Mantle convection

I am following (slowly) a path towards my version of how mantle convection works, at least in the Earth. Basically, the Turcotte/Oxburgh 1967 boundary layer theory of convection is very successful, but for the wrong reason. They had constant viscosity, but for realistic variable viscosity, the top boundary layer will be thick, cold and...stagnant: no subduction. Subduction is therefore the fundamental problem in understanding plate tectonics. However, this stagnant lid convection develops huge stresses in the lid, such that the yield stress can be exceeded. The resulting plastic flow does allow subduction, and (in the Earth) for realistic values of the yield stress. Future work will therefore develop the boundary layer theory of convection at high Rayleigh number, strongly variable viscosity, and with a yield stress. I've also been applying this to Venusian plate tectonics, to explain the apparent episodicity of that planet's tectonics. Another interest is the idea that for strong pressure dependence as well, the internal temperature may become isoviscous rather than adiabatic. Even if true, this is unlikely to become fashionable.

Dynamics of the Earth's core

Well, I haven't done anything here, though I'm interested in extending the laboratory scale studies of alloy convection to the compositional convection above the Earth's mushy inner core (solidifying from an iron/sulphur or iron/oxygen alloy). The main problem would seem to be to develop a coherent theory for the self evolution for the dendrite microscale, a problem also for the laboratory scale system, and one which I don't know how to do.

Solidification processes in magma chambers

The phenomenon I am principally interested in, layered magma chambers, was brought to me by Ron Elsdon in Dublin in 1979. Layered igneous intrusions, such as the Skaergaard in East Greenland, reveal chemical and petrological banding over vertical scales of order centimetres (inch-scale layering) to metres. Explaining this has to be one of the worst problems you could have. You have double diffusive convection of a multi-component melt, and accumulation of a crystal pile at the floor which grows in situ slowly via nucleation and crystal growth kinetics, creating a mush/slurry with its own convective flow. For crystal fractions above 55% (at the base), this mush will be essentially rigid, like a porous medium, whereas above this it will probably behave as a non-Newtonian fluid, thus partaking in the chamber convection. I've thought about this off and on, without significant progress. The main idea is an Avrami type model for nucleation and crytal growth, but for two or even three (on the cotectic) component fluids, and including some kind of mush convection to flush the pore fluids out; the aim is to vindicate Maaløe's oscillation mechanism based on supercooled nucleation. Anyhow, it didn't work.

Magma transport

Volcanic eruptions occur when magma generated by pressure release melting in the Earth's mantle reaches the Earth's surface. Both at mid-ocean ridges and at mid-plate hot spots, this magma is formed at grain boundaries in the upwelling mantle rock, and forms a permeable porous medium through which it can rise through buoyancy. Just as in other situations where an erosive fluid flows through a porous medium (e.g. the formation of river networks via ground surface erosion, or the formation of caves by acidic groundwater in karst regions), it is to be expected that the percolating melt will form a network of channels which drain the porous region. To understand this, one needs to understand how the channels can be maintained open against viscous closure; this is presumably by melt back, in the same way that subglacial channels are. Then one needs to describe the possible geometry of the melt paths. This seems a much harder problem, and may require a validated model of river network formation, before one can make serious quantitative predictions of the nature of the melt drainage from the asthenosphere.

Frost heave

Frost heave occurs when ground is frozen. The capillary suction between ice and water in the soil pores causes a migration of pore water towards the freezing front, where it freezes to form segregated ice lenses. The phenomenon is responsible for serious damage every year to roads and pavements, and is also associated with the formation of various forms of patterned ground, such as earth hummocks and stone circles. The best model is the Miller model, but it is fearsomely complicated. The Fowler/Krantz paper below, and also Chris Noon's 1996 thesis, savagely reduce the model to an easily solvable form, and we have been able to analyse the instability responsible for patterned ground, and also extend the model to include the effects of saline pore water and compressible soils. Future work will treat unsaturated soils, and also we need to develop a numerical method to solve the fully 3-d version.

Relevant publications

  1. Fowler, A.C. 1982 The depth of convection in a fluid with strongly temperature and pressure dependent viscosity. Geophys. Res. Letts. 9, 816-819.

  2. Fowler, A.C. 1983 Preferred aspect ratios of convection in a strongly temperature and pressure dependent viscous fluid. Phys. Earth Planet. Int. 31, 83-90.

  3. Fowler, A.C. 1983 On the thermal state of the earth's mantle. J. Geophys. 53, 42-51.

  4. Fowler, A.C. 1983 Reply (to Dr Christensen's comments). J. Geophys. 53, 203-205.

  5. Fowler, A.C. 1985 Fast thermoviscous convection. Stud. Appl. Math. 72, 189-219.

  6. Fowler, A.C. 1985 A mathematical model of magma transport in the asthenosphere. Geophys. Astrophys. Fluid Dynamics 33, 63-96.

  7. Fowler, A.C. 1985 Secular cooling in convection. Stud. Appl. Math. 72, 161-171.

  8. Fowler, A.C. 1985 A simple model of convection in the terrestrial planets. Geophys. Astrophys. Fluid Dynamics 31, 283-309.

