GOME THiXEAL
T0N$MJB:s"
OF A THEORY OF 00NTINElT G OuTH
by
JAMS P. DOWNI
3.58, GOREIA INSTIUTE Ot fEOWOGT
(196)
S7MITElD IN PARTIAL FOIJIIMNT
OF M, axnuusm"ne ?OR
onI
mmiR
N'
o7 I0T-OR0 O
PHILOSOPYT
st the
MfASACOUS2TTS INSTITUTE OF TSOHNQLOGT
June, 1960
,/1
S
Aooptt
tur
Of AntbOtlP
by
..s...-e e
.
-*
*
.*
*
* .* * * *
*tant of 6o0otg aM Gophyso
May 11, 1960
,,........
,,......,,,..,,,,..
?hairman, Departxental Committee
on Graduae stadents
TABLIE OF QONTHNTS
Title Page................
6**. **#**d****a**940*4
Table of Otate.........
.
Table of
#
2
we***Cee*****b*...*.**
Tables...............
.............
3
......
4
Table of Figures..e,.......c...*..e..........eee
Abs
tract..........
.............................
Aokneawedgementa.. ...... *, *.........
5
.
*.................
6
Chapter I - Intreduction.........................
7
12
Chapter II - Formalation of tMdl......,,....,..,,,.
12
Mathesatioal Model....,,.,....,.................
15
Pressure and Temperature Dependoene
on the
Yieosity..
...........
**
Iu.merical Approximation..............,...
............
.....
22
24
Parameters Used in Theas,..........i*.,,.,..
Chapter III - General Features of Model and
arly Differentiaton of Earth,.,,...
Ras to Study EarlyX iffehntiation..........,.
29
29
Possible Yarly Hitory of
arth......,.........
36
Initial Distribution of Aoid Frsstion for
Near surface Differentiation.....................
40
Time Scale of Initial Separation................,
41
Qomparison of Model to Other Modele,...........
43
45
45
oaputer
run*s...
.....
* ......
*...
* C *... **...
*,
Disassion of Results..................,.... ,.e
47
49
Chapter V - 0onalusions and fStgestioas for
er Work,..............*............
Sa
54
estions for ?urther Work...............,.....
Appeanit A -
Visosity,*,..*.
..
,,
,....*,*.*..**.
Pro ears and Taemprat re D pendonoe.....'e
The Effelt of Volatiles on tb*
ises
.... .....
Appendix s *- er
Po.sible
.........
67
w ,a.....
70
83
SQea1S.*...,,.,................
.....
iour*oes.................
e...a*
Appendix a - Nmerlal ApproxiAto...... ... ... ..
genegrl..
****
...............
#...
4,......
**.,..
$tability and
oaergeneoe
onXitioaes.,........
Aocuracy..9.9.............
...
Differonoe Eouations... ,,a..,.a
4
Ciphioal
...
83
92
92
93
#.#............,.96
q..
a
a e••.. 96
.......................
*tde.
956
56
e,..
Value of Vieesity Parasters Used in
*
OSalulations..............
51
1...
100
TABLE OF TABL"S
3.1
Desription of \sna Usd in Study of
3.2
IRune to Ztudy Effect of 1ha"des in Uo,***......
3.3
Goera Ieatures
MIeR to Illstrat#
of MOeLl.............,..,.e
Low 4asity layer Pfeattare...
31
32
35
3.4
Euns to Determine Distribrtion of I for
Near Surface Differentiation Study*............
42
4.1
R=us to Stdy lear surface Differentiationa.....
48
5.1
Dpth to Top of Low D*nstty Laer for
Several
ases....................,.........
52
A.1
Viseosit
Glases...,*
A.2
Parameters for Some Silioate
+
*#4
e
*6*4*9*0*4*06*4*
...
44
A.5
A.6
*
Least S~uares Fit of Vie
siVty
. **........**
..
.,..eoo
73
74
omparison of lit of Viscosity vs 1/T for
Two Natural Avaes by Two Yorms of the
Visoosity as a unotion of Tmp eraSure.........
75
Differeance Between Activation Energy for
Wiet and Dry elt........,.....,,,,,......
69
Data from Stewart (Unpublished) Used in
Study of Effot of Water on Visaosity in
Silicate elts.............................
.i
62
1/T of
Earth-Like Materials.o........,.....
A,4
4**
Least Squaresoo Fit of Visoosity vs 1/T
by Three Cures. .*.*.........
A,3
*
69
Omparison of Two i'una to Indicate Aocuray
of
Numerioal
Solution....................0*
*6
97
TABLIE O1 FfIGURES
2,1
Temperatue vs Depth Used as Initial
Temperature
211DisribPion............
3.1
Melting Point
38
Ad
Energy of ACtivation for Visaos flaw in
A.2
A43
A.4
ve Depth for Iron and Diopside...
Binary Liquid Silioates.....................
76
Relation Between the Preexponential Constant
and Composition for iLinary Liquid Sfilioates...
77
Visoosity vs 1/T, 1/T-273, 1/T--473 for a
Silioate GlaSe s..o...
... .. ,....,........
78
Visoosity va 1/2 of Silioate Glass Fitted
by Three Anotion Forms of the Visosity.......
79
A.5
Viscosityvs I/T -for System Diopside
Albite-Anorthite,........
.
4 6644 44 90
4 4 ...,,..,... 80
A.6
Viscosity vs /IT for System OrthoolaseAlbite...............*.....,,............,..
81
Viscosity vse 1/T for Two Natural Laas.........
82
A.7
ABSTRACT
Titles
Author:
Some Thermal Consequences of a Theory of Continent
Growth
James P. Downs
Submitted to the Department of Geolo
and
Geophysics on Mayi11 1960 in partil fulfillaent of the requirements for the degree of
Dotor of Philosophy at Xassanusetts Institute
of Teohnology.
A nthaMatoial model is
formlated whioh is thought
to contain the essential features of the process whtich
would be active in an earth with *ontimusly growing
continents. The model is based on partial melting leading
to the formatio
of a "glassy" or molten "acid fration"
with a visoosity given by q qo ezp AI + XZ)A/k(T-C).
The form of the viscosity ahd the value of the parameters
used are based on experimental information reported in
the literature for rooks and artificial
elts.
Once te
the acid fraction melts, it is suggested that
flcw will result due to the action of the buoyanoy fore
resulting from the density difference between the solid
and glass if the melted sones are eoneoted over a saffioient depth.
The flow regime is approximated by a steady
state
oiseuile flow through tubes.
The equations of the mathematical model were solved
numerically on a omuuter for several values of the parameters involved in the formalation of the model.
The oelolations indicate that the hypothesis of
slowly growing sontinents requires an uIalted and comparastively unbroken near surface region in the early history
of he earth. Unless these oonditions are met, differentiation will be completed early in the earth's history. If
these conditions are met, a "low density layer" seems to
be a consequaene of this hypothesis. The depth to the top
of this layer is between 30 and 90 kms.
In the model, the lowest energy density neoessfy
for later differentiation to oqTw in avolume 10' km
(down to 100 kas) is 7.7 x i0 "
ol/omv**a
This is
equivalent to dissipation of the amount of energy carried
in seiamio waves for the whole earth in this volume. If
this energy is provided, it seems that differentiation in
this volme would ocour with a time seale between one to
fifty million years after the onset of the energy release.
Thesis
upervieor:
H. Hughes
Assistant Professor of Geophysics
ACKNO L1I4METS
The author wishes to express his sincerest thanks
to Irofessor Harry judhes who asuggested the problm and
offered valuable suggestions in the formulation and in
the writing of the thesis.
The many conversations with the staff and students
of the Department of Geoloy and Geophysics at M.I.T.,
in particular with Professors S. U. Simpson, T. R. Madden,
and
. F. 1raee were extremely helpful at critical moments.
Professor 1iadden, among other things, read the final draft.
Professor Simpson helped in the earlier stages of the
programming.
Professor Brace made valuable suggestlons
of source material.
The assistance of the staff at the M.I.T. Computational Center was invaluable in getting the program
to work.
The oaloulations were made on the IBM 704 at
the M.I.T. Computational Center at no cost to the author.
The author is
indebted to M.I.T. for awarding him
a tuition scholarship for his first two years of study
at M.I.T. (1956 and 1957).
The National Jcienee Founda-
tion awarded the Fauthor a fellowship for his last two
years in the Graduate
.ohool at '.I.T.
(1958 and 1959).
Credit for the spelling in this thesis being
conventional is
due to the author's wife, Cecole H. Downs,
who typed, proofread and
d helped organize the thesis.
Ui1.i1 I
INRODUJTION
Theories have been advanced sugkgestii
that the
earth was formed by a process of cold aoretion during
lthe eaxth has ntver been much, if
which the temperature of
any, in excess of the temperature that is thought to pervail
in the earth today.
These theories have gained considerable
acceptanoe in spite of the many unsolved problems associated
with them.
One of these unsolved problems is how the conti-
nents could have segregated out of an earth that was never
molten, at least not near the surface,
Wilson (1954) has
suggested that the continents are now iyowing and have
grown throughout geological time by a process of adding
aoidia material at the continental margins.
The continents
are thought to be growing at the expense of the ocean basins.
Wilson states that "Iach oontinent oan be divided
into structures o
three tyes-
ranges, and continental shelves
old atuble shields, mountain
and it
is becoming apparent
that there is a mountain building prooess which oonvets
shelves into mountains while mountains, by a process of
erosion,
are in
time converted into provinces of shields.
There seems to be a process by which the continents expand
and encroach upon the ocean basins.
"There is a suggestion that the ocean floors represent those parts of the original crust of tthe earth that has
been least altered and that they owe their eeneral laevel to
that cause.
They will be oonsidered first and then
shelves, mountains, and shields which seem to represent
suocessive stages in the growth o
the continents over
former ocean floors."
The above quotation contains the main sequence of
events that seem to be assooiated with the bypothesis of
the continuing growth of the oontinents.
"irst sediants
are laid down ocf the shore of the current continents.
These sediments are altered by the addition of acidio
material from within the mantle and by heat and pressure
into gneisses and elevated into mountain ranges.
These
mountain ranges are then reduced by erosion to shield
areas and provide the sediments for the next stage of
mountain building.
The inarease in the mass of the conti--
nents is due to the addition o the acidic material from
within the mantle.
The addition of the acidio material is
not continuously occurring in any partioular region, such as
the shelves off the coast of any continent, at aall times,
but rather, it ocours in a particular region at a particular
time.
A consequence of this type of a process would be that
the mantle under the oceans should be less differentiated
than under the continents.
The geolocical arguments concerning the above prooess
will not be discussed here and the reader is referred to
the above paper by Wilson and the references listed there.
urrent seismological information oan=ot resolve
the question oaf whether the material under the Qoeans is
-
11111
11111
YI
leose differentiated than the material under the continents
for it
is
a question of being able to determine chemical
oomposition at depth (i.e., being able to distinguish between say cologite
and forsterite).
The use of the "coamic
abundances" of the elements in oonjunction with the seismological information can limit speculations but not solve
the Problem (Birch 1952).
tecent measurements have revealed that the heat flow
in oceanio regions is not significantly different from what
is
observed in continental regions.
Due to the laok of a
granitic layer under the oceans, the heat souroes needed to
give the observed heat flow are thought to be distributed
to depth in the mantle.
In other words, the radioactive
material under the oceans seems to be less differentiated
than under the continents where, except for potassium, most
of the radioactive materials are thought to be conentrated
in the acidio material of the orust.
This lack of segrega-
tion of the radioaotive materials under the ocean may be
taken as an indication thatthe the mantle
under the ocean is
ntleunder
less differentiated than the material under the continents.
In the more or less qualitative geological arguments
such as the above,
it
iis
desirable to abstract the physical
processes that are thought to be important and include them
in
a mathematical model from which numerioal values of the
quantities involved can be obtained and evaluated as to
their plausibility.
Some arguments can be shown to be
-Li
implausible for the values of the quantities required to
make the model work are in disagreement with the measured
values by many orders of magnitude.
Other arguments are
found to be consistent with the laws of physics and the
measured values of the quantities involved.
The class of mathematical models which can be treated
by numerical techniques using high speed computers include
rather involved nonlinear systems.
This permits considera-
tion of many geological arguments which have a mathematical
formulation that is beyond the reach of analysis alone.
By
solving the mathematical model for several values of the
parameters involved,
of values if
that is
it
is
possible to determine the range
any for which the model gives a prediction
realistic.
These values can be compared with existing
experimental values or they can be taken as suggestive of
experiments that may be performed to check the argument,
Rubey (1951)
and Wilson (1954)
have argued that the
volume and composition of the volatiles and lava that is
escaping from the interior of the earth today is
if
sufficient,
taken as typical of what has existed over geological time,
to explain the composition and volume of the atmosphere,
oceans and continents.
All the previous examinations of the
hypothesis of the slow srowth of the continents do not show
any serious conflict with the observational data available.
Consequently,
a further examination of the ideas contained
in the hypothesis is
desirable.
In particular,
the ideas
-U
involved in the addition of acidic material from within
the mantle to the near
asurface regions are uaseeptible
to a mathematical model, and it
iis with this part of the
above theory of continent formation that this thesis is
concerned.
