Since lavas are far from thermal equilibrium with ambient surface temperatures and are erupted relatively close to their solidus temperatures, cooling induces very large changes in rheology that will eventually bring flows to a halt (Griffiths, 2000). In contrast to the isothermal Bingham models, in which the rheology of the lava is responsible for stopping the lava, a second group of models takes into account the effects of cooling and solidification on flow length. The non-isothermal models can be further divided into the conductive models (Pinkerton and Sparks, 1976; Head and Wilson, 1986, Pieri and Baloga, 1986) and the thermally mixed models (Crisp and Baloga, 1990, 1994). However, by expressing the limiting equation for radiative cooling in terms o f conductive cooling and initial chilling time, Kilbum (1996) has demonstrated that the limiting conditions for radiative and conductive cooling are different forms of the same physical limiting condition. Hence, both conductive and radiative cooling models may be used to assess the maximum lengthening time of a basaltic aa flow, although neither can be used to identify the limiting style of cooling and solidification.
In addition to cooling, a flow may crystallise and solidify via another mechanism, known as undercooling. This is a typical phenomenon in basaltic lavas and has the effect of lowering the crystallisation temperature of the lava, generating higher crystallinity for its temperature. Prior to emption, the magma is approximately in thermal equilibrium, and crystal formation occurs along a well- defined Temperature-Pressure curve for its dissolved volatile content (mostly dissolved H2O). By dismpting the Si-0 bonds, the dissolved water in the magma delays crystallisation until lower temperatures are reached. Thus, when the magma is empted, it rapidly loses its volatiles by exsolution, and the magma is forced to rapidly crystallise to return to equilibrium. However, the temperature at which this rapid crystallisation occurs is that of the volatile-rich liquid in equilibrium, i.e. a temperature tens of degrees higher. By this mechanism the temperature o f crystallization is lowered allowing crystals to form at lower temperatures.
As described above, cooling affects the surface of the flow forming an insulating cmst. The thermal evolution of this cmst has an important effect on the development of lava flows (Pinkerton and Sparks, 1976; Guest et al., 1987; Kilbum and Lopes, 1988b; Pinkerton and Wilson, 1994). Hence, understanding the properties of this cmst and its interaction with the underlying molten core is vital for the understanding of flow dynamics.
C h a p t e r 2- L a va Flan's a n d M o d e l li n g Their F in p l a c e ia e n t
Radiation
Convection
Natural Forced
Crust Thickness,
Conduction Terrain Slope,
a
Figure 2.15 - Schematic diagram of cooling processes during lava advance. (Diagram from Head and Wilson, 1986).
The first to propose a model based on heat loss as a control o f lava flow dimensions were Harrison and Rooth, (1976) and, later, Pieri and Baloga (1986) and Head and Wilson (1986). They all modelled heat loss by conductive heat transfer to the ground and by radiation to the atmosphere. Based on previous studies (Pinkerton and Sparks, 1976) of conductive flow cooling, Wilson and Head (1986) have related the maximum flow length to the dimensionless Gratz number (Prandtl, 1952) by.
_
kM
k W L
[2.6]
where Gz is the Gratz number and, upon flow halting, is generally in the order of 100 and never more than 300 (Guest et al., 1987), Q is the effusion rate, d is the flow depth, k is the thermal diffusivity, n is the ratio o f equivalent diameter to flow depth, and Wc is the theoretical channel width given by.
[2.7]
where r| is the Bingham viscosity, (3 is the mean gradient and Sy is the yield stress. By combining equations 2.6 and 2.7, the flow length may be obtained from.
C l m p t i ’v 2 - L a v a F la n s a m i M o d e l l i n g T lic ir E m p l a c e m e n t
The limiting Gratz number is taken to be constant for all basaltic flows. Field measurements of rheology suggest that the value (5y/r|)'^^ may also be considered a constant for most basalts and lies between 0.2 and 0.6 s*'"^ (Pinkerton and Wilson, 1988). The factor n is also a constant for any given flow geometry. Hence, equation 2.8 can be usefully used to forecast flow lengths. Pinkerton and Wilson (1988) have modified this equation to take into account the changing rheological properties of the flow as they advance by including a time parameter (equation 2.9).
