Contrib Mineral Petrol (2002) 142: 653–665
DOI 10.1007/s00410-001-0323-8
S. Maaløe
Physical behavior of the plume source
during intermittent eruptions of Hawaiian basalts
Received: 1 May 2000 / Accepted: 28 February 2001 / Published online: 14 November 2001
Springer-Verlag 2001
Abstract The plume surrounding a source region for
magma may either compact by elastic or viscous deformation when magma leaves the source region. It is
shown that the compaction is elastic. The elastic deformation implies that only a small volume fraction of the
melt can leave the source region. On the other hand, the
isotopic secular disequilibria show that the fraction melt
must be much larger. This discrepancy is solved if the
source region is veined and consists of intercalated veins
and residuum, with the veins forming about 1% volume
of the source region. The approximate width of the
source region is estimated to be 10–20 km and its height
is 1–2 km. An eruption starts when a dike is formed
above the source region because of overpressure of the
melt within the source region. During the repose time,
the overpressure of the melt in the source region increases because of the replenishment of melt from the
plume situated below.
Introduction
The petrogenesis of plume-derived basaltic magmas occurs in four consecutive stages. In the first stage, melt is
formed interstitially in partially molten lherzolite (Waff
and Bulau 1979). During the second stage, the lherzolite
matrix becomes permeable after which the interstitial
melt begins to migrate by percolation, or in veins. The
extruded volumes of basaltic magma show that the interstitial melt somehow must accumulate into large
batches of melt within a source region before eruptions
can take place. The third stage is, therefore, an accumulation of the melt. Finally, a dike is formed above the
S. Maaløe
Geologisk institutt, Allegaten 41, 5007 Bergen, Norway
E-mail: sven.maaloe@geol.uib.no
Editorial responsibility: I. Carmichael
source region, which conducts the melt towards the
surface and an eruption can take place.
Although the very initial melt generation in interstices
is well understood, different models have been proposed
for the dynamics of melt accumulation. Melt-accumulation models can be divided into matrix percolation
models, where the flow of melt is entirely interstitial
(McKenzie 1984), and conduit models, where the plume
source consists of intercalated conduits and residuum
(Wood 1979; Shaw 1980; Quick 1981; Maaløe and
Scheie 1982; Ceuleneer and Rabinowicz 1992; Richardson et al. 1996; Kelemen et al. 1997; Maaløe 1998).
Conduits have both been suggested to be veins that only
contain melt and channels with a high porosity.
Some insight into the physical aspects of the accumulation processes within a mantle plume can be
obtained by considering the eruption dynamics, the
volumes of magma extruded, the duration of the eruption episodes, and the deformation of a magma source
within a plume. These features are considered below.
The seismic activity and tilt of the flanks of a volcano
preceding tholeiitic eruptions have been extensively
monitored on Hawaii. Hawaiian models for eruption
dynamics have been represented by Eaton and Murata
(1960), Aki et al. (1977), Aki and Koyanagi (1981),
Chouet (1981), Dvorak and Okamura (1987), Klein et al.
(1987), Decker (1987), and Klein and Koyanagi (1989).
The tholeiitic eruptions of the Kilauea Volcano are
preceded by an increasing tilt of the volcanic flanks and
a harmonic tremor caused by the ascent of magma. The
tiltmeter measurements show that the subvolcanic
magma chamber of the Kilauea Volcano is continuously
fed by magma between eruptions, and that eruptions
occur at different tilts of the flanks of Kilauea (Koyanagi
et al. 1987). This variation suggests that the state of
stress in the crust above the subvolcanic magma chamber exerts control on when eruptions occur. So far, no
post-shield or rejuvenated eruptions have occurred
recently in the Hawaiian chain; hence, there are no
detailed eruption data available for these types of
intermittent eruptions.
654
The repose time of intermittent eruptions may either
be related to the pressure relations within a subvolcanic
magma chamber, or to a variation in the supply rate of
melt from the plume source. The phenocryst content of
the Hualalai basalts is mostly less than 5%, and there is
no systematic variation in the phenocryst content with
time (Moore and Clague 1991). This shows that the
magmas have undergone a small degree of cooling, and,
consequently, there is no evidence for a long-lasting
subvolcanic magma chamber. If such a magma chamber
was present, eruptions would require a supply of melt
from the plume source. The tiltmeter measurements for
Kilauea show that eruptions are preceded by swelling of
the flanks of the volcano (Decker 1987), which shows
that the subvolcanic magma chamber is supplied from
below. The density of basaltic magma is similar to, or
less than, the density of the volcanic pile (Eaton and
Murata 1960), so that any excess pressure in a subvolcanic magma chamber must arise from the influx of
magma from the plume. Therefore, it is considered that
repose time is related to the dynamics of the plume
source.
The Hawaiian post-shield and rejuvenated eruptions
differ from the Kilauean eruptions by being intermittent,
with repose times from some hundred to several thousand years (MacDonald and Abbot 1979; Langenheim
and Clague 1986). The repose time for the Hualalai
Volcano varies from a few hundred to 2.5 ka, and a
representative repose time is taken as 1.0 ka (Moore and
Clague 1991). In comparison, the repose time for Mauna
Kea is 4–5 ka (MacDonald and Abbot 1979; Sharp et al.
1996). The rejuvenated eruptions on Oahu have occurred at different vents at time intervals of 10–20 ka
(MacDonald and Abbot 1979; Clague 1987). The average time intervals between the rejuvenated vents on
Kauai are even longer at 80 ka (Clague and Dalrymple
1988). The repose time between the rejuvenated eruptions on Sawaii, Samoa are much shorter; historic
eruptions took place in 1780, 1902, and 1905–1911
(Thomson 1921; Sapper 1927).
The alkalic post-shield eruption episode of Hualalai,
Hawaii lasted for about 1 year, with an erupted volume
of lava of 0.41 km3 (MacDonald and Abbot 1979; Bohrson and Clague 1988). The volume of the two lava
flows on Mauna Kea is about 0.8 km3 (Wolfe et al.
