- Monday 11am
- Tuesday 10am
- Wednesday 11am

Modules I am involved with teaching are:

- ECM3707: Fluid Dynamics (module leader),
- ECMM731: Waves, instabilities and turbulence (module leader).
- PhD MAGIC module: Topological Fluid Mechanics (module leader)

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

- I am an Editor (one of three) for the journal Fluid Dynamics Research, journal of the Japanese Society of Fluid Mechanics (published by the Institute of Physics).
- I am a member of the Editorial Board for the journal Geophysical and Astrophysical Fluid Dynamics (published by Taylor and Francis).

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

- Modelling of microscale pumps and swimmers. This is a collaboration with colleagues in Physics: see the swimmers web page.
- Magnetic field generation in fluid flows. Motion of an
electrically conducting fluid can create currents and magnetic fields,
and this
*dynamo*process is at the origin of the magnetic fields of the Earth, Sun, planets, stars and galaxies. Topics include investigating generation of field in idealised fluid flows, and the modelling of fluid flows relevant to the Sun. - Vortex dynamics and mixing: the atmosphere and oceans are dominated by vortices, regions of rotating fluid such as hurricanes, tornadoes and gulf stream rings. Understanding the dynamics of such structures and the mixing that can take place within them links stability, waves and chaotic mixing.
- Mixing in complex fluid flows: mixing of chemicals, heat, pollutants, and even plankton raises a whole range of interesting mathematical problems that may be addressed by simulations in idealised flows. The aim is to explore the subtle interaction of mixing by the fluid flow and other processes such as diffusion, chemical reaction, population changes, or predator-prey interactions.

See the swimmers web page

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.

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.

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.

A related interest of mine is the

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:

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:

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.

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.

Childress, S. & Gilbert, A.D. 1995

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):

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).

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.

- Riedinger, X. & Gilbert, A.D. 2013 Critical layer and radiative
instabilities in shallow water shear flows.
*J. Fluid Mech.*, submitted. - Gilbert, A.D., Riedinger, X. & Thuburn, J. 2013 Note on the form
of the viscous term for two dimensional Navier-Stokes flows.
*QJMAM*, submitted.

- Jones, S.E. & Gilbert, A.D. 2013 Dynamo action in the ABC flows using symmetries.
*Geophys. Astrophys. Fluid Dynam.*, in press.

