Energetics of the Minoan Eruption

The maximum eruption column height during the initial plinian phase was about 29 km; this phase lasted about 8 hr and produced a 360 m diameter vent at the surface connected to a magma reservoir at a depth of about 3.4 km. A short period (about 1 hr) of phreatomagmatic activity followed. The final phase involved the eruption of up to 35 km³ of magma in the form of ash-flows leading to ignimbrite deposits. This activity, lasting for about 9 hr, enlarged the vent to a diameter of about 2.2 km, and was accompanied by violent precipitation to produce flood deposits. The total duration of activity was at least 18 hr, during which up to 3.2 X 10¹⁹  J of energy were released into the atmosphere.

INTRODUCTlON

The present author has recently been involved in the development of new models of explosive volcanic processes; these are applied here for the first time to the analysis of the deposits from the Minoan eruption.

The geological history of the volcano Santorini has been documented by a number of workers, and the important facts are summarised by Bond & Sparks (1976) who present a detailed analysis of the thickness and grain-size characteristics of the Minoan deposits. Their data are used as the basis of the present study. They show that the eruption consisted of the following sequence of events:

-(a)    the formation of an air-fall deposit of plinian type as defined by Walker (1973);

(b)    a fairly rapid transition to phreatomagmatic activity as water gained access to the vent, leading to a series of base surge deposits interspersed with fine air-fall beds;

(c)    the at least partial collapse of a tuff ring (built up by the base surge deposits) to form mud flows; and

(d)    the emplacement on the outer flanks of the volcano of ignimbrite deposits, made up of a large number of flow units interbedded with which are coarse breccia beds identified as the products of flash floods occuring during the pyroclastic flow phase;

(e)    the later collapse of the central parts of the volcano.

The above time sequence seems well established from the stratigraphic relationships in the deposits, including the fact that at least some of the caldera collapse must have taken place after the pyroclastic flow phase since the flood deposits contemporaneous with the ignimbrites require a source of material above the level of the present caldera rim.

Various types of analysis can be carried out on the data from each stage of the eruption, and these are now treated in turn.

PLINIAN PHASE

I have already described (Wilson 1976) a method of finding the eruption velocity of gas (and small particles) in and just above the vent in plinian eruptions. The force balance equation for a large clast just supported against gravity by the gas flow in the vent, and just deposited at the edge of the vent is

4/3 πr³_o σ_οg = 1/2 ρ_g C_D πr²_o u²_o                                             (1)

where r_o is the mean radius of the clast, σ_o is its density, g is the acceleration due to gravity, ρ_g is the density of the volcanic gas (assumed to be steam at a magmatic temperature of about 1200 K and decompressed to atmospheric pressure so that r_g = 0.18 kg/m³), u_o the gas velocity and C_D is a drag coefficient with value close to unity. This equation simplifies to

u²_o = (8r_o σ_οg) / (3ρ_g C_D ) = K r_oσ_o                                (2)

where the numerical value of K is 145 m⁴ kg ^-1 S ^-2. In practice it is rarely possible to measure r_o, since caldera collapse or the emplacement of later deposits near the vent prevent access to the largest clast erupted. One therefore measures the maximum radii, r, of the clasts (pumice and accompanying lithic fragments) of various densities σ as a function of distance R from the vent and plots a graph of the product rσ against R. The extrapolation of this graph to R = Ο yields r_oσ_o.

If an appreciable wind is blowing, there is a complication in that while larger clasts (rσ > about 500 kg/m²) are scarcely affected by the wind, smaller clasts falling out from the eruption column are displaced in a downwind direction.

In such cases, the range R is measured at right angles to the downwind axis, which can be located from the symmetry of the deposit.

The rσ/R graph has a second use. At some large value of R, say R_e, rσ is found to decrease rapidly, corresponding to the location of the edge of the eruption cloud. Both examination of photographs of plinian-type eruption columns and calculations based on the fluid dynamic theory of high-speed jets (Wilson 1976), show that the total height, H, of an eruption column should be very close to 2R_e.

Fig. 1 shows the results of carrying out the above analysis for the Santorini plinian deposit and also a number of other plinian and subplinian deposits documented in the literature. R_e is estimated to be 14 to 15 km, so that the cloud height is H = 28 to 30 km. Extrapolation of the rσ curve to R = Ο yields u_o = 300 m/s. Both values are maxima, since the largest values of rσ were measured at each range R. It is not easy to predict how u_o will have changed with time during the eruption (Wilson & Sparks 1977).

The maximum column height, H, can be used to derive the maximum mass eruption rate, since Wilson et al. (1977) have shown that cloud height in plinian eruptions is directly related to mass eruption rate, m, by

H_m = 8.2 (ms ΔΘ)^¼

                                                  (3)

where H_m is H in metres S is the specific heat of the eruption products (essentially the rock), ΔΘ is the difference between magma temperature and ambient temperature and the term in parentheses is the rate of release of thermal energy.