  9. Fowler, A.C. 1986 Thermal runaway in the earth's mantle. Stud. Appl. Math. 74, 1-34.

  10. Fowler, A.C. 1987 Theories of mushy zones: applications to alloy solidification, magma transport, frost heave and igneous intrusions. In: Structure and Dynamics of Partially Solidified Systems, ed. D. Loper, NATO ASI series, Martinus Nijhoff, Dordrecht: pp. 161-199.

  11. Fowler, A.C. 1989 Generation and creep of magma in the earth. SIAM J. Appl. Math. 49, 231-245.

  12. Fowler, A.C. 1989 Secondary frost heave in freezing soils. SIAM J. Appl. Math. 49, 991-1008.

  13. Fowler, A.C. 1990 A compaction model for melt transport in the earth's asthenosphere. I. The basic model. In: Magma transport and storage, ed. M.P. Ryan, John Wiley, pp. 3-14.

  14. Fowler, A.C. 1990 A compaction model for melt transport in the earth's asthenosphere. II. Applications. In: Magma transport and storage, ed. M.P. Ryan, John Wiley, pp. 15-32.

  15. Audet, D.M. and A.C. Fowler 1992 A mathematical model for compaction in sedimentary basins. Geophys. J. Int. 110, 577-590.

  16. Fowler, A.C. 1993 Asymptotic analysis of the O'Neill/Miller model for secondary frost heave. In: Free boundary problems in fluid flow with applications, eds. J.M. Chadam and H. Rasmussen, Pitman Research Notes Vol. 282, pp. 204-214.

  17. Fowler, A.C. 1993 Towards a description of convection with temperature and pressure dependent viscosity. Stud. Appl. Math. 88, 113-139.

  18. Fowler, A.C. and C.G. Noon 1993 A simplified numerical solution of the Miller model of secondary frost heave. Cold Reg. Res. Technol. 21, 327-336.

  19. Dewynne, J.N., A.C. Fowler and P.S. Hagan 1993 Multiple reaction fronts in the oxidation/reduction of iron-rich uranium ores. SIAM J. Appl. Math. 53, 971-989.

  20. Fowler, A.C. 1993 Boundary layer theory and subduction. J. Geophys. Res. 98, 21997-22005.

  21. Fowler, A.C., C.G. Noon and W.B. Krantz 1993 A computationally feasible reduction of the O'Neill-Miller model of secondary frost heave. Proc. VI Int. Conf. Permafrost, Beijing, China, Vol. 2, pp. 1100-1104. South China University of Technology Press; Wushan Guangzhou, China.

  22. Fowler, A.C. and W.B. Krantz 1994 A generalised secondary frost heave model. SIAM J. Appl. Math. 54, 1650-1675.

  23. Fowler, A.C. and S.B.G. O'Brien 1996 A mechanism for episodic subduction on Venus. J. Geophys. Res. 101, 4755-4763.

  24. Fowler, A.C. and D.R. Scott 1996 Hydraulic crack propagation in a porous medium. Geophys. J. Int. 127, 595-604.

  25. Fowler, A.C. and C.G. Noon 1997 The formation of massive ice in permafrost. In: Ground freezing 97: frost action in soils, ed. Sven Knutsson. Proc. Int. Symp. Ground freezing and frost action in soils, Luleå, Sweden. A.A. Balkema, Rotterdam, pp. 81-85.

  26. Fowler, A.C. and C.G. Noon 1997 Differential frost heave in seasonally frozen soils. International symposium on Physics, Chemistry, and Ecology of Seasonally Frozen Soils, Fairbanks, Alaska. CRREL special report 97-10, Hanover, New Hampshire, pp. 247-252.

  27. Fowler, A.C. and Xin-she Yang 1998 Fast and slow compaction in sedimentary basins. SIAM J. Appl. Math. 59, 365-385.

  28. Fowler, A.C. and C.G. Noon 1999 Mathematical models of compaction, consolidation, and regional groundwater flow. Geophys. J. Int. 136, 251-260.

  29. Fowler, A.C. and Xinshe Yang 1999 Pressure solution and viscous compaction in sedimentary basins. J. Geophys. Res. 104, 12,989-12,997.

  30. Fowler, A.C. and Xinshe Yang 2002 Loading and unloading of sedimentary basins: the effect of rheological hysteresis. J. Geophys. Res. 107, B4, ETG1-1 -ETG1-8.

  31. Fowler, A.C. 2002 Compaction and diagenesis. In: Resource recovery, confinement, and remediation of environmental hazards, eds. J. Chadam, A. Cunningham, R. Ewing, P. Ortoleva and M. Wheeler, IMA Volumes in Mathematics and its applications, vol. 131, pp. 247-262, Springer-Verlag, Berlin.

  32. Fowler, A.C. 2003 A mathematical model of differential frost heave. Proc. 8th Int. Conf. Permafrost, eds. M. Philips, S.M. Springman and L. Arenson, volume 1, pp. 249-252, A.A. Balkema, Lisse.

  33. Fowler, A.C. and S.B.G. O'Brien 2003 Lithospheric failure on Venus. Proc. R. Soc. A459, 2663-2704.

  34. Fowler, A.C. and Xinshe Yang 2003 Dissolution/precipitation mechanisms for diagenesis in sedimentary basins. J. Geophys. Res. 108, B10, 2509, doi: 10.1029/2002JB002269. EPM 13 1-14.