## School of Mathematics and Statistics |

MSc(Kyoto), DSc(Kyoto), FInstP

Professor of Applied Mathematics

School of Mathematics and Statistics

The University of Sheffield

Hicks Building, Hounsfield Road

Sheffield S3 7RH, U.K.

office: H07 Hicks Building

email: K.Ohkitani[at]sheffield.ac.uk

tel: +44 (0)114 222 3861

fax: +44 (0)114 222 3739

- Surface quasi-gostrophic (SQG) equations

- We will revisit the open problem of regularity issues for (generalised) SQG equations, mathematically and numerically. The key involves an extention of Poincare-Bertrand formula for exchanging singular integrals. Also of interest is a comparison of probability distributions of active scalars, in particular, characterisation of rearrangement of active scalars in SQG and 2D Eular with a control theoretic method.

- Quasi-geostrophic (QG) and surface quasi-geostrophic (SQG) equations

- We will study the motion of a surface bounding a domain of uniform potential vorticity in 3D. This may be regarded as a 3D extention of "contour dynamics". We will start seeking a governing equation. We will also consider a 3D analogue of SQG equation as a mathematical interest.

- Statistics of line vortices

- We will study statistics of vortex filamants in 3D by regarding energy, helicity and enstrophy as topological invariants. We will study how we may incorporate their constraints imposed by integral geometry into dynamical evolution of the system.

- Turbulence on general manifolds

- We will study fluid flow on mainfolds and how the curvature affects fundamental properties of turbulence, such as Kolmogorov's 4/5-law. Specific applications are in sight.

- Functional integration approach to turbulence

- We study the action integral in a path-integral representation of the Hopf functional on the basis of control theory. We try to deduce non-trivial information regarding the probability distributions by examining the action integral. This project is inevitably heuristic in nature to some extent.

- Fluid Dynamics Research, Advisor (current)