Energetics of the Minoan Eruption: Some Revisions

The combined data have been interpreted by Sparks and Huang (1978) and Sparks (personal communication, 1978); the new values for deposit volumes and masses, together with improvements to some parts of the theoretical analysis, are the basis for the present reassessment. The notation is the same as that in Paper I, and new material is presented here only where significant improvements have been made.

PLINIAN PHASE

Re-examination of the measurements (Bond & Sparks 1976) of the radii of the largest pumice blocks on land, and re-calculation of the likely values of supersonic drag coefficients show that the value of K used in equation (2) of Paper I should be reduced from 145 to about 110 m⁴ kg^-1 s^-2 and that the implied value of u₀, the eruption velocity in the vent, should be raised from 300 to 330 m/s. The maximum height of the eruption cloud, 29 km, and the implied maximum mass eruption rate at the end of the plinian phase, 2.1 x 10⁸ kg/s, are unchanged.

Better estimates can be made of the vent radius, x_m, at the end of the plinian phase and also of the depth of origin of the magma, L, than are given in Paper I. New calculations of the fluid mechanics of eruption of rhyolitic liquids exsolving gas have been made (Wilson, Sparks & Walker, unpublished) which show that u₀ is roughly proportional to the square root of n, the exsolved magma gas weight fraction. For u₀ = 330 m/s, n is deduced to be about 0.023 (i.e. 2.3 wt %). Equation (4) of Paper I then shows that x_m should have been about 160 m, slightly smaller than the value 180 m originally found. The new estimate (Sparks & Huang 1978) of the total mass erupted in the plinian phase is between 5 and 10 X 10¹² kg of which 8 wt % is lithic debris: thus the estimate of the mass of eroded vent and conduit wall material is 4 to 8 X 10¹¹ kg (the estimate in Paper I was 3 X 10¹¹kg). Using an inverted conical vent shape, the implied vertical extent of the conduit, and hence depth of origin of the magma, is in the range 6 to 11 km.

Finally, the duration of the plinian phase can be deduced from the mean mass eruption rate and the total mass of pumice erupted. In Paper I it was assumed that the mean mass eruption rate was one half the maximum value of 2.1 X 10⁸ kg/s. This ratio, one half, applies, for example, to the well documented example of plinian eruption cloud height development at Hekla in 1947 (Thorarinsson 1954). However, simple calculations show that if the conduit and vent radius increase linearly with time, the mean eruption rate is one third the maximum; also, if the radius increases exponentially with time (the condition which leads to lithic debris being distributed uniformly in the pumice as is commonly observed), the mean eruption rate is about one tenth the maximum. As a compromise, it is assumed that the ratio in the case of Santorini was one third. The new range of masses given by Sparks and Huang (1978) for the plinian pumice then implies a duration for this phase of 20 to 40 hours - much longer than the estimate of 8 hours given in Paper I.

IGNIMBRITE PHASE

The new data of Sparks and Huang (1978) show that the mass of the ignimbrite deposits was not less than 6 X 10¹³ kg, of which about 12 wt%, i.e. about 7 X 10¹² kg, consisted of lithic material. A substantial fraction of the lithic debris in the pyroclastic flows may have been picked up after the flows left the vent (C.J.N. Wilson, personal communication), and so a conservative estimate of the lithic debris derived from conduit wall erosion in this phase is taken as 4 X 10¹² kg.

The acid magma body of mass about 6 X 10¹³ kg could be accommodated in a sphere of diameter about 3.6 km. It is more likely that the horizontal dimensions of the magma body were similar to those of the caldera produced by collapse after the eruption, some 8 by 11 km, in which case a vertical thickness of the magma body of order 400 m is implied. This value shows that the vertical extent of the magma body was small compared to the inferred depth of its top, 6 to 11 km, deduced above. The removal of 4 X 10¹² kg of lithic material from the walls of the conduit would have led to an enlargement of the vent radius from the 160 m found for the end of the plinian phase to a value at the end of the eruption of about 380 m if the conduit length were 11 km or about 520 m if the conduit length were 6 km.

In Paper I, an attempt was made to deduce the duration of the ignimbrite phase from the observation that the proportion of lithic debris in the ash flow deposits is roughly constant. A mistake appears in equation (5) of Paper I: the term 2πxLα(dx/dt) for the mass rate of removal of wall material should read 2π (x/2) Lα (½ dx/dt), since the average radius of the conduit wall over its vertical extent will be about half of the vent radius at the surface for the assumed conical shape. The result is that while equation (6) of Paper I is unchanged, equation (7) should read:

τ = nαL /  2ρ_gu₀q

Inserting the latest estimates of n = 0.023, L = 6 to 11 km, u₀ = 330m/s and q= 0.07 implies τ = 11 to 20 hours. Using equation (6) of Paper I, the duration, t, of the ignimbrite phase is found to lie in the range 13 to 17 hours, rather longer than the 9.5 hours initially estimated.

THE AIR-FALL/PYROCLASTIC FLOW TRANSITION

Sparks and Wilson (1976), Sparks et al. (1978) and Wilson, Sparks and Walker (unpublished calculations, 1978) have investigated the factors that cause transitions from plinian activity (a high eruption column producing an air-fall deposit) to ignimbrite formation (a low, "collapsed" eruption column feeding pyroclastic flows). If it is assumed that the magma volatile phase is water, it is possible to specify the combinations of vent radius (or, as an approximate equivalent, mass eruption rate) and eruption velocity (or, equivalently, exsolved magma water content) at which column collapse is likely to occur. Fig. 1 shows this relationship: the shaded zone marks the boundary between the two types of activity. 

Also plotted in Fig. 1 is point (x = 160 m, u₀= 330 m/s) corresponding to the end of the plinian phase and the zone with x = 480 m and u_o of order 300 m/s which defines the end of the eruption. The progress of the eruption with time, dictated largely by the enlargement of the vent and conduit system, can be represented by the line of arrows. Clearly, the available data do not define the track very accurately; but the point corresponding to the end of the plinian phase is quite close to the theoretically expected boundary. As noted in Paper I, a phase of phreatomagmatic activity occurred between the plinian and ignimbrite forming phases: this may well have complicated the transition between the two.

SUMMARY

The results of the new calculations given above, together with the appropriate unmodified results from Paper I, are summarised in Table 1, where the vent radius, and the accumulated erupted masses, are shown as a function of time. Assuming that the total mass of erupted juvenile magma was about 6 X 10¹³ kg, and that the eruption velocity stayed close to 300 m/s, then the total kinetic energy liberated was 2.7 x 10¹⁸ J. The total thermal energy represented by the heat loss from the magma was about 6 X 10¹⁹ J if the latent heat of vaporisation of the magmatic water is included.

The potential energy liberated by the subsequent caldera collapse can be found by assuming that crustal material filled the evacuated magma chamber: a mass equivalent to that of the erupted magma (6 x 10¹³ kg) would have to be displaced vertically by about 400 m (the approximate vertical extent of the chamber). The implied energy is 3 X 10¹⁷ J; it is not clear how quickly this energy was released, however. Thus, as must commonly be the case in eruptions of this sort, the thermal energy deposited in the atmosphere was the major energy release of the eruption. The minimum total duration of the whole event must have been about 35 hours if there were no breaks in the activity.

Table 1

Addendum:

For reference 'Wilson, Sparks and Walter, unpublished' read 'Wilson, Sparks and Walker 1980'.

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