Transcript Effects of Cooling and Solidification

The Dynamics of Lava
Flows
R. W. Griffiths
Outline
• Motivation and Methods
• Flow without Cooling
– Viscous Flow
– Viscoplastic Flow
• Flow with Cooling
• Summary and Conclusions
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
Methods
• Review of theoretical and experimental studies of
flow dynamics
• Compare:
– Field Observation
– Numerical Solutions
– Experimental Results
 Towards more physically consistent models
Flow without Cooling
• Isothermal models
• Horizontal + vertical momentum equations
• Render equations dimensionless
Assumptions
• 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 Regimes
• Significance of yield stress set by Bingham Number
– B=0 for Newtonian behaviour
B=
– 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
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
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
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
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
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 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
• 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)
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Cooling
Changes in rheology
Flow stops
Important to investigate thermal effects in flow models
–
Laminar vs turbulent
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QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
thermally and rheologically stratified
mixing of surface boundary layer will cool interior
Rheological change
Rate of cooling
Rate of spreading of flow
Dimensionless numbers
–
–
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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
We know that dramatic rheological changes occur with
changing temperature!
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
• 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?
Basaltic lavas : 0.6
Rhyolitic lavas : 0.8
Proximity of eruption temp to solidification temp
Now, let’s define a dimensionless parameter to describe the
extent and effects of solidification…. (i.e. a dimensionless
solidification time)
Submarine lavas : 0.1s
Subaerial basalt: 100s
Subaerial rhyolite: 60s
Time for solidification
Defined for constant volume flux Q
Advection timescale
“Provide general indication of
whether crust thickens quickly or
slowly relative to lateral motion.”
When flow is plastic…
-------Ratio of the rate of lateral advection to the rate of solidification!------
Creeping flows with cooling
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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.
Folding flows
Leveed flows
c) Thick solid, plates
form and buckle or fold,
jumbled plates.
Pillow flows
Rifting flows
d) Crust only seen around margi
of the flow, forming levees
Creeping flows with cooling
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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
Isothermal and rheologically
uniform (viscous or plastic)
Thin surface layer with larger
viscosity or yield strength.
• Flow driven by: gravity or overpressure
• Flow retarded by: basal stress and crustal stress
Extending previous solutions for homogeneous
flows…
…to cases involving balance between:
1.
2.
3.
Buoyancy and crust viscosity
Buoyancy and crust yield strength
Overpressure and crustal/interior retarding forces
viscous flow, no crust
viscoplastic
crust with yield strength
buoyancy and viscosity
buoyancy, viscosity, yield strength
buoyancy, viscosity, yield strength (crust)
H ~ Q^1/4
BIG Q : deep flow!
independent of R
H ~ (Qt)^1/5
for the crust
strength control
H ~ Q^-1/3 R^2/3
BIG Q : thin crust!
small flow height!
H ~ t^1/4
Growth of dome height with time for PEG wax
Overpressure: sudden
increase in height
No cooling
Growth of dome height with time for solidifying kaolin/PEG slurry with yield stress
Trends consistent with: H ~ t^1/4
Solid only at margins
Encapsulated a thick solid
and grew threw upward
spines
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