Mixing of stratified liquids by the motion of gas bubbles: application to magma mixing

Earth and Planetary Science Letters, 115 (1993) 161-175

161

Elsevier Science Publishers B.V., A m s t e r d a m

[CH]

Mixing of stratified liquids by the motion of gas bubbles: application to magma mixing Nathalie Thomas a Stephen Tait a and Takehiro Koyaguchi b a Laboratoire de Dynamique des Syst~mes Gdologiques, Institut de Physique du Globe et Uniuersitd de Paris 7, 4place Jussieu, 75252 Paris, France b Earthquake Research Institute, University of Tokyo, Bunkyo Ku, Japan Received August 6, 1992; revision accepted January 11, 1993

ABSTRACT Mafic m a g m a underlying more silicic m a g m a in a reservoir can exsolve volatiles as it crystallizes, forming bubbles that segregate upwards to the interface with the overlying silicic magma. We describe laboratory experiments designed to investigate how gas bubbles migrate into the more viscous upper layer. In one regime, bubbles move individually across the interface, entraining a small quantity of lower layer fluid and the two layers become progressively stirred into a h o m o g e n e o u s mixture. In a second regime, bubbles form a thin foam layer at the interface which becomes gravitationally unstable, giving rise to two-phase plumes which rise and produce a coarse mixture of the liquids. For a given viscosity ratio, the transition between these regimes occurs when the gas flux is greater than a critical value, which we found to decrease with increasing viscosity ratio. We estimate the critical gas flux for magmatic conditions, and how vesiculation may vary with pressure, degree of crystallization and the composition of volatile components. Mafic inclusions, which are often found in silicic to intermediate lavas, can form in the plume regime; analysis of the instability of the foam predicts wavelengths of between 1 cm and 1 m, which is consistent with the observed sizes of inclusions. Volatiles from the mafic m a g m a may be efficiently transported by this mechanism into the viscous silicic m a g m a as bubble plumes despite low Stokes' velocities for individual vesicles. At small viscosity ratios or low gas fluxes, the bubbling regime can produce hybrid magmas.

1. Introduction

The injection of mafic m a g m a into a reservoir containing silicic m a g m a can lead to mixing of the magmas and play a fundamental role in the compositional evolution of volatile components in silicic magmas. Anderson et al. [1] provide evidence that the silicic m a g m a of the Bishop Tuff was enriched in volatile components that were probably acquired from a mafic magma. Many geological observations show that small inclusions of more mafic m a g m a are very common in intermediate and silicic lavas [2,3]. Lavas also occur in which mixed phenocryst populations indicate that two distinct magmas have been thoroughly stirred to produce hybrid magmas that are homogeneous on a microscopic scale. However, the mechanisms by which mixing generates inclusions and hybrids, and by which vapour phase is supplied from mafic

to silicic magma, are not well understood from a physical viewpoint. In a conceptual model, Eichelberger [4], envisaged that an influx of hot, volatile-rich mafic m a g m a may pond at the bottom of a silicic chamber as a separate, denser layer. The mafic layer loses heat across the interface and crystallizes, becomes saturated with respect to volatile components, and exsolves vapour bubbles. The bubbles rise and collect at the interface between the two magmas to form a foam layer, pieces of which then detach and rise to form vesicular mafic inclusions in the silicic m a g m a (Figs. 1a-c). H u p p e r t et al [5] suggested, in contrast, that vigorous convection in the lower layer could keep the bubbles in suspension such that accumulation of a separate bubble-rich layer would not occur. They argued that the bulk density of the m a g m a of the lower layer decreases as the bubble frac-

0012-821X/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

162

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AL.

the critical value. At low concentration, the particles were uniformly distributed by vigorous convection in the body of the fluid, but sedimented in the boundary layer adjacent to the base of the tank [6], because the convective velocity must decrease to zero at the boundary of the fluid. Convection involves continuous exchange between the bulk of the fluid and the boundary layer and hence there was a flux of particles lost from the vigorously convecting fluid. Intermittent behaviour occurs if the initial particle concentration is greater than a critical value [7]. At the start of Koyaguchi et al's. experiment [7], convection in the fluid was essentially suppressed; sedimentation took place throughout the fluid leading to a particle-free upper layer which thickened with time as the underlying particle-bearing layer became thinner. The temperature gradient across the particle-bearing layer increased until a sudden overturning event occurred which mixed some of the particles back into the fluid. A proportion of the particles, however, did not become remixed, having formed a thin sediment at the bottom of the tank, and the new concentration was less than the initial value. This process was repeated, with the particles being intermittently homogenized by overturning events, until the critical value was attained, after which the concentration decreased continuously. The critical value of particle concentration separating the continuous and intermittent regimes is thought to be = 1 wt.% [8]. The above results can be applied to the geological situation of a stratified magma chamber

tion increases and becomes equal to that of the overlying magma, at which point convective overturn occurs. In this paper we make use of recent fluid dynamic results which show that even when convection in a layer of mafic magma is vigorous, segregation of bubbles can occur. This provides a mechanism by which a foam layer such as envisaged by Eichelberger [4] can form. Vesicles that segregate from the lower mafic layer must somehow cross the interface and rise into the silicic magma, and here we describe new fluid dynamic experiments designed to investigate the physics of bubble migration in a system stratified in density and viscosity. We show how the dynamics of the interface, and hence of gas transport, vary with the viscosity ratio between the layers and the gas flux in the lower layer. We provide a framework to interpret our observations, and constrain the fluid dynamic behaviour of a magma chamber given the initial state assumed, i.e., discrete layers of silicic and mafic magma.

