G 3. AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES Published by AGU and the Geochemical Society. Caldera collapse and the generation of waves

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1 Geosystems G 3 AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES Published by AGU and the Geochemical Society Article Volume 4, Number 2 15 February , doi: ISSN: Caldera collapse and the generation of waves J. P. Gray School of Mathematical Sciences, Monash University, P.O. Box 28M, 3800 Australia ( James.Gray@astro.le.ac.uk) Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK J. J. Monaghan School of Mathematical Sciences, Monash University, P.O. Box 28M, 3800 Australia [1] The aim of this paper is to begin a study of the waves produced by the collapse of a caldera connected to the sea. An example is the bronze age collapse of the caldera of Santorini (Thera), which is thought to have involved an area of approximately 70 km 2 subsiding to a depth close to the present 390 m. In this paper, we concentrate on the purely mechanical aspects of the flow and adopt a simple geometry that replicates some of the features of the pre-bronze age caldera of Santorini. By combining laboratory experiments with computer simulations, we have been able to determine the amplitude of the waves for a wide range of cavity parameters. For cavities with a width comparable to the depth of water entering the cavity, we have determined a scaling relation for the amplitude in terms of the geometry of the system. In the case of wider cavities, the flow begins like a breaking dam flow; it then becomes similar to a classical bore before breaking up into waves. The computer simulations agree well with experiment and will allow us to simulate more complicated geometries. Components: 10,488 words, 17 figures, 1 dataset. Index Terms: 3210 Mathematical : Modeling; 4255 Oceanography: General: Numerical modeling. Received 23 July 2002; Revised 25 October 2002; Accepted 31 October 2002; Published 15 February Gray, J. P., and J. J. Monaghan, Caldera collapse and the generation of waves, Geochem. Geophys. Geosyst. 4(2), 1015, doi:, Introduction [2] Volcanic eruptions near a marine environment can generate tsunamis by pyroclastic flows, submarine explosions, landslides or caldera collapse [Latter, 1981]. In some cases all these processes might occur in the one eruption. In this paper we consider how waves are generated as a caldera collapses allowing entry of the sea into the resulting cavity. The example we have in mind is the bronze age eruption of Santorini (the official name is Thera but we use the alternative name Santorini because of its common use). [3] Santorini is situated in the southern Aegean sea between Greece and Crete (Figure 1). It currently consists of two major islands Thera and Therasia together with the smaller islands of Nea Kameni, Palaea Kameni and Aspronisi (Figure 2). The major islands form the rim of the volcano. The islands of Nea and Palaea Kameni are young, slowly growing islands in the center of the caldera Copyright 2003 by the American Geophysical Union 1 of 28

2 Figure 1. Crete. Location map of Santorini. Santorini lies in the southern Aegean Sea, approximately 100 km north of which has an area of approximately 70 km 2. The southern part of the caldera consists of three basins with a typical depth of 290 m while the northern part consists of a single basin with a maximum depth of 390 m [Perissoratis, 1995]. The caldera is connected to the sea by a south western channel nearly 4 km wide which shallows to a typical depth of 20 m along a line connecting Therasia with Cape Akrotiri in the south of Thera. In the north of the caldera there is a channel to the sea of width 1 km and depth 400 m, which is connected to the sea along a 4 km long, 100 m deep contour. [4] Prior to the bronze age eruption, an aerial view of Santorini would have appeared much as it does today. However, there were significant differences. The northern half of the bronze age caldera is thought to have been shallow and more like a lagoon. The evidence supporting this are the widespread fragments of stromatalite which are found on Thera and Therasia, and are distributed in such a way to suggest that they were blasted from the northern part of the caldera during the eruption [Friedrich et al., 1988; Eriksen et al., 1990]. The southern part of the caldera was a shallow bay 2of28

3 Figure 2. Present-day map of Santorini showing the islands of Thera, Therasia, Palea Kameni, Nea Kameni and Apronisisi. Also illustrated are the Kameni line and approximate bathymetry of the region. Map taken after Graves [1848] and Heiken and McCoy [1984]. 3of28

