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Stresses within an active volcano — with particular reference to kilauea
Tectonophysics, 22 (1974) 355-362 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
STRESSES WITHIN AN ACTIVE VOLCANO - WITH PARTICULAR REFERENCE TO KILAUEA P.M. DAVIS, L.M. HASTIE and F.D. STACEY Physics Department,
University of Queensland, Erisbane (Australia)
(Accepted for publication
December 12, 1973)
ABSTRACT Davis, P.M., Hastie, L.M. and Stacey, F.D., 1974. Stresses within an active volcano - wit.? particular reference to Kilauea. Tectonophysics, 22: 355-362. The absence of an observable piezomagnetic effect correlated with rapid deformations of Kilauea Volcano, Hawaii, necessitates critical reappraisal of the conventional model of a dilating magma chamber in an elastic half-space. The introduction of a plastic or readily deformable pseudo-chamber around the magma chamber itself has the required effect of reducing the volcano-magnetic anomaly in model calculations. It also gives an increasing value of the ratio of the maximum horizontal displacement of the surface to the maximum vertical displacement during progressive inflation, and this accords better with available observations than do simple elastic models. In the case of Kilauea it appears that the ovoid pseudo-chamber is substantially larger than the true magma chamber. The volcano is visualised as an essentially cold structure, highly fragmented and penetrated by rainwater, with only localised magma channels.
INTRODUCTION
Measurements of surface displacements and tilts on Kilauea Volcano, Hawaii (e.g. Fiske and Kinoshita, 1969) are more extensive and detailed than for any other volcano and it is therefore natural to use these measurements for comparison with mathematical models of the eruption mechanism(s). The simplest approach is to assume that the volcano is essentially a spherical magma chamber, containing fluid under hydrostatic pressure, enclosed within a perfectly elastic half-space. The surface deformation accompanying changes in the pressure of the magma chamber (due to flow of magma from a much greater depth) is then calculable and the chamber depth is deduced by comparison of the scales of the observed and calculated displacement fields. Calculations of this type by Mogi (1958) are widely used to model Kilauea in particular and volcanoes in general. Using the available data on vertical displacements of the summit region, Mogi deduced that Kilauea deformation was due to an essentially spherical magma chamber at a depth of 2-3 km below the summit. More complicated models obtained as syntheses of combinations of solutions of the Mogi type (e.g. Walsh and Decker,
1971) produce somewhat different patterns of surface displacement, while retaining the concept of a chamber in an elastic half-space. Yokoyama (1971) presented a quite different model based upon a dipolar stress nucleus, but for which he offered no physical justification. The concept of a deformed elastic half-space allows a straightforward mathematical representation of the stresses within the medium, and these can to be used to calculate the piezomagnetie effect (the effect of stress on magnetic minerals) and the consequent time-dependent magnetic anomaly at the surface. Calculations of this type were reported by Stacey et al. (1965) for Caribbean volcanoes, Although based upon a quite different stress model, which we do not now accept, the application of piezomagnetic equations applies equally well to any stress pattern and the appearance of a volcanomagnetic anomaly is an inevitable consequence of any elastic model. Observations of striking magnetic anomalies preceding and accompanying eruptions of New Zealand volcanoes (Johnson and Stacey, 1969a, b) were taken to confirm the validity of the argument that the deformation of a volcano is elastic. It was of some interest therefore to make magnetic measurements on Kilauea, the volcano to which most of the elastic-model calculations were specifically applied. But, in spite of what appear to be ideal conditions for these observations, in particular the strongly magnetic rocks of Kilauea, Davis et al. (1973) found no systematic magnetic anomalies to accompany a short-term inflation-eruption sequence of Kilauea. There is now evidence of a longer-term magnetic trend, which is still being evaluated, but the reported observations suffice to cast serious doubt on elastic models of the eruption mechanism. These observations have provoked the present paper. CHAMBER SHAPE - THE EFFECT OF INTROlXJCING
A ‘PLASTIC’
ZONE
For a volcano, such as Kilauea, in which the magma is very fluid, the ~sumption of a spherical magma chamber can hardly be more than an extreme simplification of the real (but unknown) chamber shape. It is more realistic to imagine a complex array of tubes and magma lenses in a much larger volume of rock, which is multiply fractured, so that its response to chamber dilation is essentially plastic. The pattern of defo~ation in the surrounding elastic region is then- determined by the size and shape of the ‘plastic’ region rather than by the magma chamber itself. We term this plastic region the pseudo-chamber; increases in stress within it are essentially hydrostatic. To theoreticians this is an important advantage, because the shape of the pseudo-chamber is necessarily fairly simple and is susceptible to plausibility arguments or, with certain assumptions, to quantitative estimates, whereas the true shape of a magma chamber is subject only to gumsea with varying degrees of either im~l&usib~ty or mathematics in~ctabi~ty. If the true magma chamber (and thermal centre) is small (in the sense that the mechanical properties of the pseudo-chamber are independent of depth over the range which the true chamber occupies) then regardless of the shape of the magma chamber itself, the surrounding pseudo-chamber is close to
