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LAVA FLOWS 137

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times πr divided by 2πr, which is 2r. But 2r is, square roots (κt) 1/2 = d /4.65, i.e., d = 4.65(κt) 1/2 ,
c
c
of course, the actual diameter of the tube, so in this essentially identical to the above expression for the
case the equivalent diameter is equal to the actual total depth of cooling. Thus the idea that an open
diameter. Now consider lava flowing with a depth channel flow stops moving when its Grätz number
d in an open rectangular channel of width w . The reaches 320 appears to be perfectly consistent with
c c
lava wets the sides and the base of the channel but the idea that it has lost heat by conduction through
the upper surface is not in contact with anything both its upper and lower boundaries.
fixed to the ground, and so the wetted perimeter In fact, however, there are a number of com-
is (w + 2d ). The cross-sectional area is equal to plications not considered in the above analysis.
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(w d ), and so d is equal to [(4w d )/(w + 2d )]. Figure 9.1 shows that open channel flows tend to
c c e c c c c
Now assume the commonest case for flows, that form a “raft” of relatively cool lava in the middle of
the channel width w is much greater than its depth the channel, with zones of shearing on either side
c
d . Then the 2d part of the denominator can be where fresh, incandescent lava is exposed. This
c c
neglected so that d is approximately equal to 4d . lava will be radiating away heat much faster than
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Some exact values are d = 3.33d when the chan- it can be conducted through the rigid cooled cen-
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nel width is 10 times the depth, d = 3.64d when tral raft, and so the above calculation based only on
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the width is 20 times the depth, and d = 3.85d conduction of heat will underestimate the heat
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when the ratio is 50, a typical value for basaltic loss, and we might therefore expect flows to stop
flows in Hawai’I. moving sooner than anticipated on the basis of
With this definition of d , it is possible to define conductive cooling alone. Also, the factor 2.3 which
e
a dimensionless number called the Grätz number, appears in eqn 9.1, although the one commonly
Gz, as used in treatments of heat flow, is nevertheless not
unique. It represents the depth below the cooling
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Gz = d /(κ t) (9.9) surface where 90% of whatever total amount of
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cooling will eventually take place has already hap-
Clearly when a flow is first formed, no cooling pened. One might equally well choose, say, 85%,
has occurred, t is very small and the Grätz num- in which case the factor in eqn 9.1 would be 2.07
ber is extremely large. As the flow advances with instead of 2.3 and we would be comparing the
time, Gz decreases. After compiling information on value 4.65 obtained from the Grätz number with
cooling-limited lava flows with a range of different 4.14 instead of 4.6 – not such an impressive match.

compositions, Pinkerton & Wilson (1994) found The fact that numerous open channel lava flows
that individual flows stop moving when Gz has observed in the field do stop when their Grätz num-
decreased to a critical value Gz close to 320. In the ber reaches about 320 suggests that many of the
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case of volume-limited flows it may happen that, details neglected in the simple model must cancel
although the vent has ceased to release lava, the out. Understanding exactly why this is requires a
central channel of the flow drains out to extend the more detailed model of lava flow motion and cool-
front of the flow, and in these cases too the process ing than has yet been developed. This is even more
is found to cease when the Grätz number decreases true for flows in which the cooled raft in the central
to a value equal to about 320. channel becomes connected to the levées creating
This finding about the Grätz number underlines a fully insulated lava tube.
the basic limitation on lava flow advance. Equation
9.1 implies that in a time t conductive cooling will
have penetrated a distance ∼2.3(κt) 1/2 into a flow. 9.8 Surface textures of lava flows
Cooling occurs at the base as well as the top, so
the total thickness of cooled lava is ∼4.6(κt) 1/2 . So far the large-scale structures of lava flows have
But eqn 9.9 implies that a flow stops when been considered. If one is faced with having to walk
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(κt) = d /320, which, with d =∼3.85d , typical across a lava flow, then other, small-scale issues
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of Hawaiian flows, means (κt) = d /21.59. Taking become important. The first is that, even if the flow
c