Download thermal state of continental and oceanic lithosphere

THERMAL STATE OF CONTINENTAL AND
OCEANIC LITHOSPHERE
by
Derrick P. Hasterok
A dissertation submitted to the faculty of
The University of Utah
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Geophysics
Department of Geology and Geophysics
The University of Utah
August 2010
c Derrick P. Hasterok 2010
Copyright All Rights Reserved
THE UNIVERSITY OF UTAH GRADUATE SCHOOL
STATEMENT OF DISSERTATION APPROVAL
The dissertation of
Derrick P. Hasterok
has been approved by the following supervisory committee members:
, Chair
21 May 2010
John Bartley
, Member
21 May 2010
Barbara Nash
, Member
21 May 2010
Philip E. Wannamaker
, Member
21 May 2010
Richard C. Aster
, Member
21 May 2010
David S. Chapman
and by
the Department of
D. Kip Solomon
Geology and Geophysics
and by Charles A. Wight, Dean of The Graduate School.
, Chair of
ABSTRACT
The thermal state of the continental and oceanic lithosphere is reassessed on the
basis of new databases for global heat flow and lithospheric heat production, recent
advances in thermophysical properties measurements of minerals at high pressures and
temperatures, and a better understanding of convective heat loss in young seafloor.
The updated global heat flow database incorporates >60,000 records with >44,800
heat flow determinations. The update significantly increases the quantity and spatial
coverage of global heat flow data since the last update in 1993. A new family of
continental geotherms is proposed that is parametric in surface heat flow and takes
advantage of thermophysical property data. The range of geotherms is constrained
by xenolith P –T estimates; a cratonic geotherm consistent with a surface heat flow of
40 mW/m2 is particularly well constrained. Upper crustal heat production represents
∼26% of the total surface heat flow. Average heat production for the continental lower
crust and mantle are 0.4 µW/m3 and 0.02 µW/m3 , respectively. Recent controversy
about the interpretation of heat flow observations in young seafloor is resolved by
careful filtering of data based on sediment thickness and distance from seamounts
and weighting marine studies where the environment of heat flow measurements is
carefully documented. Oceanic geotherms, fit to bathymetry and heat flow data, are
produced for a plate model with 7 km thick crust, a plate thickness of 95 km, and
mantle potential temperature of 1425◦ C. While the current estimate of global heat
loss (44 TW) is reasonable, these new reference models will be instrumental in refining
and estimating uncertainty in the solid Earth’s global heat loss.
CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
CHAPTERS
1.
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2.
AN UPDATED GLOBAL HEAT FLOW DATABASE . . . . . . . . . .
4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Updated Global Heat Flow Database . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
HEAT PRODUCTION AND GEOTHERMS FOR THE
CONTINENTAL LITHOSPHERE . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
11
12
13
14
16
21
27
38
3.
4.
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geotherm Sensitivity to Heat Production . . . . . . . . . . . . . . . . . . . . . .
Previous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Observational Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
OCEANIC HEAT FLOW: IMPLICATIONS FOR GLOBAL HEAT
LOSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Thermal Model of Sea-floor Spreading . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Observed Heat Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Global Filtering—Sediments and Seamounts . . . . . . . . . . . . . . . . . . . .
4.7 Environmental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Corrections to Global Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Global Heat Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
40
41
44
45
48
57
61
64
66
5.
6.
PLATE COOLING MODELS FOR THE OCEANIC
LITHOSPHERE: ARE COMPLEXITIES NECESSARY? . . . . . . .
68
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Note on McKenzie et al. [2005] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
68
69
76
82
96
97
CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
APPENDICES
A. RADIATIVE DIFFUSIVITY AND CONDUCTIVITY OF
OLIVINE REVISITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B. THERMAL CONDUCTIVITY OF AMPHIBOLES:
INFLUENCE OF COMPOSITION . . . . . . . . . . . . . . . . . . . . . . . . . . 112
C. MODELS OF THERMAL CONDUCTIVITY FOR
INDIVIDUAL MINERALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
D. EMPIRICAL CONSTANTS USED TO MODEL
PHYSICAL PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
E. SIMPLIFIED CONTINENTAL GEOTHERMS . . . . . . . . . . . . . . . . 135
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
iv
LIST OF FIGURES
2.1 The global, continental and oceanic heat flow distributions. . . . . . . . . .
6
2.2 Heat flow locations in the updated global heat flow database. . . . . . . . .
7
3.1 Temperature sensitivity to 25 and 50% variations in heat production. . .
14
3.2 Heat production models of the lithosphere. . . . . . . . . . . . . . . . . . . . . . .
15
3.3 Heat production from granulite terranes and mantle xenoliths. . . . . . . .
19
3.4 Estimated P –T conditions for mantle xenoliths using the PBKN and
TBKN barometer and thermometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.5 Misfit of heat production models to compositionally adjusted elevation.
28
3.6 Best-fitting thermal isostatic models for each of the heat production
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.7 Fits of preferred geotherm family (P = 0.74) to xenolith P –T estimates
for the Kalahari craton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.8 Lithospheric thickness and heat loss. . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.1 Oceanic datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2 Thermal isostasy of the oceanic lithosphere. . . . . . . . . . . . . . . . . . . . . .
45
4.3 Observed oceanic heat flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.4 Effect of sediment cover on hydrothermal circulation and heat flow
through the oceanic lithosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.5 Median heat flow versus age in 2 m.y. bins divided into groups with
50 m increments of sediment thickness. . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.6 Metrics for improvement in globally filtered datasets as a function of
minimum sediment thickness and minimum distance to seamounts. . . .
52
4.7 Global filtering results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.8 Heat flow case study from Juan de Fuca flank. . . . . . . . . . . . . . . . . . . .
58
4.9 Filtered heat flow adjusted for sedimentation and thermal rebound. . . .
60
4.10 Estimated heat flow response to sedimentation. . . . . . . . . . . . . . . . . . . .
62
4.11 Estimated fraction of thermal rebound as a function of sediment cover
and time since cessation of hydrothermal circulation. . . . . . . . . . . . . . .
63
4.12 Estimated global advective power loss from Monte Carlo analysis with
106 realizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
5.1 Effective thermal expansivity model. . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.2 Influence of plate thickness variations on subsidence and heat flow. . . .
83
5.3 Influence of mantle potential temperature variations on subsidence and
heat flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.4 Influence of crustal thickness variations on subsidence and heat flow. . .
86
5.5 Slices through the misfit surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.6 Preferred plate cooling model with a crustal thickness of 7 km, plate
thickness of 90 km and potential temperature of 1425◦ C including heat
production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5.7 Model sensitivity to selected parameters. . . . . . . . . . . . . . . . . . . . . . . . .
92
A.1 Thermal diffusivity of olivine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2 Radiative diffusivity models and data for olivine. . . . . . . . . . . . . . . . . . 107
A.3 Comparison of geotherms computed from several radiative conductivity
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B.1 Thermal conductivity of amphiboles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B.2 Thermal conductivity of solid solution minerals . . . . . . . . . . . . . . . . . . . 120
B.3 Influence of amphibole composition on lithospheric temperatures. . . . . . 121
C.1 Effective thermal conductivity of quartz. . . . . . . . . . . . . . . . . . . . . . . . . 124
C.2 Effective thermal conductivity of muscovite. . . . . . . . . . . . . . . . . . . . . . 125
C.3 Effective thermal conductivity of orthoclase. . . . . . . . . . . . . . . . . . . . . . 125
C.4 Lattice thermal conductivity of plagioclase. . . . . . . . . . . . . . . . . . . . . . . 126
C.5 Lattice thermal conductivity of orthopyroxene. . . . . . . . . . . . . . . . . . . . 127
C.6 Lattice thermal conductivity of olivine. . . . . . . . . . . . . . . . . . . . . . . . . . 127
C.7 Lattice thermal conductivity of garnet. . . . . . . . . . . . . . . . . . . . . . . . . . 129
C.8 Lattice thermal conductivity of spinel. . . . . . . . . . . . . . . . . . . . . . . . . . . 130
vi
LIST OF TABLES
2.1 Number of heat flow measurement sites . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2 Example records to the updated global heat flow database. . . . . . . . . . .
9
3.1 Compostional model used to compute geotherms. . . . . . . . . . . . . . . . . .
22
4.1 Heat flow from environmental analysis. . . . . . . . . . . . . . . . . . . . . . . . . .
58
5.1 Parameters for sediment correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5.2 Global bathymetry and heat flow data. . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.3 Compositional model for the oceanic lithosphere. . . . . . . . . . . . . . . . . .
87
5.4 Compositional model for the oceanic lithosphere. Values as molar fraction of approximate end-member mineralogy. . . . . . . . . . . . . . . . . . . . . .
94
B.1 Amphibole compositions and conductivity. . . . . . . . . . . . . . . . . . . . . . . 115
B.2 Fitting constants for estimating conductivity coefficients . . . . . . . . . . . . 117
D.1 Physical properties and empirical constants for mineral end-members. . 132
D.2 Empirical constants for estimating conductivity. . . . . . . . . . . . . . . . . . . 133
D.3 Empirical constants for computing heat capacity of mineral end-members.134
E.1 Empirical conductivity constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
E.2 Empirical expansivity constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
ACKNOWLEDGEMENTS
Many people helped make this work possible by contributing their own data and
compilations to this database. The following researchers made substantial contributions to this end. Therefore, I would like to thank Francis Lucazeau, Jeffery Poort,
Bruno Goutorbe, Dallas Abbott, Carol Stein, Valiya Hamza, Maria Richards, Jacek
Majorowicz, Graeme Beardsmore, Niels Balling, Trond Slagstad, Andrea Förster,
Bruno Della Vedova, Roman Kutas, Georg Delisle, Karla Bojadgieva, and Shaopeng
Huang.
Xenolith P –T databases provided by Roberta Rudnick and Derek Bell helped
tremendously in modeling continental geotherms.
Discussions with Juan Carlos
Afonso, Jun Korenaga, Tom Shankland, and Christopher Grose were helpful in modeling rock properties used in the continental and oceanic studies.
During the course of this research, I needed to collect hundreds of references for the
global heat flow database, the xenolith database, and a crustal thickness database
currently under construction for global heat flow modeling. The document delivery
service and interlibrary loan at the University of Utah were indispensible and saved
me weeks of hunting down articles and reports.
I would like to thank Lucy Flesch at Purdue University for use of computing
resources which proved invaluable at the end stages.
The members of my committee, Barbara Nash, John Bartley and Rick Aster,
provided excellent guidance that helped shape my scientific understanding of many
subjects covered and related to this work, both through coursework and discussion.