4 [Druitt and Francaviglia, 1992]. No precise depths are known and for the purpose of this paper we consider a shallow bay to be 20 m deep. The present-day northern channel was produced by the eruption [Heiken and McCoy, 1984; Druitt and Francaviglia, 1992]. [5] The pre-bronze age caldera is thought to have evolved into the present-day caldera by four stages [Heiken and McCoy, 1984; Sparks and Wilson, 1990; Druitt and Francaviglia, 1992]. In phase one a 36 km high plinean column erupted from a subaerial vent. Water entered the vent leading to the violent phreatomagmatic activity and pyroclastic surge discharge of phase two overlain by the massive poorly sorted depositions (up to 35 m thick) of phase three. Druitt and Sparks [1996] suggest that water access probably occurred as caldera subsidence began and fractures propagated out to the NE and SW. Deposits from phase four are fine-grained ignimbrite laid down by relatively high temperature pyroclastic flows. After, or during phase four, when the magma chamber supplying the eruption was empty, the major collapse of the caldera occurred. For a discussion of caldera collapse mechanisms see Smith and Bailey [1968], Druitt and Sparks [1984], or McBirney [1990]. In the present case the collapse was probably facilitated by the large scale fracturing produced by the violent phreatomagmatic explosions of the early phases, combined with natural faulting along the Kameni line. [6] The stage of the eruption central to the present paper is the formation of the present-day basins. The properties of the waves which are generated will depend on the formation sequence. If the northern basin formed first then water from the southern part of the caldera, and the northern channel, will pour into it. The waves will then be those generated by flow from one basin to another. The later rapid deepening of the southern basins to a depth of 290 m would then result in water pouring back from the northern basin into the newly formed cavity. If the southern basin formed first then water would enter it from the southern channel filling it to the present depth of 290 m. The later formation of the northern basin would then result in water pouring into the newly formed northern basin. [7] These processes will probably never be known in the detail we need to form a comprehensive model of the fluid dynamics associated with the formation of the basins. Nevertheless, it is clear that a common process is the surge of water from one filled basin to another empty basin. This paper is concerned with the waves generated by this process. We assume that, by the stage of magma chamber collapse, it is reasonable to neglect the thermal effect of the hot rocks on the large volume of water entering the basins. [8] We also assume that the formation of the basins was rapid. In the present context rapid means that the collapse occurs in a time less than the time it takes for water to flow across the caldera. p At a depth D = 100 m the flow speed ð ffiffiffiffiffiffi gdþ is typically 32 m/s, and the crossing time will be 6.5 min. This is a very short time but a collapse is possible on this timescale. [9] As far as we are aware there are no direct observations for the start and end times of a caldera collapse event. Despite extensive monitoring of the 1991 eruption of Mt Pinatubo, caldera subsidence can only be inferred to have occurred at some stage during a six hour period [Wolfe and Hoblitt, 1996]. However, it is known that seismic activity, for example in earthquakes, can cause large scale faulting in very short times, often less than a minute. This short timescale is comparable to the timescale for fractures in a stressed medium which propagate at speeds close to those of shear waves (approximately 2000 m/s) [Scholz, 1990, p. 169]. If the roof of a partially evacuated magma chamber is substantially weakened by fracture itp will collapse on a gravitational timescale (T grav ffiffiffiffiffiffiffiffiffi X =g, where X is the distance the roof falls). If X = 1 km this timescale is T grav 10 s. On this basis we assume that it is possible for subsidence to reach (say) the 290 m depth of the southern basin before water from the northern basin has traveled far across the caldera. [10] Whether or not collapse did occur quickly in the case of Santorini is not known. If the basins formed on a scale of many minutes or hours then the formation of large waves would have been impossible, just as filling a bucket very slowly 4of28

5 Figure 3. Schematic diagram of piston caldera collapse. (a) The existence of a magma chamber with an overlying block of rock surrounded by a ring fracture. (b) As the magma erupts it can lead to a partially evacuated chamber. (c) The decrease in pressure due to the evacuated chamber leads to the block subsiding down into the chamber. only produces negligible waves. Note that our results would not be significantly altered if collapse occurred in a stepwise manner, say 20 m very slowly, followed by a large rapid collapse of several hundred meters. [11] In order to study the formation of waves from flow into a newly created cavity we use a combination of laboratory experiments and computer simulation (note that video examples of the experiments and simulations described in this paper are available as an electronic supplement). Although the cavity formed by the collapse of a caldera is expected to have a complicated topography, we concentrate here on cavities which have very simple shapes since these can be easily set up in a laboratory wave tank and they allow us to focus on features of the flow which are unaffected by complicated topography. Accordingly we consider a piston subsidence [Lipman, 1997] in which the chamber roof collapses as a single block, as seen in Figure 3. In our experiments the walls are vertical and the base horizontal. Even for cavities with such a simple geometry the problem does not appear to have been studied previously. [12] Our experiments provide not only useful information about the wave making processes, but they also enable us to establish that our computer simulations give a good description of the wave making processes. Because the problem involves time dependent free surfaces changing their topology as jets of water collide, the approximation of shallow water waves [see, e.g., Nomanbhoy and Satake, 1995] cannot be used. Instead we use the particle method smoothed particle hydrodynamics (SPH, for extensive references, see Monaghan [1992]) which can simulate arbitrary flow. This technique has the further advantage that it can be easily extended to handle complicated topography. The extensive series of tests described in this paper show that SPH accurately simulates a wide range of complicated fluid dynamical phenomena. 2. Experiments [13] In this section we describe laboratory experiments where the motion is essentially two dimensional. The two dimensional model (illustrated in Figure 4) replicates the boundary effects that would occur in the rapid collapse of a caldera connected to the sea. It consists of a tank of width 2L, with water in two outer compartments separated by a central cavity of width 2W. The initially empty central cavity models the cavity formed by the sudden collapse of the roof of the magma chamber. The walls defining the inner cavity are of height D. We wish to study the effects of barriers between the basins so the bottom sections of the inner walls are fixed to a height D F, the remainder of the two inner walls is removable. The removable section has height D R = D D F, and is denoted by the thinner vertical lines in Figure 4. The collapse is modeled by the removal of these upper sections of the wall, allowing water to flow into the cavity and potentially produce waves. [14] We make use of the symmetry of the model in our second experiment by placing a wall in the middle of the tank and only using one of the water filled compartments. In this second experiment the cavity is of width W, and the tank is of length L, making the dimensions consistent between the two experiments. We distinguish between the two cases by referring to the width of the cavity as being 2W in the case depicted in Figure 4, and W 5of28

