Bull Volcanol (2000) 62 : 153±170
Springer-Verlag 2000
R ESEAR CH A RT IC L E
O. Melnik
Dynamics of two-phase conduit flow of high-viscosity gas-saturated
magma: large variations of sustained explosive eruption intensity
Received: 25 March 1998 / Accepted: 11 December 1999
Abstract The ascent of gas-saturated magma in a conduit can lead to the transition from the laminar flow
of a bubble-rich melt to the turbulent flow of particlerich hot gas in upper part of the conduit. This process
is investigated with a help of a disequilibrium twophase flow model for steady and unsteady conditions.
For the description of different gas-magma flow
regimes in the conduit separated sets of equations are
used. The main difference from previous conduit flow
models is the consideration of the pressure difference
between a growing bubble and the surrounding melt.
For a bubbly liquid to evolve into a gas-particle flow a
critical overpressure value must be exceeded. This
transition region is simulated by means of a discontinuity, namely by a fragmentation wave. The magma
flow calculations are carried out for a given conduit
length and overpressure between the magma chamber
and the atmosphere. The steady solution of the
boundary problem is not unique and there are up to
five distinct steady regimes corresponding to fixed
eruption conditions. Within the framework of the quasi-steady approach (governing parameters being varied
monotonically) the transition from one regime to
another can take place suddenly and is accompanied
by the fundamental restructuring of the conduit flow,
resulting in rapid or abrupt changes in the intensity of
explosive eruptions. Abrupt intensification of an
explosive eruption occurs when the chamber pressure
becomes sufficiently less than saturation pressure, and
therefore corresponds to the case of shallow-depth
Editorial responsibility: D.A. Swanson
Oleg Melnik
Institute of Mechanics, Moscow State University, 1-112-b,
Mitchurinski pr., 119192, Moscow, Russia
Fax: +7-095-939-01-65
e-mail: [email protected]
Present address:
Oleg Melnik
Department of Earth Sciences, University of Bristol,
Wills Memorial Building, Queen's Road, Bristol, BS8 1RJ, UK
magma chamber and high initial water content. A
regime with minimum flow rate may relate to the
growth of a lava dome following the explosive phase
of eruption. These models can explain geological
observations that imply large and sudden changes of
discharge rate in large-magnitude explosive eruptions,
particularly at the transition between plinian phase
and ignimbrite formation.
Key words Volcanic eruption ´ Conduit flow model ´
Magma chamber ´ Bubbly melt ´ Gas-particle
dispersion ´ Fragmentation ´ Bubble overpressure ´
Discharge rate
Introduction
During ascent of gas-saturated high-viscosity magma
from a magma chamber to the Earth's surface exsolution of dissolved gas leads to bubble formation and
subsequent growth. Under some conditions the viscous
bubbly flow transforms into a turbulent gas-particle
mixture accelerating to velocities up to several hundred metres per second. Figure 1 illustrates the flow
regimes and pressure distribution, corresponding to
this process and the nomenclature used in this paper.
The typical range of physical parameters for such sustained explosive eruptions is wide. Key factors that
must be considered are the mass concentration of dissolved gas and the high viscosity of the magma. In this
paper both dimensional and dimensionless variables
are used. We highlight dimensional variables with
apostrophe to distinguish them from dimensionless.
The parameters considered in this paper are: initial
weight concentration of dissolved gas (mostly H2O)
co = 3±7 wt. %, magma ascent velocity V9 = 0.1±5 m/s
(recalculated for the density of unvesiculated magma),
magma chamber depth L9 = 3±20 km and viscosity of
magma m9 = 104 ± 108Pa s, with higher values of viscosity corresponding to lower concentration of dissolved
gas. In the case of a volcanic conduit being a fissure,
154
Fig. 1 Schematic view of the flow in the conduit of the volcano,
corresponding to sustained explosive eruptions. On the left ±
typical pressure distribution along the conduit (thick curve), mixture weight (dotted curve) and conduit resistance (dashed curve).
For the homogeneous flow the weight and conduit resistance are
constant; therefore, pressure drops down linearly with depth. In
bubbly flow weight decreases but conduit resistance increases
dramatically due to the exsolution of dissolved gas, and pressure
drops down faster than in homogeneous regime. In gas-particle
dispersion flow weight is small and conduit resistance is negligible. Pressure drops down weakly. Position of the fragmentation
level determines average weight and conduit resistance, and,
therefore, the discharge rate
the typical surface vent width, d9, is approximately a
few metres to a few tens of metres. For a cylindrical
conduit d9 is in a range of tens to hundreds metres.
The values of governing parameters in some well-characterised eruptions are reproduced in Table 1.
Table 1 Dimensional parameters for eruptions of Mount
St. Helens and Santorini
The simplest steady-state one-dimensional conduit
models of sustained explosive eruptions (Wilson et al.
1980; Wilson 1980; Slezin 1983; Woods 1995; Sparks et
al. 1997) consist of the equations of mass conservation
for liquid and gaseous components, the momentum
equation for the mixture as a whole, and the equations
of state in the bubble melt and gas-particle dispersion
zones. The principal differences between published
models arise concerning the pressure distribution
along the conduit and description of the transition
mechanism from a bubbly fluid to a gas dispersion.
Thus, in Wilson et al. (1980) and Wilson (1980) lithostatic pressure distribution is assumed. These studies
also assume that fragmentation occurs when the gas
bubbles coalesce on reaching some critical close packing state. In the model of Slezin (1983) the walls of
the conduit are assumed to be rigid; therefore, pressure depends only on flow dynamics. Instead of a fragmentation surface Slezin (1983) introduces a region of
so-called ªbreaking down foamº, which is simulated
by a liquid porous medium, in which gas and melt
have they own velocities. The frictional resisting force
within the conduit in this foam is calculated using the
Poiseuille law where the viscosity is that of the melt.
The slip velocity between the particles and a gas is
determined from the condition of dynamical equilibrium, as in dilute gas dispersions.
In Barmin and Melnik (1990) a more complete
model was developed which takes into account disequilibrium effects in mass transfer between a growing
bubble and surrounding melt due to the low value of
gas diffusion coefficient. The model also takes account
of magma viscosity dependence on concentration of
dissolved gas, the temperature and the volume fraction of bubbles, as well as heat transfer processes. The
inertia of the particles in the gas-dispersion regime
was also considered. The slip velocity of free gas in
the fragmentation zone is calculated from Darcy's law
for a porous medium with variable permeability.
In Slezin (1983), Slezin (1991) and Barmin and
Melnik (1990) the dependence of magma discharge
rate on the eruption parameters was considered for a
fixed chamber pressure and conduit length. Up to
three different regimes of magma ascent were recognised corresponding to the same values of chamber
pressure and other fixed parameters of eruption. In a
Eruption
co
L9 (km)
d9 (m)
m9 (Pa s)
Q9 (kg/s)
Mount St. Helensa
(18 May 1980)
4.6
7
130
2.3 ” 105
~ 2 ” 107
Santorini
(BC 1650)
5.5b
5±6c
112
2 ” 107 d
2.5 ” 108 e
a
Carey and Sigurdsson (1985)
Sigurdsson et al. (1990)
c
Cottrell et al. (1999)
d
Kaminski and Jaupart (1997), value calculated for 1.8 wt. % H2O, in the approximation (Eq. (3))
m90 is 108 Pa s
e
Sparks and Wilson (1990), value of discharge rate for initial Plinian phase
b
155
low-intensity regime the conduit is filled by bubbly
melt. The effect of frictional resistance can be neglected in comparison with that of the mixture weight,
so the pressure drop between the chamber and the
atmosphere is controlled by the high average weight
of the mixture. In an intensive regime the bubbly liquid occupies only the lowermost portion of the conduit and the conduit resistance is large, due to the
high mixture velocity. In an intermediate regime both
conduit friction and mixture weight are important.
The gradual variation of the parameters can lead to a
sudden intensification or weakening of the eruption,
due to the transition from one regime to another.
These studies demonstrated nonuniqueness in the
solutions for conduit flows. Nonunique solutions for
conduit flow were also obtained in Jaupart and
Allegre (1991) and Woods and Koyaguchi (1994)
where the effect of gas loss to the surrounding rock
was investigated.
In the numerical model of Dobran (1992) the distribution of key parameters along the conduit was
obtained for constant cross section of the conduit and
sonic flow velocity at the outlet of the conduit
(choked condition). The velocity disequilibrium
between gas and liquid were taken into account both
in bubble-melt and gas-particle dispersion regimes.
Dobran (1992) demonstrated that the pressure distribution could be far from the lithostatic, especially in
the gas-particle flow regime.