357
being spherical. Mogi’s (1958) model is then the appropriate description of surface displacements, which would indicate a highly symmetric source of the deformation. However, more violent inflation or a larger magma chamber cause extension of the ‘plastic’ zone. Then the pressure-dependence of the properties of the medium must be taken into account. At a critical shear stress the material ceases to respond elastically, becoming ‘plastic’, but the plastic deformation is due substantially to the presence of multiple fractures
RADIAL
DISTANCE
(Km)
Fig. 1. Progressive enlargement of a pseudo-chamber of the type proposed to represent the behaviour of Kilauea Volcano. The criterion for the transition from elastic (outside) to plastic (inside) deformation is that of Coulomb-Navier, whereby the shear strength of the material is linearly related to overburden pressure, so that the pseudo-chamber enlarges more upward than downward during inflation of the much smaller magma chamber. In this model the magma chamber is centred at 1.5 km depth, the shear strength is assumed to be 20 bars at the surface, increasing with depth by 30 bar/km (corresponding to a coefficient of friction of 0.1). Successive chamber sizes represent changes due to 8 bar increments in hydrostatic pressure.
358
(in the presence of some fluids) and therefore occurs more readily at low overburden pressures. Thus the ‘plastic’ zone extends upwards, more readily than outwards or downwards, becoming ovoid with its long axis vertical and smaller end uppermost (Fig. 1). This has two effects which are significant in the present context. The maximum horizontal displacement of the surface becomes larger, relative to the maximum vertical displacements, than in the spherical chamber model and the deviatoric stresses are greatly reduced, so reducing co~espondingly the piezomagnetic anomaly. The conclusion regarding surface displacements appears to be in accord with recent observations (Fiske and Kinoshita, 1969) which therefore provide data from which we can estimate the size and shape of the ‘plastic’ pseudochamber, Such a calculation does, of course, fall short of a convincing demonstration that the model is valid. But there is a further detail of the model which provides an acid test of its vahdity. During progressive inflation, the plastic region grows at the expense of the elastic region, as the stress which the material can support elastically is exceeded in an increasing volume, Enlargement of the plastic pseudo-chamber thus involves increasing vertical elongation. The initial deformation may correspond nearly to a spherical centre of dilation, for which the ratio of maximum horizontal to maximum vertical displacements is 0.4, as in Magi’s (1958) model. As deformation proceeds, so the horizontal displacements become proportionately greater until for extreme elongations of the pseudo-chamber, the vertical displacement is no longer greatest immediately above the magma chamber. Thus, according to our model, the effective size and shape of the centre of dilation change progressively during the inflation. TREATMENT OF AN ELLIPSOIDAL CAVITY
The stresses produced in an elastic medium by inflation of a spherical cavity may be represented by three equal mutually perpendicular doubleforces, which are equivalent to the centre of dilation. To calculate the stresses and displacements due to an ovoid pseudo-chamber, we approximate the chamber by a prolate ellipsoid with its major axis vertical and this in turn is represented by three double forces as before, but with the vertical double force diminished by the factor (l-e), for a chamber with ellipticity e. General equations for deformation of an elastic half-space, presented by ~~yarna (1964), can then be used to find surface displacements, u (horizontal) and u (vertical):
uzag[(1+&L-?pT P3
P5 1
359
where r is the radial distance from the coordinate centre, vertically above the centre of the ellipsoid, c is the depth of the centre of the ellipsoid, p = (r2 + c* )%, a, b are the major and minor semi-axes of the ellipsoid, E = (a+)/~, P is the pressure increment causing the displacements and /J is the Lame constant (rigidity). For simplicity it is assumed that the Lame constants A, p are equal (i.e., Poisson’s ratio = 0.25). In the case of a spherical cavity (E = o) these equations reduce to the solution of Mogi (1958). Surface displacements determined from these equations are plotted in Fig. 2 for various pseudo-chamber ellipticities. Since it is an essential point of our model that the effective ellipticity of the pseudo-chamber (that is the bounclary of the zone which behaves plastically) increases with increasing inflation, we predict that the initial displacements may have the form approximated by Fig. 2(a), but that with progressive inflation the displacement pattern shifts toward Fig. 2(c) or even 2(d).
