Andrew D. Gilbert

OFFICE HOURS: SEMESTER 1, 2013-14

• Monday 11am
• Tuesday 10am
• Wednesday 11am

TEACHING: 2013-14

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

ADMINISTRATIVE ROLE:

CV

EDITORSHIP:

• 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).

Miscellaneous

• Open day talk materials.
• General research talk.

PhD PROJECTS

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.

RESEARCH

Swimming (collaboration with Physics)

Fundamental dynamo mechanisms

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.

Fast dynamos

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.

For more information on fast dynamos see the monograph:

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

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:

• 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.

In press:

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

In print:

• 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.

Presentations :