It
is
sugiested that any process which is
advanoed
to explain the upward migration of material from within
the mantle mast be based on a process involving the
partial melting of material within a
energeties of such a meohanism is
. cal region.
discused,
The
and the
processes that are found to be important are inaluded in
a mathematioal model which is
treated as des e ibed above.
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conclude that "Diffusion rates would become appreciable
only at temperatures well within the ma
tie
range,
and
large scale tratnsformations (e.g., granitination) at
temperatures around 400
GC. would require a span of time
considerably in excess of reasonable geologic estimatee.
This confliots with what some petrologists would consider
irrefutable evidence from field observations.
The only
alternative seems that either granitization effeotively
o0curs only at temperatures within the maz atic range
(>70000.) or that the phenomenon is metasomatio."
onsequently, it seems asaif any process which
attempts to explain the formation of the crust must be
based on a mechanism involving bydrodynamie flow in a
region where at least partial melting has occurred,
In the
following, the possibility of diffusion in the liquid is
neglected.
In regions where a prooess of partial melting has
occurred,
the acidic phase would be molten and oocupy the
pore spaes amon4gst the basic phase.
aue t tthe density
differenoe between the two phases and their presence in
the earth's gravitational field, a buoyancy force exists
which tends to drive the lihter acid phase towards the
surface.
This
~ilration is possible if the pore spaces are
sufficiently well connected.
The solid material in such a region would be ina
masses of various sizes and shapes, and would
sink to the
_
_~
bottom of the melted region.
if
the basti
material had
infinite strength, these masses would settle until they
touched and supported bhemselves.
This would force part
of the acid material towards the surface and retain part
in the enolose4 pore apace.
However,
has a finite strength, and if
the solid portion
the stress differences
resulting from the material attempting to support itself
exsceeds this strength, the materiul will tail
either
elastically or plastically forcing the remaining fluid
part towards the surface if
connected.
If
the strength of material in
of order 108 dynes/m
1/2
3
ya/o
the pore spaces are still
2
the matle is
and the density difference is
, the alidie phase would be squeezed toward the
top of any column with connected pores extending over 2 km.
factors that govern the rate at which the fluid flows
towards the surfaoe are the cross sectional area of the
pore spaoes,
the viaoosity of the acid phase,
and the
length of time it takes the oasio phase to adjust to the
stresses built up in
settling.
It sems that the basic
solid should adjust as rapidly as the acidio material can
flow out of the voids.
A simple model is
formulated which shows the relation-
ships between the important paxameters and allows limits to
be set upon the values of the parameters in the earth for
reasonable time scales -or the misration of the acidio
material.
The justification for the simplifying assumptions
L^~~^Il---~I"Lllllllllil
lili
15
is found in the unoertainty in the values ot the parameters
appropriate to the earth.
a
Ca MO$Idel
tthaot
The simplest one dimensional model which retains the
physics of the problem is
to assume that the pore spaces
can be approximated by uniorm tubes of length L and ratdius
'a'
The eoordinate
orientated in a vertical direotion,
system is selected so that the Z axis is vertical with
inoreasing positive Z into the interior.
viscosity
Assuming that the
of the material oan be represented by some
average value over the length L, the pressure gradient
P= 6fI
where & is the density difference between the
At
phases and g is the acceleration of gravity, the average
stady state Poiseuille velooity of the acidio phase in a
tube is
61
2.1
The velocity of the acidio material is
toward the surface.
on the average always
U is negative in the coordinate system
selected.
GPsersaot
f
as
EQuation
Let N be the number of pores per unit area and $ be
the fraction of the unit volume oeoupied by the aoidle
phase, then S = Nua 2, N is a funotion of depth in order to
allow for varying concentration of the acidic phase.
By
-----~-----
---
-----
I
-~
IIIY
16
considering volume elements with linear dimenasions large
the density of such a two phase model is
compared to 'a*,
:P
The ohanb
2.2
-
e in the density of the individual phases with
depth are neglected for they are
maller than the ohange in
the gross density, p, due to ohanges in the ohemical
composition.
The onservation of massan the system can be written
straightaway by reelizing that %he assumption that the density of the phases is independent of depth requires in the
model that the volume of the basio phase which moves down be
equal to the volume of the acidie phase moved up.
condition, it
is
With this
only neaessary to eonaider the oonservation
of the aidio phase.
The not influx of mass of the sidio phase into a
unit volume is
whioh oet equal the time rate of change of the mass of the
aoidio phase per unit volume
a
Pq ) f-1
2.4
_I IL-~-(-----l^---
~-I~1--~
1_III~--
111
11
1
-C-
17
Equating equations 2.3 and 2.4 giVr
e
of
onservation
ohe
massa equation as
)
-Q
2.5
where U is given by equation 2.1
The boundary condtione are that there is no mass
transport across the surfaces Z = 0 and Z a 1 where R is
the greatest depth to which the acidio material was distributed before the segregation prooess started.
to be the a
H is assaumed
af depth at which melting ooeurs.
In Appendix B, it is argued that it is only necessary
to consider the radioaotive heat production and the strain
energy released locally as heat in the energy balance.
The
oonversion of potential energy into heat energy is negligible in the energy balanoe, but is included for this was
not realized when the program was written.
in the one
dimensional model, the oonoervation of energy is expressed
by
(heat/unit volume) = rate heat laded by
oonduotion + by oveation + by oonversion of
potential energy + by heat generation per unit
volume.
-
2.6
The rate heat is added by oonduction is well known as
R
3- T
--
where R is the theral conductivity,
2.7
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19
U
2.9
The he&t generation per unit volume consiste of two
parts.
One is
the contribution from radtoaotive deay and
the release of strain energy as heat.
the other is
strain energy released locally as heat is
treated as a constant.
obtained by assumin
The
called Q and is
The radioactivity contribution is
that the radloactive souroes have a
distribution proportional to the as*i4o material.
If A is
the rat* of radioaotive heat produotion/ce for a granite,
the souree term appropriate to the model under the above
sawmption in
2.o
qr, An 7
The total heat contained in
a volume daV containing a
two phase mtixture is
cT (euwhere QO is
A(dff
a constant.
A
The time rate of change of this
quanttty is
cT
ej-~.cffa A tc(e e
N4 a
_--~Lsll-II~LII-F
IX----.1 L-^.II ~,DI^I
20
The conservation of mas
-f
equation (equation 2.5) gives
a At
where as usual p
a2.11
t
b - 0 wa2 N.
Substituting equations
2.7, 2.8, 2.9. 2.10, 2.11 and QE in equation 2.6 gives the
oonservation of energy equation as
I"
) T/
- c
a-
-WEa2-2,l
-
2.12
The boundary oonditlons are
-/ (6) = 7o
where the first condition reflects a constant surface temperature, and the second neglects any heat flow at the bottom of
the partially melted region.
The selection of the initial temperature distributions
used in the various models is rather arbitrary due to the
uncertainty in the temperature that exists under the oceans.
In the runs to study the general features of the model, the
nethod used to select the initial temperature distribution
was to assume a steady state temperature under the oceans
(i.e., a parabola down to H, the bottom of the region
--
TEMPERATURE
IN Cg
5S&
MODEL 4
aooo t
MODL 6
1750 t
1500-
1000.
Figure 2.1. Plot of steady state temperature vs. depth for the two cases reported
by MacDonald (1959) used as initial
temperature distributions in the
calculations.
750-
500
DEPTH I K(M
?50
I
/00oo
I
/50o
/00
Zoo
SO2co
?0
ao
I
300
30
-
I
3 0
.36
400
40
22
containing heal sources, a constant below this depth and
no heat flow between these regions).
the
The specification of
uerface temperature, To, and the tnperature at the
depth H, TE,
in addition to
gives the initial
0
temperature T(Z,O) as
2.13
4
Using the suZrface temperature, and the value of the tmapera,
The
*treat the depth H specifies the initial tea perature.
distribution of heat souroes
which gives this distribu-
tion is
2/
For the cases to study near surf ao
the initial temperature was taken fro
2.14
differentiation,
a plot (figure 2.1)
of either model 4 or 6 in MacDonald's (1959) paper.
The
particular mdel used to
the initial temperatare for
'ive
a given oase is ~iven with a
escoription of the case.
rEsare0d
and Te
eept
rs eedence of the V s1ositi
A discussion of the preessuare and temperature dependenoe
of the Viscosity of silioate melts given in Appendix A shows
that the viscosity of the acid traction which must be a
polysomponet allioate can be represented by two forms
(l
- 7o exp EkT
t)
2.15
k-c)
2.16
These two formulas describe the Viscosity with thee
necessary in this thesis.
ouracy
E, E' are activation energies
for the procesaes, Q6 and 7 are the visoosity at infinite
temperature, k is Boltzman's constant, and X (=1.5 pgdL
where K is the bulk modulus) gives the pressure dependence
on the proceas.
The hydrostatic approximation to the
gressure has been made.
it was not possible to neglect
volume chancre in the formulation of the viscosity (included
in X) as in the energy balance equation.
The form 2.15 is shown to be applicable only when
temperatures near the solidus point are not considered,
for in this region, this form predicts values of the
viscosity much lower than are observed.
The form 2.16 is
shown to be valid over a wide range of temperature inaluding
temperatures near the solidus point.
Both physical argu-
ments and experimental evidence requires zhe equation 2.14
to be used for complex melts consisting of
cany
omponents
which melt over a wide rane of temperature when temperatures near the solidues
oint are considered.
-4119r--~--~~-7
~-LPI
LI - -l~P
-a~--
24
Num ri al
arximation
The equations 2.1,
2.5,
and 2.12 with the appropriate
and boundary conditions
form of the visoosity, initial
This set of nonlinear
oonstitute the mathematical model.
equations cannot be solved in closed form.
Jonsequently,
they are approximated by a set of finite difference
equations which were solved on the ~lA 704 at the M. I.
Computational center.
T.
The dixLerence equations ares
for the velocity equation
t-x
~U
( II+
I
2.17
J
where
d6,=p~~
t.ndj
for the conservation of mass equation
4/
A4
~~A
-0;)
(/ j +
n
/
2.18
and for the conservation of energy equation
TI
I-4
,I Ui
M
,r' -e 0
7jsk
4__
PI
r
7
c
y V
4 Jr
'
7
+4 47c
+
-+.l-AJrjL
2.19
~_I_
____
25
where m and
A refer
to the time and *saceloeation of a
point in the differenoe net respetively,
The boundary conditions for the energy equation are
'o= To
at upper surfaoe
X + 1
temperature is
The initial
-
at lower surface
t-
t----
or taken from Figure 2.1.
The boundary oonditions for the oonservation of mass
equation are
U
e,
at upper surface
= 0Q
at lower surface
and the h0 must be speciifed.
The requirements which the parameters must satisfy
for stability are
2.20
2.21
26
The lengtL of h is
selected by
h = B/L
where H is the depth to which the acidio phase extends
and L is the number of space net points desired.
time interva
The
is selected during the ruaing so that
equations 2.20 and 2.21 are satisfied.
The compatational
marohing type.
shame used was a siaple forward
In all of the cases run, 100 time inere-
ments were used.
?sraneters U94d
4a
Tkheis
The parasters involved in the formulation of the
problem ares
k
K
g
a
N
b
pb
x
qt
E',E
r
0
S
k, 1, ,
= Boltzmn oonstant
= bulk modulus
a=oeleration of gravity
= radius of tubes
= number of tabes per unit area
= density difference between the phases
= density of basio phase
= pressure dependenoe of the visoosity
= rseexponential constant
= activatlon energy for viseous flow
= thermal oonduotivity
= speolfic heat
= distribution of acidit
,
phase
and a need no special disoussion.
The
values used whenever they oeour in the salculations ares
k =1.38 x i10 1
6
27
erge
g 1033 /see
S= .003 oal/o/so/ 0 0.
o a .25 oal/ga/ 0 .
For detailed studies, the aoitatio
energy and
the preexponential constant should be empiricaly determined for a melt of a given chemical composition.
For
the order of magnitude studies made in this thesis where
the ahemical composition of the melt is uncertain, E and
f are estimated from Maexensie (1957) which gives the
activation energy and the preexponential constant
appropriate to equation 2.15 as a funation of ohemical
composition for dry melts.
These values are then adjusted
for the estimated effect of the presence of volatiles.
this is done in Appendix A.
The values of the activation energy used in the
caluwlations are E = 20, 35 sad 50 koal/mel, the value of
the preexponential oonstant is
S= 35 koal and
= 10
4
poises.
The values
0- 4 polses.re thought to be appro-
priate to a "wet" plutonio granite, and the other two
values are intended to allow for errors in the estimation
of E.
Less information is available to aid in the seleotion of Ef,q' and O.
Viscosity as a function of tempera-
ture for some natural rooks (Biroh et al 1942) and
glasses (Shartsis and Spinner 1951) was plotted and ourves
~-i_
--------.-11_1~-
1
~- ---
I~-_
L-L-L
L1(_-.-~I~P~LI
--
28
were fitted in order to determine typical values of E',
Qoand c.
The values of TO determined by Dienes (1953)
was considered in determining 0.
The values used in
the thesis are E' = 10, 20 keal/mol,
/a 10- 2 poises
and 0 = 7000 K.
