, and it takes account of several different strands in the existing literature.

A geometric interpretation of coherent structures in Navier–Stokes flows

The pressure in the incompressible three-dimensional Navier–Stokes and Euler equations is governed by Poissons equation: this equation is studied using the geometry of three-forms in six dimensions. By studying the linear algebra of the vector space of three-forms Λ3W* where W is a six-dimensional real vector space, we relate the characterization of non-degenerate elements of Λ3W* to the sign of the Laplacian of the pressure—and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, Δp, satisfies Δp>0, the three-form associated with Poissons equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When Δp<0, a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence.

Families of lift and contact transformations

A study of the contact transformation is carried out, with special emphasis on the lift transformation. The latter is the more basic concept, because it applies to a single curve, rather than to pairs of them, in the first instance. From that viewpoint the lift transformation can also transform an unfamiliar system of partial differential equations to a familiar one, without reference to contact. We illustrate this by showing how the semi-geostrophic equations of meteorology transform to hamiltonian form. General and special lift transformations are distinguished. New and very wide families of such transformations are derived. These give a desirable perspective to particular cases such as the Legendre transformation and the geostrophic transformation. A brief study of generalized duality is described. The distinction between regular and singular lifted curves is discussed in general terms, and their participation in a contact transformation is illustrated by an accessible physical example.

Weak constraints in four-dimensional variational data assimilation

The formulation of four-dimensional variational data assimilation allows the incorporation of constraints into the cost function which need only be weakly satisfied. In this paper we investigate the value of imposing conservation properties as weak constraints. Using the example of the two-body problem of celestial mechanics we compare weak constraints based on conservation laws with a constraint on the background state. We show how the imposition of conservation-based weak constraints changes the nature of the gradient equation. Assimilation experiments demonstrate how this can add extra information to the assimilation process, even when the underlying numerical model is conserving.

Hyper-Kähler geometry and semi-geostrophic theory

We use the formalism of Monge–Ampère operators to study the geometric properties of the Monge–Ampère equations arising in semi-geostrophic (SG) theory and related models of geophysical fluid dynamics. We show how Kähler and hyper-Kähler structures arise, and the Legendre duality arising in SG theory is generalized to other models of nearly geostrophic flows.

A dynamical systems analysis of the data assimilation linked ecosystem carbon (DALEC) models

Changes in our climate and environment make it ever more important to understand the processes involved in Earth systems, such as the carbon cycle. There are many models that attempt to describe and predict the behaviour of carbon stocks and stores but, despite their complexity, significant uncertainties remain. We consider the qualitative behaviour of one of the simplest carbon cycle models, the Data Assimilation Linked Ecosystem Carbon (DALEC) model, which is a simple vegetation model of processes involved in the carbon cycle of forests, and consider in detail the dynamical structure of the model. Our analysis shows that the dynamics of both evergreen and deciduous forests in DALEC are dependent on a few key parameters and it is possible to find a limit point where there is stable sustainable behaviour on one side but unsustainable conditions on the other side. The fact that typical parameter values reside close to this limit point highlights the difficulty of predicting even the correct trend without sufficient data and has implications for the use of data assimilation methods.

Mathematical and Algorithmic Aspects of Atmosphere-Ocean Data Assimilation

The nomenclature “data assimilation” arises from applications in the geosciences where complex mathematical models are interfaced with observational data in order to improve model forecasts. Mathematically, data assimilation is closely related to filtering and smoothing on the one hand and inverse problems and statistical inference on the other. Key challenges of data assimilation arise from the high-dimensionality of the underlying models, combined with systematic spatio-temporal model errors, pure model uncertainty quantifications and relatively sparse observation networks. Advances in the field of data assimilation will require combination of a broad range of mathematical techniques from differential equations, statistics, probability, scientific computing and mathematical modelling, together with insights from practitioners in the field. The workshop brought together a collection of scientists representing this broad spectrum of research strands. Mathematics Subject Classification (2000): 65C05, 62M20, 93E11, 62F15, 86A22, 49N45. Introduction by the Organisers The workshop Mathematical and Algorithmic Aspects of Atmosphere-Ocean Data Assimilation, organised by Andreas Griewank (Berlin), Sebastian Reich (Potsdam), Ian Roulstone (Surrey), and Andrew Stuart (Warwick) was held 2 December – 8 December 2012. The meeting was attended by over 45 participants representing a broad range of mathematical subject areas as well as practical aspects of atmosphere-ocean data assimilation. 3418 Oberwolfach Report 58/2012 A total of 23 talks were presented during the workshop. The talks were selected both to cover novel mathematical developments and to point towards practical advances and challenges in atmosphere-ocean data assimilation. Talks relating to mathematical developments include those on, e.g., Lagrangian data assimilation (Chris Jones, Amin Apte), particle filters (Chris Snyder, Dan Crisan, Wilhelm Stannat), ensemble Kalman filters (Georg Gottwald, Lars Nerger, Tijana JanjicPfander, Roland Potthast), statistical inference (Youssef Marzouk, Andreas Hense, Illia Horenko), variational techniques (Eldad Haber, Philippe Toint, Jim Purser, Arnd Rösch, Michael Hinze), model/representativity errors (Nancy Nichols, Alberto Carrassi) and mathematical fluid dynamics (Edriss Titi). Talks relating to practical advances and challenges in atmosphere-ocean data assimilation include those by Chris Jones, Roland Potthast, Andreas Hense, Chris Snyder, Hendrik Elbern, Georg Craig, Alberto Carassi, Patrick Heimbach). A poster session was held on Tuesday evening which gave the attending PhD students and postdocs the opportunity to present and discuss their work. Throughout the workshop a large number of spontaneous discussion groups arose triggered by the many different facets of data assimilation presented during the talks. The following discussion groups in the central lecture hall of the MFO shall be mentioned in particular: (i) filter stability (inspired by the talks by Wilhelm Stannat and Roland Potthast), (ii) Bayesian inference and optimal transportation (inspired by the talk by Youseff Marzouk), and (iii) high-dimensional particle filters (inspired by the talks by Chris Snyder and Dan Crisan). These examples also reflect the actuality and broad scientific appeal of data assimilation. Aspects of Atmosphere-Ocean Data Assimilation 3419 Workshop: Mathematical and Algorithmic Aspects of Atmosphere-Ocean Data Assimilation