Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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48
S.
Brianzoni,
C.
Mammana, and
E.
Michetti
(a) (b)
1
1\~~~\
-1L-
______________
~
o
60
o 60
Fig.
1.
State
variable
Wt
versus time
for
an i.c. Xo = 0.8,
rna
= 1 and Wo = 1 and
parameter values
a = 0.5,
)..
=
1,
(J2
=
1,
r = 0.02, C = 0.5,
(3
=
1.
(a)
a = 0.1.
Periodical fluctuations.
(b)
a = 0.5. Convergence .
• for a
~
1 + r the fundamental steady state E
f
is unique
2.
Let a = 1, then:
• EveTY point E = (1,
tanh
{ -
Cf},
ill)
is a fundamental equilibrium.
Now we move
to
the
study
of
the
asymptotic
dynamics
of
the
system
by
using
numerical
simulations.
We first
consider
the
extreme
case
in
which,
at
the
initial
time,
all
agents
are
fundamentalists
and
they
own
the
total
wealth.
Moreover, we focus
on
the
more
interesting
a's
range
of
values, i.e.
a < 1 + r.
In
Figure
1
(a)
we
present
the
trajectory
of
the
state
variable
Wt
while
assuming
that
at
the
initial
time
'rna
=
Wa
= 1
and
the
price
is
above
its
fundamental
value.
We
choose
two
different values
of
a
and
observe
that
if a is low
enough
the
system
fluctuates
(a
12-period
cycle is observed) while,
as
a increases
the
long
run
dynamics
becomes
simpler
(Wt
--+ 1
after
a long
transient)
.
Numerical
simulations
show
the
existence
of
a value ii
such
that
for all
a E
(ii,
1
+r)
the
system
converges
to
Ef,
while
more
complex
dynamics
is ex-
hibited
if a < ii, i.e. for a low
enough
(see
Figure
2). Notice
that
the
system
preserves
similar
features, i.e.
the
stabilizing
effect
of
a
in
the
long
run,
also
in
the
other
extreme
case
Wa
=
'rna
=
-1,
i.e.
at
the
initial
time
the
market
is
dominated
by
chartists
who
own
all
the
wealth.
In
Figure
2 we observe a
succession
of
periodic
windows
which
occur
at
border
collision
bifurcations
(see
Hommes
and
Nusse[5]
and
Nusse
and
Yorke[6]).
These
periodic
windows
have
the
following
properties:
(i)
the
width
of
the
periodic
window
reduces
monotonically
as
the
parameter
a increases; (ii)
the
windows
are
character-
ized
by
the
same
periodicity;
(iii)
for a > ii,
the
periodic
windows
terminate
in convergence.
In
order
to
consider
also
the
role
of
the
parameter
(3,
in
Figure
3 we
present
a
two-dimensional
bifurcation
diagram
in
the
parameters'
plane
(a,
(3)
in
the
extreme
case
'rna
=
Wa
= 1
and
the
price
.TO
above
the
fundamental
value.
Complex Dynamics
ill
Asset Pricing Model with Updating
Wealth
49
a
a
a
Fig.
2.
Bifurcation
diagrams
of
the
state
variables
w.r.L a.
Initial
condition
and
parameter
values as
in
Figure
1.
Rkh
dynamics is exhibited
and
the
final behaviour increases
in
complexity
as
(3
increases (according
to
what
happens
in asset pricing models
with
het-
erogeneous agents
and
adapti
veness).
Other
numerical simulations confirm
the evidence
that
the
systems is lIlore cornplicated for high values
of
/3
and
low values of a.
.
~jg~
I
I.
Jj:;
.
G-~'
c--';r.
~<
";
. .fl
r.;.
.
.',
(~.~
(:,,~
cc...o
C-~
c;.
.
.';;
'J
Fig.3
.
Two
dimensional
bifurcation
diagran:l
in
parameters'
plane
(a,
,(1)
for i.e.
Xo
= 0.8,
?no
=
'Wo
= 1.
The
other
parameter
values
as
in
Figure
1.
In
order
to
focus
on
the
role
of
the
initial condition, we
fix
a low enough.
