Second AGM Meeting 2019 on Stochastic geometric mechanics: fluid models and uncertainty quantification
2 December 2019, Imperial College London, Mathematics Department, ICSM Building Room 402 (entrance via Sherfield Building)
Timetable:
Titles and Abstracts
So Takao
Stochastic advection by Lie transport: the past, present and future
When trying to predict the weather or climate, adding noise to your model seems to be a good idea considering the complexity of fine scale fluid motion and the uncertainty it generates. The question is, how though? One idea would be to add noise in such a way that the underlying Hamiltonian structure of inertial fluid motion is preserved. This is called stochastic advection by Lie transport (or SALT for short). By doing so, important properties of fluids such as the conservation of circulation come out for free and furthermore, it allows for a data-driven approach to modelling fluids. After giving an overview of SALT and comparing it with other stochastic models, I will also talk about what happens when dissipation is added to the model and discuss its long time dynamics.
Golo Wimmer
A Poisson bracket framework for compatible finite element methods in numerical weather prediction
An important aspect of discretisations in numerical weather prediction, particularly for climate simulations, is conservation of quantities such as mass and energy. Conservation of energy requires a careful discretisation of all prognostic equations, ensuring that the energy losses and gains are balanced between the resulting discretised terms. One way to guide this process is to consider the equations in a Hamiltonian framework, where the Hamiltonian is given by the system’s total energy, and the equations are inferred by a Poisson bracket. Conservation of energy then follows easily by the bracket’s antisymmetry, and any space discretisation maintaining this property will then also conserve energy. In this presentation, the Hamiltonian framework is applied to the compatible finite element method, which is the underlying space discretisation for the UK Met Office's next generation dynamical core. It includes a discussion on the combination of this framework with another fundamental aspect of the discretisation of fluid dynamical equations: the use of transport schemes that are numerically stable and avoid the creation of spurious oscillations.
Oana Lang
Well-posedness analysis for the stochastic transport 2D Euler equation
The analysis of stochastic partial differential equations with stochastic advection by Lie transport (SALT) is becoming a flourishing topic within both theoretical and applied communities. However, the study of analytical issues like existence, uniqueness, or continuity with respect to initial conditions for this type of equations is still in its incipient phase. In this talk we will present a global well-posedness result for the stochastic 2D Euler equation driven by Lie transport noise.
Zhongmin Qian
Geometric structures in Navier-Stokes equations
In this talk, I will review a few geometric structures arising from the study of Navier-Stokes equations, such as the relation between the natural curvature and the rate-of-strain tensor, Bochner identity and vorticity equation, the helicity and Chern-Simon invariant, and etc.
The talk is based partly on the joint work with S. Fang.
Oliver Street
Stochastic modelling of ocean debris
The issue of ocean plastics has recently been much discussed by academics, policy makers, and environmental campaigners. Some studies into this issue use an entirely statistical approach, and those which attempt to model the dynamics of the problem explicitly have largely ignored key factors such as sub-grid-scale dynamics and the mass of the debris. This raises the interesting question of how inertial particles move in a fluid governed by a SPDE. Using recent developments in stochastic fluid equations [Holm, 2015], as well as an idealised model describing the movement of a sphere in a fluid [Maxey & Riley, 1983], we will attempt to understand the dynamical properties of this problem. Successfully improving the accuracy of existing models would have a sizeable impact on a number of crucially important scientific questions, and the formulation of this new approach has raised a number of interesting mathematical problems.
James-Michael Leahy
On the equations of incompressible fluids driven by rough transport noise
The need for robust and computationally efficient stochastic parameterization schemes that model the effects of fast sub-grid scale physics and other unresolved processes is well-understood in computational fluid dynamics. We believe the preservation of structure and physicality of deterministic fluid dynamics should serve as a guiding principle in designing robust SP schemes.
Inspired by this challenge, we present on well-posedness of the rough partial differential equations (RPDEs) associated with structured parameterizations of the fast scales of fluid motion by a temporally rough vector field. We work within the framework of geometric rough paths (GRPs) in order to define the rough temporal dependence of the vector fields. Examples of GRPs include Stratonovich Brownian motion, fractional Brownian motion, and more general Gaussian processes. The parametrizations manifest themselves in the equations for the modeled coarse-scale Eulerian velocity as a perturbation of the advecting velocity (in the covariant-derivative or Lie-transport form) of the deterministic Navier-Stokes or Euler equations. The corresponding equation is a system of RPDEs with a rough transport term that models the effects of the sub-grid scale on the coarse-scale velocity.
We also present corollaries in two-dimensions for Brownian driven equations, which include a Wong-Zakai approximation, support, and large deviations theorem, and that the corresponding unique solution induces a random dynamical system. These corollaries illustrate that the analysis of the equations in the general GRP setting leads to significant consequences in the Brownian case.
This talk presents on joint work with Dan Crisan, Martina Hofmanova, Darryl Holm, and Torstein Nilssen.
Aythami Bethencourt de Leon
Stochastic fluid dynamics: fundamental chain rules and applications
We show how to extend the Itô-Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to k-form-valued stochastic processes. The result is a chain rule emulating Cartan’s formula, which we call the Kunita-Itô- Wentzell (KIW) formula for k-forms. We discuss how to use this formula in order to derive stochastic extensions of fundamental deterministic fluid equations like conservation of mass or the momentum equation. We also briefly discuss one of its applications to well-posedness by noise problems.
Travel Expenses
Travel expenses can be reimbursed by completing a Visitor's Expense Claim Form, and mailing to
Paul Skerritt
Mathematics Department
University of Surrey
Guildford, GU2 7XH
Please remember to include receipts. For train/bus tickets, either an email or physical receipt will work (also include the train ticket if you still have it). Please do not send scans of train tickets, since they will not be accepted by our Finance Department.
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Main contact: Erwin Luesink (e.luesink16@imperial.ac.uk)
Secondary contact: Paul Skerritt (p.skerritt@surrey.ac.uk)