I am happy to call each a mentor and look forward to many fruitful discussions in
the future. I would especially like to thank my advisors David Chapman and Phil
Wannamaker for being encouraging, understanding and excellent role models for a
young scientist.
Most of all, I would like to thank my lab mates, Paul Gettings, Michael Davis,
Melissa Mursbach, Imam Raharjo, Bryce Johnson and Christian Hardwick, and my
wife Kristine Nielson for fielding numerous questions and giving excellent suggestions
which helped improved the quality of this research.
ix
CHAPTER 1
INTRODUCTION
My Ph.D. research seeks a better understanding and description of the thermal
state of the continental and oceanic lithosphere. Principal components of this study
of the continental lithosphere involve assembling a comprehensive, expanded database
of heat flow observations, a vertical heat production profile for the continental lithosphere, a family of continental geotherms parametric in surface heat flow, and
constraining geotherms using thermal isostasy and xenolith P –T estimates. For
oceanic lithosphere, I focus on filtering heat flow data to reveal the background
conductive heat loss of the oceanic lithosphere, estimating the spatial extent and
magnitude of redistribution of heat in young seafloor by hydrothermal processes, and
using plate cooling models with P –T dependent thermophysical properties to model
heat flow– and subsidence–age patterns for the seafloor.
Chapter 2 describes the expanded global heat flow dataset assembled for this study.
Since the last update, significant advances have been made in the experimental study
of mineral behavior under a large range of P –T conditions. Using this updated
database and improved estimates of thermophysical properties, the distribution of
heat loss within the lithosphere can be revisited in a substantial and substantive way.
Chapters 3 through 5 illustrate the beginning stages of this reassessment. These
reference thermal models calibrate observables, such as elevation/bathymetry, which
can then be used as predictors of surface heat flow in data-poor regions.
Chapter 3 explores the average contributions of radioactive heat generation to
temperature profiles and heat loss out of the continental lithosphere. Because temperature governs many physical and chemical states and thermophysical properties
that influence other geologic processes, it is important to establish reference geotherm
2
models; radiogenic heat production is shown to be one of the most important properties for computing lithospheric temperatures and heat flow across the asthenosphere/lithosphere boundary.
Chapter 4 examines a systematic bias in oceanic heat flow data resulting from
a fluid infiltration in sediments where measurements are collected. The background
conductive heat loss of the oceanic lithosphere is estimated by filtering heat flow both
globally and in several high resolution studies. Emphasis is placed estimating the
redistribution of heat in young seafloor due to hydrothermal circulation of seawater
through the oceanic lithosphere.
Chapter 5 computes a number of plate-cooling models, which explore the importance of physical complexities and the addition of a crustal layer not commonly
included in these models. The focus is on developing an average cooling model for the
oceanic lithosphere that fits observed heat flow– and bathymetry–age relationships.
The families of geotherms developed in this study incorporate several recent advances in describing pressure and temperature effects on thermal conductivity, thermal diffusivity, expansivity, density, and specific heat capacity. Because data and
information are drawn from multiple sources, the thermophysical information is compiled in five appendices.
Appendix A examines the radiative effect on thermal conductivity and diffusivity
of olivine. This corrects an error in the model that is most commonly employed.
Appendix B proposes a method for determining a relationship between composition
and thermal conductivity for amphiboles.
Appendix C includes figures showing the pressure and/or temperature sensitivity
of thermal conductivity for a number of minerals used to calibrate the model used
throughout this work. Model fits are shown with the thermal conductivity data used
to calibrate them.
Appendix D lists the empirical constants necessary to estimate P –T dependence
of thermophysical properties used to model continental and oceanic geotherms and
isostasy.
3
Appendix E provides empirical constants that can be used to more easily compute
the geotherm family derived in Chapter 3.
CHAPTER 2
AN UPDATED GLOBAL HEAT FLOW
DATABASE
2.1
Introduction
Advances in understanding the physical and chemical state of the Earth and its
evolution have often followed the growth and increased spatial coverage of global
geophysical data sets. For example, recognition of the geometry of plate boundaries followed the accumulation of earthquake locations in a catalog, and the details
of sea-floor spreading were revealed after accumulating vast quantities of marine
magnetic data.
The global heat flow dataset has likewise been instrumental in
identifying relationships between heat flow and tectonics on land, the importance
of heat production for analysis of heat flow patterns, relationships between heat flow
and age on the sea-floor, and the role of hydrothermal circulation in redistributing
heat in young oceanic lithosphere.
2.2
Updated Global Heat Flow Database
The first version of the global heat flow database contained a mere 63 measurements,
43 on the continents and 20 on the oceans [Birch, 1954]. The most recent update by
Pollack et al. [1993] contained 20,201 heat flow determinations with roughly equal
fractions on the oceans and continents. The database used in this study now contains
60,398 records (58,008 heat flow determinations), with approximately 25% oceanic
and 75% continental data (Table 2.1). About 9,300 sites are located on the continental
shelves which nearly triples these data. Many of the new continental data come from
14,483 BHT (bottom hole temperature) estimates in the Gulf of Mexico oil states and
Cordillera foreland basin [Majorowicz et al., 1999; Blackwell and Richards, 2004].
5
Table 2.1. Number of heat flow measurement sites
Reference
Birch [1954]
Lee [1963]
Lee and Uyeda [1965]
Horai and Simmons [1969]
Lee [1970]
Jessop et al. [1976]
Chapman and Pollack [1980]
Chapman and Rybach [1985]
Pollack et al. [1993]
This study
Continentala
43
73
131
474
597
1,699
2,808
3,601
10,337 (13,249)
33,693 (43,060)
Oceanic
20
561
913
2,348
2,530
3,718
4,409
5,181
9,684 (6,952)
24,315 (14,948)
Total
63
634
1,044
2,822
3,127
5,417
7,217
8,782
20,201
58,008
a
Values in parentheses breakdown continental and oceanic sites when marine data
on continental shelves are reclassified as continental.
The most frequent occurence of heat flow is between 50 and 55 mW/m2 for the
global distribution as well as the continental and oceanic subsets (Figure 2.1). The
global median heat flow is 65 mW/m2 . Heat flow in the ocean basins (79 mW/m2 ) is
higher than the global average with a broader distribution and a long tail containing
several observations in each bin up to 1000 mW/m2 . The resulting intraquartile
range is large for the oceanic observations (∼139 mW/m2 ). Continental heat flow
shows a sharper peak than the oceanic data with a rapid decline in the number
of observations above 150 mW/m2 . The intraquartile range for the continents is
therefore much smaller (37 mW/m2 ). It should be noted that these statistics are not
necessarily the true continental or oceanic heat flow because spatial bias inherent in
this dataset occludes the true spatially averaged heat flow.
The updated global heat flow database significantly improves the data coverage
on the continents, particularly in west Africa, China, South America and in the
Canadian shield (Figure 2.2). Improved spatial coverage in the oceans is most evident
on the Antarctic Plate, north-central Pacific, Atlantic east of Argentina and the Arctic
Ocean. However, significant gaps in data still persist, with few or no data reported in
central Africa, Middle East, Brazilian rainforests, Argentina, and Himalayas. Gaps
in oceanic data occur over much of the southern Oceans, Arctic Ocean, and Pacific.
6
3500
All Data
N: 44861
Quartiles: [48.0, 65.1, 97.6] mW/m2
Number of Observations
3000
2500
Continental
N: 29868
Quartiles: [46.0, 62.3, 83.2] mW/m2
2000
Oceanic
N: 14993
Quartiles: [50.6, 79.0, 189.0] mW/m2
1500
1000
500
0
0
0
0
0
150
0
100
50
100
150
Heat Flow [mW/m2]
200
150
250
200
200
250
250
Figure 2.1. The global, continental and oceanic heat flow distributions. Median
values are shown in bold and indicated by solid lines on histograms. BHT data from
Majorowicz et al. [1999] and Blackwell and Richards [2004] are excluded to prevent
strongly biasing the histogram. Each histogram shift is 500 observations.
7
Figure 2.2. Heat flow locations in the updated global heat flow database. Data
from Pollack et al. [1993] (blue) and this update (red). Plate boundaries shown as
thin black line. Oceanic regions are shown in white with light grey and dark grey
regions for the subaqueous and subaerial continental regions, respectively. Data are
presented using a Robinson projection.
Only a handful of heat flow determinations exist on Greenland and Antarctica
where rocks are not covered by ice sheets. The extensive cover by ice sheets makes
determinations of heat flow logistically more challenging. I add three additional heat
flow determinations to the global dataset that are made in ice-boreholes on slow
moving ice sheets that may reflect the underlying lithospheric heat flow.
I removed several heat flow estimates from isotopes in hot springs so that only heat
flow estimates made directly from temperature measurements are included. A number
of duplicates are also removed. About 50% of the database has been checked against
the original references and several typographical and position errors are corrected.
Beginning with the database version by Jessop et al. [1976], database records were
fixed to 80 character widths, which resulted in several compromises. In previous
versions, letter codes were used to indicate metadata, such as the type of instrument
used to measure temperature, method of estimating thermal conductivity, and the
country where the site is located. These metadata are now explicitly given to make the
database more readable and some metadata has been removed because GIS programs
can easily serve the same function. A single letter code was used to indicate that
8
a correction was applied to the heat flow, although multiple corrections could not
be listed individually. The new update seeks to fix this issue by giving each of the
corrections and value if possible. Examples of three of sites, are given in Table 2.2.
Continental data are often recorded in “holes of opportunity” in boreholes drilled for
mining, petroleum, or water resource exploration. The vast majority of newly added
continental data are from BHTs collected in oil exploration and are generally lower
quality than well-sampled and isolated boreholes. BHTs are temperatures measured
at the bottom of exploration wells and may be measured at several stages during and
after drilling. However, the depth of the temperature measurement, typically more
than 3000 m, and the number of BHTs collected within a region add confidence to
average heat flow estimates (see spatial clustering in Figure 2.2).
About the time of the last global update, oceanic heat flow studies benefited from
a combination of an improved apparatus used to measure temperatures in ocean
sediments and high resolution positioning using GPS. These improvements led to a
paradigm shift in most survey designs away from regional heat-flow mapping to high
resolution transects and three-dimensional surveys that focus less on background heat
flow and more on individual processes. Thus, despite the large growth in the oceanic
dataset, many of the new data cluster in locations where few sites already existed.
The updated global heat flow database compiled as part of this research represents
considerable improvement to past updates and represents a significant contribution
to the thermal geophysics and solid Earth geoscience communities. This improved
and updated database, along with recent laboratory studies of pressure and temperature dependence on thermophysical properties, present an opportunity to revisit the
thermal state of the continental and oceanic lithosphere.
9
Table 2.2. Example records to the updated global heat flow database.
