Andrew D. Gilbert

At Llanthony Priory in 2008.

OFFICE HOURS: SEMESTER 1, 2013-14


or feel free to email me for an appointment.

TEACHING: 2013-14


Modules I am involved with teaching are: See the Exeter Learning Environment (ELE) for module resources, etc.

ADMINISTRATIVE ROLE:

College Director of Postgraduate Researchers

CV


CV in pdf format, produced using the CurVe class in Latex. Last updated 10.4.12.

EDITORSHIP:



Miscellaneous



PhD PROJECTS


Potential PhD students are welcome to discuss possible projects in any of the areas of my research interests including:

Most of my work involves simulations on desk-top scale machines, in parallel with analytical calculations/approximations of models or very idealised problems.

RESEARCH


Swimming (collaboration with Physics)


See the swimmers web page

Fundamental dynamo mechanisms


The magnetic fields in the Earth, Sun, planets, stars and galaxies, are generated by the flows of electrically conducting fluid. With Andrew Soward (Exeter), Yannick Ponty (Nice) and Pu Zhang (formerly at Exeter), I am studying fundamental dynamo mechanisms: the generation of magnetic fields in convective fluid flows.

Magnetic field in convective fluid flow

The top two panels show a convective fluid flow: the arrows in the second panel show the flow in the (x,z)-plane depicted, while the first panel shows the magnitude of the velocity in the y-direction, into the screen. The third panel shows a magnetic field generated with a sheet-like structure.

Magnetic field 3d image

This shows a visualisation of sheets of field in another run, using the 3-d visualisation package vis5d. Such sophisticated packages are needed to gain an understanding of magnetic field twisting and folding in complex fluid flows.


Fast dynamos


A related interest of mine is the fast dynamo problem. Here is a picture of magnetic field evolving in a fast dynamo:

Magnetic field in chaotic flow

This picture shows the kinematic evolution of magnetic field in a Kolmogorov flow u =(sin z, sin x, sin y), first investigated by D.J. Galloway and M.R.E. Proctor (Nature 356, 691-693, 1992). These authors found numerical evidence for fast dynamo action: growth of field on an advective time-scale independent of molecular diffusion, when the diffusion is very weak, but non-zero. Such studies are relevant to the Sun, where the diffusivity is extremely small, as measured by a magnetic Reynolds number. The diffusive time-scale is millions of years, and yet the field evolves on an advective time-scale of months and years. Clearly diffusion has no role in controlling the evolution of the solar cycle, yet to prove this mathematically, in any but the most idealised models, is an open and challenging problem. The next picture shows the field at later time, with finer structure emerging:

Magnetic field in chaotic flow

In the run used to obtain these pictures, the magnetic diffusion is set identically zero, and field is evolved from a smooth initial condition for a time. Vertical field is shown on a section in the flow and is coloured yellow/red for positive values, green for near-zero values, and torquoise/blue for negative values. As the chaotic streamlines stretch and fold field at zero diffusivity, complicated patterns are generated, dominated by spiralling of trajectories near hyperbolic stagnation points. Although the field grows ever more complicated, average measures of the field such as its flux through a fixed surface show clear exponential growth (with oscillations). This constructive folding of magnetic field lines is suggestive of fast dynamo action. Little is known about the amplification mechanism in this flow, and almost nothing has been proven mathematically about dynamo action in the limit of vanishing diffusion for flows of this complexity (or simplicity, depending on your point of view --- Lagrangian or Eulerian!).

To study such fully three dimensional flows is very difficult, even numerically, and so theoretical approaches involve studying simplified mappings such as the `stretch-fold-shear map' of Bayly and Childress:

Stretch fold shear map

The stretch--fold--shear map. (a) Magnetic field depending on z is stretched and folded with a baker's map in the (x,y)-plane to give (b). In (c) the field orientation is shown in the (x,z)-plane, which after the shear operation gives (d). The effect of the stretch--fold--shear operations from (a) to (d) is to double the magnitude of field vectors and partially bring like-signed field together.

Magnetic field eigenfunctions become very complicated for small diffusion, but they have well-defined growth rates.

Growth rates for stretch fold shear map

The above picture shows growth rates for the stretch-fold-shear map as a function of the shear parameter for zero diffusion. Our aim is to understand such dynamos and their growth rates for zero and weak diffusion. Related problems involve the decay of passive scalars.


For more information on fast dynamos see the monograph:


Childress, S. & Gilbert, A.D. 1995 Stretch, Twist, Fold: The Fast Dynamo. Springer-Verlag Lecture Notes in Physics: Monographs, volume 37.

Magnetic field in chaotic flow


Review of dynamo theory:


See also my review of dynamo theory below. This covers a range of topics from the basic derivation of the inducton equation from Maxwell's equations, through anti-dynamo theorems and upper bounds, to asymptotic models, alpha effects and fast dynamos.

Gilbert, A.D. 2003 Dynamo theory. In: Handbook of Mathematical Fluid Dynamics, volume 2 (ed.\ S. Friedlander and D. Serre), pages 355-441 (Elsevier).

Copyright does not allow me to put this review on the web, but I am allowed to post a preprint version (which in fact has a couple of minor errors corrected since the printed version):

dynamo.ps.gz.


Vortex dynamics and mixing


I am currently working on problems of vorticity wind-up (with Konrad Bajer, University of Warsaw, Andrew Bassom, now of the University of Perth), and Matt Turner (Exeter).

Vorticity wind-up

The picture shows a vortex (small red circle) in a weak background flow with a vorticity gradient (colours show vorticity strength). The fluid rotates around the vortex causing the background vorticity to form a spiral structure around the vortex. This gives a feedback on the vortex causing it to move right and upwards in the flow.

We have analysed such vortex motion with a combination of analytical tools, and numerical simulations. Vortices are important in the atmosphere and oceans, and our research programme involves studying the fundamentals of vortex dynamics, vortex stability and mixing properties.



Recent publications:



Submitted:



In press:



In print:



Presentations :


All were written using the Apple software `keynote' but stored here in .pdf form. Some are quite large, but don't contain movies!
  • Dynamo action in flows with cat's eyes, Exeter 2006, Warwick 2006.
  • Dynamo theory, Warwick 2006.
  • Nonlinear dynamos driven by shear flows, St. Andrew's 2006.
  • Dynamics and diffusion in coherent vorties, closing the stability loop Exeter 2009.
  • Mixing and quasi-modes in two-dimensional planar vortices Sheffield 2009.
  • Mixing, quasi-modes, tripoles and vorticity staircases in 2-d fluid flows Leicester, Cambridge, City 2010.
  • Theory of ferromagnetic microswimmers Exeter, 2010.
  • Complex singularities in 2-d Euler flows (followed by a little on swimming and mixing) Bozeman, Montana, 2010.

    Collaborators/supervisors/students/visitors, over the years:



    Other links:




    Family pictures


  • Kate, Patrick and Nina: some mug shots.


    Address:

    Prof. A.D. Gilbert,
    Mathematics Research Institute,
    College of Engineering, Mathematics and Physical Sciences,
    University of Exeter,
    Harrison Building,
    North Park Road,
    Exeter,
    EX4 4QF, U.K.

    e-mail : A.D. Gilbert at ex . ac . uk
    Tel. : +44-1392-269222
    Fax. : +44-1392-264067