Through laboratory experiments, we examine the transport, settling and resuspension of sediments as well as the influence of floating particles upon damping wave motion. Salt water is shown to enhance flocculation of clay and hence increase their settling rate. In studies modelling sediment-bearing (hypopycnal) river plumes, experiments show that the particles that eventually settle through uniform-density fluid toward a sloping bottom form a turbidity current. Meanwhile, even though the removal of particles should increase the buoyancy and hence speed of the surface current, in reality the surface current stops. This reveals that the removal of fresh water carried by the viscous boundary layers surrounding the settling particles drains the current even when their concentration by volume is less than 5%. The microscopic effect of boundary layer transport by particles upon the large scale evolution is dramatically evident in the circumstance of a mesopycnal particle-bearing current that advances along the interface of a two-layer fluid. As the fresh water rises and particles fall, the current itself stops and reverses direction. As a final example, the periodic separation and consolidation of particles floating on a surface perturbed by surface waves is shown to damp faster than exponentially to attain a finite-time arrest as a result of efficiently damped flows through interstitial spaces between particles - a phenomenon that may be important for understanding the damping of surface waves by sea ice in the Arctic Ocean (and which is well-known to anyone drinking a pint with a proper head or a margarita with rocks or slush).

# Past Mathematical Geoscience Seminar

In this talk, I show how concepts from non-equilibrium statistical physics can be employed in the study of climate. The specific problem addressed is the geophysical-scale evolution of Arctic sea ice. Using an analogy with Brownian motion, the original evolution equation for the sea ice thickness distribution function by Thorndike et al. (J. Geophys. Res. 80(33), pp. 4501 — 4513, 1975) is transformed to a Fokker-Planck-like conservation law. The steady solution is $g(h) = {\cal N}(q) h^q \mathrm{e}^{-~ h/H}$, where $q$ and $H$ are expressible in terms of moments over the transition probabilities between thickness categories. The solution exhibits the functional form used in observational fits and shows that for $h \ll 1$, $g(h)$ is controlled by both thermodynamics and mechanics, whereas for $h \gg 1$ only mechanics controls $g(h)$. We also derive the underlying Langevin equation governing the dynamics of the ice thickness $h$, from which we predict the observed $g(h)$. Further, seasonality is introduced by using the Eisenman-Wettlaufer model (Proc. Natl. Acad. Sci. USA 106, pp. 28-32, 2009) for the thermal growth of sea ice. The time-dependent problem is studied by numerically integrating the Fokker-Planck equation. The results obtained from these numerical integrations and their comparison with satellite observations are discussed.

Numerical simulation provides an important contribution to the management of oil reservoirs, and the ‘reservoir simulator’ has been an essential tool for reservoir engineers since the 1970’s. I will describe the role of the ‘well model’ in reservoir simulation. Its main purpose is to determine the production and injection flows of the reservoir fluids at the surface under a variety of operating constraints, and to supply source and sink terms to the grid cells of the reservoir model.

Advances in well technology (horizontal, multilateral, and smart wells containing flow control devices) have imposed additional demands on the well model. It must allow the fluid mixture properties to vary with position in the well, and enable different fluid streams to comingle. Friction may make an important contribution to the local pressure gradient. To provide an improved representation of the physics of fluid flow, the well is discretised into a network of segments, where each segment has its own set of variables describing the multiphase flow conditions. Individual segments can be configured to represent flow control devices, accessing lookup tables or built-in correlations to determine the pressure drop across the device as a function of the flow conditions.

The ability to couple the wells to a production facility model such as a pipeline network is a crucial advantage for field development and optimization studies, particularly for offshore fields. I will conclude by comparing two techniques for combining a network model with the reservoir simulation. One method is to extend the simulator’s well model to include the network, providing a fully integrated reservoir/well/network simulation. The other method is to run the reservoir and facility models as separate simulations coupled by a ‘controller’, which periodically balances them by exchanging boundary conditions. The latter approach allows the engineer to use a choice of specialist facility simulators.

In contemporary ecology and mathematical biology undergraduate courses, textbooks focus on competition and predation models despite it being accepted that most species on Earth are involved in mutualist relationships. Mutualism is usually discussed more briefly in texts, often from an observational perspective, and obligate mutualism mostly not at all. Part of the reason for this is the lack of a simple math model to successfully explain the observations. Traditionally, particular nonlinearities are used, which produce a variety of apparently disparate models.

The failure of the traditional linear model to describe coexisting mutualists has been documented from May (1973) through Murray (2001) to Bronstein (2015). Here we argue that this could be because of the use of carrying capacity, and propose the use of a nutrient pool instead, which implies the need for an autotroph (e.g. a plant) that converts nutrients into living resources for higher trophic levels. We show that such a linear model can successfully explain the major features of obligate mutualism when simple expressions for obligated growth are included.