6 Figure 4. Initial setup for our simple two dimensional caldera model, simulating the sudden collapse of an island into the sea. Depths and lengths are described in the text. in the case where only one water filled compartment is used Small Width Cavity [15] In the left side of Figure 5 we show an experiment with parameters D = 30 cm, D F =12 cm, 2L = 200 cm and 2W = 40 cm. The SPH numerical simulation shown on the right side of Figure 5 is conducted with the same parameters as the experiment and will be described in more detail in a later section. [16] Although the model is quite simple, the resulting flow turns out to be fairly complicated. Once the wall is removed water flows into the central cavity where it collides with the flow from the other compartment and splashes up to form a large central column of water. The central column grows, filling the cavity before attaining a height greater than that of the surrounding water. The column then breaks up into two waves traveling left and right, and a column over the cavity which subsides and disperses over time. The two traveling waves reach and run-up the end walls before they reform into two waves, and return to the cavity. [17] The flow in the two water filled sections is laminar, with the initial flow over the barriers resembling the flow across a weir (weir flow is discussed by Dias and Tuck [1991]). The fluid motions become highly disordered when the jets collide, mix and rebound off the cavity walls. We note that the flow is not quite symmetric, but that as the variations are only minor it is reasonable to treat the experiment as being symmetrical. [18] The breadth (measured perpendicular to the front wall seen in Figure 5) of the experimental wave tank was 40 cm. The experimental setup contained guides to support the removable walls that each protruded 2 cm into the tank (the width of these guides measured parallel to the front wall was 2.6 cm) and perturb the flow slightly. The effects of the guides are small because the initial depth is 30 cm and the front and back walls of the tank are 40 cm apart, meaning that this experiment is largely two dimensional Very Wide Cavity [19] The previous experiment is for a cavity with W D. We now consider the case where W D, noting that this latter scenario is more likely to be the case at Santorini. We mentioned earlier that the setup is altered from the previous case, we make use of symmetry in the experiment by placing a wall in the middle of the cavity and removing one of the water filled compartments. This experiment consists of a tank of length L = 160 cm, with a cavity of width W = 80 cm, the remainder of the tank is filled with water to a depth D = 10 cm. We take the case of no fixed wall D F = 0, so that water flows along the ground into the cavity. The flow is illustrated in Figure 6, where we present the left 100 cm of the tank. [20] After the wall is removed, water proceeds to flow across the tank as a breaking dam [see Martin and Moyce, 1952; Nichols and Hirt, 1971], the front overturns upon reaching the end of the tank, before forming into a plunging wave. The maximum height reached by the overturning wave is approximately equal to the initial water depth D. [21] Water continues to surge across the tank toward the left hand wall, while a turbulent bore traveling toward the right hand side of the tank forms. Bores occur when there is an abrupt increase in fluid depth associated with an accompanying change in flow rate [Rayleigh, 1908, 1914; Simpson, 1997]. 6of28

7 Figure 5. Waves produced by our simple caldera collapse model, with D F = 12 cm, D = 30 cm, 2W = 40 cm and 2L = 200 cm. The frames on the left are taken from the wave tank experiment, those on the right correspond to an SPH computation. Particles in the simulation are shaded according to their speed, with lighter shades denoting higher speeds. [22] Taking H 1 and H 2 to be the depths of fluid before and after the bore front, such that H 1 > H 2, the velocity of propagation of a bore is given by Lamb [1932], sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gh 2 ðh 1 þ H 2 Þ V t ¼ : ð1þ 2H 1 [23] From the experiment at t = 1.32 s, the bore of height H 1 = 7.5 cm is moving into a fluid H 2 =3.0 cm high, giving a theoretical bore speed V t = 0.45 m/s. The experimental bore speed is V e = 0.40 m/s. The depth of the fluid into which the bore is traveling increases toward the right side of the tank. We note in the final frame of Figure 6 that after the turbulent bore has reflected off the right hand wall waves are initiated on the surface of the bore. [24] The guides which hold the removable wall have a greater effect in this case than in the 7of28