1997). Historically, Hawaiian rejuvenated eruptions are
few. Haleakala erupted at least 3.5 · 107 m3 lava in
1790, but the duration of the eruption is unknown
(MacDonald and Abbot 1979). The rejuvenated 1905–
1911 Matavanu eruption episode of Savaii, western Samoa lasted for about 6 years, and yielded at least 4 km3
of alkali olivine basalt (Jensen 1907; Sapper 1927;
Stearns 1944). The rate of outpouring of lava from
Matavanu Volcano varied during the 6 years, but there
was a constant flow of lava into the sea from 1905 to
1910 or 1911 (Klautzsch 1907; Thomson 1921). The
durations of these eruption episodes show that a plume
source is able to deliver magma for a year or more. This
finding suggests that some feature must prevent eruption
for the duration of the repose time. There is magma
available for eruption in the plume source, but eruptions
do not occur until a certain critical stage has been attained. The most obvious control of eruptions is dike
formation, which could be related to the overpressure of
the magma in the plume source.
The modeling of the dynamics of intermittent eruptions is based here on the 1800–1801 Hualalai eruption,
Hawaii. This erupted 0.41 km3 of lava during the 1800–
1801 eruption episode, which is the minimum amount of
melt that would have left the plume source (MacDonald
and Abbot 1979; Moore and Clague 1991) . The tiltmeter measurements on Kilauea show that it is only
about one-third of the magma entering the volcano that
erupts, whereas the remaining two-thirds is intruded into
sills and dikes (Dzurisin et al. 1984). It is not known if a
similar feature applies to the post-shield eruptions. The
erupted volume of lava reported from Kilauea is considered to be representative of the volume of melt leaving a plume source.
The presence of peridotite nodules in alkali olivine
basalts, nephelinites, and melilitites shows that the ascent of the magma must have been rapid, and, therefore,
it is considered that the ascent through the lithosphere
takes place in dikes (Jackson and Wright 1970). The
presence of garnet-bearing nodules suggests that the
ascent occurred from depths of at least 75 km (Sen
1988). In comparison, the thickness of the lithosphere
beneath the central Pacific is estimated to be 86 km
(Leeds et al. 1974).
The region of the plume that supplies magma during
an eruption is called the source region, in order to distinguish it from the entire partially molten region within
the plume. The source region is supplied with melt from
the partially molten plume during the repose time.
The overpressure, DP, of the melt is defined as the
difference in pressure of the melt Pm and the lithostatic
pressure Pl:
DP ¼ Pm Pl
ð1Þ
The necessary condition for dike formation above a
source region is that the melt has an overpressure. The
melt cannot ascend if its pressure is less than the lithostatic pressure. Further, because intermittent eruption
episodes may last for a few years, an overpressure must
prevail in the source for this length of time. The density
of peridotite and melt is taken as 3.3 and 3.0 g/cm3,
respectively (Maaløe 1998). The buoyancy gradient of
the melt is then 30 bar/km. If the height of a source
region is given by z km, then the overpressure of the melt
as a result of buoyancy varies from zero at the bottom
where z=0 to 30z at height z.
It is not possible to obtain geophysical evidence for
the shape and dimensions of the source regions in mantle
plumes. The resolution of seismic tomography allows an
approximate estimate of the shape of a plume, but not
its detailed structure. These features must be estimated
by combining the sparse evidence available. The present
work attempts to estimate some aspects of the dynamics
655
of the intermittent post-shield eruptions. The purpose of
this study was to show which parameters are involved
and how the problem can be approached. It is emphasized that the quantitative results obtained in the modeling are based on several assumptions; however, the
dependence of the results on the different parameters is
demonstrated.
Various possible models for a source region are first
considered. Thereafter, the implications of viscous and
elastic deformation of the source region are dealt with,
the conclusion being that the plume surrounding the
source region must react mainly by elastic deformation
and pressure changes in the source region. For elastic
deformation, the fraction of melt extracted during an
eruption must be very small, and less than 1% if the melt
accumulates in a chamber that only contains melt. This
result is compared with the volume constraints exerted
by the activity ratio (226Ra/230Th). This activity ratio
shows that the volume fraction extracted from the
source must be much larger than 1%. This apparent
inconsistency is solved if the source consists of intercalated veins and residuum, the residuum forming the
major part of the source region.
Types of source regions
The repeated eruption of lavas with similar compositions during the post-shield stage suggests that the
plume source is reacting in a cyclic manner. An eruption
is followed by a repose time where magma accumulates
within the source region until a new eruption takes
place. This behavior is not particular to Hawaiian volcanoes, but a typical feature of volcanic eruptions. The
repeated post-shield eruptions of Hawaii have vents
situated near the summits of the central volcanoes,
which suggests that the melt is focused towards the same
region within the plume. Therefore, it is assumed that
the post-shield eruptions have the same source region
and that the dynamic features of the source region
should result in a cyclic variation of the dimensions of
the source region. During an eruption, the cross sectional shape and size of the source region should revert
to the same shape and size it had previously.
Four basically different types of source regions are
evaluated here. Considering the structure, the source
region may be a sill filled with melt or a veined source
region that consists of intercalated veins and residuum.
Both these source regions may loose all melt during an
eruption and be entirely renewed during each repose
time. Alternatively, these two source regions may only
loose a fraction of the melt during an eruption. The
likelihood of each of these four source regions is dealt
with, and it is concluded that a veined and partly
replenished source region is preferred.
The conditions for repeated eruptions during eruption episodes that last for 1 year or more is that the
mantle surrounding the source is able to exert pressure
on the source region. If the mantle remains stiff, so that
the walls of a source remain in the same position during
an eruption, then the lithostatic pressure will not be
transferred from the surrounding mantle to the source.
In that case, only the compressibility of the magma will
result in outflow of magma from the source. Assuming
stiff walls, the volume of melt in the source may be
estimated from:
ðP2 P1 Þj ¼ ln
V1
V2
ð2Þ
where V1 and V2 are the initial and final volumes, respectively. P1 and P2 are the initial and final pressures,
respectively, and j is the compressibility. For a pressure
decrease of 100 bar, an erupted volume of 0.41 km3, and
j=2.1 · 10–6 bar–1 (Scarfe et al. 1979), the volume of
melt is V1=2,050 km3, which appears excessive.
The mantle surrounding the source exerts a lithostatic
pressure on the source, which leads to either elastic or
viscous deformation. The propagation of seismic waves
through the mantle shows that the mantle reacts elastically to instant stress. On the other hand, the ascent of
plumes shows that the mantle reacts more or less viscously to long-term stresses. It may be noted that plume
models generally assume that the plume behaves like a
Newtonian fluid, but the textures of lherzolite nodules
suggest that lherzolite has a yield point, and, therefore,
displays an elastic behavior at stresses below the yield
point (Ave’Lallement et al. 1980). The type of deformation that the mantle undergoes during the repose time
is not obvious, and both types of behavior are considered below.
