Convection in Earth’s Mantle · 2016. 12. 1. · Convection in Earth’s Mantle: impacts of solid-solid phase transformations and crystalline rheology Doctor of Philosophy, 2001,

Convection in Earth’s Mantle:impacts of solid-solid phase transformations

and crystalline rheology

by

Samuel Leonard Butler

A thesis su b m itte d in conform ity w ith th e requ irem en ts for th e D egree o f D octor of Philosophy

G ra d u a te D epartm en t of Physics U niversity o f Toronto

© C opyrigh t by Sam uel Leonard B u tle r 2001

R e p ro d u c e d with perm iss ion of th e copyright ow ner. F u r th e r reproduction prohibited without perm iss ion .

Convection in Earth’s Mantle: impacts of solid-solid phase transformations

and crystalline rheology

D octor of Philosophy, 2001, Samuel Leonard Butler, Departm ent of Physics, University of

Toronto

Abstract

I first present a detailed analysis of the linear stability of an in ternal therm al boundary

layer. In the absence of effects due to convective motion, this boundary layer is shown to be

extrem ely unstable. W hen effects due to the background flow are param eterized through a

Peclet number and an endotherm ic phase transition is present, it is shown th a t the boundary

layer is stabilized to the extent seen in numerical simulations of m antle convection.

A series of detailed numerical calculations are presented in which I perform a broad

survey of the effects of internal heating, depth-dependent viscosity, and differing Clapeyron

slopes of the endothermic phase transition . The size distribution of mass flux events tha t

cross the 660-km depth horizon is exam ined. It is determined th a t as the degree of layering

increases, the number of small mass flux events increases while the num ber of large events

decreases. I also determine th a t when the core-mantle boundary tem peratu re is set such

as to be in accord w ith high pressure experiments, the calculated surface heat flow is sig

nificantly greater than tha t of the real E arth . Earth-like surface heat flows were calculated

only when convection was very strongly layered or when the mean viscosity was significantly

greater th an the viscosity inferred on the basis of post-glacial rebound.

I also derive a simple param eterized m odel of convection and dem onstrate th a t its pre

dictions are very close to those o f the full dynamical model. This param eterized model is

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further used to calculate possible E a rth therm al histories. It is determ ined th a t for models

which have a constant degree of layering, the E a rth ’s initial core tem perature m ust be im

probably high in order to m atch the constraint of the observed surface heat flow today. If

the degree of layering is allowed to vary w ith the system Rayleigh number, a novel mecha

nism occurs which buffers the upper mantle tem perature allowing for long periods of time

w ith constant, Earth-like, surface heat flow w ith reasonable initial core tem peratures. All

acceptable models also require high viscosities which argues tha t the viscosity of the mantle

th a t controls convection may be greater th an th a t for post-glacial rebound.


ACKNOWLEDGEMENTS

Thanks are required for a number of people and organizations who made my time spent at

the University of Toronto both productive and enjoyable. I would first like to thank my

supervisor, W .R. Peltier, for his guidance, enthusiasm, and generosity. I would further like

to th a n k Giovanni Pari, Guido V ettoretti, Rosemarie Drummond, and Larry Solheim who,

among others, provided valuable technical and scientific advice. Thanks also to my extended

and immediate family for their support and nourishment and to various ultim ate frisbee

teams and the U. of T. nordic ski team for providing much needed diversions. I would also

like to acknowledge financial support from the N atural Sciences and Engineering Reseaxch

Council, the D epartm ent of Physics a t the University of Toronto, the Ontario G raduate

Scholarships in Science and Technology, and the W alter Sumner Foundation. Finally, I

must thank my beloved Erica.

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Contents

1 Introduction 1

2 Internal thermal boundary layer stability in phase transition modulated

mantle mixing 13

2.1 In tro d u c tio n ...................................................................................................................... 13

2.2 Theoretical F o rm u la tio n .............................................................................................. 16

2.3 Boundary C o n d itio n s .................................................................................................... 20

2.3.1 Rayleigh-Taylor Instability ............................................................................ 21

2.3.2 P hase Transitions ............................................................................................. 22

2.4 Numerical M e th o d o lo g y .............................................................................................. 23

2.5 R esu lts ................................................................................................................................ 26

2.5.1 T herm al Boundary Layer S ta b i l i ty .............................................................. 27

2.5.2 A pplication to Convection in the E arth ’s Mantle: In itial Estim ates . 34

2.5.3 Rayleigh-Taylor Analysis W ith a Phase T r a n s i t io n ................................. 38

2.5.4 Process of Dynamical S ta b iliz a tio n .............................................................. 41

2.6 Discussion an d C o n c lu s io n s ....................................................................................... 48

3 On scaling relations in time-dependent mantle convection and the heat

transfer constraint on layering 52

3.1 Model D e s c r ip t io n ......................................................................................................... 57

3.1.1 M odel P a r a m e te r s ............................................................................................ 57

3.1.2 D escription of the Numerical M o d e l ........................................................... 58

3.2 Results 1: S ta tis tica l Analyses o f the Avalanche Effect and the SOC Scaling

A n a lo g y ............................................................................................................................. 67

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CONTENTS________________________________________________________________________________jc

3.3 Results 2: Param eterized Model for “W hole M antle” and ’Layered M antle’

C o n v e c t io n ......................................................................................................... 76

3.3.1 Model Derivation and Implications for the Timescale of Convective

A d ju s tm e n t............................................................................................ 78

3.3.2 Interior T e m p e ra tu re s ...................................................................................... 82

3.3.3 Heat F lo w ............................................................................................................. 85

3.3.4 Effects of L a y e r in g ............................................................................................ 89

3.4 Results 3: Scaling of Time-Averaged Q uantities in Tim e-Dependent Convection 91

3.4.1 Scaling o f Surface Heat F l o w .......................................................................... 93

3.4.2 Scaling of Surface V elocity................................................................................ 97

3.4.3 Scaling of the 660-km Mass F l u x ................................................................... 99

3.4.4 Scaling of the Surface Boundary Layer T h ick n e ss ................................... 102

3.4.5 Relationships Between Scaling E x p o n e n ts ................................................. 102

3.5 D iscussion .............................................................................................................. 107

3.5.1 Tem perature Constraint a t 660 km D epth . . . '....................................... 107

3.5.2 Quantifying Degrees of L a y e r in g .................................................................. 108

3.6 C o n c lu sio n s .......................................................................................................... 116

4 The Thermal Evolution of the Earth: Models with Time Dependent Lay

ering of Mantle Convection which Satisfy the Urey Ratio Constraint 121

4.1 In tro d u c tio n .......................................................................................................... 121

4.2 Model D e riv a tio n ............................................................................................................ 127

4.3 Choice of P a ra m e te rs ..................................................................................................... 134

4.4 R esu lts .................................................................................................................... 141

4.4.1 One Layer M o d e l s ............................................................................................ 141

4.4.2 W hole M antle C on v ec tio n ............................................................................... 148

4.4.3 Strongly Layered C o n v e c tio n ........................................................................ 150

4.4.4 Interm ediate m o d e ls ........................................................................................ 153

4.4.5 Rayleigh-number dependent la y e r in g .......................................................... 153

4.5 Discussion and C o n c lu s io n s ............................................................................ 159


C O N T E N T S __________________________________________________________________________________________________3d

5 Conclusions 164

5.1 Detailed Description of Rayleigh Num ber Dependent Layering Therm al His

tories ................................................................................................................................... 165

5.2 Future C o n s id e ra tio n s ................................................................................................. 166

6 References 171

R e p ro d u c e d with p erm iss ion of th e copyright ow ner. F u r the r reproduction prohibited without perm iss ion .

List of Tables

2.1 Representative Values of the Parameters of the 660-km Phase Transition . . 35

2.2 Effects Included in the Analysis of Convective Instability for Earth-Like Ge

ometry and Com parison w ith the Numerical Experim ents of Solheim and

Peltier [1994a, b ] ............................................................................................................ 51

3.1 Summary of S im u la tio n s ............................................................................................... 65

3.2 Comparison of Heat Flow From Numerical and Param eterized Models . . . 88

3.3 Summary of Scaling P a ram ete rs ................................................................................. 94

3.4 Relations Between Scaling P a ra m e te r s .................................................................... 106

4.1 Param eter Values a t Boundary D e p th s .................................................................... 135

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List of Figures

1.1 Temperature contour plots of mantle convection, ju s t before, and during, a

mantle “avalanche” .......................................................................................................... 4

1.2 Post-glacial rebound constrained viscosity inference of radial viscosity of

Peltier and Jiang [1996] and seismic tomography and geoid constrained radial

viscosity inference of Forte et al. [1993].................................................................... 9

2.1 Time series of m ass flux and boundary layer param aters from detailed nu

merical modeling............................................................................................................... 15

2.2 Temperature profiles for which the linear stability is in v e s t ig a te d ................ 18

2.3 Neutral curves for various layer depths w ith the boundary layer in the center

(a) and adjacent to an outer boundary (b)............................................................... 27

2.4 Minimal Rayleigh num bers and critical wave-numbers for various layer thick

nesses (a) and various positions of the boundary layer (b )................................. 29

2.5 Minimal Rayleigh num bers for various layer thicknesses and using different

forms of the tem peratu re gradient.............................................................................. 30

2.6 Temperature and velocity eigenfunctions for delta-function gradients (a) and

the Benard problem (b)................................................................................................. 32

2.7 Neutral curves (a) and eigenfunctions (b) for various velocity convergences . 36

2.8 Growth rates (a) and velocity eigenfunctions (b) for Rayleigh-Taylor insta

bilities w ith various velocity convergence intensities............................................. 39

2.9 Schematic illu stra ting the process of dynamical stab ilization ........................... 43

2.10 Neutral curves (a) and eigenfunctions (b) for various velocity convergences

for which phase-transitions are ac tiv e ....................................................................... 45

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T.TST O F FTOURES________________________________________________________________________ ix

2.11 N eutral curves when velocity convergence is active for various Clapeyron

slopes of the phase tran sitio n in (a) cartesian and (b) spherical geometry . . 46

3.1 (a)Depth-dependent viscosity profiles (VPs) used in this study, (b) The radial

variation of the non-dim ensional param eters used in the m odel........................ 59

3.2 (a) Two choices for the function X (r ) th a t defines the rad ia l resolution.(b)

The corresponding resolution for the same two cases as a function of radius. 64

3.3 (a) Color contour plots o f the tem perature field in the m odel m antle demon

stra ting mantle avalanches............................................................................................. 68

3.4 (a)400 and 660 km dep th m ass flux tim e series.(b) The mass flux as a function

of the polar angle a t the position of the 660-km dep th ......................................... 69

3.5 Mass flux distributions for various Clapeyron slopes of the endotherm ic phase

transition ............................................................................................................................. 71

3.6 Mass flux distributions for various mean viscosities and (b) in ternal heating

rates (b)............................................................................................................................... 74

3.7 Mass flux distributions for very high Rayleigh number calculations................. 75

3.8 Analogy between the phase-transition-induced m antle avalanche process and

sandpile avalanche m odels............................................................................................. 77

3.9 Tim e series comparing th e average internal tem perature from the numerical

model with predictions o f the param eterized model.............................................. 81

3.10 Comparison of geotherm s from the parameterized and num erical models. . . 84

3.11 Heat flow as a function of the dep th variation of viscosity (a), and heat flow

tim e series comparison betw een the numerical and param eterized models (b). 87

3.12 Comparison of layered geotherm s from the numerical and parameterized

models.................................................................................................................................. 92

3.13 Heat flow as a function o f the m ean viscosity from our num erical model. . . 95

3.14 Surface velocity as a function of m ean viscosity from the num erical model. . 98

3.15 660-km depth mass flux as a function of the mean viscosity from the numerical

m odel................................................................................................................................... 100

3.16 Surface boundary layer thickness as a function of the m ean viscosity from

our numerical model........................................................................................................ 103


LIST OF FIGURES________________________________________________________________________ x

3.17 Geotherms calculated for various m ean viscosities in the presence of (a) in

ternal heating, and (b) a very large Clapeyron slope............................................ 104

3.18 Geotherms from the numerical model and the bound on the tem perature at

660-km dep th .................................................................................................................... 109

3.19 Nondimensional numbers characterizing layering and their dependence on

the Rayleigh number and Clapeyron slope.............................................................. 110

3.20 Correlation between the non-dimensional numbers quantifying convective

layering............................................................................................................................... 112

3.21 Correlation between heat flow w ith and without layering (a), and the cor

relation of the exchange tim e between the upper and lower m antle and the

mixing tim e of the upper m antle (b)......................................................................... 115

4.1 Tem perature contour plots of (a) strongly layered and (b) whole m antle con

vection................................................................................................................................. 125

4.2 Geotherms for layered and ’whole-mantle’ convection........................................... 126

4.3 /3 as a function of Rayleigh num ber for various Clapeyron slopes..................... 133

4.4 Radio-active heating in the m antle as a function of tim e..................................... 140

4.5 The variation of internal tem peratures as function of time for various strengths

of the tem perature-dependence of m antle v isco s ity ............................................. 143

4.6 Tim e evolution of mantle tem perature in a model w ith no layering and in

which the core simply follows the tem perature in the m antle............................ 144

4.7 Tem perature (a) and surface heat flow (b) in a model w ith no layering and

in which the tem perature of the core simply follows the tem perature in the

m antle................................................................................................................................. 146

4.8 Therm al history for ’whole-mantle’ convection with a PG R constrained lower-

m antle viscosity. ........................................................................................................... 147

4.9 Therm al history for ’whole-mantle’ convection with an inefficient lower bound

ary layer............................................................................................................................. 149

4.10 Therm al history calculation w ith layered mantle param eters.............................. 151

4.11 Layered therm al history with an inefficient boundary layer at 660-km depth. 152


T.TST O F FTGTTRES JC1

4.12 Layered, therm al history calculation w ith an inefficient CMB therm al bound

ary layer.............................................................................................................................. 154

4.13 Partially layered therm al history calculation w ith boundary layers of inter

mediate efficiency. ......................................................................................................... 155

4.14 Therm al history calculation in which the degree is Rayleigh num ber depen

dent, 7 = 8.5...................................................................................................................... 156

4.15 Therm al history calculation w ith constant layering for com parison withFig-

ure 4 . 1 4 ............................................................................................................................. 158

4.16 Therm al history calculation w ith Rayleigh num ber dependent layering and

7 = 7.5................................................................................................................................. 160

5.1 Geotherms from a param eterized convection calculation w ith variable layering 167

5.2 Geotherms from a param eterized convection calculation w ith constant layeringl6 8


Chapter 1

Introduction

The E a rth ’s m antle comprises the 2890 km thick region between the core and crust. Com

posed of silicate material, the m antle is known to be solid as it allows the transm ission

of seismic shear waves. Fluid-like motion ensues, however, when lateral density hetero

geneities, caused by therm al and perhaps chemical variations, result in buoyancy forces

which drive convective motions which are supported by the m igration of crystal defects

w ithin the polycrystalline mantle m aterial. In w hat follows, I will first provide a brief

overview of the im portance of the m antle convection process and its effects on the history

and future of the E arth . I will then describe th e composition of the mantle and its effective

viscosity together w ith the phase transitions th a t occur w ithin it as these are the prim ary

control variables of convection th a t will be investigated in the main body of this thesis. I

will then give an overview of the m ain results of convection theory and discuss the way in

which these apply to therm al convection in the E a rth ’s mantle. The specific elements of

the work to be reported in what follows will th en be introduced.

Perhaps the most directly observable consequence of convection in the E a rth ’s mantle

is the relative motion of different points on E a rth ’s surface. The E arth ’s surface is divided

into tectonic plates which are quasi-rigid and translate relative to one another a t an average

speed of 4 cm /year (Demets et al. 1994). Oceanic plates are also constantly created and

destroyed as new plate m aterial rises from the m antle a t mid-ocean ridges and old oceanic

plates descend into the m antle a t deep ocean trenches. T he theory of plate tectonics, first

cham pioned by Wegener [1929], Holmes [1931] and others, was eventually found to explain

a num ber o f previously unexplained phenom ena including the pa ttern of m agnetic stripes

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2

on the ocean floor ( Vine and Mathews, 1963), the relatively young age of all ocean floor

materials, and the apparent relative m otion of continents inferred from paleo-magnetic mea

surements (Blackett et al., 1960). Geological and biological observations, and the apparent

“fit” of the outline of the continents (when one includes the continental shelves) (Bullard

et al., 1966) also m ade a strong argum ent th a t the continents must at one time have all

been connected so as to form a single supercontinent. Today, relative plate motions can

be measured directly using the global positioning system (GPS) and very-long baseline

interferometry (VLBI) methods.

Moving surface plates also have some very direct consequences for humanity. At plate

boundaries where there is relative motion between two plates, plates can remain locked for

significant periods of time, resulting in significant stra in in the plate regions close to the

boundary. W hen local elastic forces exceed the friction th a t is available to bind the plates

together, slippage can occur. Sometimes local slippage can trigger further sliding along a

significant length of the plate boundary resulting in a dram atic instability known as an

“earthquake” . T he num ber distribution of earthquakes as a function of their m agnitude

exhibits a power-law form over a large range of m agnitudes (e.g., Main, 1996) and it has

been argued th a t earthquakes are am example of a self-organized critical phenomenon (Bah

et al, 1988). The theory of plate tectonics has provided a very useful framework for studying

the motions and deformations tha t occur on the E a rth ’s surface. The mechanism responsible

for driving surface plates and creating their particu lar pa ttern on the surface, however, is

to be found in convection w ithin the E arth ’s mantle.

As hot, upwelling, mantle m aterial reaches the E a rth ’s surface, it is deflected horizon

tally. W hereas therm al energy in the interior of a convection cell is transported mostly

by advection, a t the upper boundary it must be transported entirely by conduction. As a

result, there exists a very steep therm al gradient near the E arth ’s surface which is known

as a therm al boundary layer. Since the tem peratures in the boundary layer are significantly

colder th an in the interior of the mantle, and the viscosity of the mantle is therm ally acti

vated, the viscosity characteristic of the surface is many orders of magnitude greater than

tha t of the interior. This therm al boundary layer, high viscosity, region of the m antle is

known as the lithosphere and comprises roughly the upper 100 km of the mantle. This

relatively th in rigid “skin” which forms the “roof” of individual convection cells is the stuff


A

of which tectonic plates are comprised. Embedded w ithin the lithosphere, and having an

average thickness of 6 km in oceanic regions and 40 km in continental regions, is th e E a rth ’s

surficial crust. This crust is chemically distinct from the mantle in terms of composition.

The process by which the materials tha t make up continental and oceanic crust were ex

tracted from the E a rth ’s interior is referred to as differentiation and it too is a consequence

of convection in the E a rth ’s mantle, operating in conjunction with near surface m elting, as

will be described in w hat follows.

The current rate of crust formation is approximately 20 km ?/yr and is supported through

the action of volcanism. Volcanoes occur when partia l melting of mantle m aterials takes

place and the buoyancy of the resulting liquid is sufficient to carry it to the E a rth ’s surface.

Elements whose valencies and ionic radii are significantly different from those o f M g and

F e are preferentially melted, and are added to the crust. Volcanoes a t mid-ocean ridges

occur when a parcel of hot material from deep in the m antle is carried upward by the action

of convection; the resulting decrease in pressure experienced by the parcel can be sufficient

to allow for partial m elting and the melted m aterial forms the oceanic crust. W hen oceanic

plates descend back into the mantle a t deep-ocean trenches, sea water is entrained which

has the effect of reducing the melting tem perature of mantle materials. As a result, when

water is released into m antle materials in the vicinity of the descending oceanic crust, fur

ther melting can occur resulting in back-arc volcanism. This material is then added to the

continental crust at a ra te of 1 km ? /yr (Hofmann, 1997). Some continental m aterial is also

likely carried by erosion to deep ocean trenches where it is resubducted and itself repro

cessed (Armstrong, 1984). This processing of m aterial results in continental crust which is

highly concentrated in many of the more exotic elements of the Earth. As an example, the

E arth ’s continental crust currently contains radioactive elements producing roughly 1/3 of

the E arth ’s to ta l heat production (Hart and Zindler, 1986) while the continental crust rep

resents only 0.6 % of the E a rth ’s volume. A further type of volcanism which is found in the

middle of plates is thought to be related to narrow, hot, mantle plumes tha t are responsible

for the creation of surface hot-spots. Although the back-arc volcanism described above is

the main process through which continental crust is produced today, it is likely th a t in the

geological past, large-scale melting of the upper m antle occurred, adding large quantities

of material to the continental crust (e.g, Stein and Hofmann , 1994). The geological record


Figure 1.1: Tem perature contour plots of mantle convection, ju s t before, and during, a m antle “avalanche” .

suggests th a t a num ber of such events have occurred (e.g., Gastil, 1960) and these correlate

well w ith times in which the continents were sutured together. The dynamical processes

w ithin the m antle th a t may be responsible for these events o f super-continent construction

are m antle ’avalanches’ associated with the action of m antle phase transitions on the con

vective flow (P eltier et al., 1997; Condie, 1998) and these will be the subject of much of

chapters 2 and 3. In Figure 1.1 we show contour plots of th e tem perature in our numerical

model ju s t before a n “avalanche” takes place when flow is very strongly layered and during

an avalanche event. D uring the avalanche, a very large cold down-welling at the equator

has burst through th e 660-km depth phase boundary which is indicated by the inner black

line.

T he energy th a t drives m antle convection derives from a combination of gravitational

energy of planetary form ation (2.199 x 1032 J ), the energy o f core accretion (1.394 x 1031 J)


R

the ongoing radioactive decay of K 40, T /i232, Z7235 and C/238 (7.8 x 1030 J ), as well as a

sm all contribution from the freezing of the inner core (7.289 x 1028«7). All estim ates axe

those of Stacey and Stacey, [1999]. Much of this energy, particularly th a t arising from

planetary form ation and accretion of the core, has been carried to the surface and rad iated

away although the am ount remains unknown. The current surface heat flow, m easured a t

drilling sites a t the E a rth ’s surface is estim ated a t 44 T W (Pollack et al, 1993). This heat

flow is not evenly d istributed w ith the average oceanic and continental heat fluxes being

100 m W /m 2 and 50 m W /m 2, respectively. The to ta l radioactive heat production today is

estim ated by geochemists to be approxim ately 19 T W (Zindler and Hart, 1986). Clearly,

the E arth ’s interior is cooling. However, the current tem peratures inside the m antle and

core depend on the efficiency w ith which convection has removed heat from the p lanet’s

interior in the past. Factors th a t affect this efficiency include the viscosity of the E a rth ’s

m antle and possible layering caused by phase transitions occurring w ithin the m antle. It

is the study of these effects tha t will form the central theme of this thesis. Determ ining

the nature of mantle convection, and hence its efficiency, will in tu rn shed light on such

issues as to how long we can expect convection and p late tectonics to continue to act, the

tim e dependence of convection and its surface expression, and the tim e at which the E a rth ’s

crust was formed.

Further consequences of convection in the mantle derive from its effects on the E a rth ’s

magnetic field. Convection in the E a rth ’s outer core, which is composed prim arily o f liquid

iron, sustains the E a rth ’s magnetic field. As motions in the solid m antle are roughly 6

orders of m agnitude slower than m otions in the outer core, the mantle controls the ra te of

heat loss from the core and sets the upper boundary condition for core convection (e.g.,

B uffett, 2000). It has fu rther been postulated tha t periods in which avalanches are taking

place, and descending cold m aterial from the upper m antle is vigorously cooling the core,

are responsible for periods in which relatively few reversals of the geomagnetic field occurred

{Sheridan, 1997).

The upper m antle is composed prim arily of Olivine whose chemical form is

(M g i- xFex )2 S i 0 4 which represents roughly 60 % o f the upper m antle by volume and

G arnet, M g \- xF exSiQz representing roughly 40 % {x = 0.1 for both minerals). As pressure

-and to a lesser extent tem perature- increase w ith depth, a number of phase transitions


&

occur in these m aterials. The phase diagram of these m aterials is quite complicated due

to the presence o f F e as well as small amounts of A l and Ca, and due to uncertainties

in upper m antle tem peratures ( Weidner and Wang, 2000) . Phase transformations can

be detected in the m antle as they represent strong reflectors of seismic waves due to the

associated ab ru p t density changes. A lthough a num ber o f seismic reflections are detected

{Shearer, 2000), the most significant from a m antle dynamics perspective axe those tha t

bracket the transition zone a t 660 and 400 km depths. Results from high pressure mineral

physics (e.g., Ringwood and Major 1966, 1970; Akimoto and Fujisawa 1966; Boehler, 2000)

indicate th a t a transition from the Olivine phase to a spinel phase occurs at pressures

corresponding to those a t 400 km depth in the mantle. Similarly, a transition from spinel

to a m ixture of magnesiowiistite and pervoskite occurs a t pressures corresponding to those

a t 660 km depth in the mantle. An estim ate of the mean tem perature a t 400 and 660 km

depths of 1850 and 1900 K can be obtained from the tem perature a t which each of these

transitions occur a t the pressures appropriate for these {Boehler, 2000). The Clapeyron

slope of each of these reactions has also been measured and found to be 2 M Pa/K and -3

M Pa/K , (Chopelas, 1994) respectively. The negative Clapeyron slope of this second reaction

results in it being of particular significance to m antle dynamics. In order for the phases to

remain in equilibrium , the 660 km depth phase boundary will always be deflected in the

direction in which a fluid parcel is traveling, w hether it be a hot up-welling or a cold down-

welling {Busse and Schubert, 1971; Schubert et al., 1975). Due to the large density difference

between the upper and lower phases, buoyancy forces result which impede the convective

flow and can result in layered convection. Deflections of the 400-km phase transition result

in an enhancem ent of the convective flow. There is a further effect due to latent heating,

which causes therm al expansion or contraction of the fluid parcel, resulting in buoyancy.

These effects act in the opposite sense to those due to phase boundary deflection. They

have been shown to be less significant, however, when convection is as vigorous as it is in

the mantle {Solheim and Peltier, 1994a).

The debate as to the degree to which m antle convection is layered has been long

standing. Early researchers advocated either completely “whole-mantle” convection or

completely layered convection. Recently, high-resolution seismic tomographic images (e.g.,

van der Hilst, 1997) have indicated tha t cold p late m aterial descending from deep-ocean


z

trenches appears to penetrate directly into the lower m antle in some places and is deflected

a t 660-km depth a t others. Numerical modeling of convection w ith Earth-like param eters

(e.g., Christensen and Yuen, 1985; Machetel and Weber, 1992; Solheim and Peltier, 1992;

Tackley et al., 1993; Solheim and Peltier, 1994a,b; Butler and Peltier, 2000) indicates th a t

convective flow becomes highly tim e dependent. Periods exist w ith relatively little mass

flux across this boundary, interspersed w ith short periods of rap id mantle mixing. D uring

periods of mantle layering, a significant therm al boundary layer exists at a depth of 660 km

in the mantle. The instability of this boundary layer results in the occurence of vigorous

mantle mixing events which are associated with the development of avalanches of cold ma

terial from the transition zone into the lower mantle. The detailed study of this instability

is the main topic of chapter 2. Geochemical analyses of m aterials from mid-ocean ridges,

ocean island volcanoes, and estim ates of bulk E arth concentrations of various elements and

detailed box models indicate th a t there must be some mixing between the lower and up

per mantle (e.g., Hofmann , 1997). Hence, it is no longer a m atte r of determining w hether

mantle convection is ’whole-mantle’ or layered bu t ra ther of determining the extent of con

vective layering and its tim e dependence. Geochemical analyses, particularly those based on

noble gas systematics (e.g., O ’nions and Tolstikhin, 1996), indicate tha t the mantle should

be strongly layered, while seismological observations indicate more modest layering. The

resolution of this apparent impasse is a very im portant issue in solid-Earth geophysics and

is one in which the results obtained on the basis of detailed numerical modeling will play a

significant role.

The viscosity th a t is characteristic of the E a rth ’s m antle is a primary control variable

of the convection process. T he flow of solid m aterials takes place due to the migration of

crystal defects under the influence of applied stresses. If the defects are vacancies, then flow

results when vacancies diffuse towards regions of high stress. This type of flow is known as

diffusion creep. A further type of flow can occur in which a line defect may move through a

crystal under an applied stress. This process is known as dislocation creep (Poirier, 1991).

The microscopic creep m echanism is of macroscopic im portance as it determines the stress-

stra in rate relation for a m aterial. In materials in which diffusion creep is taking place,

stra in rate is linearly proportional to stress. In m aterials which are undergoing dislocation

creep, strain rate is generally proportional to stress raised to a power greater than 1 (3


&

is common). Fluids of the first type axe referred to as N ew tonian. One cannot uniquely

define a viscosity for a non-Newtonian fluid as the apparent viscosity changes w ith the strain

ra te . Phenom ena which take place on long time-scales, and hence have low strain rates,

such as m antle convection, will have high characteristic viscosities if the mantle is a non-

Newtonian fluid. Phenom ena th a t occur on shorter tim e scales, such as post-glacial rebound

(PG R ), will have a sm aller apparen t viscosity. Arguments as to w hether the viscosity th a t

chaxacterizes P G R is the same as th a t which is characteristic o f convection axe im portant as

they will determ ine w hether m antle viscosity is Newtonian or non-Newtonian. This will in

tu rn determ ine the m agnitude of the viscosity th a t is characteristic of the mantle convection

process which governs the dynam ical state of the mantle.

The determ ination o f the viscosity of mantle m aterials in laboratory experiments is

difficult since the high pressures and low strain rates associated w ith m antle convection are

difficult to reproduce {Karato, 1993). There is evidence from seismology and from rocks at

mid-ocean ridges th a t the viscosity of the upper 200 km of the m antle is non-Newtonian

(Kendal, 2000; Karato, 1993). D islocation creep results in m aterials w ith significant seismic

anisotropy. Seismic observations indicate that near surface shear waves travel faster when

their polarization is parallel to the direction of the tectonic p la te which is the direction

in which the mean m antle flow has deformed the crystals th a t make up mantle material.

Seismic anisotropy is also seen in the region just above the core-m antle boundary (CMB),

and there is some evidence for anisotropy in the vicinity of 660-km depth (e.g., Vinnik et

al, 1998). Most of the m antle is observed to be seismically isotropic, however. This may

be caused by unresolvable, short wavelength, flow geometries in these regions rather than

non-Newtonian viscosity, however.

One set of observations th a t axe used to infer the viscosity an d its radial variation in the

m antle comes from the P G R process already alluded to. Since th e end of the last ice age

10 kyrs ago, continental regions th a t were previously glaciated have been bouncing back to

their unloaded equilibrium position. The rate at which this process has been taking place

is a function of the viscosity of the m antle and can be inferred from the current altitude of

ancient beaches as well as by direct measurements using G PS an d VLBI techniques (e.g.,

Peltier, 1998). The m ean viscosity inferred on this basis is close to 1021 P as. An example

of an inference of the radial dependence of mantle viscosity which best accounts for the


5.

2 2 .5

COia 0.

22.0

2 1 .5

cnOom 21.0

00O^ 2 0 .5

20.0

Dark — P e lt ie r & J ian g (I9 9 6 a .b ) P e lt ie r (1 9 9 6 )

Light — F o rte e t a l. (1 9 9 3 )

R adius (km)

Figure 1.2: Post-glacial rebound constrained viscosity inference of rad ia l viscosity of Peltier and Jiang [1996] and seismic tom ography and geoid constrained radial viscosity inference o f Forte et al. [1993].

rebound rates (Peltier and Jiang, 1996) is shown in Figure 1.2 (dark line). An additional

m ethod of inferring the dep th variation of viscosity th a t is characteristic of the mantle

convection process was first devised by Hager [1984]. Seismic tomographic work (e.g., Su

et al, 1994; L i and Romanowicz, 1996), provides a low resolution three-dimensional map

of seismic velocity in the mantle. If a particu lar scaling from seismic velocity to density

is assum ed, the long wavelength com ponent of convective velocities can be inferred for

a particu lar radial viscosity profile. Knowing the convective velocity and density fields,

the geoid (surface of constant gravitational potential a t sea-level) can be calculated and

com pared w ith the m easured geoid for the E arth . The radial variation of viscosity can

then be varied until an optim al fit o f the calculated and m easured geoids is achieved. The

geoid is sensitive only to the radial variation of the viscosity and not its absolute value,


IQ

however. An example of such an inversion from Forte et al. [1993] is also shown in Figure

1.2 (light line) and it can be seen th a t when appropriately scaled, this profile can be made

very similar to the PG R inference. This argues tha t the viscosity of the two processes may

be the same and hence, mantle viscosity may be Newtonian. In chapters 3 and 4, we will

examine whether these radial viscosity profiles allow for appropriate heat flow a t the E arth ’s

surface in a priori models of m antle convection and parameterized therm al history models.

Careful experiments observing the onset of therm al convection in a fluid layer were first

performed by Benard [1900]. Theoretical analysis of this problem and the determ ination

of a minimal criterion for convective instability were first carried out by Lord Rayleigh

[1916]. The dimensionless num ber tha t is used to characterize the strength of convective

forcing is hence known as the Rayleigh number. The critical Rayleigh number at which

instability sets in is a function of the geometry and boundary conditions of the fluid volume

but is generally of order 1000 (e.g., Chandrasekhar, 1961). Based on estim ates of m aterial

properties in the mantle, the Rayleigh number for the E arth is of order 107. Hence, the

E arth ’s mantle is well into the unstable regime. The wavelength of the initial instability can

be predicted from linear stability analysis. If the Rayleigh number of convection is further

increased, there are changes in the planform of convection, rolls and hexagonal patterns axe

common, and there exist param eters for which multiple steady solutions exist(Busse 1975).

Generally shorter wavelength solutions occur as the Rayleigh num ber is increased. The

velocity and tem perature fields, however, rem ain time-independent until a second critical

Rayleigh number is reached and tim e dependence sets in (Krishnam urti, 1970; Solheim

and Peltier, 1990). As well as tim e-dependent flow, high Rayleigh number convection is

characterized by radial tem perature profiles whose azim uthal average is adiabatic in the

interior and has steep tem perature gradients a t the top and bottom surfaces (e.g., Jarvis

and Peltier, 1982). These are the therm al boundary layers tha t were alluded to previously.

The time-dependence of the convective flow sets in prim arily due to instabilities in the

thermal boundary layers (Howard, 1966). W hen convection is layered a t the depth of 660

km, an internal therm al boundary layer occurs a t this depth as well. It is the stability of

this therm al boundary layer th a t is the subject of the detailed analysis in chapter 2. As the

Rayleigh number is increased, the boundary layer region becomes increasingly thin. The

scaling of the boundary layer thickness, as well as the heat flow and convective velocity with


I I

the Rayleigh number, can be predicted by boundary layer theory. Boundary layer theory

was first applied to the problem of therm al convection in the E a r th ’s m antle by Turcotte

and Oxburgh, [1967]. The results of boundary layer theory, particu larly the scaling of heat

flow w ith Rayleigh num ber have been used to develop param eterized models of therm al

convection. These simple models have been used for some tim e in order to investigate the

therm al history of the E arth (e.g., Sharpe and Peltier, 1979). In chapter 3, I derive such a

model in order to compare its predictions with those of our num erical model in which we

solve finite difference approxim ations to the full Navier-Stokes equations. In chapter 4 I

further employ this model in order to do a new set of therm al h isto ry calculations in which

the effects of incomplete m antle layering are included and m an tle layering is a function of

the time-dependent system Rayleigh num ber as is observed in num erical simulations. W hen

this process is included, a novel buffering mechanism arises which allows for models in which

the surface heat flow has changed very little over geological history, despite the diminution

of the internal radioactive heat sources.

Chapters 2, 3 and 4 are largely self-contained documents. C hapters 2 and 3 have already

been published, m ostly in th e form in which they occur here (B utler and Peltier, 1997,

2000) while chapter 4 has been subm itted for publication. C h ap te r 2 contains a number of

calculations in which I investigate the stability of internal th e rm a l boundary layers. It is

determ ined that the background flow plays a crucial role in th is stability. In chapter 3, a

large number of numerical calculations of the mantle convection process are presented in

which the parameters are set such as to be as Earth-like as possible and control parameters

such as the degree of internal heating, the depth variation of viscosity, and the Clapeyron

slope of the endothermic phase transition a t 660-km depth are varied. It is determined tha t

an excess of surface heat flow occurs unless mantle viscosity is m ade significantly greater

than th a t which is characteristic of the PG R process or if m antle convection is very strongly

layered. I present the size d is tribu tion of mass flux events w hich cross the 660-km depth

horizon and I characterize how these change with the aform entioned control variables. I

also demonstrate the variation of the heat flow, surface velocity, mass flux, and therm al

boundary layer thickness as a function of the mean viscosity. A simple parameterized

model of convection is also developed in order to explain a nu m b e r of the trends seen in the

results of the numerical model. In chapter 4, I use a considerably extended version of the


1 2

param eterized model in order to investigate possible therm al h istory scenaxios. I determ ine

th a t more realistic therm al h istory scenaxios can be achieved if the degree o f m antle layering

is a function of the tim e-dependent Rayleigh num ber in a way th a t has been dem onstrated

by a num ber of numerical sim ulations of the m antle convection process. These results also

argue th a t the viscosity in the m antle may be greater for convection processes than it is for

P G R processes which in tu rn argues th a t m antle viscosity is most probably non-Newtonian.

This is the most significant conclusion of the sequence of analyses o f the m antle convection

process th a t form the basis of th is thesis.