The form of A is based on Frankel (1946) and the
value (A = 5 z 10" 9 c
-1
) is based on Hughes' (1955)
measurement on the pressure dependence of the activation
energy for cations in the ionic semi-conduction mechanism
in peridot.
Value for A p
An estimation of
p is obtained by using jadeite
phase.
(pa = 3.3 ga/or 3 ) as representative of the acidic
Hypersthene (b = 3.4 gm/e 3 ) and andradite
(Pb = 3.5 g1n/m
3)
are used as typical of the basic phase.
For these two examples of the basic phase, #p
.2 gm/a
3.
=
.1 and
The presence of volatiles with a density of
2 gm/cm 3 would lower the density of the acidic phase to
say 3 gm/cm 3 . This givesp = .4 and .5 gm/on3 . The
3.5 m/icn 3 are used in
values 6 p = .5 g/cm3 and Pb
the calculations.
To determine the initial values of a, N and S for
the study of the near surface differentiation, it is
necessary to obtain a better understanding of the
general features of the model.
Chapter III.
This is the subjeot of
29
(HA1ERT.TU
III
GERBLAC LiTURtY OV MODEL
AND
ARLY DIFERETIATION OF v;ARTH
AUI$
to £ttyz£srIlytrtierentiation
£4sription of
une
Before an estimation of the distribution and value
of a, N and 3 for use in the study of near surface
differentiation can be made,
it is necessary to study the
solution of several cases run on the oomputer using
arbitrary values of the parameters in order to make olear
certain general features of the model and the nature of
some approximations.
These cases give some insight into
what the concept of the slow growth of the continents
implies about the early history of the earth and will
allow the initial distribution of the acid fraction to be
determined.
For the runa,
oexp
of
E/kT
.
the form of the viscosity used is
The form was used beosase the values
and B are better known than
<and E', and only
general features of the model are wanted.
S was taken as a constant (S = .1)
Z = H = 300 km.
This is
down to
to approximate what would happen
in an earth that
hat had an initial uniform composition.
value A is
arbitrary.
It
The
was selected by assuming the
continents to be 30 km thick at the end of differentiation.
If
,
.14a
ma111
IM
30
If the 30 ka of material was initially distributed throughout
300 km, S m=st be .1.
Due to the uncertainty in the gross
ohemical composition of the initial earth, no more meaningful selection of 8 *an be made.
The computations
show that the value of 8 used in the model is only important in the oonveotive transport of heat and the conversion of potential energy terms.
The effeot in these
terms do not change the general features of the model.
The heat procation per unit volume was selected by
equation 2.14 to keep the temperature oonstant in time.
This is
done to make the effect of the temperature distri-
bution clear.
It negleots the change in the distribution
of radioaotive sources due to ohanges in the distribution
of the aoldio phase.
This is treated later,
The cases
are desoribed in Table 3.1.
Effoot of Changes in Uo
Oases 1, 2, and 3 examine the effeoot of chages in
Ut
on the results.
The only difference between the three
oases are the radius of the tubes and the initial value
of the number of tubes per unit area which aeast be changed
to aeooomodate the change in the radius.
The extreme
values of the radius used differ by a factor of 106.
The
relative number of tubes per unit area, the temperature,
and the elapsed model time after a hundred time steps are
given in Table 3.2.
The ratio (number of tubes/rit
area)final/(number of tubes/unit area)inital is
the
T BLE 3.1
DESCRIPTION OF RUNS USED IN STUDY OF GEN2RAL FEATURES OF MODEL
Run
To o
H
a (cm)
2400
10-6
kcal/ml
al #*see
0ooK
2
300
3
300
4
300
6
300
300
7
8
9
28004
300
16-4
2800
300
10-
2800
---
For these cases to
0
3.18 x 106
3.18 x 106
6
3.18 x 10
3.18 x 106
'-
100
million years
er
10mlo
300
10
1.4
35
o-"
1.4 x i0
104
35
1.4 x 10-14
10-4
20
1.4 x 10-14
50
1.4 x 10"
20
1.67 x 10- 1 4
300
35
1. 67 x 10
300
50
"
1.67 x 10
300
106
zx x 1$0
3.18
3.18 x 106
L
35
10l
les
100
o10
10o-4
-I
NOTEt
10
2400
2400
x 10 - 1
H(im)
104
1-4
10-
- -I~ ~'
- ~-I
300
300
300
10
300
10
10
TiBL 3.2
RINSs
TO sToDT a
-To OTF MOAVs TN 100s
Aol
tCTITVA'IOT
Inttl
2,
t,nI
$$ 7, Tas1
IN
X
m TInn
1 (z, )/ ( o), **str emD
ta
n1hr0Y,
= 35 KcAI/OL Fm ALL THRi? CASn.
Case
RO4,o)
2
r
W7,
OaCe 2
t a 10r
.18 lx 1 0 0tuobes/ea 2
-
c
S
10,0)=C 3(8O
Case 3
t = 5.1
x 10yr
3o
700o
6
1 oo
1370
16t4
1875
180
300o
69
3
70o
1.035
106
4.*027
r370
4.880
164
1875
0.057
o0.o000
.000
2060
o.00oo
2210
0.000
2311
000
.000
2315
0.000
2379
24o
.000oo
.000
2380
24~o
o.oo
o.oo
300
1.000
1.000
1.000
1056
13
1643
1873
1056
1370
1603
187
1.0oo
1.035
4.035
4.872
0.057
.000
2060
1.0oo
2060
1051
2059
210
2210
1.000
2206
1.01
29
20
2315
1.000
230
1.018
270
S3o
2380
Ahoo
1.0oo
1.ooo
2371
2392
.979
.831
6
90
120
150
.0oo
1.00
tab
oo/
n
1.000
1.oo0
1.0o0
1.009
1.026
1.0k
30o
.
o038
aT
T
0
=1=5 ts 3.A8o
699
1.000oo
2 yr
_~___
~L _1
111111
1~----
33
relative number of tubes.
There is no difference between
case 2 and 3 in the relative number of tubes or temperature at the end of the same
number of iterations.
How-
ever, there is a signifioant difference in the elapsed
model time.
There is a stradghtforward explanation of the
difference in model tine.
In all cases, except where the
velocity is so slow that the time interval is given by
equation 2.20 as in case 1, the change in the time seale
is inversely proportional to the change in the velooity.
This is expected from equation 2.21.
That the numerical
solutions keep this relationship is another indication
that the program is free of any serious errors.
A change
in U0 oauses a change in the time scale for the process.
Effeet of Initial TeaOr are and Activato EuurA
Qasea 2, 4, 6, 7, 8, and 9 are used to show the
effect of changes in the initial temperature distribution
and the activation energy on the final distribution of
the acid fraction.
The final distribution is said to
occur when there has been no significant change in N for
a 100 million years period.
To cover a sufficient amount
of time, it was necessary to make suooessive computer runs
for the same ease.
The reason is that there are only a
hundred time steps programmed for each run.
The length
of a step can be anywhere from a fraction of a second to
a million years depending on the velocity with a
corresponding spread in the elapsed model time.
--
-
NNM1I'116I1
L
-"~I
-^-
34
The initial conditions for a later run are the
final conditions from the previous run,
In order to
lengthen the time interval in the later runs, the extent
of the region, H, is redefined at the start of each run
by dropping all points where the fraction of the acidic
material is
.01 of its initial value.
This removes the
points where the temperature is highest thus lowering
(
)')
giving a longer time interval in succeeding runs.
The same boundary oonditions are applied to the new region.
Uo was taken to be the same in all six models to
eliminate the effects of changes in this quantity.
Table 3.3 gives the temperature, number of tubes per
unit area, and the elapped model time for each run.
A feature of the model revealed by the sixeases
described in Table 3.3 is
in
that there is
a marked inarease
the concentration of the acid fraction in
depth Z1 forming a low density layer.
a layer at
The depth Z1 is
seen to depend on the temperature distribution, and the
value of the activation energy used.
depends on the selection of Uo ,
the value of Uo used.
Since the time scale
this depth also depends on
In fact, there are selections of
Uo which allow the differentiataion tO be almost instantaneous
all
the way to the surface, and others which make the separa-
tion negligible at all depths
If
the 1/(T - 0) form of
the visoosity had been used in the study, a seleotion of Uo
TABLE 3.3
RUNS
TO ILLUSTRATE LOW DENSITY LAYER
DEPTH, Z, IN KM AND TEMPERATURE IN*K.
INITIAL
1 m'y. = 106yr,
CASE 2
FEATURE AND TIME SCALE OF MODELS. IN THE TABLE,
MODEL TIME IN PARENTHESES AFTER RUN NUMBER.
CASE 4
DATA
=35kcallmol, a= 10 cm
E=20 kcal/mol, a =10" cm
N =CONS. N=3.67x 1dtu beslcrri
N= 3.67x 10Otubes/ cm
conditions at end of run
1 (5.2 my) 2 (100 my) 3 (100 my)
1 (.14 my)
2
(2.6 my)
Z
T
0
300
30
699
60 105
90 137
120 164
150 1875
180
210
240
270
3001
N()
N(0)
IT T
1.00 300 1.00 300
1.04 699 2.91
691
4.04 1056 7.09 1021
4.87 1370
.581643
N()T
NO)
1.00
309
6.91
300
561
667
conditions
1
T
N(t)
N (0
0
300 1.04
30
60
90
775 6.19
1200 3.76
1575
.01
120 1900
150
18
210
240
270
N()
N (0)
1.01
4.49
5.45
300
700
1056
1.94
9.01
2175
2400
2 570
270(
277
300 280
E = 50 kcal mol, a= 10 cm
IN= 3.67x 10" tubes/ cm a
1
(100 my)
N(t)
N (0)
300
700
20¢
2210
2315
238
240
CASE 7
E=20kcall/mol, a=10 cm
N =3.67x1d* tubes/cm'
Z
m
NT
N ()
CASE 6
1.00
1.00
1. 09
3.84
3.98
.097
300
699
1056
1371
1643
1873
2
( 100my)
N(t)
N (0)
T
1.00
1. 00
1.18
4.35
3.37
300
691
1021
1258
1348
CASE 8
4
E= 35kcal/mol, a =10 cm
N= 3.67 x 10otubes/cm
CASE 9
E=5Okca/mol,
a=10 cm
N=3.67 x10'0 tubes/cm'
1 (1.5myv)
1
t end of run
(.O 07 my)
T
2 ( 8.1 my )
N(t)
-~(0)
T
300
9.53
775
1201
1575
300
2.68
772
1120
N(t)
1.00
T
2 (100 my) 3 (100 my
N( t )
N(0)
300 1.00
1.12
775
5.66 1200
3.21 575
.01 1900
T
N(t )
N
300 1.00
7.15
730 7.69
2.84 1044 2.29
1165
T
300
611
737
( 34 my)
N(t )
N(0)
1.00
1.00
1.71
641
T
2 (100m )
N(t )
FW-)
30011.00
T
3 (100my)
N(t)
N
T
300 1.00 300
775 1.01
765 1.02 765
1200 3.40 11 59 4.74 1159
1575 5.43 1441 4.21 1441
.88 1900
.43 1549
.29 1549
NOTE Thitial temp. used for these runs. 4
was not increased when H was shorted.
3b
oould be made that would allow the differentiation to be
instantaneous to the depth where the temperature equaled
0.
This is the shallowest depth in
this model to which
differentiation can ocaur.
The uncertainty in
the value of 0 appropriate to
the acid fraction coupled with the unoertainty in the
temperature under the oceans rule out an accurate estimation of the depth at which the motion ceases.
If 0 is
in the range 400-8000 0, and MaoDonald's estimation of
the temperatures under the oceans, plotted as figure 2.1
is
assumed as being representative,
the minimum depth to
the top of the low density layer would be between 30-90 km.
These oaloulations show that if partial melting
occurs in a region, the mobile material must move toward
the top of the melted region in a comparatively short time.
The shape of the masses in
which this fraction would
collect should have a larger horizontal dimension than
vertical.
eor as argued in
Chapter II, the vertical
dimension of thee masses can be no longer than the
maximum height that the basic material can support.
msib le Bar& Hiater of
rth
Now that the formation of the low veloeity layer
in
the model has been described and some understanding
gained as to the parameters which govern its
is
depth, it
possible to further develop the physical description
of the model.
The next few paragraphs are a description
_I_ __~
~I
~--L-~-~PL---~
37
of a process which might have occurred in the earlier
history of the earth.
The disoussion is
given in
order
to fix ideas for a later section and to sug6gest possible
extensions of this work.
No one is
more aware of the incompletenesa of the
following discussion than this author.
For a complete
disoussion, considerably more experimental data on the
flow properties of rooks particularly near their melting
points is necessary.
it
is also necessary to extend the
work of this thesis to cover a three phase system including
the terms involving pressure which were omitted in
the model
formulated for the upper part of the earth,
Assume that the earth was initially
of uniform
chemIal composition including radioactive elements and
that the material eonsisted of three fractionsi
basic, and acid.
Aarthermore,
iron,
assume that the tempera-
ture at the time of formation was everywhere below the
melting point of the material.
D4e to the burial of the radioactive material,
would be a general warming up of the earth.
This process
would continue until one of the fraotions melted.
(e.g. 4acDonald 1959,
Studies
p. 1979) of the melting point gradient
of basic materials and iron have been made,
point of iron is
there
and the melting
2000-20000 C, less than that of diopside
for depths greater than 300 km (see figure 3.1).
