Past Mathematical Geoscience Seminar

13 July 2018
14:00
Abstract

In the Boussinesq framework, velocity couples to density fluctuations whereas in magnetohydrodynamic turbulence, the velocity field is coupled to the magnetic field. Both systems support waves (inertia-gravity in the presence of rotation, or Alfvén), with anisotropic dispersion relations. What kind of turbulence regimes result from the interactions between waves and nonlinear eddies in such flows? And what is delimiting these regimes?

I shall sketch the phenomenological framework for rotating stratified turbulence within which one is led to scaling laws in terms of the Froude number, Fr=U/[LN], which measures the relative celerity of gravity waves and nonlinear eddies, with U and L characteristic velocity and length scale, and N the Brunt-V\"ais\"al\"a frequency. These laws apply to the mixing efficiency of such flows, indicating the relative roles of the buoyancy flux due to the waves, and of the measured kinetic and potential energy dissipation rates. Various measures of mixing are found to follow power laws in terms of the Froude number, and may differ for the three regimes that can be identified, namely the wave-dominated, wave-eddy balance and eddy-dominated domains [1]. In particular, in the intermediate regime, the effective dissipation varies linearly with Fr, in agreement with simple wave-turbulence arguments. This analysis is inspired by and corroborates results from a large parametric study using direct numerical simulations (DNS) on grids of 1024^3 points, as well as from atmospheric and oceanic observations.

Such scaling laws can be related to previous DNS results concerning the existence for the energy of bi-directional constant-flux cascades to both the small scales and to the large scales, due to the presence of rotation in such flows, as measured for example in the ocean. These dual energy cascades lead to an alteration, and a decrease, of the mixing and available energy to be dissipated in the small scales [2]. Some perspectives might also be given at the end of the talk.

 

[1] A. Pouquet, D. Rosenberg, R. Marino & C. Herbert, Scaling laws for mixing and dissipation in unforced rotating stratified turbulence. J. Fluid Mechanics 844, 519, 2018.
[2] R. Marino, A. Pouquet & D. Rosenberg, Resolving the paradox of oceanic large-scale balance and small-scale mixing. Physical Review Letters 114, 114504, 2015.

  • Mathematical Geoscience Seminar
15 June 2018
14:15
Nathalie Vriend
Abstract

A granular material forms a distinct and fascinating phase in physics -- sand acts as a fluid as grains flow through your fingers, the fallen grains form a solid heap on the floor or may suspend in the wind like a gas.

The main challenge of studying granular materials is the development of constitutive models valid across scales, from the micro-scale (collisions between individual particles), via the meso-scale (flow structures inside avalanches) to the macro-scale (dunes, heaps, chute flows).

In this talk, I am highlighting three recent projects from my laboratory, each highlighting physical behavior at a different scale. First, using the property of birefringence, we are quantifying both kinetic and dynamic properties in an avalanche of macroscopic particles and measure rheological properties. Secondly, we explore avalanches on an erodible bed that display an intriguing dynamic intermittency between regimes. Lastly, we take a closer look at aqueous (water-driven) dunes in a novel rotating experiment and resolve an outstanding scaling controversy between migration velocity and dune dimension.

  • Mathematical Geoscience Seminar
18 May 2018
14:15
Abstract


A Surstseyan eruption is a particular kind of volcanic eruption which involves the bulk interaction of water and hot magma. Surtsey Island was born during such an eruption process in the 1940s. I will talk about mathematical modelling of the flashing of water to steam inside a hot erupted lava ball called a Surtseyan bomb. The overall motivation is to understand what determines whether such a bomb will fragment or just quietly fizzle out...
Partial differential equations model transient changes in temperature and pressure in Surtseyan ejecta. We have used a highly simplified approach to the temperature behaviour, to separate temperature from pressure. The resulting pressure diffusion equation was solved numerically and asymptotically to derive a single parametric condition for rupture of ejecta. We found that provided the permeability of the magma ball is relatively large, steam escapes rapidly enough to relieve the high pressure developed at the flashing front, so that rupture does not occur. This rupture criterion is consistent with existing field estimates of the permeability of intact Surtseyan bombs, fizzlers that have survived.
I describe an improvement of this model that allows for the fact that pressure and temperature are in fact coupled, and that the process is not adiabatic. A more systematic reduction of the resulting coupled nonlinear partial differential equations that arise from mass, momentum and energy conservation is described. We adapt an energy equation presented in G.K. Batchelor's book {\em An Introduction to Fluid Dynamics} that allows for pressure-work. This is work in progress.  Work done with Emma Greenbank, Ian Schipper and Andrew Fowler 

  • Mathematical Geoscience Seminar
9 March 2018
14:15
Bruce Sutherland
Abstract

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).

  • Mathematical Geoscience Seminar
23 February 2018
14:15
Srikanth Toppaladoddi
Abstract

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.

  • Mathematical Geoscience Seminar
9 February 2018
14:15
Jonathan Holmes
Abstract

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.

  • Mathematical Geoscience Seminar
26 January 2018
14:15
Abstract

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.

  • Mathematical Geoscience Seminar
1 December 2017
14:15
Lawrence Percival
Abstract

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.

  • Mathematical Geoscience Seminar
17 November 2017
14:15
Vassillios Dallas
Abstract

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.

  • Mathematical Geoscience Seminar

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