The shape of the source region is unknown, but its
horizontal dimensions are probably larger than its
height. During an eruption, the overpressure will decrease rapidly with time in a source region that has a
dike-like shape (cf. Pollard and Muller 1976), whereas
the decrease in overpressure with time will be small in a
source region with a relatively large horizontal extension. Because the eruption episodes may last a year or
more, a shape with a larger horizontal than vertical dimension appears likely. The repeated post-shield eruptions near the summits of Hualalai and Mauna Kea
show that the melt in the plume is focused towards a
source region, which could suggest a source region with
radial symmetry, perhaps an oblate ellipsoid source
region. One the other hand, the divergent ascent of the
Hawaiian deflected plume could result in a source region
elongated in a SE–NW direction (Ribe and Christensen
1999). The shape of the source region chosen here is a
compromise between a central and elongated source
region. The source region is modeled by an elongated
ellipsoid with horizontal axes of a and b=2a and vertical
axis of c=ka, where k is the aspect ratio. The model
source region is thus twice as long as it is wide. The
variation in the shape of the cross section of such a
source region with varying overpressure may be
approximated by the variation of a cylinder with an
elliptical cross section.
656
The change in R with time is then given by:
Viscous deformation
The change in volume of a 2-D elliptic cylinder during
viscous deformation may be determined using the
method of complex potentials. The equations for viscous
deformation of an elliptic body under biaxial stress was
derived by Berg (1962) using the methods of Muskhelishvili (1963). Using the complex potentials of Muskhelishvili (1963) for hydrostatic pressure and the
calculation procedure by Berg (1962), the following
equations are derived. Let the elliptic, horizontal long
axis along the x axis be a, and the vertical short axis
along the y axis be c. Further let:
R ¼ ða þ cÞ=2
ð3Þ
m ¼ ða cÞ=ða þ cÞ
ð4Þ
R is the mean radius of the ellipse, and 0<m<1 depends
on the shape of the ellipse. For m=0, the ellipse becomes
a circle (a=c), and for m=±1, the ellipse becomes a
straight line (a=0 or c=0). The conformal transformation from physical z-space onto the unit circle is given by:
m
z ¼ x þ iy ¼ R f þ
f
ð5Þ
where f=qeiu. The angle u is the angle between the x axis
and a point on the unit circle and q=1 on the unit circle.
The complex potentials are given by (Muskhelishvili
1963):
/ðfÞ ¼
DPR
f
ð6Þ
wðfÞ ¼
DPR DPRm 1 þ mf2
2
f
f
f m
ð7Þ
The velocities u and v at the free surface of the ellipse are
estimated from (Berg 1962):
2gðu þ ivÞ ¼ j/ðfÞ xðfÞ 0
/ ðfÞ wðfÞ
0 ðfÞ
x
ð8Þ
where g=viscosity and j=3–4m=1 for m=1/2, and m is
Poisson’s ratio. The bars above the complex functions
indicate the conjugate complex function. Using Eqs. (6),
(7), and (8) the result is that:
u þ iv ¼
DPR
m
f
2g
f
xðf; tÞ ¼ RðtÞ f þ
mðtÞ
f
ð10Þ
mðtÞ ¼ mo expðDPt=gÞ
dxðf; tÞ dR
dðRmÞ 1 DPR
m
¼
fþ
¼
f
dt
dt
dt f
2g
f
ð13Þ
where Ro and mo are the initial values of R and m, respectively. DP is the difference between the melt pressure,
Pm, during viscous creep and the lithostatic pressure, Pl,
at time zero:
DP ¼ Pm Pl
ð14Þ
The overpressure of the melt increases with height so that
there is an asymmetric pressure distribution within the
source region. The overpressure applied subsequently is
the maximal overpressure of the upper part of the source
region. This approach results in an estimate of the maximal deformation. From these equations, the values of
the axes a and b at time t can be estimated using:
c ¼ ð1 mðtÞÞRðtÞ
ð15Þ
a ¼ ð1 þ mðtÞÞRðtÞ
ð16Þ
When m has values near 1.00 the length of the horizontal
axis a hardly varies during creep, and almost all the
variation in a cross-sectional area is caused by a variation
in the vertical axis c.
Equations (15) and (16) only apply to the relatively
short eruptive stage, where the pressure changes may be
considered hydrostatic. During the long repose time, the
source region may be under stress because of the divergent flow of the plume, and the stress field could have
a tensional component in addition to the hydrostatic
component. The change in cross-sectional shape by the
stresses P1 and P2 and the hydrostatic pressure P0 is
estimated from:
ð1þjÞðP1 þP2 þ2P0 Þt
8g
R ¼ R0 e
ð17Þ
T1 ¼ ð1 þ mo Þ cosðuÞ
ð18Þ
P1 P2
eu ðP1 P2 Þ
T2 ¼
ðcosð2aÞ cosðuÞ þ sinð2aÞ sinðuÞÞ
P1 þ P2 P1 þ P2 þ 2P0
ð19Þ
u¼
2ð1 þ jÞðP1 þ P2 þ 2P0 Þ
8g
x ¼ RðT1 þ T2 Þ
ð20Þ
ð21Þ
Similarly, for the y coordinate:
S1 ¼ ð1 mo Þ sinðuÞ
where t=time. The variation in R and m is then estimated
from the differential with respect to time in Eq. (10):
ð12Þ
The variation in m with time is obtained from:
ð9Þ
The time-dependent mapping function that continuously
transforms the elliptical surface onto the unit circle has
the form:
RðtÞ ¼ Ro expðDPt=2gÞ
S2 ¼
ð22Þ
P1 P2
eu ðP1 P2 Þ
ðsinð2aÞ cosðuÞ cosð2aÞ sinðuÞÞ
P1 þ P2 P1 þ P2 þ 2P0
ð23Þ
ð11Þ
y ¼ RðS1 þ S2 Þ
ð24Þ
657
where u is the angle between a point on the ellipse and
the x axis, g is the viscosity, and j=1 for m=0.5. For
tensional and compressional stresses, P1 and P2 are
positive and negative, respectively. P0 is positive when the
hydrostatic pressure exceeds the lithostatic pressure.