In the models described above pressures in the
growing bubble and the surrounding melt were
assumed to be the same. The applicability of these
models is limited to relatively slow melt motions and
low magma viscosity. The pressure in the growing
bubble can be obtained from the Rayleigh-Lamb
equation (see Scriven 1959):
2s0
3
a_ 0
…1†
p0g ˆ p0m ‡ 0 ‡ 4m0 0 ‡ r0m a_ 02 ‡ a0 a0
a
a
2
Here pg9 and pm9 are internal bubble and ambient pressures, respectively, m9 is the dynamic viscosity, s9 is
the interfacial tension coefficient, a9 is bubble radius
and rm9 is the density of surrounding melt. The inertia
term (fourth) is negligible in almost all cases of volcanological interest, and the surface tension term (second) is large only for bubble radius close to an
embryo radius. Therefore, the main cause of pressure
disequilibrium is the viscous resistance of surrounding
melt.
McBirney and Murase (1971) discussed the possibility of accumulating the potential energy of a compressed gas during the process of bubble growth in
magma. Sparks (1978) investigated the ascent of a single bubble in magma and demonstrated that the viscous term in bubble growth becomes large in sustained explosive eruptions for viscosities of 108 Pa s or
above. In Barclay et al. (1995) the analytical solutions
for single bubble growth in suddenly decompressed
magma both in cases of infinite volume of surrounding
melt and a bubble surrounding by a viscous shell. It
was shown that pressure relaxation time depends
strongly on the relative thickness of the shell. For viscosity 107 Pa s an analytical solution gives the relaxation time approximately 0.8 s for the ratio of bubble
radius to the shell thickness equal 20 and approximately 103 s in cases of infinite shell thickness. The
shell solution is based on a finite velocity in the external shell boundary; therefore, it is not applicable to
the case of high bubble volume concentration,
because, from the condition of symmetry, the liquid
velocity on the external shell boundary must be equal
to zero. As a result, the response times predicted by
shell theory are smaller than real times. Only solution
of the three-dimensional problem can clarify the situation completely. There are several more complicated
bubble growth models (Toramory 1989; Proussevitch
et al. 1993; Hurwitz and Navon 1994). The nonequilibrium diffusion profiles and melt viscosity dependence
on the concentration of dissolved gas were taken into
account there in the case of radial symmetry of flow.
Bennett (1976) proposed that the eruption process
be considered qualitatively as the propagation of a
rarefaction wave into the volcanic conduit by analogy
with processes taking place in the high-pressure
chambers of shock tubes. Alidibirov (1987) considered
the propagation of a fragmentation wave through a
solidified magma containing gas bubbles at high pressure, following the approach of Khristianovich (1979),
who applied it to blasts in coal banks. In Alidibirov
(1988) the conditions leading to the formation of a
magma containing overpressurised bubbles and its
explosive fragmentation were examined. In the first
stage viscous magma ascends rapidly (in comparison
with gas diffusion time) to refill a part of conduit,
emptied in the previous explosion. Then solidification
of the interstitial melt occurs due to the loss of gas to
growing bubbles. As the interstitial melt solidifies, gas
flows into the bubble leading to an increase in pressure in the bubble. When the pressure rises to a level
higher than a critical value determined by the tensile
strength, fragmentation occurs and propagates down
along the conduit as a shock wave. The potential energy, accumulated in overpressured bubbles, is released
by fragmentation and leads to rapid evacuation of the
disintegrated magma. This process then repeats itself.
This theory predicts separate volcanic explosions alternating with pauses during which the potential energy
of gas accumulates in pressuring bubbles. Although
this model is more applicable to vulcanian eruptions
than sustained explosive eruptions, some of these principles can be developed in the context of sustained
explosive activity. Decompression of the ascending
magma can result in overpressures developing in the
bubbles and a similar kind of fragmentation wave.
Recently, a new approach to fragmentation was
developed by Papale (1999). He assumed Maxwell
behaviour of magma and obtains a fragmentation
156
criterion that the elongation strain rate must be higher
than structural relaxation time of the fluid. This criterion can be written in the form:
dV 0
1
G01
>
k
ˆ
k
r
r
dx0
m0
t0r
Here dV9/dx9 is the elongation strain rate; tr9 magma
9 the elastic modulus at
structural relaxation time; G?
infinite frequency; and kr is a proportional coefficient
determined from experiments by Webb and Dingwell
(1990). Mathematically this criterion represents the
same relation between the critical value of the elongation strain rate and magma viscosity as those, developed by Barmin and Melnik (1993), but uses a different physical approach to fragmentation. Papale (1999)
shows the possibility of different steady-state solutions
which correspond to the same set of governing parameters with discharge rate difference of more than two
orders of magnitude. The difference rises from different assumption on boundary condition on the outlet
of the conduit: low-intensity regime corresponds to
subsonic flow conditions; for high-intensity regime
flow is choked.
In a series of papers presented by Papale and coauthors (Papale and Dobran 1994; Papale et al. 1998;
Neri et al. 1998; Papale and Polacci 1999) significantly large variations in magma discharge rate were
explained as a consequence of gas content and chemical composition variation in ascending magma. For
example, in Papale and Dobran (1994) change in
composition of erupting magma resulted in changing
magma viscosity and volatile solubility. Consequently,
the discharge rate changes from 4.4 ” 107 to
1.6 ” 107 kg/s without changes in conduit diameter. In
Papale and Polacci (1999) the role of carbon dioxide
in the dynamics of magma ascent was studied. They
showed that increase in CO2 content from 0 to 50 %
from total volatile content decreases discharge rate
by 2.8 times as solubility of CO2 and H2O mixture is
strongly different from pure water solubility. In
Papale et al. (1998) the role of magma composition,
water content, and crystal content on the dynamics of
explosive eruptions was investigated. The results of
the modelling show complex dependence of the flow
variables on governing parameters. The common
compositional trend in explosive eruptions is characterised by chemically evolved, water-richer and crystal-poorer magma erupted first followed by more
mafic crystal-rich magmas. The increase in mass flow
rate results from the increasing density of erupting
products.
The role of chemical composition and amount of
dissolved volatiles on discharge rate and other parameters of explosive eruption is studied by Papale et al.
(1998) in detail. The increase in amount of initially
dissolved water from 1.5 to 6 wt. % leads to an
increase in discharge rate of approximately an order
of magnitude. Change of the chemical composition
from rhyolite to dacite causes an increase in the discharge rate of more than two times. The influence of
crystals is in decreasing discharge rate for fixed chemical composition. Neri et al (1998) took input parameters from calculations done by Papale et al. (1998)
and studied the influence of magma chemical composition on dispersion of pyroclastic in the atmosphere.
The new contribution of the model presented in
this paper is the consideration of pressure disequilibrium between growing bubbles and interstitial melt. A
new condition for fragmentation is proposed based on
the critical overpressure between bubbles and liquid.
Both steady state and unsteady cases are investigated.
The implications of nonunique character of the solutions for interpreting sustained explosive eruptions
and pyroclastic sequences are discussed.
Formulation of the problem of high-viscosity
gas-saturated magma flow
Physical properties of magma
As magma ascends through the conduit, the pressure
falls down from a value of the order of several kilobars to atmospheric. This process causes exsolution of
a dissolved gas, bubble nucleation and subsequent
growth. The relation between the equilibrium dissolved gas (mainly H2O) concentration c and the pressure p9 is approximated by the following experimental
formula:
p
…2†
c ˆ k0p p0 ; k0p ˆ …4:11 6:33†10 6 Pa 1:2 ;
The coefficient kp9 depends on the type of magma and
the composition of the dissolved gas. The temperature
dependence of the solubility is much weaker for the
pressures up to 200 MPa (Lebedev and Khitarov 1979;
Burnham 1979; Stolper 1982). General discussion on
the role of solubility law on the eruption dynamics is
given by Tait et al. (1989).