Fig. 2. Normalised vertical (u) and horizontal (u) displacements of the surface corresponding to pseudo-chamber ellipticities (a) E = 0, (b) E = 0.15, (c) f = 0.18, (d) E = 0.25. This sequence represents the trend of changes during progressive inflation, which causes enlargement and elongation of the pseudo-chamber.
360 LABORATORY
MODELS
In an attempt to model Kilauea Volcano in the laboratory we have experimented with balloons which can be inflated with water while immersed in gelatine or in sand. When the gelatine is sufficiently concentrated to behave elastically its surface deformation is accurately described by Mogi’s (1958) equations (or those presented here with e = 0 for the case of a spherical balloon). Inflation of a spherical balloon in sand gives a surface deformation with a maximum horizontal to maximum vertical displacement ratio of 0.6,
independently of the degree of inflation. We are unable to scale the variation of strength with depth which gives the variable pseudo-chamber elongation and consequent variable displacement ratio which are important features of our theoretical model. CONCLUSIONS
The stimulation for this calculation was the unexpected lack of success in observing a volcanomagnetic effect accompanying rapid inflation of Kilauea Volcano (Davis et al., 1973). This was interpreted as evidence that, by virtue of its fragmented structure the volcano can support only very limited stresses, insufficient to permit the development of a significant piezomagnetic anomaly. However, this is not the only possible explanation. If most of the structure of the volcano is very hot, i.e. above the Curie temperature of magnetite (58O”C), then no magnetic effects would be seen whatever the internal stresses are. Thus an independent test of our model is essential. Surface-d~placement me~urements provide such a test. But the adjustable parameters of an elastic volcano model are sufficiently accommodating that conclusive evidence of their inadequacy cannot be offered in terms of the presently available data (R.M. Wilson, quoted by Magi, 1958; Wright et al., 1968; Fiske and Kinoshita, 1969; Swanson and Jackson, 1970). Thus the finer points of our model are offered as predictions of effects which should be sought as essential evidence for the behaviour of Kilauea Volcano. Our suggestion that the effective centre of inflation is a vertically elongated pseudo-chamber is not in any way surprising; rather it would be surprising if the centre of inflation were perfectly spherical, as assumed in the simple, elastic models. Thus the ratio of horizontal and vertical displacements, which is characteristic of ~udo~h~ber e~ipticity, can be fitted to an elastic model. But an acid test of our model is that it predicts that the displacement ratio should change progreasiveiy and characteristically during inflation, in the manner represented by the sequence in Fig. 2, although not necessarily extending to the extreme of Fig. 2(d). If such behaviour is found, then it will be clear evidence of the an&&city which we assumed as a constraint imposed by the magnetic observations. The necessity for a large pseudo-chamber is emphasised by a calculation of the pressure increments required to produce 10 cm vertical surface d&lacements above chambers or pseudo-chamba of different radii centred at 2 km
3151
‘.
,b z -
f
x
w
s
x
,
, I
2
3
4
5
I RADIAL
2
.)-z-y x
3 DISTANCE
4
5 (Km)
I
x
.