8 previous experiment. A discontinuity in the water level occurs where these guides are positioned, with the water level being shallower in the cavity than the initially water filled compartment. In Figure 6 the discontinuity can be clearly seen in the second, third and fourth frames of the experiment (the guides are positioned just to the left of numbers which give the time). The discontinuity arises because the initial depth of 10 cm is now only twice the breadth of the guiderails (4 cm). Note also that the top of the time indicator obscures part of the tank behind it and that the tank has a flat bottom across both compartments. 3. Numerical Technique [25] The experiments in the previous section involved jets of water which impact with themselves and/or with walls to form overturning waves within the cavity. For problems of this type the full Navier Stokes equations can be solved in a straightforward way using the Lagrangian particle method smoothed particle hydrodynamics (SPH). For a review with extensive early references see Monaghan [1992]. In this method the continuum equations are replaced by equations for a moving set of interpolation points which we refer to as particles. The forces between the particles are due to pressure terms, they take the form of central forces which vary with distance according to an interpolation kernel. The calculations in this paper employ a ratio of smoothing length h, to initial particle spacing Dp of h/dp = 1.2. The pressure is determined from a suitable stiff equation of state which is chosen to ensure the relative density variation is around 1%. With this method, overturning waves can be followed easily [see, e.g., Monaghan and Kos, 1999]. [26] Boundaries can be treated either by using boundary forces similar to those used by Peskin [1977] for membranes, or by using ghost particles [Randles and Libersky, 1996; Morris et al., 1997]. Boundary forces are less accurate but allow for the treatment of complicated topography easily. The calculations described in this paper use the ghost particles of Morris et al. [1997] since they give greater accuracy for the simple geometry of the experiments. These ghost particles are stationary fluid particles which are given appropriate properties to mimic viscous boundary conditions. We find that this works satisfactorily, except that in some cases (see for example Figure 5) after fluid moves away from a wall some particles remain attached to the wall. However, this only involves a very small mass of the fluid and the effect on the rest of the flow should be negligible. [27] SPH has been used for a wide variety of flows with nearly incompressible fluids. Examples include simple elliptic flows without boundaries, bores and wavemakers [Monaghan, 1994], low Reynolds number flows [Morris et al., 1997], gravity currents [Monaghan et al., 1999], moving solid bodies impacting water [Monaghan and Kos, 2000]. SPH has also been used for metal-metal impact [Randles and Libersky, 1996] and fracture [Benz and Asphaug, 1995; Gray et al., 2001]. 4. Water Wave Code Validation [28] Before applying SPH to the simulation of our experiments, we first discuss three tests of our current code relevant to the problem of waves in a caldera. Other authors have carried out similar tests using SPH [Monaghan, 1994; Morris et al., 1997; Monaghan and Kos, 2000]. The first test involves the propagation and run-up of solitary waves. The problem of a breaking dam has been simulated to show that we can correctly model the collapse of a column of water and its motion as it surges across a boundary. Thirdly, we consider weir flow in the case of zero gravity Solitary Wave Propagation [29] Simulations were performed to determine the extent of run-up of a solitary wave against a vertical wall, with the results compared to similar computations using a Marker and Cell (MAC) method and laboratory experiments [Chan and Street, 1970]. The agreement with experiment is excellent. [30] Small amplitude solitary waves are known to satisfy the Kortweg de Vries (KdV) equation [Lighthill, 1980]. The profile h of a solitary wave 8of28

9 Figure 6. Flow of water into a cavity where W D, actual parameters are D F = 0 cm, D = 10 cm, L = 160 cm, W = 80 cm. Frames on the left show a wave tank experiment, those on the right a corresponding SPH simulation. Particles in the simulation are shaded according to their speed, with lighter shades denoting higher speeds. The large numbers in the left hand frames (experiments) show the time. Note that only the left 100 cm of the tank are shown to better highlight the variation with depth. propagating in the x direction at time t has the form, hðx; tþ ¼ H qffiffiffiffiffiffi ; ð2þ cosh 2 3H ðx UtÞ where H is the wave amplitude, D is the depth of water p on which the wave is propagating and U ¼ ffiffiffiffiffiffi gd is the wave speed. Small amplitude waves imply that H D. [31] In the simulations the particles were initially set up on a Cartesian lattice with those fluid particles above the free surface of the wave omitted. Particles were initially assigned with zero vertical velocity p ffiffiffiffiffiffiffiffiffi and a horizontal velocity V x ¼ hðx; 0Þ g=d. This gives an initial particle setup which is not in equilibrium because the particles are initialized with constant density and pressure. However, they quickly adjust to the solitary wave configuration. [32] Figure 7 shows a sample simulation, from the initial state we see the wave moving toward the 4D 3 right and running up the wall. Particles are shaded according to their speed, with lighter shades denoting higher speeds. The lower velocities (darker shades) found in the third frame of Figure 7 are due to the loss of kinetic energy of the wave as it climbs the wall, gaining in gravitational potential energy. [33] Figure 8 compares a series of SPH results for the dimensionless extent of run-up against a wall (R/D) with the experiments and MAC simulations of Chan and Street [1970]. The run-up R is the maximum height (measured from the bottom of the tank) reached by the water on the wall and the depth D is the initial fluid depth also measured from the bottom of the tank. It is clear that both numerical methods are in excellent agreement for low amplitude waves H/D 0.3, with good agreement for higher amplitudes. [34] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The horizontal length scale of these waves is 4D 3 =3H (see equation (2)). This quantity decreases as the amplitude is increased. The smallest 9of28

10 Figure 7. Solitary wave running up a wall. The tank is 160 cm long with the same scale being employed in the vertical direction. The wave propagates on water of depth 20 cm, with the wave amplitude being half the undisturbed depth. Particles are colored according to their speed, with the key at the bottom referring to all three frames. The times of each of the above frames are 0.0 s, 0.28 s and 0.55 s from top to bottom respectively. amplitude wave we modeled had H = 2 cm, the depth of water on which our waves propagated was D = 20 cm, and the length of the tank was 160 cm. The horizontal length scale in this case is 73 cm, or approximately half the length of the tank so boundary effects are negligible. [35] The initial setup used by Chan and Street [1970] is based on a power series expansion given by Laitone [1960], this is different to the initial setup used for the SPH computations. Because the SPH simulations are in excellent agreement with experiment even though the initial state (2) is only an approximation, it is clear that the wave in the simulation quickly adjusts to a form consistent with the equations of motion Breaking Dam [36] The collapse under gravity of a uniform depth water column is known as the breaking dam problem. The classic dam break problem involves supporting a column of water of depth D behind a barrier, before removing the barrier. An analytical solution based on shallow water wave theory is given by Acheson [1990] for an infinitely long dam. From this solution the height H DB of fluid at the point where the barrier was positioned is given by, H DB ¼ ð4=9þd ð3þ and the horizontal p ffiffiffiffiffiffi speed of the flow at the same position is ð2=3þ gd. [37] The breaking dam problem has been investigated experimentally by Martin and Moyce [1952]. The experiments involved tracking position of the surge front of a collapsing water column. The experiments were three dimensional, though the behavior is largely only two dimensional. [38] Our SPH results are presented pffiffiffiffiffiffiffiffi in the dimensionless units; z = Z/A and T ¼ t g=a where, Z is the distance of the surge front from the left hand 10 of 28