The rheology of a magmatic melt is complex and
depends on many factors: temperature; pressure; composition; and dissolved gas content. The magma is
assumed to be a Newtonian fluid with the viscosity
depending on dissolved gas concentration and temperature:
0
A
0
0
0
m ˆ m0 exp 0 0 …exp… B c† 1† ;
RT
…3†
0
7
9
m0 10 10 Pa s
Here m09 is the viscosity of the ªdryº magma, T9 temperature, R9 gas constant, A9 and B9 constants,
depending on the magma type. In particular we
choose A9/R9T9 = 15.6, B9 = 11.3 as typical of rhyolitic
melts. Equation (3) expresses that the logarithm of
viscosity has an exponential dependence on dissolved
157
gas concentration. Equation (3) was obtained as an
approximation of viscosity measurements data for
rhyolitic gas-rich magma (Lebedev and Khitarov
1979). More recent data (Dingwell et al. 1996; Richet
et al. 1996) show that melt viscosity increases rapidly
when the concentration of dissolved gas c decreases
from 1 wt. % to zero. However, when c>1 wt. % (typical concentration before the fragmentation; Eq. (3))
has a good agreement with these data. Equation (3)
does not take into account the influence of crystals
and bubbles on the viscosity of mixture. High crystal
content in magma can increase the viscosity by several
orders of magnitude (Lejenue and Richet 1995), but
the influence of crystals on volcanic eruption dynamics
is beyond the scope of this paper. The influence of
bubbles is much smaller. The increase in volume concentration of bubbles from 0 to 50 % decreases viscosity only by a factor of 3 (Lejeune et al. 1999).
Physical approach to magma flow description
The problem of magma ascent along a volcanic conduit from a chamber located at a depth L9 is considered. At the chamber outlet pressure, temperature
and the bubble volume concentration in the escaping
magma are given. If the chamber pressure is greater
than the saturation pressure p09, then nucleation occurs
in the conduit; otherwise, the bubble volume concentration in the chamber is calculated assuming equilibrium. During magma ascent along the conduit the
pressure decreases and the velocity of the bubble-rich
melt tends to increase due to expansion of decompressing bubbles. Magma viscosity also increases due
to the decrease of dissolved gas (see Eq. (3)). As a
result, the time for pressure relaxation between growing bubbles and magma increases with distance up the
conduit; therefore, magma pressure decreases faster
than the pressure in growing bubbles. It is assumed
that the bubbly mixture fragments, when the overpressure in the bubbles exceeds a critical value Dp9*. The
particular values of Dp9* is discussed below. Melt partitions between bubbles break down and the medium
transforms into a gas-particle dispersion. It is shown
later that the length of the region where the pressure
difference between bubbles and surrounding melt is
significant is small in comparison with the conduit
length; thus, fragmentation is visualised as taking
place in a thin flow zone, which is investigated as a
discontinuity: a fragmentation wave.
After the fragmentation, the gas-particle mixture
accelerates to velocities comparable to the velocity of
sound. If the outgoing flow is subsonic, the outlet
pressure is equal to atmospheric pressure; otherwise,
for conduit of constant cross-section area the equality
of the flow velocity to the local velocity of sound must
be taken as the boundary condition (the choked flow
condition; see Shapiro 1954; Woods and Bower 1995).
For conduits becoming larger with height the transi-
tion to a supersonic regime can occur inside the conduit (Shapiro 1954; Wilson et al. 1980).
The bubbly flow model
The following assumptions are made after comparison
of values of the terms in general multiphase flow system (Nigmatulin 1979):
1. The velocity difference between magma and gas is
negligible in comparison with the ascent velocity.
2. Temperature variations in the bubbly regime are
small due to the high thermal capacity of magma
and low expansion velocity.
3. Nucleation is assumed to be instantaneous (Sparks
1978; Hurwitz and Navon 1994) and no new bubbles appear after an initial nucleation event.
4. The mass transfer between dissolved gas and bubbles maintains the system at equilibrium. This
assumption is not applicable to the conditions close
to fragmentation level, because diffusion coefficient
becomes small when the melt viscosity increases
dramatically. However, the contribution of diffusion to the volume changes in growing bubble is
small in comparison with decompression in the
upper part of the bubbly flow. Pressure nonequilibrium between bubble and melt phase also changes
the intensity of mass transfer. As is shown below,
pressure difference is negligibly small in almost all
flow except the vicinity of fragmentation level;
therefore, it does not contribute significantly in
mass transfer rate.
The one-dimensional equations for bubbly flow are
developed in dimensionless form. As a characteristic
density we take the density of magma without bubbles
r09; as a characteristic pressure, the saturation pressure
p90 as a viscosity, viscosity of dry magma, and as a
velocity the typical ascent velocity of unvesiculated
magma V09 = Q09(r09 Sc9) ± 1, where Q09 is average mass
discharge rate, Sc9 is the cross-sectional area of the
conduit.
The following dimensionless variables are introduced as:
rm ˆ
Vˆ
r0g
r0m
p0
x0
;
r
ˆ
; pˆ 0 ; xˆ 0 ;
g
0
0
r0
rg0
p0
x0
V0
a0
n0
m0
;
a
ˆ
;
n
ˆ
;
m
ˆ
V00
a00
n00
m00
p90 = (c0/kp9)2; r9g0 = p09/R9T9; x09 = p09/r09g9;
a09 = (3n09/4p) ± 1/3
Here g9 is gravitational acceleration, and the characteristic length of the conduit x09 corresponds to the
depth equivalent at which the saturation pressure is
reached for a column of unvesiculated magma.
In these notations the system of bubble flow equations is written in the following form:
158
@rg @rg V
@rm @rm V
@n @nV
ˆ 0;
ˆ 0;
‡
ˆ 0 …a†
‡
‡
@x
@x
@t
@x
@t
@t
2
@rV @rV ‡ Eups
ˆ Eu…r ‡ lArmV †
‡
@x
@t
@a
@a
a
‡V
ˆ Ca
pg
@t
@x
4m
ps ˆ apg ‡ …1
rm ˆ … 1
Ar ˆ
a†…1
m00 V 0
r00 g0 d02
pm
…b†
…c†
p
a†pm ; pg ˆ r0g ; c ˆ c0 pg ; a ˆ a3 n; …d†
c†; rg ˆ dr0g a ‡ …1
; Eu ˆ
p00
02 ; d ˆ
r00 V0
p00
r00 R0 T00
a†c;
; Ca ˆ
p00 x00 …e†
;
m00 V00
(4)
This system of equations (Eq. (4)) consists of the continuity equations for the components of the mixture
and number density of bubbles (a), the momentum
equation for the mixture as a whole (b), as well as the
Rayleigh-Lamb equation for bubble growth (c); (d)
represents definitions for pressures and densities and
(e) the notations for nondimensional parameters.
Gravity forces and the conduit resistance (in Poiseuille
form, l is a shape factor of the conduit equal 32 for
cylindrical conduit and 12 for a fissure) in momentum
equation are taken into account. As a consequence of
the bubbles and viscous melt having equal velocities,
it is possible to express densities of components in
such a way as to avoid mass transfer terms in the continuity equations (a).
All simplifications and assumptions in the development of Eq. (4) are quite usual for conduit bubbly
flows, except the usage of Reyleigh-Lamb equation
for bubble overpressure growth. This equation is
obtained from equations describing spherically symmetrical individual bubble growth in an infinite incompressible liquid. In Hurwitz and Navon (1994) and
Navon and Lyakhovski (1998) the modification of this
equation is proposed to model the presence of other
neighbouring bubbles for the case when foam has a
high volume concentration of bubbles. They solve 1D
spherically symmetric problem of the growth of a bubble surrounded by a shell of viscous liquid. Unfortunately, this solution does not represent the real situation, because the radial velocity on the outer shell
boundary does not tend to zero, which is needed from
the conditions of symmetry of the bubbly grid; therefore, this modification gives a wrong relation between
the bubble overpressure and its growth rate. To get
more accurate pressure distributions around the growing bubble the full 3D equation for momentum should
be solved and averaged.
Gas-particle mixture flow
Writing the system of equations for the gas-particle
dispersion the following simplifying assumptions are
made.
1. The particle temperature is assumed to be constant
and equal to the temperature of the surrounding
gas. This is justified by a high heat capacity, large
mass concentration and small size of particles. Second, t
2. The variation of the gas temperature is neglected
since there is an efficient heat exchange between
the gas and the particles; therefore, the flow in the
conduit is assumed to be isothermal. At the outlet
of the volcanic conduit, where the particle concentration is low and the expansion rate of the gas is
substantial, the gas temperature will have decreased
by not more than 3±5 % (Barmin and Melnik
1993). A single particle size dispersion is considered.