,
1
2
3
4
5
Fig. 3. Surface-displacement curves for pseudo-chambers with ellipticities, E = 0.15, 0.18, 0.25 centred at depths 1.4 km, 1.25 km and 1.0 km, respectively, with scales normalised to fit the vertical disptacement data of Fiske and Kinoshita (1969). These data are seen to be insufficient to distin~ish between models. Critical to further tests of the theory are simultaneous measurements of horizontal displacements and measurements close to the centre of inflation. The ratios of maximum horizontal displacement/maximum vertical displacement for these cases, 0.64, 0.75 and 0.96, are all larger than the value 0.4 obtained with the spherical chamber (Mogi’s 1958 model).
depth. A chamber of radius 0.5 km requires a pressure increment of 800 bars, 1 km requires 100 bars and 1.5 km only 30 bars The higher values are obviously improbable and even 30 bars appears unacceptable in terms of the magnetic data (Davis et al., 1973), so that the anelastic pseudo-chamber may well extend to the surface itself in the manner indicated in Fig. 1. The strength of the magnetic constraint on possible stresses within Kilauea Volcano depends essentially upon its internal temperatures. It is therefore important to note that in this connection our model offers a further prediction which may be sensitive to observational test. The development of an inelastic pseudo-chamber of the type we propose necessarily supposes that the volcano is highly fragmented and therefore readily penetrated (and so cooled) by rainwater (the rainfall of the area being high). It is not a requirement of the model that the structure of the volcano be hot, but rather cold, so that the blocks or grains do not weld together under the overburden pressure. We envisage the hot region immediately surrounding the magma chamber (if indeed there is a significant chamber at all) to be extremely limited in size and that the structure of the volcano is essentially cold. Indeed the only chamber of consequence is then the pseudo-chamber of
362
anelastic deformation of cool material. The depth of the centre of dilation (which becomes the pseudo-chamber) is then determined simply by the depth at which rising magma has sufficient pressure to overcome the limited (depth-dependent) shear strength of the volcano structure. Also we note that some lateral migration of the centre of dilation is permitted, or even expected. ACKNOWLEDGEMENTS
This work is supported by the Australian Research Grants Committee. It arose from a collaborative experiment with the Hawaiian \iolcano Observatory of the U.S. Geological Survey and discussions of these ideas with D.W. Peterson, D.B. Jackson, W.A. Duffield and C.J. Zablocki, and also with J. Field of this department, have been important to our recognition of the problem. REFERENCES Davis, P.M., Jackson, D.B., Field, J. and Stacey, F.D., 1973. Kilauea volcano, Hawaii: a search for the volcanomagnetic effect. Science, 180: 73-74. Fiske, R.S. and Kinoshita, W.T., 1969. Inflation of Kilauea volcano prior to its 1967-1968 eruption. Science, 165: 341-349. Johnston, M.J.S. and Stacey, F.D., lQ69a. Volcanomagpletic effect observed on Mt. Ruapehu, New Zealand. J. Geophys. Res., 74: 6641-6644. Johnston, M.J.S. and Stacey, F.D., 196Qb. Transient magnetic anomalies accompanying volcanic eruptions in New Zealand. Nature, 224: 1289-l 296. Maruyama, T., 1964. Statical elastic dislocations in an infinite and semi-infinite medium. Bull. Earthquake Res. Inst., 42: 289-368. Mogi, K., 1958. Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them. Bull. Earthquake Res. Inst., 36: 99-134. Stacey, F.D., Barr, K.G. and Robson, G.R., 1965. The volcano-magnetic effect. Pure Appl. Geophys., 62: 96-104. Swanson, D.A. and Jackson, D.B., 1970. 1968-70 Kilauea deformation: summit. EOS (Trans. Am. Geophys. Union), 51: 440 (abstract). Walsh, J.B. and Decker, R.W., 1971. Surface deformation associated with vulcanism. J. Geophys. Res., 76: 3291-3302. Wright, T.K., Kinoshita, W.T. and Peck, D.L., 1968. March 1965 eruption of Kilauea volcano and the formation of Makaopuhi Lava Lake. J. Geophys. Res., 73: 3181-3205. Yokoyama, I., 197 1. A model for the crustal deformations around volcanoes. J. Phys. Earth, 19: 199-297.