11 wall, A is the length of the base of the column, and g is the acceleration due to gravity. [39] In Figure 9 we show an SPH simulation of the collapse of a two dimensional water column. In this case the fluid is of equal height and width D = A = 5.72 cm, with the tank being cm long. In this simulation we have a resolution of 120 particles across the height of the fluid. The shades again represent a particles speed, with lighter shades corresponding to higher speeds. [40] Good agreement is found between the SPH and Martin and Moyce [1952] experimental results (Figure 10). In the SPH results the surge front travels faster than that found experimentally. Nichols and Hirt [1971] found a similar error in their MAC computations and thought that it may be due to uncertainty of the time at which motion began in the experiments, although Martin and Moyce [1952] normalized the experimental data to have the same dimensionless time T = 0.8, when the surge front reaches the point Z/A = This should not be a major source of error, provided the numerical data is also normalized in this way (the results in Figure 10 are presented in this way). Excellent agreement is obtained up until Z/A = 3.0 after which a slight divergence occurs. The agreement is within 6% of the experimental results. Agreement is also found with the analytic solution (3) for the depth of fluid at the point where the wall was positioned. [41] In Figure 11 we track the (unnormalized) motion of the bore front for a series of SPH simulations with different particle resolutions. The SPH solutions are seen to converge as the particle resolution is increased Zero Gravity Weir Flow [42] Another important class of flows that have been extensively studied are those over weirs. Their motion is similar to the initial flow in our experiments when D F > 0. Weir flows are steady nonuniform flows, where velocity is constant over time, but varies over the spatial extent of the channel due to obstructions (weirs) in the channel. They involve free surfaces which present difficulties for analytic solution, except for simple geometries and steady flow [Dias et al., 1988]. Dias and Tuck [1991] present results for a variety of weir flow configurations. Their solutions are found numerically using a complex variable technique [Vanden-Broeck and Keller, 1987]. [43] If gravity is neglected Dias and Tuck [1991] show that it is possible to find a solution for the angle q of the jet, which forms as fluid passes over a barrier (weir) of different heights B. The problem involves injecting fluid at a constant velocity and height, then determining the angle made with the horizontal by the jet which passes over a wall at the opposite end of the domain. The angle q, is plotted against the dimensionless barrier height B/D (where D is the downstream fluid height) in Figure 12. Obviously the B =0 solution should be a horizontally propagating jet (q = 0). With the limiting solution for high walls being a perpendicular propagating jet (q = p/2) where the fluid is unable to pass over the boundary. [44] A tank 420 cm long and 30 cm high with fluid being input with velocity U in = 0.5 m/s was simulated and the results compared to the Dias- Tuck solution in Figure 12. The dimensionless weir height B/D was varied between 0.25 and 3.0. Calculations proceeded until a steady state was reached, at which time the angle of the jet over the weir was measured. The scatter in the SPH results is due to errors in the measurement of the angle q. 5. Simulations of the Caldera Experiments [45] In the previous section we have seen that the SPH method is able to accurately model features seen in the experiments of section 2. In this section we present comparisons between these experiments and SPH simulations, as well as describing a numerical investigation into the effects of different cavity widths and heights. [46] In the simulations the removable sections of walls are replaced by particles, which are instantaneously removed when the fluid reaches static 11 of 28

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13 Figure 9. Collapse of a column of water, note the front surging out from the bottom of the tank, where hydrostatic pressure is greatest. The water column quickly evolves from a square profile as the front approaches the right end of the tank. Shades represent the speed of the particles, with lighter shades denoting higher speeds. Figure 8. (opposite) Comparison of SPH results for the run-up against a wall to the MAC numerical and experimental results of Chan and Street [1970]. H is the amplitude of the wave, D is the depth of water on which the wave propagates and R is the maximum extent of run-up on the wall. The run-up is seen to increase with increasing wave amplitudes. 13 of 28