Inserting H_m = 29000 m one finds m = 2.1 Χ 10⁸ kg/s. This is again the maximum value. Now m is given by 

m = βu_o πx²  =   (ρ_gu_o πx²) / n                                     (4)

where x is the vent radius and β is the bulk density of the eruption products, given adequately by β = ρ_g/n in which ρ_g is the exsolved gas density defined earlier and n is the weight fraction of exsolved gas in the erupting mixture. One must expect x to have increased during the plinian phase (as evidenced by the presence of lithic clasts derived from the vent and conduit walls) from some small value to a maximum x_m, with a corresponding increase in m with time. For the present purpose it will be assumed that the average value of m, say m_a, was half the maximum value found above, say 1 X 10⁸ kg/s. The estimated mass of the plinian deposit is about 3 X 10¹² kg (Bond & Sparks 1976), implying that the duration of the plinian phase was rather more than 8 hours. In view of the uncertainty in the exact value of m_a, this should be regarded as about 8 ± 3 hours of equivalent continuous activity.

The maximum mass eruption rate m can also be used to estimate the maximum vent radius x_m at the end of the plinian phase. The calculations of Wilson & Sparks (1977) show that for u_o= 300 m/s in a plinian event, we must have n≥0.02. Consideration of the solubility of water in rhyolite (Hamilton et al 1964) makes it unlikely that n was greater than 0.08. Adopting n = 0.05 ± 0.03 in (4), x_m = 246 m with an error of about a factor of 2. This value is in satisfactory agreement with the observed sizes of other rhyolitic vents (Walker, 1966).

Bond & Sparks (1976) found that the lithic content of the plinian deposit varied from 4 to 15 % by weight. Adopting an average of 10 %, the total mass of lithics erupted during the erosion of the vent to 246 m radius was 3 X 10¹¹ kg; this must have occupied a volume of 1.2 X 10⁸ m³ (using a density of 2.5 x 10³ kg/m³). If we assume that the vent and conduit system had an inverted conical shape, with a negligible radius at the top of the magma source region compared with the radius x_m at the surface (a simple but probably adequate approximation to the true shape), then the required length, L, of the conduit would be 1.9 km. Of course, the error of a factor of 2 in x_m implies an error of a factor 4 in L. Fortunately we can improve the estimate greatly. The calculations of eruption velocity by Wilson & Sparks (1977) show that for a fixed exit velocity, the implied depth of origin increases as the assumed gas content of the erupting mixture is increased. So for a set of assumed gas contents between, say 0.02 and 0.10, the implied depth of origin L_i can be noted. For each of the selected gas contents, equation (4) can be solved for x_m and the above calculation for L repeated.

The correct solution is that for which L = L_i. In this case the result is L = L_i = 3.4 km, n = 0.035 and x_m = 180 m. Of course there is still considerable uncertainty in these values since the total lithic mass is only an estimate, and a very simple conduit geometry and a constant gas content have been assumed. None the less, it appears that we are dealing with a fairly shallow magma source.

BASE SURGE PHASE AND MUDFLOWS

Using the data provided by Bond & Sparks (1976), an estimate of the total volume of material contained in the base surge beds can be made at about 1.5 X 10⁹ m³; using a mean density of 0.8 X 10³ kg/m³ for these deposits, the mass is 1.2 X 10¹² kg, to be compared with some 3 X 10¹² kg for the plinian deposit. Presumably, vent and conduit erosion continued during this stage of the eruption, and so the average mass eruption rate must have been rather larger than the value at the end of the plinian phase (which was estimated above to be 2.1 X 10⁸ kg/s). Adopting, say, 2.5 X 10⁸ kg/s, the duration of this phase must have been about 1.3 hr.

As mentioned earlier, Bond & Sparks (1976) proposed that the numerous mud flows lying above the base surge beds represent the collapse of a tuff ring built up during the base surge phase. Such a process of collapse could, presumably, have happened quickly as a single event or could have been episodic; in either case, there is no obvious way of determining the duration.

IGNIMBRITE PHASE

The transition from air-fall to ash-flow activity, though complicated by the intervening phreatomagmatic phase, was almost certainly associated with the progressive enlargement of the vent, as indicated by the calculations of Sparks & Wilson (1976).

Bond & Sparks (1976) noted that the ignimbrite deposits contained about 50 % by weight of lithic material. It will be assumed that the ignimbrites on the presently observable islands are representative of all the ash flow material deposited (a reasonable but untestable assumption). The volume of the present caldera is some 60 km³, representing a missing mass of 7.5 X 10¹³ kg of dense rock of density 2.5 X 10³ kg/m³. Since the total mass in the pre-ignimbrite deposits is some 4.5 X 10¹² kg, the mass of ignimbrite required to explain the missing volume is about 7 X 10¹³ kg, of which, therefore, 3.5 X 10¹³ kg should consist of lithic material from the erosion of the conduit system. This would have occupied a volume of 1.4 X 10¹⁰ m³ (vastly greater than the volume of the conduit eroded during the plinian phase which was found above to be 1.2 X 10⁸ m³).