STRESSES WITHIN AN ACTIVE VOLCANO - WITH PARTICULAR REFERENCE TO KILAUEA P.M. DAVIS, L.M. HASTIE and F.D. STACEY Physics Department,
University of Queensland, Erisbane (Australia)
(Accepted for publication
December 12, 1973)
ABSTRACT Davis, P.M., Hastie, L.M. and Stacey, F.D., 1974. Stresses within an active volcano - wit.? particular reference to Kilauea. Tectonophysics, 22: 355-362. The absence of an observable piezomagnetic effect correlated with rapid deformations of Kilauea Volcano, Hawaii, necessitates critical reappraisal of the conventional model of a dilating magma chamber in an elastic half-space. The introduction of a plastic or readily deformable pseudo-chamber around the magma chamber itself has the required effect of reducing the volcano-magnetic anomaly in model calculations. It also gives an increasing value of the ratio of the maximum horizontal displacement of the surface to the maximum vertical displacement during progressive inflation, and this accords better with available observations than do simple elastic models. In the case of Kilauea it appears that the ovoid pseudo-chamber is substantially larger than the true magma chamber. The volcano is visualised as an essentially cold structure, highly fragmented and penetrated by rainwater, with only localised magma channels.
INTRODUCTION
Measurements of surface displacements and tilts on Kilauea Volcano, Hawaii (e.g. Fiske and Kinoshita, 1969) are more extensive and detailed than for any other volcano and it is therefore natural to use these measurements for comparison with mathematical models of the eruption mechanism(s). The simplest approach is to assume that the volcano is essentially a spherical magma chamber, containing fluid under hydrostatic pressure, enclosed within a perfectly elastic half-space. The surface deformation accompanying changes in the pressure of the magma chamber (due to flow of magma from a much greater depth) is then calculable and the chamber depth is deduced by comparison of the scales of the observed and calculated displacement fields. Calculations of this type by Mogi (1958) are widely used to model Kilauea in particular and volcanoes in general. Using the available data on vertical displacements of the summit region, Mogi deduced that Kilauea deformation was due to an essentially spherical magma chamber at a depth of 2-3 km below the summit. More complicated models obtained as syntheses of combinations of solutions of the Mogi type (e.g. Walsh and Decker,
1971) produce somewhat different patterns of surface displacement, while retaining the concept of a chamber in an elastic half-space. Yokoyama (1971) presented a quite different model based upon a dipolar stress nucleus, but for which he offered no physical justification. The concept of a deformed elastic half-space allows a straightforward mathematical representation of the stresses within the medium, and these can to be used to calculate the piezomagnetie effect (the effect of stress on magnetic minerals) and the consequent time-dependent magnetic anomaly at the surface. Calculations of this type were reported by Stacey et al. (1965) for Caribbean volcanoes, Although based upon a quite different stress model, which we do not now accept, the application of piezomagnetic equations applies equally well to any stress pattern and the appearance of a volcanomagnetic anomaly is an inevitable consequence of any elastic model. Observations of striking magnetic anomalies preceding and accompanying eruptions of New Zealand volcanoes (Johnson and Stacey, 1969a, b) were taken to confirm the validity of the argument that the deformation of a volcano is elastic. It was of some interest therefore to make magnetic measurements on Kilauea, the volcano to which most of the elastic-model calculations were specifically applied. But, in spite of what appear to be ideal conditions for these observations, in particular the strongly magnetic rocks of Kilauea, Davis et al. (1973) found no systematic magnetic anomalies to accompany a short-term inflation-eruption sequence of Kilauea. There is now evidence of a longer-term magnetic trend, which is still being evaluated, but the reported observations suffice to cast serious doubt on elastic models of the eruption mechanism. These observations have provoked the present paper. CHAMBER SHAPE - THE EFFECT OF INTROlXJCING
A ‘PLASTIC’
ZONE
For a volcano, such as Kilauea, in which the magma is very fluid, the ~sumption of a spherical magma chamber can hardly be more than an extreme simplification of the real (but unknown) chamber shape. It is more realistic to imagine a complex array of tubes and magma lenses in a much larger volume of rock, which is multiply fractured, so that its response to chamber dilation is essentially plastic. The pattern of defo~ation in the surrounding elastic region is then- determined by the size and shape of the ‘plastic’ region rather than by the magma chamber itself. We term this plastic region the pseudo-chamber; increases in stress within it are essentially hydrostatic. To theoreticians this is an important advantage, because the shape of the pseudo-chamber is necessarily fairly simple and is susceptible to plausibility arguments or, with certain assumptions, to quantitative estimates, whereas the true shape of a magma chamber is subject only to gumsea with varying degrees of either im~l&usib~ty or mathematics in~ctabi~ty. If the true magma chamber (and thermal centre) is small (in the sense that the mechanical properties of the pseudo-chamber are independent of depth over the range which the true chamber occupies) then regardless of the shape of the magma chamber itself, the surrounding pseudo-chamber is close to
