1. The Dynamics of Lava Flows R. W. Griffiths

2. Outline Motivation and Methods Flow without Cooling –Viscous Flow –Viscoplastic Flow Flow with Cooling Summary and Conclusions

3. Motivation Assess hazards –Rheology, Effusion Rate, Topography, Flow front, Stability of lava domes Interpret ancient flows –Understand Ni-Fe-Cu Sulphide ore formation Interpret extraterrestrial flows –Morphology --> Rheology + Eruption Rates

4. Methods Review of theoretical and experimental studies of flow dynamics Compare: –Field Observation –Numerical Solutions –Experimental Results  Towards more physically consistent models

5. Flow without Cooling Isothermal models Horizontal + vertical momentum equations Render equations dimensionless

6. Assumptions Lava domes --- Re 10 -10 -10 -4  Spreading of very viscous Newtonian fluid creeping over horizontal / sloping planes Hawaiian channel flows --- Re 1-10 2 Komatiites --- Re 10 6  For long basalt flows assume well-mixed flows with uniform properties in the vertical

7. Dynamical Regimes Significance of yield stress set by Bingham Number –B=0 for Newtonian behaviour –B --> infinity for large yield stresses –B=1 is critical Bingham Number (viscous-plastic transition)) Silicic domes grow very slowly –viscous stresses >1 ) –balance between gravity / yield stress Large basaltic channel flows - B<<1 balance viscous forces / gravity After onset of yield stress, plastic deformation dominates in cooler / slower areas B=

8. Axisymmetric Viscous Flow Solution by Huppert (1982) gives: –speed of flow front advance –relation between height and radius  Rate of advance of the front slows - dome height decreases (under constant source flux)  Dome height increases under increasing source flux Good fit with experiments involving viscous oil Discrepancies with La Soufriere data due to Non-Newtonian properties

9. Viscous Flow on a Slope Solve for flow outline + 3-D depth distribution –Add dependence on slope of angle  Solution by Lister (1992) shows: – flow becomes influenced by slope after a certain time or volume –followed by width and length increase Flow becomes more elongated for larger viscosity and larger volume flux Grows wider compared with its length as volume increases

10. Axisymmetric Viscoplastic Flow Introduction of a yield strength Assume fluid only deforms at base Solution by Nye (1952) implies central height and radius always related Good agreement with experiments involving kaolin/water slurries except at origin and flow front

11. Scaling Analysis Solution Based on force balance Agrees well with experiments involving slurries of kaolin + polyethylene glycol wax Static Solution:  dome does not continue to flow when vent is stopped  H and R independent of time Same material remained at flow front

12. Newtonian vs. Plastic? Model gives larger shear rates at the vent –Viscous forces important at early times Transition from viscous flow (B >1) at later times Good agreement with height and radius data from La Soufriere Plastic model describes lava domes better

13. Viscoplastic flow on a slope Levees imply non-Newtonian flow Consider Bingham fluid flow down slope  (Hulme, 1974) Lateral flow would stop when pressure-gradient balanced by yield stress –Implies a critical depth below which there will be no downslope motion –With a width of stationary fluid along the edge of the flow Free viscoplastic flow between stationary regions

14. Kaolin/Water experiments show stationary levees bounding long down-slope flows –Height consistent with numerical solution –Levees can be explained in terms of isothermal flows having a yield strength Lava domes more challenging –Need to predict full 3-D shape inc. up-slope Equation for flow thickness normal to the base

15. When the dome volume is normalized by  Dynamical regimes can be identified: –For V<<1, minor influence of slope, close to axisymmetric with quadratic thickness profile –For V>1, strong influence of slope = displacement from vent –For V>>1, down-slope length of dome tends to infinity Experiments with slurries of kaolin in polyethylene glycol wax consistent with theoretical solution –Departure from circular as V increases –Development of levees for long flows V>10