The angle a is the angle between stress P1 and the x axis.
The stress P2 is perpendicular to stress P1. If the direction
of stress is 0 for P1 it is 90 for P2. These equations do
not apply for pure shear where P1+P2=0, but equations
for pure shear have been derived by Berg (1962).
The viscosity of the plume surrounding the source
region is estimated using Griggs (1939) equation:
g ¼ r=3e
ð25Þ
where r=stress and =strain rate. The strain rate is
estimated from the experimental data of Hirth and
Kohlstedt (1995) for an isostatically compressed olivine
matrix containing 2% melt. The strain rate s–1 is estimated from:
Q
e ¼ Arn d p exp
RT
ð26Þ
where A=1.82 · 10–6, a materials constant for 1–3%
melt, r=deviatory stress in bars; n=1 for diffusion
creep; d=grain diameter in cm; p=–3 for diffusion
creep; Q=315 kJ/(g · mol · K), the activation energy;
R=8.314 J/(g · mol · K). For a grain diameter of
0.1 cm, T=1,873K, and a deviatory stress of 1 bar, the
strain rate is calculated as =3 · 10–12 s–1, and the viscosity as 1.1 · 1017 poise. Hirth and Kohlstedt (1995)
observed that the strain rate is enhanced when the
samples contained more than 5% melt. At 7% melt, the
strain rate is enhanced by a factor 25 relative to that of
melt-free samples. A viscosity within the range of 1016–
1018 poise is here considered relevant. This range applies
only to the partially molten part of the plume, and not to
the entire plume, which has larger viscosities at its
margin and in the zone below the partially molten zone.
The change in cross section with the gradual hydrostatic increase in pressure from 0–30 bar during a 100year repose time is shown in Fig. 1 for a viscosity of
1018 poise and a source region with an initial aspect ratio
of 0.1. The change in shape is independent of the size of
the elliptical cross section. The cross section becomes
more circular, and the c axis increases more than the a
axis. When the overpressure of the melt in the source
decreases, the source becomes exposed to a compressive
stress from the surrounding plume. As long as the source
retains a certain height, the overpressure cannot become
zero because the overpressure is given by 30z at height z
above the bottom. A maximal estimate of the decrease in
height of the source, therefore, is obtained by assuming
the overpressure equal to zero, in which case the compressive stress equals 30 bar. With this stress, the height
of the source region will decrease by less than 0.3%
during a 1-year eruption episode for g=1018 poise.
Hence, the dimensions of the source region will not
revert to the values they had before the repose time
started, i.e., after the last eruption. The same conclusion
Fig. 1 The viscous deformation of an elliptic source region for a
viscosity of 1018 poise and a 1,000-year repose time. The inner
ellipse shows the initial shape. The change in shape for a gradual
increase in hydrostatic pressure from 0 to 30 bars during 1,000 years
is computed assuming a constant intermediate pressure of 15 bar.
With hydrostatic pressure, the height of the ellipse increases
whereas its width changes little. The height of the ellipse remains
nearly constant when a tensional stress of 30 bars is applied for the
same period of time, but its volume increases
is evident if other overpressures other than 30 bar are
applied. During successive repose periods, the source
region will retain most of its melt and continue to grow
in size. The volume of lava erupted, therefore, should
increase with time, but, instead, the post-shield stage is a
declining stage with an decreasing eruption frequency.
The divergent plume flow may exert a tensional stress
on the source region. It is possible that this tensional
stress can restrict the vertical increase of the source region during the repose time. The effect of a tensional
stress of 30 bar that is equal to the maximal hydrostatic
pressure is shown in Fig. 1; the actual value of the tensional stress is of lesser importance. As is evident, this
tensional stress diminishes the increase in height during
the repose time. However, both the horizontal length
and cross-sectional area have increased substantially.
With an overpressure equal to zero during a 1-year
eruption episode, the height of the elongated source region will decrease by less than 1% because of compression. During the next repose period, the source
region will be stretched even more. Hence, the presence
of a prevailing tension does not result in cyclic variation.
Therefore, it is concluded that viscous behavior of the
source region cannot cause cyclic variations. If other
viscosities other than 1018 poise are used, then the same
conclusion is reached. The basic feature of viscous deformation in the present context is that the deformation
that occurs during the long repose period cannot be
reversed during a short eruption episode. This relationship indicates that elastic behavior might be predominant because it depends only on pressure and is
independent of time.
Elastic deformation by gravity and overpressure
The deformation of an elliptic cross section is here estimated by applying the method of complex potentials.
658
By using this approach, two complex potentials are
estimated from the stress distribution and the shape of
the body under stress. Thereafter, the two potentials are
used for to estimate the deformation and stress fields
(Savin 1961; Muskhelishvili 1963).
The lithostatic stress field in the mantle and crust as a
result of gravity is given by (Jaeger and Cook 1979):
t
ry ¼ qgy; rx ¼ 1t
ry ; sxy ¼ 0
ð27Þ
where rx and ry are the stresses in the x- and y-directions, respectively, and m is Poisson’s ratio. For m=0.25,
a body behaves like a perfect elastic material, whereas
for m=0.5, the stress is hydrostatic and a body behaves
like a fluid. The y axis is here taken with the positive
direction upwards. Further, the tensional stress is positive and the compressive stress is negative. Using the
approach given by Savin (1961), the potentials for a
gravity field given by Eq. (27) are estimated as:
iqg 2
z
ug ðzÞ ¼
8ð1 mÞ
ð28Þ
ijqg 2
z
8ð1 mÞ
ð29Þ
wg ðzÞ ¼
The ‘‘i’’ indicates an imaginary part in z=x+iy. These
potentials yield the exact values for rx and ry given by
Eq. (27), but the shear stress sxy is not zero because
sxy=–x. Consequently, these potentials do not represent a lithostatic stress field where sxy=0. The shear
stress can be made equal to zero if the term
i(x2+y2)ln(z)/2 is added to Eq. (29). However, this
term is not suitable for estimates of stress and deformation when using complex potentials because these
should depend only on z. This may explain why Savin
(1961) did not use the complex potentials of Eqs. (28)
and (29) for an estimate of the gravity-controlled deformation of a circular cross section. The elastic deformation for bodies subject to gravity may instead be
estimated using the approach of Biot (1935), who
suggested that the gravity field is replaced by a
boundary pressure given by qgy, where q is the density
and g the gravity constant.