Chapter 2

Internal thermal boundary layer

stability in phase transition

modulated mantle mixing

2.1 Introduction

A lthough it is now well established th a t a thermally induced convective circulation exists

in the E a rth ’s mantle, there rem ain many unanswered questions as to its detailed phys

ical characteristics. Foremost am ong these is the issue as to w hether the circulation is

“whole m antle” in style or w hether a two-layer pattern exists th a t could be enforced by

the endotherm ic spinel to postspinel phase transition a t 660 km depth (or perhaps by a

chemical discontinuity). A num ber of nonlinear simulations have recently been described

[e.g., M achetel and Weber, 1991; Peltier and Solheim, 1992; Tackley et al., 1993; Honda et

al., 1993; Solheim and Peltier, 1994a, b; Tackley et al., 1994; Peltier, 1996], which suggest

th a t convection might, in fact, be layered by the influence of the phase transition alone.

These layered states are in term ittent, however, with avalanches consisting of cold down-

wellings breaking through the 660-km phase transition and causing episodes of brief but

intense m ixing to take place between the upper mantle and transition zone and lower mantle

regions. It has also been suggested [Peltier et al., 1996] th a t this source of interm ittency of

the circulation may be im portant to understanding the supercontinent cycle.

13


2.1. In tro d u ctio n __________________________________________________________________________________ 14.

The question as to the criterion th a t m ust be satisfied for an avalanche to occur has

come to be seen as im portant. Tackley [1995] has usefully studied this problem in the case

of a single local upwelling or downwelling interacting with an endothermic phase boundary,

while Davies [1995] has employed a param eterized convection model of the kind introduced

by Sharpe and Peltier [1979] to investigate possible behaviors of time dependent phase

change m odulated convection. In the investigation to be reported herein, I will address the

circumstance in which an internal boundary layer is established in the azim uthally averaged

tem perature field. Motivation for this analysis is provided by Figure 2.1, which is reproduced

from Solheim and Peltier [1994a]. The diagnostic analysis of the axisymmetric spherical flow

presented in this figure was performed on a statistically stationary sim ulation of convection

heated from below at an Earth-like Rayleigh num ber of 107. The analysis dem onstrates tha t

if a local Rayleigh number is defined for the internal thermal boundary layer th a t develops

at 660 km dep th during the layered phase, Rae6 0 , then this Rayleigh num ber reaches a peak

just prior to the occurrence of a typical avalanche (indicated by the peak in the 660-km

mass flux tim e series), whereupon it drops sharply and then rises again to an apparently

critical value near 700, whereafter the next avalanche occurs. In this diagnostic analysis

the tem perature difference across the boundary layer is denoted by AT6 6 0 - The w idth of

the therm al boundary layer, ^660: significantly influences the variation of the boundary

layer Rayleigh number {Raeeo = 9 &&-Ts6o8 qqQ/k.l>), suggesting (which Solheim and Peltier

[1994a] did suggest on this basis) that the avalanche phenomenon is controlled by a thermal

instability of the boundary layer and th a t a linear stability analysis might be devised to

explain the onset of such events. The purpose of this chapter is to provide a detailed

assessment of the ability of an analysis of this kind to explain the observed “avalanche

effect” .

In the following section I briefly discuss the formalism that I shall employ to analyze

the stability of mean states tha t are characterized by the presence of an internal ther

mal boundary layer. The formalism'will be presented for both Cartesian plane layer and

spherical geometry, it being im portant to establish whether or not the results obtained are

sensitive to this characteristic of the physical problem. Subsequent sections include a dis

cussion of the results obtained through application of the formalism and a sum m ary and

conclusions.


2.1. Introduction

Figure 2.1: From Solheim and Peltier [1994a], illustrating the high mass flux events (avalanches) tha t occur across the 660 km phase transition following maxima in the boundary layer Rayleigh num ber. The system Rayleigh num ber R a = 107. the 660 km phase tran sition has a C lapeyron slope -2.8M P a / K , and th e convective circulation is heated entirely from below. The b o u ndary layer Rayleigh num ber is defined as R aô = (gaAT660^66o) / ( KU) in which AT660 and ^660 are respectively the tem peratu re difference across the internal boundary layer and th e boundary layer thickness.


2,2. Theoretical Formulation Iff

2.2 Theoretical Formulation

Subject to the usual Boussinesq approximation, th e nondimensional equations for mass,

momentum, and energy conservation, along w ith a linearized equation of state, assum e the

following respective forms:

V - u = 0, (2.1)

Ra D u _ pk + V p P r D t 8

V u, (2 .2)

(2.3)

p = 1 - 5 { T - T 0). (2.4)

The nondim ensionalization employed in deriving th is system is one in which length, tim e, ve

locity, tem perature, pressure, and density are expressed as: mdim — dx, ^dim = (v f gocATd)t,

Udim = (g a A T d 2/u )u , Tdim = A T T , pdim = gadp, and pdim = pop, respectively. Here the

subscript dim refers to a dimensional quantity, g is the acceleration due to gravity, po is a

reference density, d is the characteristic length scale, A T is the tem perature change across

the boundary layer and the characteristic tem perature scale, u is the kinematic viscosity, k

is the therm al diffusivity, and a is the thermal expansivity. The thermodynamic and trans

port coefficients are herein assumed to be constant. Ra = (gaATd?/ vk) is the Rayleigh

number, P r = v / k. is the P ran d tl number, 8 = a A T , and k is a unit vector in the vertical

direction. The P ran d tl num ber is effectively infinite in the E a rth ’s mantle, but effects due

to finite P r will be shown to be of interest in m ore general circumstances to be discussed

later.

Expanding th e dependent variables in the system (2.1)-(2.4) as the sum of a basic state

field plus a sm all-am plitude perturbation as u = {u fgaATd?)w k.+ eu ', p = p+ep', p — p+eir.

and T = T + ed, where e is an, assumed small, ordering param eter and w is w ritten as a

R e p ro d u c e d with p e rm iss ion of th e copyright ow ner. F u r th e r reproduction prohibited without perm iss ion .

2.2. Theoretical Formulation 11

dimensional quantity, I obtain the set of linear field equations (2.5)-(2.8). In my analyses,

T will be taken to b e a boundary layer tem perature profile, some examples of which are

shown in Figure 2.2. Internal heating Q is a function of dep th and is taken to have the

distribution required to maintain the boundary layer tem perature profile in a steady state.

Although the therm al boundary layer in the large-scale, nonlinear simulations is m aintained

by the background flow, this assumption will allow us to evaluate the thermal part of the

boundary layer instability in isolation. In the following system the variable w is an assumed

background variation of vertical velocity tha t I will employ to capture the boundary layer

stabilizing effect of convergence within the large-scale flow in which the boundary layer is

embedded.

V • u ' = 0, (2.5)

R a d u ' wd „ , (p 'k-FVyr) _ 2 /P r d t v+ — (k -V )u ' = - - (2 .6 )

q / j — »

R a — + — (k • V)0 + R a w (k - V )T = V 20,C/t AC

(2.7)

p' = - 5 9 . (2.8)

In terms of the above described scaling, momentum advection scales like the Reynolds

number (wd/i/), which can be expected to be less th an 10-22 for mantle convection and

hence can be safely neglected. Temperature advection, however, scales like the Peclet

number (tDd/re) which might be as large as 200 and will be retained in the equations. (In

both cases, surface p la te velocities are taken as upper bounds on w). In these analyses the

above defined Peclet num ber plays a critical role and will hereafter be referred to by v. It

must be noted tha t the Reynolds number scales like 1 / P r , and, as such, calculations at

finite P rand tl num ber correspond to nonzero Reynolds num ber or have v = 0.

Substituting (2.8) into (2.6) and eliminating the pressure term in the linearized momen-


2.2. Theoretical Formulation m

890

1890

CMB 2890

Temperature (Dimensionless units)

Figure 2.2: Tem perature profiles used to approximate the boundary layer tha t develops a t 660 km depth. The m athem atical forms of the profiles (a)-(e) are as follows with z w ritten as a dimensional quantity: a)T = 0 for z > zq, T = A T for z < zq, b )T = 0 for z > zo +0 .5bw/L, T = A T / 2 — (z — zo )A T (lfbw ) for \z — zq\ < 0.5bw, T = A T for z < zq — O.obw/L, c)T = Q.z>AT{l — tanh{2.l%{z — ZQ)/bw)), d )T = 0.5AT(1 — tanh(4.o(z — zo)/bw)), e)T = {1.57&AT/ y/irbw) Jq exp{—((z' — zo)l-578/friw)2) d z ' .


2.2. Theoretical Formulation Iff

t urn balance equation by applying the operator V x V x in Cartesian coordinates results

in

(2.9)

From the vertical com ponent of this equation and from the previous form of th e energy

equation, subjecting bo th to Laplace transform ation in tim e and Fourier transform ation of

the horizontal space coordinates, I ob ta in the coupled set of ordinary differential equations

In th is system, D denotes d /d z , k 2 = k 2 -+- k 2 and a is the growth rate. W and © are the z

dependent amplitudes of the pertu rb a tio n vertical velocity and tem perature, respectively.

Substitution for © from (2.10) into (2.11) yields the following modified form of the usual

sixth-order ordinary differential equation in W alone in which the Peclet num ber v appears

as a parameter.

In spherical coordinates th e system of ordinary differential equations th a t replaces (2.10)

and (2.11) may be simply shown to comprise the following:

(D 2 - kr){D 2 - k2 - — a ) W = k2Q (2.10)

(D 2 - k 2 - R a a - vD )Q = W R a D T . (2 . 11)

(D2 — k2 — R a a - v D ){D 2 - k 2) (D 2 — k 2 — — a ) W = D T k 2 W R a (2 .12)

(2.13)

(2.14)

In th is system I have employed the no ta tion Di = d 2/ d r 2 -+- (2/ r ) d /d r , in which I is

spherical harmonic degree and r ^ is th e radius of the boundary layer a t m idpoint. I will


2.3. Boundary Conditions 20

also find it useful in w hat follows to have equations relating th e norm al stress 7r to 0 and W ,

the vertical structure functions for tem perature and vertical velocity, respectively. These

relations may be obtained by tak ing the divergence of (2.6) which, in Cartesian coordinates,

delivers

D Q = (D2 — k2)% (2.15)o

D {D 2 - k 2 ~ ^ a ) W = &2t - (2-16)P r o

2.3 Boundary Conditions

O uter boundaries of the dom ain of analysis to be employed in what follows will always be

taken to be isothermal in space and time, as well as ffee-slip. and impermeable. These

conditions imply th a t 0 = 0 , D ~ W = 0, and I f = 0 on these boundaries. In many cases

the isothermal condition m ay be satisfied by requiring th a t D AW = 0 on boundaries in

the usual way. For some purposes it will be found interesting to consider circumstances in

which the outer boundaries are placed a t infinity. In these cases, W , D 2W , and 0 were

required to tend asym ptotically to 0 as a function of increasing distance from the internal

boundary layer. This condition will be satisfied in what follows by matching an inner

solution to decaying solutions of the governing equations w ith D T = 0 a t a point sufficiently

d istan t from the region of strong vertical tem perature gradient. I will also find it useful in

w hat follows to consider basic s ta te vertical tem perature variations characterized by a delta

function in vertical tem peratu re gradient (see Figure 2.2a). On the basis of continuity of

mass, horizontal velocity, tangen tial stress, normal stress, an d tem perature, it can be shown

th a t W , D W , D 2W , 7r/5, and 0 m ust be continuous across a delta function tem perature

gradient. To find the approp ria te six th boundary condition required in this situation, I

integrate (2.11) w ith D T = —5{z — zq) across an infinitesimally th in layer containing the

delta function. Given the previously stated five continuity conditions, I thereby obtain a

jum p condition on D© such th a t


2.3. Boundary Conditions_______________________________________________________________ 21

D Q i - D O 2 = - R a W ( z 0). (2.17)

In this expression the subscript 1 refers to the upper layer, while the subscript 2 refers to

the lower layer, and z q refers to the vertical position of the boundary layer.

2.3.1 Rayleigh-Taylor Instability

I have found in several of the analyses to follow that results from a simpler Rayleigh-Taylor

instability analysis are instructive when compared w ith the results of the thermal instability

analysis. The Rayleigh-Taylor stability equations may be derived from the governing equa

tions in the usual way, following Chandrasekhar [1961]. They can also be derived from the

above discussed therm al stab ility model by taking the therm al diffusivity to vanish, resulting

in an infinite Rayleigh num ber. Subject to this assumption, (2.11) becomes simply

-0 -0 = W D T . (2.18)

W hich implies tha t

p' = - W ^ - . (2.19)a

Substitu tion of this result into (2.10) then delivers

(D 2 - k 2)(D 2 - k 2 = k2 W ^ . (2.20)v* a d

The growth rate may next be rescaled as a' = <5cr in order to elim inate the param eter 6 and

this results in the following equation for the onset of Rayleigh-Taylor instability:

{D2 - k 2){D2 - k2 - ^ - a ’) W = k2 W ^ . (2.21)v *■ a


2.3. Boundary Conditions 22

2.3.2 Phase Transitions

Univaxiant phase transitions will be included in my analysis by em ploying the formulation

first developed by Busse and Schubert [1971] and employed by Sczhubert and Turcotte [1971]

and Peltier [1972] in application to the mantle convection problem— In this analysis the phase

change is assumed to occur a t thermodynamic equilibrium and s o it must exist at a mean

depth where the Clapeyron curve intersects the mean pressure am d tem perature profiles in

the mantle. Such an equilibrium phase transition exerts its influ.ence on convective mixing

through two physical effects, namely, latent heat release and p-hase boundary deflection.

Latent heat will be released or absorbed by material passing thr-ough the phase boundary,

depending upon whether the transition is exothermic or endotlierm ic, causing a vertical

motion inhibiting or vertical m otion enhancing effect on the m a te r ia l owing to the influence

of therm al expansion. Furtherm ore, since the phase transition rm ust be assumed to remain

on the Clapeyron curve if it is to rem ain in thermal equilibrium , heating or cooling of the

phase boundary due to laten t heat release and/or tem peratures advection will cause the

phase boundary to be deflected up or down. Because of the d e n s ity difference between the

shallower and deeper phases, a local vertical buoyancy force w il l result, which will again

tend either to favor or to h inder instability. In the following forrmulation these effects will

be taken into account using effective boundary conditions acro ss the equilibrium position

of the phase transition, assum ing the phase transition to be uni v a ria n t.

The additional param eters or, k , Cp, and \ j will be assumed ffor present purposes to be

the same in both phases. A t the level of approximation a t whfich I shall work, the den

sity difference between the phases will be taken into account o*nly when considering the

buoyancy force resulting from the phase boundary deflection. VW, D W , D 2W , and © are

taken to be continuous across a phase boundary owing to the com straints of conservation of

mass, continuity of tangential velocity and tangential stress a n d the assum ption that the

background tem perature field is in equilibrium. The latent heat re lease per unit time at the

phase boundary must be balanced by a discontinuity of the pertm rbation tem perature gra

dient. When appropriately nondimensionalized, this delivers th e following jum p condition


2.4. Numerical Methodology 23

on DO:

D 0 l - D 0 2 = R QW. (2.22)

In th is equation, subscript 1 denotes the region above the phase boundary and subscript 2

denotes the region below the phase boundary. In (2.22) the phase change Rayleigh number is

ju s t R q = (gacP'yTAp/Kiscpp2) . P hase boundary distortion effects axe taken into account by

im posing a discontinuity in pertu rba tion pressure tha t is sufficient to balance the buoyancy

induced by the phase b o und ary deflection. In nondimensional form th is balance yields the

second jum p condition

ILL— !!?. = S O , (2.23)o

in which S = ([Ap/p\/ctd[gp/j + DT]) is the ratio of the phase change density contrast to

the therm ally induced density contrast. In the case of the Rayleigh-Taylor instability this

phase boundary deflection equation may be w ritten in the more appropria te form

71-1 ~ - = S ~ W . (2.24)

2.4 Numerical Methodology

Two different methods will be employed in w hat follows to solve for the critical Rayleigh

nu m b ers, growth rates, and eigenfunctions required to characterize the stability of the

basic states th a t will be of interest to us. The first of these is a shooting method in

which a Runge-Kutta-Verner scheme (as implemented in the In ternational M athematics and

S tatistics Libraries, Inc. software package) is used to integrate th ree linearly independent

solutions satisfying the lower boundary conditions from the lower boundary to the middle of

the region of strong radial tem peratu re gradient, while three additional linearly independent

solutions satisfying the upper boundary conditions are integrated dow n to the center of the

region of strong tem perature gradient. In order to implement this conventional “shooting”

m ethod, the sixth-order system was w ritten as a set of six sim ultaneous first-order equations,


2.4. Num erical Methodology 24

in the form

/ = A y, (2.25)

where y is the solution vector, y ' is its vertical derivative, and A is a m atrix of coupling

coefficients. W hen phase transitions axe included in the model it is most convenient to

include 7r and 0 explicitly in y. The solution vector y is then taken to be

y = ( W w , w", f , e, ©') -

If boundaries axe at finite distances, appropriate linearly independent s ta rtin g vectors sat

isfying the boundary conditions a t the top and bottom boundaries may be taken to be

yi = ( 0, 1, 0, 0, 0, 0 ) ,

y 2 = ( 0 , 0 , 0 , 1, 0 , 0 ) ,

y3 = ( 0 , 0 , 0 , 0 , 0 , 1 ) .

W hen the outer boundaries are at infinity, s tarting vectors must contain decaying solutions

of the equations(2.10,2.11,2.16) w ith D T — 0, since the DT profiles axe always assumed to

be localized to an internal boundary layer region.

The m atrix A is simply derived from (2.10),(2.11), and (2.16) and in its most general

form (in Caxtesian coordinates) is

( 0 1 0 0 0 0

0 0 1 0 0 0

0 k 2 + -ppcr 0 k2 0 0

(k2 + 0 1 0 1 0

0 0 0 0 0 1

Ra D T 0 0 0 k 2 + Ra a V

\

(2.26)


2.4. Numerical M ethodology 25

In order to calculate the locus of neutral stability in Ra-k space, a is set to 0 based

on the assumed validity of the exchange of stabilities principle and Ra is varied w ith k

fixed until a linear combination of the solutions tha t match across the inner boundary is

found. The lowest such value of Ra is the critical Rayleigh number for tha t wavenumber.

In circu m sta n ces in which growth rates are calculated explicitly, Ra is fixed and cr is varied.

In spherical coordinates the coupling m atrix required to calculate the neutral curve is

f 0 1 0 0 0 0 N

0 0 1 0 0 0- L7 3 -

Lr2

_3r

Lr 0 0

—Lr 3

2r-

ir 0 1 0

0 0 0 0 0 1

V R a ^ D T 0 0 0 Lp -

= 1 + v ( U u l )2 ,

In (2.27), L = (1(1 -1- 1)). In spherical coordinates, W r is employed in place of W .

The shooting m ethod, based on (2.25), has the advantages that it is relatively simple to

implement and tem perature gradients of any functional form can be investigated in either

Cartesian or spherical geometry. For certain problems, however, the system of equations

becomes numerically stiff, and it is found advantageous to circumvent the need to employ

the numerical ordinary differential equation (ODE) solver.

In these cases a finite region of constant tem perature gradient will be used to characterize

the basic state (Figure 2.2b). Since it was found tha t the exact form of the gradient is not

im portant, the necessity to employ this assum ption to combat “stiffness” is not a major

drawback. U nder the assumption of a piecewise constant temperature gradient, the sixth-

order system has constant coefficients and solutions of the form exp(qz) can be found both

inside and outside the boundary layer region. T he solutions outside the region of nonzero

gradient are the same as those employed as starting vectors when using the shooting method

with boundaries a t infinity. The equation to be solved for q inside the region of nonzero

gradient is simply


2.5. Results 26

(q2 — k2)(q2 — k2 — R a a — vq)(q2 — k2 — ~^~a) = —k2 Ra (7—)- (2.28)P r bw

In this algebraic equation the param eter bw is ju s t the dimensional thickness of the boundary

layer. W hen there is no background flow(-u = 0), this equation reduces to a cubic in <f> upon

substitu tion oi<p = (q2 —k 2) and the six complex roots can be determined analytically using

C ardan’s formula. W hen v is nonzero, the roots may be simply found numerically using

Laguerre’s m ethod. W hen this m ethod was employed, v was required to be a constant.

As previously discussed, I will be employing the param eter v to model the influence of a

basic s ta te flow convergence onto the internal boundary layer, and for this purpose I will

simply assume v to be a negative constant for z > zq and a positive constant for z < zq. A

m atrix equation containing the boundary conditions may then be constructed and critical

Rayleigh num bers and growth rates calculated by finding the zeros of the determ inant of

this m atrix. Very sim ilar procedures to these are used to solve for the most unstable modes

in the Rayleigh-Taylor problem. The only difference in this case is th a t the system is then

fourth order, and only growth rates can be determ ined because density inversions are always

unstable in the absence of the influence of therm al diffusivity effects.

In w hat follows I will first employ the above discussed theoretical methodology to present

a num ber of issues concerning the problem of therm al boundary layer instability in general.

Following this, I w ill address in successive sections the problem of the stability of the

internal boundary layer th a t appears to develop in the m antle convective circulation under

the influence of th e endothermic phase transition a t 660 km depth.

2.5 Results

Although linear s tab ility analysis is most often applied to the therm al convection problem

in the analysis of th e stability of the state of rest, the same methods have also been applied

to investigate the stab ility of more general boundary layer tem perature profiles in order to

b e tter understand the onset of secondary instabilities in the therm al boundary layers th a t

develop in high R ayleigh num ber convective flows [e.g., Howard, 1964; Yuen, et al., 1981].

As originally po in ted ou t by Howard [1964], however, a th in boundary layer is unstable for


2.5. R esu lts------------------------------------------------------------------------------------------------------------------------- 2Z

40

30

1 20 •10

10

40

30

20

1010

0.5

Figure 2.3: a) N eutral curves with, quantities scaled by bw and differing distances to outer boundaries w ith the boundary layer placed a t th e center of the fluid region. From top to bo ttom L /b w = 4,6,8,10 and oo. b) N eutral curves w ith quantities scaled by bw, and differing distances to the far outer boundary w ith th e boundary layer adjacent to one outer boundary. From top to bottom L/bw = 4,6,8,10 and oo.

a vanishingly small local Rayleigh number and at very large wavelengths.

2.5.1 Thermal Boundary Layer Stability

In order to explicitly validate Howard’s [1964] comment, neutral curves were calculated for

a boundary layer of fixed thickness located in the middle of a fluid region as a function

of the to ta l layer depth L. The results of these analyses axe shown in Figure 2.3a for the

sequence (L /b w ) = 4 ,6 , 8 ,10, oo. For the purpose o f these analyses the governing equations

were scaled by the thickness of the boundary layer bw. Curves of neutral stability do not


2.5. Results 2&

depend on the P ran d tl number, and the Peclet num ber is set to 0. Effects due to finite

Peclet number will be discussed in a later section. In this calculation and for all o ther

calculations to be presented, the tem perature gradient is taken to be piecewise constant

unless otherwise s tated . The region above a neutral curve denotes the region of Rayleigh

number-wavenumber space in which the growth rate is positive and in which a fluctuation

will therefore grow in time. The minimum in this curve R amjn denotes the lowest Rayleigh

number above which instability is possible, and the horizontal wavenumber fcmin at which

this m in im um is realized denotes the horizontal wavenumber of the so-called fastest growing

mode of linear instability. It will be noted from Figure 2.3a th a t as the outer boundaries

are moved progressively farther from the region of nonzero temperature gradient, i2amin

and fcmin both rapidly approach zero. The lowest curve, corresponding to the circumstance

in which the boundaries are placed at infinity, never recovers for small k. This may seem

surprising in th a t it appears tha t a vanishingly sm all tem perature gradient can cause the

onset of a very large scale instability in a thick layer of fluid. However, it is im portant to

realize that in a thick layer there will be considerable gravitational potential energy available

to drive instability owing to the large thickness of fluid of higher density overlying an equally

great thickness o f fluid of lower density. W hen R a min and km;n are plotted as a function

of the ratio bw/L, (long-dashed line and short-dashed line in Figure 2.4a, respectively), an

interesting scaling emerges. Ramin is seen to scale like (b w /L )3 for L sufficiently greater

than bw , while k scales like (bw/L) itself. This implies tha t the natural length scale for

the system is, in fact, the entire layer depth L and th a t if this nondimensionalization is

chosen, the dynamics do not vary with bw for b w /L sufficiently small. Nondimensional

quantities will henceforth have subscripts indicating the length scale by which they axe

nondimensionalized. Figure 2.5 shows the vaxiation of Rai,m\n with bw/L for a boundary

layer placed in the middle of the fluid region. The three curves shown are for three different

mathematical forms of the boundary layer tem perature gradient, including a step function,

a hyperbolic secant squared, and a Gaussian. These are, respectively,

D T = —Lfbw for \z — zo\ < 0 .5 (bw/L),

D T = 0 |z - zQ\ > 0.5(bw/L) (2.29)


2.5. Results 2 9

c

a - 2 6i - 4

» - 6 - 1-2- 3log(bw/L)

12

10

- 3 -2-5 - 4l°g(l/L)

Figure 2.4: a) Variation of m in im um critical Rayleigh, num bers a n d wavenumbers w ith log(bw/L). Long dashed line represents Rayleigh numbers for a boundary layer placed in the middle of a fluid region. The short dashed line represents wavenumbers for a boundary layer placed in the middle of a fluid region. Dotted line represents critical minimum Rayleigh numbers for the case in which the boundary layer is adjacent to an outer boundary. Solid lines are m inim um wavenumbers for the case in which the boundary Layer is placed adjacent to an outer boundary, b) Variation of log(Rai,min) (solid line) and log{ki,min) (dashed line) w ith log{l/L) using a de lta function tem perature gradient.


2.5. Results 2 0

800

600

S, 400

200

0.4 0.80.2 0.6bw/L

Figure 2.5: a) Variation of m inim um R a l with bw /L based upon the following alternative choices for the profile of tem perature gradient: D T = —1.578L/(bwi/W)exp(—((z — zo)1.578L/im;)2)(dotted line), D T = —2.18L/(2bw)sech2(2.18(z — zo)L/bw)(dashed fine), and D T — —L/bw for \z — z q \ < 0.5bw /L and D T = 0 for \z — z q \ > 0.5bw/L(solid line).


2.5. Results

D T = -2 .lS L /{2 b w )sech 2[ 2 .1 S { z - z 0)L/bw\ (29b) (2.30)

D T = —1.578L f (bwy/7r)exp(—{[1.578(z — zo)L]/bw}2). (2-31)

The corresponding tem perature profiles are shown in Figures 2.2b, 2.2c, and 2.2e. In

spection o f the results shown in Figure 2.5 dem onstrates that for b w /L less th an 0.1, RaLmin

is independent both of bw and of the exact form of the tem perature gradient. W hen bw = L

for the step function case, I recover the Rayleigh-Benard critical value of 657.51, and when

bw=0, the m inim um Rayleigh num ber o f 327.39 obtains, which is the critical Rayleigh

num ber for a delta function tem peratu re gradient situated in the middle of a layer of unit

thickness. This quantity was found to be quite useful in that it provides a convenient check

on any th in boundary layer in the m iddle of a plane layer. Since most boundary layers of

interest will be considerably th in n er th an the entire fluid region in which they axe found,

it is seen tha t a therm al boundary layer in the middle of a deep fluid region may be well

approxim ated by a delta function tem peratu re gradient. It will also be shown presently

th a t a th in therm al boundary layer a t a rb itra ry depth in a layer of finite thickness is well

approxim ated by a delta function a t the appropriate depth. Also of interest is the fact that

Ra-Lmin is a monotonically increasing function of bw/L, implying tha t th in regions of strong

tem perature gradient in the m iddle of deep layers are more unstable th an thick regions.

This makes good physical sense since th in regions of strong gradient are associated with

g reater tem perature contrast between the upper and lower regions of the fluid, resulting

in greater buoyancy forces. M inim um critical horizontal wavenumbers kj_ were found to

vary only slightly w ith bw /L , decreasing monotonically from 2.227 to 2.221 as bw /L was

increased from 0 to 1. Figure 2.6a shows vertical velocity W (solid line) and tem perature

pertu rba tion © (dotted fine) eigenfunctions and eddy heat transport < Q W > (dashed

line) for the delta function tem peratu re gradient in the middle of a plane layer. We see that

the vertical velocity eigenfunction is v irtually unchanged from the solution of the Benard

(constant gradient) case (Figure 2.6b), and all of the vertical structu re functions are seen

to fill the entire region; they are in no way lim ited to the boundary layer region of space.

Also, because the horizontal wavenumber does not change significantly, the aspect ratio of


2.Ft. Results 2 2

1.2

ft 0.8I 0.6

0.4

0.2

0.25 0.5z

<DT33

f tscBOO0)>

1.2

0.8

0.6

0.4

0.2

0.50.25z

Figure 2.6: a) Eigenfunctions for the case in which the governing equations are scaled by th e to ta l layer depth, and a de lta function tem perature gradient is assumed: Solid line is the vertical velocity eigenfunction, do tted line is the tem perature pertu rbation eigenfunction, and the long dashed line is the eddy heat transport, b)Eigenfunctions for the case in which the governing equations are scaled by the total layer dep th for the case of a constant tem perature gradient (Benard configuration): Solid line is the vertical velocity and the tem perature perturbation eigenfunctions (they axe identical in this case), dashed line is eddy heat transport. Note th a t all eigenfunctions have been normalized so tha t their peak am plitudes are unity.


2.5. Results____________________ 33.

the convection cells changes very little w ith b w /L . For the case of a localized gradient in the

middle of a plane layer, I note tha t Howard’s [1964] predictions are sim ply a consequence

of the fact th a t the natural length scale of the system is the entire layer depth . Since

ktnu = k d b w / L ) for a given physical configuration and R a ^ = Ra-îbw/ L ) z and given that

ki, and R a l b o th tend to constants for bw /L « 1, it is clear th a t k ^ m ^ and Ra(,wmin

m ust be vanishingly small for bw « L. T h a t the tem perature pertu rbation eigenfunction

develops a cusp and, as such, is narrowed by the narrower boundary layer will be seen to

be im portan t in determining the stability of a boundary layer in the presence of a phase

transition.

Referring to the previously discussed Figure 2.3b, I present a series o f neu tra l curves for

variables scaled by the boundary layer thickness, when the boundary layer is located adja

cent to one of the impermeable boundaries for various to tal layer depths. As in Figure 2.3a,

I present results for the sequence L/fan=4,6,8,10 and oo. Again, the critical Rayleigh num

ber and wavenumber go to zero as the outer boundary is moved progressively farther away.

Yuen et al. [1981] presented growth rate curves nondimensionalized by the boundary layer

dep th for a boundary layer at the surface of an infinite half-space. Their results for constant

viscosity were qualitatively reproduced in the course of this investigation. O f interest is the

fact th a t their growth rate curve does not have a long-wavelength cutoff, cleaxly indicating

th a t jRflfrmmin m ust go to zero as k goes to zero. One might expect th a t a boundary layer

located adjacent to an impermeable boundary would be less unstable th an a boundary layer

well removed from walls because most of the fluid region would be isotherm al, w ith only a

sm all region of density contrast on one extremity. This is found to be th e case in th a t the

Rayleigh num ber is seen to approach 0 as b w /L in this case (dotted line in Figure 2.4a)

ra th e r th an as (bw /L)z , which is the scaling for a boundary layer in th e m iddle of a deep

layer. T he solid line in Figure 2.4a shows the scaling of fcfru,min when a boundary layer is

located adjacent to an outer boundary. This result implies th a t if the equations were scaled

by the entire layer depth, the critical Rayleigh num ber would, in fact, diverge to infinity as

(L /bw )2 for bw « L. This was found to be the case. From Figure 2.4b it will be noted

th a t if a de lta function gradient is moved toward the edge of a unit layer, R a £ min diverges

inversely as the square of the distance to the closest outer boundary, the sam e scaling tha t

obtains when a finite thickness boundary layer adjacent to a single horizontal boundary


2,5, Results________________________________________________________________________________ 34

is decreased in thickness. This indicates th a t it is not the depth o f the region of strong

gradient th a t is im portant but, ra ther, the distance from the effective center of the region

of strong gradient to the nearest im perm eable boundary. This length scale I will henceforth

call I. T he results shown in Figure 2.4b m ay be employed to determine the m inim um critical

Rayleigh num ber for any th in boundary layer in a deep fluid region by locating the relative

position o f the boundary layer in the layer, determ ining the corresponding Rayleigh num ber

and then scaling the result appropriately. Also of interest is tha t the eigenfunctions rem ain

“space filhng” when the therm al boundary layer is located adjacent to a n im permeable

boundary and, furthermore, since the horizontal wavenumber (dashed line Figure 2.4b) is

seen to vary very little w ith l / L , the aspect ra tio of the cells remains of order unity.

Some calculations of growth ra tes w ith supercritical Rayleigh numbers were perform ed

w ith equations scaled by bw and w ith vaxious distances to outer boundaries an d w ith various

values of the P rand tl number. In each case the Peclet number v was set to 0. In particular,

it was observed that at infinite P ran d tl num ber the wavelength of the most unstable mode

once again scaled with the distance to the nearest outer boundary and diverged if the outer

boundaries were placed infinitely fax away from the therm al boundary layer region. If a finite

P ran d tl num ber was used and, as such, the inertial term was included in the equations, it

was observed th a t even for outer boundaries a t infinite separation, the disturbance occupied

only a finite region of space. The new length scale imposed under these circum stances was

seen to be (v2/g)5.

2.5.2 Application to Convection in the Earth’s Mantle: Initial Estimates

In order to compare the critical boundary layer Rayleigh numbers inferred from the non

linear, tim e dependent simulations o f Solheim and Peltier [1994a, b] to those arising from

a purely therm al instability, the equations were scaled by 140 km, a typical boundary layer

thickness obtained in the Solheim and Peltier [1994a] simulations of the influence of phase

transitions on convective mixing. T he boundary layer was then placed so as to be centered

on 660 km depth, and the free-slip outer boundaries were placed in accordance w ith the

scaled positions of the core-mantle boundary (CMB) (dimensional dep th o f 2890 km) and

the E a rth ’s surface. The minimum Rayleigh num bers th a t were obtained in C artesian and

spherical coordinates were 0.0773 and 0.0819, respectively, for this circum stance and these


2.5. Results----------------------------------------------------------------------------------------------------------------

Table 2.1: Representative Values o f the Param eters of the 660-km Phase Transition

35.

Param eter Symbol Value

Density difference between lower and upper phases A p 450 kg /m 3

Average density P 4200 kg /m 3

Characteristic length scale d 140 km

Therm al expansivity a 2.5 x 10“ 5 K ~l

G ravitational acceleration 9 10 m /s2

Clapeyron slope 1 -2.8 x 106 kg/(m s2 K)

Tem perature gradient a t the phase boundary D T ( zq) -2.14 x 10_3K /m

Tem perature a t the phase boundary T 2300 K

K inem atic viscosity u 1018m2/s

Therm al diffusivity K 9.5 x 10-7 m2/s

Specific heat capacity a t constant pressure °P 1250 J /(K kg)

obtained a t dimensional wavelengths of 8000 and 7800 km, respectively, (see the lowest neu

tra l curve in Figure 2.7a for the Cartesian case w ith v = 0). The eigenfunctions were seen

to fill the entire space between the lower and upper boundaries (Figure 2.7b shows the ver

tical velocity (solid line) and temperature pertu rbation (dotted line) eigenfunctions). This

solution clearly represents what one might refer to as an avalanche since it does involve the

entire depth of the fluid and its horizontal wavelength is of the order of the wavelengths

of the observed avalanche disturbances, which were typically seen to be about 8000 km.

The minimum Rayleigh number, however, is many orders of magnitude smaller than those

observed by Solheim and Peltier [1994a]. Although a very small boundary layer Rayleigh

num ber is to be expected for this geometry based on the discussion of the previous section,

the question as to why the thermal boundary layer in Figure 2.1 is not unstable a t much

lower Rayleigh num bers clearly arises.

An im portan t issue certainly concerns the influence of the phase transition itself. The

param eters used to estim ate S and R q for the spinel to postspinel phase transition a t 660

km depth for this purpose are listed in Table 2.1 T he resulting parameters, S and R q , based


2.5. Results____________________________________________________________________________3£

3

1.221.0

0.81 0.6

0.40.2_

0 2010wavelength (1000km)

1.2

a, 0.8I 0.6^ _£ 0.4

0.2 - (b)

-2 0 0 0 -1000-3 0 0 0depth (km)

Figure 2.7: a)N eutral curves for the case in which the boundaries are placed to coincide w ith the earth ’s surface and CMB, with no phase change present bu t with a background velocity convergence onto the boundary layer a t 660 km depth. From bottom to top w=0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4. The stabilizing influence of a velocity convergence is clearly demonstrated, b) Vertical velocity eigenfunctions w ith v=0 and v=1.4 (solid line and short dashed line respectively) and tem perature perturbation eigenfunctions for u=0 and u=1.4 (dotted line and long dashed line respectively).


2.5. Results---------------------------------------------------------------------------------------------------------------------------3Z

upon these param eters, were calculated to be -1.8 and -95, respectively.

W ith this phase transition positioned a t the center of the therm al boundary layer, the

system was found to have a negative m inim um Rayleigh num ber, indicating th a t there would

need to be a stabilizing tem perature gradient for th e system to be even marginally stable.