- Zaggout, F. & Gilbert, A.D. 2012 Passive scalar decay in chaotic
flows with boundaries.
*Fluid Dynam. Res.***44**, 025504 (26 pages). Link to preprint version of paper. Link to paper. - Gilbert, A.D., Ogrin, F.Y., Petrov, P.G. & Winlove, C.P. 2011
Motion and mixing for multiple ferromagnetic swimmers.
*European J. Phys. E***34**, 121 (9 pages). Link to preprint version of paper. Link to paper. - Gilbert, A.D., Ponty, Y. & Zheligovsky, V. 2011
Dissipative structures in a nonlinear dynamo.
*Geophys. Astrophys. Fluid Dynam.***105**, 629--653. Link to preprint version of paper (arXiv:1005.5259v2 [nlin.CD]). Link to paper. - Gilbert, A.D., Ogrin, F.Y., Petrov, P.G. & Winlove, C.P. 2011
Theory of ferromagnetic microswimmers.
*QJMAM*,**64**, 239-263. Link to preprint version of paper. Link to paper. - Gilbert, A.D. & Pauls, W. 2011
Complex manifolds for the Euler equations: a hierarchy of ODEs and the
case of vanishing angle in two dimensions.
*Fluid Dynamics Research*,**43**, 025505 (27 pages). Link to preprint version of paper. Link to paper. - Turner, M.R., Thuburn, J. & Gilbert, A.D. 2009
The influence of periodic islands in the flow field on a
scalar tracer in the presence of a steady source.
*Phys. Fluids*,**21**, article 067103 (12 pages). Link to paper. - Hall, O., Hills, C.P. & Gilbert, A.D. 2009
Non-axisymmetric Stokes flow between concentric cones.
*QJMAM*,**62**, 131-148. Link to paper. - Turner, M.R., Bassom, A.P. and Gilbert, A.D. 2009
Diffusion and the formation of vorticity staircases in randomly
strained two-dimensional vortices.
*J. Fluid Mech,***638**, 49-72. Link to paper. - Turner, M.R. and Gilbert, A.D. 2009
Spreading of two-dimensional axisymmetric vortices exposed to a
rotating strain field.
*J. Fluid Mech,***630**, 155-177. Link to paper. - Hall, O., Gilbert, A.D & Hills, C.P. 2009
Converging flow between coaxial cones.
*Fluid Dynamics Research*,**41**, 011402. Link to paper. - Peyrot, M., Gilbert, A.D. & Plunian, F. 2008
Oscillating Ponomarenko dynamo in the highly conducting limit.
*Phys. Plasmas***15**, 122104 (1--8). Link to paper. - Turner, M.R., Gilbert, A.D. & Thuburn, J. 2008
Effective diffusion of scalar fields in a chaotic flow.
*Phys. Fluids***20**, 107103 (1-14). Link to paper. - Turner, M.R. and Gilbert, A.D. 2008
Thresholds for the formation of satellites in two--dimensional
vortices.
*J. Fluid Mech.*,**614**, 381-405. Link to paper. - Turner, M.R. and Gilbert, A.D. 2007
Linear and nonlinear decay of cat's eyes in two-dimensional vortices,
and the link to Landau poles.
*J. Fluid Mech.***593**, 255-279. Link to paper. - Hall, O., Hills, C.P., and Gilbert, A.D. 2007
Slow flow between concentric cones,
*QJMAM***60**, 27-48. Link to paper. - Blockley, E.W., Bassom, A.P., Gilbert, A.D. & Soward, A.M. 2007
Wave-train solutions of a spatially-heterogeneous amplitude equation
arising in the subcritical instability of narrow-gap spherical
Couette flow,
*Physica D*,**228**, 1-30. Link to paper. - Zhang, P. & Gilbert, A.D. 2006
Nonlinear dynamo action in hydrodynamic instabilities driven by shear.
*Geophys. Astrophys. Fluid Dynam.***100**, 25-47. - Gilbert, A.D. 2006
Advected fields in maps:
III. Passive scalar decay in baker's maps
*Dynamical Systems*,**21**, 25-71. - Zhang, P., Gilbert, A.D. & Zhang, K.
2006 Nonlinear dynamo action in rotating convection and shear
*J. Fluid Mech.*,**546**, 25-49. - Courvoisier, A., Gilbert, A.D. & Ponty, Y.
2005 Dynamo action in flows with cat's eyes
*Geophys. Astrophys. Fluid Dyn.*,**99**, 413-429. - Gilbert, A.D. 2005
Advected fields in maps: II. Dynamo action in the
stretch--fold--shear map.
*Geophys. Astrophys. Fluid Dyn.*,**99**, 241-269. - Bajer, K., Bassom, A.P. & Gilbert, A.D. 2004
Vortex motion in a weak background shear flow.
*J. Fluid Mech.*,**509**, 281-304. - Ponty, Y., Gilbert, A.D. & Soward, A.M. 2003
The onset of thermal convection in Ekman--Couette shear flow with
oblique rotation.
*J. Fluid Mech.*,**487**, 91-123. - Hall, I.M., Bassom, A.P. & Gilbert, A.D. 2003
The effects of diffusion on the stability of vortices with fine
structure.
*QJMAM*, 56, 649-657. - Hall, I.M., Bassom, A.P. & Gilbert, A.D. 2003
The effect of fine structure on the stability of planar vortices.
*Eur. J. Mech./B Fluids*,**22**, 179-198. - Gilbert, A.D. 2002
Advected fields in maps:
I. Magnetic flux growth in the stretch--fold--shear map.
*Physica D***166**, 167-196. Link to paper. - Macaskill, C., Bassom, A.P. & Gilbert, A.D. 2002
Nonlinear wind-up in a strained planar vortex.
*Euro. J. Mech B/Fluids*,**21**, 293-306. - Gilbert, A.D. Magnetic helicity in fast dynamos. 2002
*Geophys. Astrophys. Fluid Dyn.*,**96**, 135-151. - Childress, S., Kerswell, R.R. & Gilbert, A.D. 2001
Bounds on dissipation for Navier-Stokes flow with Kolmogorov
forcing.
*Physica D*,**158**, 105-128. - Ponty, Y., Gilbert, A.D. & Soward, A.M. 2001
Kinematic dynamo action in large magnetic Reynolds number flows
driven by shear and convection.
*J. Fluid Mech.***435**}, 261-287. - Bajer, K., Bassom, A.P. & Gilbert, A.D. 2001
Accelerated diffusion in the centre of a vortex.
*J. Fluid Mech.*,**437**, 395-411. - Gilbert, A.D. & Ponty, Y. 2000
Dynamos on stream surfaces of a highly conducting fluid
*Geophys. Astrophys. Fluid. Dyn.***93**, 55-95. - Bassom, A.P. & Gilbert, A.D. 2000 The relaxation of vorticity
fluctuations in locally elliptical streamlines.
*Proc. Roy. Soc. A*,**456**, 295-314. - Bassom, A.P. & Gilbert, A.D. 1999 The spiral wind-up and
dissipation of vorticity and a passive scalar in a strained planar vortex.
*J. Fluid Mech.*,**398**, 245-270. - Maksymczuk, J. & Gilbert, A.D. 1998 Remarks on the equilibration
of high conductivity dynamos,
*Geophys. Astrophys. Fluid Dyn*,**90**, 127-137. - Bassom, A.P. & Gilbert, A.D. 1998 The spiral wind-up of vorticity
in an inviscid planar vortex,
*J. Fluid Mech.,***371**, 109-140. - Bassom, A.P. & Gilbert, A.D.
1997 Nonlinear equilibration of a dynamo in a smooth helical flow.
*J. Fluid Mech.***343**, 375-406. - Gilbert, A.D., Soward, A.M., & Childress, S.
1997 A fast dynamo of alpha-omega type.
*Geophys. Astrophys. Fluid Dyn.***85**, 279-314.

All were written using the Apple software `keynote' but stored here in .pdf form. Some are quite large, but don't contain movies!

- Alex Archibald
- Konrad Bajer
- Andrew Bassom
- Bruce Bayly
- Ed Blockley
- Steve Childress
- Alice Courvoisier
- Uriel Frisch
- Ian Hall (now at the Health Protection Agency)
- Oskar Hall
- Chris Hills
- Sam Jones
- Richard Kerswell
- Isaac Klapper
- Charlie Macaskill
- Jan Maksymczuk (now at the Met Office)
- Krzysztof Mizerski
- Keith Moffatt
- Marine Peyrot
- Francisco Pla Martos
- Walter Pauls
- Yannick Ponty
- Annick Pouquet
- Andrew Soward
- Matthew Turner
- Fatma Zaggout
- Keke Zhang
- Pu Zhang
- Vlad Zheligovsky

- Gallery of birds, painted by John Gale, our neighbour.

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