If the density of the magma and mantle is 3.00 and
3.3 g/cm3, respectively, then the buoyancy gradient is
30 bar/km. In a 2-km-high intrusion, the overpressure
of the magma will vary from 0 bar at the bottom to
60 bar at the top. This overpressure causes deformation
of the host rock and stresses within the host rock
surrounding the body of magma. The complex potentials for a varying overpressure is here estimated by
first estimating the potentials for an linear pressure
variation from –|yDP|to +|yDP|. Secondly, the complex potentials are applied to a constant overpressure
of –|yDPc|.
The complex potentials for a linear pressure variation
from –|yDP| to +|yDP| are estimated using the methods
of Muskhelishvili (1963):
uðfÞ ¼
iDPR ð1 m2 Þ lnðfÞþðm m2 Þf2 Þ
4
ð30Þ
"
#
iDPR
2ð1 þ mf2 Þðm2 mÞ
2
2
ð1 m Þ lnðfÞ þ ð1 mÞf wðfÞ ¼
4
f2 ðf2 mÞ
ð31Þ
The potentials for a constant overpressure were derived
from Muskhelishvili (1963):
uðfÞ ¼ PRm
f
ð32Þ
wðfÞ ¼ PR PRmð1 þ mf2 Þ
f
fðf2 mÞ
ð33Þ
These two sets of potentials allow estimates for the deformation and stress fields of elliptic bodies in a gravity
field. The deformation is estimated from (Muskhelishvili
1963):
2Gðu þ ivÞ ¼ juðfÞ xðfÞ
x; ðfÞ
u; ðfÞ wðfÞ
ð34Þ
where G=shear modulus, and j=(3–4t) and t=Poisson’s ratio. The deformations in the x and y axes are
given by u and v, respectively. The bar above the functions denotes the complex conjugate functions.
The deformation of an elliptic cross section is estimated by adding the three deformations caused by these
pressure variations:
• Constant overpressure, Pc:–30(1 – m)R
• Linear pressure variation, –30y<Pl<–30y, [–(1 –m)R<y<+(1 – m)R]
• Lithostatic pressure variation, 330y<Pg<330y
Note here that the signs are based on the convention
that a compressive stress is negative. The center of the
ellipse is at x=0 and y=0, so that y varies from negative
to positive values. For x=0, y=±R(1–m), so that by
adding the Pc and Pl one obtains a pressure variation
from zero to the compressive stress – 60(1–m)R. If the
deformations estimated for the y-direction be uc, ul, and
ug, then the total deformation ut from linear elastic
deformation is obtained by the linear addition:
ut ¼ uc þ ul þ ug
ð35Þ
The variations in the shape of the elliptic cross sections
are illustrated in Fig. 2 for the very small G-value of
0.2 kbar in order to exaggerate the deformations. The
actual deformations are about 3,000 times smaller
because a relevant G-value is about 730 kbar.
The calculations show that the change in cross-sectional area caused by these pressure changes only depends on the overpressure, Pc. The lithostatic pressure
raises the floor of the cross section by the same amount
as the roof is elevated, and the same relationship holds
for Pl. Therefore, the variation in the cross-sectional
area only depends on Pc. The x and y coordinates with a
varying Pc is given by:
x ¼ Rð1 þ mÞ cosðuÞ þ
DPc R
ð1 jmÞ cosðuÞ
2G
ð36Þ
659
Fig. 2 a The change in shape of an elliptic cross section with a
hydrostatic increase in pressure of 30 bar. The shear modulus
applied is only 0.2 kbar in this figure, in order to show the variation
in shape. b The deformation caused by gravity. The bottom and
roof regions are raised. c The total deformation caused by
overpressure, gravity, and a linear increase in pressure from 0 to
60 bar
y ¼ Rð1 mÞ sinðuÞ þ
DPc R
ð1 þ jmÞ sinðuÞ
2G
ð37Þ
The variation in the cross-sectional area with pressure is
estimated from these two equations.
The modulus of shear, or the modulus of rigidity is
given by G=E/[2(1+m)], and the bulk modulus is given
by K=E/[3(1–2m)], where E is the modulus of elasticity.
Poisson’s ratio is estimated to be 0.25 for most of the
mantle (Masters and Shearer 1995). The ‘‘parametric
Earth model’’ (PEM) of Dziewonsky et al. (1975) estimates the P and S wave velocities at a depth of 100 km
to be 8.0 and 4.36 km/s, respectively. These values yield
a bulk modulus of 1,260 kbar and a shear modulus of
756 kbar. This applies to a temperature of 1,100 C at a
depth of 100 km of average lithosphere (Watson and
McKenzie 1991). The temperature of the plume surrounding a source region must be higher, and so a
temperature of 1,300 C is adopted here. Using the
variation in bulk modulus with the temperature estimated as (dK/dT)T=–0.214 (Graham 1970), the bulk
modulus is 1,217 kbar and the modulus of shear is
G=730 kbar. Poisson’s ratio increases with increasing
plasticity and equals 0.5 for a fluid. For m=0.364 and
K=1,217 kbar, G equals 365 kbar and is thus half the
value for m=0.25. It is possible that m for the hot central
part of the plume is larger than 0.25, so that 730 kbar is
too large a value for G. The pressure required for a given
deformation is inversely proportional to G so that the
heights estimated for the source region can easily be
changed if warranted.
The elastic behavior of a sill and a veined source region will be considered next. The pressure of the melt
within a dike cannot be less than the lithostatic pressure.
If the pressure in a dike decreases below the lithostatic
pressure then the walls of the dike close up. The supply
of melt from a sill to a dike will terminate when the
pressure of the melt in the sill equals the lithostatic
pressure. Hence, the maximal possible pressure decrease
during an eruption is 30y, where y is the height of the
sill. The change in cross-sectional area caused by a
pressure change from 30y to zero for a 200-m-thick sill is
0.031% for an aspect ratio of 0.01 [Eqs. (36) and (37)].
For a 2-km-thick sill, the change in volume is 0.31% for
the same aspect ratio. The width of a 2-km-thick sill with
an aspect ratio of 0.01 is 200 km, which obviously is too
large. The width of a 2-km-thick sill is 20 km for an
aspect ratio of 0.1, and the change in cross-sectional area
is 0.014%. Because a 20-km-wide sill is the probable
maximum width of a sill, the maximal possible volume
leaving a sill during an eruption is about 0.02%. This
implies that most of the melt will remain in the sill,
which is contradictory to the estimated activity ratios.