This clearly indicates tha t a t onset of instability, for Earth-like geometry, the destabiliz

ing latent heat effect of the endothermic phase transition dom inates the stabilizing phase

boundary deflection effect since the critical Rayleigh num ber is seen to decrease. This was

first dem onstrated by Peltier [1985]. Buffet et al. [1994] obtained a similar result for the

case of a linear tem peratu re profile throughout the mantle. In order to test the robustness

of this result, the stabilizing param eter S was increased in m agnitude, w ith fixed R q , until

the Rayleigh num ber became positive. The critical value of S was thereby shown to be near

-12, indicating th a t one could not achieve stabilization by tun ing the param eters. Again,

this result is not surprising given tha t Peltier [1985] pointed out that when the 660-km

endothermic phase transition is placed in a layer o f the dimensions of the E arth ’s mantle,

the destabilizing influence of latent heat release dom inates because it scales like L 3, while

the stabilizing phase boundary deflection effect scales like 1 /L . Although the equations

in this case are explicitly scaled with the narrow boundary layer thickness bw, it seems

clear th a t the stabilizing or destabilizing nature of a phase transition cannot depend on

an arb itrary choice of nondimensionalization. Since the na tu ra l length scale of the system

has been dem onstrated to be the “outer scale” L it would seem th a t an endotherm ic phase

tra n sitio n in a fluid region becomes increasingly destabilizing w ith the depth of the fluid

region. In order to test this, an endothermic phase transition w ith the same param eter

values for S and R q as above was placed in the middle of a boundary layer of dep th bw

in a fluid layer o f dep th L. The equations were nondimensionalized by bw. As (L /b w ) was

increased, the critical Rayleigh number varied from being well above the critical Rayleigh

number in the absence of a phase transition for {L/bw) = 1 to negative for {L/bw) = 3.

The nondim ensional distances from the phase transition to the surface and CMB are 4.79

and 15.9, respectively. This clearly indicates th a t endotherm ic phase transitions become

increasingly destabilizing w ith the depth of the to ta l layer and the 660-km phase transition

is very strongly destabilizing a t onset for E a rth geometry. This destabilization was verified

for spherical geom etry as well. The additional influences of the phase transition a t 400 km


2.5. Results M

dep th were also added to the model, as in th e work by Peltier [1972], bu t these were found

to be only slightly stabilizing.

It has been argued [e.g., Tackley , 1995] th a t the effect of latent heat in high Rayleigh

num ber convection is considerably less significant than in a quiescent background state.

This is because a convecting fluid tends to adopt an adiabatic tem perature profile, and as

a result, the tem perature change induced by latent heating is felt everywhere beneath the

phase boundary and, as such, no buoyancy-producing lateral density gradients are created.

In one series of calculations, the latent heating param eter R q was set to 0. For S = —1.8

a critical Rayleigh number of 0.1 obtained a t a wavelength of 8000 km. Evidently, even

in the absence of latent heating effects, the endotherm ic phase transition a t 660 km depth

will not stabilize a thermal boundary layer to the extent tha t stability is observed in the

numerical sim ulations of Solheim and Peltier [1994a, b]. Clearly, effects other than phase

transitions m ust be included in a linear stab ility calculation in order to account for the

relative stab ility of the thermal boundary layers seen in the numerical simulations and for

the im portance of the boundary layer thickness in determ ining this stability.

2.5.3 Rayleigh-Taylor Analysis W ith a Phase Transition

A necessary criterion for penetration through an endotherm ic phase boundary was recently

suggested by Bercovici et al [1993], based upon a param eter Fr = (p2agd)/(-jAp), repre

senting the ratio of gravitational to phase boundary deflection forces, and it was suggested

th a t the value of this parameter should exceed 1 for penetration to occur. Since bo th of

the forces whose balance is represented by this ratio increase linearly w ith A T , the pa

ram eter itself is clearly independent of the tem peratu re contrast across the boundary layer

bu t, im portantly, increases with the length scale d. Fr is also the negative reciprocal of my

param eter S if tem perature gradient effects are neglected. In order to test the u tility of

this param eter in the context of a complete Rayleigh-Taylor analysis, an endotherm ic phase

change was placed in the middle of a boundary layer and growth rates were calculated as a

function of wavelength for the Rayleigh-Taylor case w ith equations scaled by the boundary

layer thickness bw. Latent heating effects were neglected as these cannot be assessed in the

absence of therm al diffusion. This process was repeated for different values of the boundary

layer thickness, which resulted in different values of the param eter S. In Figure 2.8a I


2.5. R esu lts 3 a

1.81.61.41.2

—0.943— (a)

-0 .962

—0.90'- 1.0

- 1.020.80.60.4

S=—1.04

wavelength. (1000km)

1.2(b)

0.6 S=—0.943-0.962-

-0 .98- 1.0

I -0-60)>

-1 0 0 0-2 0 0 0depth(km)

Figure 2.8: a) Vaxiation of growth rate w ith wavelength for Rayleigh-Taylor instabilities w ith outer boundaries a t positions corresponding to the earth ’s surface and CMB and with an endotherm ic phase transition at 660 km depth. Different curves correspond to different values of the stabilizing param eter S where S = - l is the critical value. From top to bottom S=-0.943, -0.962, -0.98, -1, -1.02, -1.04 and we see the transition from a short wavelength maximum for S < —1 to a long wavelength maximum for S > —1. b) Vertical velocity eigenfunctions w ith S = —0.943(solid line), S = —0.962 (dotted line), S = —0.98(short dashed line), and S = — l(long dashed line). Note th a t below 5 = - ! the eigenfunctions all have the same layered form.


2.5. Results 4Q

show results for a sequence o f calculations w ith 5=-0.943, -0.962, -0.980, -1.0, -1.02, and

-1.04. Vertical velocity eigenfunctions, corresponding to the most unstable modes for this

sequence of analyses, are shown in Figure 2.8b. Inspection of these results dem onstrates

that the eigenfunctions change from avalanche type (layer filling) to layered type when bw

is fixed such as to set the param eter S to a value close to -1. This analysis differs from that

performed by Bercovici et al. [1993] in tha t these au thors considered an isolated descend

ing plug, while here I assume that the boundary layer has been built up uniformly by the

background flow. T he length scale used in the analysis of Bercovici et al. is the average

thickness of th a t p a rt of the plug which has accum ulated above the phase boundary and

represents the thickness of an imposed density fluctuation. In the work reported herein

the length scale used to calculate S is the thickness of the region of significant background

density gradient bw. However, in the absence of therm al conductivity, density fluctuations

are proportional to the density gradient for an avalanche solution (see (2.18)) and so their

thickness is fixed by the thickness of the boundary layer. The thickness of the boundary

layer is thus dem onstrated to be critically im portant in determining whether an avalanche

will occur in the case of a Rayleigh-Taylor instability.

It is seen then th a t a mechanism does exist w ithin the Rayleigh-Taylor formalism to

turn avalanches on and off: avalanches will occur if the boundary layer is sufficiently thick.

If not, the instability develops as a layered mode. T he question remains, however, as to

how appropriate such an analysis, based upon the neglect of thermal conduction effects,

could be in describing the boundary layer instability events tha t could occur w ith in the

Earth’s mantle. One im portant problem clearly arises in consequence of the fact th a t

the growth rate calculated for the avalanche slightly below the critical value of S is such

that risetimes are 2 orders of magnitude shorter th an those observed in the full nonlinear

simulations of Solheim and Peltier [1994a], which are operating in the statistical equilibrium

state. Clearly, another problem is that in the Rayleigh-Taylor analysis, density inversions

always drive instability; no stable regime exists of the kind th a t is observed in the nonlinear

simulations. W hen in the layered regime, the instability would erase the boundary layer,

which would destroy the possibility of further avalanches. As avalanches are observed in

the nonlinear simulations, interm ittently separated by significant periods of stably layered

flow, it seems clear th a t therm al diffusion effects cannot be neglected. The Rayleigh-Taylor


2.5. Results Al

criterion is insufficient to explain the avalanche process.

As a consequence of finite therm al diffusivity, the p e rtu rb a tio n tem perature eigenfunc

tion of the unstable m ode becomes nonlocal. As penetration through the phase bound

ary was dem onstrated to increase w ith the width o f the pertu rba tion density (which is

proportional to the p ertu rb a tio n tem perature), therm al instabilities rem ain avalanches for

much more negative values of the stabilizing param eter S . W hen tem perature pertu rbation

eigenfunctions were p lo tted along w ith velocity eigenfunctions (not shown) and when phase

transitions were present in the therm al instability case, it was observed that tem perature

perturbation eigenfunctions were always nonzero over a significant region surrounding the

phase boundary whenever avalanches occur. This dem onstrates th a t although the pene

trative or layered properties of convection through a phase boundary are not fixed by the

boundary layer thickness and the param eter S in the therm al instability case as they are in

the Rayleigh-Taylor case, the dep th extent of the tem perature fluctuation is vitally im por

tan t in determining w hether or not an avalanche will occur. For this reason, any process

th a t results in a localization of the perturbation tem perature eigenfunction will result in a

stabilization of the boundary layer against the occurrence of an avalanche.

2.5.4 Process of Dynamical Stabilization

One im portant factor neglected in the previous sections concerns effects due to the back

ground flow. More heat is advected into a boundary layer region by convective flows th an

is advected outward [Jarvis and Peltier, 1982]. Thus effects due to the background flow

might quite reasonably be expected to result in a localization of a density perturbation and

hence stabilize an in ternal therm al boundary layer in the presence of an endothermic phase

transition as described in the previous section. In order to sim ply capture effects due to

advective heat convergence by the background flow, a constant positive Peclet num ber will

be assumed below the phase transition and an equal in m agnitude constant negative value

will be assumed above. As previously discussed, m om entum advection due to the back

ground flow may be neglected for mantle circulation and, as such, this velocity convergence

will affect the stability of the boundary layer only through its influence on the tem perature

structure. This incorporation o f mean flow effects through in troduction of a constant ve

locity convergence m ight be expected to be justifiable on the basis of a separation of length


2.5. Results________________________________________________________________________________ 42

scales in two limiting cases: (1) If the w idth of a therm al plum e in the layered state is much

larger than the wavelength o f a boundary layer instability and (2) If the spacing between

therm al plumes in the layered sta te is much less th an the wavelength of a boundary layer

instability. It is this second lim it which I expect to be appropriate in the present context;

in the simulations of Solheim and Peltier [1994a], avalanche instabilities axe seen to have

long wavelengths com pared w ith the internal spacing of upper mantle therm al plumes. A

schematic representation o f th is scenario is presented in Figure 2.9. The dashed lines repre

sent the upper and lower extrem ities of the boundary layer, while the sinusoid represents an

incipient instability of the boundary layer itself. One wavelength of the incipient instability

extends over many upper and lower convection cells. A lthough there is an outward flow

th a t is necessary to conserve mass, there is a net influx of advected heat, and it is the effects

of this heat advection th a t I wish to capture through the Peclet number param etrization.

An upper bound on the Peclet number u, based upon surface plate velocities of the

order of 4 cm /yr would be approxim ately 200. For the purposes of the linear stability

analysis, v is expected to be much smaller however, since it is intended to represent only the

effective convergence onto the boundary layer and not the to ta l background flow. Referring

once more to Figure 2.7a, a series of neutral curves are presented for various values o f the

nondimensional velocity convergence (Peclet number) in the absence of phase transitions

for the Earth-like geom etry described above. The m inim um critical Rayleigh num ber is

seen to rise monotonically w ith v. In order to a tta in Rayleigh numbers similar to those

recorded in the Solheim and Peltier [1994a] analysis, v was required to be near 100. It

seems unlikely th a t the effective background convergence for boundary layer stabilization

could be of the same order of m agnitude as the mean flow speed. The role of the dynamics in

stabilizing the internal therm al boundary layer is, nevertheless, clearly illustrated, and it is

rather likely th a t effects due to deform ation flows th a t apply effective convergence across the

region of strong radial tem peratu re gradient will stabilize in ternal thermal boundary layers

in high Rayleigh num ber convection. Also of interest are the perturbation vertical velocity

and tem perature eigenfunctions, which are shown in Figure 2.7b as the short-dashed line

and long-dashed line, respectively, for the case v=1.4. It will be observed th a t the velocity

structure is shifted som ewhat from the zero background velocity case, but more importantly,

the tem perature pe rtu rb a tio n eigenfunction becomes strongly localized into the region of


2.B. Results A l

3 1445

2890

Figure 2.9: Schematic illustrating the process of dynamical stabilization of the thermal boundary layer. The influence of tem perature advection onto the boundary layer(dark arrows) in convergence is taken to exceed the influence of advection of the re tu rn flows(dashed arrows). The spacing between therm al plumes in the layered s ta te is considerably shorter th an the wavelength of a boundary layer instability(sinusoid) supporting a separation of length scales argument whereby a simple param etrization of the background flow as a cons tan t convergence may be employed.


2.5. Results-------------------------------------------------------------------------------------------------------------------------- 44,

the boundary layer itself.

W hen the endothermic phase transition is also placed in th e center of the boundary layer

and a background velocity convergence is active, it is found th a t the combination of these

effects may be very strongly stabilizing indeed. Figure 2.10a shows a series of neutral curves

for which the param eters of the endotherm ic phase transition are chosen to be the same as

those employed above and w ith various values for the velocity convergence. Two distinct

m inim a are now observed to be characteristic of the n eu tra l curves. Figure 2.10b shows

eigenfunctions for critical Rayleigh numbers and wavelengths corresponding to the long

wavelength (8800 km,u=1.3) and short-wavelength (1375 km ,u=1.4) minima. The long-

wavelength solution represents an avalanche, while the short-w avelength solution represents

a split mode of instability w ith no penetration of the phase boundary. On the basis of

the neutral curves, it is seen th a t if the background convergence onto the boundary layer

is increased, the low-wavenumber minimum disappears and th e fastest-growing instability

is switched from avalanche to layered style. In nonlinear sim ulations an increase in the

system Rayleigh number is seen to decrease the number o f avalanche events, leading to

more complete layering. It seems reasonable that larger ex te rn a l Rayleigh numbers will

correspond to higher average convergence onto the phase boundary, and so this result agrees

well w ith the numerical sim ulations. My analysis clearly predicts th a t for very large external

Rayleigh numbers the flow would be more nearly perfectly layered, a circumstance that I

expect would most probably have obtained in the early E a rth . I t has often been suggested

[e.g., Peltier, 1996] th a t such phase transition induced layering in the early E arth could

have led to the development o f a chemical discontinuity across the phase change interface,

thus strongly increasing the degree of layering.

Figure 2.11a shows a series of neutral curves for a constan t value of the background

convergence (u=1.4) bu t w ith different values of the C lapeyron slope of the phase transi

tion using a tem perature gradient of the form D T = — 2.25sech2[4.5(z — zo)](Figure 2.2d).

For smaller values of the m agnitude of the Clapeyron slope we see th a t the long-wavelength

avalanche solution corresponds to the most unstable mode. As th e m agnitude of the Clapey

ron slope is increased, however, the most unstable mode changes from one with a wavelength

near 8000 km to one w ith a wavelength near 1300 km. Clearly, there is a critical Clapeyron

slope below which layering is enforced. For this particular value of the background velocity


9 !v R psnlts AR

800

600 v=1.5

1.4O 400«oeaos200 1.3

1.2

10 15 20wavelength. (1000km)

1.2

0.6

2 - 0.6

- 1 .-2000 -1000000

depth(km)

Figure 2.10: a)N eutral curves w ith a phase transition of C lapeyron slope -2.8M P a /K placed at 660 km depth and varying values of velocity convergence. From top to bottom v= 1.5, 1.4, 1.3, 1.2. b) The solid line is vertical velocity corresponding to the long wavelength minimum w ith u=1.3. The short dashed line is vertical velocity corresponding to the short wavelength minimum w ith u=1.4. The d o tted line is tem perature pertu rbation of the long wavelength solution. The long dashed line is tem perature perturbation of the short wavelength solution.


2.5. R esu lts

2000

1500

« 1000

500

10 15 20wavelength (1000km)

2000

1500

1000 ■ a

500

010 15 20

wavelength (1000km)

Figure 2.11: a)Neutral curves w ith u= 1.4 w ith varying Clapeyron slopes in Cartesian coordinates and D T = —2.25sech2(4.5(z — z q ) ) , showing the change from an avalanche type instability to the layered mode for decreasing values of the endotherm ic Clapeyron slope. T he Clapeyron slope corresponding to the curve represented by the long dashed line is 7 = - 2.55 M Pa/K , for the dotted line is 7=-2.56 M Pa/K , for the short dashed line is 7 = -2 . 5 7

M P a/K , for the solid line is 7=-2.58M Pa/K , for the short dashed-dot line 7 = - 2 .5 9 M Pa/K , and for the long dashed-dot line 7 = - 2 .6 M P a/K . b)N eutral curves w ith u=1.4 for varying values of the Clapeyron slope in spherical coordinates and D T — — 2.25sech2(4.5(r — ro)). T he Clapeyron slopes corresponding to the curve marked by the x, dot, solid square, hollow square, s tar and triangle are respectively -2.51, -2.52, -2.53, -2.54, -2.55, and -2.56 M Pa/K .


2.5. Results------------------------------------------------------------------------------------------------------------------ 4Z

and for this form of the tem perature gradient, the critical Clapeyron slope is found to be

between -2.58 and -2.59 M Pa/K . A critical value of the Clapeyron slope is found to exist

for every value of the background velocity convergence. Note also tha t all of the curves axe

identical in the short-wavelength limit and th a t once the Clapeyron slope is sufficiently sta

bilizing, the minimum critical Rayleigh num ber does not increase w ith a further decrease in

the Clapeyron slope. This value of the Rayleigh number is an upper bound on the possible

value for the m inim um critical boundary layer Rayleigh number for an avalanche solution

for a particular velocity convergence w ith a boundary layer of a particular vertical structure.

It m ust be noted, however, th a t when a phase transition and a converging background

flow are present, the dynamics axe very sensitive to the exact values of the param eters and

to the w idth and shape of the boundary layer. It was found tha t for u=1.3 a Clapeyron slope

of -2 M P a/K is unstable at negative Rayleigh numbers, while a value of -3.2M Pa/K shows

only completely layered solutions with Rayleigh numbers of 328. Small variations in the

background velocity show similar sensitivity as do variations in the boundary layer thickness

and shape, and I have no method of determ ining an appropriate velocity convergence for a

particular value of the system Rayleigh number. However, given the sim ilarity between the

nonlinear model results and those found here, it seems likely tha t the Earth, or at least the

nonlinear models of the E arth tha t have been explored numerically, is (are) operating in

a region of param eter space in which the Clapeyron slope is subcritical but the minimum

critical Rayleigh number for the avalanche solution is positive. There are then seen to be two

ways of “triggering” an avalanche. E ither the background velocity convergence decreases,

decreasing the critical Rayleigh number required to cause the instability, or the boundary

layer Rayleigh number increases owing to an increase in the tem perature contrast or to an

increase in boundary layer thickness. Once a boundary layer is destroyed, flow across the

phase transition once more becomes inhibited and it takes time for the boundary layer to be

built up to a critical state again by the background flow, resulting in the quasi-periodicity

(interm ittency) observed in the simulations.

Calculations in spherical coordinates have also been performed w ith a phase transition

and background convergence, and in all cases the results were very similar to the results

obtained for the calculations performed in Cartesian geometry. Figure 2.11b shows the

analogous neutral curves to those in Figure 2.11a for the sphere. We see th a t they are


2.6. Discussion and flonclnsinns A&

qualitatively extrem ely similar. O f course, in the spherical case the wavelength spectrum is

discrete and thus results are shown only for the wavelength corresponding to discrete values

of spherical harm onic degree. One sm all difference is tha t the instab ility in the spherical

case becomes layered for slightly less negative values of the Clapeyron slope. In Figure 2.11b

the critical C lapeyron slope is seen to be between -2.54 and -2.55 M P a/K .

2.6 Discussion and Conclusions

Therm al boundary layers in the absence of the influence of phase transitions and dynamical

effects were shown to be highly unstable when situated in a deep layer. Boundary layers

situated a t the center of deep fluid regions were shown to be particu larly unstable, and

stability increased as the region of strong vertical tem perature gradient was moved closer

to an im perm eable boundary. The length scales controlling the dynam ics were shown to be

the entire layer d ep th and the distance to the closest impermeable boundary. The depth of

the therm al boundary layer was shown to exert very little influence on th e dynamics as long

as it was shallower th an one ten th of the full layer depth. In this case the therm al boundary

layer was shown to be well approxim ated by a delta function tem peratu re gradient a t the

appropriate depth . As a result, the spatia l extent of convection a t the critical Rayleigh

num ber arising from thermal boundary layers is not limited to the region of the boundary

layer bu t, ra ther, it fills the entire fluid layer in which the boundary layer is contained.

In the absence o f background flow and phase transitions the therm al boundary layer

a t 660 km th a t was shown to develop in the simulations of Solheim and Peltier [1994a,

b] was observed to be unstable a t extrem ely small Rayleigh n u m b ers, bu t the instability

tha t ensued was consistent w ith the spatia l extent of an avalanche. T he presence of the

endotherm ic phase transition alone a t 660 km dep th was seen to fu rther destabilize the

layer, and this was shown to be a result of the fact that endotherm ic phase transitions are

increasingly destabilizing the deeper the layer of fluid in which they exist, in accord with

argum ents of P eltier [1985].

A criterion for determ ining w hether a Rayleigh-Taylor instability will be avalanche-like

or layered was discussed, and this was shown to depend critically on the boundary layer

thickness. In the considerably more com plex therm al instability this criterion was shown


2.fi- Dismission and Conclusions____________________________________________________________ 49.

to offer insight into the importance of boundary layer thickness and velocity convergence in

d e term in in g the stability of the boundary layer.

A converging flow was seen to stabilize a therm al boundary layer, possibly explaining

the region of stability o f the therm al boundary layers observed in high Rayleigh num ber

numerical simulations of the convection process. The values of the velocity convergence

required to achieve critical boundary layer Rayleigh num bers comparable to those in the

sim ulation of Solheim and Peltier [1994a] were unrealistically high in the absence of the

influence of the endothermic phase transition itself. W hen coupled w ith the influence of

the endotherm ic phase transition, however, the applied velocity convergence was shown to

be very strongly stabilizing. Stabilization increased w ith decreasing Clapeyron slope and

increasing velocity convergence. A critical C lapeyron slope for a given value of the velocity

convergence was shown to exist, below which, only layered convection solutions exist, and

above which, avalanche solutions occurred. T he effects of sphericity were shown not to

affect these results significantly. The results obtained through application of linear stability

analysis to an internal therm al boundary layer in the Earth-like geometry and an assessment

o f the extent of the agreement of these results w ith the simulations of Solheim and Peltier

are sum m arized in Table 2.2. This stabilization of a therm al boundary layer by a uniform

strain field is quite sim ilar to the stabilization o f a vorticity strip against Kelvin- Helmholtz

instability by a stra in field [e.g., Dritchel et al., 1991] in infinite Reynolds num ber flow.

A num ber of potentially im portant effects have not been dealt w ith in the analysis

reported herein, however. The influence of in ternal heating, in particular, has not been

considered, although it is likely tha t this will sim ply accentuate the downwellings from the

upper surface, increasing the convergence onto the boundary layer and hence should be

stabilizing, as has already been dem onstrated to be the case in the numerical simulations

of Solheim and Peltier [1994a]. Also, non-Boussinesq effects have been neglected. A lthough

these will probably affect the results quantitatively, it is quite clear tha t in their presence a

converging flow onto an endothermic phase transition will still be stabilizing. Effects due to

variations in viscosity, therm al conductivity, and the thermodynamic coefficients have also

been neglected. I t remains to be seen w hether these will influence the ability of velocity

convergence to stabilize the boundary layer. Also, the Clapeyron slope of the 660-km phase

transition is only constrained to he w ithin the range -2 to -6 M Pa/K according to Ito and


2-fi. Dismission and C!rmcTnsinns____________________________________________________________511

Takahashi [1989], although Chopelas et al. [1994] found it to be more strongly constrained

to a value near -3 M P a/K . Although my model is quite sensitive to the exact value of the

Clapeyron slope, it has been shown tha t for any value of the Clapeyron slope there is a veloc

ity convergence for which the avalanche solution obtains at the m inim um Rayleigh number

and where if the velocity convergence is sufficiently high, layered convection will be the only

possible form. Clearly, a constant velocity convergence is an enormous simplification of the

influence of the real, tim e dependent, background flow; horizontal and vertical variations

in vertical velocity have been neglected, while horizontal velocity has been neglected en

tirely. Also, continuity is not satisfied at the phase boundary or a t the outer boundaries.

The Peclet number param etrization is intended, however, to capture only the effect of the

heat advection convergence, and it is the intention of my calculation to represent this in

the simplest possible fashion in order to gain physical insight into the avalanche process.

Since this simple param eterization appears to explain many of the features seen in the full

nonlinear simulations, it would appear that much of the im portant physics governing the

avalanche process has been captured.


2.6. Discussion and Conclusions 5 1

Table 2.2: Effects Included in the Analysis of Convective Instability for Earth-Like Geometry and Comparison w ith the Numerical Experim ents of Solheim and P eltier [1994a,b]

Included physical effects axe denoted by the following numbering scheme: 1, ther-

mal conductivity; 2, endotherm ic phase transition a t 660 km depth; 3, velocity convergence.

Included

Effects

Agreement w ith

Numerical Experiments

Disagreement with

Numerical Experiments

1 Velocity eigenfunction structure is

compatible w ith observed avalanches.

R a min is many orders of m agnitude

smaller than in numerical experiments.

No layered solutions axe found. The

boundary layer width does not play a

significant role in the instability.

2 Avalanche and layered solutions are

observed. Layering increases with

decreasing Clapeyron slope and de

creasing boundary layer width.

Timescale for instability is 2 orders

of m agnitude too short. No stable

regime exists.

1,2 Avalanche solutions are seen. Ramin is negative; no layering is

observed.

1,3 Avalanches axe seen; Ramin is

higher th an in the absence of velocity

convergence.

Ramin is similar to values observed

in the numerical experiments only if

an unrealistically large velocity

convergence is used. No layered

solutions axe observed.

1,2,3 Avalanches and layered regimes are

observed; Layering increases with

decreased boundary-layer width, w ith

decreasing Clapeyron slope, and

increased velocity convergence.


Chapter 3

On scaling relations in

time-dependent mantle convection

and the heat transfer constraint on

layering

The effects of phase transitions, internal heating, and depth-dependent viscosity axe now

quite routinely included in models of convection in the E a rth ’s mantle. Owing to the nonlin

ear nature of the convection process, however, there remains a great deal to be understood

in term s of the quantitative effects th a t these various influences have on bulk properties

of the mantle, such as the average tem perature as a function of depth (the geotherm ), the

m ean surface heat flow, and the boundary layer thicknesses, as well as the surface velocity

and the mass flux transiting the 660 km endotherm ic phase transition. There is also much

to be learned as to the values o f the param eters o f the model tha t are required to deliver

agreement with the geophysical observables. M any analysts have forgone such comparisons,

arguing, with some justification, tha t current models are still oversimplifications o f the real

E arth . I argue, however, th a t when large discrepancies exist between reasonable Earth-like

m odel results and E a rth observations, then potentially im portant dynamical constrain ts

may be placed on E a rth processes.

In what follows, I present a broad survey of the effects of depth-dependent viscosity,

52


fifl

phase transitions, and internal heating rates on mantle bulk properties in order to accom

plish two goals: (1) to improve physical understanding as to how these effects modify bulk

properties of the mantle and where possible to quantify the ir specific influences and (2)

to compare model predictions w ith real E arth observations in order to identify regions of

p a r a m e te r space in which E arth may be operating and to identify Earth observables tha t

cannot be explained by such simple model calculations and hence require the inclusion of

more detailed physics. T he tools th a t I will employ in order to accomplish these goals

will include a simple, param eterized model which is particularly useful in gaining physical

insight as well as a more detailed numerical model of the convection process.

Many numerical simulations of therm al convection in the planetary mantle have demon

s tra ted the im portance of the endotherm ic phase transition a t 660 km depth through which

spinel is transform ed to a m ixture of magnesiowiistite and perovskite in controlling the

exchange of m aterial between the upper and lower m antle [e.g., Christensen and Yuen,

1985; Machetel and Weber, 1991: Peltier and Solheim , 1992; Honda et al., 1993; Solheim

and Peltier 1993; Tackley et al., 1993; Solheim and Peltier , 1994a,b]. Relatively little is

known, however, about the spectrum of scales of different events tha t transit this horizon

and the way in which these vary w ith the control param eters of the convection model (e.g.,

C lapeyron slope, Rayleigh number, internal heating rate , viscosity variation). In section

3.2 I will therefore begin by presenting an analysis of the size spectrum of such mass flux

events, and I will dem onstrate th a t in certain param eter regimes the number of avalanches

is seen to scale w ith m agnitude. Bak et al. [1988] introduced the concept of self-organized

criticality (SOC) to explain the ubiquity of distributions th a t display such power law be

havior. As the size spectrum of mass flux events exhibits power law behavior, w ithout

requiring tuning of the control param eters, there is evidence of scale invariance th a t may

be related to SOC. I point out an analogy th a t may be draw n between the avalanche pro

cess in phase-transition-m odulated convection and the avalanche processes th a t occur in a

sandpile.

Further motivation for calculating the size d istribution of such mass flux events comes

from geochemical studies which suggest tha t the m antle should be viewed as being divided

into chemically distinct reservoirs [e.g., Hart and Zindler, 1986]. The observations rely on

the m easured compositional differences between mid-ocean ridge and ocean island basalts.


A n im portant issue in in terpreting such geochemical evidence is the size distribution of

heterogeneities in the m antle. I f the chemical reservoirs are interpreted to be the lower

m antle and upper mantle and transition zone, respectively, then the flux of undepleted

lower mantle material into the depleted upper mantle and transition zone (and vice versa)

is an im portant source of chemical heterogeneity, and hence the size distribution of mass flux

events should be closely related to the size distribution of mantle chemical heterogeneity.

Determ ining the spectrum of mass flux events tha t transit the 660-km phase transition

may also be im portant for de term ining the impact of transform ational superplasticity on

the viscosity in the vicinity o f the endotherm ic phase transition [Sammis and Dein, 1974;

Karato et al., 1995]. If mass is carried prim arily by small events tha t are fairly evenly

d istributed and which do not e rup t into vigorous plumes, then one might expect the region

near the endothermic phase transition to be softened globally. Similarly, if mass is carried

prim arily by large, localized events, then one might not expect a large reduction of the

spherically averaged viscosity. Peltier [1998] has recently suggested tha t this low-viscosity

region a t the 660-km depth horizon tha t is required to best reconcile nonhydrostatic geoid

d a ta in terms of a mantle tom ography constrained convection model [e.g., Pari and Peltier,

1995] may be best understood as a consequence of such superplastic behaviour engendered

by the grain size reduction th a t occurs when m aterial changes phase. It is usually thought

th a t transform ational superplasticity is a Newtonian Theological phenomenon because of the

im portan t role tha t grain size reduction plays in the mechanism. However, modern research

on the rheology of polycrstalline m aterials (metals, ceramics, ice etc.) cleaxly dem onstrates

the existence of a broad regime in differential stress w ithin which the creep mechanism is

bo th grain size sensitive and non-Newtonian (see Peltier, [1998] for a recent discussion). On

this basis it is clear tha t the superplasticafly softened layer associated with the endothermic

phase transform ation could creep via a non-Newtonian mechanism, thus explaining the

seismic anisotropy tha t is inferred to exist over this range of depths in the mantle [e.g.,

Vinnik et al., 1998]. This would most easily explain why this soft layer is required by the

convection model bu t not by models of glacial isostatic adjustm ent.

A number of recent models of E a rth evolution have included the effects of m antle

avalanches triggered by instabilities a t the endothermic phase transition boundary. Stein

and Hofmann [1994] invoke phase-transition-induced, time-dependent mixing between the


55

upper and lower mantle to explain the episodicity observed in records o f crust formation.

The W ilson cycle and the quasiperiodic production of crustal m ateria l have been explained

in models that include catastrophic m antle overturns by Condie [1995, 1998] and Peltier et

al. [1997]. Sheridan [1997] included m antle avalanches th a t occurred on shorter timescales

in models to explain periods of rapid seafloor spreading and periods in which few reversals

of the geomagnetic field occurred.

Boundary layer theories have been employed for many years to predict the scaling of the

Nusselt number (nondimensional heat flow), velocity, and therm al boundary layer thickness

as a function of the Rayleigh num ber (therm al forcing). These theories all follow essentially

the formulation of Turcotte and Oxburgh [1967], who m atch solutions in the interior of a

steady, Boussinesq, convective flow w ith those in the boundary layer regions which leads to

power law scaling relations for the above mentioned quantities w ith the Rayleigh number.

For the case of isothermal boundaries in Cartesian geometry and in the absence of internal

heating, the Nusselt number, velocity, and boundary layer thickness are found to vary as

the 1/3, 2/3, and -1/3 power of the Rayleigh number, respectively. These exponents were

shown [e.g., Jarvis and Peltier, 1982] to agree well w ith numerical calculations, although the

multiplicative prefactors were found to show considerable disagreem ent. Londe and Davies

[1985] extended boundary layer theory to include the effect of an exponential increase of

viscosity with depth and found th a t the exponents were essentially unchanged, although the

Nusselt number decreased substantially when a large increase in viscosity w ith depth was

employed. This is to be expected, however, since they held the surface value of viscosity

constant, and, as such, an increase in the viscosity w ith depth resu lted in a considerable

increase in the average viscosity in the domain. The predictions o f this boundary layer

theory were borne out by the num erical calculations of G um is and Davies [1986]. T hat the

Nusselt number varies roughly as the 1/3 power of the Rayleigh num ber for compressible

convection in axisymmetric spherical geometry with constant viscosity was dem onstrated

by Solheim and Peltier [1990]. T heir calculations showed th a t th e re was a break in the

scaling as the flow crossed the transition from steady to tim e-dependent behavior, but even

in the time-dependent regime, the Nusselt number continued to scale w ith roughly a 1/3

exponent. Their calculations dem onstrated the need to consider a tim e average of the heat

transfer effected by convective flows th a t had attained a statistically steady sta te in order


flfi

th a t the Nusselt number have a unique value.

In section 3.3 I present a new param eterized model of convection. Parameterized con

vection models have been popular for performing therm al h istory simulations [e.g., Sharpe

and Peltier, 1978, 1979; M cKenzie and Richter, 1981] as they represent significant savings

in com putational expense over num erical simulations based on solution of the com plete

set of hydrodynamic equations. Mine differs from previous models in tha t I employ it to

predict and explain a num ber o f features observed to be characteristic of numerical model

predictions ra ther than for the purpose of com puting therm al histories. I also include the

effects of compressibility and depth-dependent properties, and my model recovers a num ber

o f the scaling results of boundary layer theory. It is also useful to compare its predictions

to those of the numerical model in order to gain insight into such issues as the timescale for

convective adjustm ent, the heat flow due to secular cooling o f the mantle, and the influence

of depth-dependent viscosity on heat transfer. Honda and Iwase [1996] have also com pared

the results of param eterized and numerical models, b u t they were interested in testing the

validity of param eterized models for use in therm al history analysis.

I further investigate scaling issues using the numerical m odel in 3.4 and therein demon

s tra te th a t even in tim e-dependent simulations w ith depth-dependent viscosity, internal

heating, and phase transitions, heat flow, surface velocity, and boundary layer thickness all

continue to scale w ith the m ean viscosity. This clearly highlights the critical role played by

the la tte r param eter. In establishing these scaling relations I perform an extensive search

o f the region of param eter space in which I believe the real E a rth to be operating. Cer

ta in assum ptions concerning the nature of convection in the m antle have been required in

generating these results, and these will be discussed in detail in section 3.1.2.

In section 3.1, I will provide a brief description o f my m odel assumptions and of the

num erical model and introduce modifications th a t allow us to employ an arb itrary grid

spacing in the radial coordinate. In section 3.2 I will employ the numerical model to

investigate the distribution of mass flux events th a t transit th e 660-km endothermic phase

transition. I b o th derive and investigate the previously m entioned parameterized model in

section 3.3 and, where feasible, compare its output w ith th a t of the full fluid dynam ical

model. In section 3.4 I employ the numerical model to investigate, using Rayleigh-number-

dependent scaling relations, the effects of depth-dependent viscosity, phase transitions, and


3.1. Model Description 57

internal heating on a num ber of bulk properties of the circulation. In section 3.5 I examine

my models in terms of constraints on the tem perature at 660 km depth and in terms of the

mass flux crossing this dep th horizon.

3.1 Model Description

O f critical importance in this study are my assumptions concerning the parameters tha t

govern the nature of convection in the E arth ’s mantle. I will summarize these in section

3.1.1. In section 3.1.2 I will describe the numerical model th a t I will use to investigate a

num ber of issues concerning scaling of convection in the E a rth ’s mantle.

3.1.1 Model Parameters

I will assume that the core-mantle boundary (CMB) and surface are free slip and isothermal

in b o th space and time. The surface and CMB tem peratures are 300 K and 4000 K in all but

four of my simulations and are param eterized in my model through the variables Ts and A T,

which are the surface tem perature and tem perature drop across the mantle, respectively.

The CMB tem perature is chosen based on the best estim ate o f Boehler [1996] and will be

seen to be one of the most im portan t underpinnings of the argum ents to be presented.

In considering the surface boundary to be free slip w ith a tem perature of 300 K I am

neglecting the complex dynamics associated with tectonic plates and identifying the ocean

floors as comprising the surface boundary layer of convection cells in the mantle. Many

previous analyses have been based upon the assumption of a surface tem perature of 1200

K in the apparent belief th a t the lithosphere should not be assum ed to be fully involved

in the convective circulation. This may be reasonable for the 30% of the surface th a t is

covered by continents, bu t it is clearly difficult to justify in oceanic environments which

display age-1/2 dependence o f heat flow, which is typical of boundary layer behavior [e.g.,

Turcotte and Schubert, 1982]. In the analyses to be presented herein I will entirely neglect

the influence of the continents. On the basis of the lim ited area of the surface covered by

them , their effects on the bulk quantities of interest are expected not to be of first-order

significance although it is im portan t to recognize, as discussed in some detail by Pari and

Peltier [1996, 1998], th a t th e ir role in the circulation may be active ra ther than passive.