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15 equilibrium. In the experiments these walls are pulled up delaying the experimental flow compared to the simulation. The results (see Figure 5) show that this delay is 0.1 s Small Width Cavity [47] The SPH numerical results assume that the flow is two dimensional, this is consistent with the experimental behavior which was earlier seen to be largely two dimensional. [48] The right hand side of Figure 5 illustrates an SPH simulation with 2L = 200 cm, 2W = 40 cm, D F = 12 cm and D = 30 cm (the same as the experiment shown on the left side of Figure 5). The numerical system was damped, to allow a stable hydrostatic particle setup to be attained. The walls separating the cavities were then instantaneously removed. The SPH computation is qualitatively the same as the experimental behavior except that the flow in the experiment is delayed compared to the SPH simulation because the walls are instantaneously removed in the simulation. [49] A series of simulations and experiments were performed in a tank of parameters 2L = 200 cm, 2W = 40 cm and total depth D = 30 cm. Four different sizes of fixed wall were employed, these being D F = 0 cm, 5 cm, 12 cm and 17 cm. In Figure 13 we plot the extent of wave run-up R/D at each end of the tank against the dimensionless height of the cavity D F /D for these simulations and experiments. R and D are measured from the bottom of the tank so that in the undisturbed state R/D =1. [50] The extent of run-up (wave amplitude) is seen to increase as the height of the fixed wall D F is decreased. To ensure that the experimental results were correct, and not merely a result of experimental error, the experiments were conducted three times at each height D F. The run-up on both the left and right hand walls was measured giving six experimental points for each wall height. The spread in the experimental results was 40% of the amount of run-up above the base water level in each case. Minor variations in the run-up at each end of the tank were found in the simulations. The SPH results are at the lower limit of those experiments which give the largest run-up (D F /D < 0.2) because in these experiments the run-up involves a thin wedge of fluid which cannot be resolved adequately without using a prohibitively large number of particles Different Cavity Widths [51] We now explore how varying the cavity width W, affects the size of the waves that are produced. We again use the extent of run-up as a measure of the size of a wave. In these calculations we make use of the symmetry in our model by placing a wall in the middle of the cavity and only following fluid motions in one of the compartments, this allows us to increase the length of the tank without increasing the number of particles and computational time of the simulation. The advantage of an increased tank length is that waves are less affected by boundaries. [52] We conducted a series of numerical simulations in a 200 cm long tank, which included a cavity of width W at one end of the tank. The simulations were carried out for different widths, 2.5 cm W 75 cm, for a range of cavity depths, 5cm D F 20 cm. The depth of the water was D = 30 cm, with the removable wall of height D R, being instantaneously removed. [53] When W = 0 no wave will be produced. If W is increased but W D the waves produced will be small because the perturbation is small. At the other extreme, if we have a wide cavity W > D we again expect only small amplitude waves because the energy released is dissipated by the disordered flow. [54] For cavity widths in between we get significant waves. Beginning with a narrow cavity we find that the initial jet of water impacts with the Figure 10. (opposite) Comparison between SPH and the experimental results of Martin and Moyce [1952] for a collapsing column of water with A = 5.72 cm and resolution of 120 particles across the width of the fluid. The dots indicate experimental results. The solid line denotes the SPH computation. 15 of 28

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17 sidewall before reaching the cavity floor. Increasing the width, we find the extent of run-up to increase until the initial point of impact of the jet occurs at the bottom of the sidewall. When the width is increased further the jet strikes the floor first, then breaks into two flows which move in opposite directions, rebound off the sides of the cavity and run back into each other producing disordered motion and reducing the run-up. [55] As the tank length L is fixed as we vary the width of the cavity we find that the average water level in the tank varies due to the flow of water into the cavity. If the cavity width is much smaller than the length of the tank W L, the water level outside the cavity is not significantly affected by the flow of water into the cavity, as only a small percentage of the total amount of water is required to fill the cavity. As the cavity width increases, the amount of water required to fill the cavity increases and the average water level outside the cavity decreases. If W =((D D F )/D)L the water throughout the tank at the end of the flow will have depth D F, and motion will be largely contained within the initially empty cavity. [56] In Figure 14 we demonstrate the wave of maximum amplitude for the D F = 12.5 cm series of experiments which occurs for a cavity of width W = 15 cm. The initial jet is seen to strike the floor of the cavity near the intersection with the left hand wall. A vortex forms quickly filling the cavity, with a significant wave being produced. The maximum run-up on the right hand wall is found to be R/D = 1.17, the run-up of this wave was 5.1 cm above the initial reference level D, and 7.4 cm above the average (reduced by flow into the cavity) water level. [57] In Figure 15 we see the W = 40 cm case, in which the jet breaks up into a disordered flow inside the cavity and minimal amplitude waves are produced, the maximum value of run-up is found to be R/D = 1.0, however this does not mean that no wave was produced because, as mentioned above the average water level in the tank is reduced as water flows into the large cavity. In this case the cavity is so wide that the vortex formed as the cavity fills is on a length scale much less than the width. A column of water builds up over the cavity and forms a wave before the cavity is filled, however, this wave is unable to propagate out of the cavity. Eventually a second column builds up over the cavity (although not as high as in the previous W =15 cm case) and develops into a low amplitude long wavelength wave. [58] The factors described above are illustrated in the results of the series of numerical simulations shown in Figure 16, each line represents a different cavity height D F. We find that initially the extent of run-up increases with increasing cavity width up to the point where energy is trapped in the cavity and the size of the run-up decreases. The hypothesis breaks down at the lowest cavity height D F = 5 cm, where violent disorder dissipates a large proportion of the energy. For a cavity of zero width there is no displacement of the water and we expect R/D = 1, indicating zero run-up. We note that R/D may be less than 1 due to a decrease in the initial depth of the water as fluid flows into the cavity. [59] As with the simulations in section 5.1 we find that the run-up of the waves increases as D F decreases and more water is allowed to flow into the cavity at a higher initial velocity (due to the larger hydrostatic pressure at the top of the fixed wall with smaller values of D F ). The dissipation of energy when water flows into the cavity is such that a simple energy argument cannot be constructed which relates the loss of gravitational potential energy to the energy of the wave and run-up height. [60] We find that the R/D values in the parameter space of D F and W fall on curves so that it is theoretically possible to define a scaling relation Figure 11. (opposite) Convergence of SPH results with increasing particle resolution for a collapsing water column with A = 5.72 cm. The number of particles refers to the number of particles across the width of the initial fluid column. The simulations are seen to converge to a solution as the resolution is increased. 17 of 28