The activity ratio (226Ra/230Th) for such a sill will be
almost 1.00 because most of the melt remains stored in
the sill. However, the estimated activity ratios for the
post-erosional Hualalai lavas are within the range of
1.18–1.364 (Sims et al. 1999). Therefore, a sill filled with
melt is not a possible model for the source region within
a plume. A veined source region is, therefore, considered
next.
A veined source region consists of veins distributed
throughout a residuum. The vein model for the mantle
was first suggested by Wood (1979), and later considered
by Sleep (1988). When the veins are disconnected,
the overpressure of the melt within the upper part of the
veins equals 30y, where y is the vertical extension of the
veins. For disconnected veins with a limited height, say
10 m, the overpressure is 0.3 bar, so the pressure within
veins of limited height is essentially lithostatic. Within a
source region, the veins are connected and the melt is
able to flow freely between the veins. When melt intrudes
a source region, the melt will flow from the lower part of
the region towards its upper part. In addition, the melt
may intrude veins in a horizontal direction. Both the
height and horizontal extension of a veined source region will, therefore, increase when it is supplied with
melt. During an eruption, the melt flows radially towards a feeder dike (Maaløe 1998). The region can deliver melt through a dike to an eruption as long as the
overpressure is sufficient to keep the walls of the dike
660
apart. A limiting overpressure is unknown, but the
lowest possible overpressure within the source region is
almost zero. During an eruption, the veins loose melt
and their thickness decreases. As a result, the veins either
become disconnected, or the flow resistance between the
veins becomes so large that the flow rate of melt between
the veins is negligible. Subsequently, an approximate
size of a veined source region is estimated by assuming
that the overpressure of the melt within the veins approaches zero during an eruption.
The variation in height of a source region under
pressure depends on the size and aspect ratio of the
source region. For the ellipsoid shape adopted here, the
volume is given by:
4
V ¼ pabc
3
ð38Þ
With the values suggested above for a, b, and c, V is
estimated by:
8
V ¼ pka3
3
ð39Þ
The fractional change in volume F of the source region is estimated for a given aspect ratio k and a reference size of the source region. The volume, V km3, of the
source region is then given by:
V ¼ 0:41=F
ð40Þ
assuming that 0.41 km3 of melt leaves the source region
during an eruption. Using Eq. (32), the half height c
of the source region is estimated by:
3V 1=3
c¼k
8pk
ð41Þ
The overpressure within the source region is obtained
from:
DPc ¼ 60c
ð42Þ
The value of the overpressure DP in Eqs. (36) and (37)
was initially chosen arbitrarily and will not generally
equal the value DPc obtained from Eq. (42). The overpressure in Eqs. (36) and (37), therefore, is iterated until
it equals DPc. The estimated variation in overpressure
with the aspect ratio is shown in Fig. 3a. The overpressure shown is the increase in pressure required for a
volume increase of the source region of 0.41 km3, with
an initial overpressure of zero. The length of the horizontal a axis decreases with increasing aspect ratio, but
the variation is small for aspect ratios between 0.02 and
0.1 (Fig. 2b). For an aspect ratio of k=0.1, the source
region is estimated as 16 km wide, 32 km long, and
1.6 km high. These dimensions appear acceptable when
considering the size of the plume (Ribe and Christensen
1999), but must be considered only indicative because
the accurate values of Poisson’s ratio, the shear modulus, and the aspect ratio are all unknown. In addition,
the volume that has left the source is taken to be equal to
the erupted volume, and this volume could be larger
than the volume erupted. However, a horizontal
extension of several kilometers is in agreement with the
Fig. 3 a The variation in overpressure with aspect ratio by elastic
deformation for a given volume increase of 0.41 km3, i.e., the
volume of lava erupted during the 1800–1801 Hualalai eruption
episode. b The change in horizontal width ‘‘a’’ of the source region
with aspect ratio for the same volume increase. There is a large
change in horizontal width for small aspect ratios, but less change
for larger aspect ratios. c The variation in erupted volume of melt
as a percentage of the total volume of the source region, for a
source region that only contains melt. The percentages are small for
elastic deformation and are about 0.1%
dimensions suggested by the flow dynamics of a veined
source region (Maaløe 1998). The volume change of the
source region cannot be estimated accurately from
the elastic volume changes, but some evidence for the
change in volume is obtained by considering the activity
ratio (226Ra/230Th). The volume constraints implied by
this activity ratio will be considered to give an estimate
of realistic erupted volume fractions.
Volume constraints from (226Ra/230Th)
Estimations of the activity ratio (226Ra/230Th) depend
on the ascent rate of the plume, the relationship between
the permeability and porosity, and, in particular, on the
661
applied distribution coefficients (Iwamori 1994; Sims
et al. 1999). At present, it is not possible to accurately
estimate activity ratios, but a detailed study of these
parameters and the Hawaiian activity ratios by Sims
et al. (1999) suggest that the initial activity ratio of alkali
basaltic melts that accumulate within the plume could be
about 2.00 or less. The measured (226Ra/230Th) activity
ratios for the post-shield 1800–1801 Hualalai eruption
range from 1.18 to 1.364, with an average value of 1.27
(Sims et al. 1999), which is similar to the ratios of alkali
olivine basalt from two other oceanic islands (Chabaux
and Allègre 1994). The same ratio for Hawaiian tholeiites is smaller and ranges from 1.101 to 1.194 (Cohen
and O’Nions 1993), whereas Reinitz and Turekian
(1991) obtained even lower ratios that ranged from 0.797
to 1.079 for the 1983–1985 flank eruptions of Kilauea.