New data number
Old data number
Fluids
Bottom water variation
Refraction
Topographic
Climatic
Sedimentation/Erosion
Compaction
Other
Grade
Site Name
Latitude
Longitude
Elevation
Bottom water temp.
Bottom hole temp.
Minimum depth
Maximum depth
No. of temp.
Temperature gradient
Gradient uncertainty
Corr. temp. gradient
Corr. gradient uncertainty
No. of conductivity
Conductivity
Conductivity error
Conductivity method
18429
XM 44
18772
CAN129
-19
8.8
-9
9.8
-4.7
B
CT-29
39.3333
13.7333
-3497
Sivakasi SV-3
9.4619
77.7950
150
366
6.3
5
175
1.1
186
21
0.86
0.06
transient (needle
probe)
No. of heat production
Heat production
Heat production error
Heat production method
Heat flow
Heat flow error
Corr. heat flow
31690
151
17.15
160
18.9
0.03
A6-A7
48.5917
-123.4967
-228
2.5
6
43
29
4
2.4
0.04
6
0.72
transient (needle
probe)
25
0.6
0.14
gamma-ray spectrometer
45
0.7
21
10
Table 2.2. continued.
Corr. heat flow error
No. of determinations
Publication year
Reference
Verified
Borehole type
Environment
Comments
Age
Lithology
Sediment Thickness
1
1970/1977/1984
Erickson etal1977;
DellaVedova etal1984;
Erickson1970
*
Ewing probe
1
2003
Ray etal2003
13
1
1976
Hyndman1976
borehole
(conventional)
*
lake
(Bullard
probe)
corrected
heat flow was
recomputed
charnockite
CHAPTER 3
HEAT PRODUCTION AND GEOTHERMS
FOR THE CONTINENTAL
LITHOSPHERE
3.1
Abstract
Accurate estimates of heat production are necessary for computing lithospheric
temperatures and heat flow into the base of the lithosphere. Heat production, however, is one of the most variable and difficult parameters to estimate using surface
geophysical exploration methods. I propose a generalized continental lithospheric
heat production model that partitions crustal heat production into upper crustal and
lower crustal contributions and is constrained by thermal isostasy observations from
33 North American tectonic provinces. An average heat production for the lower crust
is estimated as 0.4 µW/m3 from exposed granulite terranes and for the lithospheric
mantle as 0.02 µW/m3 from chemical analyses of xenoliths. The best-fitting partition
model suggests upper crustal heat production accounts for ∼26% of the observed
surface heat flow. Results are relatively insensitive to mantle composition and thickness of the upper crustal heat producing layer. Continental geotherms are computed
using the generalized heat production model and incorporating thermal conductivity
results from a number of recent laboratory studies. Estimated P –T conditions of
xenoliths provide constraints to ensure that my geotherms are reasonable. Fits to
P –T conditions of 10 Precambrian regions suggest surface heat flow is ∼40 mW/m2
with a lithospheric thickness of ∼200 km. My average model for North American
heat production can be used as a reference model from which observed anomalies can
be identified.
12
3.2
Introduction
Radiogenic heat generation, created by the decay of K, Th, and U, accounts
for an estimated 30–40% of heat loss through the continents [Pollack and Chapman, 1977; Vitorello and Pollack , 1980; Artemieva and Mooney, 2001; Hasterok and
Chapman, 2007a]. Accurate estimates of radiogenic heat production are important
in computing lithospheric temperatures and heat flow across both the Moho and
the asthenosphere/lithosphere boundary, as well as many physical parameters that
depend upon temperature (e.g., density, seismic velocity, viscosity, elevation) all
of which have important geodynamic implications [Jaupart and Mareschal , 1999;
Flowers et al., 2004; Hyndman et al., 2005; Sandiford and McLaren, 2002]. However,
reliable estimates of heat production are difficult to obtain at depths greater than a
few hundred meters using common geophysical and geochemical exploration methods,
making it one of the least constrained physical parameters within the lithosphere.
Therefore, it is desirable to have a general heat production model that can be used as
both a reference for comparison, and a starting model for thermal and geodynamic
studies of the lithosphere.
Several general models for heat production exist, but they are based on crustal
age [Jaupart and Mareschal , 2003], which poorly correlates with heat production,
or are calibrated only for shield and cratonic regions [Rudnick et al., 1998; Rudnick
and Nyblade, 1999]. Neither of these model types adequately reflect the chemical
differentiation of the crust. Models based on heat production–heat flow relationships
derived from a collection of geologic/tectonic provinces show promise because they
implicitly assume both a decrease in heat production with increasing juvenile crustal
age and an increase in mantle heat flow in active tectonic regions [Chapman, 1986;
Artemieva and Mooney, 2001]. These partition models are generally applied as a
function of a single observable, surface heat flow. However, sampling of upper crustal
heat production is generally insufficient to characterize a given province with sufficient
accuracy.
Continental elevation, much like ocean bathymetry, responds to lateral differences
in the average thermal state of the lithosphere [Hasterok and Chapman, 2007b].
13
Using a set of compositionally normalized elevation data I estimate the average
contribution of heat producing elements to the observed heat flow. While the exact
distribution of heat production is somewhat difficult to determine using this method,
estimates of pressure and temperature from mantle xenoliths place limits on the
vertical distribution of heat production.
In this study, I develop an average heat production model for the North American
continental lithosphere using compositionally normalized elevations from improved
geotherm computations. The range of geotherms is constrained using estimated
xenolith P –T conditions.
3.3
Geotherm Sensitivity to Heat Production
The sensitivity of lithospheric temperatures and estimates of lithospheric thickness
to heat production is illustrated by perturbing a standard model. Consider a three layered lithosphere, with a high heat producing upper crust (1 µW/m3 ), a depleted lower
crust (0.4 µW/m3 ), and a low heat producing mantle (0.02 µW/m3 ). For this test,
I define a reference geotherm with an intermediate surface heat flow of 60 mW/m2
(see Section 3.6.1 for geotherm construction). This reference geotherm reaches the
1300◦ C adiabat at ∼90 km (Figure 3.1). By varying only the heat production within
each layer individually by ±25 and ±50% I investigate the sensitivity of temperatures
and lithospheric thickness to the depth of the heat production anomaly.
A ±50% variation in upper crustal heat production produces a difference in lithospheric thickness of ∼50 km and >400 K temperature variation at 75 km. The effect
of ±25 and ±50% lower crustal heat production variance results in 10 km and 20 km
lithospheric thickness estimates, respectively. The temperature difference is significantly smaller (∼175 K at 75 km for ±50% variation) than differences resulting from
similar percentage variations the upper crustal heat production. For the same relative
variation in heat production, the effect on temperature and lithospheric thickness is
more than twice as sensitive to changes in the upper crustal layer compared to changes
in the lower crustal heat production. As mantle heat production is varied, one can
see that the effect is negligible for both ±25 and ±50% variation. In order to rival the
14
0
0
500
1000
1500 0
Temperature [°C]
500
1000 1500 0
500
1000
1500
1 µW/m3
25
0.4 µW/m3
Depth [km]
50
75
0.02 µW/m3
100
Reference
±25%
±50%
125
150
Figure 3.1. Temperature sensitivity to 25 and 50% variations in heat production
of the (left) upper crust, (middle) lower crust, and (right) mantle. The reference
geotherm is computed with 1 µW/m3 upper crust, 0.4 µW/m3 lower crust and
0.02 µW/m3 mantle. Geotherms are computed with a surface heat flow of 60 mW/m2 ,
upper crustal thickness of 16 km, and 39 km depth to the Moho.
lithospheric thickness variation seen in the lower crustal case, mantle heat production
would have to vary by greater than ±400%.
It is apparent from these tests that unless upper crustal heat production is precisely
known, deviations from average lower crustal and lithospheric mantle heat production
can be masked by upper crustal uncertainties. Therefore, I focus my analysis only on
variations in upper crustal heat production.
3.4
Previous Models
Several models for crustal and/or lithospheric heat production exist, each using different constraints (Figure 3.2). Allis [1979]; Rybach and Buntebarth [1984] suggested
using seismic velocities to predict heat production, but heat production for any given
rock is highly nonunique. Attempts to estimate crustal heat production using xenolith
P –T conditions provide reasonable averages (0.5–0.8 µW/m3 ) [Rudnick et al., 1998],
but are only calibrated to Precambrian provinces. Rudnick and Nyblade [1999] used
15
0
0
1
2
1 2
3
Heat Production [µW/m3]
3 0
1 0
1
2 0
1
2
3
4
5
10
Depth [km]
20
30
Ar
40
Ph
Pt
(a)
(b)
(c)
(d)
50
Figure 3.2. Heat production models of the lithosphere. (a) Heat production model
from Allis [1979] is compositionally dependent; models are shown for 1, greenstone; 2,
gneiss terrane; and 3, granite. (b) Rudnick et al. [1998]; Rudnick and Nyblade [1999]
estimated heat production in Precambrian terranes from xenolith P –T –conditions and
lithospheric thickness constraints. (c) Jaupart and Mareschal [2003] estimated the
heat production within the crust of Archean (Ar), Proterozoic (Pt), and Phanerozoic
(Ph) terranes by using surface heat flow constraints on the maximum mantle heat
flow. (d) The heat production models of Chapman [1986] (black) and Artemieva and
Mooney [2001] (dashed) are derived from reduced heat flow, surface observations, and
xenolith P –T –conditions.
xenolith P –T conditions to test mantle as well as crustal heat production. However,
the results are less useful, as three out of five of the best-fitting grid search parameters
lie on their boundaries of the allowed range, including mantle heat production. This
suggests that either the ranges of allowed parameters were too restrictive or some
of the physics necessary to describe system accurately is not included (e.g., P –T
dependent thermal conductivity).
Jaupart and Mareschal [2003] use surface heat flow constraints to make an elegant
estimate of crustal heat production and find a significant variation in crustal heat
production with juvenile crustal age. They suggest the present-day bulk-crustal heat
production increases from ∼0.65 µW/m3 in Archean to ∼0.87 µW/m3 in Phanerozoic
terranes. However, the correlation between age and heat production may be weak
[Rao et al., 1982]. For example, the Wopmay Orogen and much of Precambrian
Australia have anomalously high heat flow due to high upper-crustal radioactivity
16
[Lewis et al., 2003; McLaren et al., 2003]. Additionally, their analysis does not
distinguish between upper and lower crustal heat production.