The gas dispersion dynamics is described by the
continuity and momentum equations for each component with allowance for the force of interaction Fgm
between gas and particles and the equation of state
for the gas phase. Introducing the partial densities rm
and rg of the particles and the gas, respectively, this
system of equations is given in the following form:
@rg @rg Vg
@rm @rm Vm
‡
ˆ 0;
‡
ˆ0
@t
@x
@t
@x
2
@rm Vm @rm Vm
‡
ˆ Eu Fgm rm
@t
@x
@rg Vg @rg Vg2 ‡ Eu p
ˆ EuFgm
‡
@x
@t
2
V
rg
Vg
m
pˆ
; Fgm ˆ Cm rm ; rg
d…1 rm †
2
rm ˆ … 1
…5†
a†; rg ˆ ar0g
No further gas exolution occurs from particles formed
in the fragmentation process. This is reasonable
because the residence time of particle in the conduit is
small in comparison with gas diffusion time. The interaction force between gas and particles depends on the
volume concentration of particles (Nigmatulin 1979).
Fragmentation wave
The thickness of the transition zone between the bubbly melt and the gas-particle dispersion is estimated as
follows: the lower boundary of fragmentation zone is
defined as the level at which the pressure difference
between growing bubbles and melt exceeds a critical
value. The upper boundary of the fragmentation zone
is defined by a gas velocity such that particles become
just supported by the upward gas flow (Fgm = rm).
After the excess pressure in the bubble exceeds the
159
critical threshold, some of the partitions between the
bubbles are considered to break down, and gas pressure falls down to a value between bubble pressure pg
and melt pressure pm. Consequently, gas density
decreases and gas velocity increases. Gas acceleration
causes additional partitions to break. As a result, the
medium evolves into a gas-particle dispersion.
From Eq. (4) we obtain an equation which
describes the growth of overpressure between a bubble and the magma:
da
@V
1 4m
a
r
…7†
p ˆ Ca
3a
dr
@x
Coefficient Ca is large, so in most of flow the pressure
difference is small. During magma ascent along the
conduit viscosity increases strongly with exsolution of
dissolved gas. Therefore, most of the pressure drop is
accommodated in a narrow region just below the fragmentation zone. In this region magma has large accelerations and bubble overpressures become significant.
The fragmentation condition is defined by Eq. (6)
when the value of Dp reaches some threshold value
and shows that fragmentation occurs when the mixture
reaches a critical value of acceleration. Papale (1999)
suggested a mechanism for fragmentation based on a
threshold strain rate which is similar mathematically
to Eq. (6). It is based on consideration of Maxwell
type magma rheology and experimental data of the
critical strain rate for silicate melts of different composition from Webb and Dingwell (1990), and it
describes the transition form viscous to brittle behaviour.
The choice of a critical Dp will be at a value when
the magma breaks apart due to Dp exceeding the tensile strength (Alidibirov and Dingwell 1996) or when
the accelerating forces tear the magma apart in a ductile manner. The details of the fragmentation mechanism are still not well established (Mader 1998), but
the calculations are not strongly sensitive to the choice
of Dp9* due to the narrowness of the region where
large bubble overpressures develop in comparison
with the total length of the conduit. For example, in
this paper Dp9* is always chosen as 1 MPa. The results
of calculations give an exponential increase in Dp9
over a distance of approximately 10 m. A choice of
10 MPa would only increase the distance by a few
metres, so the changes of the overall flow resistance
would be small. The experiments by Alidibirov and
Dingwell (1996) imply that a few MPa is a typical tensile strength of high-temperature magmatic materials.
Formally, the vertical velocity gradient can be estimated from the momentum equation because in the
steady case the velocity gradient is proportional to the
inertia term. The contribution of the inertia term is
given by 1/Eu. Its value for the magma flow in the
region close to saturation is only approximately 10 ± 4,
so the inertia of the medium is not important. Near
the fragmentation level Eu is approximately 10 ± 2.
Therefore, during the ascent of gas-saturated magma
the overpressure value increases both due to the rise
in melt viscosity and due to increase of vertical acceleration. The results of the calculations show that the
overpressure in a growing bubble increases exponentially with height, and that the length of the zone,
where most of the overpressure increase takes place,
is only approximately 10 m for Dp = 1 MPa.
After the start of fragmentation, particles and gas
have the same velocities. We now consider the equation of momentum for a single particle with radius rp9
in dimensional form to estimate the length where the
drag force of gas just supports the particle.
2
0
0
0
V
‡
V
3Cm r0g 0
g
m
dVm
ˆ g0 ‡ z 0
; z0 ˆ
…7†
dt
2
8r0 r0
0 p
0
t ˆ
0; Vg0
ˆ
0
Vm
;
Vg0
ˆ const
Here Cm is a drag coefficient, for high Reynolds
number of particle Cm " 0.5. The solution of this equation is given by the following formula:
s
r!
0
2g
z0 g0 0
0
…t0 † ˆ Vg0
t
Vm
…8†
0 tanh
2
z
We now estimate the characteristic length over which
particle velocity reaches a value of 90 % of the velocity V9F = V9g ± (2g9/z9)1/2, needed to keep particle in
dynamic equilibrium with gas flow. The characteristic
time t9F from Eq. (8) is equal to
s!
s
2
z0
0
0
…9†
tF ˆ
0 0 atanh 0:9 ‡ 0:1Vg
2g0
zg
The estimate for r9p = 1 cm, r9g0 = 10 kg/m3,
r9m = 2500 kg/m3 V9g = 2 m/s gives t9F approximately
1.8 s; therefore, the characteristic length L9F = V9g t9F is
approximately a few metres.
Since the thickness of the transition zone between
the bubbly liquid and the gas-particle dispersion is
small in comparison with the characteristic dimension
of the problem, this region is simulated by a discontinuity, namely a fragmentation wave.
Under the assumptions made above concerning the
flow in the bubbly liquid and the gas-particle dispersion, the conservation laws for the fragmentation wave
can be written in a following form:
(1-a ± )(S ± V ± ) = r+m(S ± V+m)
rg0 ± a ± (S ± V ± ) = r+g(S ± V+g)
Eu ps± + r± V±(S ± V ± ) = Eup+
(10)
+r+gV+g(S ± V+g) + r+mV+m(S ± V+m)
±
F+gm = r+m; pg± ± pm
= Dp*
Here superscripts ± and + correspond to bubbly liquid and gas-particle mixture, respectively, and S is the
velocity of the fragmentation wave. The first two rela-
160
tions express the mass conservation laws for the liquid
and gas phases. The third expression is the momentum
conservation for the mixture as a whole, and the
fourth is the particle support condition at the outlet of
the fragmentation zone (written in place of the
momentum equation for one of the phases). The fifth
expression is the condition for the fragmentation onset
(the overpressure in the bubble reaches the critical
value).
The boundary conditions and numerical methods
As the lower boundary condition, either the chamber
pressure or the chamber pressure variation with time in
the unsteady case must be specified. At the conduit outlet we assign the pressure to be equal to atmospheric
pressure if the flow is subsonic, or the velocity to be
equal to the local velocity of sound for chocked conditions. Supersonic regimes are not considered in this
paper. It is necessary to define the sound velocity for
the gas-particle dispersion described by Eq. (5). Equation (5) is a hyperbolic fourth-order system with charq
q
@p
@p
;
V
acteristic velocities: Vg ‡ @p
g
@pg ; Vm ; Vm .
g
Therefore the velocity of sound for this system is
(d(1±rm)) ± 1/2.
In the steady case the calculations were carried out
using the ªcut and tryº method. In each iteration the
set of differential equations (Eqs. (4) and (5)) were
solved by means of fourth-order Runge-Kutta method
with automatic step correction. For a given flow rate
and the calculated conduit length was compared with
the chosen conduit length. Depending on the magnitude and sign of this difference, a new value of the
flow rate was chosen.
For unsteady cases the numerical method was
described in Barmin and Melnik (1996). In each zone
Eqs. (4) and (5) were solved on a uniform mesh using
the purely implicit compact difference code proposed
by Tolstiykh (1990). Then the conservation laws
(Eq. (10)) were solved at four points in the neighbourhood of the disintegration wave, and values of variables and the velocity of the fragmentation wave were
determined. For the purely implicit code the calculation accuracy is of the order of [Dt, h3], where Dt and
h are mesh steps of time and coordinate, respectively.
The code is stable for any time-step value. The particular time step was chosen to keep the Curant-FridrixLevi number less then unity. See the Appendix for the
details of the numerical method.
Magma flow rate for the steady case
Influence of chamber depth and conduit resistance
on eruption dynamics
By varying pressure in the chamber we obtain a
sequence of different steady-state regimes and thus
simulate the quasi-steady process of eruption. This is
valid when considering gradual changes of chamber
pressure, i.e. for a prolonged eruption. In a magma
chamber which is closed to new influxes the chamber
pressure must fall with time (Druitt and Sparks 1984);
thus, the chamber pressure can be regarded here as a
proxy for time.