[n
Figure
4 we
present
the
basins
of
attraction
in
the
plane (rno,1uo), being
the
initial price below
the
fundamental. Observe
that
if we move
parameter
3,
the
structure
of
the
basin
also undergoes
to
a change. More precisely,
it
seems
to
become
more
complex
aN
(] increases, confirming
the
evideJlc:e previously
obtained.
In
fact
the
struetme
of
the
basin becomes
fractal
providing
the
strong
dependence
of
the
final d.ynamics
w.r.t.
the
initial conditions (in
50
S.
Brianzoni,
C.
Mammana,
aml
E.
Michetti
panel
(b)
ouside
the
stability
region
the
system
may
converge
to
a IO-period
cyele
or
to
a
more
complex
attractor).
(a)
(b)
Fig.4
. Basins
of
attraction
in
the
plane
(mo,100) for :ro = 1.3, a
,=
0.1
and
the
other
parameter
values
as
in
Figure
1.
(a)
(j
= 1.8,
(b)
;3
=
2.
In
summmary,
our
model characterizes
the
evolution
of
the
distribution
of
wealth
in
a
framework
with
heterogeneous beliefs
and
adaptiven€ss. Analyti-
cal
and
numerical results show coexistence
of
attractors,
sensitive dependence
on
initial eonditions
and
complicated dynamics. Complexity is mainly
due
to
border
collision bifurcations whieh
are
involved by
the
wealth dynamies.
References
l.'vV.A. Brock,
and
C.H. Hommes, Heterogeneous beliefs
and
routes
to
chaos
in
a simple
a..'lset
pricing model.
JO'li.rnal
of
Economic
Dynam'ic8
and
Co'nt1'01,
22: 12:'\5-1271.1998.
2.0. Chiarella,
and
X. He, Asset price
and
wealth
dynamies
under
heterogeneous
expc;ctations.
Q1UJ,TI,titati'Ve
P'i'fl,a1l.ce,
1(5):509-526, 2001.
3.C. Chiarella,
and
X. He.
An
adaptive
model
on
asset pricing a.nd
wealth
dynamics
with
het.erogeneous
trading
strategies. vVorking
paper
no. 84, School
of
Finance
and
Economics,
UTS,
2002.
4.C. Chiarella,
R.
Dieci,
and
L.
Gardini. Asset price
and
wealth
dynamics
in a
financial
market
with
heterogeneous agents. Jov,7'nal
of
Ec07l0'lnic DynaTYIiic8
and
Control, :30:1755-1786, 2006.
5.C.H. Homme",
and
H. E. Nusse.
Period
three
to
period
two
bifurcation
for
piece~
wise linear modeL
JO'U.r'7I,al
of
Econormcs.
54:157-169, 1991.
D.H. E. Nmlse,
and
.J.
A. Yorke. Border-collision Bifurcations for Piecewise
Smooth
One-dimensional J\faps.
International
Journal
of
B~f1trCl1t.ion
a.nd Chaos:
.5:189-
207, 1995.
51
Chaotic Mixing in the System Earth: Mixing Granitic
and Basaltic Liquids
Cristina De Camposl, Werner Ertel-Ingrisch
l
, Diego Perugini
2
,
Donald B. DingweU
l
and Giampero Poli
2
lEarth and Environ. Sc., LMU - Univ.
of
Munich
Theresienstr.
411III,
D-80333, Munich, Germany
(e-mail:
campos@min.uni-muencehn
.de)
2Dep
t.
of
Earth Sciences, Univ.
of
Perugia
Piazza
Universita, 06100 Perugia, Italy
Abstract:
A widely debated question in Geosciences is
if
and how viscous magmas with
extreme viscosity contrast mix under natural conditions. Chaotic mixing
in
magma
chambers
is
thought to play a central role not only in determining the timing and
dynamics
of
volcanic eruptions but may be
of
equal relevance for the evolutionary
history
of
our planet. To date the dynamics
of
chaotic mixing have been investigated
mostly both
in
analogue systems and numerical simulations. Here we report the first
experimental simulation
of
chaotic mixing dynamics in molten silicates
of
geologic
relevance and at geologically relevant temperatures (up to 1,700°C). A newly developed
device for the in-situ experimental simulation
of
chaotic dynamics has been successfully
developed and employed for this purpose. This device was designed in full awareness
of
the importance
of
chaotic dynamics for mixing processes; and earlier studies evidencing
that chaotic dynamics equally control magma mixing processes in nature.
