Kinematic significance of morphological structures generated by mixing of magmas: a case study from...

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ters 222 (2004) 1051–1066
Earth and Planetary Science Let
Kinematic significance of morphological structures generated by

mixing of magmas: a case study from Salina Island (southern Italy)

D. Peruginia,*, G. Venturab, M. Petrellia, G. Polia

aDepartment of Earth Sciences, University of Perugia, Piazza Universita, I-06100 Perugia, ItalybOsservatorio Vesuviano, Istituto Nazionale di Geofisica e Vulcanologia (INGV), Via Diocleziano, 328, I-80124 Napoli, Italy

Received 7 November 2003; received in revised form 26 February 2004; accepted 24 March 2004

Abstract

Morphological features of mixing/mingling structures of heterogeneous juveniles from the 13-ka-old Upper Pollara eruption

(Salina Island, southern Italy) are studied. These heterogeneous rocks result from the mixing between an andesitic and a

rhyolitic magma. Concentration patterns generated by magma mixing are analyzed on digital images and results have been

compared with those obtained from numerical simulations of mixing processes coupling chaotic advection and chemical

diffusion. Results show that the Upper Pollara rocks were produced by mixing 24–49% of andesite and 76–51% of rhyolite.

Multifractal analysis of the mixing structures is performed and the mixing intensity, i.e. the degree of interaction between the

two end-member magmas, has been deduced from the analysis of the singularity spectrum. Results show that narrower is the

multifractal spectrum, the higher is the degree of homogenization of magmas. Reynolds number (Re) during mixing has been

estimated from the geometrical analysis of mixing structures by using experimentally defined relationships between shape

parameters and Re. Results show that Re is between ca. 500 and 7000. There is a positive correlation between the estimated

initial percentage of mafic magma in the different analyzed samples and Re, in agreement with the observation that the higher is

the percentage of the mafic, lower viscosity magma, the higher is the turbulence of the mixing system. In addition, our analysis

reveals an unexpected inverse relationship between the calculated Re and the degree of magma homogeneity suggesting that

energy dissipation may have had a major role in the controlling mixing process because dissipation is inversely proportional to

the mixing efficiency. Results suggest that mixing processes between the andesitic and rhyolitic magmas mainly developed in

the conduit. It is also suggested that mixing occurred in a shear layer-type or pipe-type flow.

D 2004 Elsevier B.V. All rights reserved.

Keywords: magma mixing; morphological structures; concentration patterns; image analysis; multifractals; fluid dynamic regime

1. Introduction

Textural heterogeneity due to magma mixing/

mingling is a common feature of volcanic rocks

0012-821X/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.epsl.2004.03.038

* Corresponding author. Tel.: +39-75-585-2652; fax: +39-75-

585-2603.

E-mail address: [email protected] (D. Perugini).

(e.g. Refs. [1–5]), and the processes responsible

for them have been discussed extensively [5–10].

This heterogeneity includes enclaves, banding, and

‘streaky’ structures [4,11–13]. Although studies on

the mineralogical and geochemical features of mixed

rocks are abundant (e.g. Refs. [12,14–17]), only few

studies focus on quantitative analyses of morpholo-

gies related to textural heterogeneity [18–24]. These

D. Perugini et al. / Earth and Planetary Science Letters 222 (2004) 1051–10661052

studies mainly focus on (1) the deformation of the

interface between the two interacting magmas in

laminar flows (e.g. Refs. [20,21]) and turbulent flows

[18]; (2) the effects of the vesiculation on the

deformation of enclaves [23]; and (3) the relation-

ships between the morphology of the mingling/mix-

ing structures and the degree of magma interaction

[22,24].

Studies on the fluid-dynamics of interacting

liquids (e.g. Refs. [25–30]), as well as analogue

models on magma mixing (e.g. Refs. [31–33]), have

shown that different morphological structures (e.g.

folds, blobs, filaments, vortexes) give useful infor-

mation about the flow regime and the dynamics of

mixing.

In this study, we analyze mixing morphological

structures of heterogeneous juvenile fragments from a

13-ka-old flow deposit of the Upper Pollara eruption

(Salina Island, southern Italy). In a first step, we

show that different colours of the glass correspond to

different magma compositions. In a second step, we

perform image and multifractal analysis of the mix-

ing structures in order to determine (a) the initial

percentages of the end-member magmas, (b) the

mixing intensity and (c) the flow regime acting

during the mixing process. In a third step, we discuss

the relationships between (a), (b) and (c), and pro-

pose a kinematic model for magma interaction. We

show that the analytical approach proposed here can

be used to extract petrological and physical informa-

tion from the analysis of the concentration patterns of

mixed rocks.