[2.9]
where t is the time from the start o f eruption. Pinkerton and Wilson (1988) found a good correlation between lengths from equation 2.8 and field values for lavas from Pu’u ‘0 ’o in Hawaii. However, once again, introduction of the time parameter, makes the model unusable for forecasting purposes. This shows how practical models for hazard reduction often have to compromise a perfect model for a useful one. A later modification of this model (Pinkerton and Wilson, 1994) accounted for the dimensions of the channel in relation to the total feeder width. They assumed an incompressible laminar flow with a Bingham rheology and related flow length to channel width w^, mean thickness h, total feeder width w, effusion rate Q, lava rheology, and angle of slope p. The governing equations for this model are different for different combinations of channel and feeder width (equations. 2.10 and 2.11).
For Wr< 0.5w
^ {Q S in p f
v ' 7 /
[2.10]
For w > W f> 0.5w
{QSinP)
[2.11]where t is the emplacement time, Sy is the yield strength and t| is the viscosity. The model was tested for Pu’u ‘0 ’o aa lava flows and found to be in good agreement with the observations.
Modelling flow lengths by a cooling-limited flow model using the Gratz number has also proved successful when applied to the October 1999 Etnean flows (Wright et a l, 2001). A Gratz number between 230 and 330 was calculated for these flows, consistent with the typical Gratz number of -300 derived by others (Hulme and Fileder, 1977; Guest et a l, 1987; Pinkerton and Wilson, 1994). However, in cooling-limited flow models, it is essential to consider the duration of the eruption to indirectly take into account changes in rheology during the flow advance. For a flow to be limited by the solidification of its front, the minimum time from the start of the eruption needed for this to occur is given by (Kilbum,
C h u p lc v 2- LdVit f l o u s a n d M o d e l l i n g Their E n tp l a c c m c n t
t =
p C , s ( j rSinp
[2 .12]L = u
g s c j rSinj3
[2.13]where S is the energy per unit volume for failure in Jm'^ and is given by the product of the strain at failure 8t and the tensile strength Ot, p is the density of the lava crust in kg m '\ g is the acceleration due to gravity in ms'^, Cp is the specific heat capacity in Jkg'% 8 is the surface emissivity, and ct is Stefan-Boltzmann constant, t is the eruption temperature, p is the ground slope in degrees and u is the mean velocity in ms '. By substituting typical values for S, Cp and T and the constants 8, ct and g the length may be expressed in terms of velocity and slope (equation 2.14).
L
17 X
1 0'
Sinp
[2.14]This model was further was tested by application to the October 1999 Ema flow and yielded a solidification time of -17 hours (Wright et al. 2001). This is consistent with the emplacement time of 15- 21 hours obtained from field observations and ETM (Enhanced Thematic Mapper) data.
Although the inclusion o f cooling in lava flow modelling has refined greatly flow models, the Gratz number only expresses heat loss only due to conduction and neglects any type of radiation due to exposure of the core of the flow to the atmosphere through cracks in the aa flows.
The effect of radiative cooling was considered by Crisp and Baloga (1990) with a two-component thermal model. The majority of the existing models investigating lava flow emplacement styles and consequent morphologies (Fink and Griffiths, 1992; Gregg et al., 1998) envisage the flow as a single thermal component, while recognising the presence of two mechanically separate components, leaving core and crust as two independent units. In their two-component model. Crisp and Baloga (1990) suggested that the mechanical and thermal interaction between crust and core is a key factor determining which processes produce the observed surface morphologies. The basic concept on which the model is constructed is that radiation influences the cooling history and, therefore the morphology and dimensions of a flow.
Understanding thermal losses during emplacement is fundamental in flow modelling due to the control on the lengths o f lava flows, on their morphologies and on the tendency to form breakouts upstream of the advancing flow front (Crisp and Baloga, 1990). The first theoretical investigations of thermal losses on
d u t p t i ’v 2- L a va F lo w s a m i ModcUin;^ T h eir E in p ia c e iiic iit
lava flows (Shaw and Swanson, 1970; Danes, 1972; Harrison and Rooth, 1976) assumed that the flow is isothermal in each cross section of the flow perpendicular to the direction of the flow advance. However, for this to be true, the single thermal component models implicitly assume that the flow must be mechanically mixed in each vertical cross section of the flow. This however would lead to an unrealistically rapid cooling of the entire flow (Pieri and Baloga, 1986). The two-component model, on the other hand, envisages a crust that cools by radiation and thickens with time, and an inner core that is vertically isothermal and partially exposed through crustal cracks at the top surface (Figure 2.16). The core is able to cool by radiation or by thermal interaction with the boundary layer between core and crust.