Chapman [1986]; Artemieva and Mooney [2001] take a different approach, estimating upper crustal heat production from an empirically derived partitioning of
surface heat flow between an enriched upper crustal radiogenic heat flow (qrad ) and
reduced (or basal, qb ) heat flow [Pollack and Chapman, 1977]. Partition estimates
range from 60:40 (qb :qrad ) [Pollack and Chapman, 1977; Vitorello and Pollack , 1980]
to 67–71:33–29 [Artemieva and Mooney, 2001]. A major advantage of the partitioning
model, compared with the other models discussed above, is a dependence on surface
heat flow, an observable that directly responds to changes in the upper crustal
radioactivity. Thus the partition model can be applied generally to any region where
surface heat flow is observed or can be estimated. Because of the reliance on surface
heat flow, the partition model implicitly assumes that tectonically active regions
are frequently have higher than average mantle heat flow and a decrease in heat
production with age.
3.5
Observational Constraints
In order to derive a general heat production model for the continental lithosphere,
it is important to know the range of heat production as well as the general variation
with depth. Heat production observations and estimates are summarized below for
each of the lithospheric layers.
3.5.1
Upper Crust
Direct measurements of heat production are generally high in felsic rocks (∼2
µW/m3 ), low in mafic rocks (∼0.2 µW/m3 ) and very low in ultramafic rocks (∼0.02
µW/m3 ). Variations of heat production within individual and well-sampled terranes
can vary by at least an order of magnitude over small spatial scales (<1 km) [Kukkonen
and Lahtinen, 2001; Ray et al., 2003]. Individual terranes commonly have standard
deviations 50–100% of the sample mean or greater, and frequently exhibit nonGaussian distributions (e.g., Ketcham [2006]; Jõeleht and Kukkonen [1998]; Brady
17
et al. [2006]; Jaupart and Mareschal [2003]). Few terranes, however, have been
subjected to sufficiently detailed heat production investigations to be helpful to heat
flow modeling.
Airborne radiometric surveys provide good spatial averages, but the depth of penetration is ∼30 cm and often does not see through sedimentary cover [Bodorkos et al.,
2004]. Even when good surface spatial averages can be obtained, heat production
variation with depth is still problematic. Heat production with depth is frequently
modeled using one of three functional forms: constant value layers, linearly decreasing, and exponentially decreasing [Lachenbruch, 1970]. The exponential model was
developed to explain observations of heat flow–heat production patterns observed
in granitic batholiths with differing levels of surface erosion [Roy et al., 1968; Birch
et al., 1968; Swanberg, 1972]. I test these models for vertical heat production variation
using deep boreholes and exposed crustal cross-sections, which provide insight into
the nature of heat production with depth.
Some constraints on vertical heat production variation come from deep boreholes
and exposed crustal cross-sections. The research wells KTB in Germany and Kola
SG-3 in Russia are the deepest scientific boreholes at 9 and 12.2 km, respectively.
KTB is drilled into a series of layered gneiss and metabasites and SG-3 is drilled into
a Precambrian volcanic and metamorphic terrane. The heat production for each well
shows a reasonable correlation to rock type [Clauser et al., 1997; Popov et al., 1999]
but an irregular pattern with depth. The Chinese scientific borehole (CCSD-MH) also
exhibits a correlation to lithology [He et al., 2008]. However, heat production is quite
variable within any given lithologic unit. Even within boreholes drilled predominantly
into granite, heat production can be highly variable [Lachenbruch and Bunker , 1971;
Balling et al., 1990; Vigneresse and Cuney, 1991]. No single functional form can
explain the patterns of heat production with depth for these boreholes.
Exposed crustal sections that can be related to pseudo-depth profiles provide greater
spatial coverage, extending knowledge to greater depths than boreholes. Measurements of heat production have been collected in metamorphic core complexes (Catalina and Harquahala Mountains [Ketcham, 2006]), impact structures (Vredefort and
18
Sudbury [Nicolaysen et al., 1981; Schneider et al., 1987]), thrust sheets (Zentralgneis,
Wawa-Foleyet, Pikwitonei-Sachigo, Hidaka Arc, and Arunta-Muscrave [Hawkesworth,
1974; Ashwal et al., 1987; Fountain et al., 1987; Furukawa and Shinjoe, 1997; McLaren
et al., 2003]), structural folds (Egersund-Bamble, Erzgebirge [Pinet and Jaupart,
1987; Förster and Förster , 2000]), and batholiths (Idaho Batholith, and Sierra Nevada
Mountains [Swanberg, 1972; Brady et al., 2006]). These exposed sections also show
little correlation of heat production with estimated depth beyond a general pattern
of higher values near the surface where felsic rocks are dominant and lower values in
mafic rocks of the lower crust. In regions where thrust sheets duplicate parts of the
crustal section, heat production often exhibits a repeating pattern [McLaren et al.,
2003; Ketcham, 2006; He et al., 2008; Clauser et al., 1997].
Given the poor correlation of heat production with depth in the upper crust, I
suggest a reference model with a constant upper crustal heat production. A constant
heat production with depth is also the least complicated functional form, making it
easy to compute anomalies.
3.5.2
Lower Crust
Studies of seismic velocity and equilibrium conditions from xenoliths suggest that
the lower crust is more mafic on average than the upper crust and best described by
granulite metamorphic facies [Christensen and Mooney, 1995; Rudnick and Fountain,
1995]. To estimate lower crustal heat production, observations from exposed granulite terranes and xenoliths provide the best insight [Rudnick and Fountain, 1995].
Geochemical estimates of heat producing elements are converted to heat production,
A, by
A = 10−5 ρ [3.5CK2 O + 9.67CU + 2.63CTh] ,
(3.1)
with concentrations of K2 O in wt.%, and of U and Th in ppm. I assume densities, ρ, of
2800, 2850, and 3000 kg/m3 for felsic, intermediate, and mafic granulites, respectively.
Heat production measurements from 31 exposed granulite terranes range from 0.1 to
2.7 µW/m3 with a mean of 0.68±0.62 µW/m3 and a median 0.45 µW/m3 (Figure 3.3).
Many of the higher heat production terranes are mainly felsic rather than mafic
19
10
# of Granulites
8
6
N: 41 (31 terranes)
Mean: 0.68±0.62 µW/m3
Median: 0.45 µW/m3
4
2
0
0
0.5
1
1.5
2
Heat Production [µW/m3]
2.5
20
# of Xenoliths
16
12
N: 97 xenoliths
Mean: 0.031±0.024 µW/m3
Median: 0.022 µW/m3
8
4
0
0
0.02
0.04
0.06
0.08
Heat Production [µW/m3]
0.1
Figure 3.3. Heat production from granulite terranes (top) and mantle xenoliths
(bottom). Locations are shown by squares on inset maps. Mean values are given
± one standard deviation. Granulite data from Ashwal et al. [1987]; Fountain et al.
[1987]; Pinet and Jaupart [1987]; Ray et al. [2003]; Kukkonen and Jõeleht [1996];
Förster and Förster [2000]; Brady et al. [2006]; Garrido et al. [2006]; Del Lama et al.
[1998]; Attoh and Morgan [2004]; Owen et al. [68]; Martignole and Martelat [2005];
Hölttä [1997] and references therein. Xenolith data from Ackerman et al. [2007]; Bjerg
et al. [2005]; Ionov et al. [1993]; Ionov [2004]; Ionov et al. [2002]; Peltonen et al. [1999];
Rudnick et al. [2004]; Wiechert et al. [1997]; Ionov et al. [2005]; Bianchini et al. [2007];
Xu et al. [1998].
20
granulites. If only mafic granulites are considered, the mean heat production is
0.36±0.50 µW/m3 with a median of 0.15 µW/m3 . However, it is difficult to derive
completely independent estimates of mafic and felsic granulites using this dataset
because the reported heat production values are commonly terrane averages rather
separated into mafic and felsic compositions.
Xenolith-derived heat production values are lower on average than from exposed
terranes with means (medians) of felsic, intermediate, and mafic granulites of 0.85
(0.57), 0.20 (0.09), and (0.13) 0.06 µW/m3 , respectively [Rudnick and Fountain,
1995]. This difference could be due to metasomatic processes that affect surface
exposures or xenoliths during exhumation and/or near surface groundwater flow
[Jaupart and Mareschal , 2003].
I do not include a middle crust in my modeling because the temperature sensitivity
to middle and lower crustal heat production variations is small relative to the upper
crust. The mid-crustal layer is not pervasive globally [Rudnick and Gao, 2003], and
where it exists, tends to have heat production more similar to lower rather than to
upper crust. My model includes a value of 0.4 µW/m3 for the lower crust, which
is slightly higher than typical for mafic granulite and lower than most intermediate
rocks like amphibolite and tonalite.
3.5.3
Lithospheric Mantle
Xenoliths provide the best estimates of mantle heat production on continents.
Average heat production from compilations of continental lithospheric peridotites
varies from 0.006 µW/m3 in exposed off-craton massifs to 0.044 µW/m3 in cratonic
xenoliths [Rudnick et al., 1998]. However, xenoliths, particularly kimberlites, are
subject to significant disturbances by metasomatic processes during ascent. Excluding
kimberlite xenoliths, the median heat production for cratonic peridotites is estimated
to be 0.019 µW/m3 [Rudnick et al., 1998]. In general, bulk U and Th concentrations
are rarely measured in xenoliths. The above heat production estimates are based on
assumed abundances of U and Th relative to K concentration [Rudnick et al., 1998].
21
Heat-producing elements U and Th are typically found in monazite, a phosphate
mineral similar to apatite. Studies focusing on apatite in mantle peridotites suggest
that even at very small concentrations, apatite can dominate heat production [Ionov
et al., 1996; O’Reilly et al., 1997; O’Reilly and Griffin, 2000]. For example, 1% by
weight apatite in a mantle rock can increase the total heat production by 0.3 µW/m3 .
Since apatite is a common accessory mineral but rarely reported, it unknown whether
these high heat production values are widespread or isolated locally [Ionov et al., 1996;
O’Reilly et al., 1997; O’Reilly and Griffin, 2000]. However, the minor to negligible
curvature in xenolith P –T estimates suggests mantle heat production is on average
very small.
I compiled heat production for several xenolith localities that include K2 O, U,
and Th in the bulk chemical analysis (Figure 3.3). Heat production is computed
using Equation 3.1 with an assumed density of 3300 kg/m3 . Total heat production
for these xenoliths ranges from 0.003 to values greater than 1 µW/m3 with two
exceptionally high values ≫1 µW/m3 . In general, the values >0.1 µW/m3 come
from localities showing signs of metasomatism (e.g., Jericho xenoliths on the Slave
craton from Russell et al. [2001]). A strong peak in the estimated mantle heat
production histogram occurs below ∼0.02 µW/m3 . When restricted to values less
than 0.1 µW/m3 , average heat production is 0.041±0.030 µW/m3 and a median of
0.025 µW/m3 . In my models, I assume a heat production of 0.02 µW/m3 for the
mantle lithosphere.