During the post-shield stage, melt may initially accumulate and form a source region. The variation in the
average activity ratio for a source region that continuously accumulates melt is shown in Fig. 4 for an initial
activity ratio of. 2.00. The melt that enters the source
region has (226Ra/230Th)=2.00. After a while, the melt
becomes a mixture of old and new melt. With time, the
proportion of old melt increases so that the average
activity ratio decreases with time. If the source region
looses a certain fraction of melt and is replenished by the
same fraction every 1,000 years, the ratio initially decreases slightly, but thereafter it remains constant
(Fig. 4). After a number of eruptions, the source attains
a steady state where the average age of the melt remains
constant, and the activity ratio will, therefore, become
constant and is independent of the initial development of
the source. The period of time needed for a constant
activity ratio varies with the fraction of melt erupted,
but is less than 10 ka (Fig. 4), whereas the post-shield
period lasts for about 200 ka (Langenheim and Clague
1987). The calculation of the activity ratio with repeated
replenishments is shown in the Appendix. The steadyFig. 4 The variation in the
activity ratio (226Ra/230Th) of a
continuous influx of melt with
an activity ratio of 2.00 into a
source region that gradually
increases in volume. The
activity ratio decreases with
time because the average age of
the melt increases with time.
With repeated eruptions, the
source looses a fraction f of the
melt and receives the same
amount of melt during the
repose time. After several
eruptions and repose periods,
the average age of the melt
becomes constant and, therefore, the activity ratio becomes
constant. The activity ratio is
shown for the erupted fractions
0.05, 0.1, and 0.2
state activity ratio depends on the activity ratio of the
melt that enters the source and the fraction erupted. For
initial activity ratios between 1.5 and 2.0, the observed
range of activity ratios can be obtained for erupted
fractions between 0.05 and 0.7 (Fig. 5). These fractions
are equivalent to percentages between 5 and 70%, which
are two orders of magnitude larger than the percentages
estimated from the elastic deformation of a sill. Because
the initial activity ratio of the melt that enters the source
is unknown it is not possible to estimate the exact
fraction erupted. The important feature evident from
Fig. 5 is that the erupted fraction must be a large fraction of the melt present in the source region.
A veined source region
The relatively large volume fractions for the erupted
volumes of melt, which are obtained from the activity
ratio, show that the volume of melt within the source
region must be much smaller than the total volume of
the source region estimated from the elastic deformation. Taking the volume decrease by elastic deformation
as DVe=0.1%, the volume percentage of melt erupted as
DVa=20%, and a value consistent with an average activity ratio of 1.27 (Fig. 4), then the percentage of melt in
the source region is estimated as 100DVe/DVa=0.5%.
This percentage is of the same magnitude as the porosities estimated using the chromatographic porous flow
model of Sims et al. (1999). Therefore, the assumed
presence of veins does not result in an excessive amount
of melt. A percentage of melt of 0.5% is lower than the
3–10% volume percentage of melt in a veined source
that is estimated from the flow dynamics of a veined
source region (Maaløe 1998). However, considering the
approximate nature of the parameters used, the agreement is satisfactory. As mentioned above, the size of the
source region is also similar to the size estimated for a
662
Fig. 5 The variation in the
activity ratio (226Ra/230Th) with
the fraction of melt erupted
from a source. The variation is
shown for three different initial
activity ratios, 1.50, 1.75, and
2.00. The horizontal lines show
the range of activity ratios for
the 1800–1801 Hualalai eruption (Sims et al. 1999). The
possible range of erupted
fractions of melt for this range
of activity ratios is from 0.05
to 0.7. The range is from about
0.15 to 0.4 for an average
activity ratio of 1.27
Fig. 6 A hypothetical model
for the source region of a
plume. The formation of veins
is initiated at some depth
beneath the source. At a higher
level and degree of melting, the
veins becomes partly connected.
Above this level, the veins
becomes fully connected and a
source region is formed. The
curves labeled ‘‘s’’ show the
streamlines of the plume. A dike
is formed above the source
region when the overpressure
within the source region attains
a critical value
veined source region that supplies melt during an eruption.
The maximal height of the source region is probably
controlled by the overpressure needed for dike generation. According to Fig. 3a, this overpressure could be
within the range of 20–60 bars. This range is similar to
the range of overpressures estimated for subvolcanic
dike propagation, i.e., 20–80 bars (Aki et al. 1977;
Gudmundson 1983).
A necessary condition for melt accumulation, in
general, is that the ascent of melt is somehow prevented
for an extended period of time at some depth. A likely
region for magma accumulation within a plume is where
the flow of the plume becomes divergent and changes
from a vertical ascent to horizontal flow (Fig. 6). The
change in flow may be caused by a cooling effect of the
lithosphere on the upper part of the plume, which increases the viscosity of this part of the plume. The cooler
upper part of the plume may form a barrier for magma
ascent because the potential for fracture propagation
decreases with decreasing temperature.
If the plume displays elastic behavior then the yield
point, i.e., the stress at which plastic deformation is
initiated, must be larger than the maximum deviatory
stresses caused by the overpressure in the source region.
Evidence about the yield stress can be obtained from the
grain size of a rock, which decreases with increasing
deviatory stress. Estimates based on the grain sizes of
lherzolite nodules suggest that deviatory stresses of
about 40 bars at 100–200 km depth have an equilibration temperature of <1,400 C (Ave’Lallement et al.
1980). The yield point was thus at about 40 bars, which
indicates elastic behavior at smaller stresses. The suggested overpressures within the range of 20–60 bars for
an elastic deformation model may, therefore, appear
realistic.
663
Conclusions
By combining the results obtained for elastic deformation of a source region with the constraints of isotopic
secular disequilibria, it is concluded that the source region must be a veined source region that contains a small
volume fraction of melt and a large volume fraction of
residuum. The computed size of the source region is
consistent with the size estimated from the flow dynamics of a source region. The flow rate within the
source region required by the extrusion rates of eruptions is so large that the flow must take place in veins
rather than porous channels because the flow rate in
porous channels is too small (Maaløe 1998). Therefore, it
is suggested that the source region contain veins, but this
model does not exclude the presence of porous channels
beneath the source region. There may be a continuous
transition from interstitial migration via porous channels
to melt-filled veins with increasing degree of melting. The
present results suggest that the following model for
magma accumulation and eruption dynamics is possible.