2. Geological setting and the Upper Pollara

eruption

The Pollara depression is located in the north-

western sector of Salina Island (Aeolian Archipela-

go, Southern Tyrrhenian Sea), which consists of five

main volcanoes (Corvo, Rivi-Capo, Fossa delle

Felci, Porri and Pollara) formed between 170 and

13 ka (Fig. 1A; Ref. [34]). Volcanism within the

Pollara depression started with the emission of the

Punta Perciato lava flow (30 ka [35]) and ended

with two explosive eruptions known as Lower

Pollara (24 ka [12]) and Upper Pollara (13 ka

[36]). The Lower Pollara deposits consist of three

main fall units. Magma compositions are andesitic

(lowermost fall unit), andesitic and dacitic (interme-

diate unit) and rhyolitic (uppermost unit). Such

variability has been interpreted as the result of the

syn-eruptive turbulent mixing between an aphyric

rhyolitic magma and an andesitic, partly crystal-

lized, magma [12].

The Upper Pollara deposits (Fig. 1C) consist of a

70-m-thick sequence of fall beds (lithofacies A1 in

Fig. 1C) followed by a rhythmic alternation of

massive (lithofacies B) to crudely stratified (lithofa-

cies A) coarse ash and lapilli layers. In the upper part

of the sequence, thick, inversely graded beds of

medium-coarse ash and lapilli (lithofacies C) occur

[37,38]. The sequence ends with a massive bed of

fine to medium ash (lithofacies D). According to Ref.

[39], the Upper Pollara sequence mainly resulted

from the emplacement of highly concentrated pyro-

clastic flows.

3. The Upper Pollara juvenile component:

evidence of magma mixing processes

On the basis of petrographic investigations on the

coarse (2 mm) to medium (1/16 mm) ash fraction of

the deposits, the juvenile glass population consists of

four main classes [38] (Fig. 1C): (i) dark glass, (ii)

colorless glass, (iii) heterogeneous glass and (iv)

gray glass with microlites. Both (i) and (iv) glasses

are less than 20 vol.% in the sequence. Glass (ii)

decreases with increasing stratigraphic height, where-

as glass (iii) increases. Detailed observations (Fig.

2A–C) show that glass (iii) consists of micrometer-

to centimeter-sized filament-like structures, plane-

parallel and poorly folded bands to vortex- and

strongly folded-like structures. In detail, inside the

same mixing system, it is possible to observe the

contemporaneous occurrence of (i) filament-like

regions in which magmas are in contact along wide

interfaces and (ii) coherent regions, showing a glob-

ular shape, and occurring between filament-like

regions (Fig. 2A–C). It is noteworthy that the

occurrence of such structures has been widely docu-

mented in mixed volcanic rocks and it is interpreted

as the result of chaotic dynamics of the magma

mixing process [22,24]. As shown by Ref. [34],

glass compositions range from high-potassium

Fig. 1. (A) Structural scheme of the Aeolian Islands. (B) Simplified geological map of Salina Island (from Ref. [37]) with the distribution of the

Pollara pyroclastics. (C) Stratigraphic column of the Upper Pollara deposits (from Ref. [38]). The vol.% variation of the different types of

juvenile fragments with stratigraphy is also reported (data from Ref. [38]).

D. Perugini et al. / Earth and Planetary Science Letters 222 (2004) 1051–1066 1053

Fig. 2. (A–C) Pictures of representative magma mixing structures occurring in the Upper Pollara sequence; (D) SiO2 vs. grey level plot for the

Upper Pollara juveniles (data from Ref. [22]). The best fit line among the data and the 0.95 confidence is also reported.


(HK) andesites (SiO2f 63%) to high-silica rhyolites

(SiO2f 76 wt.%). According to Refs. [22,39], the

geochemical data indicate, together with the macro-

scopic and petrographic features of the juvenile

component, that the Upper Pollara magmas are the

product of the mixing of a HK andesitic magma with

a high-silica rhyolite before the magma fragmenta-

tion in the conduit. The whole eruptive sequence has

been interpreted as the result of the emptying of a

normally zoned magmatic reservoir.