Keywords: chaotic flow simulation, chaotic mixing, journal bearing system, newly
designed experimental device, high viscosity silicate melts.
1.
Introduction
Magmatic rocks are known to be formed by cooling
of
natural silicate
melts or magmas
of
wide compositional diversity. Primitive Earth is
thought to have been covered by at least one giant magma ocean
of
more than 2,000 km depth, at temperatures far exceeding 2,000°C
(Righter and Drake, 1999)[1]. At those temperatures magmas are
predominantly multi-component viscous fluids behaving according to
fluid dynamics (Reese and Solomatov,
2006)[2]. Since Hadean times,
different compositional magma layers must have been constantly
stirred and mingled together by thermal and/or compositional
convection, at different scales (Abe, 1997)[3]. With decreasing
temperature, magmas started to differentiate under wide range
of
crystallization rates and conditions. Even to the present day, this
process controls heat budget and changes in chemical composition
during magma segregation and crystallization. From this point
of
view,
magma mixing
is
one
of
the most active processes in the Earth, its
52
C.
De Campos et al.
range
of
activity spanning from the core-mantle boundary via mantle
plumes up to the Earth's surface, with volcanic eruptions, under
varying length- and timescales.
From the physics
of
mixing we know that mixing is mainly two
variables dependent: convection and diffusion (Ottino, 1989)[4]. Since
Aref (1984)[5] demonstrated that processes involving chaotic mixing
may run under absolutely laminar conditions, this is a decisive
theoretical foundation for the mixing
of
high viscosity molten silicate
melts in nature. As a consequence, viscous fluids flowing along simple
periodic patterns can generate just enough chaos for efficient mixing
(Ottino, 1990[6]; Funakoshi, 2008[7]).
The first purpose
of
this study was to build a new high temperature
experimental device allowing the first in-situ chaotic mixing
experiments at geologically relevant temperatures
(1,300 to 1,700
0c)
and experimentally reproducible conditions. Applying this device, a
systematic study
of
the chaotic mixing between synthetic basalt inside a
granitic starting glass was initiated. Full chemical analysis
of
the
resultant solidified materials was performed by electron microprobe
analysis (EMPA). The obtained results point towards a clear increase in
mixing efficiency due to combination
of
convection and chaotic
dynamics, despite very low diffusion coefficients. This process can be
an efficient way
of
mixing molten silicate melts over filament
or
lamellae dimensions.
2. The Journal Bearing System (JBS)
2.1. The original model, set up
and
starting materials
The journal bearing or eccentric cylinder geometry proposed by
Swanson and Ottino (1990)[8] is shown in Figure
1.
It consists in
principle
of
a cylindrical crucible, holding the melt
of
interest, and a
cylindrical spindle, which is located off centre to enable stirring. The
basic geometry is determined by two parametrers:
a)
the ratio
of
the
radii
of
the two cylinders, r =
rinlrout
and the eccentricity ratio
to
the
outer cylinder
£ = elr
out
; b) the discontinuous stirring protocol and its
velocities. In order to generate chaos in a flow, the streamlines, and not
just the velocity must be time dependent. This is the reason why
Swanson and Ottino (1990) proposed to move both cylinders in
opposite directions. Final experimental conditions are r
=
1/3,
£ = 0.3, and the viscosity contrast between glycerin and dye is
;::::
O.
Chaotic Mixing
in
S.vstem
Earth
53
In this Oliginal model the velocity field
is
derived from the stream
function:
~
a~
~
a~
-=V
=--'-=V
dt
x
ay
, dt Y
ax
And the final equation for the mixing in the flow is then determined
by
the integrated velocity field (the so-caned motion):
d"
a(
rp
I v
nu)
dy
a(
tp
I v
out)
--,
- =
-'--'-'-;
--
=
--'---_.::=.-
dB
ay
dB
ax
where e
IS
the angular rotation and V
out
the velocity
of
the outer
cylinder.