3.1. M odel Description 58

In Figure 3.1a I display the sequence of depth-dependent viscosity profiles (VPs) th a t I

will explore in this study, and for each, the volume average viscosity (770) for the profile

th a t is drawn is indicated. Also shown are two radial viscosity profiles (dotted lines) from

postglacial rebound inversions of Peltier and Jiang [1996] (curves 1 and 2). The issue as to

w hether or not these profiles axe consistent w ith data related to the convective circulation

will be a prim ary focus of interest in what is to follow.

I will investigate three different rates of internal heating (IH) of to tal intensity 20 TW ,

10 TW , and an entirely bottom heated configuration. These span a range of intensities

bracketing what is likely the current heating rate of 14-19 T W based on geochemical argu

ments [Hart and Zindler, 1986]. IH is related to the variable which is the internal heating

ra te per unit mass, by division by the mass of the mantle and to p, the nondimensional

internal heating rate that I will define in section 3.1.2.

I will also investigate the effects of significantly varying the magnitude of the Clapeyron

slope (CLAP) of the endothermic phase transition. The value of this quantity will be taken

to be —2.8MPa K - 1 based on the experiments of Chopelas et al. [1994] except for a small

num ber of simulations in which I will increase it substantially.

In Figure 3.1b, I display the nondimensional depth dependencies of density pTl accel

eration due to gravity g, adiabatic bulk modulus K s, therm al conductivity k and therm al

expansivity a. These parameters are chosen to be as Earth-like as possible. The depth

dependencies of pr , <7, and K s are fit to Prelim inary Reference E arth Model (PREM) d a ta

[Dziewonski and Anderson, 1981]. The value of cp is determined by a therm al model due

to Stacey [1977], and k is fit to the experim ental measurements of Osako and Ito [1991] in

the lower mantle and Kieffer [1976] in the upper mantle and transition zone, while a is fit

to the results of Chopelas and Boehler [1992]. (Please see Solheim, and Peltier [1994a] for a

more detailed description.)

3.1.2 Description of the Numerical Model

The governing equations of the numerical model consist of those for mass conservation

(continuity),

V • (pr u) = 0, (3.1)



10GO<CJa,No

GOOoW>

1.5CO

GE3COaoCOGV6 0.5

1400 4 000 4600 5200 Radius (km)

5800 6400

Figure 3.1: (a) D epth-dependent viscosity profiles (VPs) used in this study (solid lines) and two viscosity profiles of Peltier and Jiang [1996] inferred from postglacial rebound observations (dotted lines). T h e volume average 770 of each of the VPs is indicated, (b) The radial variation of the non-dim ensional param eters used in my model. These will be m ultiplied by their reference values in order to obtain their dimensional variations w ith depth. These are density (solid fine) po = 4000 kg m -3 , therm al conductivity (long-dashed-dotted fine) ko = 10 W (m K)-1 , ad iabatic bulk m odulus (short-dashed-dotted line) K s = 4.624 10u Pa, therm al expansivity (long-dashed line) ao = 2.5 10-o K -1 , specific heat capacity (short-dashed fine) Cpo = 1250J (k g K )-1 , and acceleration due to gravity (dotted line) po = 10 m s -2 .



momentum conservation,

0 = —pgr — Vp + o;oAT{—V x (77V x u) + 4 /3V (pV • u)

+ 2 V (u • V 77) - 2(u - V )'Vrj - 2V pV • u}, (3.2)

energy conservation,

D T /D t - (T o /a o A T c ^ D /D tih T i 4- l2T2) = (k / R o)[V2T + ( l/k ){d k /d r )(d T /d r)}+

fi/(c pRa) + r 0$ / ( p rCp) - rru r(T -F Ts) (3.3)

and an equation of state

p = pr{ 1 - a 0A T a (T - Tr) + l / K s (p - P r ) } + A t ( r t - r rJ + A 2( r 2 - Tr2). (3.4)

In the above system, which is an extension of th a t employed previously by Solheim and

Peltier [1994 a,b], subscript r denotes a depth-dependent quantity, whereas r itself represents

the radial coordinate, u is the velocity field, and p, p, and T are the pressure, density, and

tem perature fields, respectively. Subscript zero is employed to denote a constant reference

value used for nondimensionalization purposes. $ represents the heat generated by viscous

dissipation. I point out th a t (3.2) differs from (10) of Solheim and Peltier [1994a] in tha t

it correctly accounts for a spatially varying viscosity field. The nondimensional parameters

in the above system of equations include the Rayleigh number,

R a = aoATgod?po/K0r]o, (3.5)

the dissipation numbers

tq — goa0d/Cpo; Tr = T0g a /c p, (3.6)



and the nondimensional constant in ternal heating ra te per unit mass

ft = P o x d 2/ k 0A T . (3.7)

In the definitions of these parameters, d represents the thickness of the mantle.

Effects due to the phase transitions are included through the phase density functions

Tj = 1/2{1 + tanh[(rPi — r ) h i / d \ } (3.8)

in which a subscript i = 1 denotes the phase transition at 400 km depth while subscript

i = 2 denotes the 660-km phase transition; the hi (i = 1 , 2 ) are dimensionless param eters

used to set the thickness of the divariant phase loops and I use h \ = /12 = 400. In (3.3)

the li represent latent heat per unit mass, while in (3.4), A t represents the difference in

density between the upper and lower phases. I use A i = 200 kg and A2 = 440 kg. The

symbol r pt- represents the position of the phase boundary and is a function o f the polar

coordinate of the axisymmetric spherical model geometry and of time. Its value must be

determ ined in each angular grid column and for each time step. This may be accomplished

by assuming constant Clapeyron slopes. If the transition is in thermodynamic equilibrium,

then the phase boundary must occur a t a depth where the pressure is given by

P = Poi + 7 i { T + Ts). (3.9)

In (3.9), the poi are the zero-tem perature pressure of the coexistence curves for the phase

transitions and these are set so as to keep the mean depth of the phase boundaries a t 400

and 660-km. The dimensional values o f the Clapeyron slopes, 7 *, for the 400- and 660-

km phase transitions will be taken to be 3 M Pa K - 1 and —2.8 M Pa K -1 , respectively [e.g.,

Chopelas et al., 1994] unless otherwise stated . The latent heats are related to the Clapeyron

slopes through the Clausius-Clapeyron relation

l i = p % /[ A i (T + Ts)}. (3.10)

Equations (3.9) and (3.10) are nondimensional. It is also useful to ensure th a t only a single



time derivative appears in (3.3), and this may be achieved by introducing a transformed

tem perature A such th a t

A = T — r0/ ( a 0A T c p)(ll r l + l2V2). (3.11)

Equations (3.9) then reduce to the following coupled set of nonlinear algebraic equations

under the assum ption tha t the pressure is hydrostatic:

7 i{A + Ts + T0/ ( 2 a QA T c p)[li + l 2 -f- J2 ta n h ((rp2 — r p l)/i2)]} +Poi — Pr = 0, (3.12)

7 2 {A + Ts + T0/(2 a 0ATcp)[li + l2 + Z itan h ((rpi — r p2 )/ii)} + P 02 — Pi- = 0. (3.13)

These equations must be solved for the r pi- a t each tim e step and at each spatial position.

In the usual way, the momentum equation for the two-dimensional flow may be trans

formed into a pair of Poisson equations in the stream function ?/> and the vorticity w. It

was found necessary in some instances to employ an uneven grid spacing in the radial di

rection. The multigrid numerical solver th a t I employ to solve the pair of Poisson equations

allows only for constant spacing. As such, following Sakai and Peltier [1996], I modify the

equations so th a t the evenly spaced model coordinate, which I will call X , maps into an

unevenly sampled “real world” coordinate r , while 6 is the usual polar angle in spherical

polar coordinates. F irst and second derivatives in the radial direction must now be replaced

in the model equations with the following forms:

and

d2 . d X 2 d 2 , ( ? X x d , o l c ,dr2 dr d X 2 + d r2 d X ' ^


3.1. M odel Description 63

The stream function and vorticity equations then become

. 6 21p . . d X . 2 2 - dlf) ,d ? X 1 dpr d X . 2 - n ■ 2 - n( d x * H ~ d F 1 r s m e + a x ( l Z - s m e + s , n e a m ~ c o s e a e = - u P r r s m e '

(3.16)

and

2 • 0 d 2u , d X \ 2 d u 2 - a / t P X /•2c?77 g p r d Xr W 5 * T ( * > + v c r s ln e ll p r + <? 5 : + ^ ) ^ r )

, - ado' 2 I d 2p gpr I dp+smew t ~ cos9Se + [sm9r W + *s v ^ )!“

prg g r2sin29 d A 2 h iF i(l - F v) IjTp _ A L 3 rpl 2fr2r 2(l - T2) l2r0 _ Ax 3 rp2 rj 66 qq A T Cp pr a 30 otqAT Cp pr a 66

Ag d p .d 2ip dip 2g 1 dg . d 2ip . ^ dip dtp .-( ■ a9-sin0 — — cos0) + —— — (— 9 sin0 + -— cosd — — r s m d )3K s pr d r dd2 66 K s g d r 692 66 dr

2 dpT 1 dr) d 2ip dip dip . 2 1 d 2g d 2ip dip dipZ ^ z ( ^ Z T smd - ~ ^ cosd + -HZrsind) + SwB + cosd - — r s in d )p2 d r rj d r dd2 66 dr pT g d r2 dd2 66 d r

2sin6 dr) f dip _ d2ip , f<>'( r ) (3-17)prr) dr d r d r2

In (3.17) lj = u;rsin0. In the results tha t follow, I will assume that viscosity is a function

of radius only. T he above transformation results in only minor modifications to the energy

equation. The rad ial resolution depends on the form of d r /d X (r ) . The constraints on

X and r axe th a t they must have the sam e values a t their end points and X ( r ) m ust

be a m onotonically increasing function. In Figure 3.2a I display two possible choices of

X (r), and in Figure 3.2b I show the corresponding resolution. The solid line represents

regular spacing, while the dotted line corresponds to a choice of X (r) th a t results in a local



6400

5800

5200a4600X

4000

3400

o

om<DPZ 0.5

§400 64005200(km)

58004000 4600 Radius, r (km)

Figure 3.2: (a) Two choices for the function X (r) th a t defines the radial resolution. The solid line represents the case of a constant grid spacing, while the dotted line represents the case th a t has increased resolution in the region of the phase transition, (b) The corresponding resolution for the same two cases as a function of radius. The arrow indicates the position of th e endotherm ic phase transition.



resolution enhancem ent in the vicinity of the 660-km phase transition, which I found to be

necessary in certain calculations.

The equations were solved using centered finited difference approximations. The stream

function and vorticity equations were solved using the MUDPACK elliptic equation solver

[Adams, 1991] while the energy equation was tim e stepped using a modified Crank-Nicolson

scheme. The zeros of (3.12) and (3.13) were obtained using the Levenberg-Marquardt

method. Some calculations were performed using 513 by 1025 radial and azim uthal grid

points, respectively, while others used half of this resolution. I list all of the simulations

that I will discuss in Table 3.1, and I indicate their resolution by h (513 by 1025) and 1 (257

by 513). I also employ the letters c and v to indicate constant or variable radial resolution,

and I indicate the dimensional values of the param eters A T, 770, VP, IH and CLAP for each

of the simulations. I also indicate the m easured surface heat flow (Qs), surface velocity

(us), surface boundary layer thickness (Ss), and 660-km depth mass flux ( / ) .

Table 3.1: Summ ary of Simulations

Name A T % VP IH CLAP Resolution Qs U s /

A1 1657 2.96 2 0 0 h c 26.5 7.6 40 77.7

A2 1657 3.305 3 0 0 h c 22.0 5.8 40 52.7

A3 2485 3.305 3 0 0 h c 37.64 6.3 40 78.4

A4 2485 4 1 0 0 h c 31.78 1.9 79 43.1

B1 3700 4 1 0 0 1 c 62.2 4.0 56 65.3

B2 3700 15.3 1 0 0 1 c 38.75 1.8 90 26.3

B3 3700 28.0 1 0 0 1 c 32.61 1.1 123 18.6

B4 3700 88 1 0 0 1 c 22.8 0.65 180 8.0

B5 3700 880 1 0 0 1 c 10.6 0.16 462 2.2

C l 3700 4 1 0 -2.8 h c 50.8 1.9 62 21.3

C2 3700 15.3 1 0 -2.8 1 c 34.96 0.86 101 14.1

C3 3700 28 1 0 -2.8 1 c 29.27 0.78 124 11.7

C4 3700 88 1 0 -2.8 1 c 21.2 0.36 180 6.25

C5 3700 880 1 0 -2.8 1 c 10.8 0.13 406 1.57


3.1. Model Description. 66

Name A T Vo VP m CLAP Resolution!. Qs U s 4 /

D1 3700 28 1 20 -2.8 1 c 35.6 0.66 135 10

D2 3700 44 1 20 -2.8 1 c 33.4 0.48 158 8.8

D3 3700 88 1 20 -2.8 1 c 28.4 0.31 181 6.0

D4 3700 880 1 20 -2.8 1 c 18.3 0.07 395 1.6

E l 3700 29.1 3 20 -2.8 h c 42.6 1.87 86 13.1

E2 3700 145.42 3 20 -2.8 1 c 30.0 0.74 147 6.2

E3 3700 290.84 3 20 -2.8 1 c 27.7 0.54 182 5.4

E4 3700 2908.4 3 20 -2.8 1 c 18.3 0.10 406 1.3

F I 3700 57.82 4 20 -2.8 1 c 40.8 0.90 102 11.0

F2 3700 289.08 4 20 -2.8 1 c 27.9 0.44 192 4.3

F3 3700 578.16 4 20 -2.8 1 c 24.5 0.29 226 3.2

F4 3700 5781.6 4 20 -2.8 1 c 17.2 0.056 440 0.88

G1 3700 35.25 5 10 -2.8 1 c 34.4 1.2 113 19.4

G2 3700 58.75 5 10 -2.8 1 c 29.7 1.0 124 9.6

G3 3700 164.5 5 10 -2.8 1 c 23.4 0.54 181 5.6

G4 3700 470 5 10 -2.8 1 c 18.0 0.29 257 3.2

H I 3700 15.3 1 0 -6.5 1 c 26.2 0.59 115 6.0

H2 3700 88 1 0 -6.5 1 c 18.4 0.25 199 2.8

H3 3700 880 1 0 -6.5 1 c 10.6 0.11 412 1.4

11 3700 15.3 1 0 -9.25 1 c 20.74 0.43 116 3.2

J1 3700 4 1 0 -12 h c 19.6 0.43 80 1.5

J2 3700 15.3 1 0 -12 h c 16.8 0.42 114 0.9

J3 3700 88 1 0 -12 1 c 11.1 0.13 178 1.3

J4 3700 880 1 0 -12 1 c 8.9 0.052 410 1.0

K1 3700 3.305 3 0 -2.8 h v 58.7 4.15 42.0 27.5

K2 3700 3.305 3 0 -6.5 h v 27.1 2.0 45.9 4.8

K3 3700 3.305 3 20 -2.8 h v 69.2 3.8 41.2 24.4

Values for A T, 770, IH, and CLAP axe in units of K, 1021 P a s, TW , and M Pa K —1 re

spectively; 1 refers to a resolution of 257 by 513, while h refers to a resolution of 513 by

1025; c and v refer to constant and variable grid spacing. Heat flow, us, Ss, and / are in


3.2. Results 1: Statistical Analyses of the Avalanche Effect and the SOC Scaling Analogy 67

units of TW , cm /y r, km and kg (m2yr)_1. In ternal heating (IH) is uniformly distributed

throughout the m antle except for K3 where it is restric ted to the region below the 660-km

dep th horizon.

3.2 Results 1: Statistical Analyses of the Avalanche Effect

and the SOC Scaling Analogy

T he discovery of the “avalanche” effect [Machetel and Weber, 1991; Peltier and Solheim,

1992: Solheim and Peltier, 1994a,b; Peltier, 1996], w hereby the endotherm ic phase transition

a t 660 km dep th induces a layered style of convection th a t is episodically interrupted by

dram atic overturns, has dem onstrated the possible im portance of the endothermic phase

tra n sition in con tro llin g the nature of the circulation in the mantle. Figure 3.3 shows a

sequence of color contour plots of tem perature in my numerical model. These are taken

from run K3 as defined in Table 3.1. In the first frame (6168 Myr into the simulation), the

m antle is in a layered s tate . At 6295 Myr, a weak downwelling is occurring and a t 6393

Myr, a large avalanche is seen at roughly 65° from the n o rth pole. A smaller downwelling

quickly follows a t 70° from the pole (6465 Myr) an d this, in turn , is followed by another

large avalanche a t the equator (6557 Myr). This avalanche has essentially subsided and

the system has re tu rned to the layered state by 6618 M yr. Also indicated on the contour

plots in Figure 3.3 axe the positions of the 400- an d 660-km phase transition boundaries

(black lines). In Figure 3.4a I show a time series o f the azim uthal average of the absolute

mass flux transiting the endotherm ic phase boundary horizon also taken from simulation

K3 (arrows indicate the tim es corresponding to those shown in Figure 3.3). The solid and

do tted fines indicate th e azim uthal average of th e mass flux at 660 and 400 km depth,

respectively. The transition from a layered sta te to a regime in which three large mass flux

events occur and back to a layered state can be clearly seen. In Figure 3.4b I display the

signed value of the mass flux as a function of the polar angle in the model a t the same

tim es as the contour plots. Associated with the strong downwelfings a t 6393 Myr and 6557

M yr are isolated strong re tu rn flows. W hat can also be seen, however, is that there are a

num ber of other, sm aller events occurring and little is known as to the relative contribution

of these different events to the to ta l mass flux as well as their spectral dependence on the


3.2. Results 1: Statistical Analyses of the Avalanche Effect and the SOC Scaling Analogy 68

3876 K

3652 K

3429 K

3205 K6168 Ma 6 2 9 5 Ma2982 K

2758 K

2535 K

2311 K

2088 K

1864 K

1641 K

1417 K646 5 Ma 6 5 5 7 Ma 1194 K

970 K

747 K

523 K

Figure 3.3: (a) Contour plots o f the tem perature field in my model mantle. The black lines indicate the positions of the 400- and 660-km phase change boundaries. The model is initially strongly layered, followed by a period in which three successive avalanches occur, followed by a re tu rn to the layered s tate . T he model ru n is K3.


3.2. Results 1: Statistical Analyses o f the Avalanche Effect and the SOC Scaling Analogy 69

15001

^ 100

50GO

§048 67006374 Time (Ma)

01

00

Xj3

tnmasS

1000

500

0 6465 Myr 6557 Myr ^- - 6618 Myr

- — 6168 Myr 6295 Myr 6393 Myr

- 5 0 0

-100018060 120

Polar Angle (Degrees)

Figure 3.4: (a) The azimuthally averaged absolute value of the radial mass flux at 400 km (dotted line) and 660 km (solid line) dep th as a function of time. The arrows indicate the tim es corresponding to the contour plots in Figure 3.3 (b) The mass flux as a function of the polar angle a t the position of the 660-km dep th endothermic phase transition corresponding to the same times as the contour plots in Figure 3.3.


3.2. Results 1: Statistical Analyses o f the Avalanche Effect and the SOC Scaling; Analogy 70

various control param eters (i.e., Rayleigh num ber, C lapeyron slope, etc). In this section I

will therefore begin by describing a series of num erical experiments in which I characterize

the size d istribution of heterogeneities and describe their variation as a function of the

Rayleigh num ber, internal heating rate , dep th dependence of viscosity and m agnitude of

the C lapeyron slope a t 660 km depth. These results are all obtained in my spherical

axisym m etric numerical model, and the d istribution of mass flux events in three dimensions

m ay be somewhat different. Tackley [1997] com pared two- and three-dimensional Cartesian

models and found tha t although the to ta l mass flux and degree of layering were sim ilar in

b o th cases, the two-dimensional case resulted in more catastrophic m antle overturns and

greater tim e dependence in many flow diagnostics. Similarly, Machetel et al. [1995] found

th a t the degree of layering in three-dim ensional spherical geometry was very similar to the

degree of layering in spherical axisym m etric geometry.

In order to calculate the size d istribution of mass flux events, a t each time step I seek the

zero crossings of mass flux along the position of the endotherm ic phase transition and then

integrate the to ta l mass flux between points delim iting the angular extent of a given event.

I then compare the mass flux events to the sign and position of the mass flux events of the

previous tim e step, and if the peak of the current event is between the limits of a previous

event and if the event has the same sign, I add the mass flux of the current event to th a t

of the already existing one and update the positions. An event is assumed to have ended if

e ither the mass flux at its peak falls below 0.03 kg(m 2yr)-L or its w idth is less than three

grid points. In this way, I have counted thousands of mass flux events and performed the

statistical analyses described below. T he num ber of mass flux events has been normalized

to the num ber which would occur in a period of 10 G yr and the m agnitude has been plotted

in units of the mass of the mantle. The mass flux event sizes are binned such th a t there

are eight d a ta points for every decade. Figure 3.5a, displays a logarithmic histogram of

the num ber o f mass flux events of different sizes for simulations which differ only in the

m agnitude of CLAP (runs B2, C2, H I, and II ) . These simulations were chosen because

although the m agnitude of the viscosity is very high compared w ith the mean viscosity

estim ated from postglacial rebound inversions, a heat flow very sim ilar to tha t of the real

E a rth is obtained for the case when a C lapeyron slope of the endotherm ic phase transition

is employed th a t is in accord w ith high-pressure experim ental measurements [e.g., Chopelas



i n ri f f l f i i n m if i i n n ii|— i m mikj— i i in inj— i 11 iiiij| i i in in ] i 1111114 1 i i m u | 1 1 m nij— 1 111111^— m n |

9? 100

X XX

*** B2, C2, HI, IIuiul I I m in i ' ■ »nm l ■ ■ ■ im J • i i i i . n l■ iiniJ lnI>s.fft.iiil 1 1 mud 1 1 mini

0 .0 0 1 0 .0 1 0.1

1000

1 0 -7 1 0 -e 1 0 -5 1 0 -Mass (MM)

Figure 3.5: (a) The number of mass flux events of a given size as a function of the event size. Each model has constant viscosity w ith r/o = 15.3 x 1021P a s, and no internal heating. The squares, triangles, stars, and crosses indicate runs w ith C LA P=0, —2.8, —6.5, and —9.25 M Pa K -1 respectively, and the runs are listed in c. (b) The cumulative mass flux distribution (the number of m ass flux events of th a t size or g reater as a function of mass flux event size). The runs a re the sam e as in a. (c) The to ta l am ount of mass carried by mass flux events of a given size. T he runs are the same as in a and are listed a t the bottom.


3.2. Results 1: Statistical Analyses o f the Avalanche Effect and the SOC Scaling A nalogy 72

et al., 1994; Ito and Takahashi, 1989]. I will discuss issues concerning heat flow in greater

detail in section 3.4.1. I t can be seen th a t the first-order effect of increasing the m agnitude

of CLAP is to increase the to ta l num ber o f mass flux events. This is to be expected since

for the case without phase transitions, it is only the inherent tim e dependence of the flow

which will cause an event to end. I point out th a t as the magnitude of CLAP is increased, a

high-m agnitude cutoff is imposed th a t decreases w ith increasing magnitude of CLAP. W hen

no phase transitions are present, the largest events are almost the scale of the entire mantle,

whereas when using a Clapeyron slope of —9.25 M Pa K -1 the laxgest event is only 4/100

the scale of the mantle. The to ta l integrated mass flux across the 660-km depth horizon

decreases with increasing CLAP despite the larger number of mass flux events due to the

very large mass carried by the extrem ely large events tha t are allowed with weaker phase

transitions.

A nother feature th a t can be clearly seen is th a t the phase transition acts to give definite

structu re to the mass flux event d istribution. There axe three quite well-defined regimes for

the case in which the phase transition is present. The number of mass flux events w ith a

given m agnitude increases initially. This is likely the least reliable aspect of this diagram

as the sm all size cutoff is likely affected by the resolution of the numerical model. T he

d istribution is then quite flat over m any orders of m agnitude and then there is a scaling

regime in which the number of mass flux events decreases roughly linearly w ith m agnitude

until the large-magnitude cutoff is reached. This general shape for the mass flux distribution

was found in every case tha t I investigated in which the phase transitions were present. In

the absence of phase transitions it can be seen th a t the mass flux events of all sizes occur

w ith roughly equal frequency. In Figure 3.5b I display the cumulative mass flux distribution

for the same runs as in 3.5a (the cumulative d istribution is a plot of mass flux events th a t

axe of th a t magnitude or greater). T he reduced laxge event cutoff and the imposed scaling

regimes can be more clearly seen in this cumulative distribution.

As can be cleaxly seen in Figures 3.5a and 3.5b, another effect of increasing the Clapeyron

slope is to increase the number of sm aller events. One consequence of this is to shift

the region of the mass flux spectrum th a t carries the most mass. In Figure 3.5c I show

the to ta l mass carried by events of a given size as a function of size. Owing to the flat

natu re of the mass flux event d istribution when no phase transitions axe present, the to ta l


3.2. Results 1: Statistical Analyses of the Avalanche Effect and the? SOC Scaling Analogy 73

mass flux is dom inated by the largest events. As the m agnitude o f the Clapeyron slope

is increased, however, the mass flux event size distribution falls off m ore sharply, resulting

in there being roughly an equal amount of mass carried by ev en ts of sizes from 1 0 ~ l to

10~ 4 the system size w ith a Clapeyron slope of —6.5 M Pa K _ l . F o r a Clapeyron slope of

—9.25 M Pa K _I there is actually a local maximum near 5 x 10- 4 mantle masses in the

d istribution of to ta l mass flux carried by an event of a given size. Ln Figure 3.6a I display

mass flux event size distributions for four cases, all of which h av e the phase transitions

present, no internal heating, and constant viscosity (rims C2-C5). Tine viscosity in each case

is different, however. One trend th a t is clearly visible is tha t the num ber of small mass flux

events decreases considerably as the viscosity increases. The num b er of the largest events

remains relatively unchanged, however. There are two competing effects th a t modify the

total mass flux across 660 km depth. The ability of a phase tran s itio n to induce layering

increases w ith the effective Rayleigh num ber [Christensen and Yu en, 1985: Solheim and

Peltier, 1993, 1994a], as measured by the decreasing viscosity, w h ile the background flow

becomes more vigorous, tending to increase the to tal mass flux acro=ss the 660-km horizon.

In the case shown in Figure 3.6a the to ta l integrated mass flux decreases w ith increasing

viscosity, indicating th a t the la tte r effect is more significant for th is model configuration.

Another trend is th a t the structure of the distributions, w ith thr-ee well-defined scaling

regimes, becomes less well defined as the viscosity increases, as w ould be expected since the

phase transition effects become less significant for low Rayleigh num bers. Similar analyses

were done on rim series D, E, F, and G, and in all cases the same tren d s as observed above

were found.

In Figure 3.6b I show three different mass flux event d is tribu tions w ith different degrees

of internal heating (runs C2, G l, and D l). Although these sim ulations have different mean

viscosities and viscosity profiles, they all have surface heat flows sirniilar to th a t of the real

Eaxth. All three are remarkably similar. The large event cutoff is ver-y similar, and all cases

display three well-defined scaling regimes w ith similar slopes. In F ig u re 3.7 I display three

mass flux distributions for the highest Rayleigh num ber cases th a t I calculated (runs K l,

K2, and K3). It can be seen th a t in these low-viscosity calculations t i e avalanche spectrum

is shifted to smaller mass flux events. I t is also seen th a t as in the caJculations presented in

Figure 3.6b, the addition of internal heating does not significantly affect the shape of the


3.2. Results 1: Statistical A nalyses o f the Avalanche Effect and the SOC Scaling Analogy 74

a>XiBp'z,

* - C 2

■ C 3

x C 4

* C 5

*■*4.'4*4

- X *

■ x x

* * 4* * 4* .

x x X X X X X X X ^ xX X X X x

X V ■■

x fc

1000

100

- = 10

1 0 0 0 r -

S-.0)Xspz

100

* * * * * « 4 X

*■ C 2

* G 1

X D 1

4*44* 44,*44u**44 **44.

;X X X X X X X . A AaAaA!a

4 X

.a6*☆a.

X X

10 r- X

X

■tr x f t

X *

*»* x iu

xX* ** -dX =*

| 0 - 1°I lAIIIlL I I I J

1CT9 101-8uml

10u u a L

X X

i m i i t i t m i l i m

1-7 10-® 1CT6 10- Mass (MM)

0 .0 0 1 0.01 0.1

Figure 3.6: (a) The num ber o f mass flux events as a function o f the event size for runs w ith constant viscosity, CLA P=-2.8 M Pa K _l and no internal heating. The triangles, squares, crosses, and stars indicate runs performed with 770= 1 5 .3, 28, 88, and 880 1021Pa s (runs axe listed), (b) The num ber of mass flux events as a function of the event size for runs w ith three different values of internal heating but th a t have the same surface heat flow and CLAP=-2.8 M Pa K _1. Triangles have tjo = 15.3 x 1021P a s, VP 1 and no internal heating, stars have 770 = 35.25 x 1021P a s, VP 5 and 10 T W of internal heating. Crosses have 770 = 28 x 1021P a s, V P 1 and 20 TW of internal heating. Rims are indicated.



r r n IUI| I I I m l ^ i I i i i i i i j I I i u i i i j i > i n !I | I lU M ^ 1 I II I IU ] 1 I I i l l l l j 1 I 1111 Rj

1000

1000

0.001

i n J I i m in i LLLuBL_LULLU'-1210_u10-10 10-9 10-8 10-7 10-6 10-5 lOÔ.OO 10.01 0.1

Mass (MM)

Figure 3.7: (a) The num ber of mass flux events as a function of the event size for VP 3, 770 = 3.305 x 1021P a s and different values of internal heating and of CLAP (the runs axe indicated on Figure 6c). (b) T he cum ulative distribution for the sam e da ta as in (a), (c) T he to ta l mass carried by mass flux events of given sizes for the same d a ta as in Figure (a).


3.3. Results 2: Parameterized M odel for “W hole Mantle” and ’Layered M antle’ Convection 76

mass flux distribution. Finally, we see th a t the increase of the m agnitude of the Clapeyron

slope of the endothermic phase transition to —6.5 M P aK - 1 has a very pronounced effect

on the mass flux distribution for these low viscosity calculations. Above an avalanche size

of 10- 4 mantle masses, there is a very rapid decline in the num ber of avalanches. This

decrease is rapid enough th a t there is a maximum in the d istribution of to ta l mass flux

carried as a function of the event size for avalanche size 1 0 - 4 m antle masses.

Since the mass flux event d istribu tion displays power law scaling for many different

param eter values, it is likely th a t the avalanche effect in phase-transition-m odulated con

vection is an SOC phenomenon. As described by Solheim and Peltier [1994a] and Butler

and Peltier [1997], avalanches occur as a result of an instability of the internal thermal

boundary layer tha t develops across the phase boundary. This boundary layer is built up

by hot rising plumes and cold sinking plumes th a t are blocked by the endothermic phase

transition. An avalanche will occur if the internal thermal boundary layer Rayleigh num

ber exceeds a critical value. There is an interesting analogy w ith models of avalanches in

sandpiles, the analysis of which led to the introduction of the SOC concept by Bak et al.

[1988]. This analogy is illustrated schematically in Figure 3.8. In the sandpile model a

particle will slide whenever the slope locally exceeds a critical value. These slides can then

trigger further downhill slidings. Sandpile models tend to be driven naturally to critical

states in which sand avalanches of all sizes occur and their size spectra display power laws,

indicating scale invariance of the avalanche process. The addition of new grains of sand to

the sandpile and of therm al plumes to the therm al boundary layer are the forcing mecha

nisms in the two models. The critical slope in the case of the samd-pile models is analogous

to the critical boundary layer Rayleigh num ber in the case of the avalanche instability in

phase-transition-m odulated therm al convection. I will further discuss the implications of

these results for the E arth in section 3.6.

3.3 Results 2: Parameterized Model for “Whole Mantle”

and ’Layered Mantle’ Convection

In w hat follows, I derive a param eterized model of the convection process and compare

its predictions for the timescale, geotherm , and heat flow for whole m antle and layered


3.3. Results 2: Parameterized Model for “W hole M an tle” and ’Layered M antle’ Convection 77

SAND PILE IN TERN AL THERMAL BOUNDARY LAYER

Forcing: grains o f sand T (r)

Forcing: thermal plumes

Ra, c

0C = critical angle

e > 6C .avalanches occur

SELF - ORGANIZED CRITICALITY ?Ra,c = critical Rayleigh number

Ra > Ra,c, avalanches occur

Figure 3.8: Analogy between the phase-transition— induced mantle avalanche process and sandpile avalanche models.


3.3. Results 2: Param eterized Model for “W hole Mantle” and ’Layered M antle’ Convection 78

convection w ith those of my num erical model.

3.3.1 M odel Derivation and Implications for the Timescale of Convective

Adjustment

I am most interested in applying resu lts to the mantle. As such, I will refer to the lower

all quantities axe dimensional. I begin by simply asserting th a t the tim e ra te of change in

internal energy of the mantle must be balanced by the sum of the heat entering through the

core-mantle boundary (Qc) , th a t generated internally (xM m ), and th a t leaving the surface

(Qs), namely,

Subscript s c and s are used to indicate th a t a quantity is evaluated a t the CMB and surface,

respectively, M m represents the mass o f the m antle, t represents time, and the integral is

over the volume of the mantle. I will use angle brackets to indicate th a t a quantity has

been averaged over a surface whose norm al is in the radial direction. I next assume that the

radial derivative of < T > is adiabatic in the region between the surface and CMB thermal

boundary layers. This condition will hold as long as the effects of internal heating, viscous

dissipation, la ten t heating of phase transitions, and heat conduction are small compared

w ith vertical heat advection and the effects of adiabatic compression and expansion. As I

will show explicitly, < T > is indeed precisely adiabatic in the absence of internal heating

and strong phase-transition-induced layering. This condition of internal adiabaticity can be

expressed as

The following discussion applies quite generally to infinite P ran d tl num ber convection, but

and upper boundaries as the core-m antle boundary (CMB) and surface w ith the under

standing th a t these results could be easily generalized to other situations. In this section,

J pcpTdV = QC- Q s + xM , (3.18)

d < T > dr

— < T > Cp

(3.19)


3.3. Results 2: Param eterized Model for “W hole M antle” and ’Layered M antle’ Convection 79

Equation (3.19) m ay be simply solved in order to obtain the internal tem peratu re in the

mantle in term s of a single unknown param eter c, which represents the tem perature a t the

top of the CMB therm al boundary layer (say), and a known function cf>(r), as

r

< T > = c (f>(r), 4>{r) = exp(— J dr'), (3.20)T'c~3C&c

in which Sc is th e thickness of the CMB therm al boundary layer and r c is the position of

the CMB. I next assume th a t shifts in the in ternal tem perature dominate changes in the

internal energy. This allows us to write, for the case of spherical polar coordinates,

4 t t | J pep < T > r2dr = Air J p Cp cf>(r) r 2 dr ^ = Cp (3.21)

The depth dependencies of p and Cp are inputs in my numerical model, and hence the

integral in the m iddle term of (3.21) can be evaluated to give an effective heat capacity Cp

whose value is 4.235 x 1027J K -1 .

The heat flow across the CMB and surface are then calculated from

Qc = kcA T cA c/5 c, Q s = ksA T sA s/Ss, (3.22)

in which A T i (i = c or s) is the tem perature drop across the thermal boundary layer a t tha t

horizon, while ki and Ai are the therm al conductivity and area of horizon £, respectively.

The tem perature difference across the entire layer is given by AT = ATS + ATad + ATC, in

which ATa(i is the tem perature drop across the m antle due to the adiabatic gradient. Given

my above s ta ted solution for the adiabatic in ternal tem perature, I can identify

ATC = AT + Ts - c, A T S = c<t>s - Ts, (3.23)

in which <ps is the value of 4> a t the base of the surface therm al boundary layer. Substitution

of (3.20), (3.22), and (3.23) into (3.21) leads to th e following first-order ordinary differential


3.3. Results 2: Parameterized Model for “W hole Mantle” and ’Layered Mantle’ Convection 80

equation for c:

+ 4 > s ^ ) c = % ^ ( A T + Ta) + Ts + XMm. (3.24)U 6 O q O s O q 0 ^

Equation (3.24) has a solution of the form

c = c (o o )+ R exp(—t/X ), (3.25)

in which c(oo) is the final, horizontally averaged, tem perature at the top o f the basal

thermal boundary layer and A is the time constant for convergence to thermal equilibrium,

or thermal adjustm ent time, and has the form

A = Cp/{kcA c/5 c + 4>sksA s/5 s). (3.26)

If the therm al boundary layers are assumed to be 100 km thick, as is reasonable for the

mantle, then the e-folding time is found to be 4.6 Gyr. This is much longer th an a single

convective overturn tim e (around 100 Myr) and is, in fact, roughly the age of the Earth.