In order to estimate the typical nondimensional
values (see Table 1), dimensional parameters are
chosen from the well-documented 1980 eruption of
Mount St. Helens and of the Bronze Age Minoan
eruption at Santorini. In order to illustrate the process
of calculation of nondimensional parameters we consider the case of the Mount St. Helens eruption. The
initial concentration of dissolved gas in the melt phase
c0 " 4.6 wt. % (the magma contains ca. 30 % of crystals) for this eruption. Taking into account the value
of the saturation coefficient k9p = 4.1 10 ± 6 Pa ± 1/2 we
obtain a saturation pressure p90 " 125 MPa. For an
average melt density r90 = 2.35 ” 103 kg/m3 (melt density 2.2 ” 103 kg/m3, crystal density 2.7 ” 103 kg/m3 and
30 wt. % of crystals) the value of the characteristic
length in this model x90 " p90 /r90 g9 is 5.4 km. As the
chamber depth L for this eruption was estimated at
approximately 7 km (Carey and Sigurdsson 1985),
H = L9/x90 = 1.28. The ratio of densities d = r9g0/r90 =
8.5 ” 10 ± 2
using
r90 /(R9T9r90) = r90 r9a /(r9a r90) r90 r9a =
r9a = 0.16 kg/m3 (for pure H2O vapour at 1000 C) and
p9a = 105 Pa ± atmospheric gas density and pressure,
respectively. The estimate of Eu number is 5.3 ” 104
for a magma ascent speed V90 " 1 m/s, Ar = 4.7 10 ± 1 for
a pre-exponential viscosity coefficient m90 = 107 Pa s
(see Eq. (3)) and conduit diameter d9 = 30 m. The
parameter Ca which corresponds to the pressure disequilibrium is approximately 6.75 ” 104. For the
Bronze Age Minoan eruption at Santorini dimensional
parameters are also summarised in Table 1. Using the
same procedure as described above, the set of nondimensional parameters represented in Table 2 are
obtained.
The sets of steady solutions for discharge rate as a
function of chamber pressure are investigated. Dimensionless parameters depend on velocity and therefore
on the discharge rate. To fix values of the eruption
parameters along the solution curves dimensionless
parameters are calculated using a characteristic velocity V90, the value of which gives the characteristic discharge rate Q90 = V90S9cr90. For each point of solution
curve we can calculate the actual set of dimensionless
parameters using Q9/(S9cr90) as a characteristic velocity.
However, all these parameters can be reduced to ini-
161
Table 2 Values of dimensional parameters for eruptions of Mount St. Helens and Santorini
Parameter
Responsible for
Mount St. Helens
Santorini
c0
Eu
Ar
d
Ca
H
Initial mass concentration of dissolved gas
Inertia terms scale in momentum equation (reciprocal value)
Viscous conduit resistance
Initial ratio of densities
Pressure nonequilibrium (reciprocal value)
Non-dimension chamber depth
4.6 wt. %
5.3 ” 104
4.7 ” 10 ± 1
8.5 ” 10 ± 2
6.75 ” 104
1.28
5.5 wt. %
7.2 ” 103
1.73
0.11
2.88 ” 103
0.625±0.75*
Notations
a
Dp*
r
rg0
l
m
t
A,B
a
Cm
c
d
Fgm
g
HF
kp
L
LF
n
p0
pa
pch
p
Q
R
rp
S
Sc
T
t
tF
V
x
Volume concentration of gas
Fragmentation overpressure
Density
Density of pure gas
Conduit resistance coefficient, dimensionless
Viscosity of magma
Pressure relaxation time in magma chamber
Coefficients in viscosity equations
Bubble radius
Particle drag coefficient
Mass concentration of the dissolved gas
Conduit diameter or width of fissure
Gas drag force in gas-particle dispersion regime
Gravity acceleration
Coordinate of fragmentation level
Solubility coefficient
Conduit length
Particle dynamical equilibrium length
Numerical concentration of the bubbles
Solubility pressure
Atmospheric pressure
Chamber pressure
Pressure
Magma discharge rate
Gas constant (J kg ± 1 K ± 1)
Particle radius (m)
Fragmentation wave velocity
Conduit cross section area (m2)
Temperature
Time
Particle relaxation time
Velocity
Vertical coordinate, dimensionless
Dimensional parameters
c0
Eu = p90/r90V902
Ar = m90V90/r90 g9d92
d = p90/r90R9T 9
Ca = p90x90/m90V90
H = L9/x90
Initial mass concentration of dissolved gas
Inertia terms scale in momentum equation (reciprocal value)
Viscous conduit resistance
Initial ratio of densities
Pressure nonequilibrium (reciprocal value)
Non-dimensional chamber depth
Subscripts and superscripts
9
Dimensional parameter, all units in CI
m
Magma property
g
Gas property
0
Initial or characteristic value
a
For chamber depth 5 and 6 km, respectively
tial (calculated with velocity V90) parameters by multiplication on the corresponding power of (Q9/Q90). For
example, Aractual = Ar0(Q9/Q90), Euactual = Eu0(Q9/Q90) ± 2,
etc. Therefore, along the whole solution curve all the
dimensionless parameters with subscript ª0º remain
constant. This guarantees that eruption parameters are
also constant except the chamber pressure and the discharge rate. In both figures and text, however, the
subscript ª0º is omitted for simplification of notations.
We focus on the influence of the Ar number, the relative length of the conduit H = L9/x90 and initial gas
concentration c0 on the eruption dynamics. The ratio
of the real discharge rate Q9 to Q90 is shown on all
plots. We have fixed the basic set of dimensionless
parameters as following: c0 = 5 wt. %, Ar = 1.5,
Eu = 104, Ca = 105 and d = 0.08. Using these parameters
wide range of dimensional values can be studied. For
example, to obtain the set of dimensional parameters
162
listed above the following set of dimensional values
can be used: p90 = 147 MPa, m90 = 3.47 ” 106 Pa s,
T9 = 1458K, V90 = 2.31 m/s, r90 = 2738 kg/m3, d9 = 28.3 m
and x90 = 5.47 km. Corresponding value of discharge
rate Q90 is 4 ” 106 kg/s. In order to compare results of
calculations with observational data, dimensional
values corresponding to this set of dimensional governing parameters are shown simultaneously with
dimensionless data, but dimensionless results cover
much wider area because they correspond to a wide
range of possible dimensional parameters.
With reference to Fig. 1 the aim of the calculations
is to establish the relationship between chamber pressure, pch, and the different regimes of flow. Of principal interest are the magma discharge rate Q, the position of the fragmentation level, and how these
parameters vary with chamber pressure. The position
of the fragmentation level determines the average
mixture weight and conduit resistance resulting in a
major influence on magma discharge rate. When the
chamber pressure pch decreases in a sustained explosive eruption and falls below the saturation pressure
p0, the bubbly zone extends into the chamber. The
proportion of bubbles at the top of the chamber
increases as the pch decreases with the consequence
that the overall magma density decreases in the bubbly zone. For a given flow rate the velocity of the mixture increases together with the melt viscosity due to
gas exsolution. These factors lead to an increase in the
pressure relaxation time between growing bubbles and
surrounding melt, and therefore the bubble overpressure increases more rapidly.
The interplay of this process is now explored for
Ar = 1.5 and H = 0.6. The fragmentation level position
and the discharge rate as a function of chamber pressure are shown in Fig. 2 (a and b correspondingly) for
this shallow chamber case, when chamber pressure
pch < p0. Real chamber depth for this case is 3.28 km,
lithostatic pressure is 83 MPa and corresponding
dimensionless pressure is 0.57. At point A chamber is
overpressurised with overpressure value of 33 MPa,
the discharge rate is approximately 3.2 ” 108 kg/s
(1.2 ” 105 m3/s). Each point on this diagram represents
a steady solution of the boundary problem. For the
same values of the chamber pressure there can be several (up to five) steady solutions, and the topology of
the diagram indicates that there can be sudden jumps
from one solution to another. The physical consequences of these multiple solutions can be best illustrated by following a curve from the left to the right
in the direction of decreasing chamber pressure. In
this case all other eruption parameters are fixed and
chamber pressure decreases and can be considered as
a proxy for the time in an eruption. The eruption
parameters modelled in Fig. 2 correspond to typical
plinian eruption conditions from a shallow chamber.