The last 500 million years of Earth’s history have been punctuated by numerous episodes of abrupt climate change, some of them coincident with mass extinction events. Many of these climate events have been associated with massive volcanism, occurring during the emplacement of so-called Large Igneous Provinces (LIPs). Because of the significant impact of small modern eruptions on the Earth’s climate, a link between LIP volcanism and past climate change has been strongly advocated. Geochemical investigations of the sedimentary records which record major climate changes can give a profound insight into the proposed interactions between volcanic activity and climate. Mercury is a trace-gas emitted by modern volcanoes, which are the main source of this metal to the atmosphere. Ultimately atmospheric mercury is deposited in sediments, thus if enrichments in mercury are observed in sediments of the same age across the globe, a volcanic cause of these enrichments might be inferred. Osmium isotopes can also be used as a fingerprint of volcanic activity, as primitive basalts are enriched in unradiogenic 188Os. However, the continental crust is enriched in radiogenic 187Os. Therefore, the 187Os/188Os ratio can change with either more volcanic activity, or increased continental weathering during climate change. Changes in sedimentary mercury content and osmium isotopes can thus be used as markers of volcanism or weathering during climate events. However, a possible future step would be to quantify the amount of volcanism and/or weathering on the basis of these sedimentary excursions. The final part of this talk will introduce some simple quantitative models which may represent a first step towards such quantification, with the aim of further elaborating these models in the future.

The existence of planetary and stellar magnetic fields is attributed to the dynamo instability, the mechanism by which a background turbulent flow spontaneously generates a magnetic field by the constructive refolding of magnetic field lines. Many efforts have been made by several experimental groups to reproduce the dynamo instability in the laboratory using liquid metals. However, so far, unconstrained dynamos driven by turbulent flows have not been achieved in the intrinsically low magnetic Prandtl number $P_m$ (i.e. $Pm = Rm/Re << 1$) laboratory experiments. In this seminar I will demonstrate that the critical magnetic Reynolds number $Rm_c$ for turbulent non-helical dynamos in the low $P_m$ limit can be significantly reduced if the flow is submitted to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor more than 1000. Our finding thus points into a new paradigm for the design of new liquid metal dynamo experiments.

I will describe our research on numerical methods for atmospheric dynamical cores based on compatible finite element methods. These methods extend the properties of the Arakawa C-grid to finite element methods by using compatible finite element spaces that respect the elementary identities of vector-calculus. These identities are crucial in demonstrating basic stability properties that are necessary to prevent the spurious numerical degradation of geophysical balances that would otherwise make numerical discretisations unusable for weather and climate prediction without the introduction of undesirable numerical dissipation. The extension to finite element methods allow these properties to be enjoyed on non-orthogonal grids, unstructured multiresolution grids, and with higher-order discretisations. In addition to these linear properties, for the shallow water equations, the compatible finite element structure can also be used to build numerical discretisations that respect conservation of energy, potential vorticity and enstrophy; I will survey these properties. We are currently developing a discretisation of the 3D compressible Euler equations based on this framework in the UK Dynamical Core project (nicknamed "Gung Ho"). The challenge is to design discretisation of the nonlinear operators that remain stable and accurate within the compatible finite element framework. I will survey our progress on this work to date and present some numerical results.

Following several decades of development by applied mathematicians, models of ocean wave interactions with sea ice floes are now in high demand due to the rapid recent changes in the world’s sea ice cover. From a mathematical perspective, the models are of interest due to the thinness of the floes, leading to elastic responses of the floes to waves, and the vast number of floes that waves encounter. Existing models are typically based on linear theories, but the thinness of the floes leads to the unique and highly nonlinear phenomenon of overwash, where waves run over the floes, in doing so dissipating wave energy and impacting the floes thermodynamically. I will give an overview of methods developed for the wave-floe problem, and present a new, bespoke overwash model, along with supporting laboratory experiments and numerical CFD simulations.

Dedalus is a new open-source framework for solving general partial differential equations using spectral methods. It is designed for maximum extensibility and incorporates features such as symbolic equation entry, custom domain construction, and automatic MPI parallelization. I will briefly describe key algorithmic features of the code, including our sparse formulation and support for general tensor calculus in curvilinear domains. I will then show examples of the code’s capabilities with various applications to astrophysical and geophysical fluid dynamics, including a compressible flow benchmark against a finite volume code, and direct numerical simulations of turbulent glacial melting

In my research I model three components of the Earth system: the ice sheets, the ocean, and the solid Earth. In the first half of this talk I will describe the traditional approach that is used to model the impact of ice sheet growth and decay on global sea-level change and solid Earth deformation. I will then go on to explain how collaboration across the fields of glaciology, geodynamics and seismology is providing exciting new insight into feedbacks between ice dynamics and solid Earth deformation.