4. Analysis of the concentration patterns of

heterogeneous juveniles

Digital images of heterogeneous juveniles show

that the dispersion of the mafic magma into the felsic

one generated complex morphological structures

whose gray value is highly variable (Fig. 2A–C).

[22] selected poorly vesiculated (vesicles < 5 vol.%)

and subaphyric (crystalsV 5 vol.%) heterogeneous

juvenile clasts from the Upper Pollara sequence in


order to calibrate the gray value of digital images with

the glass composition. Glass composition has been

determined by electron microprobe on homogeneous

glass portions with variable gray intensity. Details

about the analytical methods can be found in Ref.

[22]. Results of the coupled image/chemical analysis

of the selected Upper Pollara clasts are reported in

Fig. 2D, where a positive correlation between gray

values and chemical composition (SiO2 wt.%) of

glasses is observed. The best fitting curve is the

polynomial curve reported in the graph [22]. This

result indicates that differences in the gray value of

images reflect different chemical compositions. It

should be noted that the gray value equal to 20

(SiO2f 63 wt.%) corresponds to the most mafic

magma and that gray values below 20 are associated

to the few vesicles occurring in the samples. Since the

focus here is on the analysis of concentration patterns

of samples, gray values below 20 will be not consid-

ered. Gray values above 250 (SiO2f 76 wt.%) are

not detected in the analyzed samples meaning that this

gray value corresponds to the most felsic magma.

Hereafter, results based on analysis of digital images

of samples will be presented considering the SiO2

Fig. 3. Frequency histograms of concentration for the three representative m

level (GL) and corresponding SiO2 values are reported. Light gray areas o

vesicles (GL < 20) and not detected (GL>250) in the mixing structures, re

composition of glasses calculated by using the equa-

tion reported in Fig. 2D.

5. Determination of initial percentages of magmas

We selected 10 heterogeneous clasts from the Upper

Pollara sequence. These samples have been collected

within two inversely graded beds (lithofacies B) at 25–

30 m from the base of the stratigraphic column, where

the heterogeneous clasts represent over 70 vol.% of the

total juvenile component. All samples have vesicles

< 5 vol.% and a crystal content V 5 vol.%. Polished

thick slices from these samples have been acquired in

gray scale by means of a high-resolution optical

scanner with resolution 1 pixel = 0.075 mm (1 mm=

13.35 pixels) and gray values have been extracted from

digital images using the National Institute of Health

(NIH) image analysis software.

Fig. 3 reports frequency histograms for SiO2 calcu-

lated from digital images of some representative sam-

ples (Fig. 2A–C). Histograms are bell-shaped and

encompass all compositions between SiO2 = 63%

(mafic magma) and SiO2 = 76% (felsic magma) with

agma mixing structures shown in Fig. 2. Along the x-axis both gray

n the left and right side of the graph represent gray levels related to

spectively.


intermediate compositions having the highest frequen-

cy. The histograms for the different images are different

because the maximum frequency of composition for

each image is placed at different SiO2 values (Fig. 3).

The presence of all the intermediate compositions

between the mafic and felsic end-member, as also

shown by the continuous variation of SiO2 vs. grey

level in Fig. 2D, indicates the occurrence of chemical

diffusion processes. In fact, if diffusion processes

between the end-member magmas were not operating,

only two popula t ions of composi t ion, one

corresponding to the mafic (SiO2 = 63%) and the other

to the felsic (SiO2 = 76%) magma, should be observed

in the histograms and in Fig. 2D.

As introduce above, the morphological structures

of the Upper Pollara juvenile fragments are similar to

those generated by numerical and analogue models

of chaotic mixing of fluids (Ref. [30] and references

therein). Chaotic mixing results from the dispersion

of one fluid in the other fluid by stretching and

folding processes coupled with chemical diffusion.

The process of stretching and folding can occur in

both laminar and turbulent flow regimes and has

been observed in a large variety of flow types (pipe-

flows, shear flows, convective flows). In other

words, chaotic mixing describes a set of processes

(stretching–folding–fragmentation–diffusion) that

occur in a variety of flow regimes and flow types

[30,40]. It is to note that numerical models of chaotic

mixing have been successfully applied to reproduce

morphological structures generated by mixing of

magmas [24,41]. Taking into account that the mor-

phological structures of the Upper Pollara pumices

are analogous to those produced by chaotic advec-

tion [30], a numerical model of chaotic mixing

involving both advection (i.e. stretching and folding)

and chemical diffusion is here used.