3.6
Methods
3.6.1
Geotherms
One-dimensional steady-state conductive geotherms are computed using a bootstrapping method, which requires thermal conductivity, heat production and surface
heat flow as inputs (Appendix E). Because I use a complex P –T –composition-ally
dependent thermal conductivity model, a Newton-Raphson iterative scheme is employed to solve for temperature. Pressure is computed using a logarithmic equation
of state [Poirier and Tarantola, 1998].
22
Table 3.1. Compostional model used to compute geotherms.
Mineral
quartz
orthoclase
albite
anorthite
phlogopite
hornblende
diopside
hedenbergite
enstatite
ferrosillite
forsterite
fayalite
pyrope
almandine
Upper
27
15
32
8
5
13
Crust
Middle
15
5
35
20
20
2
3
Lower
2
10
10
18
47
1
1
1
1
9
Archon
Mantlea
Proton
Tecton
1.96
0.14
23.22
1.78
64.04
4.96
3.19
0.81
5.47
0.53
15.47
1.53
63.65
3.35
5.58
1.42
9.97
1.03
15.37
1.63
54.20
5.80
9.57
2.43
Mantle compositions are derived from garnet xenocrysts and approximate Archean
(Archon), Proterozoic (Proton), and Phanerzoic (Tecton) mantle [Griffin et al., 1999].
I assume a compositional model equivalent to a granodiorite upper crust (0–16 km),
tonalite middle crust (16–23 km), and mafic granulite for the lower crust (23–39 km)
(Table 3.1). Layer thicknesses for the crust correspond to estimates discussed by
Rudnick and Gao [2003]. Average mantle compositions are based on garnet xenocryst
estimates by Griffin et al. [1999]. Their peridotite compositions are derived from
garnet xenocrysts and correspond approximately to average Archean, Proterozoic,
and Phanerozoic continental lithospheric mantle. Garnet peridotites are converted to
spinel-peridotites via 2 orthopyroxene + spinel ↔ garnet + olivine. The pressure of
the spinel-garnet transition is estimated using the empirical relationship,
Psg (T ) = 1.4209 + exp(3.9073 × 10−3 T − 6.8041),
(3.2)
where T is in Kelvins. This curve is calibrated by inversion of the data reported
by Robinson and Wood [1998]; Walter et al. [2002]; Klemme and O’Neill [2000] and
references therein.
23
3.6.1.1
Heat Production
Assuming steady-state conditions, surface heat flow results from a combination of
heat flow into the base of the lithosphere and the integrated heat production within
the lithosphere. As discussed above, temperatures are very sensitive to variations in
upper crustal heat production (Figure 3.1). Therefore, I focus only on variations in
upper-crustal heat production and fix all other layers at constant values. Thus the
middle crustal, lower crustal, mantle heat production, and sublithospheric heat flow
can be combined into a single parameter, basal heat flow (qb ). The surface heat flow,
qs , can then be written,
qs = qb + AUC D
(3.3)
where AUC is the upper crustal heat production, D is the thickness of the upper
crustal heat producing layer, and the product AUC D is the radiogenic heat flow in
the upper crust.
I use equation 3.3 to define three classes of heat production models. Class I –
invariant upper crustal heat production where heat production is independent of the
surface heat flow and all variations in surface heat production result from differences
in lithospheric thickness and variation in sublithospheric heat flow. Class II – constant
basal heat flow, where all variations in surface heat flow result from changes in upper
crustal heat production. Hence the upper crustal heat production can be computed
by
AUC = (qs − qb )/D.
(3.4)
The invariant heat production model would be most applicable in regions such as
rifts where significant variations in basal heat flow occur, but the average crustal heat
production may be relatively constant [McKenzie, 1978]. Models with constant basal
heat flow may best describe individual shields and cratons where the basal heat flow is
believed to be relatively constant and variations in surface heat flow are still observed
[Jaupart and Mareschal , 2003]. Globally, the patterns of surface heat flow are likely
to result from a combination of these end-member models, hence the basal heat flow
24
represents only a fraction of the total heat flow (i.e., qb = F qs ). Thus, Class III – the
partition model can be written,
AUC = (1 − F )qs /D,
(3.5)
where F is the partition coefficient. This partition model has been used with great success in describing global heat flow/heat production patterns [Pollack and Chapman,
1977; Vitorello and Pollack , 1980; Artemieva and Mooney, 2001] and in developing a
thermal isostatic relationship for North America [Hasterok and Chapman, 2007a].
I model geotherms using constant heat production within the upper crustal heat
producing layer for two reasons. First, the commonly employed exponential decreasing heat production models with depth can exceed 5 µW/m3 for partition models
with high heat flow. Heat production values this high are rarely observed regionally
except in shields with anomalously high heat production, but can be identified by
their associated low elevation anomalies such as the Wopmay orogen and North and
South Australian cratons [Lewis et al., 2003; McLaren et al., 2003]. The second rationale for using constant heat production stems from highly variable heat production
observations in exposed crustal sections and deep boreholes as discussed in Section
3.5.1.
Heat production in the middle to lower crust is assumed to be 0.4 µW/m3 , consistent
with granulite terranes. Mantle heat production is set to 0.02 µW/m3 as suggested
by chemical studies of mantle xenoliths.
3.6.1.2
Thermal Conductivity
Thermal conductivity is computed using a P –T –composition dependent model
resulting from a combination of lattice and radiative components. Once the total
conductivity of each mineral component is estimated, the effective thermal conductivity is computed using a geometric mixing model [Clauser and Huenges, 1995].
The lattice contribution is computed using a simplified form of the Equation 10 by
Hofmeister [1999a],
λL (P, T ) = λ
◦
298
T
n
K′
1+ TP ,
KT
!
(3.6)
25
where temperature is in Kelvins, λ◦ is the conductivity at 0 GPa and 298 K, KT
and KT′ are the isothermal bulk modulus and its first pressure derivative, and n is an
empirically derived fitting constant. At lithospheric temperatures and pressures, the
additional exponential factor included by Hofmeister [1999a] is very near unity and
thus ignored [Beck et al., 2007]. Constants used to compute conductivity are given
in Appendix D.
The radiative contribution to thermal conductivity, λR , is negligible at room temperature but represents a significant fraction of the effective conductivity at high
temperatures. I use an empirically derived relation for the radiative conductivity
developed in Appendix A,
λR (T ) = λRmax [1 + erf (ω(T − TR ))] ,
(3.7)
where λRmax is the maximum radiative conductivity, ω is a scaling factor and TR is
the temperature at 0.5λRmax . For olivine, this estimate is significantly different from
previous estimates of the radiative contribution (Shankland et al. [1979] and references
therein, Hofmeister [1999a, 2005]). A simplified thermal conductivity model is include
in Appendix E for the specific compositional model used in this study.
3.6.2
Thermal Isostasy
Much like the subsidence patterns of the oceanic sea-floor that result from the
integrated cooling of the lithosphere (e.g., Parsons and Sclater [1977]), continental
elevations record a thermal isostatic effect that can be estimated by examining the
relationship between compositionally corrected elevation and surface heat flow [Han
and Chapman, 1995; Hasterok and Chapman, 2007b,a]. Elevation responds to variations in temperature as a result of thermal expansion which changes the density of
the lithosphere. In this section, I explore how continental elevation, much like oceanic
bathymetry in the marine case, can be used as a constraint on continental geotherms,
and consequently on heat production models for continental lithosphere.
The elevation difference, ∆εT , between two regions can be computed by
∆εT =
Z
0
zmax
[αV (z, T )T (z) − αV′ (z, T )T ′ (z)] dz,
(3.8)
26
where T (z) represents a lithospheric geotherm and αV (z, T ) is the P –T dependent
volumetric expansivity computed using the method of Afonso et al. [2005]. The
primed and unprimed symbols represent the reference and observed columns, respectively. The maximum depth of integration, zmax , is the depth at which the
coolest geotherm reaches the mantle adiabat. Elevation data are normalized for
variations in crustal thickness and density and represent 36 tectonic provinces from
North America [Hasterok and Chapman, 2007a]. Three regions (Pacific Coast Ranges,
Central Valley of California, and the Wopmay Orogen) are excluded from the fit due
to large extraneous influences on the elevation and or heat flow (see Hasterok and
Chapman [2007a] for discussion). Constants used to estimate expansivity and density
are given in Appendix D.
3.6.3
Xenolith P –T Conditions
While thermal elevation contributions can be used to estimate the average difference
between geotherms, it is not possible to estimate absolute temperature. However, I
can use xenolith P –T estimates as constraints on absolute temperature. Estimated
temperatures and pressures for 1449 mantle xenoliths are shown in Figure 3.4. Temperatures range from 650 to 1500◦ C over a range of pressures equivalent to depths of
25 to 225 km. A few xenoliths from the Dabie-Sulu belt appear to be well outside
the range of other localities, which reflects rapid subduction of material that thus did
not have time to equilibrate to ambient mantle conditions.
While there is some variation in high temperature and high pressure xenoliths
>100 km, temperatures rarely exceed the 1300◦ C adiabat. In contrast to the high
pressure–high temperature xenoliths, at low pressures there appear to be very few
high temperatures above the 50 ppm melting curve (Figure 3.4). Either temperatures
are buffered by melting, restricting maximum geotherm temperatures to the 50 ppm
solidus and not the adiabat, or P –T conditions are poorly sampled. Alternatively,
these xenoliths may be sampled but P –T estimates may not be possible using the Brey
and Köhler [1990] thermobarometers. As melting continues to higher melt fractions,
clinopyroxene is one of the first minerals to be exhausted [Katz et al., 2003]. The
27
0
sp-peridotite
gt-peridotite
50
dr
y
50
UHP
150
m
pp
Depth [km]
100
200
300
0
400
Adiabat
~2σ uncertainty
wet
N = 1449
250
800
1200
Temperature [°C]
1600
Figure 3.4. Estimated P –T conditions for mantle xenoliths using the PBKN and
TBKN barometer and thermometers (circles). Conditions for the ultra-high pressure
Dabie-Sulu belt shown in black [Zheng et al., 2006]. Estimated uncertainty (1-σ) is
±0.3 GPa (∼9 km) for pressure and 20 K for temperature [Brey and Köhler , 1990].
Melting curves by Katz et al. [2003]. The heavy black adiabat is represented by a
potential temperature of 1300◦C with a gradient of 0.3◦ C/km. The spinel-garnet
transition is given with a dashed line.
removal of solid clinopyroxene makes T estimates from the 2-pyroxene thermometers
such as TBKN used in this study impossible, suggesting that the lack of data does
not preclude the higher temperatures at low pressure. This effect assumes a water
concentration >50 ppm.