No studies
l
litll
milYllil
IIlI
W
nI
lt01II
lI
hi, huAYIIY
W
I'
II2Y I10=__mll
I I mlI
38
1000
DEPTH IN KM
2000
2900
5000
,-
4000
5 3000
. 2000
I-
1000
1.O
0.5
PRESSURE X 1 0 BARS
1.5
Figure 3.1 Adapted from MacDonald (1959) to give the extrapolated melting relations for iron and diopside within
the earth, Iron I curve is Strong's (1959) estimate.
Iron II curve is extrapolated by using the initial slope
obtained by Strong and Gruneisen's constant as appropriat high pressureso
31--~~
1--~*1~
r~rr.
39
of the melting point of acidio silicates with depth have
been made but a reasonable assumption is
that the melting
point of the acidio traction at depths lies between that
of the iron and basio phases,
Due to its lower melting point, the iron fraction
would be the first fraction to melt in the warming earth.
Once the iron fraction melted, differentiation would
begin with the dense iron phase settling towards the
center of the earth,
In settling, aravitational potential
energy would be released and be converted into heat energy
through viscous dissipation.
The temperature rise aooam-
panying this release of heat would help in the melting of
the acidic fraction,
Once melted, the acid fraction containing a large
fraction of the radioactive elements and volatiles would
be forced towards the surface.
The upward migration of
the acid fraction would continue until an unelted zone
(i.e., low temperature zone) was invaded.
The upward
motion of the acid fraction of this material would stop
in this zone forming a low density layer at the bottom
of the uzmelted region.
The amid fraction would cool and
solidify unless the unmelted region was fractured so that
the acid fraction could move adiabatically upwards
through the channels opened by fracturing or else the
near surface material was melted by some prooess.
Geophysical evidence indicates that there are no
large scale (i.e., of oceanic dimensions) fluid chambers
JIQI*-UI-Q-"~-l"1~4LUs*-C-LL
-- -*
40
in the upper part of the mantle.
The assumption that
differentiation was not completed in the early stages
of the earth's history implies that the near surtfaoe
region was to a very large extent unbroken and unmelted
and that the acid fraction has solidified.
The existance of a low velocity layer is implicit
in theories of alow accretion of the continents.
It
pro-
vides a physical feature that may be found by selmic or
gravity methods.
This investigation would determine if
the ideas of incomplete differentiation are plausible.
The teohnique to be used in such a study is
straightforward
and need not be elaborated on here.
Initial Distribution of Acid ?racti n for Near Siarfac
To determine the initial distribution of the acid
fraction for use in the following study of near surface
differentiation, it
was necessary to run several cases
on the computer using the ftonmof the viscosity given in
equation 2.16.
The initial temperature distribution was
taken from model 6 in figure 2,1.
The requirement oif no
significant change in N in 100 million years was used as
a criterion of when a stable configuration was reached.
The initial value of N makes use of the observation that
differentiation in the model below 100 km is
so swift
that the distribution after the initial run can be approximated by a function of the form
_
LIIII_--~
-.-111 1~11-~--21_
?b
0s
o
o
N=
6o0km
A
S3
60 kam < Z 4 90 km
C
Z >90 km
The strain energy contribution was taken as zero.
The
runs are described and the results presented in Table 3.4.
The ooncentrations for the last run in the cases listed
in Table 3.4 are used as initial distributions for the
study of near surface differentiation in Ohapter IV.
Larger values of the radius were used but the time
scale for the separation was so short that it was not
msaningful,
The time scale for a = 1 cm can be used as a
lower limit for larger values of the radius.
the time scale of such a.process could be estimated
from the equations given if the value of the radius of the
tubes appropriate to the melted region could be estimated.
However,
this is not possible from aailable
The time scales given in table 3.3 for
information.
a= 10
an upper limit to the possible time scale.
4
amS give
The ohange in
the time scale caused by assuming a larger radius can be
estimated as argued previously.
The previous runs indi-
eate that if the effective radius of the pore spaces in
the melted regions are contimaters or larger, the initial
differentiation would be very swift, requiring 100,000
years or less once the partial melting of the acidio
fraction occurred.
&
~ ---~-
_--LI~T=-~--
~-L-slL3~CIIII~~
~-~-f~--
TAB LE 3.4
RUNS TO DETERMINE DISTRIBUTION OF N FOR
ATURE, T, IN OK.
n= N(Z,t)/N(O,O). QE= 0 FOR
GIVEN IN PARENTHESES AFTER RUN NUMBER.
INITIAL
CASE 10
DATA
E'=10 kca/lmol,
a = 10 cm
N(O,0)=3.18x10 tubes/cm7
conditions at end of run
1
(16my)
2
(100my)
T
0
15
30
45
60
75
90
300
523
693
980
1238
1493
1693
n
n
n
1. 00 1.00
1. 00 1. 00
1. 00 1.01
1.0 13.99
5.00
1.69
5.00
5.00
NEAR SURFACE DIFFERENTIATION STUDY. DEPTH, Z,IN KM, TEMPER ALL CASES. A = 5.3 x 10 callcmr.sec. ELAPSED MODEL TIME
imy =106 yr
CASE 12
CASE 14
E'= 20 kcallmd, a =10 cm
E'- 10 kcallmol,
a = lcm
N(O,0)=3.18x10' tubes/cma
N(0,0)= .0318 tubes/ cm'
1
T
n
T
n
300
517
742
1000
1237
1.00
1.00
1.02
14.96
.98
300
483
659
799
845
1.00
1.00
1. 00
1.00
5.44
6.80
2.72
CASE 17
E'= 20 kcallmol, a =lcm
N(0,0)=.0318 tubes/cm
(100 my)
T
1
300
525
742
942
1118
1242
1299
CASE 25
E'= 10kcall/mol,
N(O,0)=3.18x 10
(1.lyr)
n
T
(7.8yr)
n
T
3 (.44my)
n
T
1.00
1.00
1.28
15.68
300 1.00
523 1.00
693 1..9
CASE 28
E'= 20 kcal/mol,_
N(O,O)= 3.18x 10
a = 10 , cm
tubeslcmz
1.00
1.00
1.00
12.07
a= 10 cm
tubes/crm
300
523
693
980
2
300
523
693
conditions at end of run
(2Oyr) 2
1
Z
0
15
30
45
60
75
90
T
300
523
693
980
1238
1493
1693
n
n
1.00
1.00
1.00
1. 00
5.00
5.00
O
5.00
1.0
1.00
1.00
1.26
15.13
T
300
523
694
98
1238
(.17my)
n
1.00
1.00
1.00
2725
T
3 (100my)
n
300 1.00
523 1.00
6941.00
980 27.25
T
300
401
483
528
1 (1.1xlOyr)
n
T
2 (.7
n
1.00 300 1.00
1.00 523 1.00
1.00 693 1.28
12.02 980 1560
3.86 1238
my)
3 (100my)
n
300 1.00
523 1.00
694 6.47
980 10.41
T
300
402
484
529
(. O my)
1
2
(100 my)
n
T
n
T
1.00
1.00
1.00
1.26
14.57
300
523
694
980
1238
1.00
1.OC
1.00
4.56
11.32
300
456
594
697
752
43
At the end of this stage of differentiation,
earth would have the iron core,
the
a basic mantle and a
mixed near surface region containing excess acid material
collected in masses buried at some depth,
radioactivity would be in
Most of the
the low density layer so that
the heat production in the deep interior from this source
would be negligible.
Compris*on oftodel
to Other ,odels
This model is based on the aCrrent ideas of partial
melting as contrasted to the earlier earth models which
were based on total melting of the earth.
The essential differences between models based on
partial melting and those involving total melting are
that partial melting permits the retention of a solid
outer shell at all
times.
It
is only necessary to supply
energy to melt a small fraction of the earth in order that
differentiation occur.
The completely fluid earth requires
differentiation to be more or less completed early in its
history.
This model requires the existence of an inter-
mediate undifferentiated near surface region.
The feature of a solid outer shell allows the consideration of the formation of the earth by a process in
which the temperature of the outer shell is the equilibrium
temperature resulting from a balance between the absorption
and radiation of the sun's energy at the surface.
only a small fraction of the earth to melt,
Lequiring
reduces the
44
energy needed to cause differentiation in an earth formed
by a cold process.
Both of these features are helpful in
explaining differentiation in an earth formed by the
current cold accretion hypotheses.
In the models where the earth is
asmed to be
fluid at some stage in its history in order that differentiation can occur, it is still necessary to appeal to
"fluid filaments" in the final stages of the separation
process. The following quotation from Jeffreys (1959)
illustrates this.
".
.
.
the fluid filaments left at
any stage carried up the latent heat and the new heat
by convection, and that as solidification proceeded the
residual fluid was forced up by deformation of the crystals,
so that solidification and the raisinc of the radioactive
material to the upper layers were part of the same process."
This mechanism suggested by Jeffreys is
the mechanism used
in this thesis to explain the differentiation of a partially
melted earth.
Consequently,
the mathematics in this thesis
can be used to describe the last stages in the differentiation of an initially molten earth.
------
--rr~---^-I~--~lru~ ~
II- - ------L-Lsl-~...
---- CI'I--- ..~.-------I~~.
.--------
01111011aII
I
45
OWIPTER IV
NEAR SURFAGE DIFFERlETIATION
From the disssion
in Ghapter III, it
that
seeas
the hpothesis of an incomplete differentiation of the
mantle requires that the said fraotion be more or less
onacentrated in masses at depths smebere between 30
and 90 km.
The question is how oan this material be
brought nearer to the aurface so that it
lated into the continents.
It
can be assimi-
was suggested in Chapter I
that any process which attempts to bring this material up
from these intermediate depths must be based on a process
of partial melting, and tbus, another question is
this melting occur.
how can
Melting is used to mean that the
acid fraction has been oonverted from the crystalline
state into the glassy state.
Once glassy, equation 2.16
describes the viseosity of the aaid fraction.
The partial melting of material for this near
surface differentiation is vi aalized as the result of
the conentration of strain energy released as heat in
a rather localized region.
Some meohanism similar to
that described by Orowan (1960) and discussed in Appendix B
of this thesis or to Treitel's (1958) sugestion that
shook waves might be very important in the region of
failure may provide a mechanism for this ooneontration
of the release of strain energy.
It
is
not certain that material in the mantle
behaves as Orowaen suglests, but after studying his
arguments, it
is
snees possible that some similar mechanism
ative in the earth.
If the acid material under the oceans is
is
trated in masses at some deapth, it
onn-
should be closer to
its melting point and oentain a larger percentage of
volatile material than basio material at the same depths.
It seems plausible that the masses of acid material would
be the weakest regions in the mantle.
The release of
strain energy should occur in these masses when sufficient
stress differences have been set up.
This leads to their
reuelting,
Using 300 cal/em3 as a tyical
heat,
it
value of the latent
would require 4 x 10 5 or 4 x 10 6 years to melt
the acid fraction in 10
Q w 7.7 x l
13
or 7.7
km3 (10,x 1000 x 100) for
10
Ix
14
cal/ca3.sec respectively.
The acid fraction occupies a third or less of this volume.
This sets a lower limit on the time required for differentia-
tion after the onset of intense asismic activity in a
region.
This time
amst be added to the model times obtained
in the runs reported in this chapter, for no provision for
latent heat was included in the model,
The model assumes
that the material is in a glassy state at the temperatures
existing and that flow will ocur for T> 0.
_ _ II~_~_C__C _^__~_j_______
_III___I__
47
If the shear zones where some melting has oocurred
extend from these masses up toward the surface, the
buoyance force would cause the upward motion to restart.
The shear zones are not visualied as voids, but rather
as regions that have become viscous.
These fluid ohannels
permit upward motion to ooaur.
a process of this
type that the equations in
approximate.
It
is
Chapter I are intended to
The radius of the tubes in the mathematical
model may be associated with an effective radius of the
channels in the shear zones.
The cases run to determine the time soale of
differentiation after remelting as a function of the
visoosity parameters,
strain energy is
cases reported in
radius of the channels, and the rate
fed into the system are extensions of the
Table 3.4.
The description of the cases
and the results are reported in
Table 3.4 that the initial
Table 4.1.
The ease in
conditions were taken from is
indicated by the number in parentheses beside the Cs.se
heading.
Cases 18, 26, and 29 indioate that QE
=
7.7 x 10- 1 4
oal/cm3 osec is the lowest value of this term for which near
surface differentiation would oour in this model.
This
is the value of ig which is given by n = 1 in equation B.l
and is equivalent to dissipating an amount of energy in
TABLE
4.1
RESULTS OF RUNS TO STUDY SURFACE DIFFERENTIATION.
ELAPSED MODEL TIME GIVEN IN PARENTHESES AFTER
RUN NUMBER. lmy= 10 YEARS. INITIAL CONDITIONS TAKEN FROM CASE IN TABLE 3.4 WITH THE NUMBER GIVEN
T IN OK
IN PARENTHESES BY CASE HEADING.