The system we consider for advection is the ‘‘sine

flow’’ map (hereafter map [42–44]). The flow is

defined by two motions:

xnþ1 ¼ xn þk

2� sinð2p � ynÞ ½mod1�

ynþ1 ¼ yn þk

2� sinð2p � xnþ1Þ ½mod1� ð1Þ

It forms a two-dimensional system where the

[mod1] statement means that the domain of the system

is periodic between zero and one [i.e. 0V (x, y)V 1].

(xn, yn) and (xn + 1, yn + 1) are the coordinate of each

fluid particle at the dimensionless time t = n and

t = n + 1, respectively, and k is the parameter of the

map. The flow is defined on the 2D torus meaning

that whenever a particle exits the unit square, it re-

enters the box through the opposite side. As pointed

out by several authors (e.g. Refs. [24,42,43,45]), this

flow scheme is useful for simulating advection during

chaotic fluid mixing and, in particular, during mixing

of magmas because a variety of flow behaviors can be

obtained by varying parameter k. As indicated by

several authors (e.g. Refs. [24,42,43]) mixing induced

by this map generates, within the same system,

filament-like regions and coherent regions analogous

to those observed in the Upper Pollara structures (Fig.

2A–C). The parameter k regulates the relative pro-

portions of filament-like regions and coherent regions

in the mixing system. In particular, increasing k values

correspond to increasing amount of filament-like

regions and, hence, to increasing efficiency of stretch-

ing and folding processes (e.g. Refs. [24,42,43]).

Details on the applicability of this system to magma

mixing processes, as well as on the detailed meaning

of parameter k can be found in Refs. [24,45] and here

are not repeated. Note that changing velocity profile,

i.e. using another map consisting of stretching and

folding processes, does not influence the structure of

the results.

Diffusion is simulated by defining a continuous

concentration field c(x,y) on the plane, and spatially

discretizing it to a regular grid cij = c(xi, yj) (e.g.

Refs. [24,26,45]). For the initial configuration, we

consider the mafic magma as represented by dark

colored blobs with composition, in gray value, equal

to 20 (i.e. the gray intensity of the Upper Pollara

mafic magma) dispersed within a light colored felsic

magma with a composition equal to 250 (i.e. the

gray intensity of the felsic magma) (Fig. 4A). The

‘‘sine flow’’ map was first used to rearrange c(x,y)

through mapping (x,y) and re-interpolating to the

grid. This iteration was followed by the diffusion

step:

cij ¼ ð1� AÞ � cij þ1

4� A � ½cðiþ1Þj þ ciðjþ1Þ

þ cði�1Þj þ ciðj�1Þ� ð2Þ

Fig. 4. Example of simulation of magma mixing performed using the sine-flow map (Eq. (1)) with k= 0.3. The figure shows the initial

configuration (A) and the mixing system at 3 (B), 6 (C) and 12 (D) iterations (n).


where 0 < A < 1 plays the role of the diffusion

coefficient. This numerical scheme is a typical

finite difference system (e.g. Ref. [46]) and corre-

sponds to the discretization of the diffusion Laplace

equation. cij is updated to new colors ranging from

20 to 250 according to the diffusion that each cell

underwent calculated using Eq. (2). After several

iterations of the map coupled with the diffusion

step, the concentration field c(x,y) will have

changed. Bearing in mind that the simulation is

carried out using shades of gray, the concentration

of each single cell will range from 20 to 250. Fig.

4 reports the result for a simulation performed with

k = 0.3 and A= 0.5, and 30% and 70% of the mafic

and felsic magma, respectively. Since changing k

and A only consists in changing the relative time of

mixing and the structure of results does not change,

results are presented by using the above values of k

and A, but varying the relative proportions of mafic

and felsic magma.

Fig. 5A shows the variation of frequency histo-

grams for the system reported in Fig. 4 with the

passing of time (iterations n of the system). The graph

shows that as the mixing time (n) increases, histo-

grams evolve to a bell-shape and all compositions

between 20 (mafic magma) and 250 (felsic magma)

are present in different amounts, with intermediate

compositions having the highest frequency, as ob-

served for natural samples (Fig. 3).