A similar expression has been previously derived by Jeanloz and Morris [1986], and the

fact tha t the therm al adjustm ent time is longer than a single turnover time in numerical

simulations was also discussed by Daly [1980]. This separation of timescales was also the

basis of the early param eterized convection models of Sharpe and Peltier [1978] th a t were

employed to investigate the nature of planetary therm al histories controlled by the convec

tion process. T hat the timescale for adjustm ent is of the same order of m agnitude as the

age of the E arth argues th a t the present-day E a rth is unlikely to be in a sta te o f therm al

statistical equilibrium. As such, a significant fraction of the heat th a t is escaping the sur

face today is likely to be due to the secular cooling of the planet. Note tha t for the case

of an incompressible, constant-property cube (3.26) reduces to dS/2K. Figure 3.9 shows

four tim e series of the internal tem perature from my numerical model (solid lines). I also

show exponential decay curves of the form of (3.25) where I have calculated A from (3.26)

using the average values of 5C and Ss from the numerical simulation and c(oo) and R are

arrived a t by least squares fitting. I define 5C and 5S as the distance from the boundary a t

which point the derivative of the geotherm falls to 10% of its value a t the boundary. The


Aver

age

Tem

pera

ture

(K

)

3.3. Results 2: Parameterized Model for “W hole M antle” and ’Layered Mantle’ Convection 81

2500

2300

B2

2 10 0B3

- B4-

B11900 2.5x10'5000 1.5x10 2x10'10

Time (Ma)

Figure 3.9: The volume average tem perature as a function of time in my numerical model (solid lines) and the predictions of ( 3.25) (dotted lines) for the four indicated m odel runs.


3.3. Results 2: Parameterized M odel for “Whole Mantle” and ’Layered Mantle’ Convection 82

simulations from which, these average tem peratures are derived are indicated in Figure 3.9.

I t can be seen th a t in all cases, (3.26) does a very reasonable job of predicting the tim e

for the models to reach equilibrium . The e-folding tim es for the curves corresponding to

simulations B1-B4 are 3.78, 6.48, 7.32, and 10.33 Gyr respectively, and it can be seen th a t

the tim e to equilibrate increases as the mean viscosity increases.

3.3.2 Interior Temperatures

Solutions of the form o f (3.25) implicitly assume th a t the boundary layer thicknesses do not

vary significantly w ith tim e which is not entirely accurate. Also, it is far more satisfying

to proceed with an a prio ri expression for 5C and 5S than to rely on typical values from the

numerical model or from the real Earth. In order to do th is, I will appeal to the boundary

layer nature of high Rayleigh num ber convection [e.g., Howard, 1966], in which I assume th a t

the boundary layers become unstable and plumes form w hen the boundary layer Rayleigh

num ber reaches a critical value, Racr\t . If this is the case then the two boundary layer

Rayleigh numbers are likely to be the same and hence I will assume that

R ac — gcC*cPcATcSc/ (KcVc) = R as = 9s&sPs^ /Rs8s / (kst}s ) = Racrh - (3.27)

The boundary layer thicknesses can then be com puted in terms of the boundary layer

tem perature contrasts and the unknown Racrit which will be dem onstrated to cancel out in

many im portant situations. This assumption of the equality of the two therm al boundary

layer Rayleigh numbers has been used previously to predict the internal tem perature for a

constant-property Boussinesq fluid as a function of curvature for bo th cylindrical [Jarvis ,

1993] and spherical [ Vangelov and Jarvis, 1994] geometry and was shown to give excellent

results in both cases.

Substitution for 5S and 8C into (3.24) leads to

= A c (A T + T s ~ c ) 4 /3 “ As{-° 4)3 ~ T s ) 4 / 3 + xM m ■ (3-28)

In the above, Ai = kiAi[(giOiPi) / (KirjiRadt)]1 3 ■ The steady sta te tem perature, for the


3.3. Results 2: Parameterized M odel for “W hole M antle” and ’Layered Mantle’ Convection 83

case w ith no internal heating (x = 0 ) can then be obtained analytically and has the form

T(oo) = { A T + Ta[ 1 + {As/A cf /*}}/[ 1 + 0s (As/A c)3/4] (3.29)

Only the ratio of A c to As appears in (3.29) and as such the unknown f?aCnt as well as

the mean values of m antle properties cancel out. The only param eter that is not an input

param eter in my numerical model is <ps which depends on the thickness of the two therm al

boundary layers. Fortunately, the function <f> is relatively slowly varying and any reasonable

estimates of 5C and 5S yield sim ilar results. In Figure 3.10 I compare internal tem peratures

calculated from (3.29) w ith geotherms calculated a t the end of long simulations using my

spherical axisymmetric num erical model with differing dep th dependencies of viscosity and

CMB tem peratures. The effects of phase transitions axe absent, and I indicate the model

rim s in Figure 3.10. For the case of the right-most geotherm (in which the viscosity is

constant and the core-mantle boundary has a tem perature of 4000 K) I have included four

geotherms from the numerical model which vary only in the m agnitude of the average

viscosity. It is dem onstrated th a t the internal tem peratures do not vary strongly w ith

the magnitude of viscosity in the absence of internal heating and phase transitions, as is

predicted by (3.29). One can also see in Figure 3.10 th a t run A4 w ith VP 1 has the same

CMB tem perature (2785 K) as run A3 with VP 3 bu t has a considerably warmer internal

tem perature. T h a t an increase of viscosity w ith depth results in a lower internal tem perature

has been previously noted by G um is and Davies [1986], Hansen et al. [1993], and Tackley

[1996] and is predicted by (3.29). I can see th a t this is due to the lower boundary layer being

considerably more viscous. In order for it to be equally unstable and to allow the same heat

flow as the top boundary layer, the tem perature drop across the bottom boundary layer must

be s ig n ifican tly greater. Note th a t although the num erical models were rim for many billions

of years of dimensional tim e, the very long therm al adjustm ent tim e implied by (3.26) makes

it difficult to ru n for m any e-folding times, and hence the model may still not be in perfect

statistical equilibrium. This may account for some of the discrepancy between the curves.

For the most paxt, the agreement is very good. V ariation of interior tem peratures due to

other depth-dependent properties can be sim ilarly predicted using (3.29). I also point out

th a t the interior average tem perature gradient is very close to adiabatic, even in cases when


Radi

us

(km

)


6400

5400

4400

B1-B5A3A1 A4

34003000 40001000 2000

Temperature (K)

Figure 3.10: Geotherms from my numerical model (solid lines) and predictions of ( 3.29) (dashed lines). The VPs corresponding to the different calculations are indicated.



there is a significant increase in viscosity with depth (VPs 2 and 3). This result appears

to be in conflict w ith some previous studies [e.g.. van den Berg and Yuen, 1998]. Tackley

[1996] also used a boundary layer critical Rayleigh num ber criterion to develop the scaling

of boundary layer thicknesses and tem perature drops for depth-dependent, compressible

convection in three-dim ensional, Cartesian convection. Tackley found tha t the results of

the scaling arguments and th e numerical simulations differed significantly. This difference

may be due to the use of lower system Rayleigh numbers in those calculations.

Although I cannot explicitly solve for c when there are in ternal heat sources present,

I can nevertheless ob ta in some information as to how the in terior tem perature varies w ith

the mean viscosity in the presence of internal heat sources. I f I w rite the surface and CMB

viscosity as the product of a n average (fj = (rjc + Vs)/^) and a variation (si = (rjs — t}c) / ( t]s -t-

77c)) and I assume a steady s ta te solution, I can write (3.24) in the form

(i - T*>4/3 = d - t o ^ (Ar + T ' ~ c)4/3 + <3-3°)

Here, A' = Ai (ViR^crit) ^ ■ I t will be noted that when x *s nonzero, c depends on the m ean

viscosity and on the boundary layer critical Rayleigh num ber. Numerical solutions of (3.30)

indicate tha t c increases w ith fj when there is a nonzero Xi in qualitative agreement w ith

results to be presented in section 3.4.5.

3.3.3 Heat Flow

Using (3.22), (3.23), and (3.29), I can derive an expression for the heat transferred by a

convecting system in s ta tistica l equilibrium in the absence o f in ternal heat sources, namely

Q = Qc = Qs = {[A Tcf>s - T s( I - 4>s)}/[1/A zJa + ( l / A 3c/*)<f>s]}4/3. (3.31)

Expanding the viscosity as in the previous section I arrive a t the following result:

Q = {[AT<j>s - Ts( 1 - <£s) ] /[ ( l + s 0 1/4 /(A s/3/4) + (1 - s l ^ / i A S ^ c P S ^ / m a c r i t ) 1' 3.

(3.32)


3.3. Results 2: Parameterized M odel for “W hole Mantle” and ’Layered Mantle’ Convection. 86

From (3.32) I can see th a t in the absence of internal heating and phase transitions the

heat transferred varies roughly as the 4 /3 power of the tem perature difference as long as

A T » Ts and varies as the —1/3 power of the mean viscosity. This is in agreement with

boundary layer theory which predicts th a t the Nusselt num ber should scale like the Rayleigh

number to the 1/3 power [e.g.. Turcotte and Oxburgh, 1967; Turcotte and Schubert, 1982].

W hat is new in (3.32) is a prediction of the effects of depth-varying parameters and

finite compressibility on heat transfer. The two terms in the denom inators of (3.31) and

(3.32) can be thought of as th e im pedance to heat flow of the surface and CMB therm al

boundary layers and the heat flow takes on its minimum value when the two impedances

are equal. In Figure 3.11a I show plots of the heat transfer as a function of the depth

variation of viscosity com puted from (3.32) for four different cases. T he heat flow is de

fined only up to a m ultiplicative constant (due to the unknown R a crn) so I normalize each

of these curves such th a t the m inim um value is 1 . The solid and long-dashed and short-

dashed and dotted fines represent the relative efficiency for models w ith and without the

effects of compressibility in spherical and Cartesian geometry, respectively. The compress

ible model includes the effects o f all depth-dependent quantities as well as the effects due

to an adiabatic tem perature gradient in the interior. We see th a t for the constant-property,

incompressible C artesian m odel the minimum heat flow occurs when viscosity is constant

with depth and convective efficiency increases symmetrically when viscosity increases or

decreases with depth. A lthough the effects of the adiabatic tem perature gradient act to

increase the surface impedance, the effects of the Earth-like depth-dependent quantities are

greater and act to decrease th e surface impedance. This results in the heat flow being a

minimum for viscosity th a t decreases strongly with depth. The effect of spherical geometry

is to decrease the surface im pedance (since the surface boundary has a much larger area than

the CMB), and hence the spherical incompressible model has its minimum heat flow when

viscosity increases strongly w ith depth. Finally, when bo th the effects of compressibility

and spherical geometry are included, the result is a model w ith a minimum heat flow when

viscosity decreases slightly w ith depth; hence the effects due to compressibility are slightly

greater than those due to spherical geometry. W hat is most significant in Figure 3.11a is

th a t convection in which the viscosity increases strongly w ith dep th is significantly more ef

ficient in the radial tr a n sfer of heat th an constant viscosity convection if the mean viscosity


3.3. Results 2: Parameterized M odel for “Whole Mantle” and ’Layered M antle’ Convection 87

£o

(0<DEC

1.8 Spherical, Compressible— — Spherical, Incompressible Cartesian, Compressible Cartesian, Incompressible

1.3

V.P.2 V.P.3,4 V.P.10.8

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20

10

-1 0 0 100 200 300 400 500 600 700 800Excess Temperature (K)

Figure 3.11: a) Convective efficiency as a function of the dep th variation of viscosity for the cases of spherical and C artesian geometry, with and w ithout the effects of finite compressibility as predicted by (3.32). Arrows indicate the viscosity variations corresponding to model calculations, (b) The heat flow in a numerical model w ithout phase transitions or internal heating as a function of the tem perature above the final equilibrium tempera tu re (solid lines) for model ru n B3. The arrows indicate the direction in which time is progressing. The dotted lines are the predictions of (3.33).

R e p ro d u c e d with perm iss ion of th e copyright ow ner. F u r the r reproduction prohibited without perm iss ion .

3.3. Results 2: Parameterized M odel For “W hole Mantle” and ’Layered M antle’ Convection 88

Table 3.2: Comparison of H eat FLow From Numerical and Param eterized Models

R un VP AT, K si 77, IO 21 P a s Parameterized, T W Numerical, TW

A1 2 1657 -0.955 4.80S 25.0 26.48 ± 1.1

A2 3 1657 -0.903 5.58-4 2 2 . 0 2 2 . 0 ± 0.26

A3 3 2485 -0.903 5.58-4 39.3 37.64 ± 2.19

A4 1 2485 0 . 0 4.0 35.2 31.78 ± 1.18

B1 1 3700 0 . 0 4.0 61.4 62.2 ± 0 .6

B2 1 3700 0 . 0 15.3 39.3 38.75 ± 0.25

B3 1 3700 0 . 0 28.0 32.1 32.61 ± 0.6

B4 1 3700 0 . 0 8 8 . 0 21.9 2 2 . 8 ± 0 . 2

B5 1 3700 0 . 0 880.0 1 0 .2 1 0 . 6 db 0 .6

is the same in both cases. This fact wilil be born out in the sim ulations presented in section

3.4.1.

Table 3.2 summarizes a com parisom of the heat flow calculated from (3.32) and from

my numerical model. The com puted haeat flow from the numerical model is arrived at by

averaging the last several billion years o f a heat flow time series th a t is close to equilibrium.

An average of both the CMB and the surface heat flow is perform ed and then these two

num bers axe in turn averaged in order t o arrive a t the final heat flow. T he difference between

the averages of the surface and CMB haeat flow is a measure of how far the system is from a

stationary state and is used as the m easurem ent error. Run A2 in tab le 3.2 is used to tune

the value of the unknown i?acrit in th«e parameterized model. /?acrit is found to be 14.8,

and all of the other parameterized m odlel calculations were then perform ed w ith this value.

These results are compared w ith those- of the numerical model. T he param eterized model

reproduces the observed trends and giwes quite reasonable quantita tive agreement.

I t is also useful to obtain an estim arte of the contribution to the surface heat flow of the

heat release due to the seculax cooling *of the mantle. I can obtain a n estim ate of the extra

surface heat flow delivered by a m antle “th a t is above the statistical equilibrium tem perature


3.3. R esults 2: Parameterized M odel for “W hole Mantle” and ’Layered M antle’ Convection 89

by w riting the surface heat flow as

QM + Q's = A s (A T /9 + ATS')4/3 (3.33)

in which Q%q and AT/*7 are the tim e-averaged statistical equilibrium values of Qs and A Ts,

and Q's and A T ' represent deviations from these. When linearized, th is takes on the form

Qs = (4/3) /A T /^ A T /. (3.34)

If I use 36 TW as an estim ate o f Q%q and 1400 K as an estim ate o f A T /9, I obtain the

result th a t the surface heat flow increases roughly 3 TW for every 100 K increase in the

tem perature near the surface of the m antle. Figure 3.11b compares the predictions of (3.33)

(dotted lines) with the results of a num erical calculation (solid line). In Figure 3.11b we

can observe the progress of the num erical model toward statistical equilibrium where there

is a "basin of attraction” in heat tram sport-tem perature space. The param eterized model

captures the long timescale variation o f heat transport with tem perature.

3.3.4 Effects of Layering

Let us next consider the statistical equilibrium state, in the absence o f internal heating, in

which

Qs = Q e/P = Qc■ (3.35)

A subscript e indicates a quantity evaluated a t the position of the endotherm ic phase

transition or, more generally, a level a t which the flow is impeded. Q e represents the heat

transpo rted across the 660-km horizon by conduction, while the param eter (3 is used to

investigate the effects of incomplete layering and is equal to the ra tio of the heat carried

by conduction to the to ta l heat tran sp o rted across this boundary. T he param eter (3 takes

on th e value 1 when the flow is com pletely blocked by the phase transition and is very

close to 0 in the whole m antle scenario. In section 3.5.2 I will discuss the variation of (3

w ith CLA P th a t is observed in my num erical simulations. The regions above and below

the phase transition will have ad iaba tic tem perature profiles. The adiabatic p a rt of the


3.3. Results 2: Parameterized Model for “W hole Mantle” and ’Layered M antle’ Convection 90

geotherm is given by

< T > = ci <f>i(r), r < r e — 5e, 0 j(r) = e x p (- f — dr'),J r c+5c °P

< T > = C u (f)u(r), r > r e + 5e, d>u (r) = e x p { - f — dr'), (3.36)Jre+6e Cp

in which subscripts I and u correspond to quantities associated with the lower and upper

m antle and re and 5e are the equilibrium position of the 660-km depth endotherm ic phase

transition and half of the internal therm al boundary layer thickness. In this case, the

tem perature drops across the lower, internal, and upper thermal boundary layers are given

by

ATc - A T + TS — ci, A T e = (q <pie — Cu), A T s - Cu 4>us ~ Ts , (3.37)

where (f>ie = <t>i(re — Se) and (pus = <pu{rs — Ss). Following a similar procedure to th a t used

to derive (3.29), I can solve for q and Cu to ob tain

Q = ( A r [ I + (j3As/ A e)V U u s}+ T s [ I + (A s/ A c)3/4 + W A s/ A e)z'U us]}

{1 + (As/ A c)V 4<f>iect>us + {!3As/ A ef ' 4<j>us} - 1, (3.38)

and

C„ = {A T fre + Ts[4>le + ({3AS/ A ef / 4 + {As/A c)z/ 4<t>le}}

{1 + (As/ A c)3/Uie<ûs + (/5AsM e)3/V u J _1- (3.39)

W hen (3 has the value zero, (3.38) and (3.39) reduce to the whole mantle case (3.29) since

<Plexus = 0s bi th a t case. In Figure 3.12 I display geotherms generated from (3.38) and

(3.39) and compare these with results from num erical models tha t were ru n w ith the same


3.4. Results 3: Scaling o f Time-Averaged Quantities in Time-Dependent Convection 91

parameters. It is emphasized tha t /? is a param eter th a t is measured from, ra ther than

employed in, the numerical model, and so oue would have to estimate f3 in order to use (3.38)

and (3.39) to predict the internal temperature. Figures 3.12a and 3.12b show geotherms

for values of (3 of 0.2 and 0.9, respectively. The ratio of the surface heat flow w ith /3 7 0

(Q l , layered convection) to th a t with /3 = 0 {Q\v, whole mantle convection), assuming the

critical boundary layer Rayleigh numbers to be the same in both cases is given by

^ = {[(1/AS)3/4 + ( l/^ c)3/4 e us] /[ ( l / ls)3/4 + (/? / A ef f ^ us + (1 / A c)3^leCf>us]}A/3.

(3.40)

I will discuss the effects of layering on heat flow in detail in section 3.5.2.

3.4 Results 3: Scaling of Time-Averaged Quantities in Time-

Dependent Convection

In order to investigate possible scaling in my numerical model I will vary the volume average

of viscosity 770 w ith all other parameters held fixed for a number of different viscosity profiles,

internal heating rates, and slopes of the endothermic phase transition. By varying 770 I am

then varying the effective Rayleigh number of the system when all other parameters are kept

constant. This is done because there is ambiguity as to how an effective Rayleigh num ber

should be defined when interned heating is present w ith an isothermal lower boundary

and there are variable properties. The appropriate definition of a nondimensional heat

flow, or Nusselt num ber, is also ambiguous and as such I will investigate the scaling of the

dimensional surface heat flow. I will similarly use dim ensional quantities when investigating

scaling in surface velocity, surface boundary layer thickness, and 660-km depth mass flux.

Using dimensional quantites gives the added advantage of easy comparison with real E a rth

observations. In order th a t a unique value be obtained for each of these quantities it is

im portant to first achieve a statistical equilibrium sta te and then average quantities over

long time periods. The dimensional quantities will be fit to functions of the form

Y = m YrfQY, (3.41)


3.4. Results 3: Scaling o f Time-Averaged Q uantities in Time-Dependent Convection 92

1000 2000 3000 40006400

5400

a5013

4400

H2 J1

34001000 2000 3000 4000Temperature (K) Temperature (K)

Figure 3.12: (a) Geotherms calculated from my num erical model (solid line, ru n H I) and from ( 3.38) an d ( 3.39) (dashed line) when 0 = 0.2. (b) Geotherm from my numerical model w ith constant viscosity and i / o = 4 x 1021P a s w ith 0 = 0.9 (solid lines, rim J l ) , and the prediction o f (39) and (40) for 0 = 0.9 (dashed line).



where p y and m y are the scaling exponent and prem ultiplier of the scaling relation for a

particular quantity Y . I will indicate the scaling exponents and model runs in the figures,

and in Table 3.3 I list all of the scaling exponents and premultipliers calculated in this

study.

3.4.1 Scaling of Surface Heat Flow

For the case of surface heat flow, using the tim e average statistical equilibrium value has

the effect of reducing the contribution due to secular cooling to zero. The E a rth ’s m antle

is likely to be cooling, based on the long timescale for equilibration derived in section 3.3.1,

and as such, my m easured heat flows should be seen as lower bounds. For the purpose of

comparison w ith th e convective heat flows the conductive surface heat flows for the 20, 10,

and 0 TW internal heating cases are 10.9, 6.3, and 1.67 T W , respectively. I point ou t th a t for

the cases tha t include the influence of internal heat sources the conduction solution results

in a heating of the core, as do the high-viscosity, strongly internally heated convection rims

D4, E4, and F4. I choose to measure the internal heating rate as a constant dim ensional

heating rate (in T W ) ra ther than use the Urey ratio (the ratio of internal heating to surface

heat flow). The U rey ratio can be easily calculated from Table 3.1 and varies from 0 to 1 for

different calculations. Figure 3.13a displays the results of calculations using VP 1, 3, and 4

with 20 TW of heat d istributed uniformly throughout the mantle and using four different

values of tjq in each case. T he effects o f the phase transitions at 400 and 660 km dep th axe

also present w ith the Clapeyron slope of the 660-km phase transition taking on a value of

—2.8 M Pa K _1. I include the least squares best fit th rough the four data points and as will

be observed, the lines are very close to being parallel w ith slope of -0.19 which is, in tu rn ,

the scaling exponent (p q 3 , see equation (3.41)). The scaling exponent is a m easure of how

convective efficiency changes w ith rjo- Since the three viscosity profiles are quite different,

this indicates th a t for the same internal heating ra te and Clapeyron slope of the 660-km

depth phase transition, the scaling exponents are not sensitive to the depth dependence of

viscosity, in agreem ent w ith Londe and Davies [1985] and Gumis and Davies [1986], who

investigated this dependence in the circumstance in which phase transitions are absent. The

lines are clearly offset from one another, however, indicating tha t the prem ultipliers (m ga),

which axe measures of the heat transport efficiency of a given viscosity profile, axe different.


3.4. Results 3: Scaling o f Time-Averaged Quantities in Tim e-Dependent Convection 94

Table 3.3: Summ ary of Scaling Param eters

The values of the prem ultipliers are for values of 770 m easured in units of 1021Pa s. The

units of the mQ3, m Usl rrif, and m s3 are TW . cm /yr, kg(m2y r)-1 , and km respectively.

Series PQs Pus PI PSsB -0.320 ± 0.005 -0.62 ± 0.02 -0.61 ± 0.02 0.341 ± 0.009

C -0.288 ± 0.002 -0.49 ± 0.02 -0.49 ± 0.04 0.346 ± 0.004

D -0.195 ± 0.006 -0.652 ± 0.004 -0.55 ± 0.03 0.310 ± 0.02

E -0.18 ± 0.1 -0.63 ± 0.04 -0.49 ± 0.05 0.338 ± 0.006

F -0.19 ± 0.02 -0.61 ± 0.05 -0.54 ± 0.02 0.31 ± 0.02

G -0.248 ± 0.005 -0.56 ± 0.02 -0.6 ± 0.1 0.33 ± 0.02

H -0.23 ± 0.01 -0.42 ± 0.04 -0.36 ± 0.04 0.315 ± 0.001

I -0.16 ± 0.02 -0.42 ± 0.07 -0.04 ± 0.07 0.30 ± 0.02

m Q* mus r r i f m s.

B 87.4 ± 1 8.5 ± 0.4 123.7 ± 8 42 ± 0.4

C 76.7 ± 0.4 3.5 ± 0.3 51 ± 7 38.9 ± 0.4

D 69 ± 2 5.75 ± 0.09 66 ± 8 48 ± 3

E 77 ± 4 17 ± 3 74 ± 17 27.2 ± 0.7

F 83 ± 6 12 ± 4 98 ± 7 30 ± 3

G 82 ± 2 9.1 ± 0.8 155 ± 77 34 ± 3

H 49 ± 2 1.8 ± 0.3 15 ± 2 40.5 ± 0.2

I 24.4 ± 0.6 0.9 ± 0.2 1.2 ± 0.3 50 ± 3


3.4. Results 3: Scaling o f T ’ime-Averaged Quantities in Time-Dependent Convection 95

observed

postglacial rebound

□ Runs D l—D4. p .=—0.195 • Runs E l— E4. pa-=—0.18

Runs FI—F4, p<t=—0.1910

<e<u« x Runs Cl—C5, P(^=—0.288

a Runs Gl—G4, p4= —0.248 a Runs Dl—D4. p<4= —0.195

10

100* Runs B l— B5, p^=—0.318 x Runs Cl—C5, p4= —0.288

10■ jRuns HI—H3, p< =—0.23 * Buns J l—J4, Pu>= —0.15

10;

Figure 3.13: (a) Logarithm ic variation of surface heat flow as a function of the volume average of viscosity (770) for three different radial viscosity profiles, all with the same ra te of internal heating (runs D1-D4, E1-E4, and F1-F4). The observed surface heat flow and an upper bound on viscosity based on postglacial rebound (PGR) are indicated by the do tted and dashed lines, respectively, (b) Logarithmic variation of heat flow as a function of 770 for three cases w ith different ra te s of internal heating (runs C1-C5, D1-D4, and G1-G4). (c) Logarithmic variation of h e a t flow as a function of t]q for four cases with different C lapeyron slopes of the endotherm ic phase transition (runs B1-B5, C1-C5, H1-H3, and J1-J4).


3.4. Results 3: Seeding o f Time-Averaged Quantities in Tim e-Dependent Convection 96

In particular, the two profiles w ith the same depth increase of viscosity (VPs 3 and 4)

deliver very similar heat flows for the same 770, while the constant viscosity case (VP 1)

delivers considerably less, in agreement w ith the results in Figure 3.11a. The m agnitude of

the scaling param eter is also less than that predicted for a Boussinesq fluid w ith no in ternal

heating which is 1/3. In Figure 3.13a we also indicate an upper bound on the mean viscosity

from postglacial rebound inversions (see Peltier [1998] for a recent review) as well as the

present-day observed surface heat flow which I take to be 36 T W when the component due

to radioactive sources in the crust is subtracted (see Pari and Peltier, [1998] for a detailed

discussion).

Figure 3.13b displays the logarithmic variation of the heat flow w ith 770 for three different

rates of internal heating. Two different viscosity profiles were used in calculating the points

in Figure 3.13b. Since the d a ta presented in Figure 3.13a indicate th a t the scaling exponent

is not sensitive to dep th variations in viscosity, the difference in the depth dependencies

of viscosity will not affect conclusions regarding the variation of the scaling exponent w ith

internal heating rate. It is seen th a t for each internal heating ra te the heat transported by

the system does scale w ith 770 and the magnitude of the scaling exponent is seen to decrease

w ith the addition of in ternal heating. This is in agreement w ith the results of van den Berg

and Yuen [1998], who dem onstrated tha t for steady sta te convection in Cartesian geometry,

and using tem perature and pressure dependent rheology, the scaling param eter decreased

in m agnitude in the presence of internal heat sources. Similarly, Sotin and Labrosse [1999]

dem onstrated tha t the dependence of the heat flow on the Rayleigh num ber should decrease

w ith the addition of in ternal heating. Also of interest is th a t the lines tha t represent the

best fits to the data for the two calculations tha t were perform ed w ith constant viscosity

intersect for 770 = 3.24 x 1021P a s. This implies tha t for values of 770 less than 3.24 x 1021P a s,

addition of internal heating actually results in a decrease in surface heat flow. I have not run

my model w ith internal heating a t sufficiently low values of viscosity to determine w hether

the scaling holds for very low viscosity, however, so it is possible th a t there is a break in the

scaling law before this happens. The results of Sotin and Labrosse [1999] suggest th a t the

internally heated case should converge to the case w ith no internal heating for sufficiently

low values of the viscosity. In all cases I attem pted to have the highest heat flow case a t

roughly 36 TW as is appropria te for the heat flow delivered to the surface by the present


3.4. Results 3: Scaling o f Time-Averaged Quantities in Tim e-Dependent Convection 97

day convecting mantle.

Figure 3.13c sum m arizes the m easured heat flow in a constant viscosity, purely basally

heated model, w ith no phase transitions, as well as with both phase transitions present with

values of the endothermic phase transition Clapeyron slope of -2.8, -6.5, and —12 M Pa K-1 .

T he la tte r values of the C lapeyron slope are considerably greater th an is indicated by high-

pressure experiments. These sim ulations are intended, however, to investigate effects due to

very strong layering. I will argue fu rther in section 3.6 tha t o ther physical effects tha t axe

not included in these sim ulations may act to increase the effectiveness of an endothermic

phase transition of a given C lapeyron slope in inducing layering. In Figure 3.13c I can

see th a t strong layering has the effect of reducing the dependence of the heat flow on the

m agnitude of the viscosity. T h e scaling exponent is seen to decrease roughly linearly with the

C lapeyron slope of the phase transition . Very strongly layered flow is also seen to be capable

of significantly reducing the heat flow through the mantle and hence is one mechanism by

which an Earth-like heat flow can be obtained for mantle viscosity low enough to satisfy

the glacial isostatic ad justm ent constraints. For many of the solutions w ith large values

of the Clapeyron slope the flows are characterized as being extrem ely time-dependent.

Some show almost perfect layering for extended periods of tim e punctuated by intense

avalanche events. I also poin t out th a t the heat flow is not strongly affected by a phase

transition w ith a Clapeyron slope of —2.8 M Pa K - i which is a value th a t is consistent with

high pressure experim ental observations [Chopelas et al., 1994] of th e olivine component of

m antle mineralogy.

In section 3.4.5 I give an explanation for the variation in the heat flow scaling parameters

w ith the internal heating ra te and the Clapeyron slope of the phase transition.

3.4.2 Scaling of Surface Velocity

In Figure 3.14a I show a p lot o f the azim uthally and tem porally averaged surface velocity

as a function of the m ean viscosity for three cases with different viscosity profiles bu t all

having 20 TW of internal heating distributed evenly throughout the mantle. I t can be seen

th a t as in the plots for heat flow, the surface velocity scales w ith 770 and the exponent does

not vary significantly w ith th e dep th dependence of viscosity and has a value near 0.63. For

the same average viscosity it can be seen th a t the two profiles w ith the same to ta l depth


3.4. Results 3: Scaling of Time-Averaged Quantities in Time-Dependent Convection 98

10

0.1 fa Runs Dl— D4, p^=—0.652 Runs El—E4. p ^ —0.63

- a . Runs FI—F4, Po.=—0.61 a-£\ 0.01 Bo

10

o'aJ>a>o<a

- 0.1fx Runs Cl—C5, p^=—0.49 Runs G1-G4, Pa.=—0.56

-□ Runs Dl—D5, p^=—0.6520.0110

* Runs B l—B5, p^=—0.62 x Runs Cl—C5, ptJi=—0.49

0.1Runs HI—H3, p^=—0.42

- a Runs J l—J4, p^——0.42

Figure 3.14: (a) Logarithmic variation of surface velocity as a function of tjo for three different VPs (runs D1-D4, E1-E4, and F1-F4). (b) Logarithmic variation of surface velocity as a function of ijo for three cases w ith different rates of internal heating (runs C1-C5, D1-D4 and G1-G4). (c) Logarithmic variation of surface velocity as a function of rjo for four cases w ith different Clapeyron slopes of the endotherm ic phase transition (runs B1-B5, C1-C5, H1-H3, and J1-J4).


3.4. Results 3: Scaling o f T im e-A veraged Quantities in Time-Dependent Convection 99

variation have very similar velocities, while the constant viscosity case is considerably more

sluggish. This is quite intuitive sincae for the same average viscosity, V P 3 and 4 have a

considerably lower surface viscosity tbaan does VP 1. Figure 3.14b shows th e surface velocity

as a function of the mean viscosity f o r models w ith different internal heating ra tes. It can

be seen th a t the magnitude of the sca lin g exponent for the surface velocity increases with

internal heating. Of interest in Figure 3.14b is th a t the fits to the lines for V P 1, w ith 20 TW

of internal heating and no internal heaating cross over around 1.5 x 1022P a s, dem onstrating

th a t for viscosity lower than this, surrface velocities increase w ith the add ition of internal

heating. Figure 3.14c displays the sca lin g of surface velocity with 770 using different values

of the 660-km Clapeyron slope. It csan be seen tha t the scaling exponent decreases quite

rapidly as a function of the strengtlh of the endothermic phase transition and saturates

near —6.5 M Pa K_l at an exponent around -0.42. In the absence of phase transitions, the

surface velocity is seen to have an expoonent which is reasonably close to the boundary layer

theory value of 2/3.

3 .4 .3 S c a lin g o f t h e 6 6 0 -k m PM ass F lu x

In Figure 3.15 I present similar plots for the absolute value of the mass flux transiting 660

km depth. It will be noted th a t limear fits are not quite as good for mass flux as they

are for surface heat flow and surface velocity. This is probably due to th e extrem ely time-

dependent nature of this quantity. Mlany trends are still clearly observable, however. In

Figure 3.15a it can be seen th a t the mass flux crossing the endotherm ic phase transition

horizon is considerably less for the ca.se w ith constant viscosity than for the two cases with

an increase of viscosity w ith dep th (W Ps 3 and 4). This does not indicate th a t th e flow is

more strongly layered in this case, herwever (I dem onstrate this more fully in section 3.5.2);

it simply reflects the fact th a t for the sa m e m ean viscosity the flow is less vigorous a t 660 km

depth for constant viscosity convectiom as opposed to convection in which viscosity increases

w ith depth. It will also be noted th a t; the variation of the 660-km depth m ass flux w ith the

mean viscosity is not a strong fu n c tio n of the depth variation of viscosity. In Figure 3.15b

I present three simulations w ith d iffering rates of internal heating. I po in t ou t th a t the

two simulations performed with co n s tan t viscosity have very similar mass fluxes despite the

fact th a t the rate of internal heating is very different. This might seem to be a t odds with


3.4. R esults 3: Scaling of Time-Averaged Q uantities in Tim e-Dependent Convection 100

10

-□ Runs D l—D4, p*=—0.55

- • Runs E i—E4, Pr=—0.49 -*• Rims FI—F4, pf= —0.54

slab flux

10

-x Runs Cl—C5, Pf=—0.49

- a Runs Gl—G4, p,=—0.6 □ Runs Dl—D5, p*=—0.55

* Runs B l— B5. pf=—0.61 x Runs Cl—C5, Pf=—0.49 ■ Runs HI—H3, pf= —0.36* Runs J l—J4P pf= —0.04

10

77o(Pa s)

Figure 3.15: (a) Logarithmic variation o f th e mass flux a t the position of the endothermic phase transition as a function of 770 for three different VPs (runs are indicated), (b) Logarithm ic variation of mass flux a t th e position of the endotherm ic phase transition as a function of 770 for three cases w ith different rates of internal heating, (c) Logarithmic variation of mass flux at the position o f the endotherm ic phase transition as a function of 770 for four cases w ith different C lapeyron slopes of the endothermic phase transition.



the results of Solheim and Peltier [1994a], who showed th a t the flow becomes increasingly

layered w ith an increase in internal heating. The param eter tha t was used by them to

measure the degree of layering was P = 1 — / l / / i v , in which / l and f w are the 660-km

depth mass fluxes w ith and w ithout the effects of the endothermic phase transition. W hen

internal heating is present, the near-surface flow becomes more vigorous, and the flow near

the CMB becomes less so because of the increased internal tem peratures. For this reason,

when the effects of the phase transitions are turned off, the mass flux across the 660-km

horizon (which is much closer to the surface th an the CMB) is greater when there is internal

heating than when there is none. As such, even though the mass fluxes for these two sets

of simulations w ith different rates of in ternal heating indicate similar mass fluxes when the

phase transitions axe turned on, the case w ith strong internal heating represents a greater

decrease in mass flux from the case w ith no phase transitions and, as such, P is greater.

Since there are only two sets of simulations, it is unclear as to whether it is a general result

th a t two simulations with the same viscosity bu t different rates of internal heating will have

similar mass fluxes in the presence of axx endotherm ic phase transition. T he calculation

performed w ith VP 5 further indicates th a t w ith a Clapeyron slope of —2.8 M Pa K -L

the mass flux for a given mean viscosity is considerably greater when viscosity increases

w ith depth than when it is constant. In Figure 3.7 I also saw that internal heating does

not strongly affect the mass flux distribution. T h a t the scaling exponent for this case is

considerably greater than for the two constant viscosity sets of simulations is not significant

given the scatter of the data.

In Figure 3.15c I present four sets of calculations differing only in the m agnitude of the

Clapeyron slope of the endothermic phase transition and I note tha t the scaling exponent

decreases roughly linearly with this quantity. For the extremely large C lapeyron slope of

—12 M Pa K -1 , the exponent itself is zero to the accuracy of the calculation, indicating

th a t the increase in the ability of an endotherm ic phase transition to induce layering with

Rayleigh num ber roughly balances the increase in mass flux due to an increase in convective

vigor. It is also seen that in the absence of phase transitions, the mass flux scales with a

power law exponent —0.63 which is close to the boundary layer theory prediction of —2/3

for velocity. I have also indicated the “slab flux” for the Earth, estim ated assum ing slab

subduction of 3 km2 yr-1 , a slab thickness of 100 km and a density of 3300 kg m -3 . I note



th a t only the models calculated w ith CLAP=-12 M Pa K -1 or with extremely large viscosity

have less mass flux than the slab flux. It is of interest tha t models w ith a viscosity tha t is

sim ilar to th a t inferred from postglacial rebound observations must be very strongly layered

in order tha t the mass flux crossing the 660-km depth horizon equal the observed surface

slab flux. I will further discuss issues pertaining to the mass flux a t 660 km depth in section

3.5.2.