'
CASE 16
CASE 15
a =1cm E=10kcal mol a=lcm
E=lOkcal mol
Qo .77x10'calIcm-sec
QG.77 1'QcaIllcm-sec
NITIAL
DATA
conditions
at
end
of
1
(100my)
T
n
300
1.00
626
1.00
841
28.5
-947
(100my)
T
300
746
960
1176
n
1.00
2591
-
n"
100
100
1.00
273
27.3
!1 (lOOmy)
T
n
T
3001 1.00 300
523 1.00 626
694 1.00 841
9948
980
conditions
at
1
2 (100lmy 1 (3 5my) 2 (3.7 my)
n
T
'T
n
n
T
300
300 1.00
1.00 300 1.00
703 17.88 887
1.00 627 1.00
1142
16 87 842 16.87 907 -
(100my)
T
T
n
n
Z
300
1.00
O 1.00 300
627
1 5 100 523 1 00
842
30 1.29 694 1A8
45
60
5.5l
98
5.35
CASE
949
of
-
1136
CASE
-
n
0 1.00
15 1.00
301.00
45 1.00
5.00
7 5.00
90
5.00
T
CE=.77x1 calicm 3 sec
conditions at end of run
1 (7.7my) 2 (12my)
T
n
T
n
1.02 300
1300
30C 1 00
523 1.00 697 17.63 1061
.33 1440
693 2.39 934
1689
98 14.53 1206 1455 .01
123E
149
693
(5.1 my)
T
300
770
990
1 1201
1331
a = 10
3=1
QC-.77x10 caI/cmsec
1
n
100
8.53
9.43
-
T
n
100 300
100 523
1.00 694
1.26 980
14.6 1239,
1
n
1.00
1.00
1.00
6.16
9.72
(1 00 my)
T
300
629
874
1037
1119
1 (9.2my)
2 (4.4my)
T
n
T
n
300
100 300 1.00
100 830 11.38 1006
13.69 1095 649' 1327
316
1340
-
1538
-
13
a:16dcm E = 20kcallmol
E=10kcallmol
10kca
Z
(100 m
1
I
n
n
T
1. 00
1.00 300
1. 00
1.00 639
28.3
100 864
976
27.3
run
-
11
2
CASE 30
CASE
29
a=1cm
a=10 crr E= 20kccllmol
E=20kccl/lmol
QE=.77x10'2 call cm sec
Q =.77x10- calk.m .sec
E
E
2
ASE
27
CASE
26
-z
2
AE
2
a=10 c
emE= 10kcal/mol
E=10kcallmol a=10
OE=.77xlO-'callcr sec
QE=.77x10 cal/cm sec
E
E
end
19
CASE
CASE 18
a =lcm E=2Okcal mol a=lcm
E=20kcal mol
Q .77xlc'cal/cm. sec Q .77x-1'al/cm.sec
run
1
n
T
Z
0 1.00 300
15 1.00 523
3028.5 693
4 5
977
INITI AL
DATA
(43 my)
T
300
1218
1784
2128
THE WRONG INITIAL CONDITIONS WERE
USED IN THESE CASES. 11.6 SHOULD
BE SUBTRACTED FROM THE LARGEST
VALUE.
n = N(Z,t)/N(0,O)
where
N(O,0)=.0318 x
-?
~B~P1------------ -III-.rCIC-L
.~----r~....-ql_ _
--
--~--..---II-_----_~I~lli~--XIIII I1II
49
the region of failure equal to the annal amount of
energy carried in seismio waves.
The elupsed model times indicate that the tie
required for the material in the low density layer to
reach the near surface region (i.e., near 15 k)
is not
appreciably different from the time required to melt the
acid fraction.
-4th
million years is
acid fraction.
=
3 *See,
7.7 x 10"1 3 cal/onM
.4
the estimated time required to melt the
The time required to mov
thehe material to
the surface is 12 million years in ease 11 and one
million years for case 16.
This leads to the conlusion
that if the energy sources are available, differentiation
would require 1-50 million years after the onset of
intense seismic activity in a region.
The elapsed model time depends rather strongly on
the value of the radius and the viscosity parameters.
Very little can be said about the effective radius of
the tubes.
a = 1C-4 om is probably the aallest the
tubes can be, and if the tubes are larger than one centimeter, the time of separation is negligible compared to
the time of melting,
The visaosity parameters were deter-
mined independently and are thought to span the possible
values.
Oensequently, there is reason to believe that
the time scales are representative.
21mmeion of
esuts
The most important result of the oaloulations is
that the minimum rate of strain energy fed into a region
_ __II_~ 1_1
____ _
~I
50
of failure required for near surface differentiation maust
be at least as large as the annual rate of energy release
indioated by the energy in seismiO waves for the entire
earth.
Unless one of the suggested mechanism or some
other process can provide such conOentrations of energy,
an intermittent near surface differentiation does not
seen possible, and the major differentiation of granitio
material in the earth muat have occurred at some time in
its past history.
The existence of molten lavas indicates
that the melting of material is possible.
This implies
that either the energy sources are available or the estimation of the required energy is too high.
If the energy source is available and the low
density layer was formed, the existence of such differentiation seems possible.
The time required for the
differentiation of a local region is not unreasonable
and the process is straightforward,
ORHA TEI1 V
00NCLUSIONS A4D SUGGESTIONS eOR 'YmUETHE
This study indicates that there is
ORKi
a physical
process based on partial melting of silicate rocks that
could be responsible for the segregation of an initially
homogeneous earth formed by "oold accretion".
Differentia-
tion would not be complete in the early history of the
earth if the surface region were unmelted and relatively
unbroken.
A feature of this process is
the formation of a
low density layer containing an excess of acid material
at depths of 30-90 kas.
Table 5.1 gives the depth to
the top of the layer for the cases run on the computer.
The existence of sach a layer is
shown to be a oonse-
quence of the hypothesis that the formation of continents
is
a continuing process and any model which explains the
slow growth of continents mast contain a similar feature.
The temperature distribution required for differentiation is not in excess of the temperature thought
to exist in the ear-th today.
Computations indicate that
the time soale of this early differentiation is 100,000
years or less after the aold fraction melts.
It is suggested that the periodic addition of acid
material to the near surface region is a result of the
reelting of acid material in local regions of the low
density layer.
The reaelting is attributed to tho
52
TABLE 5.1
W
DETH TO TOP OF LO
TO TOP OF L.a
DUENITY LAYrl.
L = DBETH
DENSITY LAL IN KM.
S OISLS 2 OR ALL QAZS.
DATA IS Al~TRACT D
aROe BTABLsi
Z3.4,
Casn Iumber and Desription
II__
I____I_
_ _
Oase 10
Case 12
~___ON
-mPAm
0 i i I W- M 0_-UIII~II
Zt
a
I _I_n-*_~
a, 10 - oC
VEa 10 koal/mol
45
a 10 - 4 am
Za n 20 kinal/ml
60
E'
10 koal/aol
30
'
= 20 kal/m/el
45
Case 14
ase 17:
Oase 25:
a = 10 -
B'
Case 28
I0 koal/mol
a* 10E
2
2
45
am
= 20 koal/mol
60
53
local release of elastic energy as heat due to failure
by some mechanism similar to that suggested by Orowan
Once the acid material is
(1960) or Treitel (1958).
remelted, it would move oloser to the surface through
partially melted shear zones above the melted regions
of the lw density layer due to the action of the
buoyancy force.
The acid fraction in this stage of differentiation
does not reach the surface (i.e., volcanic phenomena),
but rather stops at depths somewhat above 30 kas.
No
attempt has been made to disuss the difficult problem of
the assimilation of this material into the continents.
7.7 x 10
14
oal/cam3ee is the minimum rate energy
can be fed into the region in order that remelting and
the near surface differentiation can occur.
This is
equivalent to dissipating the same amount of energy carried
by seismio waves in the region of failure.
Unless such
a source of energy is provided, it is doubtful that near
surface differentiation can occur.
However, the molten
lava in volcanoes indicates that perhaps such sources do
exist or that the estimation of the energy required is
too
high.
If
the sources required do exist, calculations
indicate that the tie
se
sal
for the upward movement to
be completed after the onset of intense seismic activity
is not impossibly long.
Por example,
if
the effeotive
____I__~ I
__
_~__ji_
_~__
I
rauias of the shear zones are between 1-10 -
4
____
~___~_
om, the
time scales are betneen 1-50 million years for the values
of the viscosity used,
This includes an estation
the time required to remlt
are reported iUn
the material,
of
Those results
able 4.1
the low density layer seem
to be a requ4red
feature of earth models which attempt to explain
nenat formation by a continuine prooess.
conti-
Therefore, a
thorewh inveostigation should be made to determine it
such
a layer exists.
It is well known that seisml
teohniques have not
been able to determine whether suah a layer does or does
not existE
is
Gutenberg (1959)
evidence of the exi tene
However,
and others show that there
of a loww velooity layer,
the evidenoe is not conlusive as to the existence
of a low density layer,
for the low velocity layer ay be
interpretated as due to a doorease in
reuultiw,
the elastic oonstants
frm the temperature increase.
An aproaoh using the orbital path of satellites
may yield an anawer to this ~roblem.
underneath the oceans is
or if
the mantle
undif erentiated compared to
continntal regions, there should be ant ~smetry in
jravity field ot the earth that is
of the continents.
It
the
not due to the presence
would take the form o
a mass
____
~_
detiotenoy in the oceanio regions.
Siune the sateflites
are very sensitive to large scale deviations of small
magnitude,
orbits.
this asymmetry would measurably affeet their
56
YVISOOSITY
14r4eV
Orature DepeAence
anad -T
The flow ofailicate melt
has been desoarbed very
Muoesasfully in the 41ass induetai by espirical relationships
based on the assumption that the flow is
ewtonian and that
Seperature dependence of the, viscoslty has one of the two
forms
to5 r. Lo
£
'/ (T-
A61
o +E Wi>
A,
+
and
Lo
whre
ko
and E and 16'
measured in
and
I\.
=Lo
are the viYsosity at infinfinite temperature
are activation energiese.
0 K.
The temperature is
The Kelvin seale of temperature will be
used for all tezperaturee unless other scales are specixfically
aindiated.
The eonsiderations of several
workers in
the field
will be reviewed and extended in order to determine the temperature -anre over which each of the above two expressions are
applicable and the values of the constants which are appropriate to earth materials.
-s~--
-I--
_~ II_ -
:1
Frenkel (1946) has examined the viscosity of liquids
as being the result of the movement of discrete ions and
obtains
Y
e.xp
P
where I, interatomic distance l
the
, a roelaation time; k,
n constant; 1, the activation energy for the flow;
the Boltzs
oC,
/kT
oeffiient of thermal expansion; 1i,
the bulk modul sa
a pataeaster giving the pressure
i, the pressure; and Y,
It is noted that the equation given by t*renkel
dependence.
has the temperature dependence of equation A.1 with the
additional feature of pressure dependenoe.
The following interpretation of the constant r
due to
Professor Hughes, permits an estimation of the magnitude of
the pressure dependenae.
is defined in ?reakel by
vriting
---- /
anduaing
I
T
T
0V
v
r--:
T
i s only a 4notion of the intera~-t
o
a7
which assumes that
distaus
e &ivee
4
_
_____1_C_____Ym__s__~NLII__________III
ti8
Giving
k
where V is
the Spectifio
aA/.4
olume.
The experimental value given by Hughes (1955) for the
activation energy for cations in the ionic semiconduation
meohansm for peridot
S
AA.5
is used in order to evaluate the manitude of d
O=
6
10-5/0K, k = 1.38 x 101
Orgs/*K, and
a=nx
Taking
10 " 1
2
ergse
gives
rY: ./74 1
Dane and Biroh (1938)
examined the effect of pressure
on the viscosity of borio anhydride glass for pressuare
to 2,000 bare at two temperatures.
The best
it to the data
is given by
r'
1
ex pp
p
withok = 4.6 x 10-4c2/kg at 5160G., 15 x 10-e4 M2rg at
at 35900.
up
There is insufficient data to draw ay firm
rl---l~lrl~
1 --~^-CIILY r~YIIY PII~-
conolusions from this except that the pressure dependence
does exist.
Xaoienzie (1957)
studied the effeot of chemioal
composition oonviscosity by examining polycomponent systems
of silicates.
He finds that equation Ad. is universally
obeyed for binary alkali and alkaline-earth liquid silicates,
and that the aotivation energy E i
independent of the
is
Oationia speoies at all compositions up to 55; metal oxide
in alkaline-earth systems and to 35; metal oxide in alkali
systems.
Figures A.
and A.2 reproduced from Mackenzie's
paper give the behavior of the activation energy and the
preexponential constant, respectively, on the composition,
and cationia species.
Aaakenzie (1957)
demonstrates that the behavior of
polycomponent liquid silioates can be deduced by an analysis
of the binary systems which are included,
If
the oations are
of the same group, the activation energy is dependent on the
mole fraction of the metallio oxides.
If the cations are a
mixture of different groups and the melt is
regarded as an
"ideal" mixture of the individual binary melts, the activation energy is given by
E=
where na
(E
)
+h b (Eb
and nb are the mole fractions of the two metal
exilea and (Aa)
x
and f)x
are the activation energies for
A.6
CYII~
~Cly
I~~~yy~~
for the two groups at the total metal oxide ooncentration x.
Equation A.6 oan be used to estimate the activation
energy and the preexponential constant for silicate melts
that have not been measured.