18 18 of 28

19 for the variation in run-up height R, in terms of the water depth D, cavity width W, cavity height D F and the length of the tank L. We do not present a scaling relation that picks out all the features in Figure 16, but do present one that describes some of the features. [61] From Figure 16 we find that, in general, as the height D F decreases, there is an increase in the cavity width W Max, at which the maximum run-up R Max occurs. We find that W Max is approximately given by, Max W ¼ D : D 6D F ð4þ [62] In general the extent of run-up can be given by a function, R D ¼ F D L ; D F L ; W : ð5þ L [63] However, the simulations we have conducted in this section are conducted in a tank of fixed length, L. We also note that the curves for each cavity height can be approximated by a parabola with a maximum given by (4). We find the results in Figure 16 can be fitted by an approximate relation between R, W and D F, R D ¼ 1 þ W D W : ð6þ D 3D F D [64] Substituting (4) into (6) we find the maximum run-up is given by, Max R ¼ 1 þ D 2 : ð7þ D 6D F [65] The scaling relation (6) predicts minimal runup R/D = 1, for a cavity of width W = 0, which is the case of no cavity. As the cavity width is increased we also expect minimal run-up when W/D = D/(3D F ), due to the cavity being too wide for significant wave production, this is an underestimate of the value in the simulations, but it is still a reasonable estimate. [66] A defect of (6) is that R/D!1as D F! 0, indicating a change in behavior of the system. In fact the simulations show that as D F! 0 complicated breaking waves form and the run-up is reduced from that expected from the scaling relation. The scaling relation does give reasonable agreement for larger cavity heights Very Wide Cavity [67] We now compare an SPH simulation to the very wide cavity experiment (section 2.2). Qualitatively we find the simulation and experiment in Figure 6 to be the same. When the wall is removed water surges across the tank floor like a breaking dam, striking the left hand wall and forming into an overturning wave. A turbulent bore then forms and moves toward the right hand side of the tank. [68] We noted earlier that the experiment had initial bore heights, H 1 = 7.5 cm and H 2 =3.0 cm, giving a theoretical bore speed of V t = 0.45 m/s with a measured experimental speed of V e = 0.40 m/s. The simulation has the same heights, with a measured speed V s = 0.42 m/s. After rebounding off the right hand wall, the turbulent wave motions are replaced by traveling waves as in the experiments. [69] In the first frame of Figure 6, we see the initial setups of both the experiment (left) and simulation (right). In this case we have dimensions L = 160 cm, W = 80 cm, D F = 0 cm and D = 10 cm, though only the left 100 cm of the tank are shown. [70] In the second frame we see the formation of an overturning wave after the surge front of the breaking dam has reached the left hand wall. Good agreement is found for the height of this overturning wave in both the experiment and simulation. The fluid depth to the right of the overturning wave increases from left to right in both the Figure 12. (opposite) Comparison of the angle q of the jet moving over a weir of dimensionless height B/D in the case of zero gravity. q is measured as the angle between the jet and the horizontal. Continuous line is the Dias-Tuck result, dots represent SPH results. 19 of 28

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21 Figure 14. Example of efficient filling of the cavity leading to significant wave production. W = 15 cm, D F = 12.5 cm, D = 30 cm and L = 200 cm. Shades represent the speed of the particles, with lighter shades denoting higher speeds. simulation and experiment. In the experiment however, the fluid depth begins to decrease as we approach the exit point of the dam, due to the effects of the guiderails. [71] In the third frame, we see that the bore has begun to develop, turbulent motions appear on the top of the bore in both the experimental and simulated cases. We again see that a discontinuity exists at the guiderails. An estimate of the depth of water into which the bore is propagating is given by the exit height of the breaking dam problem, H DB = (4/9)D. We find the actual exit height of 4.7 cm to agree very well with the predicted height of 4.4 cm. [72] By the fourth frame the bore has traveled a considerable distance across the tank, experimentally the height of the bore begins to diminish away from the left hand wall, whereas in the simulation the bore is level behind the front. This difference is likely to be due to the effects of the guiderails. In the simulation, water from the water filled compartment travels toward the left and allows the height of the bore to be maintained. In the experiment, the volume and velocity of this flow is Figure 13. (opposite) The increase in run-up with decreasing height of the fixed wall is seen in this comparison between SPH and experimental results for the run-up against a wall with the simple caldera model. R is the extent of run-up, D is the initial depth of water in the tank and D F is the height of the wall that remains in the tank. The solid lines join the points found from our computations. In these experiments and simulations we take 2L = 200 cm, 2W = 40 cm, D = 30 cm, and the cases D F = 0 cm, 5 cm, 12 cm and 17 cm. The error bar on the point D F /D = 0.0 and R/D = 1.22 is typical of the measurement errors in the experiments. 21 of 28