3.4.4 Scaling of the Surface Boundary Layer Thickness

In Figure 3.16 I display the logarithmic variation of the surface boundary layer thickness

w ith t]q for the same cases as in section 3.4.3. I define the surface boundary layer thickness

as I have done before to be the depth a t which the magnitude of the slope of the geotherm

falls to 10% of its value at the surface. It will be observed that for all cases, regardless

of the depth dependence of viscosity and the internal heating rate, the surface boundary

layer thickness scaling exponent (pgs) is close to the “classical” value of 1/3. Some trends

are visible, however. There is a progression toward thinning of the surface boundary layer

for viscosity profiles th a t have a smaller viscosity near the surface, as will be noted from

Figures 3.16a and 3.16b. There is also a slight decrease in pgs w ith increasing Clapeyron

slope. As can be seen in Figure 3.17 (which I will discuss in section 3.4.5), the CMB therm al

boundary layer thickness varies very strongly w ith internal heating rate, and the therm al

boundary layer in this region disappears entirely in the very viscous, strongly internally

heated calculations. T hat pg3 deviates very little from 1/3 is likely due to the fact th a t Ss

is controlled by flow dynamics in the immediate vicinity of the surface therm al boundary

layer. As such, it is not strongly influenced by either internal heating or phase transitions.

3.4.5 Relationships Between Scaling Exponents

The scaling relations for surface heat flux, surface velocity, and surface boundary layer

thickness in the presence of internal heating and phase transitions might be related to

variations in the internal tem peratures due to these effects. I test this hypothesis first for

the surface heat flow which is related to the surface boundary layer thickness Ss and surface

tem perature drop A Ts by ( 3.22). If I substitu te the scaling relations th a t I arrived a t in

sections 3.4.1 and 3.4.4, as well as assume th a t ATS = m&T, 0?o)PATs, I obtain the following



1000a Runs Dl—D4, p^=0.310 • Runs El—E4, p^=0.338

Runs FI—F4, p^=0.31

100

1000x Runs Cl—C5, p^—0.346 ^ Runs Gl—G4, p<i=0.33 □ Runs Dl—D4. pÔ.310

m«o 100

1000 E * B1-B5. p«,=0.342 I x Runs Cl—C5, p^=0.346 - ■ Runs HI—H3, pÔ.315" * Runs J l—J4, P4.=0.30

100

77o(Pa s )

Figure 3.16: (a) Logarithmic variation of the surface boundary layer thickness as a function of rjo for three different VPs (runs are indicated), (b) Logarithmic variation of surface boundary layer thickness as a function of t)q for three cases w ith different rates of internal heating, (c) Logarithmic variation o f surface boundary layer thickness as a function of rjo for four cases w ith different C lapeyron slopes of the endothermic phase transition.



64001 i

5400

4400— 77o=28 1021Pa s— 77o=44 10*’Pa s 770=88 108lPa s 770=880 108lPa s

34006400

g 5400

44007/0=4 102lPa s 77o=15.3 1021Pa s 770=88 102lPa s 770=880 1021Pa s

340042002100

Temperature (K)31501050

Figure 3.17: (a) Four geotherms calculated from, numerical simulations w ith 20 T W of internal heating and phase transitions active w ith CLA P=-2.8 M Pa K -1 and VP 1 differing in the value o f the m ean viscosity (runs D1-D4). (b) Four geotherms calculated fromnumerical sim ulations w ith 0 TW of internal heating and phase transitions active with CLAP=-12 M Pa K -1 and V P 1 differing only in the m agnitude of the mean viscosity (runs J1-J4).



relation between the scaling parameters:

P at,, - Ps, = P q , - (3-42)

For the case of a Boussinesq fluid with no phase transitions and no internal heating and when

no phase transitions are present, this is satisfied by pgs = 1/3 and p q s = 1/3 and P a t , = 0.

T h a t pat,, = 0 is a consequence of the internal tem perature not being a function of the

m ean viscosity in the absence of internal heating and phase transitions as is dem onstrated

in Figure 3.10. In Figure 3.17a I show four geotherms for simulations performed with VP 1,

20 TW of internal heating, CLA P=-2.8 M Pa K - 1 , and differing only in the value of t]q . It

can be seen tha t in the presence of internal heating, the internal tem perature becomes an

increasing function of r/o, and as such, p a t , nonzero and positive in accord w ith (3.30).

On the basis of section 3.4.1 (Figure 3.13b), it is clear tha t pqs decreases in m agnitude

w ith increasing internal heating rate, while pg, remains relatively unchanged and close to

the value of 1/3. Hence the internal tem perature increase with 770, as measured by P a t , ,

accounts for the decrease in p q s .

The variation in p q 3 as a function of the C lapeyron slope may be understood using

similar arguments. The Clapeyron slope of the phase transition becomes more effective in

blocking the flow as the value of the Rayleigh num ber increases, and hence the tem pera-

true drop across the in ternal therm al boundary layer increases which results in a smaller

tem perature drop across the surface. Figure 3.17b shows four geotherms th a t vary only

in the magnitude of 770 where a phase transition is active w ith a 660-km dep th Clapeyron

slope of —12 MPa K -1 and there is no internal heating. Note the increase in ATs w ith

770- As a result, there is again a positive value of p a t , , resulting in a value of p q s whose

m agnitude decreases as the magnitude of the C lapeyron slope increases. Table 3.4 lists the

values for pgs, P a t , , P q s as well as the discrepancy from equality of (3.42) (misfit 1). Values

of p a t , were arrived a t by least squares fitting in the same manner as the other scaling

param eters. As can be seen, the misfits are, for the m ost part, entirely negligible to the

estim ated accuracy of the calculations.

Explanation of the scaling exponents of velocity and mass flux remains significantly

more difficult, however. In boundary layer theory [e.g., Turcotte and Oxburgh, 1967], the



Table 3.4: Relations Between Scaling Parameters

Series p Qs PSs Pat, Misfit 1 Misfit2

B -0.320 ± 0.005 0.341 ± 0.009 0.05 ± 0.002 0.03 ;h 0.02 0.07 ± 0.03

C -0.288 ± 0.002 0.346 ± 0.004 0.046 ± 0.006 0.012 ± 0.01 0.084 ± 0.02

D -0.195 ± 0.006 0.31 ± 0.02 0.14 ± 0.01 0.025 ± 0.04 0.07 ± 0.06

E -0.18 ± 0.01 0.338 ± 0.006 0.189 ± 0.006 0.031 ± 0.02 0.2 ± 0.02

F -0.19 ± 0.02 0.31 ± 0.02 0.144 ± 0.005 0.024 ± 0.04 0.074 ± 0.06

G -0.248 ± 0.005 0.327 ± 0.004 0.09 dz 0.02 0.008 ± 0.03 0.08 ± 0.03

H -0.23 ± 0.01 0.315 ± 0.001 0.09 ± 0.02 0.005 ± 0.03 0.035 ± 0.02

I -0.15 ± 0.02 0.30 ± 0.02 0.14 d= 0.03 0.01 ± C.07 0.04 ± 0.09

relation between the scaling exponents arises du.e to the half space cooling model th a t is

employed to represent the dynamical balances in the boundary layers and results in the

exponent for the boundary layer thickness and h e a t flow being 1/2 the m agnitude of the

velocity exponent. It is also true, however, th a t in a vigorously convecting system, the heat

flow in the interior is carried almost entirely by ndvection ( Q ad v ) : given by

Q a d v = J PrCpU-rT d A , (3.43)

where the integration is carried out over a surface of constant radius. I have calculated an

average of this quantity over all of the radial levels in my model w ith no internal heating

and w ith no phase transitions active (run series B) and found tha t it scales w ith 770 w ith

an exponent of approxim ately —1/3 and takes o n the same average value as the heat flow

measured using derivatives of the tem perature a t th e surface and CMB. The average velocity,

however, scales w ith a —2/3 exponent with respect to 770, much as the surface velocity does.

One explanation is th a t horizontal fluctuations in tem perature (T ') scale w ith 770 w ith

exponent 1/3. W hen this quantity was com puted, however, tem perature fluctuations were

seen to be alm ost constant in 770. This implies th a t it is the correlation between u and T 1

that scales like the —1/3 power, an exponent w hich is significantly different from th a t of

either of the quantities individually.


3.5. D ism ission__________________________________________________________________________________

In order to calculate values for the heat flow In Table 3.2 I assumed th a t R a crjt was not

a function of the m ean viscosity. This implies the following relation:

P a t , + 3 p s 3 —1 = 0. (3-44)

The m isfits to the equality of this relation are listed in Table 3.4 (misfit 2 ). As will be

observed, this relation is no t as accurate as (3 .4 2 ). This suggests tha t the critical bound

ary layer Rayleigh, num ber does vary som ewhat w ith the control param eters as might be

expected.

3.5 Discussion

3.5.1 Temperature Constraint at 660 km Depth

A useful constraint on the tem perature in the m antle comes from the tem perature a t which

the phase tra n s ition from spinel to a m ixture of perovskite and magnesiowiistite is expected

to occur at pressures appropriate for 660 km depth . This has been measured to be 1800-2000

K [B o e h l e r , 1996]. In Figure 3.18 I show two geotherms for viscosity profiles 1 and 3 in which

the tem perature a t 660 km depth hits this ta rget which have values of r/o of 15.3 x 102lP a s

and 29 x 1021Pa s and which have no in ternal heating and 20 TW of in ternal heating,

respectively (solid and do tted lines). Both calculations deliver close to the observed surface

heat flow in the statistical equilibrium sta te . From this combined constraint o f heat flow

and internal tem perature an interesting interplay arises between the interior tem perature

increasing effects of in ternal heating and cooling effects of a depth dependence in viscosity.

Since these models have been integrated to a statistically steady state, I cannot rule out

models that deliver a tem perature at 660 km depth th a t is too low since I assume tha t

E a rth is currently cooling. However, I can rule out models whose tem perature a t 660 km

is too high. Hence it can be concluded th a t no model with a constant viscosity and any

significant internal heating can match the experim entally determined tem peratu re a t 660

km depth in the mantle. Similarly, models w ith a depth increase in viscosity s im ilar to tha t

of profile 3 cannot meet th is constraint and satisfy the observed surface heat flow if the

internal heating is stronger th an 20 TW . Since there is definitely a significant component

1QZ


3.5. D isrn ssinn IQS.

of internal heating in the m antle, based on this constraint, there must be an increase of

viscosity w ith dep th in the m antle. I also include a geotherm from a calculation w ith VP

3, no internal heating, and a Clapeyron slope of —6.5 M Pa K -1 . The tem perature a t 660

km depth is slightly colder th an the lower bound of 1800 K. The surface heat flow is also

too low a t 27 T W . The addition of internal heating or o f significant effects due to secular

cooling to this calculation would boost the internal tem peratu re and increase the surface

heat flow and would likely allow us to hit bo th targets.

A further constraint on the tem perature of the m antle comes from the fact th a t partia l

m elting occurs as deep as 100 km. This implies th a t th e tem perature in some rising plumes

should exceed 1700 K a t depths of 100 km [Green and Falloon, 1998]. It was found th a t

most calculations th a t satisfied the tem perature constraint a t 660 km depth also satisfy this

constraint.

3.5.2 Quantifying Degrees of Layering

It has been known for some tim e that the ability of an endotherm ic phase transition to

induce a layered style of circulation increases w ith convective vigor [e.g., Christensen and

Yuen, 1985; Solheim and Peltier, 1993, 1994a]. There rem ain a number of unanswered

questions, however, as to the effect that layering has on heat flow and the particular value

of the Rayleigh num ber above which layering becomes significant. In linear theory an

endotherm ic phase transition is actually destabilizing [e.g., Turcotte and Schubert, 1971;

Peltier, 1973, 1985; Butler and Peltier, 1997], bu t it becomes stabilizing at finite am plitude.

Solheim and Peltier [1994a], introduced the layering param eter P in order to measure

the effects of a phase transition on layering w ith P = 1 — / l / fw , where / is the mass flux

transported across the phase transition horizon and subscripts L and W indicate th a t the

effects of the phase transitions are included or neglected, respectively. In light of my interest

in the effects of layering on heat flow I have defined two new, nondimensional param eters

th a t may also be employed to measure the degree of layering and the impact on radial heat

transport. These were presented in section 3.3.4, where they were denoted by 1 — Q l /Q w

and (3, respectively, and they measure the decrease in heat flow due to layering and the ratio

of the conductive heat flow to the total heat flow a t th e 660 km horizon. In Figure 3.19

I show these three m easures as a function of the Rayleigh num ber and as a function of


3-5. D iscu ssion ------------------------------------------------------------------------------------------------------------------------------ IDS.

6400

5400

a*m3

T 3<aK

4400

C2

Dl

K2

3400 2000 400030001000Temperature (K)

Figure 3.18: Three geotherms from my numerical model in which I indicate the bound on the tem perature a t the position of the endotherm ic phase transition based on mineral physics considerations


3.5. D ism ission ULQ

-*Clap=—I S MPa/K —a Clap=—6 _ 5 MPa/K - *■ Clap=—2-8 MPa/K

Ra=2 .9 2 10aPa s Ra=5 .0 7 105Pa s Ra=5 .0 7 104Pa s

0.8

\ 0.6O '

0.4

0.2▲

0.8

0.6

0.4

0.2

0.8

0.6Ol,

0.4▲

0.2

1410Ra

10 10|Clapeyron Slope| MPa/K

Figure 3.19: The three nondimensional measures of layering as a function of the Rayleigh number and as a functi»on of the Clapeyron slope of the endotherm ic phase transition. All models have no internail heating and constant viscosity.


3.5. Discussion UL1

the Clapeyron slope of the phase transition for sim ulations with constant viscosity and no

internal heating. For the sake of com parison, the range of Rayleigh num bers represented

spans the range of 770 from 880 x 1021 P a s to 4 x 1021 P a s. Of these measures o f layering,

/3 is probably the least accurate since it requires averaging the heat conducted across the

phase transition horizon, a quantity th a t fluctuates very strongly w ith time.

One im portant observation is th a t flows may be characterized by a significant degree of

layering as measured by P w ithout having a profound im pact on the surface heat flow, as is

evidenced by the low Rayleigh num ber calculations. I t will also be observed th a t the heat

flow conducted across the 660-km horizon is largely unaffected by the Rayleigh number,

b u t it shows a very strong dependence on the C lapeyron slope of the phase transition . It

is also clear th a t for values of the C lapeyron slope th a t are in accord w ith experim ental

observations for the olivine com ponent of the mineralogy, the influence of layering on the

to ta l heat flow is not particularly significant. Figure 3.20a shows the correlation between /?

and 1 — Q l /Q w for V.P.s 1 (solid fine), 2 (dashed line), and 3 (dotted line) as calculated

from (3.40). As we can see, there is a significant am ount of scatter in the num erical data,

b u t there is some indication th a t the param eterized model gives roughly the correct shape

of the curve. The num erical experim ents indicate th a t layering can reduce the heat flow by

as much as 68%. It can be seen th a t heat transport efficiency drops rapidly initially w ith

an increase in (3 but levels off w ith continued increase in this characterization of the degree

of layering.

In Figure 3.20b I show the correlation between P and 1 — Q l /Q w - It is clearly seen tha t

there is a well defined relation between P and 1 — Q l /Q w and tha t heat flow decreases

strongly only when mass flux is quite severely impeded. The crosses indicate cases with

constant viscosity, while the triangles are cases calculated w ith VP 3. In Figures 3.20a, and

3.20b the param eterized model and scaling argum ents had to be used to infer / l and Ql

to compare simulations K1 and K2 w ith simulation A3.

The propensity to layered flow in a fluid w ith an endotherm ic phase transition an d depth-

dependent viscosity has also been a topic of considerable interest in the recent lite ra tu re [e.g.,

Monnereau and Rabinowicz, 1996; Bunge et al., 1997; Cserepes and Yuen, 1997; B runet and

Machetel, 1998]. Cserepes and Yuen [1997] dem onstrated th a t a low viscosity zone placed

beneath the endotherm ic phase transition could considerably increase the degree o f layering,


3.5 . D ism ission 112

V.P. 3 (numerical)V.P. 1 (parametrized)

V.P. 2 (parametrized) V.P. 3 (parametrized)

• * »

1

0.8

0.6

7 0 .4

0.2

0

---- 1---- 1---- 1 1 1---- 1 1 “1--- 1 1" 1 1

b J- X V.P. 1 X .— A V.P. 3 ▲x X- X J- X _

~ X X X * -

i X i i i x 1 i 1 . . . 1 t 1) 0 .2 0 .4 0 .6 0 .8

P

Figure 3.20: Decrease in convective efficiency as a function of layering as measured by /3. The lines are the predictions of (3.40) for three different viscosity profiles, while the points axe the results of numerical calculations. These corresponds to runs C1-C5 and K1 and K2. (b) The decrease in convective efficiency as a function of layering as measured by P. Data points from numerical sim ulations w ith VPs 1 and 3 corresponding to rims C1-C5 and K1 and K2.


3.5. D iscussion----------------------------------------------------------------------------------------------------------------------------- LL3.

while Monnereau and Rabinowicz [1996] and Bunge et al. [1997] found th a t a large viscosity

jum p a t 660 km depth could decrease the degree of layering. Brunet and Machetel [1998]

found, on the contrary, tha t a large jum p in viscosity a t the position of the endothermic

phase transition did not significantly affect the mean mass flux but did result in more

strongly layered states punctuated by more violent avalanches. Brunet and Machetel [1998]

compared a constant viscosity model w ith one which had the same lower mantle viscosity

bu t a lower upper mantle and transition zone viscosity, while Bunge et al. [1997] compared

theirs with a model tha t had the same upper mantle and transition zone viscosity but a

greater viscosity in the lower mantle. Hence the difference in their results can be a ttributed

to the effect of the effective Rayleigh number on layering since a lower viscosity (as in the

work by Brunet and Machetel [1998]) results in a greater effective Rayleigh number, which

is known to increase layering. Monnereau and Rabinowicz [1996] compared a simulation in

which both the lower mantle viscosity was increased and the upper mantle and transition

zone viscosity were decreased w ith a constant viscosity case, and they ensured that the

to ta l heat flow in both cases was similar, ensuring that the effective Rayleigh numbers were

similar. It seems clear tha t the best way to compare the propensity to layering of two

different viscosity profiles is to find two profiles tha t result in the same surface heat flow in

the absence of phase transitions, thus insuring th a t they have the same effective Rayleigh

number, and then compare the degree of layering effected once the phase transitions are

introduced. In Figure 3.21a I show the correlation between the heat flow measured when

phase transitions are present to tha t measured in the absence of phase transitions. The

four lines are fits derived from the scaling relations summarized in Table 3.3 for simulations

performed with constant viscosity and in the absence of internal heating. The open triangles

and squares are points from simulations performed using V P 3 w ith Clapeyron slopes of

-2.8 and -6.5 MPa K -1 . T he heat th a t would be transported in the absence of phase

transitions for this viscosity profile had to be extrapolated using (3.32). It can be seen

th a t for a Clapeyron slope o f -2.8 M Pa K_1 (solid line, triangles), the effects of using

the different viscosity profile are negligible. However, when the m agnitude of the effective

Clapeyron slope is increased to -6.5 M Pa K-1 (dotted line and squares), the radial heat

transport is significantly lower than for the constant viscosity case, indicating considerably

stronger layering. In Figure 3.7 we saw that significantly fewer large mass flux events are


3-5. D iscu ssio n ____________________________________________________________________________________ 114

allowed when using CLAP=-6.5 as opposed to -2.8 w ith VP 3. VP 3 has a considerably lower

viscosity a t the position of the endotherm ic phase transition than V P 1. I t is likely tha t

this leads to an increased velocity convergence onto the phase transition and thus to greater

stab ility of the internal therm al boundary layer as has recently been discussed in detail by

B utler and Peltier [1997]. These sim ulations axe not directly comparable to Monnereau and

Rabinowicz [1996] and Brunet and M achetel [1998] since they consider a jum p in viscosity at

the position of the phase transition, whereas I employ a smooth variation. A further question

arises as to whether any of these sim ulations are in the regime suggested by geochemistry.

Geochemical studies indicate th a t the mass flux between the upper and lower m antle may be

as low as 0.05 kg(m2yr)_1 [O’Nions and Tolstikhin 1994,1996, Tolstikhin and O ’Nions 1996].

A llegre [1997] estimates similar m ass fluxes on the basis of noble gas d a ta b u t obtains fluxes

as large as 0.37 kg(m2yr)_1 based on consideration of the rare earth elements. Alberede

[1998], using a non-steady state box m odel approach, recently calculated th a t the modern-

day noble gas ratios can be achieved w ith a model that allows 50% of the modern day

slab flux or roughly 1 kg(m2yr)-1 to be exchanged between the upper and lower mantle.

D avies [1999] dem onstrated tha t mass fluxes sim ilar to the modern day slab flux may be

allowed if K /U ratios are considerably lower th an is conventionally assumed. The most

strongly layered simulations (with C lapeyron slope —12 MPa K -L) delivered mass fluxes

of 1 kg(m 2yr)_L. One of the m ain observations of geochemistry is th a t mid-ocean ridge

basalts are quite u niform globally, indicating a well-mixed upper m antle. If the lower

m antle is to be considered as a source of heterogeneity for the upper m antle, then the

tim e for the upper mantle to be com pletely exchanged with the lower m antle due to mass

flux across the 660-km horizon m ust be very long compared with the m ixing tim e of the

upper m antle. The exchange tim e r ex in term s of the time-averaged mass flux / across the

660-km phase transition is simply given by r ex = M um/ ( f A ) , where M UTn and A are the

mass of the upper mantle and the area o f the 660-km horizon, respectively. A lthough I

have not performed the detailed m ixing calculation necessary to accurately calculate the

m ixing tim e of the simulated upper m antle, I can estim ate its value from the tim e average

o f th e surface velocity. The mixing tim e r mjx = L /u s where L is a leng th scale. Allegre

and Lewin [1995] estim ated the m ixing tim e of the upper mantle to be roughly 200 Myr

based on the dispersion of chemical heterogeneities observed a t m id-ocean ridges. Using


3 .5 D iscu ssion 115.

80

60

40

20■er ~

20 40 Qw (TW)

8060

10

9

' • A

eo . i

o . i l 10Tmix (Gyr)

Figure 3.21: a) (a) The heat flow calculated in the presence of phase transitions as a function of the heat flow for the same configuration in the absence o f the effects of phase transitions. T he long-dashed, solid, dotted, and short-dashed lines are for cases w ith Clapeyron slopes of 0, -2.8, -6.5, and -12 M Pa K -1 respectively w ith constant viscosity and no internal heating and axe calculated from the scaling relations th a t are sum m arized in Table 3. The crosses, solid triangles, solid squaxes, and staxs correspond to d a ta points from the numerical simulations perform ed w ith Clapeyron slopes of 0, -2.8, -6.5 and -12 M Pa K -1 respectively, and constant viscosity and no internal heating. T he open triangle and square correspond to a calculations w ith VP 3, no internal heating and Clapeyron slopes of -2.8 and -6.5 M Pa K -1 respectively (K1 and K2). (b) T he logarithmic variation of the tim e to exchange the upper m antle w ith the lower m antle w ith the mixing time of the upper m antle. (Explanation of the estim ation of these quantities is described in the tex t). Crosses, solid triangles, solid circles, open circles, hexagons, and pentagons axe points calculated from simulations from run series B, C, D, E, F, and G, respectively. Open triangles axe from simulations K1 and K3, while K2 is indicated by the open square and II is indicated by the open hexagon.


3.6 . C on clu sion s l l f i

4 cm y r_1 as a worldwide average for the surface velocity for the real Earth , I arrive a t

an appropriate value for L of 8000 km. In Figure 3.21b I display a plot of rex versus Tm;x

for all of the simulations th a t I performed. Points tha t he above the solid line are derived

from simulations in which r ex > r mjx and hence are in the geo chemically relevant regime.

In order tha t simulations be geochemically relevant based on this criterion, it is necessary

tha t the mass flux across 660 km dep th be small while the mixing in the upper mantle is

vigorous. Both of these criteria are more easily satisfied when a viscosity tha t increases

w ith depth is employed. We note th a t only two of my simulations are even weakly in this

regime; these are rims J1 and K2 whose geotherms are p lotted in Figures 3.17b and 3.18.

3.6 Conclusions

The results presented in section 3.2 indicate tha t as the flow becomes increasingly layered,

the spectrum of mass flux events displays a power law behavior over roughly 2 orders of

magnitude. This indicates th a t such mass flux events constitute a scale invariant phe

nomenon, and since this behavior is displayed over a wide range of param eter values, the

avalanche process is likely to be an example of SOC. (See also Butler and Peltier [1998]

for a preliminary discussion.) The mass flux across the 660-km horizon is also supported

to a greater degree by smaller mass flux events as the degree of layering increases. This

indicates tha t if the degree of layering in the mantle is as strong as has been suggested on

the basis of geochemical analyses [e.g., O ’Nions and Tolstikhin, 1996], then it is likely tha t

chemical heterogeneities th a t arise due to the mixing of undepleted m aterial into the upper

m antle will be small and thus relatively easily homogenized by convective stirring in the

upper mantle. This is in agreement w ith the global uniformity observed to be chaxacteristic

of the composition of mid-ocean ridge basalts.

The predominance of such small mass flux events may also be expected to lead to the

development of a global low-viscosity feature a t the position of the endothermic phase

transition due to the effects of transform ational superplasticity as has been suggested by

Peltier [1998]. This is the process by which the viscosity is decreased due to a change in

m aterial phase. That such a feature in the radial viscosity profile is required has been

suggested by analyses of seismic tomography based models of the convective circulation


,3.6. C onclusions__________________________________________________________________________________ 117

when aspherical geoid anomalies are em ployed to discriminate between plausible profiles

[e.g., Pari and Peltier, 1995]. The results presented in section 3.5.2 suggest tha t anomalously

low viscosity in the vicinity of the 660-km phase transition may assist in the development

of the layered state . Similarly, Cserepes and Yuen [1997] found th a t a low-viscosity region

located below the position of the endotherm ic phase transition resulted in enhanced layering.

Taken together, these results suggest th a t th e re may be a positive feedback mechanism

whereby as layering is enhanced, more small mass flux events, and fewer large ones occur

which leads through transform ational superplasticity to the development of a low-viscosity

region surrounding the phase transition horizon. The presence of this soft zone would then

lead to a further enhancement of layering. In this way, I suggest th a t significant layering

may arise for values of the Clapeyron slope of the endothermic transition tha t are measured

in laboratory high-pressure experiments.

Perhaps the most striking feature of F igure 3.13 is tha t the observed surface heat flow is

matched only for values of the mean m antle viscosity that are more th a n an order of mag

nitude larger than the values that are necessary to reconcile observations of the postglacial

rebound phenomenon when values of the C lapeyron slope of the 660-km phase transition

tha t are consistent w ith experimental m easurem ents are used. Even w hen no internal heat

ing is present and when a depth-independent viscosity is employed (which delivers the lowest

heat flow of any of the viscosity profiles), the observed surface heat flow still requires a mean

viscosity of 14 x 1021 Pa s. Since the E a r th ’s mantle must contain in ternal heat sources

and since the viscosity of the mantle is likely to increase with depth, th is m ust be seen as

a lower bound on the mean viscosity necessary to reconcile surface heat flow (to the degree

tha t my axisymmetric spherical model w hich excludes the influence o f the surface plates

can be assumed to deliver a good approxim ation to the radial heat transfer). Also, since

considerable care has been taken to in tegrate these solutions to a statistically stationary

state, the heat flow contribution due to secular cooling is not present in my model. As has

been discussed previously, the long timescale for equilibration of the m antle indicates that

there is likely more heat leaving the surface th a n is being generated in ternally and that is

entering across the CMB. Clearly, since the core is also cooling and th e heat sources in the

mantle are d iminishing in strength w ith tim e, there is no true equilibrium tem perature for

the mantle. However, an equilibrium tem peratu re based on the current conditions can be


3.6. C on clu sion s___________________________________________________________________________________11&

defined, and it is likely th a t the m antle is considerably w anner than this. In section 3.3.3

it was estim ated th a t for every 100 K th a t the E arth is above what would be its current

equilibrium tem peratu re, there should be roughly 3 T W contributed to th e surface heat

flow. Any contribution to the surface heat flow from secular cooling makes reconciling the

models w ith the observed surface heat flow more difficult.

I am therefore left w ith two possibilities: e ither I have not accurately estim ated one

or more of the param eters tha t are required to define the model, or some physical process

th a t is not included in the model acts to dram atically reduce the heat tran sp o rted to the

surface. Possibilities for the former include CMB tem perature, m agnitude o f viscosity, and

degree of layering. Possibilities for the la tte r include the effects of surface plates, continents,

tem perature-dependent rheology, and full three dimensionality. A lower bound for the CMB

tem perature, based on the melting point of iron a t CMB pressures, is 3200 K [e.g., Boehler,

1996)] although recent calculations suggest th a t this lower bound should be as high as

4500 K [A lfe et al., 1999]. Calculation A3 delivered the surface heat flow w ith a CMB

tem perature of 2785 K, which contained no internal heat sources and no contribution due

to secular cooling. This implies tha t in order to get the observed surface heat flow by

reducing the CMB tem perature, I am forced to use an improbably low value as well as

restricting m yself to a case with no internal heating and no secular cooling. My model

also has no continents. However, these cover only 1/3 of the E a rth ’s surface and it is

known th a t the m antle contribution to heat flow through the continents is roughly 5 TW

[e.g., Pollack and Chapman, 1977]. I t is also likely th a t if a continent acts to therm ally

insulate the m antle, more heat will escape through the oceans. Gurnis [1989] dem onstrated

th a t the addition of surface plates w ith tem perature-dependent viscosity d id not result

in a significant reduction in the surface heat flow. Similarly, Brunet and M achetel [1998]

presented two calculations performed using very similar depth variations o f viscosity in

which one had a tem perature dependence and one did not, and they showed th a t the flows

resulted in very sim ilar surface heat fluxes. It is known, however, th a t changing the surface

boundary conditions can have an effect on the calculated heat flow (e.g., Arkani-H am ed and

Toksoz, 1984), and it is possible that some of the complicated dynamics th a t occur near the

surface act to reduce the heat transport efficiency of therm al convection. I believe th a t the

most likely possibility is th a t I have underestim ated either the m agnitude o f th e effective


3.6. Conclusions iia

viscosity of the mantle on the convection timescale or the effectiveness of the endothermic

phase transition in inducing a layered style of convection.

If the rheology of the m antle is transien t on timescales of 103 to 10' years, it is possible

th a t the viscosity tha t controls the m antle convection process is considerably greater than

th a t which governs the postglacial rebound phenomenon. Karato [1998] argues on micro

physical grounds th a t this may be the case. This would require th a t the viscosity in the

bulk mantle is strain-rate-dependent and thus non-Newtonian. Vinnik et al. [1998] have

recently pointed to evidence of seismic anisotropy a t 660 km depth as evidence of non-

Newtonian rheology and m antle layering. Karato and Wu [1993] and others had previously

drawn attention to the apparent lack of seismic anisotropy below a depth of 200 km, a

strong indication th a t the bulk of the m antle is, in fact, Newtonian. As recently suggested

by Peltier [1998], it is possible th a t the th in soft layer at 660 km depth, which appears to be

required by convection models constrained by seismic tomography [e.g., Pari and Peltier,

1995], is a feature induced by transform ational superplasticity in which the creep mecha

nism is non-Newtonian. If this were the case, then the rheology in this region may well

be such th a t the convection process would "see” anomalously low viscosity there while the

rebound process would see “norm al” viscosity, as explicitly suggested by the recent analyses

in Peltier [1998].

The impact of the endotherm ic phase transition may also be underestim ated. The

experimentally measured pressures over which the spinel to perovskite and magnesiowiistite

phase transition takes place corresponds to a distance of only 4 km, while, for numerical

reasons, I am required to use a value of roughly 20 km. It has been shown [e.g., Peltier and

Solheim, 1992; Solheim and Peltier, 1994] th a t the effectiveness of an endothermic phase

transition in inducing a layered style o f convection increases as the phase loop thickness

decreases. I have also considered only the olivine component of the mantle mineralogy and

since a garnet transition occurs a t roughly the same pressure, it is possible that the effective

Clapeyron slope is considerably greater (or less!) than the —2.8 M Pa K -1 th a t I employ.

More importantly, I believe, and as has already been discussed, th a t it is possible tha t a

“soft zone” exists a t 660 km depth , and this m ay lead to stronger layering for a given value of

the Clapeyron slope. It will also be noted th a t the high degree of layering th a t is necessary

to reduce the surface heat flow, while no t as strong as some geochemical studies have been


3.6 . C onclusions ________________________________________________________________________________120

taken to imply [e.g., O ’Nions and Tolstikhin 1996], is in fact fully consistent w ith the upper

limits of constraints on m antle mixing from some geochemistry based analyses [e.g., Allegre

et al., 1983] and is only slightly greater than some more recent estimates [e.g., Alberede,

1998; Davies, 1999]. I t has also been recently dem onstrated th a t significant m antle layering

is required to adequately reconcile the observed geographical distribution of the surface

heat flow in terms of mantle convection models constrained by seismic tomography [Pari

and Peltier, 1998].


Chapter 4

The Thermal Evolution of the

Earth: Models with. Time

Dependent Layerimg of Mantle

Convection which Satisfy the Urey

Ratio Constraint

4.1 Introduction

Therm al history models for the E arth in which fche rate-controlling m antle-convection pro

cess is ‘param eterized’ have been effectively em ployed for some tim e as a guide to under

standing planetary evolution. Analyses of this type, which were in itiated in the work of

Sharpe and Peltier [1978], [1979], Schubert et al. [1980] and Davies [1980], have been based

upon the fact th a t therm al convection a t high Rayleigh number transfers heat a t a rate

which scales as a power of this prim ary control variable, a fact th a t follows on the basis

of boundary layer arguments (e.g., Turcotte ancd Oxburgh [1967]). By assum ing th a t the

time scale on which interior cooling occurs is lonjg compared to the tim e scale o f convective

mixing, one is able to obtain an evolution equattion for the internal energy th a t does not

require explicit solution of the dynamical equatioons. This approach to the therm al history

1 2 1


4.1 . In trod u ction -------------------------------------------------------------------------------------------------------------------------

problem has the advantage, over th a t based on solution of the full fluid dynam ical equa

tions, th a t the construction of solutions does not require intensive com putation although

some calculations of the former type have been made, (e.g., Arkani-Hamed et al., 1981).

Extensive searches of param eter space are thereby m ade possible and a num ber of physical

effects, such as th a t due to the influence of the tem peratu re dependence of viscosity, are

easily introduced. Furthermore, the construction o f three-dimensional numerical solutions

of the f ull set of governing equations for a system th a t is as vigorously convecting as is the

present-day E arth are still prohibitively expensive. Explicit modeling of convection in the

d istan t geological past, when tem peratures axe thought by many to have been higher and

viscosity significantly lower than present, is an even more formidable task. A lthough there

rem a in s some question as to the applicability o f param eterized convection models to the

therm al history problem, the model recently described by Butler and Peltier [2000] has been

com pared in detail to predictions made on the basis of complete time-dependent solutions

of the Navier-Stokes equations and shown to be in ra ther precise accord w ith these detailed

analyses in so far as the long time scale evolution o f the mean tem perature is concerned.

Due perhaps to the similarity of the chondritic heating rate with the observed mean

surface heat flow of the present day E arth , it has often been assumed th a t the surface

heat flow must have varied in lock-step w ith the ra te of internal radioactive heating. Tozer

[1972], in particular, proposed tha t the strong tem peratu re dependence of m antle viscosity,

acting in conjunction w ith internal heating, behaves in such a way as to render mantle

evolution independent of its initial conditions. I f the m antle was initially hot, according to

Tozer1 s view, the resulting low viscosity would allow therm al energy to be rapidly expelled;

similarly, if the m antle became overly cool, high viscosity and internal heating would result

in rap id m antle warming until the heat escaping from the surface roughly balanced the

heat generated in the interior. The calculations of Schubert et al. [1980] and Davies [1980],

which were based upon one layer models w ith strongly tem perature dependent viscosity,

however, suggested th a t only a narrow range of ra tio s of internal heating ra te to surface heat

flow (Urey ratios) were allowed if their models were to fit the modern heat flow constraint.

Schubert et al. [1980] were thereby led to suggest th a t allowed values of the Urey ratio

lay in the range of 0.65 - 0.85. This result is discrepant w ith the best a priori estim ate

provided by geo ch em istry which suggests th a t the present day mantle should be heated

1 2 2


4.1 . In troduction-------------------------------------------------------------------------------------------------------------------------

a t the ra te of approxim ately 13 T W (e.g., Zindler and Hard [1986]), which, when divided

by the mantle convection com ponent of the E a rth ’s surface heat flow of 36 TW , suggests

a Urey ratio near 0.4. This assum ed value of 36 TW for the ra te a t which the mantle is

currently transferring heat to the surface is arrived a t by subtracting the approxim ately 6

T W found in the continental an d oceanic crust from the roughly 42 T W of heat flowing from

the E a rth ’s interior (e.g., Pollack et aL; 1993, Sclater et al., 1980; Pollack and Chapman,

1977).

Subsequent to the above referenced early analyses of planetary therm al history based

upon the idea of param eterized convection, a large number of different physical effects have

been incorporated into models of this kind. These include the presence of therm al boundary

layers a t the CMB (e.g., Spohn and Schubert, 1982; and Honda, 1996b), as well as effects

due to inner core freezing (e.g., Mollett, 1984; Spohn and Breuer, 1993a,b). The possible

influence of layering at 660 km depth has also been discussed by M cKenzie and Richter

[1981], Spohn and Schubert [1982], and Honda [1995]. Christensen [1985] considered the

possibility that the scaling exponent of heat flow w ith Rayleigh num ber might be strongly

reduced by the fact that the m aterial in the cold surface boundary layer is highly viscous

and this could influence surface mobility. Numerical results of G um is [1989], however, imply

th a t the influence of very viscous surface plates, so long as these remain mobile, should not

significantly decrease the dependence of surface heat flow on internal tem perature. Spohn

and Breuer [1993a,b] also modeled effects due to crust formation and found th a t models

could be found tha t were in keeping w ith geochemical constraints provided th a t there was

a significant degree of in ternal heating in the core. Measurements and calculations of the

solubility of potassium in liquid iron a t outer core pressures ( Chabot and Drake [1999], Ito

et al. [1993], Sherman [1990]), however, suggest it to be highly unlikely th a t any significant

radioactive heating is actually present in the core.