2. Fluid dynamical principles Recent experiments have shown that suspensions have novel fluid dynamic behaviour because of different balances between convection and sedimentation. In these experiments, a layer of fluid with a uniform concentration of small particles was placed in a tank, and a vertical temperature gradient applied such that, in the absence of particles, the Rayleigh number was well above

magma chamber

(a)

(b)

(c)

Silicic magma

Sflicicmagma

Silicic magma

nccumu,ationl of bubbles

\

formation of plumes

/ heat transfer: o o

o

e e

o

crystallisation and nucleation of bubbles

o

~o 0

~

o

; magma 0

0 '~0

0

~o

of

bubbles

o

Mafic magma 0

0

0

0

0

ol | 0 J

I

Fig. 1. Schematic representation of the replenishment of a m a g m a chamber containing differentiated m a g m a with fresh mafic liquid. The mafic melt loses heat to the surroundings, crystallizes and vesiculates. The vesicles segregate from the mafic magma, collect at the interface because of the high viscosity of the upper layer and form plumes.

163

M I X I N G OF S T R A T I F I E D LI QUI DS BY T H E M O T I O N OF GAS BUBBLES

3. Experiments

c o n t a i n i n g convecting mafic m a g m a with susp e n d e d b u b b l e s b e n e a t h a lighter, m o r e silicic layer. T h e convective velocity d e c r e a s e s to z e r o at t h e i n t e r f a c e b e t w e e n the mafic a n d silicic m a g mas, a n d h e n c e b u b b l e s can s e g r e g a t e f r o m t h e mafic m a g m a . A n a d d i t i o n a l factor is t h a t b u b bles a r e b e i n g p r o d u c e d in the mafic m a g m a by crystallization. A s t e a d y state b u b b l e c o n c e n t r a tion is thus p o s s i b l e in which s e d i m e n t a t i o n a n d p r o d u c t i o n b a l a n c e , w h e r e a s in t h e a b s e n c e of p r o d u c t i o n b u b b l e c o n c e n t r a t i o n m u s t always d e crease. In o r d e r to u n d e r s t a n d t h e fluid d y n a m i c s of how vesicles f r o m the mafic layer m i g r a t e into t h e overlying silicic m a g m a , we c o n d u c t e d several types o f e x p e r i m e n t s . I n o n e set, a c o n s t a n t flux of gas b u b b l e s was s u p p l i e d at t h e b o t t o m of a t a n k c o n t a i n i n g two fluid layers, t h e u p p e r o f t h e two b e i n g less d e n s e a n d m o r e viscous. In a s e c o n d set, small oil d r o p l e t s r a t h e r t h a n air b u b b l e s w e r e i n j e c t e d into t h e tank. I n a t h i r d set, a h o m o g e n e o u s l y s t i r r e d layer o f fluid containing small solid p a r t i c l e s d e n s e r t h a n the fluid was e m p l a c e d a b o v e a layer o f d e n s e r , m o r e viscous fluid.

3.1 Techniques T h e e x p e r i m e n t s c o n d u c t e d with air b u b b l e s w e r e c a r r i e d o u t in a tall cylindrical Plexiglass t a n k o f 30 cm in d i a m e t e r . F o u r h u n d r e d regularly s p a c e d capillary t u b e s ( i n t e r n a l d i a m e t e r 0.3 m m ) w e r e fixed in a plate, at t h e b o t t o m of t h e tank. A c o n s t a n t flux of gas was s u p p l i e d via the t u b e s in t h e f o r m o f b u b b l e s that w e r e typically 2 m m in d i a m e t e r (Fig. 2a). U s i n g t h i n n e r capillary tubes, we injected a c o n s t a n t flux o f fine oil d r o p l e t s ( = 0 . 1 - 0 . 5 m m in d i a m e t e r ) . W e v a r i e d t h e size of t h e b u b b l e s a n d t h e i r density difference with the a m b i e n t fluid b e t w e e n t h e s e two sets o f e x p e r i m e n t s . T h e u p p e r layer was typically 2% less d e n s e t h a n t h e lower layer fluid. In a t h i r d series o f e x p e r i m e n t s a m i x t u r e of less viscous fluid a n d silicon c a r b i d e p a r t i c l e s initially lay over a layer o f viscous fluid (Fig. 2b). T h e s e e x p e r i m e n t s i l l u s t r a t e d t h e effect of a h o m o g e n e o u s initial d i s t r i b u t i o n o f p a r t i c l e s as c o m p a r e d with those in which b u b b l e s w e r e c o n t i n u o u s l y

PARTICLE EXPERIMENTS

BUBBLE EXPERIMENTS

Compressed air supply

high viscosity liquid

l o w viscosity liquid gas b u b b l e s (oil bubbles) capillary tubes

o 0 0 0

0

O o O O ° o o o 0° I

i

I

I

i

(a)

t

l

i

C

i

(b)

Fig. 2. The experimental techniques employed were (a) injection of small gas bubbles or oil droplets through a set of fine capillary tubes. The liquids used were aqueous solutions of a hydroxyethylcellulose polymer. The range of viscosity values covered for the upper layer (/x 2) and lower layer (P~I)was such that 1
164

introduced. The bulk densities of the suspensions were less than those of the viscous fluid, but the particles were denser than both fluids, and sedimented with velocities corresponding to low Reynolds numbers.