If we assume that the conduit length remained at the value 3.4 km calculated earlier, the final radius would have been 1.13 km. The mass of acid rock ejected, also 3.5 X 10¹³ kg, could be accommodated in its pre-eruption state in a spherical magma chamber of radius 1.49 km. Of course the shape may not have been so simple, but this length gives the order of magnitude of the size of the magma body. It should also, perhaps, be stressed that there is no proof that the volume of the present caldera gives a true guide to the volume of erupted material. It is known that the magma involved was a physical mixture of two types of liquid, the major component being a rhyodacite (Bond & Sparks 1976) and the minor component a basaltic andesite. Sparks, Sigurdsson & Wilson (1977) have proposed that acid eruptions are commonly, perhaps almost always, triggered by the intruson of relatively basic magma into an acid liquid body, and there is every reason to think that this happened in the Minoan event. There is no way of assessing how much subsidence may have been the result of the injection and subsequent draining of the basic magma into the surrounding area as happened, for example, in the 1875 eruption of Askja, Iceland.

The fact that the proportion of lithic material in the ignimbrite deposits was roughly constant during the eruption can be used to deduce some features of the conduit erosion process. The mass eruption rate of acid magmas given by equation (4) can be equated to the rate of removal of mass from the conduit walls which is clearly 2πxLα (dx/dt) where α is the density of the wall rock (taken as 2.5 X 10³ kg/m³) and dx/dt is the rate of recession of the wall :

ρ_gu_oπx² / n = 2πxLα dx/dt                                                (5)

which integrates to

x = x_o exp (t/τ)                                                                (6)

with

τ = 2nαL / ρ_gu_o                                                                            (7)

Thus the conduit radius increases exponentially with time from an initial value x_o, at a rate characterised by the time constant τ given by (7). We have seen that u_o≈ 300 m/s and n ≈ 0.035 for the plinian phase and that L = 3.4 km;

u_o is unlikely to have increased during the later part of the eruption unless the gas content increased significantly with depth in the magma chamber. Even if this happened, the fact that n and u_o appear as a ratio in equation (7) should ensure that their variation had a minimal effect on the value of τ . An average of 200 m/s for the ignimbrite phase is assumed and n is taken as 0.03. Thus τ  is 3.9 hr. Since the conduit must have been eroded from an average radius of about x_o = 100 m at the end of the plinian phase to x = 1.13 km at the end of the ignimbrite phase, we have t ≈ 9.5 hr for the duration of the ash flow events.

Again, this value can hardly be regarded as accurate, but must have the correct order of magnitude.

There is little that can be deduced about the mechanics of the ash flows, other than the consequence of the observation (Bond & Sparks 1976) that most flows did not surmount a 200 m high ridge at Platinamos, about 8 km from the vent. Sparks, Wilson & Hulme (1977) have shown that pyroclastic flows are likely to be so mobile near their sources that the horizontal travel velocity u_h can be found from the height, z, of hill over which the flow can just rise by equating the kinetic energy of the approaching flow, ^½u_h², to the potential energy lost in climbing the hill, g_z. For z = 200 m, u_h must be about 63 m/s. Since a few flows did cross the barrier, u_h= 60 m/s at 8 km from the vent may be a realistic average. Since the geometry of individual flows is not well documented it is not possible to deduce the eruption velocity of the material forming the ash flows.

The existence of flash flood deposits, identified by the coarse breccia they contain, interbedded with the ignimbrite flow deposits is hardly surprising.

The total mass of water vapour released into the atmosphere during the 9 to 10 hours of the ash-flow phase would have been about 10¹² kg (0.03 of the 3.5 X 10¹³ kg of juvenile magma). This corresponds to 1 km³ of liquid water.

Comparison of the expected behaviour of the convective air mass above the hot ash flows with that of large convecting systems normally encountered in the Earth's atmosphere suggests that most of the released water vapour would recondense quickly in a major storm system around the volcano.

SUMMARY

The results of the calculations presented above are summarised in Table 1, where the vent radius, and the accumulated erupted masses, are shown as a function of time. If the total mass of erupted juvenile magma was indeed 4 X 10¹³ kg, and the eruption velocity stayed mainly in the range 200 to 300 m/s, then the total kinetic energy liberated was 1.4 X 10¹⁸ J. The total thermal energy represented by the heat loss from the magma was about 3 X 10¹⁹ J; this estimate rises to 3.2 X 10¹⁹ J if the latent heat of vaporisation of the 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 (4 X 10¹³ kg) would have to be displaced vertically by about 3 km (the vertical extent of the chamber). The implied energy is 1.2 X 10¹⁸ 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 18 hr if there were no breaks in the activity.

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