Recently, Butler and Peltier [2000] dem onstrated, on the basis of integrations performed

using an axisymmetric m odel of the m antle convection process, th a t when a CMB tem per

atu re of 4000 K is assumed, which is in accord w ith the mineral physics results of Boehler

[1996, 2000], reasonable surface heat flow was obtained only when m antle viscosities were

assumed to be an order of m agnitude greater th an those inferred on the basis of post-glacial

rebound analyses (e.g., see Peltier, 1998 for a recent review) or when the convective circu

123.


4.1 . In trod u ction _______________________________________________________________________________

lation was assumed to be strongly layered a t 660 km depth. Snap shots of the tem perature

field in two such calculations are shown in Figure 4.1 and the azimuthally averaged tem

perature as a function of depth for these two end-member models is shown in Figure 4.2

(solid lines). These calculations were integrated to a statistical-equilibrium sta te and hence

there was no contribution due to secular cooling to the predicted surface heat flow. Since

the mantle is most probably cooling, however, any contribution from secular cooling would

drive up the surface heat flow resulting in a requirem ent for an even higher degree of layering

an d /o r higher viscosity.

Numerical simulations of the mantle convection process, performed w ith the effects of

an endotherm ic phase transition at 660-km depth, have shown that the effectiveness of the

phase transition in inducing layered flow increases w ith the system Rayleigh num ber (e.g.,

Christensen and Yuen, 1985; Solheim and Peltier, 1993, 1994a; Butler and Peltier, 2000).

Since the Rayleigh number th a t is characteristic of the convective circulation in the E a rth ’s

m antle has most probably decreased over the history of the E arth due to cooling of the core

and mantle, the decrease in the degree of radioactive heating, and the increase in m antle

viscosity due to decreasing tem perature, it is likely th a t the degree of layering in the mantle

is lower now than it was in the past. Davies [1995] and Honda [1996] have allowed in their

recent work on the therm al history problem for a change from fully layered to ’whole-mantle’

convection; however, no previous work has allowed for incomplete layering or for the degree

of layering to evolve w ith time. I will show th a t the incorporation of this effect in my new

param eterized model of the thermal history, in a way th a t is constrained by the numerical

simulations tha t I have performed, results in a significantly more realistic therm al history.

In what follows, I will employ this newly devised parameterized therm al history model

to further constrain the range of mantle viscosity and degree of convective layering th a t are

com patible w ith the observations. I employ a model th a t has thermal boundary layers at

the CMB, 660 km depth and a t the E a rth ’s surface. I also include effects due to freezing of

the inner core and I allow for incomplete layering a t 660 km depth and for tim e-dependent

layering. I constrain the therm al history models to deliver the modern-day surface heat

flow and to have tem peratures at the core m antle boundary and a t 660-km depth th a t are

in accord w ith mineral physics constraints. The in itial tem perature profile in the m antle

is set to be adiabatic and near the melting point for mantle material near the surface

124


4.1. Introduction 125.

3876 K

3652 K

3429 K

3205 K

2982 K

2758 K

2535 K

2311 K

2088 K

1864 K

1641 K

1417 K

1194 K

970 K

747 K

523 K

Figure 4.1: Color contour ’snap-shots’ of the tem peratu re field in my numerical, axisymmetric model of m antle convection, a) corresponds to a strongly layered case w ith a P G R viscosity profile, b) is a whole mantle case w ith viscosity th a t is significantly greater th an the PG R profile.


4 1- In trod u ction 12fi

6400

w 4800

32001200 2200 Temperature (K)

4200

Figure 4.2: Solid lines represent geotherms calculated from fields shown in Figure 4.1. The geotherm with the large tem perature drop at 660 km d ep th corresponds to the layered case. D otted and dashed lines axe calculated using the param eterized model of Butler and Peltier [2000]. For the layered calculation A e is 5.51 109 W f A:4/ 3 for the case of the dashed line and 2.52 109 W /k A/ z for the case of the dotted line. T he definitions of Tc, T], Tu, and Ts are indicated in the Figure.


4.2. Model Derivation 12Z

but I investigate the possibility th a t the core may be significantly warmer than the lower

mantle. I will dem onstrate th a t models w ith a tim e independent degree of layering, including

'whole-mantle’ models, require very high initial core tem peratures. Whole mantle models

will also be shown to require th a t the heat transported across therm al boundary layers

is very low for a given tem perature drop across the boundary layer. This might be due

to high viscosity or to plate effects a t the E arth ’s surface an d /o r effects due to chemical

heterogeneity. Completely layered models can efficiently transport heat across the surface

boundary layer but still require inefficient transport of heat across a boundary layer a t the

CMB. I will dem onstrate th a t solutions for which the degree of layering depends on the

Rayleigh num ber do not require high in itial core tem peratures, however, and generally give

much improved fits to the surface heat flow and internal tem peratures over constant degree

of layering solutions. They do require inefficient therm al boundary layers, however, arguing,

perhaps, tha t the viscosity th a t governs the mantle convection process must be significantly

greater than th a t controlling rebound.

The param eterized model th a t I will employ herein is very similar to that introduced in

Butler and Peltier [2000], to which I have, however, added a core w ith variable tem perature,

tem perature-dependent m antle viscosity and effects due to the freezing of the inner core.

I derive this model in detail in the following section, and discuss my choices of model

param eters in section 4.3. In section 4.4, I first illustrate a number of general properties of

parameterized solutions for the therm al history and then display the new solutions described

above. The implications of my results are discussed in section 4.5.

4.2 Model Derivation

By insisting th a t therm al energy be conserved in a “three-shell” spherical model consisting

of the core (quantities indicated by a subscript c), lower mantle (subscript I), and upper

m antle and transition zone (subscript u), I arrive a t the following set of coupled ordinary

differential equations.

C p u ~<it = ~ ® S + X u


4.2. M odel D erivation 128

CPl-fa = Qcm - Qe/P + XI

{Cpc + C V f) ^ - = - Q Cm. (4-1)

In the above system, Cpi (i = u, I, c) are the adiabat weighted values of the heat ca

pacities, t is time, and Xi represents the to ta l amount of internal radioactive heating in the

i th shell. C p f is the added effective h ea t capacity due to inner core freezing. Tu, T; and

Tc represent the tem perature at the top o f the 660 km and CMB therm al boundary layers

and Tc represents the tem perature on th e core side of the CMB therm al boundary layer.

Qs, Qe, and Qcm represent the heat tran sp o rted by conduction across the surface, 660 km

depth and CMB therm al boundary layers respectively, whereas p is a param eter used to

investigate effects due to incomplete layering. P represents the ratio of heat transported

by conduction across 660-km dep th to th a t transported by advection and when P is very

close to 0, I have ’whole m antle’ circulation whereas when p is equal to 1, I have completely

layered m antle convection. In all of the m odels tha t I will present, P will be taken to be ei

ther constant or a smoothly varying function of time; hence I shall be sm oothing over short

time-scale mantle mass flux events such as avalanches (see Butler and Peltier. 2000, for a

detailed discussion of the avalanche effect). It is expected th a t this should be a reasonable

approxim ation when examining the long-term variation of bulk properties such as average

mantle tem peratures and surface heat flow. I will discuss the calculation of P la ter in this

section. In Figure 4.2, I show two geothierms, one which corresponds to strongly layered

convection and one which corresponds to whole mantle convection. I indicate Ts, Tu, 7), and

Tc for the strongly layered case. I also com pare the results of a param eterized calculation of

internal tem peratures (dotted lines) and those from my numerical m odel (solid lines) and

it can be seen tha t the agreement is ra th e r good. I will delay discussion of the long-dashed

line to section 4.3.

The three assumptions th a t are used in relating the mantle and core tem peratures to

the heat flow across the boundary layers are as follows. (1) The radial tem perature profile

in m antle regions outside of therm al boun d ary layers is adiabatic. (2) The heat conducted


4.2 . M od el D erivation I2a

across a therm al boundary layer is adequately represented by

Q i = A ik iA T i /S i (4.2)

(i = cm, e, s ) where is the therm al conductivity a t the position of the boundary layer, A t-

is the area of the boundary, and AT) and Si are the tem perature drop across and thickness of

the therm al boundary layer, respectively. (3) Convection is driven sufficiently strongly th a t

boundary layer instabilities form on each boundary such th a t the boundary layer Rayleigh

num ber is m aintained a t its critical value, R a ^ u (e.g. Howard [1966]). This criterion has

been successfully used by Jarvis [1993] and Vangelov and Jarvis [1994] to predict internal

tem perature variations as a function of curvature.

These assum ptions allow us to write

Q i = A iA T * /3 , (4.3)

where

A T cm — Tc T/, ATe — J'l(f>cme Tu, A T S — Tu 4>es Ts. (4-4)

In (4.4), Ts is the tem perature a t the E a rth ’s surface and the At- are coupling coefficients

relating the tem perature drop across boundary layers to the conducted heat flow. In terms

o f m aterial properties these coupling coefficients are given by

(4.5)TJiJxG'crit

T he m agnitude of A* can be thought of as the efficiency w ith which heat is conducted

across a boundary layer w ith a given tem perature drop. (4.5) can be derived by solving for

Si in term s of R a ^ u and the other convection param eters in the vicinity of th e boundary

layer (please see Butler and Peltier, 2000, for a detailed derivation), pi, g i, a*, Cpi and r\i

represent the density, gravitational acceleration, therm al expansivity, heat capacity per unit

mass, and viscosity a t interface i. I will discuss the m agnitude of the param eters used in

th e following section. <f>cme and 4>es represent the adiabatic tem per a t rue drop between the


4.2 . M odel D erivation 13H

CMB and 660 km depth, and 660 km dep th and the surface respectively and are calculated

explicitly as

M c, L , T f and T q represent the mass of the core, the heat of inner core form ation, and

the tem peratures in the core a t the CMB a t which the inner core begins to form and the

tem perature a t which the core is completely frozen. L may be considered to be the sum

of a num ber of effects including the latent heat of core freezing, the gravitational energy

release due to core contraction upon freezing, and the energy due to the release of the lighter

elements upon inner core freezing.

The viscosity th a t is employed in the calculation of the A i is determined for each bound-

(4.6)

Substitu ting (4.3) and (4.4) into (4.1) delivers the following governing equations:

C p u -^ - = A e[Th Tu](Tl(f>cme - Tu)4^ //3 [T h Tu} - A s[Tu){Tuct>es - Ts)4/3 +

CPl- £ = AciTMTc - r,)4/3 - A e[Th TuKTz&n.e - Tu)4/3//3[T),Tu] + Xi

{Cpc + CPf ) - ^ = - A C[T,](TC - I})4/3. (4.7)

I have explicitly indicated the tem perature dependencies of the various coefficients to em

phasize the strongly non-linear nature of these equations.

I employ the functional form for C p / suggested by Mollett [1984] as

)L/2 when T f < T C < T q and Cp/ = 0 otherwise. (4.8)


4- 9. M orlpl P p r iv a tio n __________________________________________________________________________________________ 1 31

axy layer based on the local tem perature as:

Vs = vosexp[A A /T u4>es\,

r]e = T j 0 e e x p [ 2 A A / (T u + T / 0 c m e )]j

T)cm — Vocm d X p \A A /T i ]. (4-9)

Here, I regard A A as a constant and the tem perature a t 660 km dep th is assum ed to be

equal to the average of the tem peratures a t e ither extreme of this therm al boundary layer.

T he tem perature dependence of rjcm is assumed to involve only the tem perature in the lower

m antle ra ther than the tem perature in the core since it is assumed th a t the m antle side is

the ’bottle-neck’ and hence controls the convective heat transfer. The rjoi are given by

, A A . r 2AAVos = V s e x p [ - - — - ■ ], 77oe = T)ee x p [ -Ti i J 5 / U e / e w tr L rrt , rrt / J ’

u f t y u s J-uf J- l f tpcme

VOcm = V c m e x p [ - — —]. (4.10)Tu

Here, the 77i and the T2y are the present-day viscosities and the final tem peratures at the

different boundary layer depths. Clearly, the T i/ are not known a priori so models are run

w ith initial values of

Tu / = {(% ^-)3 /4 + T s} /0 es,

T if = (3800 - Tu f) / (4.11)

I then employ the final tem peratures from an initial run and use these to determ ine the

new Tuf and T \/ and iterate in this way until a stable solution is obtained. Qsf is the final

required surface heat flow and the tem perature o f 3800 K which appears in (4.11) is chosen


4.2. M odel D erivation 132

so th a t the tem perature on each side of the 660 km depth internal therm al boundary la y e r

averages to 1900 K.

In the context of a simple one-layer system, I will also examine a power law dependence

of the viscosity on tem perature in the form of

r1{ T ) = r 1o(Tn/ T ) \ (4. .12)

where T is the tem perature at the base of the layer and 770, Tn and q are constants.

In calculations in which the degree of layering, param eterized through /3, changes a s a

function of the Rayleigh number, I calculate the Rayleigh number as the sum of term s due

to the tem perature drop across the m antle and due to the internal heating in the m a n tle

as:

Ra = f ge- r -\, o{(yc - Ts) + (xi + Xu)d2/((C Pl + Cpu)Ke)}. (4-13)

P is then calculated from an empirically determ ined functional form:

P = 0.5(1 tanh[logio(Ra) — 7 ]). (4-14)

The param eter d in (4.13) is the distance from the CMB to the surface. In Figure 4. 3, I

show values of P calculated from r u n s of the full axi-symmetric convection model th a t w e re

entirely heated from below which were integrated with various choices for the m agnit ude

of the Clapeyron slope of the endotherm ic phase transition at 660-km depth. I incLude

functions of the form of (4.14) for each of the Clapeyron slopes and it will be seen thaat 7

decreases as the m agnitude of the Clapeyron slope increases. A Clapeyron slope of —2.8 iU'- >a

is w ithin the range suggested by mineral physics experiments (e.g. Chopelas, 1994) for the

spinel to perovskite and magnesiowiistite transition a t 660-km depth. It can be seen th a t; for

these calculations, this Clapeyron slope results in relatively weak layering as m easuredl by

P. A small value of P may translate into a relatively high degree of layering as m easuredl by

mass flux, however, w ith a P of 0.2 translating to a reduction of mass flux of 20-50 % baxsed

on the results of Butler and Peltier, 2000. I t is expected th a t earlier in E a rth ’s histaory,

when therm al forcing was stronger and viscosity was lower, the layering associated w i th


4.2 . M odel D erivation 123.

clap=—120.8

0.6

clap=—6.5

0.4

clap=—2.8 -

0.2

10log(Ra)

Figure 4.3: Values of /3 calculated from my numerical model w ith various values of the C lapeyron slope of the 660-km dep th endotherm ic phase transition (clap, in units of M P a /K ) . S tars indicate points w ith clap=-12, squares have clap=-6.5 and triangles have clap=-2.8. Solid lines are hyperbolic tangent fits for each value of clap.


4 .3 . C h o ice o f P aram eters---------------------------------------------------------------------------------------------------------

a Clapeyron slope of this magnitude would have been greater. The trend to increasing /3

w ith Rayleigh num ber can be seen, except for the lowest Rayleigh number case which shows

an unusually large value of /?, for all C lapeyron slopes. Although the form of (4.14) can

only be said to be suggested by the results presented in Figure 4.3, (4.14) provides a useful

param eterization of the physical effect tha t layering increases with the system Rayleigh

num ber. The value of 7 based on Figure 4.3 for a Clapeyron slope of —2 . 8 ^ ^ is 8.5. Due

to the large uncertainties in the value o f this param eter, I will investigate a range of values.

There is also a great deal of am biguity as to how to define the Rayleigh number in the

param eterized model as properties vary with depth and as heating occurs both from w ithin

and from below. I use the average of the surface and CMB viscosities in (4.13) and I include

b o th the effects of a tem perature drop and the internal heating in order to get a measure

of the to ta l forcing which drives the m antle convection process.

W ith the exception of one simple analytical result, all solutions are arrived a t by integrat

ing forward from the tim e of formation of the E arth using a sixth order Runge-Kutta-Verner

scheme. Many previous studies have integrated backwards in time from the present, since

current conditions are b e tte r known than are those from the tim e of the E a rth ’s formation.

However, I often found backward integration to lead to numerical instability and so I have

avoided this approach entirely.

4.3 Choice of Parameters

Constraints tha t I will require my param eterized convection solutions to fit are tha t models

m ust deliver w ithin 2 0 % of the m antle component of the observed, present-day, surface heat

flow, Q sf , of 36 T W (Pollack et al. 1993, Sclater et al. 1980 and Pollack and Chapman

1977). The present-day tem perature a t 660 km depth, T /e, and on the core side of the CMB,

T cM f, furthermore, m ust be similarly close to 1900 K and 4000 K (Boehler 1996, 2000). In

the case of a strong therm al boundary layer at 660 km depth, the tem peratures on either

side of the boundary layer will simply be required to bracket the required tem perature. The

tem perature in the upper mantle m ust also not have changed significantly in the last 3.8

Gyrs and I shall insist th a t the m antle has not undergone large scale melting so I require

th a t Tu and Trfcme have not exceeded 2200 and 2800 K respectively {Zerr et al., 1998,

134


4.3. Cihoirp o f P aram eters 13fi

Table 4.1: Param eter Values at Boundary Depths

Symbol Units Surface 660 km depth CMB

A m 2 5.10 1014 4.08 1014 1.52 1014

k W (K m )~ l 2.5 2.55 1 2

P kg m ~ 3 3361 4260 5566

9 m s~2 9.83 1 0 .0 1 1 0 .6 8

a K ~ l 3 10“ 5 2.5 10~ 5 1 1 0 “ 5

Cp J(kg K )~ l 1252 1142 1256

V kg(m s ) - 1 0.54 1021 0.96 1021 1 0 .6 1 0 21

A W K ~ 4/ 3 7.65 109 5.51/2.52 109 2.40 109

Boehler, 2000). T) is constrained more strongly by the melting tem perature a t 660 km

dep th than tha t near the CMB, since the melting curve in the lower mantle is steeper than

the adiabat.

Assuming characteristic values for mantle properties near the E arth ’s surface as well as

near 660 km depth and near the CMB, as well as a value for Rac-it, I can calculate values for

the A i . In a number of calculations using my axisymmetric mantle convection model with

ffee-slip surface and CMB boundary conditions, it was found that a value of R a ^ u — 14.3

gave good agreement for the surface heat flow with the parameterized convection model

described above (Butler and Peltier, 2000). This value is intermediate to the values of 8.4

and 24.4 suggested previously by Honda [1996] and Labrosse and Sotin [1999]. I t was found,

however, th a t calculations tha t included the effects of internal heating were not quite as

well represented by the param eterized model. The results should be accurate to 20 %,

however, and should be adequate for the purposes of this analysis. It will also be noted

th a t the surface boundary conditions on the E arth are significantly more complicated than

simple free slip conditions, and as such, the effective Ra^-it may be significantly different

from the value tha t was found to be characteristic when this simpler boundary condition

was employed in the a priori convection model. I will further address this point below.

Using the mantle properties listed in Table 4.1, I calculate values for As, A e, and A c

of 7.62 109 W /K 4/3, 5.51 109 W jK 4!3 and 2.40 109 W /K 4!3 respectively. Here, I have


4 .3 . C h oice n f P a ra m eters------------------------------------------------------------------------------------------------------------- 136

used viscosity profiles based on the PG R inferences of Peltier and Jiang [1996]. p and g

axe taken from the preliminary reference eaxth model (PREM ) (Dziewonski and Anderson,

1981) and Cp is from a thermal model of Stacey [1977]. The CMB value of k is based on

measurements o f Osako and Ito [1991] and the 660 km dep th and surface values are from

K ieffer [1976]. I t will be noted th a t there is significant uncertainty in this param eter and

recent calculations of Hofmeister [1999] suggest th a t the surface value may be twice as large

while the value a t the CMB may be only half as large as those listed in Table 4.1. This low

value of k at the surface is useful as the fisted value of then represents a lower bound

and I will dem onstrate th a t even this relatively low value is difficult to reconcile w ith my

calculations, a is taken from the results of Chopelas and Boehler [1992].

O f the Ai, A s is the best constrained since near surface properties are significantly b e tte r

known than the properties in the deep m antle. In particularly, the value of the viscosity

in the deep m antle is still a m atter of significant debate while there is some consensus th a t

the upper mantle viscosity inferred from PG R observations is close to 5 1020 Pa s (e.g., see

Peltier, 1998, for a recent discussion).

Knowing the present day value of A s, I can calculate the tem perature in the upper

m antle and transition zone using the present day m antle component of surface heat flow

of 36 T W together w ith equation (4.3). I ob tain a tem peratu re on the upper m antle side

of the 660 km therm al boundary layer of 915.6 K. A geotherm with this A s is shown in

Figure 4.2 from calculations with both the num erical model and the param eterized model

(the geotherms w ith the very strong internal therm al boundary layers). Clearly, this value

o f As requires an extrem ely large tem perature drop across the internal therm al boundary

layer a t 660 km dep th in order to meet the constrain t of a tem perature of 1900 K a t this

dep th (shown w ith a 20 % error bar a t this depth). Recent results of Irifune [1998], however,

indicate tha t the tem perature a t the depth of the 660 km endotherm ic phase transition may

be significantly colder th an most previous results would have suggested which would make

th is value of A s som ewhat more acceptable. If I calculate A s under the assum ption of whole

m antle convection and I require the tem perature a t 660 km depth to be 1900 K, I arrive a t a

value o f 2.09 109 W/AT4/3. Examples of this ’whole-m antle’ geotherm axe also shown from

b o th the numerical m odel and the param eterized m odel in Figure 4.2. Although there is

uncertainty in the param eters in Table 4.1 th a t axe used to calculate As, it is unlikely th a t


4.3 . C h o ice n f P a ram eters________________________________________________________________________ 137

any, except possibly axe sufficiently in error as to allow such a large change in As.

This implies th a t if the circulation in the m antle is whole mantle or only weakly layered, one

of the assum ptions th a t went into deriving As is flawed. The most likely possibilities are: (1)

The viscosity of the m antle th a t governs the convection process is significantly greater than

th a t which governs the glacial rebound process. (2) The boundary conditions appropriate to

the E a rth ’s surface deviate from free-slip conditions in such a way as to significantly reduce

the efficiency of the surface therm al boundary layer in conducting heat, hence increasing

Ra-crit and reducting As. (3) The boundary conditions appropriate to the E a rth ’s surface

cause a reduction in the scaling exponent of the surface heat flow w ith the tem perature drop

across the therm al boundary layer. (1 ) is entirely plausible if the viscosity of the mantle

is in a strain-hardening regime and if the PG R process samples a transient rheology that

is significantly less stiff th an the viscosity th a t controls long time-scale processes such as

mantle convection, as has been argued by Karato [1998]. (2) could be a consequence of

effects due to large aspect ratio convection cells th a t are forced a t the surface boundary

(Lowman et al., 2000). (3) was strongly advocated by Christensen [1984] who also showed

th a t a reduced scaling exponent dependence of heat flow on Rayleigh num ber allowed for

more plausible E a rth therm al histories ( Christensen , 1985). Later calculations, however,

dem onstrated th a t strongly tem perature dependent viscosity and plate-like behavior do

not necessarily reduce this scaling exponent ( G um is , 1989). I will consider (1 ) and (2) as

possibilities and use As values of 2.09 109 W /K A Z and 7.62 109 W /K A/Z as end-member

values.

In Figure 4.2 I indicate two geotherms for the layered case calculated w ith the param e

terized model of B utler and Peltier [2000]. In the case of the dashed line I have used values

of As, A c and A e as calculated from (4.5) and the m antle properties listed in Table 4.1 bu t in

the case of the d o tted line, I have used a decreased A e of 2.52 109 W /K 4/ z. As can be seen

in Figure 4.2, this decrease in A e leads to significantly better agreement w ith the numerical

model in term s of prediction of the mantle geotherm. This improved agreement was also

found when A e was decreased in all of my other strongly layered solutions and predictions

of the surface heat flow were also improved. A e might be expected to be smaller th an is

predicted by (4.5) since (4.5) does not take into account the fact tha t upper and lower

mantle flow is blocked and hence an internal therm al boundary layer is a double boundary


4.3. C h o ice o f P aram eters 13&

layer. I will use this decreased value of A e during the rem ainder of this investigation and I

indicate this value in Table 4.1.

The in itial conditions are chosen by setting the upper mantle temperature to be close

to its m elting point of 2200 K. The tem perature profile in the mantle is assumed to be

initially adiabatic and the lower mantle tem perature is set accordingly. W hen possible, I

look for solutions in which the initial tem perature on the core side of the CMB is 4300 K

which is the m elting point of mantle materials at CMB pressures (Boehler, 2000) and hence

I might expect th a t a recently solidified, early mantle would have left the liquid core a t this

tem perature. I also examine solutions for which the core is made significantly warmer. The

gravitational energy released during core separation is roughly 13.4 103° J (e.g., Stacey

and Stacey, 1999) which is enough energy to raise the tem perature of the core by roughly

9000 K. The heat of core separation is likely divided between the core and mantle and much

of it was likely rad iated away from the early E arth . Hence, initial conditions in which the

core is many thousand degrees warmer than the m antle axe unlikely.

In all of my calculations, unless I explicitly sta te otherwise, I will use a value of A A

of 55 500 K. A lthough the activation energy and volume for mantle materials is not well

constrained and likely varies throughout the mantle, implying th a t AA cannot be well

known, this value lies in the plausible range (e.g., Weertman and Weertman, 1975) and

gives the required very strong dependence of viscosity on tem perature.

The values of <f>cme and 4>es are calculated from (4.6) assuming boundary layer thicknesses

of 100 km. T he function varies quite slowly and hence the choice of the thickness of the

boundary layer does not significantly affect the results. I find 4>cme = 0.75 and 4>es =

0.9156. Cpu and Cpi axe obtained by integrating Cp along an adiabat and axe found to be

9.76 1Q2GJ / K and 3.26 102 7 J/AT, respectively.

Properties of the core axe significantly less well known and I take a typical value of

1.5 1027J / K for C pc. I take the sum of the various la ten t heat effects to be 1 106 J /k g

(Buffett et al. [1996]) and the mass of the core to be 1.95 1024 kg. My prefered "value of Tf

is 3200 K, based on the melting point of iron at CMB pressures (Boehler, 2000), although

the calculations of A l fe [1999] indicate tha t this tem perature may be as high as 4500 K. To

is significantly less well known; I take a preferred value to be 4200 K. This num ber is based

on estim ates of the freezing point of iron a t the E a rth ’s center by Labrosse et al. [1997].


4.3. C hoice o f P aram eters______________________________________________________________________

They arrived a t this value by extrapolating the data of Poirier and Shankland [1993]. I

then extrapolate the ir value of 5270 K down a core adiabat to arrive at an estim ate for the

tem perature a t the core side of the CMB when the inner core would have begun to freeze.

In practice, however, it was necessary to ensure tha t the final model inner core was the same

size as the inner core th a t is observed today. Since I allow a 20 % deviation from required

temperatures in the E a rth ’s core, this m eans th a t final core tem peratures can be between

3200 K and 4800 K which, if the tem perature a t which the inner core begins to form is

held fixed a t 4200 K, represents all possibilities from no inner core at all to a completely

frozen core. As a result, I first ran calculations without the effect of inner core formation.

Then, once a reasonable model was found, the effects of the inner core were turned on w ith

T0 = Tcf + 138AT. Tcf was the final core tem perature of the previous run and 138 K was

chosen so tha t the inner core had the sam e size as exists at present. T j was chosen so as

to be 1000 K less than Ta. The to tal energy released dining inner core form ation from the

time of E arth form ation to the present is only 7.3 x 1028 J , (Stacey and Stacey [1999]),

which is the energy required to warm th e core by roughly 50 K. It was not found th a t the

effects of inner core form ation were especially significant.

In Figure 4.4 I show the ra te of in ternal heating in the E a rth ’s mantle th a t I assume and

the contributions from each of the m ajor radioactive elements. The numbers were derived

assuming abundance ratios of u ranium /thorium /potassium of 1/4/10000 following Zindler

and Hart [1986]. Using a prim itive upper-m antle uranium composition of 21 parts per

billion I get 19.4 T W for the bulk silicate earth . I next estim ate the heat-energy production

due to radio-active elem ents in the earth ls crust. If I take the Uranium concentration in

the continental and oceanic crusts and in ocean sediments to be 1.2, 0.045 and 2 parts

per million, respectively, I get heat production rates o f 6.3, 0.1 and 0.1. I sub tract these

heating rates from the to ta l for the e a r th to get roughly 13 TW tha t is available to drive

convection in the E a rth ’s m antle today, which gives a present day Urey ratio of 13/36=0.36.

In most of the calculations th a t include in ternal heating, I will use the heating ra te history

tha t is shown in Figure 4.4. Because I d o not include the effects of differentiation (to be

discussed more fully in w hat follows), this assumes that the E arth ’s crust formed, and hence

the crustal component of radio-active heating was removed from the mantle, very early in

E arth history. Geochemical studies (e.g., Armstrong, 1981) indicate tha t th e continental

139.


4.3. C h o ice o f P aram eters 140

70

Total Heat Production

U238

- -U235

T23203

-- K40

(!)(DB

35

a;

~ ~t ~ ■— i-4.55x10® -3.64x10® -2.73x10® -1.82x10® -9.1x10'

Time (yrs)

Figure 4.4: The to ta l internal radioactive heating ra te as a function of time (solid-line), as well as the contribution from each of the m ajor radioactive isotopes as indicated in the figure.


4.4 . R p sn lts M l

crust has been in a roughly steady sta te for the last 2.9 Gyrs and hence the crust was formed

very early. I also perform ed some calculations, however, in which the crustal component of

radio-active heating was removed a t various rates over the course of E arth history.

In some calculations in section 4.4.1, I will vary the Urey ratio of calculations. In varying

the Urey ratio, I will sim ply vary a m ultiplier of the heating history shown in Figure 4.4.

W hen I perform layered calculations, I will include 2/13 of the heat sources in the upper

m antle and 11/13 in the lower m antle in accord w ith geochemical estimates. I do not include

any internal heat sources in the core. I also neglect effects due to exchange of radioactive

heat sources between the lower and upper mantles.

4.4 Results

In section 4.4.1 I will describe results pertaining to simple one layer models in order to

illustrate a number of the m ost fundam ental features of the therm al history problem. In

subsequent sections I will first describe the results I have obtained when the convective

circulation is characterized by therm al boundary layers a t the surface and the CMB only

(whole mantle models), and then for circulation models which also possess an internal

therm al boundary a t 660 km dep th (layered models). In the final section I describe results

in which the degree of layering varies as a function of the system Rayleigh number. As I

will show, it is this la tte r class of models th a t my analyses will lead me to prefer.

4.4.1 One Layer Models

I begin by analyzing a simple, one-layer system w ith power law rheology described by

(4.12). A therm al evolution m odel for the E arth in which the mantle convective circulation

has a boundary layer a t the surface only, and therefore w ith a core th a t simply follows the

tem perature of the lower m antle, is governed by the single ordinary differential equation:

C p ^ - = - A s T -1 /3 (Trfestcme - Ts) ^ +^ 3. (4.15)


4.4. R psnlts 1 4 2

Here, Cp is the sum of the heat capacities of the whole m antle and the core. This equation

has the simple analytic solution:

W ) = {[m(0)<f>es<f>cme-Ts) - t i +lV3 + + T s}/(4>es4>cme).

(4.16)

for the case q =£ —1. Clearly, when q = — 1 the solution involves simple exponential decay of

the internal tem perature w ith time. Unlike simple exponential decay, the general solution

(4.16) actually diverges when extrapolated backward in tim e, at an instant prior to t = —oo

when

t = -(T tiO W esfcne - TS) ~ ^ V 3/ { ^ T - « / 3<4>es<t>crne)■ (4.17)

A very similar relation, for a somewhat less complicated model, has been derived previously

by Davies [1980]. In Figure 4.5, I show T)(f) for various values of q. representing the

sensitivity of the viscosity to internal tem perature. I note tha t as q increases, the time

a t which the tem perature diverges becomes more recent. In these calculations, I have set

Tv to have the value of the tem perature drop across the upper surface thermal boundary

layer a t t = 0 and T)(0) is set such as to give a present day heat flow of 36 TW . In these

calculations I use a value of As of 2.09 109W /K Af3 th a t is appropriate for ’whole-mantle’

convection. A q value of 30 is thought to be roughly appropriate for the E arth ’s mantle

(Davies, 1980), and it is clear th a t a ’therm al-catastrophe’ occurs in the geologically recent

past in this model even for significantly smaller values of q. The tim e a t which the solution

diverges also depends on the param eter As, however, and these calculations have employed

the small end-member value which leads to the most stable solutions in the past.

In order to avoid ’therm al catastrophes’, previous workers (e.g. Schubert et al., 1980;

Davies, 1980) noted th a t the addition of internal heating resulted in regular solutions that

extended further back in E arth history. In Figure 4.6 I dem onstrate this phenomenon. The

viscosity is only m oderately dependent on the tem perature (q = 5) and I include calculations

w ith the indicated fraction of the total heating shown in Figure 4.4. It can be seen tha t

when all of the geochemically constrained internal heating is in the mantle, the solution


4 4. R psnlts

10 ‘

8000

<o 6000 u<D( Xs<DE- q=0

— q= 1.5

q=3

q=6

q=10

4000

2 0 0 0 1 1 1 1 *____1— 1— 1— 1— — 1— 1___ 1___ 1 I 1____1— 1 1 *— 1—-4.55x10® -3.64x10® -2.73x10® -1.82x10® -9.1x10'

Time

Figure 4.5: The variation of in ternal tem peratures as function of tim e for various strengths of the tem perature-dependence o f m antle viscosity as param eterized by q in a model w ith no layering and in which the core tem perature simply follows the m antle tem perature. A s is set to 2.09 109W /K ‘1 and there is no internal heating. Solutions are calculated from(4.16).


4 .4 R psnlts 144

8000

— - 0.75 0.5 0.25

6000

<o(-<DA££>B-

4000

2 0 0 0 ■__i__i___i__I___i— i— i— i— I— i— i— i— i— >— ■— >— 1— 1— ■— '—-4.55x10® -3.64x10® -2.73x10® -1.82x10® -9.1x10'

T im e

Figure 4.6: Time evolution of m antle tem perature in a model w ith no layering and in which the core simply follows the tem p era tu re in the mantle. Tem perature dependence of viscosity is weak with q=5. Lines indicate calculations performed w ith various fractions of the total heating history indicated in F igure 4.4.


4.4 . R esu lts 1 4 5

becomes regular for the entire age of the Earth.

W ith very strongly tem perature dependent viscosity, there exists only a very narrow

range of internal heating rates in a one layer system which allow for the existence of regular

solutions for the lifetime of the E arth . In Figure 4 .7 ,1 show solutions with rheology law (4.9)

and for which effects due to the freezing of the inner core axe now present bu t there are still

no therm al boundary layers a t 660 km dep th or a t the CMB. The CMB therm al boundary

layer can be eliminated by setting A c to a very large value. T he lower, ’whole m antle’,

value of -As is used. I allow solutions which neither decrease nor increase in tem perature

by more than 300 K and which come w ithin 20 % of the final surface heat flow target of

36 TW . Solid lines indicate end-member solutions for extremes of allowable surface heat

flows and the dotted line indicates the result obtained for an integration starting w ith a

mantle th a t is 300 K warmer than is needed to deliver 36 T W of surface heat flow and

with the geochemically constrained value of internal heating. This la tter solution delivers

significantly too little surface heat flow for the present day. The Urey ratio of the two

end-member solutions is indicated and it will be seen tha t, in such a model, considerably

more internal heating is required than is suggested by geochemistry. Very similar results

have been noted previously by a number of authors (e.g. Schubert et al, 1980; Davies, 1980;

Christensen, 1985; Honda, 1995) and based on these calculations it has often been accepted

th a t the Urey ratio must be very large. Christensen [1985] noted th a t if the scaling exponent

of heat flow on the Rayleigh num ber is decreased, the required Urey ratio is smaller and a

larger range of Urey ratios is allowed. Decreasing the scaling exponent of heat flow on the

Rayleigh number has the same effect as decreasing the sensitivity of viscosity to tem perature

and this result was confirmed during the course of this study (not shown).

W ith solutions in which the viscosity is strongly sensitive to the mantle tem perature

and w ith the geochemically constrained internal heating rates, and in the absence of in

ternal therm al boundary layers, I also attem pted to increase the final surface heat flow by

increasing the initial mantle and core tem peratures. Because the removal of heat is ex

tremely efficient when mantle tem peratures are high, this added internal energy is almost

immediately removed and has almost no effect on the final tem peratures and heat flows.


4.4 . R esu lts 146.

2500

2300 (-<u

B 2100astna>g 1900a>E-

1700 [-

►fc

asa)K

Ur=0.93

Ur=0.75

1500

-4.7775x10®

2x10“

1.5x10“

10

5 x l0 13

Time (yrs)

Figure 4.7: a) Tem perature in a model with no layering and in which the tem perature of the core simply follows the tem perature in the mantle. Solid lines indicate end-member models where initial tem peratures were a t most 300 K different from the tem perature required to produce 36 TW of surface heat flow and for which the final surface heat flow is within 20 % of 36 TW . The Urey ratio (Ur) for each of these crises is indicated. The dotted line represents a calculation with the internal heating history as indicated in Figure 4.4. b) The surface heat flow corresponding to the same cases as in part a ).