The melt generated by partial melting is initially situated interstitially. At some degree of melting the melt
begins to accumulate, perhaps first in porous channels
and then in veins (Fig. 5). This accumulation may be
caused by compaction or sheared flow. The veins increase in frequency and length with increasing degree of
melting. The melt in the veins intrudes upwards and
begins to form a source region where the veins become
hydraulically connected (cf. Rubin 1998). With time, the
volume and overpressure of the melt in the source region
increases. A dike is formed above the source region
when the melt attains an overpressure of somewhat less
than about 60 bars. The dike ascends to the surface and
thereafter an eruption episode begins. During the eruption episode, the plume surrounding the source region
compacts by elastic deformation because the overpressure decreases in the source region. The elastic deformation retains an overpressure within the source region
so that eruptions can occur repeatedly during an eruption episode that can last a year or more. During the
eruption, the veins loose melt and begin to close up and
become disconnected from each other. With time, the
overpressure decreases and the pressure of the melt in
the veins approaches the lithostatic pressure throughout
the source region. After 10–50% of the melt has left the
source region, the overpressure becomes too small to
keep the dike walls apart, and the eruption episode
terminates. When the flow of melt in the dike comes to
an end, the melt in the dike consolidates and the dike
becomes sealed at some height above the source region.
Subsequently, new melt is accumulated in the source
region during a new repose period, and, thereafter, a
new eruption episode can begin.
The quantitative results obtained here must be considered approximate, but the dynamic relationships are
expected to have general validity for intermittent eruptions of basalts.
Acknowledgements The author thanks K.W.W. Sims and L.M.
Larsen for their valuable suggestions. This work is part of projects
SUBMAR and 128156/410, which are supported by the Norwegian
Research Council.
Appendix
The isotope activity ratio (226Ra/230Th) of an erupted
magma depends on the initial ratio of the magma entering the source in the plume, and the mean storage
time of the magma in this source. Initially, a source region is supplied with melt from time zero to time t1,
when the first eruption occurs. During the subsequent
repose time, the source is continuously supplied with
melt from the plume until the amount of melt equals f, at
which stage a new eruption occurs.
Let the decay constant for the parent isotope Th be k,
and the initial concentration be NoTh, then the amount
NTh of Th is given by (Ivanovich 1982):
o kt
NTh ¼ NTh
e
ðA1Þ
where t is the period of time of decay. Let the decay
constant of the daughter isotope be l, and the initial
concentration NoRa. The amount of Ra is then given by
(Ivanovich 1982):
NRa ¼
o kNTh
o lt
e
ekt elt þ NRa
lk
ðA2Þ
The decay constants for 226Ra and 230Th are 4.33 · 10–4
and 9.217 · 10–6 year–1, respectively (Ivanovich 1982).
The initial concentrations are not known, but the initial
activity ratio is assumed. Let this initial activity ratio be
Ro, so that:
o
lNRa
o
o ¼ R
kNTh
ðA3Þ
Because only the activity ratio is of interest one can let
NoTh=1, so that NoRa=Rok/l.
Initially, the source is created by a constant influx of
melt into the source. Let the influx per year be constant
and equal to w m3. The volume at time t is then wt. The
present calculations is simplified without loosing generality by taking w=1. The amount of Th and Ra at time t
by a constant influx of melt is then estimated from:
in
NTh
¼
1
t
Zt
NTh dt
ðA4Þ
NRa dt
ðA5Þ
0
in
NRa
¼
1
t
Zt
0
The results is then:
in
NTh
¼
in
NRa
¼
o NTh
1 ekt
t
ðA6Þ
o
1 kNTh
1 lt
1
N0
ðe 1Þ þ ð1 ekt Þ þ Ra ð1 elt Þ
t ðl kÞ l
k
l
ðA7Þ
664
If the first eruption occurs at t1, then the initial accumulation takes place during a period of time equal to
t1, and t in Eqs. (A6) and (A7) equals t1. The total
volume accumulated then equals W=wt1=t1.
At eruption Q, the amount of isotopes in the initial
volume has decayed during a period of time of (Q–1)Dt,
where Dt is the repose time. The amounts are then
estimated from:
in
in ðQ1ÞDtÞk
NTh
ðQÞ ¼ NTh
e
in
in
ðQÞ ¼ NTh
NRa
k ðQ1ÞDtk
e
eðQ1ÞDtl
lk
ðA8Þ
in ðQ1ÞDtl
þ NRa
e
226
Ra
230 Th
¼
av
lNRa
av
kNTh
ðA19Þ
The melt accumulates for some time in the source region before the first eruption takes place. The variation
in the activity ratio (226Ra/230Th) during the initial
accumulation is estimated using Eqs. (A6) and (A7),
and is shown in Fig. 4 for an activity ratio equal to
2.00 of the magma entering the source region. The
activity ratio of the accumulated magma approaches
1.00 with time. The variation in the activity ratio for a
replenished source for an initial ratio of 2.00 is also
shown in Fig. 4.
ðA9Þ
At each eruption, a fraction f of the remaining initial
melt is surrendered, so that the remaining volume at
eruption Q becomes:
V in ðQÞ ¼ W ð1 f ÞðQ1Þ
ðA10Þ
During the repose periods of length Dt, the source has a
continuous influx of melt. At the end of each repose
time, the added batch of melt contains the following
amounts of Th and Ra:
BoTh ¼
BoRa ¼
o
NTh
ð1 ekDt Þ
kDt
ðA11Þ
o 1 kNTh
1 lDt
1
N0
1Þ þ ð1 ekDt Þ þ Ra ð1 elDt Þ
ðe
l
Dt l k l
k
ðA12Þ
The amount of melt received during the repose period is:
DV ¼ mDt ¼ fW
ðA13Þ
where f=Dt/t1. At the time of eruption Q, the batches of
melt replenishing the source during the repose time after
eruption N will have decayed during (Q – N)Dt years. The
amounts of Th and Ra at the time of eruption Q for batch
N is given by:
BTh ðNÞ ¼ BoTh ekðQN ÞDt
BRa ðN Þ ¼ BoTh
ðA14Þ
k
ðekðQN ÞDt elðQN ÞDt þ BoRa elðQN ÞDt
lk
ðA15Þ
The volumes of each batch entered decreases during subsequent eruptions to:
vðN Þ ¼ Wf ð1 f ÞðQN Þ
ðA16Þ
The total volume of source, W, remains constant, so the
average amounts at the time of eruption Q is estimated
from:
"
#
Q
X
1
in
V o ðQÞNTh
ðQÞ þ
V ðN ÞBTh ðN Þ
W
2
"
#
Q
X
1
o
¼
ðQÞ þ
V ðN ÞBRa ðN Þ
V o ðQÞNRa
W
2
av
NTh
¼
ðA17Þ
av
NRa
ðA18Þ
The activity ratio of the erupted magma is then given
by:
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