16. Effects of Cooling and Solidification Large temperature contrasts between lavas and atmosphere (or ocean) –Cooling –Changes in rheology –Flow stops Important to investigate thermal effects in flow models –Laminar vs turbulent thermally and rheologically stratified mixing of surface boundary layer will cool interior –Rheological change –Rate of cooling –Rate of spreading of flow Dimensionless numbers –Pe (rate of advection : rate of conduction) –Nu (rate of convection : rate of conduction) –S (latent heat : specific heat) Large values indicate active flows When flow involves crystallization, L can be significant

17. We know that dramatic rheological changes occur with changing temperature! Solidification Glass transition temperature (quenching) Temperature when crystallinity ~ 40-60% (slow cooling) Concerned with rapid surface quenching and glassy crust What’s the extent of solidification? Proximity of eruption temp to solidification temp Basaltic lavas : 0.6 Rhyolitic lavas : 0.8

18. Now, let’s define a dimensionless parameter to describe the extent and effects of solidification…. (i.e. a dimensionless solidification time) Advection timescale Time for solidification Submarine lavas : 0.1s Subaerial basalt: 100s Subaerial rhyolite: 60s Defined for constant volume flux Q When flow is plastic… “Provide general indication of whether crust thickens quickly or slowly relative to lateral motion.” -------Ratio of the rate of lateral advection to the rate of solidification!------

19. Creeping flows with cooling Experiments to test Viscous Fluid: polyethylene gylcol wax –Freezes at ~ 19 ºC Extruded from small vent under cold water on to horizontal (or sloping) base a) Cooling is rapid or extrusion is slow, pillows form b) Thick solid forms over surface, rigid plates, rifts form, forms ropy structure. c) Thick solid, plates form and buckle or fold, jumbled plates. d) Crust only seen around margins of the flow, forming levees Pillow flows Rifting flows Folding flows Leveed flows

20. Creeping flows with cooling Experiments to test Viscous Bingham-like fluid: mixture of kaolin-PEG –Freezes at ~ 19 ºC –Yield strength Extruded from small vent under cold water on to horizontal (or sloping) base Different sequence of morphologies suggest rheology of interior fluid plays a role in controlling flow and deformation a) Spiny extrusion b) Lobate extrusion c) Distinct lobes surfaced by solid plates. d) Axisymmetric flow, unaffected by cooling. Morphologies resemble highly silicic lava domes

21. Thin surface layer with larger viscosity or yield strength. Isothermal and rheologically uniform (viscous or plastic)

22. Extending previous solutions for homogeneous flows… …to cases involving balance between: 1.Buoyancy and crust viscosity 2.Buoyancy and crust yield strength 3.Overpressure and crustal/interior retarding forces Flow driven by: gravity or overpressure Flow retarded by: basal stress and crustal stress

24. Growth of dome height with time for PEG wax No cooling Overpressure: sudden increase in height

25. Growth of dome height with time for solidifying kaolin/PEG slurry with yield stress Encapsulated a thick solid and grew threw upward spines Solid only at margins Trends consistent with: H ~ t^1/4

26. Growth of dome height with time for 4 lava domes: La Soufriere, Mt. St. Helens, Mt. Pinatubo, Mt. Unzen Trends consistent with: H ~ t^1/4

27. If we compare theoretical scaling with available measurements for active lava domes… …Models of spreading with yield strength of crust compare to real data! Trends consistent with: H ~ t^1/4 Scaling analysis can be applied to evaluate crustal yield strengths for real lava domes! Isothermal Bingham model used to estimate internal lava yield stress Neat!

28. conclusions Theoretical solutions for simple isothermal flows provide: –Explanations for elementary characteristics of lava flows and –Demonstrate implications of viscous and Bingham flow Solutions serve as basis of comparison for more complex models Thermal effects lead to range of complexity: –Rheological heterogeneous flows –Instabilities: flow branching, surface folds, pillows, blocks, lobes, spines, etc. Difficulties with moving free surface at which thermal and rheological changes are concentrated