4.4. R esu lts 1AZ

8500

« 6800

a 5100

a 3400

1700

1.5x10

25

21.5

01

1 R 1/ / i— i i i I i i— i— i— I— i— i— i i I i i i i I i__-4.55x10® -3.64x10® -2.73x10® -1.82x10® -9 .1 x 1 0 '

Time (yrs)

Figure 4.8: a) Tc (short-dashed line), T) (dotted line), and Tu (solid line) for a model w ith no layering, = 2.09 109 W /K * /z , A c = 2.40 109 W /K Af z , w ith an initial core tem perature of 8500 K, and w ith the heating history shown in Figure 4.4. b) Q s (dotted line), Qc (solid line), the heat liberated due to the secular cooling of the m antle (dash-dotted line) and the internal radio-active heating component (long-dashed line), for the same conditions as in (a), c) The viscosity o f the lower m antle (solid line) and upper m antle (short-dashed line) for the same case as described in (a).


4.4. R esu lts IAS.

4.4.2 Whole Mantle Convection

In Figure 4.8 I show an integration using the inefficient end-member value of A s of 2.09 x

109 W /K 4/ 3, and the value of A c derived by accepting typical values for m antle properties

o f 2.40 x 109 W /K 4 / 3 starting w ith a core tem perature of 8500 K. In this figure and those

th a t follow, the solid, dotted, and dashed lines in the top plate represent Tu, Ti and Tc. In

the middle plate the dotted and solid lines represent the surface and CMB heat flows while

the long-dashed and dot-dash lines represent the internal heating and secular cooling rate for

the entire mantle. In the bo ttom plate, the solid and dashed fines represent the viscosity

a t the CMB and surface, respectively. It can be seen th a t even w ith this unreasonably

high starting tem perature for th e E arth ’s core, the final surface heat flow and internal

tem peratures axe significantly too low. This is because the CMB therm al boundary layer

is relatively efficient and hence heat can flow rapidly from the core into the mantle. This

warms the mantle which results in low viscosities and hence rapid heat removal and very

high surface heat flow very early in E arth history. If the efficiency of the CMB therm al

boundary layer is decreased by a factor of 4.6 which corresponds to the effect of increasing

the viscosity in the vicinity o f the core by a factor of 1 0 0 , reasonable final surface heat

flows and internal tem peratures are now obtained as is dem onstrated in Figure 4.9. This

is because the core now cools m ore slowly, allowing for less removal of heat early on, but

warmer mantle tem peratures today. Although there is some trade-off between decreasing A c

and increasing the core tem perature, extensive searches of param eter space did not reveal

viable solutions with significantly cooler core tem peratures or larger values of A c. It is also

possible to derive models w ith somewhat larger modern day values for A c if the dependence

o f viscosity on tem perature is decreased. This causes the values of A c to be smaller for

times representing the early stages of E arth history. However, it was still found necessary

to have a very hot in itial core tem perature and the m odern day A c was still required to

be less th an that predicted from (4.5) when reasonable values for the m antle parameters

are employed. Similarly, calculations were performed for which the crustal component of

radio-active heating was removed over the course of E arth history (not-shown). Unless this

component was not removed until very recently, however, acceptable models were not found

w ithout strongly decreased A c and very high initial core tem peratures.


Results. 149

8500

6800<U3■§ 5100Va.56 1700

3400

1.5x10

1 0

5 x 1 0

2591

<d0 ^'5oo

21.5

91OOW

^ -I.%5xl0' 3.64x10' 2.73x10®Time

-1.82x10'(yrs)

9.1x10'

Figure 4.9: a) Tc (short-dashed line), T/ (dotted line), and Tu (solid line) for a model w ith no layering, A s = 2.09 109 W /K A/3, A c = 5.17 108 W /K 4/ 3, with an initial core temperature of 8500 K, and w ith the heating history shown in Figure 4.4. b) Qs (dotted line), Qc (solid line), the heat liberated due to the secular cooling of the mantle (dash-dotted line) and the internal radio-active heating component (long-dashed line), for the same conditions as in (a), c) The viscosity of the lower m antle (solid line) and upper mantle (short-dashed line), for the same case as described in (a).


4.4. R psnlts 15Q

4.4.3 Strongly Layered Convection

I now proceed to investigate the possibility tha t the internal tem perature constraints can

be met w ith the high efficiency coupling coefficients calculated for mantle properties which

include an internal viscosity distribution th a t fits the PGR constraint if there is a therm al

boundary layer at 660-km depth. In Figure 4.10 I show a calculation w ith the A* given

in Table 4.1, using ( 3 = 1 , corresponding to completely layered convection, and w ith an

initial core tem perature of 7500 K. For calculations th a t include significant layering, I have

added a long-dashed line to the top plate indicating the tem perature a t 660 km depth,

a short-dashed line in the m iddle plate indicating Qe/P , and a dotted line in the bottom

plate indicating r}e. It will be observed th a t even w ith complete layering, heat is removed

very rapidly early in the cooling history resulting in modern tem peratures and surface heat

flows th a t are significantly too low. The value of Ae is subject to significant uncertainty,

particularly since many mantle properties are changing rapidly in the vicinity of the phase

transition and, as I have previously discussed, I am already using a value th a t is significantly

smaller than mantle properties would suggest. In order to test whether an even smaller value

of Ae might allow for more reasonable therm al histories, I decreased this quantity until the

minimal surface heat flow constraint was met. The results of this calculation are shown

in Figure 4.11 and in the figure the decrease in Ae by a factor of 2.12 is indicated by an

increase in viscosity at this depth by a factor of 9.56. Again, it must be emphasized th a t

any other factor leading to less heat conducted across a therm al boundary layer w ith a

given tem perature drop could also account for this process. The surface heat flow in the

early history is now significantly reduced and the final temperatures and surface heat flow

are w ithin the required constraints. The tem perature drop tha t is required across 660 km

depth, however, is extremely large, a t 1856 K, since Ae is so strongly reduced. Also, it

can be seen tha t the lower m antle tem perature is now very high although it is still slightly

less than the melting tem perature for lower m antle materials. Spohn and Schubert [1982],

however, in their investigation of completely layered convection, calculated lower m antle

tem peratures that were warm enough th a t the lower mantle melted. They concluded on

this basis th a t the lower m antle m ust be depleted in radio-active elements.

I next investigate the possibility th a t a reasonable therm al history can be obtained

with a model with As and A e from Table 4.1 b u t w ith a significantly reduced Ac. It was


4.4 . R esu lts

8500

£ 6800

a 5100

ft 3400

1700

1.5x10

25

BO

21.5

3.64x10' 1.82x10' 9.1x10'2.73x10'Time (yrs)

Figure 4.10: a) Tc (short-dashed line), T) (dotted line), and Tu (solid line) and the tem peratu re at 660-km dep th (long-dashed-line) for a model w ith /? = 1 corresponding to complete layering. A s = 7.65 109 W / K A/Z, A e = 2.52 109 W / K ^ z and A e = 2.40 109 W /K 4/3, w ith an initial core tem peratu re of 7500 K, and w ith the heating history shown in Figure 4.4. b) Qs (dotted line), Qc (solid line), the heat liberated due to the secular cooling of the mantle (dash-dotted line) and internal radio-active heating component (long-dashed line), and Q e/fi (short-dashed line) corresponding to the to ta l heat flow across 660-km depth, for the same conditions as in (a). c) The viscosity of the lower m antle (solid line), upper m antle (short-dashed line), and 660-km depth (dotted line) for the same case as described in (a).


4-4- R p su lts 152

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F 6800

a 5100

0. 3400

1700

1.5x10

25

21.5

1.82x10'3.64x10' 2.73x10' 9.1x10'Time (yrs)

Figure 4.11: a) Tc (short-dashed line), T[ (do tted line), and Tu (sohd line) and the tem perature at 660-km dep th (long-dashed-line) for a model w ith (3 = 1 corresponding to complete layering, A s = 7.65 109 W /K 4/3, A e = 1.19 109 W /K A' Z and A c = 2.40 109 W /K **3, w ith an in itial core tem perature of 7500 K, an d w ith the heating history shown in Figure 4.4. b) Qs (dotted line), Qc (solid line), the heat liberated due to the secular cooling of the mantle (dash-dotted line) and internal radio-active heating component (long-dashed line), and Qe/ (3 (short-dashed line) corresponding to the to ta l heat flow across 660-km depth, for the sam e conditions as in (a), c) The viscosity of the lower mantle (solid line), upper mantle (short-dashed line), and 660-km dep th (dotted line) for the same case as described in (a).


4.4 . R esu lts

found tha t a therm al history model tha t is in accord w ith the surface heat flow and internal

tem perature constraints can be found if Ac is reduced by a factor of 4.93 with an in itial core

tem perature of 7500 K. T his decrease in A c is again indicated by a decrease in viscosity in

the lower plate in Figure 4.12. Although the lower m antle tem perature is now significantly

lower than in the case illustrated in Figure 4.11, the tem perature drop at 660 km dep th is

still large at 1042 K.

I also tested models for which the crustal radioactive heating was not removed until

later in Earth history for layered convection cases (not shown). Again it was found tha t

unless the crustal component was removed very recently, no models were found for which

A c was not strongly reduced and the initial core tem perature was not very high.

4.4.4 Intermediate models

The possibility tha t the A i should be strongly reduced and tha t partial layering was occur

ring was also investigated. I t was found that for a particu lar reduction in the m agnitude of

the A{, a range of degrees of layering was allowed and the degree increased as the A i were

increased. The upper bound on the degree of layering for a particular set of A i occurred

because the core and m antle tem peratures became too high, while the lower bound came

about because the final heat flow became too low. In Figure 4.13 I show one interm ediate

model in which all of th e A i axe decreased by a factor of 2.15 and /3 = 0.5 and the initial

core tem perature is 7500 K. I t was not found possible to obtain acceptable models for which

the initial core tem perature was not much greater th an the mantle tem perature, however.

4.4.5 Rayleigh-number dependent layering

In Figure 4.14 I show a calculation for which the in itial core tem perature is 4300 K.

The Ai are all decreased from their values in Table 4.1 by 2.92 and j3 depends on the

Rayleigh number in a way described by (4.14) w ith 7 set to 8.5. I have added a dash-

dot line to the top p late representing the m agnitude of the tem perature drop across the

internal thermal boundary layer. I t can be seen th a t an acceptable solution can now be

obtained w ithout requiring an initially very hot core. O f special interest is the fact tha t

the degree of layering in this solution, as measured by /?, is not especially laxge. W hen

a similar calculation was perform ed with a constant degree of layering set to be roughly


4.4. Results_______________________________________________________________________________ 154

8500

® 6800

a 5100

9* 3400

1700

1.5x10

25

21.5

3.64x10® -2 .7 3 x 1 0 '5x10' 1.82x10' 9.1x10'Time (yrs)

Figure 4.12: a) Tc (short-dashed hne), T; (dotted line), and Tu (solid line) and the teampera- ture at 660-km depth (long-dashed-Une) for a model w ith (3 = 1 corresponding to cormplete layering, A s = 7.65 109 W /K ^ 3, A e = 2.52 109 W /K 4 / 3 and A c = 4.87 108 W / K ^ 3, with am initial core tem perature of 7500 K, and with the heating history shown i n Figure 4.4. b) Qs (dotted line), Qc (solid line), the heat liberated due to the seculax coo ling of the m antle (dash-dotted line) and internal radio-active heating component (long-cdashed line), and Qe//3 (short-dashed line) corresponding to the to ta l heat flow across 6560-km depth, for the same conditions as in (a), c) The viscosity of the lower mantle (so lid line), upper m antle (short-dashed line), and 660-km depth (dotted hne) for the same c a se as described in (a).


4.4. Results 1 5 5 .

8500

£ 6800

«S 5100

& 3400

1700 ^

1.5x10

25

21.5

xlO® -3 .64x10® -2.73x10® -1.82x10® -9 .1 x 1 0 'Time (yrs)

Figure 4.13: a) Tc (short-dashed hne), Ti (dotted hne), and Tu (sohd line) and the tem perature a t 660-km dep th (long-dashed-line) for a model w ith /3 = 0.5, A s = 3.56 109 W / K ^ /z , A e = 1.17 109 W / K ^ z and A c = 1.12 108 W /K 4/ z, w ith an initial core tem perature of 7500 K, and w ith the heating history shown in Figure 4.4. b) Qs (dotted line), Qc (solid line), the heat liberated due to the secular cooling of the m antle (dash-dotted line) and internal radio-active heating com ponent (long-dashed hne), and Qe/P (short-dashed line) corresponding to the to ta l heat flow across 660-km depth, for the same conditions as in (a), c) The viscosity of the lower m antle (solid line), upper m antle (short-dashed hne), and 660-km depth (dotted line) for the sam e case as described in (a).


4.4. R esu lts 15fL

4500

3000

1500

26

24

22

20

180.80.60.40.2

3.64x10’ 1.82x10’ 9.1x10'x lO ’ 2.73x10'T im e ( y r s )

Figure 4.14: a) Tc (short-dashed hne), T/ (dotted hne), and Tu (sohd hne), the tem perature a t 660-km dep th (long-dashed-hne), and the tem perature drop across the in ternal therm al boundary layer (dash-dot line) for a model w ith A s = 2.62 109 W /K 4/ 3, A e = 8.63 108 W /K * /3 and A c = 8.22 108 W /K 4/ 3, with an initial core tem perature of 4300 K, and w ith the heating history shown in Figure 4.4. The layering is a function of the system Rayleigh num ber w ith 7 = 8.5 b) Qs (dotted hne), Q c (sohd hne), the heat liberated due to the secular coohng of the m antle (dash-dotted hne) and internal radio-active heating component (long-dashed hne), and Qe/P (short-dashed hne) corresponding to the total heat flow across 660-km depth, for the same conditions as in (a), c) The viscosity of the lower m antle (solid hne), upper mantle (short-dashed hne), and 660-km dep th (dotted hne) for the same case as described in (a), d) The degree of layering as measured by /? as a function of tim e for the case as described in a ).


Visc

osity

lo

g(Pa

s)

Tem

pera

ture

(K

)

4.4. R esu lts I5Z

the average degree of layering of the calculation displayed in Figure 4.14, the final s ta te

was significantly too cold (Figure 4.15). The reason for this large difference in behaviour

between two seem in g ly very similar physical models is th a t in the case of the time-variable

degree of layering calculation, the degree of layering decreases over the second half o f the

calculation due to the decrease in mantle tem peratures and internal heating rates, resulting

in a smaller tem perature drop between the lower and upper mantle. A decrease in the

tem perature drop across 660-km depth results in a decrease in the tem perature of the lower

m antle which, in turn , results in energy released from the lower m antle which is available to

warm the upper mantle and sustain a significant surface heat flow. Mantle tem peratures,

and hence mantle viscosities, for these solutions do not change significantly over the course

of E arth history and it therefore follows tha t Rayleigh number dependent layering provides

a strong buffer to mantle cooling.

No solutions of this type were found when the Ai were set to the values indicated in

Table 4.1, hence, solutions of this type require a mechanism whereby the efficiency w ith

which all of the therm al boundary layers conduct heat is reduced. Solutions with Rayleigh

num ber dependent layering and acceptable thermal histories were found for a wide range

of reduced A,-. W ith 7 = 8.5, solutions were found with the A{ reduced by factors of

2.92 to 5.22. Solutions of this type were also found for values of 7 ranging from 7.5 to 9,

indicating tha t the exact form of (4.14) is not im portant. W hat is physically im portant for

the buffering effect of tem perature-dependent layering to occur is th a t as the mantle cools

and heat sources extinguish, convection must pass through a regime where the degree of

layering decreases as a result.

Calculations were also performed w ith different internal heating histories in which the

crustal component of internal heating was extracted during the course of the calculation.

Larger internal heating early in E arth history had a somewhat greater effect in calculations

w ith Rayleigh number dependent layering than in calculations w ith constant mantle layering

because the effects of Rayleigh number dependent layering make the extraction of in ternal

energy less efficient. However, solutions were found th a t showed similarly constant surface

heat flow and m antle tem peratures over long periods of time. The A i for which acceptable

solutions could be found were slightly smaller than for the cases w ith the internal heating

history shown in Figure 4.4.


4 .4 . R p sn lts 158.

4500

3000

1500

26

24

22

20

180.80.60.40.2

2.73x10® -1.82x10' Time (yrs)

xlO1 3.64x10’

Figure 4.15: a) Tc (short-dashed line), 7} (dotted line), and Tu (solid line), the tem pera tu re a t 660-km depth (long-dashed-line), and the tem perature drop across the internal therm al boundary layer (dash-dot line) for a model w ith A s = 2.62 109 W /K 4/2. A e = 8.63 108 W /K 4/2 and A c = 8.22 108 W /K 4/2, w ith an initial core tem perature of 4300 K, and w ith the heating history shown in Figure 4.4. The layering is constant in tim e w ith = 0.1. b) Qs (dotted line), Qc (solid line), the heat liberated due to the secu la r cooling of the mantle (dash-dotted line) and internal radio-active heating component (long-dashed line), and Qe//3 (short-dashed line) corresponding to the to ta l heat flow across 660-km depth , for the same conditions as in (a), c) T he viscosity of the lower mantle (solid line), upper m antle (short-dashed line), and 660-km depth (dotted line) for the same case as described in (a). d) The degree of layering as measured by /5 as a function of tim e for the case as described in a).


Visc

osity

log

(Pa

s) Te

mpe

ratu

re

(K)

4.5. D iscu ssio n and C on clu sion s

Solutions were also found in which the degree of layering was significantly different. In

Figure 4.16, I show the results of a calculation w ith 7 = 7.5 and in which A s was set to

its ’whole-mantle’ value of 2.09 109W /K '1/ 3 and A c was decreased by a factor of 1.44. In

this case, it can be seen th a t the degree of layering is much greater than tha t shown in

Figure 4.14. However, many features such as the relatively constant mantle viscosities and

surface heat flow remain the same and these features were found to be characteristic of

solutions for which the effects of Rayleigh-num ber-dependent layering buffered the decrease

of interior tem peratures w ith time.

The final core tem perature in the calculation shown in Figure 4.14 is slightly greater

than th a t shown in 4.16. Both are w ithin the 20 % range of 4000 K although both are also

somewhat low. The cause of the difference in core tem perature is that A c is lower for the

calculation shown in Figure 4.14. O ther calculations (not shown) indicated tha t if A c is

further reduced, the final core tem perature is further increased. Also, since the core cools

more slowly when A c is smaller, this requires th a t the inner core has existed for a longer

period of time. If the geodynamo in the core is driven by compositional convection, and

hence requires the existence of the inner core, then the oldest known magnetized rocks of

age 3.5 G a (e.g., Dale and Dunlop, 1984) provide a lower bound on the age of the inner core.

The oldest inner core th a t I calculated in my models was only 2 Ga. However, the inner

core was seen to come into being earlier for models for which A c was strongly reduced. This

may argue th a t models for which A c is lower are preferred. Such a reduction in A c could

be caused either by mantle viscosity which is strongly increased over tha t which is inferred

on the basis of the PG R constraint or by a significant chemical heterogeneity in the lower

m antle th a t reduces the efficiency w ith which heat is transported, perhaps similar to the

one advocated by Kellogg et al. [1999].

4.5 Discussion and Conclusions

Based on the above described sequence of analyses, the possibility tha t the mantle has been

layered by a constant amount throughout geological history must be considered unlikely. In

all such cases, a core tha t is initially much w arm er th an the mantle is required in order to

meet constraints on the therm al history. For the case o f ’whole-mantle’ convection, A s and


4 .iv D is m is s io n a n d (~!r>nrliisinns IfiQ

►O&h

as0)«

4500

3000

1500

26

24

22

20

180.80.60.40.2

0 t i i i i I i____

.55x10® -3.64x10* 9.1x10*2.73x10' 1.82x10*Time (yrs)

Figure 4.16: a) Tc (short-dashed line), T[ (dotted line), and Tu (solid line), the temperature a t 660-km depth (long-dashed-line), and the tem perature drop across the internal therm al boundary layer (dash-dot line) for a model w ith A s = 2.09 109 W / K ^ z , A e = 2 109 W /K aIz and A c = 1.90 109 W /K 4/3, w ith am initial core tem perature of 4300 K, and with the heating history shown in Figure 4.4. The layering depends on the Rayleigh number w ith 7 = 7.5. b) Qs (dotted line), Qc (solid line), the heat liberated due to the secular cooling of the m antle (dash-dotted line) and internal radio-active heating component (long-dashed line), and Qef (3 (short-dashed line) corresponding to the to tal heat flow across 660-km depth, for the same conditions as in (a), c) T he viscosity of the lower mantle (solid line), upper m antle (short-dashed hne), and 660-km depth (dotted hne) for the same case as described in (a), d) The degree of layering as measured by /3 as a function of time for the case as described in a ).


Vis

cosi

ty

log(

Pa

s) T

empe

ratu

re

(K)

4.5. D ism ission and C onclusions_________________________________________________________________ I f i l

A c must be decreased from their values in Table 4.1 by factors of 3.46 and 4.64 respectively,

which may be interpreted as requiring very large increases in viscosity in tfrese regions. For

the case of convection w ith a non-zero but constant degree of layering, A s from Table 4.1

can be employed if A e is decreased by a factor of 2.12 or if A c is decreased by a factor of

4.93.

Another common feature of the constant-layering solutions is tha t a Large amount of

heat (around 13 TW ) is flowing from the modern-day core. As discussed previously, the

gravitational energy released during core formation provides enough heat to significantly

increase the core tem perature although a core th a t was initially a few thousand degrees

above the mantle tem perature seems unlikely, since the energy of core form ation was most

probably distributed between the m antle and the core. Clearly, addition of in ternal heating

to a cooler core could give a similar result as was previously dem onstrated by Spohn and

Breuer [1993]. Evidence from the solubility of potassium in core m aterial, however, suggests

th a t it is unlikely that there exists significant radioactive heating within the core. The

calculated CMB heat flow for constant layering models is also substantially greater than

the estim ate of the heat flow conducted down a core adiabat, according to Anderson [1998],

of 4.4 TW . The heat conducted down an adiabat should be considered a lower bound,

however, as there is likely a strongly super-adiabatic therm al boundary layer region on the

core side of the CMB. Similarly, estim ates of the heat flow carried by mantle plumes, thought

to originate at the CMB, is only roughly 2.4 TW (Davies, 1988; Sleep, 1S90). However,

Malamud and Turcotte [1999] proposed tha t the observable hot-spots may make up only a

small amount of the total and tha t if small unseen plumes are added, the to tal heat flux

from plumes may be as high as 15.8 TW . Also, studies which take into account only the heat

carried by plumes ignore the heat carried by large scale convection which likely accounts

for much of the mantle heat flow carried away from the CMB. The convective heat flow

required to sustain the dynamo in the core is not an especially useful constraint. Buffett

et al. [1996] estim ated tha t very little heat flow above th a t carried down the adiabat is

necessary to m aintain the dynamo as most of the energy can be derived from compositional

convection.

Solutions with time-dependent layering do not require an initially very ho t core, however,

and the final CMB heat flows are only around 5 TW , in accord w ith estim ates from hot


4 .5 . D ism ission and C on clu sion s__________________________________________________________________162

spots and from the core adiabat. These solutions do require, however, th a t all of the A*

are decreased by a factor o f a t least 2.92. Solutions were not found for which only one Aj

was reduced, suggesting th a t the reduced efficiency w ith which hea t is conducted across a

therm al boundary layer a t each horizon may have the same cause.

Honda [1995] considered the case of a transition from com pletely layered to completely

whole m antle convection. He concluded th a t for any transition th a t occurred earlier than 1

G yr ago, the effects of early layering would be completely ’forgotten’ due to the subsequently

vigorous ’whole-mantle’ convection. I performed some calculations o f this type (not shown)

and it was found that ju s t subsequent to the transition, an extrem ely intense pulse occurred

in the surface heat flow as the therm al boundary layer a t 660-km d e p th collapsed and energy

was released into the upper mantle. I f the time of this transition was sufficiently long ago,

the effects on the final s ta te were not, in-fact, found to be significant. W ith an on-going,

gradual, decrease in m antle layering, however, internal energy from the lower mantle is

slowly released into the upper m antle resulting in the buffering effect described in section

4.4.5. T he dem onstration of the im portance of this buffering effect on the therm al history

of tim e dependent layering is the m ain result of this chapter.

As m entioned previously, there is significant evidence from A rchean continental materi

als th a t the geotherm and surface heat flow in continents has not changed significantly over

the past 3.8 Gyrs. Since the internal heating rate is significantly less a t present than it was

in the d istan t past, many authors (e.g., Lenardic, 1998) have suggested th a t the heat flow

through the ocean floors was much greater in the past while the hea t flow through the con

tinents has stayed roughly the same. As I have dem onstrated in Figures 4.14 and 4.16, the

effect o f Rayleigh-number-dependent layering, if it exerts its’ influence in the way suggested

by the analyses presented herein, make it unnecessary for the relative heat flow through the

continents and oceans to have changed to any significant degree. T h is mechanism makes it

possible th a t the heat flow in all regions has not changed significantly over the last several

Gyrs.

P artia lly layered, present-day m antle convection is also in accord w ith results obtained

on the basis of direct seismic tom ographic imaging of the circulation (e.g., Van der Hilst et

al., 1997) which show th a t although some down-going slabs p en e tra te into the lower mantle,

others appear to be arrested in the ir descent a t the depth of th e 660 km discontinuity.


4 .5 . D iscu ssion and C onclusions_______________________________________________________________

Furtherm ore, high-pressure measurem ents of the tem perature a t 400 km depth suggest a

tem perature o f 1850 K (Boehler, 2000) which would preclude the existence of a strong

therm al boundary layer a t 660-km depth . Geochemical observations, however, suggest the

existence of distinct chemical reservoirs w ithin the mantle. O ne way to m aintain such

reservoirs is through mantle layering. A llegre [1997] suggested th a t an E a rth history model

in which m antle convection is initially strongly layered but is m ore weakly layered today

may reconcile geochemical and seismic tom ographic observations. Models of the type shown

in Figure 4.16, which are initially very strongly layered and become gradually less layered,

may allow a reconciliation between seismological and geochemical observations.

Rayleigh-number-dependent layering provides a novel mechanism whereby E arth history

models th a t are in accord w ith geochemical constraints on in ternal heating rates, mineral

physics constraints on mantle tem peratures, and observed surface heat flow become both

possible and plausible. Solutions of this type, and constant layering solutions, require

th a t therm al boundary layers are significantly reduced in the efficiency w ith which heat is

conducted for a given tem perature drop from a priori estimates based on fluid mechanical

numerical modeling when m antle viscosity is fixed to a m agnitude th a t is in accord with

th a t inferred on the basis of the P G R constraints. That the heat transfer efficiency of all

o f the therm al boundary layers must be reduced may be construed to suggest tha t the

viscosity th a t governs mantle convection is significantly greater th an th a t which governs

the post-glacial rebound process. This m ay be construed to im ply th a t the rheology of

the m antle is non-Newtonian and therefore th a t the PG R process is controlled by transient

creep ra ther than the steady sta te creep which controls the convection process and which

is governed by a higher viscosity as would be expected of a stra in hardening rheology such

as m ight be expected for a polycrystalline m aterial.

163


Chapter 5

Conclusions

A significant am ount of insight into the m antle convection process has been gained by using

different theoretical approaches to the convection problem. Where possible, I have tried

to use simpler models to explain some o f the phenomena seen in the more complicated

numerical model and all results have been compared to observations of the real Earth . In

chapter 2 , linear stability analysis was used in order to explain the result observed in the

numerical model th a t avalanches occurred after a boundary layer Rayleigh num ber exceeded

a value of approximately 700. It was found th a t internal therm al boundary layers on their

own axe extremely unstable and the instabilities th a t ensue have very long wavelengths and

hence are of avalanche type. In order to get stabilization such tha t the critical Rayleigh

num ber was similar to tha t seen in the more detailed numerical model, it was found necessary

to param eterize the convective m otion in term s of a Peclet number. S tabilization was

observed to increase with the m agnitude of the Peclet number, explaining the phenomenon

th a t layering increases with the degree of convective vigour.

A large num ber of simulations using th e detailed numerical model were also performed

in which I investigated the effects of various radial viscosity profiles, internal heating rates,

and m agnitudes of the Clapeyron slope of the 660-km depth endothermic phase transition.

T he effects th a t these have on the d istribu tion of mass flux events t r ansiting the 660-km

depth horizon, the geotherm, the heat flow, the surface velocity, as well as the boundary

layer thickness were investigated. In particu lar, it was found tha t Earth-like heat flows

occurred only when the model mantle viscosity was chosen to be much greater th an the

viscosity suggested on the basis of analysis of PG R or when convection was very strongly

164


5.1. Detailed Description o f Rayleigh Number Dependent Layering Thermal Histories 165

layered. I also developed a simple parameterized m odel o f convection in order to explain a

number o f the trends seen in the numerical model concerning the geotherm and the surface

heat flow.

This param eterized model was further used to investigate various therm al history sce

narios and to try to distinguish between the possibilities of high viscosities and strong

convective layering. W hen convective layering was constant in tim e, it was found th a t the

initial core tem peratu re was required to be unreasonably high in order for the final heat

flow of the model to be the same as the heat flow observed on E a rth today. If the degree

of layering was m ade to be a function of the Rayleigh num ber of convection, in a m anner

suggested by the num erical simulations, then it was not found necessary to use a very high

initial core tem perature. Rayleigh number dependent layering results in a novel buffering

mechanism which I will discuss in greater detail in the following section. As the m antle

cools and the Rayleigh num ber decreases, the degree of layering falls which releases heat

from the lower m antle into the upper mantle. As a result, the upper m antle tem perature

and surface heat flow rem ain relatively constant for much of the age of the E arth , despite

the diminishing streng th of internal heat sources. All acceptable param eterized modeling

solutions require th a t the therm al boundary layers in the m antle be inefficient in the con

duction of heat. This argues th a t the viscosity th a t is characteristic of m antle convection

may be significantly greater th an th a t which is characteristic of the P G R process and hence

that mantle viscosity may be non-Newtonian.

5.1 Detailed Description of Rayleigh Number Dependent Lay

ering Thermal Histories

As the buffering mechanism described above and in chapter 4 is potentially a very significant

result, I will discuss the mechanism by which it occurs in some detail. If we examine

Figure 3.17b, it can be seen th a t as the degree of layering increases when the lower boundary

tem perature is held fixed, the lower mantle becomes progressively warm er while the upper

mantle becomes progressively cooler. W ith this in m ind, we can examine Figure 5.1 which

shows geotherms a t various times from solutions corresponding to late and early times

in the simulations shown in Figure 4.14. In the very early stages, radio-active heating


5.2. F u tu re C on sid eration s_______________________________________________________________________ lfifi

and heat flow from the core to the mantle result in an increase in the system Rayleigh

num ber which results in an increase in convective layering. As a result, the lower m antle

tem perature increases, despite the decrease in the tem perature of the core. This can be seen

in Figure 5.1a. In the second half of the calculation, as the radioactive heat sources decrease

in strength and the m antle tem peratures begin to cool, the system Rayleigh num ber and

the degree of layering decrease. As a result, the tem perature in the lower m antle begins to

decrease. However, heat is then given to the upper m antle as the degree of layering decreases

which warms the upper mantle. In this calculation, and many like it, this w anning of the

upper m antle due to a decrease in convective layering offsets the decrease in the upper

m antle tem perature due to the decrease in the core tem perature. The calculation shown in

Figure 5.2 shows geotherms corresponding to the sam e times as those shown in Figure 5.1

bu t for a calculation in which the degree of layering was held constant (the sam e case as in

Figure 4.15). It can be seen th a t, unlike in the variable layering case, the m antle and core

cool very rapidly early on because the degree of layering and lower mantle tem peratu re do

not increase, and as a result, the heat flow' from the core remains large. Similarly, since

the degree of layering does not decrease over the second half of the calculation, the upper

mantle tem perature continues to decrease rapidly during the second half of the calculation.

5.2 Future Considerations

A natural next step in the process of further refining our understanding of m antle dynamics

would be to dem onstrate th a t the buffering mechanism discovered in the context o f the

therm al history analysis also operates in the detailed numerical model. This would require

some modification of the numerical model in order for it to allow for tim e varying in ternal

heat sources, a floating core tem perature, and a viscosity th a t varies as a function of m antle

tem perature (and perhaps stress). If the viability o f this mechanism were to be confirmed

in this way, it would lend substantial credibility to the idea that this is an im portan t effect

in the E a rth ’s therm al history as I have argued in chapter 4.

The calculations presented in chapter 2 indicate th a t if heat can be efficiently advected

away from an internal therm al boundary layer, th en it should be stabilized against m antle

avalanches. Hence, if the viscosity of the m antle is decreased in the vicinity of the en-


5.2 . F u tu re C on sideration s----------------------------------------------------------------------------------------------------------- IfiZ

6400

5400

1=—4.55 Gyrs t= —4 Gyrs 1=—3.5 Gyrs t= —3.0 Gyrs 1=—2.5 Gyrs

T34400

8400

g 5400

1=—2 Gyrs t= —1.5 Gyrs 1=—1 Gyrs t= —0.5 Gyrs 1=—0.0 Gyrs

4400

340034402580 4300860 1720

Temperature

Figure 5.1: Geotherms calculated for the times indicated corresponding to times during the sim ulation whose results are shown in Figure 4.14.


5.2. Future C on sid eration s

6400

g 5400 JX

1=—4.55 Gyrs t= —4 Gyrs 1=—3.5 Gyrs t= —3.0 Gyrs 1=—2.5 Gyrs

COg

I 4400

8400

g 5400

1=—2 Gyrs t= —1.5 Gyrs 1=—1 Gyrs— t= —0.5 Gyrs

1=—0.0 Gyrs

w d

• H

£ 4400

3400 43002580 3440860 1720Temperature

Figure 5.2: G eothenns calculated for the times indicated corresponding to times during the simulation whose results are shown in Figure 4.15.


5.2 . F u tu re C on sid eration s______________________________________________________________________

dotherm ic phase transition, due to the effects of transform ational super-plasticity, then it

may b e that an endotherm ic phase transition of a given Clapeyron slope will be significantly

enhanced in its ability to cause layered convection. Considerable effort has already been

expended in trying to dem onstrate this phenom enon in the detailed num erical model; the

calculations have been found to be fraught w ith numerical instability, however. I t would

still be useful to dem onstrate this effect, as it is possible th a t most previous numerical cal

culations have significantly underestim ated the degree of mantle layering caused by a phase

transition with a given Clapeyron slope. A lthough the buffering mechanism described in

chap ter 4 does not require very strong convective layering, the m agnitude of m antle layering

is still a significant issue in interpreting geochemical observations.

F uture progress in m antle convection sim ulations will require more com plicated models

th a t m ore closely resemble the real E arth . At present, three-dimensional, spherical, convec

tion models in which the m antle is convecting as vigorously as the real E a rth and in which

effects of tem perature-dependent viscosity are taken into account are still com putationally

prohibitive but should become viable in the neax future. A trade-off always exists, however,

betw een the degree of realism in the m odel and our freedom to explore a range of different

m odel param eters.

O ne significant problem in mantle convection sim ulations involves m aking surface plates

in a self-consistent m a n n er. Tem perature dependent viscosity acting on its own in convec

tion simulations has been shown to form a rigid-lid in which the upper boundary becomes

im m obile (e.g., Christensen, 1984). A lthough this may be the regime in which the planet

Venus is currently operating, a planet which shows no evidence of active plate tectonics,

it is clearly not appropriate for the E arth . Progress has recently been m ade by Trompert

and Hansen [1998], by incorporating a yield stress into the rheology, however, the surface

m otions that result are not entirely plate-like as they lack true transform faults. Tackley

[1998] has also m ade progress by assum ing a rheology th a t weakens w ith stra in rate. If

num erical models could be shown to deliver plate-like surface motions in a self-consistent

m a n n er, it would go a long way toward explaining much of the complicated dynamics tha t

occurs a t the E a rth ’s surface.

Finally, it would be useful to have a num erical model in which differentiation and effects

due to mixing of trace elements are taken into account. As mentioned previously, there is

i s a


5.2. Future Considerations______________________________________________________________

an apparent contradiction in the evidence from seismic tomography and from geochemical

mixing models in term s of the degree of layering th a t is imposed upon the circulation by the

phase transform ation a t 660-km depth. Geochemists continue to employ relatively crude

box models in which all reservoirs or boxes are assum ed to be rapidly homogenized. I t would

be useful to have more detailed models available in order to investigate the consequences of

more realistic circum stances in which the various reservoirs are not completely homogenized

so th a t we might b e tte r understand the degree of spatial variability tha t might ensue. If

effects due to p a rtia l melting and differentiation were also added to such a model, th is might

also provide insight into the time of continental crust formation and mantle depletion.

In this thesis, I have argued that the E a rth ’s m antle is a t least partially layered and th a t

its rheology is non-Newtonian. These argu m e n ts are based on the results of highly idealized

calculations, however. Observations of the E a r th ’s deep interior axe by necessity indirect

and suffer s im ila r ly from uncertainties. Acceptance of a particular model concerning m otion

in the E a rth ’s in terior will require a convergence of evidence from a number of disciplines of

which geodynamics is only one. There is hope th a t as new and more powerful technologies

and techniques are brought to bear on this im portan t problem, that such a convergence will

emerge.

170


6

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Convection in Earth’s Mantle · 2016. 12. 1. · Convection in Earth’s Mantle: impacts of solid-solid phase transformations and crystalline rheology Doctor of Philosophy, 2001,