The graph of Fig. 5B shows that a variation in the

initial percentages of magmas in the simulations cor-

responds to a shift of the maximum frequency of

histograms toward different compositions. A detailed

observation of Fig. 5 evidences that the maximum

frequency corresponds to the theoretical hybrid com-

position of the mixing system, as can be verified using

the classical equation for the two end-member mixing

process:

CH ¼ CFxþ CMð1� xÞ ð3Þ

Fig. 5. Frequency histograms of concentration (in gray levels) of the mixing systems resulting from the numerical simulation of magma

mixing (Eq. (1)) with k = 0.3. (A) Mafic magma= 30%, felsic magma = 70%, at different iterations (n) of the mixing process; (B)

frequency histograms for mixing systems with different initial amounts of mafic and felsic magma at the same number of iterations

(n= 12).


where CH, CF and CM are the composition of

the hybrid, felsic and mafic magma, respectively,

and x is the proportion of felsic magma in the

mixture. For instance, the theoretical hybrid for a

system constituted by 70% of felsic magma is ca.

181 [i.e. CH = 2500.70 + 20 (1� 0.70)], which

exactly corresponds to the maximum value of the

histogram in Fig. 5A. Therefore, the maximum

frequency of compositional histograms corresponds

to the theoretical hybrid composition of the mixing

system.

These results can be utilized to estimate the

initial percentages of end-member magmas occur-

ring in the analyzed natural samples. Indeed, know-

Table 1

Parameters estimated and calculated from the analyzed digital

images of magma mixing

Sample % of

acid

magma

% of

mafic

magma

Intensity

of mixing

(Da)

Log

(Re)

Re

Poll-1 55 45 0.258 3.66 4600

Poll-2b 76 24 0.119 2.74 550

Poll-3 68 32 0.177 3.19 1550

Poll-2-4a-bis 57 43 0.221 3.56 3650

Poll-2-4a 54 46 0.158 3.65 4500

Poll-2-6a 58 42 0.104 3.20 1600

Poll-2-2d 57 43 0.240 3.86 7200

Poll-2-6a-bis 48 52 0.242 3.80 6300

Poll-2-6a-ter 51 49 0.205 3.65 4500

Poll-2-2b-2 64 36 0.064 2.81 650


ing the composition of the hybrid magma, i.e. the

maximum frequency of the histograms (e.g. Fig. 3),

and the compositions of the end-member mafic and

felsic magmas (SiO2 = 63 and 76 wt.%, respective-

ly), the percentages of magmas can be calculated

using Eq. (3). As an example, the sample Poll-2b

has a maximum frequency at SiO2 = 72.63 wt.%

(Fig. 3) and, therefore, the calculated initial percent-

age of the felsic magma is about 76% and, conse-

quently, that of the mafic magma is 24% (Table 1).

We calculated the initial percentages of end-member

magmas for all the analyzed samples. Results are

reported in Table 1 and indicate that the heteroge-

neous Upper Pollara samples resulted from the

mixing of different amounts of the mafic and felsic

magma in proportions of 24–49% and 76–51%,

respectively.

6. Multifractal analysis and determination of

mixing intensity

Several studies (e.g. Refs. [47,48]) have shown

that morphological structures generated by mixing

of fluids are multifractals. This means that they are

described by a spectrum of fractal dimensions

(singularities) at the different scales (e.g. Ref.

[49]). The singularity spectrum formalism is defined

by two indices: the singularity strength a and the

singularity spectrum f(a). These parameters can be

estimated from digital images using the Box Count-

ing Method (BCM). The method consists in cover-

ing the image with boxes of size l and defining

Pi(l ) as

PiðlÞ ¼Gi

Gtot

ð4Þ

where Gi the sum of gray values of pixels within

the cell i and Gtot is the total sum of gray values

of the whole image (Fig. 6). Pi is calculated for

the different scales using different values of l. A

family of normalized measures, l(q,l ) can be

obtained as

lðq; lÞ ¼ ½piðlÞ�q

XNl

j¼1

½piðlÞ�qð5Þ

where Nl is the total number of cells for each box

size (l) utilized in the BCM and q is a parameter

that varies from �l to +l (Fig. 6). The f(a) anda are calculated by linear regression analysis using

the following relations:

f ðaÞ ¼ liml!o

XNi¼1

lðq; lÞlog½lðq; lÞ�

logðlÞ ð6Þ

a ¼ liml!o

XNi¼1

lðq; lÞlog½pðq; lÞ�

logðlÞ ð7Þ

In practice, f(a) and a are estimated for each value of q

calculating the slope of the straight line by linear

regression analysis in aPN

i¼1 lðq; 1Þlog½lðq; 1Þ� vs.

log(l) and aPN

i¼1 lðq; 1Þlog½ pðq; 1Þ� vs. log(l) graph,respectively.