In the interval AB the decrease in bubbly zone
length is not rapid enough to offset the decrease in
pch; therefore, the flow rate decreases slightly. From a
certain value of the pressure pch the decrease of the
fragmentation level is sufficiently rapid that the flow
rate, passing through the minimum point B, begins to
increase. This increase in flow rate leads to increase in
the rate of bubble overpressure growth, and therefore
the fragmentation level goes further down to the conduit. A critical condition is reached at point C in
which there is no continuous steady-state solution as
the chamber pressure decreases further. The system
must adjust to one of three other solutions. The plau-
Fig. 2 Fragmentation level position (measured from the
chamber and related to the length of the conduit) and discharge
rate of eruption (related to Q90 = r90V90S9c) as a function of
chamber pressure pch. On the opposite axis typical dimensional
values are given. Each point represents a steady solution of a
boundary problem for the conduit flow. In the point A the discharge rate is in the range of plinian phase of Santorini BC 1650
eruption. As pch decreases, fragmentation level goes dipper in
the conduit and at critical point C the intensity of eruption
increases an order of magnitude while fragmentation level
reaches the chamber exit. At the point F eruption abruptly stops
with a possibility of caldera collapse
163
sible response is that the fragmentation level drops
dramatically and flow rate increases by an order of
magnitude to point E (Fig. 6 presents the calculation
when this transition occurs monotonically). Note that
the flow rate increases by an order of magnitude and
reach the value of approximately 1.6 ” 109 kg/s
(5.8 ” 105 m3/s) and that the fragmentation level is at
the chamber exit. Regimes CD±DE cannot be physically reached when the pressure in the chamber
decreases monotonically.
Due to the reduction in length of the bubbly zone,
the pressure drop takes place mainly in the gas-particle mixture zone at the expense of its weight (interval EF). As pch decreases further the amount of gas
contained in the bubbles before fragmentation
increases, and the average weight of the gas-dispersion
decreases. Finally, this weight becomes comparable to
the conduit resistance in the short bubbly zone. It
leads to a rapid decrease in the flow rate as point F is
approached, where critical conditions are reached.
Further monotonic decrease in flow rate for the pressures lower then pch(F) is impossible, because it
requires the increase in fragmentation level, and
therefore, sharp increase in conduit resistance. From
the point F the system may change to a low-intensity
eruption regime (with flow rate comparable to lava
dome growth flow rates). In this regime the entire
conduit is filled with bubble liquid, because the overpressure in the growing bubble does not exceed the
critical value and no fragmentation occurs. Of course,
due to the elasticity of conduit and chamber walls, the
pressures corresponding to the end of the explosive
part of the eruption will be higher and explosive activity will stop somewhere in the interval EF. It is anticipated that the very low chamber pressures near point
F are not realised, so that the conduit closes or the
roof of the chamber collapses.
It is clear from Fig. 2 that for fixed chamber pressure up to five steady-state eruption regimes are possible. In Fig. 3 the profiles of pressure along the volcanic conduit are shown for pch = 0.55. Curve 1
corresponds to the upper regime (EF). Here the
length of the bubbly zone is too small to be shown in
the plot, and the pressure drop occurs in the gas-particle dispersion. Because the outlet flow is sonic, the
outlet pressure is greater than atmospheric. Curve 2
corresponds to B±C branch of solution. In the short
liquid zone, the sharp pressure decrease occurs due to
the high resistance of the conduit. In the lowest
regime (G±H, curve 3) most of the conduit is occupied
by bubbly melt and the pressure distribution is closer
to lithostatic. As the critical overpressure in the growing bubble is used for the fragmentation condition, the
volume concentration of bubbles does not have a fixed
value at the fragmentation. Values of volume concentration on the fragmentation level are: 0.23 for the
upper regime; 0.44 for the intermediate; 0.78 for the
lowest regime.
Fig. 3 Pressure profiles for fixed chamber pressure pch = 0.6 and
different flow regimes. Jog point corresponds to the fragmentation level. Curve 1 corresponds to the highest intensity regime
(DF in Fig. 2), where all the conduit is occupied by gas-particle
dispersion; curve 2 to initial phase (AC); curve 3 to the low-intensity regime (GH)
In the case of a deeper chamber (H = 0.9; Fig. 4)
the topology of the solution is more complicated.
Three steady solutions correspond to the chamber
pressure range between pch(A) and pch(C). They form
an S-shaped curve (obtained also in Barmin and Melnik 1990, and Slezin 1991). At lower pressures two
steady regimes exist, which cannot be reached by
monotonic decrease of chamber pressure. On the
plane Q ± pch they form a cyclic curve. Discharge rate
at point A is 108 kg/s, pressure is equal to saturation
pressure (147 MPa), and chamber is overpressurised
by the value of 22MPa.
The evolution of the steady solution topology as a
function of chamber depth is shown in Fig. 5. For
deeper chamber location the loop contracts into a
point (see curves for H = 0.9, 0.95) and the S-shaped
curve moves to the higher chamber pressure values.
At H = 1.5 the S-shaped dependence of discharge rate
on chamber pressure becomes monotonic and singlevalued. These results show that condition for sudden
changes in discharge rate can occur in magma
chambers that are saturated in volatiles, and therefore
for shallow chambers and high gas content. Corresponding values of chamber depth and lithostatic
pressures for dimensionless chamber depth are:
0.95±5.2 km and 132 MPa; 1.2±6.6 km and 167 MPa;
8.2 km and 209 MPa for H = 1.5.
The effect of Ar number on the dynamics of eruption is now considered. Larger Ar values correspond
164
Fig. 5 Flow-rate dependence on chamber pressure for Ar = 10
and different chamber depth H. Ar = m90V90 /(r90g9d92) represents
the part of the pressure drop in bubbly melt flow due to the
conduit resistance. Chamber depth varies from 3.3 to 8.2 km.
For deeper chamber location maximum intensity of eruption
decreases and no intensification of eruption occurs with
chamber-pressure drop. Finally, the dependence of the discharge
rate on pch becomes monothonic as in the case of pipe flow of
Newtonian liquid. For lower Ar values the range of chamber
depths corresponding to different flow regimes is wider
Fig. 4 The same as Fig. 2 for the deeper chamber (H = 0.9).
Only the S-shaped part of solution could be physically reached
by the decrease of a chamber pressure
to large contribution of viscous conduit resistance to
the pressure drop in the liquid zone. In Fig. 6 we have
plot the flow rate as a function of chamber pressure
for Ar = 0.02, 0.15, 1.5 and 10 for chamber depth
H = 0.6. In order to keep the conduit diameter constant we varied the magma viscosity and, in consequence, the values of parameter Ca were changed
simultaneously. Corresponding values of pre-exponential coefficients are: 4.6 ” 104; 3.5 ” 105; 3.5 ” 106; and
2.3 ” 107 Pa s. When Ar = 0.02 there is no intensification of the eruption with decrease in chamber pressure. When Ar reaches a value equal to 0.15 the transition to the gas-dispersion regime (EF in Fig. 2)
occurs with an almost monotonic increase in flow rate.
The extent of the intensification of eruption at the
critical point C increases with increase of Ar for
Ar>0.15. The weak dependence of Q = Q(pch) on Ar
Fig. 6 Flow-rate dependence on chamber pressure for H = 0.6
and different Ar values. The variation of Ar corresponds to the
variation of viscosity, and therefore for fixed chemical composition of magma, to the variation of initial temperature. Values of
Ar = 10 to 0.02 correspond to the range of temperatures
750±950 C using data of Dingwell et al. 1996)
165
for chamber pressures in the regime EF confirms that
the pressure drop takes place mainly in the gas-particle dispersion zone. For the same chamber pressure
larger Ar value corresponds to a shorter liquid zone
and, therefore, the chamber pressure at point C corresponding to the sudden intensification of eruption,
increases with increasing values of Ar. As an eruption
proceeds Ar values can decrease due to a decrease in
magma viscosity or due to a conduit erosion. In this
case for constant pch a sudden transition from low-intensity regime to high-intensity regime is possible, as
also suggested by Slezin (1983). The other possibility
is shown by Papale et al. (1998) where the variation in
chemical composition from chemically evolved, waterricher and crystal-poorer magma to more mafic crystal-rich magmas results in increase in viscosity, and
therefore Ar value increases.
Influence of initial dissolved gas concentration on
eruption dynamics
The initial concentration of dissolved gas, c0, in silicic
magmas is typically in the range of 3±6 wt. %. Some
estimated values of c0 are given in Table 1. This
parameter has a strong influence on eruption dynamics. The smaller the concentration of dissolved gas, the
greater the proportion of the conduit occupied by the
flowing viscous magma, which leads to raising of the
average weight of mixture and the conduit resistance,
causing the flow rate to decrease.
Saturation pressure p90 is a strong function of the
concentration of dissolved gas; therefore, the characteristic length x90 introduced as p90 /r90g9 depends on c0.