3.2 Qualitativedescription of flow phenomena The behaviour of the bubbles at the interface depended strongly on the ratio of the viscosities of the two fluids. When the viscosities were approximately equal, bubbles crossed the interface

N. THOMAS ET AL.

individually, each one carrying with it a small amount of fluid from the lower layer, visualised by dying the lower layer (Fig. 3a and b). The result was efficient small-scale mixing of the two fluids and a relatively homogeneous upper layer in which the concentration of lower layer fluid increased with time. However, when the upper layer viscosity was substantially higher than the lower layer viscosity, the bubbles decelerated upon reaching the interface, and a foam layer formed there (Fig. 3c). The foam had a bulk density less than that of the upper layer fluid,

Fig. 3. Photographs of experiments in which gas bubbles were injected into the tank (a). (b) W h e n the viscosity of the upper layer and the lower dyed layer were approximately equal (the ratio in this case was 2), homogeneous bubbling occurred. (c) Here the viscosity ratio is = 100, a thin foam layer has formed at the interface and is just going unstable. (d) A few m o m e n t s later, plumes are generated from the foam layer.

MIXING OF STRATIFIED LIQUIDS BY THE MOTION OF GAS BUBBLES

and, after a short period of accumulation, it became unstable, generating a set of bubble plumes which rose into the upper layer (Fig. 3d). The plumes consisted of heads of many packed bubbles connected to the interface region by narrow conduits of lower layer fluid, resembling 'cavity' plumes which form when the fluid forming the plume is substantially less viscous than the surrounding medium [9]. Bubbles rose rapidly up the conduits to become incorporated in the heads, which swelled and accelerated as they rose. For a given viscosity contrast, the higher the gas flux the larger were the initial wavelengths of the instability (plume spacing) and the size of the plume heads. Plume size similarly increased with

165

increasing viscosity contrast at fixed gas flux. One feature, observed only in the experiments with oil, was the coalescence of the droplets in the emulsion at the interface. This had the effect of reducing the rate of entrainment of lower layer fluid into the upper layer. Photographs of a representative experiment with solid particles, carried out at high viscosity ratio, are shown in Fig. 4. As the particles in the upper layer settled, a particle-rich layer formed at the interface between the fluids (Fig. 4a). This layer became unstable (Fig. 4b), and descended into the lower viscous layer in the form of plumes (Fig. 4c). The size of the plume heads and their descent velocity increased with time (Fig. 4d).

Fig. 3. ( c o n t i n u e d ) .

166

N. THOMAS ET AL.

The wavelength and the growth rate of the plumes varied depending on the initial concentration and the viscosities of the two fluids. The rate of increase in plume head size and that of the velocity of each plume increased with increasing initial particle concentration for a given fluid pair. In these experiments the accumulation rate at the interface is given by the product of the initial particle concentration and the Stokes velocity aoVs, and is analogous to the gas flux. We deduce that the growth rate of the plumes increased with increasing accumulation rate. The fact that we observed phenomena similar to those seen in experiments with bubbles implies that the difference between the case of an initially homo-

geneous particle distribution and that in which bubbles are continually introduced is not a crucial one. After the inital phase of plume formation the conduits left by the initial passage of the plume heads remained in place. During the latter parts of experiments new plumes continued to form, but much lower viscosity fluid was transferred into the more viscous layer via these conduits. Unlike the experiments at low viscosity contrasts, mixing caused by plumes did not produce a mixture homogeneous on a small scale, but rather a coarse, inhomogeneous mixture. Another novel effect is the coalescence of bubbles in the foam, an important factor determining

Fig. 4. (a-d) Photographs of experiments with particle suspensions. See text for discussion.

167

M I X I N G O F S T R A T I F I E D L I Q U I D S BY T H E M O T I O N O F GAS BUBBLES

the depth and density of the foam. Coalescence was often seen in experiments with oil droplets. We observed that high viscosity ratio, high gas flux and low surface tension all favour coalescence. However, it is difficult to quantify the conditions under which this occurs. We measured the flux of low-viscosity fluid entrained by the bubbles, and observed that coalescence could greatly reduce the amount of lower layer fluid entrained into the upper layer by the bubbles. These results on mixing are beyond the scope of the present paper, but coalescence is an intriguing possibility for magmas, in which mainly gas and virtually no mafic melt would be transferred into the silicic magma.

4. The limits of the regime of plume formation We analyse the two regimes by which bubbles traversed the interface by considering the gas flux which can pass through a liquid when the bubbles are homogeneously distributed. We make use of the drift flux, which is the flux of either phase relative to a surface moving at the volumetric average velocity of the two phases. Defining Vg and v~ as the velocities of gas and liquid relative to the fixed laboratory framework, fg and fl as the fluxes per unit area of gas and liquid, f as the total flux and a as the volume fraction of gas, we have: fg = a V g , f I = (1 - oz)v I (4.la,b)

F i g . 4. ( c o n t i n u e d ) .

168

N. T H O M A S E T AL.

I

T h e drift flux is d e f i n e d as: fg, = a(Vg - f )

I

I

I

(a)

i

(4.2)

which, in t e r m s of the velocity of the gas relative to the liquid, is: fg, = a ( 1

-

a)(

Ug U,) -

-

x

(4.3)

A t small values of a , V g - C j is equal to the S t o k e s ' velocity of an i s o l a t e d s p h e r e in an infinite fluid (v S) a n d it is simple to calculate the gas flux. A t large values of ~ i n t e r a c t i o n s b e t w e e n n e i g h b o u r i n g b u b b l e s m a k e the d y n a m i c a l description of the system difficult. H o w e v e r , a large b o d y of e x p e r i m e n t a l d a t a on diverse t w o - p h a s e systems can be r a t i o n a l i s e d by plotting the observed drift flux as a function of the volume fraction o f the d i s c o n n e c t e d phase, in this case gas [10]. T h e g e n e r a l curve which successfully c o r r e l a t e s m a n y d a t a is:

0 0

0.2 0.4 0.6 bubble volume fraction

0.8

t

t

t

t

0.2

0.4

0.6

0.8

1

(b)

x

fg, = t ~ a ( 1 - a ) "

(4.4)

T h e e x p o n e n t n is a function of the R e y n o l d s n u m b e r ; for gas b u b b l e s in a liquid at low R e y n o l d s n u m b e r , n = 2 is the a p p r o p r i a t e value [10], for which the drift flux curve is shown in Fig. 5a. T h e i m p o r t a n t f e a t u r e is the m a x i m u m , which is n a t u r a l as the m o v e m e n t of the isolated bubbles at low a differs greatly from that of closep a c k e d b u b b l e s with flow of interstitial liquid. F u r t h e r m o r e , the drift flux must be z e r o for a = 0 a n d a = 1, as shown by (4.3). O u r e x p e r i m e n t s are the special case of gas b u b b l i n g t h r o u g h static liquid, i.e. v~ = 0, and for n = 2 use of (4.3) a n d (4.4) gives: fg =

v~a(] -

a)

(4.5)

This result can b e used to i n t e r p r e t our l a b o r a tory results, a n d p r o v i d e s a way of g e n e r a l i s i n g t h e m to n a t u r a l situations. C o n s i d e r a two-layer system with low-viscosity fluid u n d e r l y i n g highviscosity fluid. If the R e y n o l d s n u m b e r is low in b o t h layers, curves of gas flux versus a have the s a m e functional form b u t different n u m e r i c a l values a c c o r d i n g to the viscosities (Fig. 5b). F o r a gas flux b e l o w the m a x i m u m of the curve for the u p p e r layer (Fc), a s t e a d y flux m a y pass t h r o u g h the system, a l t h o u g h the gas fraction in the u p p e r layer must be h i g h e r t h a n in the lower layer. However, fluxes can pass t h r o u g h the lower layer which are g r e a t e r t h a n the critical flux (F~), and

Fc

0

c~ 0

bubble volume fraction c~ Fig. 5. (a) For a homogeneous bubble distribution. The dependence of the drift flux of gas relative to liquid given by eq. (4.4) with n = 2. (b). Representation of the gas flux per unit area when there is no net liquid flux (see eq. 4.5). Note the maximum at a = 0.5. The upper curve is for the less viscous lower layer and the lower curve is for the more viscous upper layer. In order to ensure constant flux at the interface, the bubble volume fraction changes from a~ to a 2. Fc represents the maximum gas flux that can pass through the upper layer with a homogeneous bubble distribution.

c a n n o t be a t t a i n e d in the u p p e r layer if the b u b b l e s are h o m o g e n e o u s l y distributed. This implies that if we i m p o s e a gas flux g r e a t e r t h a n Fc, a t r a n s i t i o n to s o m e n o n - h o m o g e n e o u s flow r e g i m e must occur in the u p p e r layer. W e p r e s e n t o u r e x p e r i m e n t a l d a t a in Fig. 6 by plotting the ratio of the gas flux p e r unit a r e a (fg) to the critical flux (Fc) against the u p p e r layer viscosity. W h e n fg exceeds F c we expect p l u m e s to form, a n d w h e n it is less t h a n F c the b u b b l e s should pass the interface individually, and we see that o u r analysis p r o v i d e s a g o o d f r a m e w o r k for

MIXING

OF STRATIFIED

LIQUIDS

BY THE

103

MOTION

'

o

-r-,:

1

-

*~x

+

×

××

10 -3 10 -3

A

@

0

10-1 10-2

o zx~

o

1o

×

+ × × I

10-2

I

I

I

10-!

1

10

102

viscosity of the upper layer (Pa.s) experiments with gas bubbles: zx plumes weak regrouping of bubbles

experiments with oil bubbles: O plumes +

169

BUBBLES

6 8'

102 e~

e.~ e~

OF GAS

isolated bubbles

× isolated bubbles Fig. 6. The experimental data to test the criterion for plume formation devised in section 4. For each experiment we show the ratio of the imposed flux of bubbles to the critical flux (Fc). © and zx show experiments in which plumes formed, and × and + show those in which the bubbles crossed the interface individually. The stippled diamonds show experiments in which some weak regrouping of bubbles occurred at the interface without clear plume formation.

interpreting the observations. Note that the absence of natural convection in our experiments does not prejudice application of the above criterion to magmatic systems. This is because we only need to know the gas flux at the interface in each case, and it is immaterial whether this flux comes from a convecting fluid or a quiescent one. This generality allows us to use our criterion to estimate the conditions under which plume formation can occur in stratified m a g m a chambers (see section 7). 5. P l u m e ascent data

Although in some cases diapir plumes were observed, the plumes were generally of the cavity type in which a large head is fed by flow in a thin conduit (Fig. 7b), causing the heads to swell and accelerate. In our experiments we have a field of plumes which interact and there is no guarantee that the plumes should be fed by a constant flux from the interface region, and we characterized our data using the following approach. In each

experiment, the lengths ( L ) of the plumes were measured as a function of time, the data were fitted to a curve of the form L = A t x and the values of x were plotted as a histogram, typical examples of which are given in Figs. 7a and c. This shows that the cavity plume model works reasonably well for these plumes, deviations from this being due to interactions between plumes, and because the volume flux from the interface region decreased somewhat with time. The latter was confirmed by measurements of the diameters of the plume heads as a function of time. As far as m a g m a chambers are concerned, these quantitative details are of secondary importance. The main point is that such plumes traverse the upper viscous layer at a rate vastly greater than would individual bubbles, and can efficiently transfer exsolved gas from the mafic m a g m a to overlying silicic magma. In short, the plume regime is a dynamic adaptation of the system which enables the upper layer to pass the same elevated gas flux as the lower layer when this is above the critical flux F c (Fig. 5b). 6. G r a v i t a t i o n a l instability o f the f o a m layer