The working
fluid in Swanson and Ottino (1990)[8J was glycerin which
has
a viscosity around 0.7 Pa. and a density
of
1.2 g/cm
3
.
The resultant
Reynold number reported by the authors is
0.8. For the basic
experiments,
fluorescent dye with a very small diff11sion coefficient
(lO-8
cm
2/
s
)
has been dissolved in the glycerin to highlight the flow
lines.
Experimental results were then recorded on color slide film.
Figure 2 depicts the resultant image for
Bill::::
270°.
Hgure
1. Basic geometry
of
JBS after
Swanson and Ottino ([990)[8J.
Figure
2.
How
lines for
Oin=270°
(Swanson and Ottino, 1990)[8).
54
C.
De Campos et al.
2.2. Our set up
and
experimental end-member compositions
Our self-made device has built according
to
the basic geometry for the
JBS, and is shown in Figures 3 and
4.
Starting materials consisted
of:
1)
hapJogranite (HPG8), and
2)
a basaltic composition due to the 1 atm
eutectic of Anorthite-Oiopside (An-Oi). Mean values
of
end-member
compositions are reproduced in
Table
1.
Experiments were performed
with the conditions given by
Swanson and Ottino (1990)[8]: (r =
1/3
and € = 0.3;
b)
, and a discontinuous velocity protocol, presented in
Table
2.
Calculated viscosity values for the stmting materials (after
Giordanno et
aI.
, 2008)[9] at 1,400°C are:
1)
haplogranite (HPG8)
;:::;
15.10
2
Pa s (log a =3.19) and
2)
An-Oi
;:::;
7.
10
3
Pa s (log a =-0.15), so
that the viscosity contrast
averages
;:::;
2.10
3
,
e.g., over three orders
of
magnitude higher than in the original JBS experiment. A representative
streamline distribution through a cross section
of
the obtained glass
cylinder observed in section 5
(20 mm above the lower end
of
the glass
cylinder) is shown in Figure 5.
Figure
3. Starting vertical geometry
of
JBS in our experiment
with sampling location.
Table 1: Composition
of
starting materials (end-members)
in
weight % (normalized)
L:
SiOz
Ah0
3
Na
2
0
K
2
0 CaO MgO (Rb+Sr+Ba %
+Zr+RE)
HPG8 70.33 12.43 7.70 9.50 >0.02 >0.02
- 0.10
9
5
AnOi
48.50 16.31 >0.1 >0.5
23
.72 11.33 -
5
Hy 69
.2
5
12.61 7.34
9.05 1.26 0.56
»0.10
*
Chaotic Mixing in System Earth
55
Table 2: Protocol for the first chaotic mixing experiment with silicate melts
Obs: final wobble
of
the platin/ceramic spindle was 0.057 mm (eccentricity
of
the
spindle). Spindle changes direction. Crucible rotates only in one direction.
Time
Tin
°C
Sin
e out
Observations
14:27
Heating started towards
1400°C full speed
15:53
1400°C± 0° +720° 2 rotations in 40 min -7 v
ou
, 0.05 rpm
16:07 1400"C± -2160" 0° 6 rotations
in
20 min -7 V
O
Ul
0.33 rpm
16:27
1400°C± -2160°
+1440°
2 rotations in 40 min
-7
V
O
Ul
0.05 rpm
17:07 1400°C± +2160
0
0
0
6 rotations in 20 min
-7
Vom
0.33 rpm
17:07
1400"C±
0
0
+1440
0
full speed down with
T.
All
rotation stopps.
17:07 1400°C± 0
0
+1440
0
full speed down with T. Rotation stopps.
Final remarks: no leaking or visible deformation
in
the crucible. Easy to remove it from
the pedestal. Total duration: 1 hour and 54 min. Error. 6 min.
Figure 4. Stmting geometry
of
JBS,
our expo (horiz. crosssection).
Figure
5. Flow Jines from this work;
6>
o
Ul=1440°.
3. Sample Preparation and Preliminary Electron
Microprobe
(EMP) Data
The experiment was stopped by termination
of
the stirring protocol,
switching off furnace power and cooling
to
room temperature. Crucible
and spindle were removed from the furnace, and a cylinder
of
the
resultant mixed glassy sample was recovered
by
coring. Horizontal
equidistant sections
(5
mm intervals)
of
this cylinder were prepared for
56
C.