This is applied to a plutonio
granite to illustrate the use of this equation,
Barth (1952)
gives the composition of a plutonic granite as being 70,
and 30
meatallic oxides.
i02
The potassium and sodium oxides
Treating the other 24.5% of the metal oxides as
are 5.5>.
alkaline-earth oxides and using figures A.1, A.2,
and
equation A.6, gives the activation energy tor a melt of this
comonpositio
ase
= 45 koal/mol.
The preexponential constant
1o is approximately 10 - 4 poises for a 30 metal oxide melt.
Brookris and Lowe (1954)
studied the aotivation energy
of the system CaO-1i0 2 as a function of temperature and per
cent metal oxide in the melt.
The interpretation of the
results xas given in terms of the flow of a discrete silicate
anions.
If the melt contained 52, or more metal oxide, the
flow unit was a silicate anion chain.
the chain was one unit long.
At 66A metal oxide,
For the ranle 12-5Z4, the flow
unit was of different structure than a chain.
less metal oxide,
molecule.
the flow unit was tu ought to be the silica
The behavior of logney.
some ourvature.
For 12/ or
1/T was found to have
It is sugested in the above paper that
the change in sativation energy at the hi4her temperatures
can be explained by assuming that the bonds of the flow unit
are weakened by the addition of thermal energy and that the
-~ -IIIIICrPIIIIII ICI
~-~LIC--~-~)IIC~*Y
IIC~L--~PIIIII
1
shearing stress is more easily relaxed by a breakdown of
the complex ion into simpler (and smaller) species than
by their flow.
energy.
The smaller anions have a lower activation
This ourvature in the plot of log r vs. 1/2
may be explained by the introduotion of an activation
energy for short range order as follows.
Dienes (1953) considers the effect of short range
order on the viscosity by using an expression given by
Frenkel for the temperature dependence of the short range
order and obtains
p
k(T-)
A.
where B is the activation energy for visass flow in
disordered liquid, U is the structure activation energy
of Jrenkel and/is the viscosity at infinite temperature.
If U = 0 or T is large, equation A.7 reduoes to equation
A,1 and if U>E and
U/k (T - To)> 1, it reduces to
equation A.2
In equation AT7, Dienes finds that To is a constant
and is approximately 5000 K. for some silicate glasses.
Table A.l which is part of Table I from the above paper
by Dienes is given to show some values of the above quantities, and the closeness of fit is obtained by the use of
equation A.7.
The range of viseosity oovered by the data
in Table A.1 is based on 102 - 1014 poises and the
temperaturea are between 5400 0.-1400O C.
LII^X~~ IIIYIIIYI
II/
_~1^
b2
45s
TABti
A.d
Parameters for some ailicate glasses fitted by
This is part of Table 1 in Dienes (1953).
Curve A.7.
AVg. devia-
0 flo T0 in
%
0J
POies
a/
2
-2.135
560
18.28
11.77
.073
i 10
-1.446
560
4,57
17.11
.029
# 330
-2.249
562
18.28
10.47
.002
Silicate
a eea
NoTs
Logl
E
U
tion in
k±l122l10
L
Visoity of gins )szaples are reported by Robinson
ad Peterson (1944).
Dienes describes a
omputational sheme for caiao-
lating the paran eters in equation A.7.
However,
the avail-
able data for earth aterials which eonsists of three or
four measurements per sample does not Justify the appioation of A.7.
in fact, even 0 in equation A.2
annot be
determined with any reliability by a oomputational *oeaso
A typioal value of C for equation A.2 is
determined later
which is used in the oalculations.
A better understanding of the effect of the parameter
C and its importance in the model can be obtained from a
study of the measurements of 3bartais and Spinner (1951)
on optical silicate glasses.
for sample #8198 which is
The results inclxded are
typoeal of all the samples studied.
Three ourves are plotted in Figure A.3 to show the effect
of the parameter C.
Iog viscosity in poises is plotted vs.
1/T, 1/T-273, and 1/2-473.
The
aurves were fitted by eye,
-Y--~~n~-~-~_-_-r
Y P---c-
-I-- lpllli
-~LIII~
lrm((
As can be seen, there is a seleation of 0 which removes
the ourvature.
The difference in the predicted vieoasity between
a curve including 0 and one omitting it
Figure A.4.
vs. I/T.
is shown in
in this figure, log visoosity was plotted
A least squares fit of this data by four curves
(
was made where(:toy
-
-I; io-3
an
td
,,.
In fitting aurve A10O,
ct
:
4
ad
i
r
Tk
3
the term b/t was found to be
so small that this form could not be distinguished from A.9.
Therefore, the form A.10 is omitted from the disnussion.
The ourves oorresponding to equations A.8, A.9, and A.11
are plotted in Figure A.4,
Table A.2 gives the values of
the parameters for the three ourves and the differences
between the measured values and the values given by these
From Pigure A.4, it i s seen that both the linear
form and the 1/T-c form give a good fit
to the data in
the
ly ___~
__1 _11__1^_ ~1
1__111^
_ _CI__
_ll___
__~__1.^_
1___~ __I_(~I I~_~_~_I_~_
interval of temperature where data was taken.
The
differences between using either of these two forms in
the calculation in
this interval of temperature away from
the melting point are much less than the error associated
with the uncertainty in the chemical composition of the
material.
However,
the agreement between equation A.8
and equations A.9 and A.11 is
low temperatures.
rather poor in regions of
The values of viscosity given by the
form A.8 are much lower.
Birch et al (1942) lists
the measurements of the
viscosity as a function of temperature for several rocks
and artificial
natural rocks.
materials with a composition similar to
The data given for seven of these samples
are plotted in Figures A.5,
A.6,
and A.7.
Values of
O7
and E determined by a least squares fit of the data are
given in Table A.3.
9.
and E'
was determined for two
natural lavas using 0=70 0 °K (the selection of the value
is
explained later) and they are given in
Table A.3.
The
curves for these two lavas are plotted in Figure A.7.
Table A.4 gives the comparison between the fit
obtained
with 1/T and l/T-C temperature dependence.
There is a definite relationship between the chemical
composition and the value of the activation energy as
expected with the more basic material having the lower
activation energy.
and
?
However, the values obtained for E
using a 1/T dependence are higher than those
r_------I~
LIII--~ - -l_
I
CL__IIIIIIIYIIIIIIII~L
__ _/I4__1C__ls_
_^IC~s~_
predicted by use of the curves in mackensie,'s paper for a
This
glas y melt with the same per cent of metal oxide.
discrapancy arises from the ourvature in plot of the Via-The high values of
cosity vs. 1/T for the natural roks.
the vis osity at the low tenperature *ad inerease the slope
of the line and decrease the interoept at infinite temperature.
The physieal interpretation of this is not certain.
There zre two Poesible explanations that have some merit.
is
The first
is
it
even if
assumed that theequation A.8 is
valid down to the solidus point of an individual component,
this relationship cannot be valid in
a polyoamponent system
due to the dependence of E and ?. on the ohemical composition of the melt.
melt is
lowered,
he temperature of the polycomponent
When the
crystale forn over a wide temperature range.
These orystals remove a greater portion of metal oxide than
The remaining fluid would contain a
silica.
cent of metal oxide.
aller per
This would lead to a higher activa-
tion energy and a smaller
the activation energy is
q. for the remaining fluid since
larger, for
more acid melts.
temperature scale of the measurements reported in
V( vs.
change in
1/T to be detected if
it existed.
slope would be continuous.
If
the litera-
the slope of
ture would not permit a discrete change in
log
The apparent
care is
not taken
to insure that equilibrium conditions were reached,
experiment would show a discrete change.
the lit
dependence of the visoosity ie
The
Hence,
no
even if
correct for a simple
~C1~_
system, It iis necessary to use a 1/T-C dependence in a
complex systsm such ae a granite to describe the viscosity
as a function of temperature
when taperatures in
the
range containing the melting point of the various components
are included.
The second explanation is
in
that there is an increase
the short range order as suggested by Dienes as the
approached even for a single component
solidus point is
system, and that the curvature is
There is
too little
due to this phenomena.
known about the behavior of the vie-
cosity as a material passes throSh its
deoide
the
melting point to
question of which of these explanations is the
most realistic.
One group of authorities look on the transition from
a orystalline to a liquid state as being analogous to the
liquid to gas transition.
While others view the orystalline
to liquid transition as being
transition.
Boreliu
of both groups,
(1953)
esaentially a discontinuous
gives a review of the arguments
and presents his arguments for accepting a
smooth transition from the crystalline to liquid state.
The question in an oversimpliied form is,
does a critical
point exist for the crystal to liquid transition.
Experi-
mental evidence on this point does not seem to be conolusive.
For the purpose of the calculations in this thesis,
it does not matter which of these two viewpoints are "correot".
-li-x)
l--rru
I---- -----~-
----* --_1..
.__^-
II~- ----- -
- -~~~-----~--
~a~-6~
~~~-*111~1
._---~-~PPIICI~II~L
6
It is enough that the viscosity is described by either
equation A.1 or A.2 for the temperature ra4ge of interest.
In either oase, the forn A.1 is a poor predictor formula
when temperatures near the solidus point must be considered.
Two forns of the viscosity are used in the thesis.
They are
ku
/
/
and
(/
where the hydrostatic approximationP = pgZ, for the
pressure has been used sad X = 1.5 pgAk.
The first relationship is used when the melt is at
teamperatures relatively far above its
solidus point, and
the second form is ued at lower temperatures.
oulations in the thesia, the first
form is
In the cal-
used in
the
study of the initial separation ad the second form is used
in the near surface differentiation.
th
ffot
of . Vat
Jose
the Visaoit y
Obervers have remarced on the effect of water on
the viseosity of silicate melts.
It seems that'et melt"
are less viscous than "dry melts" of the same chemical
composition except for the water.
A rather extensive
_YI^L~PPL
Y~
_L~PL~
63
literature search did not disclose any experimental work
oontrastine the viscosity of a "wet melt" and a "dry melt"
of the same cheaical composition from which any
aatitap-
estimation of the effect of the water could be made.
tie
Therefore, it is necessary to use what data is available
to estimate the effoot of the water.
Minakami (1951) reports that the viscosity of lava
as measured
from the 1951 eruption of the volcano Oo-sxi
in the
a. Samples of
ield was 4 x 103 poises at 11500
the same gross chemical composition when remelted in the
laboratory had a viasosity of 5 x 104 poises at the same
temperature.
The differenae in the viscosity was attributed
to the richer volatile content o the natural lava.
It it
is assumed that the effeoot of the volatiles is to chane*
the activation energy for visoous flow, the relationship
wr
where
bi
n
-dry"
?4ry
A12
wet and T1 is
the temperature in
% at
which the measurements were made. equation A.12 oan be
used to estimate the change in the aotivation energy due to
the presence of the volatiles.
several values of
is
at Ti
Table A.5 gives &E for
1423 O.
SE = 7 koal/mol
indicated by the measurement on the 0o-siza lavas.
activation enery for such basaltlc lavas is
30-40 keal/mol.
The
approximately
A tentative estimation of the effoet of
- ~
IIIIIIWLII--~LPLllsl~sl\lP
____
_l__pl__lllUI__WI____II___~-*~L3YIIOI~I;
the preeenae of the volatiles is that it can cause a
chauge of 2< in the activation energy
£.
TABUE A.5
b kcal/mol
t4
ar/
4
3.9
12
7.0
13.0
100
David B. Stewart of the United States Geological
onn the
Survey has furnished some unpublished informatio
sttling rate of crystals in hydrous silioate liquids from
which the viscosity of the liquids can be estimated. The
data and the caloulated visosity for four of the measurements are given in Table A.6, The viaeosity was calculated
using Stokes approximation for the slow fall of spheres in
a fluid (i.e., U = 26pga2 /9z where 'a' is the radius of
the aphere and the other symbols have the same meaning as
before).
T-ABLE A.6
SiSe in
Qent Microns
jj
?rosjjr#
iare
2,000
5,000
5.5
10.
7
5-10
10,000
12.5
4
10,000
12.5
4
NOTE:
Di*-
}zrtiole
bAPt
Tp
(
0 )
"
1040 An7OSi30
900
itj
Qomposi- Plaoeaent
tion
1320 AnOSi20
1190
go Vias-
"
in
Poises
.163
.6
.25
.3
.35 .233-.933
.15
.43
.367
.10
.43
.51
An stndo for anorthite and Si for silioa.
_~_
The data is not
u floicently extenslve to determine
the s4atition energy and the preexponential constant for
these sibstances.
However, the extremely low valaes of the
visoOsity found for these wet melts indicates that the
preeanoe of water vapor is a very important parameter, and
that the estimation of a 20%, ohange in the aotivation due
to this ascse is probably not too high.
The density differenoes
3
of .4 g/o#
between the
hydrous silieate liquid and the orystals indicates that
the value of .5
Wrm3 used in the thesis as the density
difference between the basic solid and the liquid aid
fraction is not too high.
The data supplied by Stewart also indicates the
Iaportanoe of crystals settling out of even suah a relatyiely
simple system as given here.
In particular, Stewart mentions
that the settling of crystals is an unwanted phenomena for
equilibrium is diffi
lt to obtain when settling oecurs.
The cheeical composition of a plutonio granite should
be typioal of the omposition of san
"acid fraction" which
can exist except for the volatile content.