For a fixed chamber depth L the relative chamber
depth (H=L9/x90), therefore, varies with c0. If the
dimensional chamber depth is fixed so that H = 0.6 for
c0 = 5 wt. %, the value of H for other values of c0 can
be calculated by multiplying by the factor (5/c0)2. Variation of c0 has influence not only on the relative
chamber depth, but on other parameters as well. To
represent only the influence of dissolved gas concentration these parameters were also multiplied by the
corresponding power of concentrations ratios.
Figure 7 shows the variations in flow rate Q with
chamber pressure pch for the case and Ar = 1.5 and
c0 = 6 to 2.5 wt. %. Corresponding chamber depth
range is 0.42±2.4, and saturation pressures range is
212±36.8 MPa; therefore, for low concentrations of
dissolved gas, magma in the chamber is undersaturated, and for high concentrations it contains significant
volume fraction of gas bubbles. In the case
c0 6 4.5 wt. % the point of transition to the intensive
regime moves to lower values of pch and the value of
Q at point E decreases as concentration of dissolved
gas decreases. The value of Q beyond E depends partially on c0 when pch is fixed, where fragmentation
level practically reaches the chamber. In this regime
the influence of bubbly zone on discharge rate is
Fig. 7 Influence of initial mass concentration c0 on the flow rate
of magma for Ar = 1.5 and H = 0.6. Chamber pressure is divided
by saturation pressure, corresponding to c0 = 5 wt. %. Variation
in c0 effectively changes the relative depth of the chamber;
therefore, for fixed physical chamber depth, the part of the conduit occupied by gas-particle dispersion is smaller for smaller c0.
Flow regime variation with the change c0 is similar to that which
occurs with the change of the chamber depth. The variation of
initial gas content in the magma even in  0.5 wt. % could cause
the catastrophic variation of the flow rate
small, and it is controlled by the weight of gas-particle
dispersion which does not changes significantly in the
range of c0 = 5±6 wt. %.
For c0 = 3.5 % no intensification of eruption occurs
like in the case H = 1.2 described above. In the case
c0 = 2.5 % the dependence Q(pch) is monotonic (similar
to the case H = 1.5).
Gas concentration can vary significantly in a zoned
magma chamber. The commonest form of this variation is an upward increase in water content related
to the processes that cause magma chamber zonation,
such as sidewall crystallisation, fluxing of CO2 bubbles
and magma mixing (Sparks et al. 1984; Hildreth 1981).
Slight variations in the initial gas content in the
magma even in  1 % can cause substantial variations
of the flow rate. For example, if the initial gas concentration decreases from 5 to 4.5 % at a chamber pressure of approximately 0.5, the decrease in eruption
rate can be more than an order of magnitude. Strong
dependence of discharge rate on water content is also
shown by Papale et al. (1998), but increase in discharge rate with increase in volume concentration
occurs monotonically in this model.
166
Transitions between steady regimes
By using an unsteady model the processes which occur
during the transition from one steady regime to
another can be investigated. The solution of the
unsteady problem is more complicated; thus, only simple cases with S-shaped curve dependence Q(pch) are
studied here. The following relation between the rate
of variation of the chamber pressure and the amount
of the material in it is considered:
dpch
ˆ
dt
Q
t
…11†
Here t is the pressure relaxation time in the chamber.
The relaxation time depends on numerous parameters
of the volcanic system such as, for example, the
magma compressibility, the elasticity of the magma
chamber walls, its dimensions. In this model the relaxation time is assumed to be constant. For a large
chamber volume and elasticity of the chamber the
relaxation time is large. Case t = ? corresponds to
pch = const. The same kind of boundary condition was
suggested by Woods and Koyaguchi (1994), where a
transition from effusive to explosive regime was investigated in the framework of a quasisteady approach.
They suggested that the value of relaxation parameter
is equal to V9ch r90 /g9 where V9ch is a chamber volume
and g9 is the effective bulk modulus of the host rock.
Converting Eq. (11) to dimensionless form using the
dimensionless variables introduced in discussion to
Eq. (4) we obtain dimensionless t = V9ch r90 g9/g9d92. For
example, g9 = 1010 Pa, V9ch = 2 ” 1010 m3, d9 = 20 m and
r90 = 2.5 ” 103 kg/m3 t = 1.25 ” 102.
At t = 0 the steady solution is assumed with fixed
pressure in the magma chamber corresponding to the
upper branch of S-shaped curves shown in Figs. 4, 5,
6, 7. The evolution of flow rate and fragmentation
level in time is investigated. Figure 8 shows the relationship between flow rate and chamber pressure pch
for values of t = 5, 10 and 100. The dashed curve represents the steady solution (t = ?). Smaller values of
tcorrespond to faster change in chamber pressure
(smaller values of V9ch /g9 ratio). For chamber pressures
larger then 0.32 the unsteady solution follows the
steady curve Q(pcb) closely for all values of t. Fragmentation level slowly moves down and flow rate variations along the conduit are small (see Fig. 9). Near
the critical point the fragmentation level reaches its
minimum value and begins to ascend rapidly. Due to
the inertia of the system, the length of bubble zone is
shorter than in the steady solution, and therefore, the
average weight of the mixture and the conduit resistance is lower. Consequently, the flow rate for a fixed
chamber pressure decrease is higher than in the steady
case, allowing lower pressures in the chamber before
the end of eruption. The duration of explosive part of
eruption depends on the value of t, and is calculated
to be 7.5 h for t = 100, 1.1 h for t = 10 and 39 min for
Fig. 8 Simulation of transition from one steady-state solution to
another. Discharge rate dependence on chamber pressure is
shown. Chamber pressure variation in time is given by Eq. (11).
The parameter of the curves is a chamber relaxation time. Duration of eruption for t = 5, 10 and 100 is 39, 70 and 450 min,
respectively. Despite different timescale of these eruptions, the
difference in the Q-pch plain is similar except for the region of
regime transition. It means that upper regime is stable
Fig. 9 Fragmentation level position (related to the conduit
length) as a function of chamber pressure for the run shown in
Fig. 8. For longer eruption (larger t) fragmentation level position follows changes in discharge rate, but for shorter eruptions
inertial delay of the bubbly fluid is essential
167
t = 5. As the end of explosive eruption, we assume
conditions when outlet gas velocity becomes insufficient to support particles (particle velocity at the exit
of the conduit becomes negative).
Implication of results
These results confirm the previous studies by Slezin
(1984, 1991) that there can be several steady solutions
for the flow dynamics in explosive eruptions for the
same set of eruptive conditions. Although some of the
steady solutions may not be realised in nature, the
topology suggests that there can be very dramatic
changes in the intensity during some explosive eruptions. Rapid or abrupt changes of magma discharge
rate are predicted to be one to nearly two orders of
magnitude and can arise if the chamber pressure
decreases below some critical value. Slight changes in
magma water content or conduit cross-section area
can also result in rapid increase or decrease in discharge rate. The example in Fig. 2 illustrates a case
for a fairly typical large-magnitude, high-intensity
explosive eruption which might generate plinian eruptive columns and pyroclastic fountains to form ignimbrite. The early phase of the eruption involves discharge rates in the range 120,000 to 240,000 m3/s. Exit
conditions are within the range of high-intensity plinian and ignimbrite eruptions (Sparks et al. 1997). At
point C a sudden jump to a discharge rate of
2.4 ” 106 m3/s is predicted. It corresponds to the conditions well within the regime where fountain collapse
is expected (Wilson et al. 1980; Bursik and Woods
1991). Inspection of the results for different values of
Ar (Fig. 6) and different water contents (Fig. 7) imply
that the eruptive conditions where the sudden jumps
can occur correspond to those encountered in largemagnitude explosive eruptions. The calculations presented here indicate that the main factor affecting the
occurrence of such jumps is the development of significantly lower chamber pressures; therefore, it is a feature of shallow chambers and gas-rich magmas.
There is good evidence that abrupt increases and
decreases in discharge rate commonly occur in such
large-magnitude eruptions. In the Taupo eruption
AD 180 in New Zealand the plinian phase involved
one of the highest intensity plinian eruptions with discharge rate of approximately 106 m3/s from column
height estimates (Walker 1980). The plinian phase was
followed by the Taupo ignimbrite where discharge
rate has been estimated as approximately 108 m3/s
(Wilson and Walker 1985). The model presented here
can explain the abrupt two order of magnitude
increase in intensity.
Other stratigraphic studies also qualitatively imply
a sudden large increase in discharge rate. Druitt et al.