De Campos et al.
electron microprobe (EMP) analyses (Figures 3 and 5). The location
of
the EMP profile is along the black line in Figure
5.
Optical and EMP micro-chemical studies revealed crystal-free
filaments
of
intermediary compositions, changing with depth in
complex patterns, with chaotic attractors showing slight variations in
vortex strength. Here we present the results for the distribution
of
CaO-
and K
2
0-contents (Figures 6 and 7) from a single analytical profile
along the black line in section 5 (sample CD-25) at
20 mm depth
depicted on Figure 5. This image is a false colour image from a mosaic
consisting
of
over 150 photomicrographs from section 5 on Figure
3.
This section crosses several filaments along a striking, stretched vortex
structure. In Figure 6 the bottom white line represents the maximum
error in CaO-content
of
the starting materials. This means that all
points above this line are due to non-ideal mixing. Two main
CaO-
enriched filaments are visible at about 2.5 mm and 3.5 mm distance. In
Figure 7 these points are not correlated with the K
2
0-distribution along
the same line. This clearly shows that the mobility
of
both elements is
not correlated.
Another way to demonstrate this lack
of
correlation is shown in Figure
8:
ratios
of
LlC
MgO
and
LlCcao
between ideal hybrid concentrations and
measured partially mixed concentrations, for
MgO and CaO are plotted
along the same line. For this plot we assumed an error
of
0.02 in the
MgO- and CaO-contents due
to
background signal and sample preparation.
0.6
~
0.4
cf
"
()
8.8
~-<
~.~~~.~~.-"
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
distance in mm
Figure
6.
CaO distribution profile plotted
against distance, along a line across slice
CD-25 on Figure
5.
distance in mm
Figure 7. K
2
0 distribution profiles plotted
against distance, along a line across slice
CD-25 on Fig. 5
To compare the mixing processes between different elements in the
melt, along the analysed profile, we used a coefficient (v
E
),
which we
called
'mixing
efficiency coefficient': this coeffcient is obtained from
Chaotic Mixing
in
System Earth
57
the following ratio:
where:
VI:
= [(C
mean
-
Chybrid)
/
(Cend-member-
Chybrid)]
.
Vm
Cend-member
= measured, corrected and normalized concentration
of
an oxide in the end-member;
Chybrid
= calculatated concentration
of
an oxide in the ideal
hybrid
(100% homogeneous mixing);
C
mean
= measured, corrected and normalized mean concentration
of
an oxide in the mixed sample, for the most frequent value
population;
V m = C
mean
frequency;
The
mixing efficiency coefficients calculated for CaO, MgO,
Na20,
and K
2
0 for section 5, from this experiment, are: CaO = 0.047;
MgO = 0.048; K
2
0 = 0.43; Na20 = 0.48. For final experimental
VI:
computation we will need the mean value obtained from the L
of
all
measured sections.
0.14
~~~~~~~~I========:=:::'===J
•
L'.C
CaOIL'.C
MgO
0.12
0.10
0.08
~
0.06
(]
<l
0.04
.
.
.
. -
......
-.... .
0.02
0.00
-0.02
I .. .
"«,.
S./,I.:
........
·•
......
,
~.,
.....
,
...
"
..
...
: e
...
::_.
~
••
,.
i.
,:
...
,.If"
.....
,~..
.'
•.
•••
__
"..
.....
__
••
1
........
.
-0.04
L-L-~~~~~L'.~C--";ty1""~g~O~~~~~~~
-0.04 -0.02 0.00
0.02 0.04
Figure
8. MgO and CaO distribution profiles plotted
against ideal hybrids, along slice CD-25 on Figure
5.
4. Conclusions
Our goal in this work was
to
present a newly built JBS experimental
device for first in-situ chaotic mixing experiments at geologically
relevant temperatures
(1,300 to 1,700
DC)
under experimentally
reproducible conditions. Applying this device, a first run, using
synthetic basalt (An-Di) inside a granitic starting glass was performed.
Due to the very high viscosity contrast and to the predominance
of
a
very high viscosity melt in the crucible this represents the worse
experimental scenario.