Consequently,
the value of the aotivation energy of a plutonio granite
previously oaleulated from Fig re A.1 reduced by 201 is
used as typical of the value of the activation energy for
an "aCid fraction".
This valne is 35 koal/mol.
Figure A.I
~~.
.---^^-.-
l-rM.111411,
.. .1il
indicates that the activation energy omanot radioally
or
differ from thiis value for my melt containing 20
Therefore,
more metal oxide,
three activation energies
are used in the calaulations where equation A.1 is used
as the form of the viscosity to include all possible aoid
The
They are E = 20, 35, and 50 koal/mol.
fractions.
range could be narrowed if
the quantitative effect of the
water vapor were better understood.
preexponenial aconstant,
1,
val
The value
of the
is taken as 104 potea.
This value is not in error by more than an order of
magnitude for melts oontaining 10-50Q
El'to
of metal oxide.
and C
ointed out, there is
As has booeen
a soaroity of
data on the temperature dependence of the visoosity ih
the region near the solidus point.
A result of this
that the values of E'
scaroity of data is
are not
and
known as well as E and fo , but as has been shown, the
1/T-C dependence
ast
be used <or temperatures in the
melting range of a complex silioate.
Therefore,
it
is
necessary to estimate as well as possible the values 2',
16 and
C for use in the thesis.
The values
' = 10 and 20 koal/mol are used as
being typical of the activation onergy of an acid fraction.
The values are typical of the activation energy of the
natural lavas and the glasses of Shartsai
I 10 -
2
and Spinner.
poises was used for the preexponential constant.
r~Y"CL"Y-YI~Cr~rrrraPIIIPslll
~----C-XI--;L---a---
0 = 500
PI~-------
O,. is a typical value for the glasses
reported in shartsis and Spinner.
For the natural laas
studied, 0 fell in the range 7000 K.-1000 0 K.
is the valio reported by DiaSes for his glasses.
m 5000 K.
The value
of 0 adopted for use in the calulations is C = 7000 K.
The
unaertainty in 0 is indicated by the preeeding discussion.
ln the calolations where C is used, the viseosity
is made infinite for 2T%
temperatures.
.
This ignores flow at very low
The main effoot of changing 0 is to change
the depth at which flow becomes negligible (i.e., the
visoosity is made infinite).
The uncertainty in the
temperatures that exist under the ooeans would not permit
determination of a partioular value for this depth even if
0 were very well known.
TAX 1E A.2
LUAEST SQUA3M P ITO DA IM SAM
esoon,
m cmv.
mC
. .670o + 10.30
8-
x
/t
.9 9-.
AM SPI Mp 1951)
.
l OmAP
x
5t.
- 1.409 + 3.710/t - .473
.11
log l
103/
=
j198 (swAt9si
INzSe
A*6
A.9
/A
m
/ A.8
* /
.9
/a-
&U.
.6398
1.961
1.859
1.968
1.995
.102
-.007
-. 034
.6789
2.306
2.256
2.286
2.301
.050
.020
.005
.7287
2.736
2.764
2.719
2.718
-.. 8
.017
.018
.7862
3.241
3.347
3.254
3.234
-. 106
-.O~k
.007
.8525
3.9
4.
3.924
3.891
-..120
-.
.011
.931
4.771
4.8t21
.78
4.761
..o5o
*.0t7
-.
.oo
1.0163
5.833
5.687
5.810
5.851
.146
+.023
-.018
--.
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Mole 5. metal oxide
Fe,. A..
40
50
2
nrgy of activation for viscous flow in binary liquid silicates.
S6
7
8
10
20
M ole
F
A.2. Relation
30
merol
40
50
oxide
sand composition for binary
between the pre-exponential constant
liquid silicates.1t
LOG ViscosITY
/
/
/
-----V5 I/T
/x
3.9
/T-"73
1.,....,
//VS
-
29.
1/T-473 where T is--- in
5.9
, */
TEMPERATURE
SCA LE
/
i'
,
i
.65
.70
.75
.o80
.75
.0
.85
.90
.90
1.00
.0
.0
.Ao
.
I
i
p
o
.85
.90
.95
I.oo
.95
.oo00
1.05
1.to
1.15 ,.20
.30
1.00
.40
1.0
.50
70
I
I/T
1.80
I T- 273
1/T-473
I
SLOG VISCOSITY
/
I..
q
-.1409 t3.710/x -. 473
7-
6
LOG .= -.554
6.61 X
LOG
-4.670o
-.
o.121
5
4
.3
Figure A.4. Plot of log viscosity in poises for a silicate
glass (sample #8189, Shartris
and Spinner 1951) vs. l/T, 'T in
oK. Three curves were fitted
by least square technique.
x- IO/T
X
0o,
3.5
.4
.5
.6
.7
.8
.9
1.0
1.1
/.2
/.3
--
80
LOG VISCOSITY
5-
4
4
4
SYSTEM DIOPS IDE-ALBITE-ANORTHITE
CURVE
1
DI-AB-AN
40-60-0
2
3
4
20-80-0
40-40-20
20-60-20
SYMBOL
x
0
C&
Figure A.5.
Plot of log viscosity in poises for indicated
Data
system vs. l/T, T in OK.
is from Birch (1942). Curves
log r = log q + E/kT fitted
by method of least squares.
2
107
S1
.536
.567
..
.599
I
.630
.662
.693
,
.756
I
.788
.81/9
"T
1
LoS VISCOSITY
SYSTEM ORTHOCLASE-ALBITE
CURVE OR-AB SYMBOL
0
60-40
1
2
0-100
7
6-
Figure A.6.
Plot of log viscosity in poises vs. 1/T, T inoK,
for the indicated system. Data
is from Birch (1942). Curve,
log r = log 1 + E/T, is fitted
by least square technique.
5-I
.56
*I
.57
.58
.s5
.60
I
.61
.62.
.63
.(o4
.65
103
l"
(2 A
LOG
V15COSI TY
47/
//
//
CURVE EQUATION
1
log
= -5.118+10.426X
3
log
= -6.289+10.645X
4
log
= -3.076+ 4.439Y
3
2
/
2
log
= -0.253+ 1.866Y
where X = 10 /T, Y= 10,-700
.700
.750
.o00
.850
.3o0
10/T
Figure A.7,
Plot of log viscosity in poises vs. 1/%
T in oK, for two natural lavas.
Curves are fitted
by least squares technique.
Curves 1 and 2 are for
a lava from Mt. Vesuvius, and curves 3 and 4 are for
a tachylite lava from Mauna Iki flow, Kilawea.
The
data is from Birch (1942).
---.--~I^--I------- --- ----
--- I---~^~*---------L-~LLIII~----~"II~
n~__
APPENDIX B
ENflRGY SOmiozs
The possible sources or sinks of energy which may
be important in the model are the release of gravitational
potential
ential energy as the material sepaates, chemical and
phase ohanges,
VdY
work,
radioactive decay,
and strain
energy which is locally converted to heat by dissipation.
An order of magnitude estimation of the relative importanae
of the various possible sources is given in order to determine which sources may be neglected and to indicate the
error introduced by their neglect.
The change in the gravitational energy as a volume
V of the basie phase is
replaced by acidic material from
depths is
where A Z is
the distance over which the transfer to made,
and M is
the not mass transferred,
that g
constant,
negligible.
3.1,
it
is
assumed
= 0 and compressibility effectse are
These last two assumptions are Justified in
the next section.
rise is
o
In
For the adiabatic oase,
/17k~ Ay'A
D - "C--~
the temperature
where c is the specifio heat.
If A Z = 90 km,d p = .5
gm/Em3 , p = O - Sp = 3.5 t/=
3,
C
=,25 aal/gM O
A T = 330 0. The other sources of energy are compared
to this source in order to determine their relative
importance,
The term 4.1 was included in the energy
balance equation and was found to be negligible for
reasonable values of the velocity of the aid fraction.
The .rdV work done by ±nterohan ing equal volumes
of acidio and basic material through a distance A Z is
(P V di - Pd V)bC
V
35
-T b)_
where Vo is the volume exchanged, 1 is the preseure,r i
is the coefficient of thermal expansion of the i phase,
anad
iT
is the isothermal compressibility.
The relative
importance of this term is found by taking the ratio of
equations 3.5 and 3.1.
Giving
-(AT(r)
i p
-,gz,
b
dA
P
4
3 X 13
'0
10
5
0/m,
2/4,
bf
-. ,b = 0
3.5 c~/=3
~ a
/*0,
d
.5 gm/ca 3 , this ratio becomes 7 x 10"4 z/m.
The
PdV work is seen to be relatively unimportant if the
changes occur above Z = 10 3 ks.
86i
Obai
Cal and Lhaae.
Ohat
In a phase change, the amount of energy involved
in the reversible conversion of mass M = pV is
A wn = pA
where
3H.2
H is the heat of reaction/gm for the transition.
Forming the ratio of equations 3.1 and 3.2 gives
p is the density difference between the aoidia and
basic phases before the phase oaanres occur,
The transitions
albite + aepheline a 2 jadeite
and
quarts = ocesite
are typical of the transitions which might oocur in a
segregation process.
If the ratio in equation B.3 is set
equal one, a value of i Z oan be found where the energy
released by both prooesses are equal.
For the first relationship, Robertson at al (1957)
gives
A
= - 3.6 keal/aml.
Ap = .5 g/on3 , and g = 10 3
Using pj 4 3.3 P/Oa/3 ,
m/sea gives A Z = 49 kn.
For the second relationship, MaoDonald (1956) gives
AH = -. 22 keal/aol and poes
A z= 9 km.
=
3.02 g./ca 3 .
This gives
a
aiai
aIa......
-...
.....
,
.
The value of A Z found for the first transition is
comparable tto the distances material is
the earth.
transferred in
The energy requirements of the phase changes
are of the same order of magnitude as the potential energy
released by the separation of the phases, and can be
neglected in the energy balance equation,
Volume changes are found to be negligible in the
energy balance consequently,
there is no provision for
volume changes in the model.
This leads to an unrealistic
final configuration, for the surface of the granitio
material would be the same as the original surface of the
ocean bottom.
There are large volume changes associated
with the phase changes that would ocur in the acid fraotion durin
separation.
A brief discussion is
given to
indicate the magnitude of the possible increases in
volume in the acid fraction.
The changes in the basic
he deep depths at which
fraction are neglected due to to
such transitions would occur.
The feldapars at depth would be in a form similar
to jadeite (Robertson et al 1957).
A 28% volume increase
is associated with the transition 2 Jadeite-*albite +
nepheline at 1 bar and 250
.
The transition coesite--
quartz gives a 14% volume increase (atoDonald
under the same eoditions.
is
little
1956)
The percent change in volume
affoected by pressure.
If 30 kas of aeidic material is converted from
the high density phase stable in the mantle to the low
density phase stable in the oraust, the resulting increase
in elevation would be between 4 and 9 kas.
The loss of
volatiles to form the oceans ad atmosphere might decrease
this by about I km,
If A is the rate of radioactive heat produotion/
m3 . see for a granite, the raPioactive source term for the
model is obtained by assuming that the distribution of the
radioactive material is proportional to tohe distribution
of the acidic phase.
This gives
A8a2N
where for a granite isa2N = 1.
A = 5.3 x
10
*
"13
3.6
Bullard (1954) 4ives
cal/ea 3 . seo, and this is the value used
in this thesis.
The question of how amuh strain energy is released
anmually is not settled.
However, it
seems certain that
all of the strain energy released in a local region is
not oarried away as seismiaenergy.
It is equally certain
that the deep and intermediate-depth seismio faulting is
not a brittle fracture.
Consequently, the energy carried
by seismic waves mast be oonsidered as a lower limit to
the elastio energy released by failure.
I_
_~~__~_L~ _CI_
___ -1_1
-1-~-1-1111
~~I_
^l~l___I~
~
~II
C8
timations of the amount of energy released each
year by earthquakes as determined by relating the measured
amplitudes of the seismic waves to the energy of the earth-
quake are not at all certain.
estimate 10 2 4 ergs/yr.
Gatenberg and Piohter (1956)
Their previous estimation was
1026 ergs/yr.
The potential energy lost in
source of energy which is
a cooling earth is
a
stored as elastic energy and
To esti-
may subsequently be oonverted into heat energy.
mate the magnitude of this souroe, a rough treatment of
the potential energy lost in a cooling earth is given.
The common assumption made in treating the energy
released in an earth which undergoes some sort of failure
is
to assume that the initial and final stress states are
hydrostatic.
With this assumaption, the potential energy
released in a cooling earth can be expressed by oaloulating
the difference in potential energy between two spheres
in a hydrostatie state of stress that have thehe
but a different radius.
ae mass
in terms of the change of radius,
the difference is
S(g)'
where A R is the difference in
(pg
4
5 R
radius and 4 R< R.
assume that the release of the potential is
continuous,
It we
more or less
this can be written as rate of energy release
by replacing A W by A W/d t and A R by ARiB
t.
Hales (1953) gives A/d t/t = 10
cooling earth.
11
O/sec for a
a
Taking
R = 6.4 x 10 8o
E = 10
2
dynras/=2a
13
p = 5.5 gm/m3
g= 103 m/seo2
gives A
IA
t a 2 x 10 2 5 ergs/yr,
This is 10 times the
observed seismic energy and 10- 2 of the total heat flow
out of the earth (8 x 1027 ergs/yr.).
