Page 103 - Caldera Volcanism Analysis, Modelling and Response

78 Roberto Sulpizio and Pierfrancesco Dellino

Figure 10 (a) Sketch of the changes in driving and resisting forces for a PDC that moves over
an inclined obstacle; (b) simple diagram showing the splitting of the initial velocity (v 0 ) into
the normal (v n ) and the tangential (v t ) components due to the steepening of slope.

(r ¼ 0) v t ¼ 0. Hence, the kinetic energy of the moving flow can be written as:
1 2
E c ¼ rhn t (7)
2
To reach a complete stop of the moving current, the work exerted by frictional
forces (W f ) and transformation in potential energy (E p ) must equal the available
kinetic energy:
E c þ E p þ W f ¼ 0 (8)
under the assumption that frictional forces are dissipated only as work (W f ) at the
flow base:
W f ¼ krhgL cos y (9)
where k is the coefficient of dynamic friction, L the travelled length on slope and
y the slope angle (Figure 10), and:
E p ¼ rhgL sin y (10)

then, we can obtain the length travelled along slope, L:
2
L ¼ n ½2gðk cos y þ sin yފ 1 (11)
t
Inspection of Equation (11) reveals that it represents a power-law function (i.e.
a parabola in L vs. v space) and that L critically depends on slope but increases
dramatically for very fast currents.