One of the main pieces of field data available is the size of mafic inclusions in intermediate and silicic lavas, and we would like to know whether the sizes of such inclusions correspond to those of plumes generated by the gravitational instability of a magmatic foam layer. In Appendix A we give an analysis of the instability of a foam, based on that of Lister and Kerr [11] for a thin buoyant layer sandwiched between two semi-infinite layers, but allowing for the growth of the foam as more bubbles accumulate (Fig. 8a). We assume that instability occurs when the rate of thickening of the buoyant layer and the rate of growth of the instability are equal. The gas fraction in the foam (E) is unknown and cannot be controlled independently of the experimental conditions. Indeed, we observed that e tended to increase with both gas flux and viscosity ratio, all else being equal. Our approach allows us to solve for the wavelength of the instability as a function of e, and assuming bounds of 0.1 < E < 0.6 we obtain satisfactory agreement with the wavelengths measured in our experiments (Fig. 8b). The quantitative application of this analysis to m a g m a cham-

170

N. T H O M A S

ET AL.

bers requires knowledge of the gas flux in the mafic layer, which we estimate in the following section.

where Q is the b u b b l e p r o d u c t i o n rate a n d h is the thickness of the lower layer. With a = O/o at t = 0, the solution is:

7. The d y n a m i c s of stratified m a g m a c h a m b e r s

a = Qh/v~ + (a o - Oh/v~) exp(-v~t/h)

I n o r d e r to apply this model, we must constrain the m a g m a viscosities a n d the gas flux in the lower mafic layer. Viscosities can be estim a t e d quite well, but the gas flux is m o r e difficult to obtain. A t low b u b b l e c o n c e n t r a t i o n s this flux is given by the p r o d u c t of a a n d the Stokes velocity (v~) of an isolated b u b b l e , a n d we now estimate a. T h e differential e q u a t i o n governing the b u b b l e c o n c e n t r a t i o n , assuming a u n i f o r m distribution in the lower layer, is: do/ h ~ - = h Q - a~'~ (7.1)

Bubble p r o d u c t i o n a n d segregation initially compete u n d e r t r a n s i e n t conditions, but the system tends to the steady state: O/steady

=

Qh/v~

(7.2)

(7.3)

for which the gas flux a z~ = Qh. D e p e n d i n g o n m a g m a composition, physical properties a n d the pressure, o% can be less or greater t h a n O/steady. T h e characteristic time r e q u i r e d to a t t a i n steady state is h / Vs. It is difficult to quantify Q for m a g m a chambers as a f u n c t i o n of pressure, cooling rate a n d the composition of volatile c o m p o n e n t s , and

40

.~ 20

0 0.5

1

1.5

range of value of x 20 (c)

:~:3333::

iiiiiiii

10 experiments

:::::::::

lO

~iiii~i~itiiiiiiiiiiiiiiii

N

i!iiiiiiii!iii!i!! 0.5

1 1.5 2 range of value of x Fig. 7. Histograms of the time exponents for sets of plumes from experiments with (a) bubbles and (c) particles. A range of values of x are obtained (mostly between 1.1 and 1.5) with a maximum at x --~1.2. Reference values are x = 1, for the ascent of an isolated sphere of fixed volume at its Stokes terminal velocity, 1.4 for an isolated cavity plume fed by a constant flux [9], and x = 1.67 for a sphere whose volume grows linearly in time. (b) Sketch of a cavity plume, which was the type most commonlyobserved.

171

MIXING OF STRATIFIED LIQUIDS BY THE MOTION OF GAS BUBBLES

associated d e f o r m a t i o n of c o u n t r y rocks [12]. However, for the p r e s e n t p u r p o s e s this is probably unnecessary. W e have m a d e calculations for volatile species of p u r e H 2 0 (Fig. 9a) a n d of p u r e C O 2 (Fig. 9b) using the solubility laws:

h e n c e c o m p u t e O/steady, b u t we c a n nevertheless estimate the gas flux by placing b o u n d s o n a. W e have e s t i m a t e d the v o l u m e fraction of gas prod u c e d in mafic m a g m a by fractional crystallization by calculations similar to those of H u p p e r t et al. [5], in which o n e assumes a c o n s t a n t pressure, a solubility law for volatiles in the melt, the total c o n t e n t of volatiles a n d e q u a t i o n s of state. Conservation e q u a t i o n s for mass a n d v o l u m e are solved to o b t a i n a as a f u n c t i o n of the a m o u n t of crystallization. M o r e c o m p l e t e solutions can allow for the c h a n g e in p r e s s u r e in the c h a m b e r a n d

H 2 0 : x d = 6.8 × 1 0 - 8 p 0"7

(7.4a)

C O 2 : x d = 4.4 × 10-12P 1°

(7.4b)

where x d is the mass fraction of volatiles dissolved in the mafic melt [13,14]. F o r H 2 0 , exsolution generally occurs after extensive crystalliza-

(a)

high viscosity liquid (I)2,

h(t)

~

~ ~=lxl ( 1 - e ~5/2

OO

c 0 0

pm=Pl ( 1 - e )

o

o 0

•

O

0 o o

•

o

• •

0

f

o~

gas bubble

0 o

low viscosity liquid (13~P-t) (b)