Due to the uncertainty in the rate of strain energy
released as heat, the value of this term (called Qg) will
be assumed onstant with depthdeph sad some muliple of the
rate of energy release as measured by seismic nwne
energy.
The expression is
QE
a
1
x 10 04/004
where n is a multiple of th
3.7
energy oontained in seismia
waves and V is the volume in which this energy is assipated.
V is taken a a cube 100 x 1000 x
ka3 where 100 and 1000 km
are the area of the region and H is the depth to the bottom
of the undifferentiated region. The values used in the
caloulations are n
QE = 8 x
1 3
10 and I and H = 90 km giving
d x 10' 14
and8
al/se/cu3 respectively.
When the dimension of the region is decreased to iengthen
the time scale as described in Chapter III, the amount of
elastic energy released in the region is kept constant by
increasing the value of
E used in the later runs.
These Valaue
of n were selected because the
calculations in Ohapter IV
minit
ind oate that they are the
values of Q which clan
differentiation.
easily explained.
ause the near surface
These values are higher than oan be
Unless
ome meOhaiaM of failure oan
concentrate such large amounts of energy locally, It
is
diffioult to see how the near surface differentiation
oould oocur.
The following two mechanioss are mentioned
in order to indicate possible means of aohieving this
concentration,
Treitel (1958) suaets that if
shock waves ocur
in the region of failure, the energy carried in amall
amplitude seismic waves is less than .1 of the energy
released with the reander dissipated as heat.
This would
provide a meohanien for conentrating such large amounts of
energy.
Orowan (1960) discusses the streosses required for
brittle failure to ooour even at relatively moderate depths
in the earth and finds that the stress differences required
for this type of failure are prohibitively large (l
bare).
400,000
He then suggests that earthquakes (i.e., faulting)
may be due to a graual conoentration and local aooeleration of creep arising from an inherent physical instability.
In any mechanism of failure similar to that suggested
by Orowan, a large part of the elastic energy released
locally is
converted into heat.
An equally interesting
feature of this type of mechanism of failure is
that it
111.
1~r
_^11~1--MOMMOMMOOMMOMWA
provides a means of concentrating the strain energy from
a much larger region into the region of failure.
It
is
difficult to visualize any meehanism of
failure under high hydrostatic confining pressure which
takes into account the observed plastio flow at high
temperatures under long period stress differences that
would not convert a large part of the released elastic
energy into local heat energy.
-II
~-_IIIIYIUlm~-IIX_-___
_ I^IYII~-QII~C-PI^
8'
~ol ~f
A PENDIL 0
NUIMEICAL iEAPROXIMATION
General
The set of simultaneous nonlinear equations (2.1,
2.5, and 2.12) which form the mathematical model cannot be
solved in closed form.
Consequently, numerioal techniques
are used to obtain a solution.
The partial differential
equations were approximated by a set of finite difference
equations using central differences to replace the
differential operators.
This set of difference equations
was solved on the IBM 704 at the M. I. T. Computational
Center.
The method of solution used is a simple forward
marching scheme in the time variable.
There is a danger that answers obtained from such
approximations will bear little or no relationship to the
solution of the differential equations.
This can oocur
due to the amplification of round off error, to excessive
trunoation error, or to blunders.
The net spacing and
the order of the differences used determine the relationu
ships that must exist between the parameters in the
problem in order that the solution to the difference
equations approximates the solution of the differential
equations.
In certain linear and quasilinear eases, an
error analysis can determine the required relationships
between the parameters for a given net and order of
differences.
For most nonlinear systems, it
is not
-rr--~l_-~_--~- -----
-D
--^^--1----^III
-'~X-II-L'I-
Ga
possible to perform an error analysis.
is
In such cases, it
necessary to experiment numerically in order to estab-
lish the required relationships between the parameters.
The set of equations that form the mathematical model in
this thesis is such nonlinear system.
Stability and Co verqne COnditiones
Prom analysis of finite difference approxiations
to a quasi-linear parabolic partial differential equation,
John (1952)
derived requirements that the space and time
net spacings and the
oefficient of the highest order
derivative must satisfy it the solution of the differene
approximation
is to be stable and convergent to the solu-
tion of the differential equation.
In the difference net,
these requirements take the forn
<
z<
In addition to this requireant, the oonservation of
energy equationin the model requires an additional constraint.
This iss due to the term giving the rate of
conversion of the gravitational potential energy into heat
energy.
Since in finite dAfference schemes, the value of
a function over an interval in
mated by its
value at a point.
8p4oe and time is
approxi-
There are selections of
the size of the time and space increments that satisfy
for which the estimation
the requirement in equation 0.3
of the concentration of the acidic phase bears little
no relation to its
or
mean value over the inIteaval selected.
This leads to dif ficulty in
the potential oner
conversion
term which involves a product of the velocity and the conThe volume selected by
centration of the acidic phase.
the sLace interval may be swept clean of the acidic
phase in a very small fraction of the time interval selected
by equation C.1,
The conversion of potential energy tera
uses the non-zero concentration for the whole time interval.
The amount of energy that is apparently converted to heat
energy by viscous dissipation can be several orders of
the potential
magnitude greater than the decrease in
energy,
In extreme cases, this can lead to apparent tempera-
order 10 5 OK.
ture rises of of
To insure that the amount of energy oonverted given
by the approximation in
equation 0.1 does not exceed the
amount of energy available,
it
is
necessary to apply a
condition which permits only one space interval to be
swept clean in a time interval.
The required condition is
For eaoh iteration, the velocity at all points of the net
is caloulated and the spaoe increment is
velocities.
The smallest quotient is
divided by these
selected.
This is
_
______I~
_L _ ^
the longest time increment that oan be used if, at most,
one apae interval is
to be swept olean in
This value of the time increment is
selected by the requixzment in
a time interval.
compared to the value
0.1, and the smaller of
these two values is used as the length of the time incre-
ment
for that iteration.
This makes the time increment
a variable which must be calculated at each step, but it
insures that both equations ;:.1 and 0.2 are satisfied
while using the longest possible time interval consistent
with the two conditions.
The above is
the condition 0.2.
the orioinal thinking which lead to
Oalculations made using the condition
0.2 indicates that the conversion of potential energy term
is negligible in the energy balance and on these grounds
could be omitted from the equation.
However, the conser-
vation of mass equation also requires a condition similar
to equation 0.2 for it involves the product NU and since
the conversion of potential energy term is very sensitive
to the violation of the constraint, it is retained in the
calculations.
An observation based on a study of the literature
conoerning finite difference approximations to linear and
quasi-linear equations is that the higher the order of the
differences used in an approximation, the smaller is the
range of the variables for which the solution given by the
approximation represents the solution of the differontial
_~
___1~X l~^~~llil
L~_~~
111_
equation.
It
is
work of this type to use
preferable in
as low order differences as possible to approximate the
differential operators and a fine mesh spaoing in
order
to cover the widest possible range of variables.
A check on the accuracy of the approximation was
to caloulate the solution for two mesh spacings and conpare
the solutions,
these runs.
Table C.
jives the solution for
The parameters used were the same except for
the number of mesh points and the length of a space
increment.
Column 1 is
Case 1 described in
The difference between the two runs is
Table 3.1.
insignificant.
The physical arguments that were applied to check
for blanders in the formulation of the problem or the
approximations used are (1) the velocity of the acidic
phase must always be toward the surface,
tare cannot be negative when
and (3)
(2)
the tempera-
ta4sured on the Kelvin scale,
the possible range of the temperature variations
can be estimated.
The subscript / refers to the spaoial location and
the superscript a refers to the temporal location in
difference net.
The space increment is
the
selected by
h =H/L
where R is the length of the space and L is the number of
97
tABLZ 0.1
__ __"_"A
" _",hl"'
"
i
I_
1".
<
/
;
"''"
L
_?
1MERATUMI AU2
,-0
.......
::: "
"
.
,
.,'''_
'
f.
RlATTV
,
:
'
: -010111
..
.
"
^::
. .
.
XWtiTE.
: m_ 1110
0. im
O'"_
ggOR,
.
.
.
.
.
01
. .
- 110-4,
. .
. L : .
.
..
.
..
UB*S
........
141
m
W..
.
. .
for spaae izur4mln't and
number of space not
are tsed.
ratis
Sm 15 kam, L = 20)
N( )/(o0)
N(*t)/N(O)
P300
300m~s
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
255
270
285
300
s0r~i~a
699
1000
1.000
1056
1371
1.009
1.026
1873
1.044
300
1.000
505
1.00
699
883
1056
1219
1371
1513
1644
1765
1874
1.000
1.000
1.002
1.012
1.021
1.031
1,041
1.048
1973
2060
2206
1.51
1.041
2309
2371
2392
B($)~(Qf
.
see parafaers exAcet
ROemit for first run of
~a
1 desribed in
Table 3.1. (Resulnts are
a
~iven in Table 3.2.)
L = 1Q)
SOk
3
(b
Depyt
" ""...
.979
.831
~ ReIatIve
1.052
2062
2140
2207
2264
2310
2346
2372
2387
2393
mber of Tabes/aem.
1.051
1.046
1.037
1.025
1.011
*.991
.935
.690
2
IIII~I~
I~- L~
t~l)
The time increment I is selected as indicated
increments.
above.
The approximations to the differential operators
used are
2C-
where f
in
is the value of the function at the point (m, Z)
the net.
The velocity equation in
/
where
-
/I
U6
when the form
used, Tm is
0
this approximation is
-
a
(-
=t e
for the viscosity is
replaced in equation 0.3 by T~ - C and the
calculations proceed as before unless t the velocity, U
=
0, is
A 0, then
used.
The conservation of energy equation becomes
'(T
-
),-T
slE-
ll~ ll^_
l~
and the conservation of mass equation is
N - N;+,'
N
"
,.
The boundary oonditions on the energy equation are at
0
a0
O
and to express the aondition of no heat flow at Z = H,
it is neoessry to extend the domain of definition of
the temperature to include the point N + 1, and define
The initial
temperature distribution is
The boundary condition on the ooneervation of mos equa-
tion at Z = 0 is
The use of the equations are slapler if the oon4ition of
no mass transport &cross Z = H is
satisfied by extendtg
the domain of definition as in the temperature case and
defining
N+mL + 1 :0
which insures that no mass is transported across the lower
surface.
Equations C.3,
0.4, and 0.5 are the equations
programed and run on the computer.
_IIX__
BIOGRAPHIAL NOTE
J. Downs was born and attended school in
Savsanh, Georgia.
He graduated from the Georgia
Institute of teohnology with highest honor in 1956,
receiving a B,
S. in Phylrsis
Institute of Teohnologyj
At the Masaehusetts
ho had a two year tuition
scholarship (1956 and 1957).
He was a National Solence
Foundation Fellow for two years (1958 and 1959).
-r~ ~-~-~--- ------- --- -L-r~L~
_~ . ~II_
^-I
II_.
II-~l-X~IJ/
BIBLIOGRAPHY
Barth, T.F.W.,
1952, Theoretical Petrology, John Wiley
and Sons, ino., 369 pp.
Birch, F., Sohairer, J.F, and Spioer, H.0.,,eds. 1942
Handbook of baysioal Constants, Geologioal
Society of Americal Spooial Papers, Number 36,
325 pp.
Birch,
f,, 1952, 2lastioity and constitution of the
Earth's Interior, Journal of Geophysical
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2,
2 p.
227.
Borelius, G., 1953, On the Solid-Liquid transformation
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Brockris J. O'M., Lowe, DG., 1954, Viscosity and ths
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*gh
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Ballard, L. . , 1954, The Interior of the Earth,
Kuiper, G., ed, The Barth as a Planet, Chicago
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P'F
Jr., and Birch, F., 1938, The Effect Of
Dane, iE.,
Eressre on the Visaosity of Boric Anhydride,
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Dienes, G. J., 1953, Activation Energy for Visous
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Bulletin Seiaaological Sooiety of America,
4f,
p. 105.
-----rrr-----r~-._ ----~--
l- I-r~-~---l- --- -~-----~..-----
I 'PI-C--~ _I. 1IYI
Rales, A. L., 1953, The Thermal Contraction Theory of
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Hughes, H., 1955, the Presuare on the Eleotrioal Ownduotivity of Peridot. Journal of Geophysical
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MaDonald G. J. if., 1956, Quartz-Goesite Stability
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Makensie, J. D., 1957, Discrete ion Theory and Visoous
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Minakami, Takeshi, 1951, On the Temperature and Visaosity
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t
, Part
Institute Balletin,
Orowan,
., 1960, Mechanism of Seiomi
Grigg,
Do and Handia J.,
d*s.,
FaPulting, p.323,
reek Deffo
Geological Sooiety of America, Mfeair
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j,
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of
IC-'1lllr i
^II -I I - ------ ..I--~ ^1IIIIIYII"'LI~-I --~-~-r--*r~--~I-- ----
1
Stewart0,D.,
1960,
ersonal 0
.ation.
Trettel, ., 1958, on the Dissipation of Seimi Energy
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Wlson, J.
T., 1954, The Devml
rast,
ohicag
UnieraG.,I
t
a
Ar,*..
siversity Press,
pp.
Structure of the
"