The Dynamics of Lava The Dynamics of Lava FlowsFlows

R. W. GriffithsR. W. Griffiths

OutlineOutline

• Motivation and Methods• Flow without Cooling

– Viscous Flow– Viscoplastic Flow

• Flow with Cooling• Summary and Conclusions

MotivationMotivation

• 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

MethodsMethods

• Review of theoretical and experimental studies of flow dynamics

• Compare:– Field Observation– Numerical Solutions– Experimental Results

Towards more physically consistent models

Flow without CoolingFlow without Cooling

• Isothermal models• Horizontal + vertical momentum

equations

• Render equations dimensionless

AssumptionsAssumptions

• Lava domes --- Re 10-10-10-4

Spreading of very viscous Newtonian fluid creeping over horizontal / sloping planes

• Hawaiian channel flows --- Re 1-102

• Komatiites --- Re 106

For long basalt flows assume well-mixed flows with uniform properties in the vertical

Dynamical RegimesDynamical 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 << yield stress (B>>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=

Axisymmetric Viscous Axisymmetric Viscous FlowFlow

• 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

Viscous Flow on a SlopeViscous 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

Axisymmetric Axisymmetric Viscoplastic FlowViscoplastic 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

Scaling Analysis Scaling Analysis SolutionSolution

• 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

Newtonian vs. Plastic?Newtonian vs. Plastic?

• Model gives larger shear rates at the vent– Viscous forces important at early times

• Transition from viscous flow (B<<1) to plastic flow (B>>1) at later times

• Good agreement with height and radius data from La Soufriere

• Plastic model describes lava domes better

Viscoplastic flow on a Viscoplastic flow on a slopeslope

• 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

• 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

• 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

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

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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

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What’s the extent of solidification?

Proximity of eruption temp to solidification temp

Basaltic lavas : 0.6Rhyolitic lavas : 0.8

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 solidificationSubmarine lavas : 0.1sSubaerial basalt: 100sSubaerial 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!------

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

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

Thin surface layer with larger viscosity or yield strength.

Isothermal and rheologically uniform (viscous or plastic)

Extending previous solutions for homogeneous flows…

…to cases involving balance between:

1. Buoyancy and crust viscosity2. Buoyancy and crust yield strength3. Overpressure and crustal/interior retarding forces

• Flow driven by: gravity or overpressure• Flow retarded by: basal stress and crustal stress

BIG Q : deep flow!

H ~ Q^1/4 independent of R

buoyancy and viscosity

viscous flow, no crust

H ~ (Qt)^1/5

buoyancy, viscosity, yield strength

viscoplastic

BIG Q : thin crust!small flow height!

H ~ Q^-1/3 R^2/3

for the cruststrength control

H ~ t^1/4

buoyancy, viscosity, yield strength (crust)

crust with yield strength

Growth of dome height with time for PEG wax

No coolingNo cooling

Overpressure: sudden increase in height

Growth of dome height with time for solidifying kaolin/PEG slurry with yield stress

Encapsulated a thick solid Encapsulated a thick solid and grew threw upward and grew threw upward spinesspines

Solid only at margins

Trends consistent with: H ~ t^1/4

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

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!

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