3.7
3.7.1
Results and Discussion
Thermal Isostatic Models
Misfits to compositionally normalized elevations of North America are shown in
Figure 3.5. The misfit is computed by,

1/2
N
N
S2 X
1 X
Misfit = 
(qi − qm )2 +
(εj − εm )2 
N i=1
N j=1
,
(3.9)
28
2.0
Reference Heat Flow [mW/m2]
40
50
60
0
Temperature [°C]
800
1600
0
Class I
1.2
0.8
1.4
0.8
0.4
0.6
0.6
1
0.8
0.4
Invariant
qref = 47.5 mW/m2
AUC = 0.7 µW/m3
300
0
Class II
0.6
25
200
0.4
Depth [km]
1
100
30
0.8
Basal Heat Flow [mW/m2]
0
35
200
Depth [km]
100
1.2
1
Invariant HP [µW/m3]
1.6
Basal
qref = 39.5 mW/m2
qb = 23.5 mW/m2
20
1.0
300
0
1.4
0.4
0.8
1.2
0.6
Partition
qref = 47.5 mW/m2
F = 0.74
Class III
40
50
60
2
Reference Heat Flow [mW/m ]
0
800
Temperature [°C]
200
Depth [km]
100
1
Partition Coefficient
1
0.8
0.6
0. 4
0.6
0.8
300
1600
Figure 3.5. Misfit of heat production models to compositionally adjusted elevation
(left). Best-fit model represented as ‘×.’ Models where the reference heat flow does
not reach the adiabat are shown in grey. Best-fitting geotherms for each associated
heat production class (right). Geotherms range from 30–120 mW/m2 .
29
where N is the number of samples, qi and qm are the observed heat flow and model
heat flow, and εi and εm are the adjusted province elevation and model elevation,
respectively. In order to give roughly equal weights to the elevation and heat flow, the
heat flow is scaled (S = 4 km/70 mW/m2 ). Results are only shown for upper-crustal
heat-production thickness D = 16 km and Proton composition mantle (Table 3.1).
Given the scatter in normalized elevations, it is unsurprising that the misfit surface
shows that each model class has a wide range of reasonable parameters. The minimum
misfit for the invariant and partition models are very similar (∼0.34) and only slightly
higher for the constant basal heat flow model (0.39). Misfits using geotherms from
this study show an improvement in fit over previous work using geotherm families
computed using the method by Chapman [1986], which have a minimum misfit >0.7
[Hasterok and Chapman, 2007a]. The best-fitting thermal isostatic curve for the
invariant model has an upper crustal heat production of 0.7 µW/m3 and reference heat
flow of 47.5 mW/m2 . The reference heat flow corresponds to an adjusted elevation of
zero. The elevation resulting from the invariant model is ∼2.5 km, and very similar
in form to the partition model (Figure 3.6).
The partition model results in the same reference heat flow as the invariant model
with a partition coefficient of 0.74. This value is slightly higher than a recent global
estimate of partition coefficient from Artemieva and Mooney [2001] and higher than
my previous estimate of 0.6 using the Chapman [1986] style geotherms. This higher
value results from my improved conductivity model and the use of P –T dependent
thermal expansivity.
The constant basal heat flow model results in a very different character to predicted
thermal elevation resulting from the invariant and partition models (Figure 3.6),
although the range of elevation for observed heat flow data is similar. Geotherms
computed using the Class II model are also radically different from Class I and III
(Figures 3.5). The minimum misfit has a reference heat flow of 39.5 mW/m2 , and
a basal heat flow of 23.5 mW/m2 . Physically, the basal heat flow is a combination
of the sublithospheric heat flow and radiogenic heat flow from the lower crust and
mantle lithosphere. For a 40 mW/m2 cratonic region, my geotherm models predict
30
Adjusted Elevation [km]
3
2
I. Invariant, AUC
II. Constant, qb
III. Partition, P
1
0
-1
40
60
80
Heat Flow [mW/m2]
100
Figure 3.6. Best-fitting thermal isostatic models for each of the heat production
models. Compositionally adjusted elevation for North American geologic provinces
from Hasterok and Chapman [2007a], open squares not used in best-fitting model.
Field of xenolith P –T estimates, range of adiabats and solidi curves from Figure 3.4.
a lithospheric thickness of ∼200 km, and the total radiogenic contribution from the
lower crust and mantle is then 12.4 mW/m2 . Using estimates of sublithospheric heat
flow from the Canadian shield (11–15 mW/m2 ) [Mareschal and Jaupart, 2004], the
predicted lower bound for basal heat flow is consistent with the best-fitting result.
3.7.1.1
Effect of Varying D
The thickness of the upper crustal heat producing layer, D, need not be the same
as the compositional upper crust as some processes such as melt migration and fluid
circulation may concentrate heat producing elements toward the surface Jaupart et al.
[1981]; Gosnold [1987]. Tests varying D between 6 and 18 km suggest that the
partition coefficient is relatively insensitive to variations in D, ranging from 0.76 at
8 km to 0.74 at 16 km. For Class I models, iso-misfit lines closely track constant values
of AUC D, implying a nearly constant upper crustal radiogenic heat flow. Best-fitting
upper crustal radiative heat flow ranges from 9.9 mW/m2 at 6 km to 11.2 mW/m2 at
16 km. These observations do not hold when D and the compositional upper crustal
31
thickness are coupled as the effect of varying conductivity can trade-off for variations
in D in misfit space.
Independent estimates of D range from 2–16 km, with an average ∼8–10 km
determined using heat flow–heat production relationships [Artemieva and Mooney
[2001] and references therein]. Heat flow–heat production relationships were originally
developed to explain a relationship between these two parameters for co-genetic
plutons [Roy et al., 1968; Birch et al., 1968; Gosnold , 1987; Förster and Förster ,
2000] and, although frequently attempted, may not work when extended to encompass
all rocks in a given region [Nielson, 1987; Furlong and Chapman, 1987; Jaupart
and Mareschal , 1999]. The value of D determined from heat production–heat flow
relationships may also be more sensitive to the lateral scale of heat production
variations as much or more than the vertical variations, representing a lower bound
on the thickness of the upper-crustal heat-producing layer [Jaupart, 1983; Sandiford
and McLaren, 2002].
3.7.1.2
Effect of Mantle Composition
Using Archon and Proton mantle compositions produce nearly identical results
whereas the Tecton composition fits to elevation result in ∼10% higher total heat
production with similar misfit. The higher heat production compensates for conductivity difference between the Archon/Proton and Tecton compositions. This result
implies that younger tectonic regions are slightly enriched in upper crustal heat
producing elements than older terranes. The reduction in heat producing elements
with age and affinity for melts that migrate towards the surface in tectonically active
regions contribute to the greater enrichment in the upper crust of young terranes.
However, the conductivities of mantle phases other than olivine and garnet are not
well constrained and require further study before these differences in model results
can be rigorously modeled.
32
3.7.2
Xenotherms
While my models fit the compositionally adjusted elevation, the scatter in modeled
elevations allows for a large range of acceptable heat production models and therefore
temperatures. Typical values for regional heat flow on the continents range from a
little less than 40 mW/m2 to ∼100 mW/m2 and can reach 120 mW/m2 in magmatic
provinces and active rift zones. However, conductive heat flow can be as low as
∼20 mW/m2 in cases of extremely low crustal heat production [Chapman and Pollack ,
1974; Mareschal et al., 2000; Roy and Rao, 2000]. The best-fitting geotherm family for
the constant basal heat flow model is cooler than the xenolith P –T field at low heat
flow and does not reach the high temperatures recorded by xenoliths at intermediate
and low pressures. Both the invariant and partition models appear to fall within the
range of xenolith P –T field. The temperatures for the 40–120 mW/m2 geotherms are
relatively similar. Neither geotherm family reaches the 1300◦ C adiabat by 300 km for
a heat flow of 30 mW/m2 . However, at 30 mW/m2 the two geotherm models diverge
considerably with a difference of ∼350 K at 300 km.
In addition to examining the full P –T field from mantle xenoliths, I model the
heat flow for 10 individual kimberlite provinces. Geotherms are computed using my
preferred geotherm model, P = 0.74, with an Archon composition. Misfit for these
xenotherms (xenolith derived geotherms) is computed using the misfit functional,
N
δTi2 δPi2
1 X
+ 2
misfit =
N i=0 σT2
σP
"
!#−1/2
,
(3.10)
where δTi and δPi are the differences between the geotherm and xenolith P–T conditions. The differences are normalized by the estimated uncertainties in xenolith P–T
results from the geobarometer PBKN (σP = 0.3 GPa) and TBKN (σT = 20 K) [Brey
and Köhler , 1990].
Xenotherm misfits range from ∼3–5 with a surface heat flow uncertainty of ±2
mW/m2 estimated from the width of the misfit trough (insets in Figure 3.7). The
xenotherm fits indicate that, at the time of xenolith equilibration, heat flow for the
regions analyzed ranged from ∼37–47 mW/m2 with an average of 40 mW/m2 .
33
0
Kalahari Craton
N: 323
Misfit: 3.4
HF: 40.2
4
25
Misfit
Pressure [GPa]
2
6
0
30
8
0
HF
50
500
1000
Temperature [°C]
1500
Figure 3.7. Fits of preferred geotherm family (P = 0.74) to xenolith P –T estimates
for the Kalahari craton. Misfit (inset) for variations in surface heat flow (HF) in
mW/m2 computed using Equation 3.10.
34
0
0
500
Ouachita
1500
Slave
Somerset
N: 60
Misfit: 4.3
HF: 40.6
N: 111
Misfit: 5.3
HF: 37.2
N: 11
Misfit: 4.0
HF: 40
4
1000
8
0
Superior
Pressure [GPa]
W. Greenland
N: 23 (27)
Misfit: 2.3 (4.8)
HF: 47 (46.2)
N: 44
Misfit: 3.3
HF: 39.8
N: 44
Misfit: 3.6
HF: 39
4
Homestead
8
0
Anabar
8
0
500
Tanzania
N: 16
Misfit: 4.1
HF: 37.6
N: 85
Misfit: 5.2
HF: 38.6
4
Baltic
1000
1500
N: 26
Misfit: 5.1
HF: 43.2
0
500
1000
1500
Temperature [°C]
Figure 3.7 continued. Fits of preferred geotherm family (P = 0.74) to xenolith
P –T estimates for selected cratonic localities. All inset misfit axes are the same as in
3.7. All xenolith P –T conditions use TBKN and PBKN thermobarometers unless noted.
Open circles in Homestead are included in dashed analysis and excluded from solid.