22 Figure 15. Example of inefficient filling of the cavity due to the breakup of the initial jet and leading to minimal wave production. W = 40 cm, D F = 12.5 cm, D = 30 cm and L = 200 cm. Shades represent the speed of the particles, with lighter shades denoting higher speeds. reduced, as water struggles to move past the guiderails, resulting in the bore height being diminished. [73] In the final frame we notice the return bore with a series of waves on its surface. The height of the first wave is lower in the experimental case, with the front also not being as sharp. 6. Application to Santorini [74] To discuss the effects of water flow at Santorini we need to make some assumptions as to how the collapse occurred. We assume that collapse occurred at the end of the eruption sequence, and was unaffected by violent explosions and phreatomagmatic events [Sparks and Wilson, 1990]. We simplify the caldera geometry (discussed in the introduction) to consist of two rectangular basins illustrated in Figure 17. We assume two basins 6 km across, a northern basin 6 km wide (from north to south) and 20 m deep, and that the three basins in the south combine to make up a single basin 6 km across and 20 m deep. These basins have widths which are very much greater than their depth. Therefore, the relevant experiments and simulations are those presented in sections 2.2 and 5.3 respectively. [75] We noted in the introduction that (at least some part of) the collapse needs to occur rapidly for significant wave production. If collapse occurred gradually, over a period of hours, the cavity would be slowly filled as the caldera subsides and significant waves would not be produced. However, if the collapse occurred rapidly, the effects of the flow would be much greater. [76] The sequence of collapse also effects the types of waves that will result. If the northern and southern basins collapsed simultaneously, the resulting waves would be very different to a case 22 of 28

23 23 of 28

24 where the northern basin collapsed and filled with water (through the northern inlet) before the southern basin collapsed and allowed water to flow into it. Alternatively, the caldera may have collapsed partially and filled with water, before collapsing fully at a later stage, in this instance the wave generation process would be similar to that due to a submarine earthquake. Although the actual sequence of collapse is unknown we discuss some possible scenarios below. [77] If the northern basin collapsed before the southern basin, flow through the northern inlet could fill the basin to a depth up to 390 m. Assuming the northern inlet fills across a 4 km length of the 100 m deep contour, the northern basin would be filled in 18 min. Subsequent collapse of the southern basin would lead to the flow of water from the northern basin and shallow south western inlet into the enlarged southern basin. Initially the flow from the north would be greater than the flow through the shallow inlet and would dominate the wave generation process. [78] We saw earlier that this flow would resemble a breaking dam with an initial depth of 290 m (the northern basin was 390 m deep, but the critical depth is that of the shallower southern basin). The surge front of this dam could be expected to have a typical depth of p 133 ffiffiffiffiffiffi m (see equation (3)), with a typical velocity ð gdþ of 54 m/s. At this speed the front would travel the 6 km to the southern cliffs in just under 2 min. In our experiments (section 2.2) this front overturned upon reaching the southern cliffs, with the overturning wave reaching a height similar to sea level, since this was the initial level of the water in the northern basin. Although, in this case the basin is being filled from both inlets, at this early stage the southern cliffs are only affected by flow from the shallow south western inlet, with the average depth only increasing by 5 m. [79] Druitt and Sparks [1996] note that the ignimbrite deposit at the Akrotiri excavations in the south are strongly eroded and covered by alluvial deposits from flash flooding. Our models show that only minimal run-up above sea level will occur at this point. It is therefore unlikely that the flow in this scenario would be able to traverse the 80 m high southern cliffs to cause this flooding. [80] Our experiment and simulation (Figure 6) show that a reflected turbulent bore would develop traveling back toward the north. The height of the bore would be 3/4 of the critical water depth, that is 218 m above the southern basin floor and 300 m below the northern cliffs. A typical speed for this flow is given by equation (1) and is found to be 33 m/s. At this speed it would take 6 min to travel the 12 km to the northern cliffs. The inflow of water through the northern inlet is quite large and is expected to fill the cavity in around 12 min, this inflow will increase the height of fluid into which the bore is traveling (compared to the model in Figure 6), but the effect is to alter the form of the bore front, the bore height is unlikely to be greater than the initial sea level. The bore would be unable to pass over the 250 m high northern cliffs, although the influx of water means that a small amplitude wave may be able to propagate outside of the caldera through the narrow northern inlet at this stage. The remainder of the bore would rebound off the cliffs, and travel back toward the south. In the simulation (section 5.3) noticeable traveling wave motions formed on the bore, which would continue to propagate toward the southern cliffs, reducing in amplitude due to viscous dissipation. By the time these waves have reached the southern end of the caldera, the influx of water from the sea (through the northern and south western inlets) would have filled the caldera and some of these waves would be able to escape the caldera through the inlets and propagate across the sea. [81] If the two basins were to collapse to their final depths simultaneously, the initial amount of water in the caldera would be insufficient to fill the cavity and produce significant effect. The flow would be controlled by the water passing through the northern and Figure 16. (opposite) The extent of run-up R/D, for different cavity widths W/D, each line represents a different cavity height D F. R is the maximum value of run-up (measured from the bottom of the tank) and D =30cmisthe initial water depth. These results are obtained from numerical simulations and are consistent with the experimental results of section of 28