357
being spherical. Mogi’s (1958) model is then the appropriate description of surface displacements, which would indicate a highly symmetric source of the deformation. However, more violent inflation or a larger magma chamber cause extension of the ‘plastic’ zone. Then the pressure-dependence of the properties of the medium must be taken into account. At a critical shear stress the material ceases to respond elastically, becoming ‘plastic’, but the plastic deformation is due substantially to the presence of multiple fractures
RADIAL
DISTANCE
(Km)
Fig. 1. Progressive enlargement of a pseudo-chamber of the type proposed to represent the behaviour of Kilauea Volcano. The criterion for the transition from elastic (outside) to plastic (inside) deformation is that of Coulomb-Navier, whereby the shear strength of the material is linearly related to overburden pressure, so that the pseudo-chamber enlarges more upward than downward during inflation of the much smaller magma chamber. In this model the magma chamber is centred at 1.5 km depth, the shear strength is assumed to be 20 bars at the surface, increasing with depth by 30 bar/km (corresponding to a coefficient of friction of 0.1). Successive chamber sizes represent changes due to 8 bar increments in hydrostatic pressure.
358
(in the presence of some fluids) and therefore occurs more readily at low overburden pressures. Thus the ‘plastic’ zone extends upwards, more readily than outwards or downwards, becoming ovoid with its long axis vertical and smaller end uppermost (Fig. 1). This has two effects which are significant in the present context. The maximum horizontal displacement of the surface becomes larger, relative to the maximum vertical displacements, than in the spherical chamber model and the deviatoric stresses are greatly reduced, so reducing co~espondingly the piezomagnetic anomaly. The conclusion regarding surface displacements appears to be in accord with recent observations (Fiske and Kinoshita, 1969) which therefore provide data from which we can estimate the size and shape of the ‘plastic’ pseudochamber, Such a calculation does, of course, fall short of a convincing demonstration that the model is valid. But there is a further detail of the model which provides an acid test of its vahdity. During progressive inflation, the plastic region grows at the expense of the elastic region, as the stress which the material can support elastically is exceeded in an increasing volume, Enlargement of the plastic pseudo-chamber thus involves increasing vertical elongation. The initial deformation may correspond nearly to a spherical centre of dilation, for which the ratio of maximum horizontal to maximum vertical displacements is 0.4, as in Magi’s (1958) model. As deformation proceeds, so the horizontal displacements become proportionately greater until for extreme elongations of the pseudo-chamber, the vertical displacement is no longer greatest immediately above the magma chamber. Thus, according to our model, the effective size and shape of the centre of dilation change progressively during the inflation. TREATMENT OF AN ELLIPSOIDAL CAVITY
The stresses produced in an elastic medium by inflation of a spherical cavity may be represented by three equal mutually perpendicular doubleforces, which are equivalent to the centre of dilation. To calculate the stresses and displacements due to an ovoid pseudo-chamber, we approximate the chamber by a prolate ellipsoid with its major axis vertical and this in turn is represented by three double forces as before, but with the vertical double force diminished by the factor (l-e), for a chamber with ellipticity e. General equations for deformation of an elastic half-space, presented by ~~yarna (1964), can then be used to find surface displacements, u (horizontal) and u (vertical):
uzag[(1+&L-?pT P3
P5 1
359
where r is the radial distance from the coordinate centre, vertically above the centre of the ellipsoid, c is the depth of the centre of the ellipsoid, p = (r2 + c* )%, a, b are the major and minor semi-axes of the ellipsoid, E = (a+)/~, P is the pressure increment causing the displacements and /J is the Lame constant (rigidity). For simplicity it is assumed that the Lame constants A, p are equal (i.e., Poisson’s ratio = 0.25). In the case of a spherical cavity (E = o) these equations reduce to the solution of Mogi (1958). Surface displacements determined from these equations are plotted in Fig. 2 for various pseudo-chamber ellipticities. Since it is an essential point of our model that the effective ellipticity of the pseudo-chamber (that is the bounclary of the zone which behaves plastically) increases with increasing inflation, we predict that the initial displacements may have the form approximated by Fig. 2(a), but that with progressive inflation the displacement pattern shifts toward Fig. 2(c) or even 2(d).