35
Locality
Kalahari
Ouchita
Slave
Somerset
Superior
Homestead
West Greenland
Anabar
Baltic
Tanzania
Reference
Bell et al. [2003a]
Grégoire et al. [2003]
James et al. [2004]
Saltzer et al. [2001]
Simon et al. [2003]
Dunn [2002]
Rudnick and Nyblade [1999] and references therein
Aulbach et al. [2007]
MacKenzie and Canil [1999]
Kopylova and Caro [2004]
Kopylova et al. [1999]
Russell et al. [2001]
Rudnick and Nyblade [1999] and references therein
Schmidberger and Francis [1999] (P estimated using
method by MacGregor [1974])
Kjarsgaard and Peterson [1992]
Zhao [1998]
Rudnick and Nyblade [1999] and references therein
Hearn [2004]
Nielson et al. [2008]
Sand et al. [2009]
Bizzarro and Stevenson [2003]
Hutchison and Frei [2008]
Agashev et al. [2008]
Roden et al. [1999]
Roden et al. [2006]
Rudnick and Nyblade [1999] and references therein
Kukkonen and Peltonen [1999]
Peltonen et al. [1999]
Lee and Rudnick [1999]
Rudnick et al. [1994]
Rudnick and Nyblade [1999] and references therein
Figure 3.7 continued. References to xenolith P –T data.
36
This result is consistent with heat flow collected in these and other Precambrian
cratonic and shield regions [Nyblade, 1999; Mareschal and Jaupart, 2004; Roy and
Rao, 2000; Roy et al., 2008; Alexandrino and Hamza, 2008]. Results using the
invariant heat production are comparable.
3.7.3
Preferred Geotherm Family
The constant basal heat flow model is excluded as a viable geotherm family on the
basis of its inability to explain global variations in mantle xenolith P –T conditions.
Elevation and xenolith P –T conditions, however, do not discriminate between the
invariant and partition models. However, there are a number of factors that suggest
a partitioning model is more likely.
In general, heat production is higher heat production in younger terranes [Jaupart
and Mareschal , 2003], making the constant heat production model unlikely. In the
Canadian Shield, much of the variation in surface heat flow can be tied to regional
differences in heat production [Mareschal and Jaupart, 2004]. The heat flow of most
Precambrian regions is ∼40 mW/m2 as shown by the xenotherm fits above. There are,
however, a number of shields and cratons with high heat flow yet little or no tectonic
activity for >1 Ga. This suggests significant variations in surface heat production
[Lewis et al., 2003; McLaren et al., 2005].
In light of these observations and my results, my preferred geotherm and heat
production model is a partition model with ratio of 74:26 between the basal heat
flow and upper crustal radiogenic heat flow, and an upper-crustal heat-producing
thickness of 16 km.
3.7.4
Lithospheric Thickness and Sublithospheric
Heat Flow
Many geodynamic processes require an estimate of lithospheric thickness and/or the
heat flow at the base of the lithosphere. Values for these parameters can be derived
from my preferred geotherm models (Figure 3.8). Lithospheric thickness ranges from
just under 200 km at 40 mW/m2 surface heat flow to ∼50 km at 90 mW/m2 .
37
Lithospheric Thickness
[km]
250
200
150
100
50
(a)
Sub-Lithospheric Heat Flow
[mW/m2]
0
100
80
(b)
60
1:1
40
20
0
Sub-Lithospheric
Surface Heat Flow
1.0
0.8
(c)
F = 0.74
0.6
0.4
0.2
0.0
40
60
80
Surface Heat Flow
[mW/m2]
100
Figure 3.8. Lithospheric thickness and heat loss. (a) Lithospheric thickness for
partition model with coefficient F = 0.74, and Proton mantle composition. (b)
Sublithospheric heat flow computed by subtracting lithospheric heat generation from
surface heat flow. (c) Ratio of sublithospheric heat flow to surface heat flow.
38
Geotherms no longer intersect the adiabat for heat flow <34 mW/m2 . Sublithospheric heat flow is estimated by subtracting the estimated upper crustal heat production from the partition model and the contributions from the lower crust and mantle
lithosphere. Minimum sublithospheric heat flow for the preferred geotherm family is
11 mW/m2 and increases to just under 65 mW/m2 for a 100 mW/m2 surface heat
flow. The sublithospheric heat flow is nearly linear despite the large curvature in the
lithospheric thickness because heat production within the mantle is very small in my
models.
Using surface heat flow models from xenotherms (Figure 3.7), one can estimate the
range of sublithospheric heat flow into shields and cratons (14.5–20 mW/m2 ). The
sublithospheric heat flow accounts for 39–47% of the total surface heat flow. Estimates
from the Homestead xenoliths suggest a slightly higher ∼24 mW/m2 ; however, the
relatively few samples at near-adiabatic temperatures make it difficult to obtain a
reliable estimate.
3.8
Conclusions
Radiogenic heat production is highly variable within the continental lithosphere
and difficult to estimate from standard geophysical techniques. Using compositionally
corrected elevations and xenolith thermobarometry, the uncertainties in upper crustal
heat production can be substantially reduced. I define three types of heat production
models: an invariant heat production (independent of surface heat flow); constant
basal heat flow; and partitioning between upper crustal radiogenic heat production
and basal heat flow. Xenolith P –T estimates suggest the constant basal heat flow
models are not generally applicable. Likewise surface observations of heat production
suggest upper crustal heat production can not be invariant.
I propose a reference heat production model for the North American lithosphere
with 26% of the surface heat flow resulting from upper-crustal heat production within
a 16 km thick layer. Lower crustal and lithospheric mantle heat production are
estimated at 0.4 and 0.02 µW/m3 . My model is calibrated using compositionally
normalized elevation for 33 individual tectonic provinces and verified with xenolith
39
P –T suites from 10 separate cratons and shields. My preferred geotherm family
estimates heat flow between 37 and 47 mW/m3 for these Precambrian xenoliths with a
corresponding lithospheric thickness between 225 and 135 km, respectively. Predicted
sublithospheric heat flow for these localities varies from 14.5–24 mW/m2 .
CHAPTER 4
OCEANIC HEAT FLOW: IMPLICATIONS
FOR GLOBAL HEAT LOSS
4.1
Abstract
Heat flow measurements in seafloor younger than 55 Ma are systematically lower
than model predictions for a conductively cooling lithosphere. Oceanic heat flow are
typically collected in sedimented seafloor. These sedimented regions also tend to be
the location of hydrothermal recharge, causing a systematic bias in measurement towards low heat flow at young ages. Hydrothermal circulation drops below detection as
sedimentation and compaction sufficiently reduce permeability, causing temperatures
to return to background conductive equilibrium. By filtering heat flow data to retain
sites with sediment cover >325 m and located >85 km from the nearest seamount,
the effect of hydrothermal circulation can be minimized. Additional adjustments
for incomplete thermal rebound and sedimentation are estimated. Adjusted and
filtered heat flow approaches the plate model estimate GDH1, although a deficit still
persists at ages <25 Ma, possibly as a result of limitations of global filtering. An
environmental analysis of heat flow co-located with seismic data suggests that heat
flow at young ages is consistent with background estimates. A heat flow deficit due to
hydrothermal circulation is also estimated, allowing for an estimate of the advective
heat loss (7 ± 2 × 1012 W). Hydrothermal circulation in young seafloor out to ∼70 Ma
accounts for ∼17% of global heat loss.
4.2
Introduction
Since the first published heat flow measurements in the oceanic crust by Revelle and
Maxwell [1952], over 13 000 determinations have been collected. Earliest estimates
of global heat loss prior to the discovery of widespread hydrothermal circulation by
41
Lister [1972] included oceanic heat flow data on ridge flanks, arriving at a loss rate
of ∼31 TW [Lee and Uyeda, 1965; Lee, 1970; Chapman and Pollack , 1975]. With the
discovery of seafloor spreading, conductive cooling models were developed to explain
the subsidence of oceanic lithosphere with age [Parker and Oldenburg, 1973; Davis
and Lister , 1974; Crough, 1975; Parsons and Sclater , 1977; Stein and Stein, 1992].
These plate cooling models predict the background conductive heat loss through
the lithosphere, highlighting a deficit in heat flow on young seafloor attributed to
hydrothermal circulation [Lister , 1972]. Global heat loss models that attempt to
account for the heat flow deficit by using modeled heat flow for young ages yield
estimates of ∼40–44 TW [Williams and Von Herzen, 1974; Langseth and Anderson,
1979; Davies, 1980; Pollack et al., 1993].
The systematically low observed heat flow results from a spatial bias in measurements. Because the most commonly employed technique to estimate heat flow in
the oceans requires sediment cover, the site of most recharge regions, the low heat
flow values are preferentially captured. To estimate the true conductive heat flux,
Sclater et al. [1976, 1980] attempt to remove the systematically low heat flow values
by filtering sites using sediment thickness and basement morphology criteria. Their
attempt focused on only a few localities and a limited dataset. I expand their analysis
using similar filters to the entire oceanic heat flow dataset.
In this study, I (1) illustrate that hydrothermal circulation is a major influence on
upper crustal heat flow; (2) demonstrate that a systematic low in observed heat flow
results from experimental design and spatial distribution of recharge and discharge
zones; and (3) expand the filters developed by Sclater et al. [1976] to remove the
influence of hydrothermal circulation on heat flow and reveal the background thermal
regime.
4.3
Datasets
Heat flow data used in this study are extracted from an updated global heat
flow database (Chapter 2). Sites are assigned ages of the nearest pixel using the
seafloor age model by Müller et al. [2008].
The heat flow data are prefiltered,
42
restricting heat flow values between 0 and 500 + qref (t) mW/m2 , where qref (t) is
the heat flow computed by GDH1. Some previous versions of the global heat flow
database use 0 mW/m2 to denote heat flow not calculated, and sites with negative heat flow are in regions where the bottom water temperatures are not stable.
Therefore, I choose to exclude all of those data. Large Igneous Provinces (LIPs),
which cover a small yet significant fraction of the seafloor and represent anomalous
volcanism, are excluded from all oceanic data prior to any binning and filtering.
The LIPs boundaries used in this study were initially defined by Coffin and Eldholm
[1994]. (A 2004 digital version of LIP polygons is available from the UTIG Plates
Project: http://www.ig.utexas.edu/research/projects/plates/). After pre-filtering,
13,501 heat flow determinations remain for global analysis (Figure 4.1).
Some measured heat flow values are in excess of 1000 mW/m2 above estimated
background. These extreme highs, most commonly on ridge flanks, are the result of
very isolated and concentrated hydrothermal discharge associated with black smokers
and mud volcanoes (e.g., Williams et al. [1979]; Becker and Von Herzen [1996];
Rona et al. [1996]; Eldholm et al. [1999]; Kaul et al. [2006]). Lateral variations
near these discharge zones can be several hundred to 10 000+ mW/m2 in the space
of a few meters. These highly anomalous regions can strongly affect the standard
deviations when included. Median values (2 m.y. bins) are also affected because many
measurements are typically taken near these anomalous regions. While the choice of
500 + qref mW/m2 is arbitrary, I compared the results with 1000 + qref mW/m2 and
see little difference in the end result.