(1989) have documented several pyroclastic deposits
related to large-magnitude, high-intensity explosive
eruptions in the Thera Pyroclastic Formation on San-
torini, Greece. Gardner et al (1996) estimated the discharge rates in these eruptions in the range
104 ± 106 m3/s for the plinian phases. These sequences
involve a plinian pumice deposit overlain by pyroclastic flow deposits. The pyroclastic flow deposits are
often represented by proximal lag breccias (Druitt and
Sparks 1982). Several features indicate sudden intensification of the eruption at the boundary between the
plinian deposit and ignimbrites. The upper surface of
the plinian deposit is sometimes severely eroded. In
four cases (Cape Riva, Middle Pumice, Lower Pumice
2 and Lower Pumice 1) large ballistic blocks up to
2 m in diameter create deep-impact structures in the
plinian deposit. The impact ballistic blocks are distributed in these four cases over most of Santorini at distance of up to 5 km from the inferred vent (R.S.J
Sparks, pers. commun.). The ballistic blocks are
approximately 10±20 times larger than lithic blocks in
the coarse upper part of the underlying plinian deposits. The lag breccias of the major Santorini explosive
eruptions extend to distances of 5 km or more from
the vent. Such deposits are generally interpreted as
forming in the deflation zone around the collapsing
central fountain (Druitt and Sparks 1982). Fountains
of a few kilometres height are implied and required
high-intensity causative eruptions. It is suggested that
the plinian to ignimbrite transitions in these eruptions
can be explained by the sudden intensification predicted in this model.
There is also the possibility implied in these models
for rapid decreases in intensity. The Middle Pumice
deposit of Santorini (Druitt et al 1989) consists of a
plinian deposit overlain by very coarse lag breccias
and than finally by a finer-grained plinian deposit.
The deposit is chemically zoned from dacite at the
base to andesite pumice at the top. The first two units
can be explained in terms of intensification of eruption, but the finer-grained Middle Pumice C unit
requires a large decrease in intensity. As demonstrated in Fig. 7 marked decreases in intensity can
occur as a consequences of a small decrease in water
content. Although other explanations can be postulated, this model illustrates that such stratigraphic
sequences implying abrupt changes in eruption intensity can be accommodated by conduit flow models
without any change in conduit dimensions.
A feature of the model is that often large chamber
decompressions are required to reach conditions
where abrupt intensification can take place. In the
illustrative example (Fig. 2) the pch drops to 55 % of
the saturation pressure and must have developed pressures well below lithostatic pressure. For example, for
Fig. 2 the chamber was located at 3.28 km depth with
an initial overpressure of 33 MPa at point A, at point
C pressure is slightly smaller then lithostatic pressure.
With sudden intensification the chamber pressure will
quickly underpressure further so that caldera collapse
is inevitable. Druitt and Sparks (1984) estimated that
caldera collapse can initiate when the underpressuring
168
reaches
the compressive
strength of
rocks
( ~ 50±100 MPa). Caldera collapse can thus abruptly
stop the eruption or increase the chamber pressure
back towards lithostatic conditions and thus switch the
eruption back to another flow regime, such as extrusion of lava. For a shallow chamber the magma in the
chamber will have vesiculated and the rapid increase
in pressure following caldera collapse may cause gas
to redissolve and the eruption to either to stop or to
be followed by dome growth.
Figure 7 illustrates that chamber pressure at which
intensification can occur is very sensitive to gas content. The model also suggests that intensification is a
feature of shallow chambers. The important parameter
is not the chamber depth itself but its relative value.
For example, Mount St. Helens in 1980 had a magma
chamber located at 7 km, but melt water content of
4.6 wt. % only. This resulted in the relative chamber
depth of 1.28. For Santorini (BC 1650) eruption
chamber depth is a slightly less (5±6 km), but the relative depth is only 0.625±0.75 due to the higher water
content. According to the calculation intensification of
explosive eruption on Mount St. Helens would not be
expected, but intensification should be expected on
Santorini. The calculations, therefore, are consistent
with the observations as no intensification occurred on
Mount St. Helens after the plinian phase, whereas in
the Minoan eruption plinian phase was followed by a
major phase of voluminous and high-intensity ignimbrite production (Sparks and Wilson 1990).
It is also a common situation that after the explosive part of the eruption lava dome growth occurs.
Lava dome growth can begin immediately after the
explosive eruption, or can be separated from it by a
period of several weeks or months or even years
(Sparks et al. 1997). This feature of an eruption can
also be explained using the results of these calculations. Low-intensity regime discharge rates are comparable with those in lava dome growth. For shallow
chamber depth the pressure corresponding to the end
of the explosive part of eruption is very small and
immediate lava dome growth is impossible. If the
chamber pressure can build up due to inflow of
magma from the deeper supply system or caldera collapse, then the chamber pressure can become large
enough to initiate dome growth. For deeper chambers
the decompression of the chamber is much smaller
and dome growth can occur immediately after the
explosive eruption. In contrast, explosive eruptions
can be initiated by dome collapse (Robertson et al.
1998). This case can be interpreted as movement
along the lower-intensity regime with increasing
chamber pressure and then a sudden transition to an
explosive regime at a critical pressure threshold. Of
course, caldera collapse or changes in chemical composition or intensification of gas loss to wallrocks can
affect discharge rate in this regime significantly.
Appendix: Numerical method in the unsteady case
To illustrate the idea of numerical method the single
linear differential equation is considered:
@U
@U
‡A
ˆ0
@t
@x
…a†
The approximation of this equation by tree nearest
points (i-1), (i) and (i+1) is written as follows:
a1 U i 1 U i 1 ‡ a2 U i U i ‡ a3 U i‡1 U i‡1
…b†
t
‡ A b1 U i 1 ‡ b2 U i ‡ b3 U i‡1 ˆ 0
h
Here U is the value of dependent variable on the previous time step. We order that Eq. (b) should approximate Eq. (a) with an error of O(h4). Then the following relationship between ai and bi must be satisfied.
1 ‡ a2 h
1 a2 h
1 ha2
;
; a3 ˆ
; b1 ˆ ‡
4
2h
2h
6
2
1 ha2
b2 ˆ ; b3 ˆ
;
4
3
6
a1 ˆ
If we also insist on conservation feature of approximation (b), i.e. satisfaction of the following integral
relationships,
U
i
U
i 1
ˆA
1
Zxi
xi
@U
dx; U i‡1
@t
i
U ˆA
1
Zxi‡1
xi
1
@U
dx;
@t
then only two sets of coefficient values are allowed.
b1 ˆ
1= ; b ˆ 1= ; b ˆ 5= ; a1 ˆ 0; a2 ˆ
12 2
3 3
12
b1 ˆ 5=12; b2 ˆ 1=3; b3 ˆ
1= ; a1 ˆ
12
1; a3 ˆ 1;
…1†
1; a2 ˆ 1; a3 ˆ 0;
…2†
The choice of concrete set of coefficients depends on
the sign of A. If A > 0, for stability of algorithm one
must use the upwind differences (set 1); otherwise,
downwind differences (set 2) must be used. If general
system is written in characteristic form, the choice of
coefficients depends on the sign of characteristic
value. Tolstykh (1990) has shown that for subsonic
velocities in Euler equations, in the approximation of
continuity equation one must use the set (1) and the
set (2) for momentum. Similar analysis for Eq. (5)
shows that in discontinuity and momentum of particles
equations set (1) and for gas momentum set (2) must
be used. This implicit approximation of Eqs. (4) and
(5) leads to block matrix equations that could be
solved by means of the Tomas method.
Another complication of numerical code comes
from existence of a moving boundary (the fragmentation wave) separating two different regions of flow.
169
To satisfy the conservation laws and to find the position and velocity of this boundary Eq. (10) is approximated with four points around the discontinuity (two
in the bubbly liquid and two in the gas-particle dispersion). The relationships between the incoming characteristics are also approximated in these points. This
makes it possible to calculate the whole flow field.
The accuracy is controlled by the norm:
k P
n
P
dy2ij , where Dy is an increment of dependkdk ˆ
jˆ1 iˆ1
ent variable, k rank of system of equation and n
number of points is the domain. The iterations were
stopped, when kdk < 10 ± 12, which guarantees the satisfaction of conservation laws with an accuracy of
10 ± 7 ± 10 ± 10 in the same norm.
Acknowledgements I thank A.A. Barmin for useful discussions
on the mechanical model and obtained results, R.S.J. Sparks and
G. Ernst for reviewing this article and help with geological interpretation of the numerical results. The work was carried out
with support from the Russian Foundation for Fundamental
Research (project no. 96-01-01082) and NERC grant
(GR3/11683).
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