105 104 .~

103 e = 0.6 L=6cm

•~

102

e = 0.1 t=6cm e = 0.6 X---lcm

;> 10

e = 0.1 l = l c m 1

10-5

10-4 Flux per unit area fg

10-3

Fig. 8. (a) The foam is subject to a gravitational instability, although the wavelength and growth rate of the fastest growing disturbance are functions of the thickness of the foam, which increases as more bubbles accumulate. (b) The points show the conditions of experiments (fg and /x2//~i) in which we measured the wavelength (A) of the instability. We obtained values of between 1 cm and 6cm. The observed range of A can be explained by assuming the gas fraction in the foam (~) to vary between the reasonable bounds of 0.1 and 0.6 as shown by the curves which give the predictions of (A.12) for the experimental conditions.

172

N. THOMAS ET AL.

regimes of the interface, is calculated with OLcrit= 0.5 (the maximum in the flux curve in Fig: 5b):

(a) 0.4 (1).'." " " ' " " "" . . . .

'

Apgd 2 F c = 0.25 18/x~-7

= 0.3

~g 0.2

where d = bubble diameter. We have the criterion (Fig. 6) that the plume regime is to be expected if:

0.l

00

(7.5)

0.2

0.4

0.6

0.8

Crystal mass fraction

(b) oA 0.08 (I)

~- 0.06 "~ 0.04

(2)

0.02

(3)

0.2

0.4

0.6

0.8

Crystal mass fraction

Fig. 9. Calculations of the volume fraction of gas formed in a mafic magma as fractional crystallization proceeds, and as a function of total volatile content and pressure. (a) For a pure H 2 0 volatile phase: (1) 3 wt% H 2 0 at 500 bar; (2) 2 wt%, 500 bar; (3) 3 wt%, 1500 bar; (4) 1 wt%, 500 bar; (5) 2 wt%, 1500 bar; (6) 3 wt%, 3000 bar; (7) 2 wt%, 3000 bar; (8) 1 wt%, 1500 bar. (b) For a pure CO 2 volatile phase: (1) 0.5 wt% CO 2 at 500 bar; (2) 1500 bar; (3) 3000 bar. CO 2 is relatively insoluble and so the magma is easily saturated but not much gas is produced. A magma rich in H 2 0 can produce appreciable amounts of gas.

fg/Fc =

(7.6)

where a is the gas fraction in the lower layer, fg depends basically on the lower layer viscosity, and rewriting (7.6) in terms of the viscosity ratio between the layers, we expect the plume regime if the viscosity ratio 0Xz//X l) is greater than 3-30. This is interesting as it implies that both regimes may occur under geological conditions. For example in the case of the intrusion of a mafic layer beneath an upper layer ranging from andesite through to rhyolite, the plume regime is most likely. When the viscosity contrast is much lower, for example if basalt were intruded beneath slightly more differentiated mafic magma such as basaltic andesite, the two could become mixed together to produce a homogeneous hybrid in the bubbling regime. 10 6

:~

10 ~ ~l~cm

=~ .~

tion, and the final gas fraction is of the order of 0.1 (Fig. 9a). Because of its much lower solubility, CO 2 exsolves after little crystallization. Indeed, for concentrations in the range 0.1-0.5 wt.% CO 2, the magma is always saturated in an upper crustal chamber. The volatile species is likely to be a mixture of H 2 0 and CO 2, under which circumstances Figs. 9a and 9b suggest that a gas phase will often be present during the entire cooling history, a o is likely to be approximately 0.01 and the total volume fraction of the order of 0.1. These arguments suggest an order of magnitude estimate for a of 0.01-0.1. The critical gas flux for the upper magma layer, corresponding to the limit of the dynamic

OLUs

~-c > 1

8

10~

X= 10cm

103

10~ 10 .9

lff 8

10-7

10 -6

Flux per unit area fg (m/s) Fig. 10. The predictions of (A.12) for the wavelength of the instability of a magmatic foam layer as a function of viscosity ratio between the magmas and gas flux. Values of gas flux were calculated assuming a lower layer viscosity of 10 Pa& middot;s, a bubble size of 0.1 mm, and Ap of 2500 kg.m 3. Wavelength contours are shown for A = 1 cm, 10 cm and 1 m. Heavy dashed curves are for E = 0.2 and lighter dashed curves for E = 0.6. Plausible magmatic conditions are shown by the shaded area. This range of values is in good agreement with the typical sizes of mafic inclusions found in intermediate to silicic lavas.