As reported by Ref. [50], the multifractals tech-

nique is suitable to determine the degree of homoge-

neity of samples and, therefore, in our case, is suitable

to estimate the degree of mixing of studied natural

samples.

The technique has been first applied on the

images obtained by the simulation of magma mixing

Fig. 6. Schematic representation of the BCM used to calculate the multifractal spectra of mixing structures: (A) the image is covered with a grid

having different cell size (l1, l2, l3, etc.); for simplicity, the procedure is shown only on one quadrant; (B) enlargement of the portion of image in

(A) evidenced by the bold square; (C) enlargement of the portion of image in (B) evidenced by the bold square. For each pixel, the gray level gj,

rescaled to the silica content (Fig. 2D), is detected; (D) equations used to estimate Gi and Gtot: Gi is the sum of all concentration values gj present

within each box (e.g. C) and Gtot is the sum of all concentration values gi present within the entire image. N = number of pixels in the entire

image. M = number of pixels in each box.


(Fig. 4). Results show that as the mixing time (n)

increases, the width of the f(a) spectrum decreases

(Fig. 7A). The narrower is the f(a) curve width (i.e.

smaller Da), the higher the degree of mixing (ho-

mogeneity). Note that, in the limit case of perfect

mixing (i.e. complete chemical homogenization), the

f(a) spectrum collapses to a single point [50]. Then,

the width of the multifractal spectrum, calculated as

the difference between the maximum (amax) and

minimum (amin) value of a (Da = amax� amin), can

be utilized to estimate the mixing intensity for the

natural samples.

Representative multifractal spectra measured for

natural structures whose gray intensity has been

converted to the silica content (Fig. 2D) are reported

in Fig. 7B. The graph shows that the analyzed

samples display f(a) spectra with different width

(Da). Da values have been calculated for the natural

samples and the results are reported in Table 1. The

analyzed samples show Da values between 0.064

and 0.258 indicating that they experienced different

mixing intensities.

7. Estimate of Reynolds number

Ref. [27] demonstrates, using experiments of

turbulent jet mixing between water and water dyed

with disodium fluoscein (viscosity ratio f 1.0), that

there is a strict relationship between concentration

patterns of mixing structures and associated Rey-

nolds numbers (Re). In particular, there is a linear

Fig. 7. (A) Multifractal spectra for the systems resulting from the numerical simulation of magma mixing (Fig. 4) at different iterations (n); (B)

representative multifractal spectra calculated for the natural structures of Fig. 2A–C.


relationship between Log(Re) and Log( Ptot/yb),where Ptot is the perimeter of contours traced con-

sidering the modal concentration value of the prob-

ability density function (PDF) of mixed structures

normalized by the bounding box (yb) in which the

structure is embedded. Ptot is rescaled by yb in order

to obtain a parameter that is independent on the scale

[27].

Fig. 8 shows how the calculation of (Ptot/yb) is

performed. Fig. 8A and B shows the digital image of a

sample and the corresponding concentration (SiO2)

histogram, respectively. The modal value (i.e. the

maximum frequency of the histogram) corresponds

to SiO2 = 72.63 wt.%. Fig. 8C shows the concentra-

tion contours traced considering this modal value; the

bounding box (yb = 3.3 4 = 13.2 cm) used to nor-

Fig. 8. Schematic illustration of the steps required to measure Ptot and yb. (A) Reference sample; (B) concentration frequency histogram (in both

gray levels, GL and SiO2); (C) concentration contours traced from the image presented in (A) considering the modal value of the histogram

reported in (B). In (C), the measured values of Ptot and yb are also shown.


malize the perimeter (Ptot) of contours is also shown

in Fig. 8C. Perimeter of contours has been measured

by image analysis using the NIH software.

