I am an applied mathematician specialising in fluid dynamics. More precisely, I investigate instabilities and turbulence for fluid flows, focusing on convection. My work combines ideas from dynamical systems, functional analysis, variational analysis and optimisation to understand fluid flows as observed on geophysical and astrophysical scales.
I am also passionate about public engagement in physics, mathematics and philosophy.
(Jun 2023 to present) Postdoctoral researcher, Earth and Planetary Magnetism group, Institute of Geophysics, ETH Zürich
(Oct 2019 - Jan 2023) PhD Fluid Dynamics, CDT in Fluid Dynamics, Department of Aeronautics, Imperial College London
(Oct 2018 - Sep 2019) MRes in Fluid Dynamics, Centre for Doctoral Training (CDT) in Fluid Dynamics, Imperial College London
(Sep 2016 - Jun 2017) MSci Astrophysics, Department of Physics and Astronomy, UCL
(Sep 2013 - Jun 2016) BSc Astrophysics, Department of Physics and Astronomy, UCL
(continue at your own peril)
Scaling Laws and instabilities in fluid dynamics
I am interested in understanding the dynamics of fluid flows with mathematical methods. This work encompasses two areas: (1) the study of fluid instabilities and (2) the investigation of turbulence on long-time scales. The stability of fluid flows involves the analysis of the fluid to perturbations. For perturbations in the form of normal modes the technique is known as linear stability analysis, whereas considering perturbations by energy is nonlinear stability analysis. In general, intermittent behaviour and global stability are far more difficult to determine analytically. In addition to stability, I look at the long-time behaviour of flows and in particular turbulence, to determine the laws governing weakly emergent properties of the system. I seek to construct proofs for the scaling of quantities like the dissipation, drag, and heat or momentum transport of turbulence, as the physical quantities of the flow vary. To this end, I utilise the modern formulation of the background flow method and leverage techniques from functional analysis, calculus of variations and convex optimisation. Then, comparison with heuristic results, simulations and experiments assess our mathematical abilities at unravelling the features of turbulence.
Examples of background temperature profiles used to construct proofs of a scaling law for heat transport.
Example of one of the theorems I have proven with the background flow method for internally heated convection.
Internally Heated Convection
I am studying convection within bounded domains, driven by internal sources or sinks of heat. Compared to the motion of a fluid heated from below and cooled from above (Rayleigh–Bénard convection), internally driven fluids exhibit fundamentally different physics and are a mathematical generalisation for convection relevant to geophysical and astrophysical fluid dynamics. For example, it is known that with uniform internal heating, subcritical convection can occur, i.e. linear and nonlinear stability do not coincide (in contrast to Rayleigh–Bénard convection). Furthermore, when the flow is turbulent, the boundary layers at the top and bottom possess different physical characteristics (see figures below), and in fact, heat transport can overcome the bottleneck of boundary layers with bulk forcing.
In geophysical and astrophysical environments, buoyancy emerges from scalar quantities like temperature or chemical concentration, injected or removed throughout the domain of a fluid. Therefore, I also seek to unravel the complexities that arise when sources and sinks vary in space within the domain. Currently, my research probes the uncharted territory where the Coriolis and Lorentz forces due to rotation and magnetic fields affect the flow features and the long-time behaviour of the turbulence of internally heated convection in geometries relevant to geophysics.
The partial differential equation used to describe internally heated convection. With a scalar (the temperature) T, a source S and an incompressible velocity field u.
(a) A direct numerical simulation of uniform internally heated convection, with isothermal and no-slip boundary conditions.
(b) Snapshot with the same conditions as (a), but for a very viscous fluid.
Rotating magnetohydrodynamics
If considering geophysical and astrophysical dynamics, like in atmospheres, the interiors and cores of planets or convection zones in stars, fluids are subject to the Coriolis and Lorentz forces due to global rotation and self-generated magnetic fields. I am interested in the effects on stability and turbulence (in the long-time) of rotating magnetohydrodynamics, split into three questions (1) the effects of rotation on turbulent convection, (2) the effects of magnetic field generation on turbulent convection, (3) the effects of both.
One model of interest is the Taylor dynamo, where the momentum equation is diagnostic for the velocity and is in so-called magnetostrophic balance, between pressure, buoyancy, Coriolis and the Lorentz force. This unique approximation of the Navier-Stokes equations is mathematically complex due to the lack of inertia and viscosity terms while still forming a system with nonlinearity present in the temperature and induction equations.
Snapshot with the same boundary conditions as above, but for a fluid subject to "mild" rotation
ih1_pr0p1_ek1e4.mp4ih1_pr10_ek1e4.mp4videos of simulations of rotating internally heated convection for fluids with low and high viscosity relative to heat diffusion
Linear and Nonlinear Dynamos
Fluid dynamos are crucial to life on Earth, with the large-scale magnetic field of our planet contributing to the favourable conditions for the development of complex life. A large-scale magnetic field is not unique to the Earth, however, the magnetic fields observed in our solar system's planets—each a product of unique internal structures—pose many puzzles. Unravelling the conditions under which fluid motion can sustain a magnetic field thus becomes an important question. Kinematic dynamos occur when a fluid generates a growing magnetic field without the magnetic field acting back on the flow. While only a toy model to fully nonlinear dynamos, kinematic dynamo studies inform our knowledge of the means to create, control and evolve dynamos. I am interested in linear, dynamic and nonlinear dynamo action and the nature of flows affected by and driving magnetic fields.
A configuration of von Karman swirling flow for which kinematic and nonlinear dynamos were sought, the plot is of the magnetic Reynolds number necessary for dynamo action, for a given rotation ratio of the discs and flow Reynolds number.
I am interested in meaning creation in language as viewed through the lens of a constructivist paradigm, contrasting it with the analytic tradition post-Wittgenstein. Of particular intrigue to me is the dialectic between measure and meaning, and how the progress of humanity's ability to measure has influenced the creation of meaning, when analysing language from a holistic anthropological approach. A measure, as it appears in mathematics, is a central concept for framing discussions and quantifying the characteristics of mathematical objects. As such, the question of a measure-theoretic approach to addressing the fundamental questions in the philosophy of language is an exciting and unique perspective. Applications of this approach to understanding language are to the philosophies of art and science. Both of which are in my area of interest.