to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

Variable-density flow and transport in porous media: approaches and challenges

Variable-density groundwater flow and solute transport in heterogeneous porous media: approaches, resolutions and future challenges.

Previous studies of fluid convection in porous media have considered the onset of convection in isotropic systems and the steady convection in anisotropic systems. This paper bridges between these and develops new results for the onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. These results are relevant to sedimentary formations where the average vertical permeability is some fraction γ of the average horizontal permeability. Linear and global stability analyses are used to define the critical time tc at which the instability occurs as a function of γ and the dimensionless Rayleigh-Darcy number Ra* for both thermal and solute-driven convection in an infinite horizontal slab. Numerical results and approximate analytical solutions are obtained for both a slab of finite thickness and the limit of large slab thickness. For a thick slab, the increase in tc as γ decreases is approximately given by (1+γ)4∕(16γ2). One important application is to the geological storage of carbon dioxide where it is shown that the use of an effective vertical permeability in estimating the critical time is only valid for low permeabilities. The time scale for the onset of convection in geological storage can range from less than a year (for high-permeability formations) to decades or centuries (for low-permeability ones).

Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions

In this paper a theoretical investigation of convection currents in anisotropic porous media is performed. The porous layer is homogeneous and bounded by two infinite perfectly heat-conducting horizontal planes. The criterion for the onset of convection is derived. The supercritical steady two-dimensional motion, the heat transport and the stability of the motion are investigated. Some of the results may be applied in insulation techniques.

/pdf/nonlinear-thermal-convection-in-anisotropic-porous-media-20pth6yool.pdf

Nonlinear thermal convection in anisotropic porous media

Summary and conclusions The present work is the first study of Rayleigh-Be'nard convection in an anisotropic porous medium with oblique principal axes. The analysis is restricted to transversely isotropic media with isotropic thermal conductivity. Qualitatively new flow patterns occur at the onset of convection. If the transverse permeability is larger than the longitudinal permeability, the planes of motion are tilted, but the cell walls are vertical as usual. On the other hand, if the longitudinal permeability is the larger one, the flow occurs in vertical planes, but the cell walls are tilted. 'The preference for these different patterns is explained as a preference for flow directions with as small a tangential permeability as possible. This preference also gives rise to a tendency of concentration of the flow along the cell boundaries when the anisotropy increases, in the case when the cell walls are tilted. The tilt is primarily of mechanical origin, because a vertical forcing due

Onset of convection in an anisotropic porous medium with oblique principal axes

The principle of large-scale anisotropy due to small-scale layering is applied to thermal convection. The motion takes place in a bounded porous medium heated from below. The medium is periodically layered with respect to permeability and thermal conductivity. The onset of convection as well as slightly supercritical convection are investigated. Anisotropic modelling proves useful even for small numbers of layers as long as the motion is of ‘large-scale convection’ type (Masuoka et al. 1978). The modelling always fails for motion of ‘local convection’ type.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112082001104

Anisotropic modelling of thermal convection in multilayered porous media

The impulsively starting motion of a circular cylinder submerged horizontally below a free surface is studied analytically using a small-time expansion. The series expansion is taken as far as necessary to include the leading gravitational effects for two cases: constant velocity and constant acceleration, both commencing from rest. The hydrodynamic force on the cylinder and the surface elevation are calculated and expressed in terms of bipolar coordinates. Comparisons are also made with earlier theoretical and experimental work. The theory is valid for arbitrary value of submergence depth to cylinder radius.

Free-surface flow due to impulsive motion of a submerged circular cylinder

It is found in this note that basic results concerning thermal convection in a rotating porous layer may be obtained from known analysis of thermal convection in an (non-rotating) anisotropic porous medium.

Peder A. Tyvand

Papers

Nonlinear thermal convection in anisotropic porous media

Onset of convection in an anisotropic porous medium with oblique principal axes

Anisotropic modelling of thermal convection in multilayered porous media

Free-surface flow due to impulsive motion of a submerged circular cylinder

Thermal convection in a rotating porous layer