Fig. 2. Normalised vertical (u) and horizontal (u) displacements of the surface corresponding to pseudo-chamber ellipticities (a) E = 0, (b) E = 0.15, (c) f = 0.18, (d) E = 0.25. This sequence represents the trend of changes during progressive inflation, which causes enlargement and elongation of the pseudo-chamber.
360 LABORATORY
MODELS
In an attempt to model Kilauea Volcano in the laboratory we have experimented with balloons which can be inflated with water while immersed in gelatine or in sand. When the gelatine is sufficiently concentrated to behave elastically its surface deformation is accurately described by Mogi’s (1958) equations (or those presented here with e = 0 for the case of a spherical balloon). Inflation of a spherical balloon in sand gives a surface deformation with a maximum horizontal to maximum vertical displacement ratio of 0.6,
independently of the degree of inflation. We are unable to scale the variation of strength with depth which gives the variable pseudo-chamber elongation and consequent variable displacement ratio which are important features of our theoretical model. CONCLUSIONS
The stimulation for this calculation was the unexpected lack of success in observing a volcanomagnetic effect accompanying rapid inflation of Kilauea Volcano (Davis et al., 1973). This was interpreted as evidence that, by virtue of its fragmented structure the volcano can support only very limited stresses, insufficient to permit the development of a significant piezomagnetic anomaly. However, this is not the only possible explanation. If most of the structure of the volcano is very hot, i.e. above the Curie temperature of magnetite (58O”C), then no magnetic effects would be seen whatever the internal stresses are. Thus an independent test of our model is essential. Surface-d~placement me~urements provide such a test. But the adjustable parameters of an elastic volcano model are sufficiently accommodating that conclusive evidence of their inadequacy cannot be offered in terms of the presently available data (R.M. Wilson, quoted by Magi, 1958; Wright et al., 1968; Fiske and Kinoshita, 1969; Swanson and Jackson, 1970). Thus the finer points of our model are offered as predictions of effects which should be sought as essential evidence for the behaviour of Kilauea Volcano. Our suggestion that the effective centre of inflation is a vertically elongated pseudo-chamber is not in any way surprising; rather it would be surprising if the centre of inflation were perfectly spherical, as assumed in the simple, elastic models. Thus the ratio of horizontal and vertical displacements, which is characteristic of ~udo~h~ber e~ipticity, can be fitted to an elastic model. But an acid test of our model is that it predicts that the displacement ratio should change progreasiveiy and characteristically during inflation, in the manner represented by the sequence in Fig. 2, although not necessarily extending to the extreme of Fig. 2(d). If such behaviour is found, then it will be clear evidence of the an&&city which we assumed as a constraint imposed by the magnetic observations. The necessity for a large pseudo-chamber is emphasised by a calculation of the pressure increments required to produce 10 cm vertical surface d&lacements above chambers or pseudo-chamba of different radii centred at 2 km
3151
‘.
,b z -
f
x
w
s
x
,
, I
2
3
4
5
I RADIAL
2
.)-z-y x
3 DISTANCE
4
5 (Km)
I
x
.