173

MIXING O F S T R A T I F I E D L I Q U I D S BY T H E M O T I O N OF GAS BUBBLES

Figure 10 shows the quantitative application to magma chambers of the analysis of the gravitational instability of the foam. Contours of the wavelength of the instability are plotted as a function of viscosity contrast and gas flux. The gas fraction in the foam (e) is unknown, but by assuming bounds of 0.2 < • < 0.6 we predict A to be in the range 1 cm to 1 m, with a most likely value being perhaps 10 cm. The wavelength (i.e., strictly the plume spacing at the moment of instability) is likely to be approximately equal to the plume size, and this result is in good agreement with the sizes of mafic inclusions in silicic to intermediate rocks [2,15,3]. This assumes that the inclusions have not undergone substantial dilation during their transport from the magma chamber to the surface. This is plausible because the inclusions are chilled by contact with the silicic magma and hence become very viscous. Indeed, the likely amount of dilation can be shown to be small at high viscosities [16]. The reason for these small values of A is that although magmatic viscosities are much higher than our experimental fluids, which tends to increase the wavelength, the gas fluxes are much lower and these two effects come close to cancelling one another out. This good quantitative agreement seems to be evidence in favour of the physical model explained here and originally proposed in qualitative form by Eichelberger [4]. For example, the model of Huppert et al. [5] in which the entire mafic layer becomes less dense and goes unstable does not obviously predict small inclusions and requires some supplementary hypothesis such as break up during convective overturn or eruption. Sparks and Marshall [17] argued that a large temperature difference between two magmas should inhibit their mechanical mixing because the mixture comes rapidly to thermal equilibrium, at which temperature the mafic end member is too viscous to be stirred in. A small thermal difference, on the other hand, tends to promote mixing. While this is generally correct, it begs the question of how mixing takes place. We have shown that a low viscosity contrast (also likely to be associated with a small temperature contrast) promotes hybridization because the bubbling regime in which the magmas are stirred together on a small scale is more likely. A high viscosity contrast promotes plume formation and hence

mixing on a more gross scale. The thermal and fluid dynamical effects thus should reinforce one another. A related point is that the temperature gradient which will inevitably form at the interface by thermal diffusion between the magmas will greatly reduce the viscosity contrast between them. In consequence, even when the initial viscosity contrast is high, in the immediate region of the interface, the magmas may become somewhat hybridized in the bubbling regime. This mixed layer can then become unstable to produce plumes because, far from the interface, the magma of the upper layer still has high viscosity. This is somewhat speculative as it depends on rates of mixing and thermal diffusion, whose quantitative investigation is beyond the scope of the present study. However, it may provide an explanation for the fact that magmas forming inclusions often show signs of having been hybridized with the host lava, presumably before forming discrete inclusions [e.g. 3,18]

8. Summary When vapour bubbles migrate upwards from a low-viscosity layer into a high-viscosity layer, there are two dynamical regimes for the i n t e r f a c e - - o n e of homogeneous bubbling, and one of formation of a foam layer and plumes; both may occur under magmatic conditions. Hence, the same basic physical mechanism could provide an explanation both for the formation of more mafic inclusions in silicic to intermediate rocks, and for hybrid magmas, depending on the gas flux in the mafic magma and the viscosity ratio. Furthermore, aggregations of vesicles from the mafic magma may ascend rapidly as two-phase plumes, or even as larger gas pockets of coalesced bubbles, through the viscous silicic magma and be mixed in, despite the small value of Stokes velocity for individual vesicles. This may show up as an unusual gas phase composition in the silicic magma such as anomalously high CO 2 (see, for example, the evidence from the Bishop Tuff [1D. An analysis of the gravitational instability of the foam layer produces good quantitative agreement with the sizes of mafic inclusions commonly observed in intermediate and silicic lavas, suggesting that inclusions observed in lava flows may indeed be recording a scale close to that of the initial instability, and not of some secondary pro-

174

N. THOMASET AL.

cess, during eruption for example, by which initially larger inclusions are broken and reduced in size.

Appendix A: gravitational instability of the foam layer The gravitational instability of a layer sandwiched between two semi-infinite layers, when all three fluids have high Prandtl numbers, can be described by using the Stokes equations for each of the three fluids, and imposing continuity of horizontal and vertical components of velocity and stress at the two interfaces [11]. They give general expressions for the wavelength (A) and growth rate (o-) of the most unstable disturbance, as a function of the viscosities and densities of the fluids, for the appropriate boundary conditions (i.e., w = 0 , C32W/0Z2-~'0 at z = + / - o o ) , where w is the vertical velocity. We have adapted the analysis of [11], in which the thickness of the intermediate layer is fixed, in order to treat the instability of the foam layer whose thickness is growing with time as more bubbles accumulate (Fig. 8a). We assume the foam to be homogeneous, with constant gas volume fraction (e), giving it a bulk density Pm and viscosity p.~ (Fig. 8). We have: Pm=Pl(l-E)

+pgE

(A.I)

For the viscosity of the foam we assume [19]: IXm/tXl

_-

1 (1 - e) 5/2

(A.2)

Disregarding the instability, the rate of thickening of the foam is: dh

d-~ = fg/E

(A.3)

where fg is the gas flux per unit area. We define a characteristic growth rate for the foam as:

0 = 1/h(dh/dt)

equal, giving: ~r = (1/~/) - & q / d t - - 0 = ( l / h ) . d h / d t

~r can be calculated as a function of h for given fluids from Lister and Kerr [11], and we know d h / d t for a given gas flux and E. Condition (A.6) closes the problem and enables us to calculate the foam thickness (h c) at the onset of instability and the wavelength of the instability as a function of the experimental conditions. Dimensional analysis of the governing equations leads to the following three dimensionless parameters [11]: V= ~2/~m, W = ~l/~m,

P=(P,-Pm)/(P2-Pm) In our experiments in which plume formation occurred, /z2> >/Xl; however, the relationship between the viscosity of the upper layer (/x 2) and that of the foam (tz m) is a function of E (A.2). This is likely to lie between 0.1 and 0.6, for which limits 1.3 > 1 even for large gas volume fractions. The limits of the dimensionless parameters relevant to this case are V>> 1, W < 1. The results are not very sensitive to P for the range of values in our experiments. In the above limits, the expressions for the wavelength and growth rate of the fastest growing disturbance are considerably simplified, and we use these to carry out the calculations. The expressions given below are not, therefore, perfectly general, but they should cover many cases of interest and allow us to understand simply how the main parameters affect the results. The wavelength (h) of the fastest growing disturbance is: A ~- 4.36h( ]./.2/]3qn )1/3

