A. C. Fowler

THE COMPLEX LORENZ EQUATIONS

We have undertaken a study of the complex Lorenz equations x = −σx + σy . y = (r − z)x − ay . z = −bz + 12(x∗y + xy∗) . where x and y are complex and z is real. The complex parameters r and a are defined by r = r1 + ir2; a = 1 − ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r2 = 0. We have been able to determine analytically two critical values of r1, namely r1c and r1c . The origin is a stable fixed point for 0 r1c, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ + 1 then this limit is only stable in the region r1c rlc, a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.

Thermally controlled glacier surging

Bakaninbreen in Svalbard and Trapridge Glacier in Yukon Territory, Canada, are two prominent examples of surging glaciers which are thought to be controlled by their thermal regime. Both glaciers have developed large bulges which have propagated forward as travelling wave fronts, and which are thought to divide relatively stagnant downstream cold-based ice from faster-moving warm-based upstream ice. Additionally, both glaciers are underlain by a wet, metres thick layer of deforming till. We develop a simple model for the cyclic surging behaviour of these glaciers, which interrelates the motion of the ice and till through a description of the subglacial hydrology, We find that oscillations (surges) can occur if the subglacial hydrological transmissivity is sufficiently low and the till layer is sufficiently thin, and we suggest that these oscillations are associated with the development and propagation of a travelling wave front down the glacier. We therefore interpret the travelling wave fronts on both Trapridge Glacier and Bakaninbreen as manifestations of surges. In addition, we find that the violence of the surge in the model is associated with the resistance to ice flow offered by undulations in the bed, and the efficiency with which occasional hydrological events can release water accumulated at the glacier sole.

A mathematical model of magma transport in the asthenosphere

Abstract We present a mathematical model for the flow of a partial melt through its solid phase. The model is based on the conservation laws of two-phase flow, which reduce to a generalization of porous flow in a permeable medium, when the solid matrix deforms very slowly. The continuity equation for the melt contains a source term (due to melting), which is determined by the energy equation. In addition, the melt fraction is unknown, and a new equation, representing conservation of pore space, is introduced. This equation may also be thought of as a constitutive law for the melt pressure (which is not lithostatic). The model is non-dimensionalized and simplified. Some simple solutions are considered, and it is suggested that the occurrence of high fluid pressures in the solutions may initiate fractures in the lithosphere, thus providing a starting-up mechanism for magma ascent to the surface.

Subglacial floods beneath ice sheets

Subglacial floods (jökulhlaups) are well documented as occurring beneath present day glaciers and ice caps. In addition, it is known that massive floods have occurred from ice-dammed lakes proximal to the Laurentide ice sheet during the last ice age, and it has been suggested that at least one such flood below the waning ice sheet was responsible for a dramatic cooling event some 8000 years ago. We propose that drainage of lakes from beneath ice sheets will generally occur in a time-periodic fashion, and that such floods can be of severe magnitude. Such hydraulic eruptions are likely to have caused severe climatic disturbances in the past, and may well do so in the future.

Modelling ice sheet dynamics

Abstract This paper surveys the problem of modelling the dynamics of large ice sheets. A simplified model for two-dimensional plane ice sheets is derived, and both isothermal and non-isothermal cases are considered. The model is not uniformly asymptotically valid at a divide or at a margin, and we suggest local (isothermal) analyses which give order of magnitude estimates for divide curvature and margin slope. We also give a uniformly valid description for small perturbations to an isothermal ice sheet, which decay diffusively. For the more interesting non-isothermal case, we are able to provide explicit approximate solutions for the surface profile, based on Lliboutrys heuristic boundary layer analysis, and give an approximate description of the temperature field.

On the transport of moisture in polythermal glaciers

Abstract We reconsider the problem of formulation of a model for polythermal glaciers, focussing attention in particular on the temperate zone where ice and water can coexist at the melting temperature. The energy equation for the ice-water mixture in this zone introduces a moisture flux, and a constitutive law for this flux is required. By analogy with the flow through a porous medium, we use Darcys law (i.e. the second momentum equation of a two-phase flow model with “porous” geometry), and then require a mechanical constitutive relation relating the water pressure p w to the average ice pressure p i . Experience in two phase flows suggests that p w =p i may be problematical, and experience in soil mechanics suggests it is inaccurate. A constitutive relation is therefore presented based on work of Nye (1976), and its effect on the well-posedness of the model is examined. Considerations of the sort presented here have clear relevance in the formulation of similar problems in other geophysical situations...

A Sliding law for Glaciers of Constant Viscosity in the Presence of Subglacial Cavitation

A method of solution for the problem of slow flow of a Newtonian viscous glacier slipping over a rough bed is constructed, for the case where cavities form when the lubricating water film pressure reaches that of the local subglacial drainage system. The treatment of Nye (Proc. R. Soc. Lond. A 311, 445-477 (1969)) is reformulated as a Hilbert problem, and the solution presented for the particular case of a periodic bedrock with one cavity per period. For such bedrocks, it is found that the basal stress has a maximum for a finite basal velocity, and the basal stress decreases towards zero as the velocity tends to infinity, in line with the suggestion of Lliboutry (J. Glaciol. 23, 67-95 (1979)). For more complicated bedrocks, with many different obstacle sizes, direct solution appears impractical and some kind of further approximation seems advisable.

An instability mechanism for drumlin formation

Abstract Drumlins are subglacial bedforms that are formed by the interaction of ice flow with an erodible basal topography. The mechanism of their formation bears resemblance to similar processes that cause the formation of dunes and anti-dunes in rivers, and sand dunes in deserts. In 1998 Hindmarsh showed that the interaction of a shearing ice flow with a deformable basal till layer could cause an instability which promotes the growth of basal topography, though he was unable to give analytic criteria for the instability. Here we analyse Hindmarsh’s model, and by using certain approximations, we are able to give concise analytical parametric criteria for this instability. The resultant instability occurs if the basal shear stress is larger than a critical value which increases with increasing basal effective pressure, and which also depends on the basal till thickness. It is hypothesized that this instability is the basic mechanism involved in the formation of Rogen moraine and drumlins.

A correlation function for choosing time delays in phase portrait reconstructions

Abstract A new method is proposed for the selection of the time delay used in creating phase portrait reconstructions of dynamical systems. Using the singular values computed from a singular value decomposition (SVD) of the embedded flow, we compute a quantity which we term the singular value fraction (SVF) as a function of the chosen lag time Δ, and denoted as f sv ( Δ ). For a given value of k , it gives the fractional power in the first k singular values. We seek to choose Δ so that folding of the attractor is minimised, but the attractor volume is maximised, and we thus choose Δ to be at the first minimum of f sv . Equivalently we use a window of size Δ w , where Δ w is apprximately the first major maximum of f sv . It is shown that this method compares favourably with some other extant methods.

The real and complex Lorenz equations and their relevance to physical systems

Abstract We summarize some recently obtained results on real and complex Lorenz equations and discuss their possible significance in relation to real fluid dynamical processes.