Using experimental data from Ref. [27], the equa-

tion relating Log(Re) and Log(Ptot/yb) is

LogPtot

yb

� �¼ �0:33LogðReÞ þ 1:90 ð8aÞ

Log (Re) can be derived as

LogðReÞ ¼1:90� Log

Ptot

yb

� �

0:33ð8bÞ

Eqs. (8a) and (8b) holds for 103 <Re < 104 and length

scales 10� 2–10� 3 m in liquid and gas jets and shear


layers [27,51]. Re for all analyzed natural structures

has been calculated (Table 1) and results show that Re

ranges from ca. 500 to 7000. Note that the values of

Ptot/yb measured on the natural structures are of the

same order of magnitude of those measured by Ref.

[27], except two samples (Poll-2b and Poll-2-2b-2)

that, however, show Ptot/yb not far from those esti-

mated by Ref. [27]. Note also that the length scale of

the experiments is 10� 2–10� 3 m, which is also the

length scale of the images of the Upper Pollara

pumices. This indicates that, in our case, Eqs. (8a)

and (8b) can be utilized to estimate Re for natural

structures. It is to note that extrapolation of Eqs. (8a)

and (8b) to higher (Re>104) or lower (Re < 103) values

of Re than those investigated by Ref. [27] should

require caution because of the possible non-linearity

of the Re vs. Ptot/yb relationship outside the experi-

mentally investigated Re values. Therefore, although

additional experimental efforts should be done to

define the limits of applicability of the results from

Ref. [27] to a wider range of Re, the use of Eqs. (8a,b)

can be considered, in our case, a useful tool to

estimate Re values during the magma interaction

process.

Fig. 9. (A) Binary graph showing the positive correlation between

the percentage of mafic magma and Log(Re); (B) binary graph

showing the positive correlation between Da and Log(Re). The lines

on the graphs are drawn to guide the eye.

8. Discussion

The morphological structures observed in the

Upper Pollara juvenile fragments, which include

plane-parallel to poorly folded structures, strongly

folded-like, vortex-like structures, result from the

mixing between a HK andesitic and a rhyolitic

magma. The values of Re estimated for the studied

structures (500 <Re < 7000) suggest that the flow

regime during the interaction between the andesitic

and rhyolitic magmas was heterogeneous, from sub-

turbulent (Re below of order 103, i.e. below the

mixing transition) to fully turbulent (Re above of

order 103). Such flow heterogeneity is inherent to

turbulence and is commonly due to the local fluctu-

ations of the velocity in the system at the different

scales (e.g. Ref. [25]).

Fig. 9A shows that there is a direct relationship

between the percentage of initial mafic magma esti-

mated in the different analyzed samples and Log(Re).

This result is in agreement with the observation that,

having the mafic magma a lower viscosity (andesite

viscosity f 103.5–4.2 Pa s) than the felsic magma

(rhyolite viscosity f 104.8–6 Pa s [12]), the higher

is the percentage of the mafic magma, the lower is the

viscosity of the whole system and, therefore, the

higher is the turbulence, since turbulence is inversely

related to viscosity by:

Returb ¼ffiffiffiffiffiq2

pL

m>> 1 ð9Þ

whereffiffiffiffiffiq2

pis the characteristic velocity in the turbu-

lent flow associated with the velocity fluctuations,

Returb is the Reynolds number, L is the characteristic

length of the system and m is the kinematic viscosity

(e.g. Ref. [52]). The results of Fig. 9A are also

consistent with those from fluid dynamic experiments


on magma mixing (e.g. Refs. [53,54]), which show

that increasing amounts of initial mafic magma pro-

voke increasing turbulence in the magma mixing

system.

Fig. 9B shows a direct relationship between Log(Re)

and Da. Recalling that Da increases as the mixing

intensity decreases, the graph shows that the higher

is the turbulence, the lower is the mixing experi-

enced by magmas. This conclusion contradicts

results from experimental and numerical models on

convective and fountain mixing in magma chambers

[53,55–57]. However, it is worth noting that increas-

ing turbulence does not necessarily imply increasing

mixing efficiency. [52], using dimensional analysis,

define a parameter (I) inversely related to the mixing

efficiency as

IflRe 1þ lnffiffiffiffiffiSc

p� 1ffiffiffiffiffiffi

Rep

� �2

ð10Þ

where Sc is the Schmidt number, which is the ratio

between m and the chemical diffusivity D, and l is the

fluid viscosity. Eq. (10) is valid for Sc larger than about

104, a value well below typical Sc values for magmas

(Sc larger than about 1015). Eq. (10) shows that the

higher is I, the higher is the turbulence (increasing Re)

and, hence, the lower is the mixing efficiency. This

result depends on the fact that higher Re values

correspond to higher energy dissipation of the mixing

system for highly viscous fluids and this provokes a

strong decrease of mixing efficiency toward higher Re

[52]. Experimental models in shear layers and pipe

flows also corroborate this result (e.g. Refs. [28,58]).