,
1
2
3
4
5
Fig. 3. Surface-displacement curves for pseudo-chambers with ellipticities, E = 0.15, 0.18, 0.25 centred at depths 1.4 km, 1.25 km and 1.0 km, respectively, with scales normalised to fit the vertical disptacement data of Fiske and Kinoshita (1969). These data are seen to be insufficient to distin~ish between models. Critical to further tests of the theory are simultaneous measurements of horizontal displacements and measurements close to the centre of inflation. The ratios of maximum horizontal displacement/maximum vertical displacement for these cases, 0.64, 0.75 and 0.96, are all larger than the value 0.4 obtained with the spherical chamber (Mogi’s 1958 model).
depth. A chamber of radius 0.5 km requires a pressure increment of 800 bars, 1 km requires 100 bars and 1.5 km only 30 bars The higher values are obviously improbable and even 30 bars appears unacceptable in terms of the magnetic data (Davis et al., 1973), so that the anelastic pseudo-chamber may well extend to the surface itself in the manner indicated in Fig. 1. The strength of the magnetic constraint on possible stresses within Kilauea Volcano depends essentially upon its internal temperatures. It is therefore important to note that in this connection our model offers a further prediction which may be sensitive to observational test. The development of an inelastic pseudo-chamber of the type we propose necessarily supposes that the volcano is highly fragmented and therefore readily penetrated (and so cooled) by rainwater (the rainfall of the area being high). It is not a requirement of the model that the structure of the volcano be hot, but rather cold, so that the blocks or grains do not weld together under the overburden pressure. We envisage the hot region immediately surrounding the magma chamber (if indeed there is a significant chamber at all) to be extremely limited in size and that the structure of the volcano is essentially cold. Indeed the only chamber of consequence is then the pseudo-chamber of
362
anelastic deformation of cool material. The depth of the centre of dilation (which becomes the pseudo-chamber) is then determined simply by the depth at which rising magma has sufficient pressure to overcome the limited (depth-dependent) shear strength of the volcano structure. Also we note that some lateral migration of the centre of dilation is permitted, or even expected. ACKNOWLEDGEMENTS
This work is supported by the Australian Research Grants Committee. It arose from a collaborative experiment with the Hawaiian \iolcano Observatory of the U.S. Geological Survey and discussions of these ideas with D.W. Peterson, D.B. Jackson, W.A. Duffield and C.J. Zablocki, and also with J. Field of this department, have been important to our recognition of the problem. REFERENCES Davis, P.M., Jackson, D.B., Field, J. and Stacey, F.D., 1973. Kilauea volcano, Hawaii: a search for the volcanomagnetic effect. Science, 180: 73-74. Fiske, R.S. and Kinoshita, W.T., 1969. Inflation of Kilauea volcano prior to its 1967-1968 eruption. Science, 165: 341-349. Johnston, M.J.S. and Stacey, F.D., lQ69a. Volcanomagpletic effect observed on Mt. Ruapehu, New Zealand. J. Geophys. Res., 74: 6641-6644. Johnston, M.J.S. and Stacey, F.D., 196Qb. Transient magnetic anomalies accompanying volcanic eruptions in New Zealand. Nature, 224: 1289-l 296. Maruyama, T., 1964. Statical elastic dislocations in an infinite and semi-infinite medium. Bull. Earthquake Res. Inst., 42: 289-368. Mogi, K., 1958. Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them. Bull. Earthquake Res. Inst., 36: 99-134. Stacey, F.D., Barr, K.G. and Robson, G.R., 1965. The volcano-magnetic effect. Pure Appl. Geophys., 62: 96-104. Swanson, D.A. and Jackson, D.B., 1970. 1968-70 Kilauea deformation: summit. EOS (Trans. Am. Geophys. Union), 51: 440 (abstract). Walsh, J.B. and Decker, R.W., 1971. Surface deformation associated with vulcanism. J. Geophys. Res., 76: 3291-3302. Wright, T.K., Kinoshita, W.T. and Peck, D.L., 1968. March 1965 eruption of Kilauea volcano and the formation of Makaopuhi Lava Lake. J. Geophys. Res., 73: 3181-3205. Yokoyama, I., 197 1. A model for the crustal deformations around volcanoes. J. Phys. Earth, 19: 199-297.