( A .7)

for which the growth rate is

(A.4)

g'(P2--Pm) "h 0.232(# 2//Xm) 1/3

Assuming that the amplitude of the instability (rl) grows exponentially, with growth rate or, we have:

o" =

d~7

Use of (A.3), (A.6) and (A.8) leads to:

__

(A.6)

(A.8)

tZ2

(A.5)

dt We assume that instability occurs when the growth rate of the foam and that of the instability are

hcefg -(0.232)g(P2-Pm)hc(iz2 ~]/~2]1/3

(A.9)

M I X I N G OF S T R A T I F I E D L I Q U I D S BY T H E M O T I O N OF GAS BUBBLES

The density difference between the two pure fluid layers is much less than between either of the fluids and the foam, so we can write:

P2

-- Prn

=

P2

--

pl(1 -- E) -- Epg = P2 -- Pl( 1 -- E)

--- Epl

(A.10)

Combining this with (A.2) and (A.9) we obtain: [ he

fg/X 2

[ 0.232gPl

1/2

1/6 (1 -- E) -5/12

~-I

(A.u) This value of foam thickness at the m o m e n t of instability (h c) can now be substituted into (A.7) to calculate the wavelength of the instability as a function of the conditions of an experiment and • . Thus:

91[fglz211/2(l-•)5/12 ( l'X211/6 ( A . 1 2 ) " tTL-p, j

--TJ

The principal unknown in our experiments and in magma chambers is the gas fraction in the foam. The full dynamical problem of how the bubbles accumulate at the interface is difficult. However, the interest of the above approach is that we obtain solutions as a function of •. The value of • is likely to vary with gas flux and fluid viscosities and we have no independent check. In our experiments, we counted the number of plumes generated by the first instability and deduced the mean spacing. (A.12) gives results which agree with the measured values of A for a range of values of gas fraction 0.1 < • < 0.6, which is plausible (Fig. 8b). This suggests that we can have reasonable confidence in the above analysis, and use it to estimate the wavelength of this instability under magmatic conditions (see section 7). References 1 A.T. Anderson, S. Newman, S.N. Williams, T.H. Druitt, C. Skirius and E. Stolper, H 2 0 , CO2, CI and gas in Plinian and ash-flow Bishop rhyolite, Geology 17, 221-225, 1990. 2 J.C. Eichelberger, Origin of andesite and dacite: Evidence

175 of mixing at Glass Mountain in California and at other circum-Paciflc volcanoes, Geol. Soc. Am. Bull. 86, 13811391, 1975. 3 C.R. Bacon, Magmatic inclusions in intermediate and silicic volcanic rocks, J. Gephys. Res. 91, 6091-6112, 1986. 4 J.C. Eichelberger, Vesiculation of mafic magma during replenishment of silicic magma reservoirs, Nature 288, 446-450, 1980. 5 H.E. Huppert, R.S.J. Sparks and J.S. Turner, Effects of volatiles on mixing in calc-alkaline magma systems, Nature 297, 554-557, 1982. 6 D. Martin and R. Nokes, Crystal settling in a vigorously convecting magma chamber, Nature 332, 534-536, 1988. 7 T. Koyaguchi, M.A. Hallworth, H.E. Huppert and R.S.J. Sparks, Sedimentation of particles from a convecting fluid, Nature 343, 447-450, 1990. 8 T. Koyaguchi, M.A. Halworth and H.E. Huppert, An experimental study on the effects of phenocrysts on convection in magmas, J. Volcanol. Geotherm. Res., in press, 1993. 9 P. Olson and H. Singer, Creeping plumes, J. Fluid Mech. 158, 511-531, 1985. 10 G.B. Wallis, One-Dimensional, Two-Phase Flow, 408 pp., McGraw-Hill, New York, 1969; 11 J.R. Lister and R.C. Kerr, The effect of geometry on the gravitational instability of a buoyant region of viscous fluid, J. Fluid Mech. 202, 577-594, 1989. 12 S.R. Tait, C. Jaupart and S. VergnioUe, Pressure, gas content and eruption periodicity of a shallow, crystallising magma chamber, Earth Planet. Sci. Lett. 92, 107-123, 1989. 13 E. Stolper and J.R. Holloway, Experimental determination of the solubility of carbon dioxide in molten basalt at low pressure, Earth Planet. Sci. Lett. 87, 397-408, 1988. 14 D.L. Hamilton, C.W. Burnham and E.F. Osborn The solubility of water and effects of oxygen fugacity and water content on crystallisation in mafic magmas, J. Petrol. 5, 21-39, 1964. 15 C.R. Bacon and J. Metz, Magmatic inclusions in rhyolites, contaminated basalts, and compositional zonation beneath the Coso field, California, Contrib. Mineral. Petrol. 85, 346-365, 1984. 16 N. Thomas, C. Jaupart and S. Vergniolle, The vesicularity of volcanic pumices, in prep, 1993. 17 R.S.J. Sparks and L.A. Marshall, Thermal and mechanical constraints on mixing between mafic and silicic magmas, J. Volcanol. Geotherm. Res. 29, 99-124, 1986. 18 T. Koyaguchi, Evidence for two-stage mixing in magmatic inclusions and rhyolite lava domes on Nijima Island, Japan, J. Volcanol. Geotherm. Res. 29, 71-98, 1986. 19 C. Jaupart and S. Vergniolle, The generation and collapse of a foam layer at the roof of a basaltic magma chamber, J. Fluid Mech. 203, 347-380, 1989.