In addition, Ref. [32] shows that, in laminar and sub-

turbulent regimes, mixing in a rising magma batch in a

conduit occurs and increases when the dimensionless

number Imix is larger than ca. 10� 1. In the sub-

turbulent–turbulent regime, Imix is defined as:

Imix ¼4ðqr � qmÞql

kDqqm

Re�1l ð11Þ

where qr is the density of the country rock, qm is the

density of the bulk magma, ql is the density of the

upper magma, Dq is the density difference between

magmas, k is a friction factor. Eq. (11) refers to a

system in which a more viscous magma overlays a

less viscous magma within a conduit and it is valid (a)

for fluids with viscosity ratio between 1 and 103, a

range of values within that estimated for the Upper

Pollara magmas (viscosity rhyolite/viscosity andesite

f 1–103 [12]); and (b) 102<Re< 106. Length scales

investigated by Ref. [32] are of the order of 10� 2 m,

which is also consistent with the length scale of

Upper Pollara structures. This equation demonstrates

that, when Imix, and then mixing efficiency, increases,

Re decreases. As reported by Ref. [32], this model of

mixing in a conduit ‘‘contrasts with fountains [7] and

plumes [59]’’, which refer to mixing in magma

chambers. The positive correlation between Re and

Da revealed by our analysis is in agreement with

results of the mixing model of Ref. [32]. This

suggests that the mixing/mingling processes in the

Upper Pollara eruption mainly developed in the

conduit and not within the chamber. This conclusion

is also supported by the relatively high estimates of

Re-values (Ref 103), which are poorly consistent

with a high viscosity magma, like the Upper Pollara

rhyolite, flowing within a chamber [32]. In addition,

on the basis of our results and those from experimen-

tal models [58], we conclude that the mixing between

the rhyolite and andesite developed in a shear layer-

type or pipe-type flow rather than in a fountain-like

flow in a magma chamber.

9. Concluding remarks

Results presented in this study indicate that it is

possible to estimate (i) the initial percentages of mag-

mas that took part in the mixing process, (ii) the mixing

intensity and (iii) the Reynolds number, from the

analyses of the concentration patterns of heterogeneous

juvenile samples. These estimates can give useful

information on the kinematics of the mixing processes

and may be used to better constraint petrological/

volcanological models of eruptions. The possibility

to estimate the initial percentages of magmas that are

involved in a mixing process cannot be easily deduced

using classical geochemical modeling (e.g. Ref. [15])

and the approach proposed here, when applicable, can

give additional information to constraint such models.

In addition, the possibility to estimate the intensity of

mixing using multifractal analysis can be an indepen-

dent method to test the reliability of whole rock

geochemical modeling (e.g. Refs. [12,60]). Moreover,


we have shown that it is possible to deduce the flow

regime associated to mixing structures.

Results obtained in this study are promising and

future researches need to be carried out in order to test

the general validity of the proposed analytical ap-

proach by using both numerical and analogue experi-

ments, as well as by studying other volcanic eruptions

characterized by magma mixing processes. This kind

of studies may provide the basis for better constrain-

ing time scales of magma mixing processes.

Acknowledgements

We thank Paola Donato and Rosanna De Rosa for

having kindly provided some of the studied samples

and for discussions on the Upper Pollara eruption. We

also thank Francois Holtz for discussions on magma

mixing and Paul Dimotakis for his help on turbulent

mixing. Antonio Costa is gratefully acknowledged for

the revision of an early version of the manuscript. We

are grateful to R.C. Kerr and H. Gonnerman for

helpful suggestions and criticisms in reviewing the

manuscript. The editorial handling of S.D. Scott is

gratefully acknowledged. This work has been sup-

ported by the Italy–Germany ‘Vigoni’ Project and by

MIUR grants. [SK]

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