Digital sediment thickness maps are used to estimate the sediment cover below
heat flow sites (Figure 4.1). I interpolate Divins [2007] 5′ × 5′ and Laske and Masters
[1997] 1◦ ×1◦ sediment thickness models to the same 2′ × 2′ grid as the age model.
The lower resolution dataset is only used where the higher resolution is not available,
specifically the northern Philippine Sea and Arctic Ocean.
I use the seamount database by Wessel [2001] to compute the distance of heat
flow sites to seamounts. The seamount database includes a height and radius fit to
a truncated conical shape. Minimum height resolution is ∼1 km, but the number
Heat flow sites:
sediment cover
≥ 325 m
all other sites
Sediment
Thickness
[m]
20000
seamounts
≥ 85 km
8192
4096
2048
1024
784
576
400
256
144
64
16
0
43
Figure 4.1. Oceanic datasets used in this study. Basemap is 5′ × 5′ sediment thickness from Divins [2007] supplemented
by 1◦ × 1◦ Laske and Masters [1997]. Seamounts (black dots) by Wessel [2001] with LIPs (grey regions) from UTIG
Plates Project. Sea-floor age isochrons in 20 Ma intervals by Müller et al. [2008] with ridges identified in bold. Heat flow
(red circles, included by preferred filter; white circles, excluded by preferred filter) is from an updated global database by
(Chapter 2).
44
of seamounts by size is reliable above >2 km. The minimum distance from a heat
flow site to a seamount is computed by determining the distance from the heat flow
site to the seamount center less the radius. No attempt has been made to globally
characterize basement exposures less than 1 km in height, which could potentially
affect some sites.
4.4
Thermal Model of Sea-floor Spreading
Before examining the effect of hydrothermal circulation on global heat loss estimates, I must first understand the background thermal regime. Continental thermal
models use heat flow estimates made from temperature measurements that extend
>100 m into the subsurface. The vast majority of oceanic heat flow measurements
are made in the upper few meters and therefore are easily disturbed by hydrothermal
circulation. Therefore, a proxy for the background thermal regime is required.
Unlike heat flow, subsidence of seafloor is related to the integrated thermal state
of the lithosphere and little affected by skin effects (i.e., hydrothermal circulation).
Bathymetry therefore is an excellent constraint on oceanic cooling models (Figure 4.2b). Numerous cooling models have been developed [Parker and Oldenburg,
1973; Davis and Lister , 1974; Crough, 1975; Parsons and Sclater , 1977; Stein and
Stein, 1992], including several recent models using temperature–pressure dependent
thermophysical properties [McKenzie et al., 2005; Afonso et al., 2005, 2007]. The most
common mathematical model used to describe ocean cooling is the plate. The initial
thermal state of the plate is a uniform temperature with depth (Figure 4.2a). As
the plate cools, the lithosphere grows in thickness and contracts, causing subsidence.
In plate cooling models, the maximum plate thickness is fixed at depth by a basal
temperature, which is held constant through time. The fixed basal temperature
causes an asymptotic flattening of the predicted subsidence and heat flow curves
(Figure 4.2b).
While the cooling models mentioned above yield different plate thicknesses and
mantle adiabatic temperatures based on choice of input parameters, the modeled
heat flow and bathymetry are relatively similar. Subsidence of the seafloor provides a
45
0
2
0.5
5
25
50
50
t = 150 Ma
75
4
5
6
(a)
100
0
3
Bathymetry [km]
Depth [km]
25
Sediment Corrected
Bathymetry
500 1000 1500
Temperature [°C]
(b)
0
50
100
Age [Ma]
150
Figure 4.2. Thermal isostasy of the oceanic lithosphere. (a) Oceanic geotherms
in 25 Ma intervals. (b) Observed (circles) bathymetry in 2 m.y. age bins with 1-σ
standard deviation and plate model bathymetry (grey line). Oceanic plate cooling
model by Stein and Stein [1992], including a static 90 m reduction in bathymetry to
fit the subsidence better, is shown in grey. Bathymetry data are isostatically corrected
for sediment loading and exclude LIPs.
reliable proxy for the integrated thermal state of the lithosphere. Since I are interested
in the deficit in heat flow created by hydrothermal circulation, not the individual
properties of plate cooling models, I have chosen to use the model by Stein and Stein
[1992] because of its widespread use and good fit to the global bathymetry (Figure
4.2).
4.5
Observed Heat Flow
Heat flow predicted from plate cooling models is high near the ridge and rapidly
decreases with age (∝ t−0.5 ), asymptotically approaching a background heat flow at
old ages (Figure 4.3a). Observed heat flow data for the oceans show a significant
deficit at young ages relative to plate cooling estimates (Figure 4.3). The data also
show considerable variability in median values at young ages. Vigorous hydrothermal
circulation through the young oceanic crust is generally invoked as the explanation
for this mismatch and variability [Lister , 1972; Williams and Von Herzen, 1974;
Anderson et al., 1977; Sclater et al., 1980; Davis et al., 1992; Stein and Stein, 1992;
Harris and Chapman, 2004].
46
300
Heat Flow [mW/m2]
Observed Heat Flow
(a)
200
100
No. observations
0
400
13501 sites
(b)
200
0
0
50
100
Age [Ma]
150
Figure 4.3. Observed oceanic heat flow. (a) Observed heat flow data in 2 m.y.
bins with 1-σ standard deviation. shown in grey. Observed heat flow excludes data
located on LIPs, with values ≤0, and > qGDH1 + 500 mW/m2 . Hatched regions are
‘reliable’ heat flow from Sclater et al. [1976, 1980]. (b) Number of heat flow sites in
each bin.
Hydrothermal circulation in the oceanic crust results from buoyancy-driven flow,
unlike the pressure-driven flow in continental groundwater systems. Figure 4.4 illustrates
how oceanic hydrothermal systems can affect heat flow. The seafloor-water interface
is at constant temperature, Ts , causing isotherms beneath high bathymetry to deflect
toward the surface. Cold dense seawater seeps into the oceanic lithosphere, generally
through sedimented basins and exposed bathymetric lows. The water is heated as
it travels through the lithosphere and then is discharged as hot buoyant seawater
beneath exposed basement highs (Figure 4.4a). In this type of environment, heat
flow is low in the sediments at intermediate distance surrounding exposed basement
and high on exposed volcanic highs. A high may even be observed in the sediments
immediately adjacent to the basement high [Stein and Stein, 1997].
47
Ts
Tb
(a) Thin/No Cover
qobs
qref ~ 0.1-0.2
q
σobs
qref ~ 0.7-1
qref
(b) Intermediate Extensive Cover
Ts
Tb
qobs
qref ~ 0.5
q
σobs
qref ~ 0.6-0.8
qref
(c) Thick Extensive Cover
Ts
qobs
qref ~ 1
q
σobs
qref ~ 0.4
qref
Tb
Figure 4.4. Effect of sediment cover on hydrothermal circulation and heat flow
through the oceanic lithosphere. Top plots in each panel illustrate observed heat flow
(qobs , heavy line) and reference heat flow (qref , thin line). Standard deviation of observed heat flow σobs . Bottom plots illustrate sediment cover (grey region), isotherms
(thin lines), and hydrothermal circulation (solid and dashed lines). Sea-floor water
interface is an isotherm, Ts , and temperatures at the sediment–basement interface,
Tb , is nearly isothermal (panel (c) only).
While some temperature measurements used to estimate heat flow in oceanic lithosphere are collected in crystalline basement (e.g., DSDP), >95%, are made in the
upper 3–9 m of sediment owing to the types of probes typically employed. The combination of cold recharge in sediment and limitations in the measurement environment
leads to a systematic bias toward anomalously low heat flow (Figure 4.3).
Beneath buried volcanic basement, hydrothermal circulation is sufficiently vigorous that temperatures at the sediment–basalt interface are relatively constant, Tb .
Isotherms in the sediments are deflected toward the surface above these volcanic highs,
setting up the necessary density gradients for circulation (Figure 4.4b). Because both
recharge and discharge zones are sediment-covered, it is possible to measure heat flow
everywhere around these anomalies. Low heat flow exists in regions of thick sediment
and high in regions of thin sediment. In a well sampled study, the integrated heat flow
48
can reach the background, qref . In general, however, heat flow in this environment is
not well sampled, average heat flow ∼50% of the background, with high variability.
As additional sediment accumulates, sediment compaction reduces the hydraulic
conductivity.
Eventually sediment permeability decreases fluid flow below levels
detectable by heat flow (Figure 4.4c). To determine the global conductive heat
flow through the lithosphere, I can exploit regions where hydrothermal circulation
is reduced below detection by filtering for regions with sufficient sediment cover and
far enough to fall outside the influence of basement highs and exposures.
4.6
Global Filtering—Sediments and Seamounts
In order to filter out the effect of hydrothermal circulation and determine the
background conductive heat flow, I need to develop criteria that can be used to
find environments where hydrothermal circulation is minimal.
Sclater et al. [1976, 1980] proposed three filters to oceanic heat flow data to limit
the effect of hydrothermal circulation and estimate the conductive heat flow through
the oceanic lithosphere. They restrict high quality heat flow determinations to regions
with flat or rolling hill morphologies, thick sediment cover (>200 m) and distances
>18 km from basement highs. These criteria can be applied to individual sites,
requiring local bathymetric information and seismic lines that are frequently collected
prior to heat flow surveys. Average heat flow values can be improved using this sort
of analysis as seen by the hatched boxes in Figure 4.3a.
The volume of information and format (many of the records are not digital or have
been lost) make it impractical make these assessments on 13 000+ oceanic heat flow
sites. Therefore, it is desirable to develop a method that can be easily automated
from global sediment cover and bathymetry models in order to extract the lithospheric
background heat flow. In this study, two filters are applied to the heat flow data: a
minimum sediment thickness filter with a range of 0 to 1000 m; and a minimum
distance to seamounts, ranging from 0 to 100 km.
Figure 4.5 shows heat flow data collected in 2 m.y. age bins with sediment thickness
restricted to 50 m intervals from 0 to 900 m. Because data are unevenly distributed in
49
200
0-50 m
50-100 m
100-150 m
150-200 m
200-250 m
250-300 m
300-350 m
350-400 m
400-450 m
450-500 m
500-550 m
550-600 m
600-650 m
650-700 m
700-750 m
750-800 m
800-850 m
850-900 m
150
100
50
0
150
100
50
0
150
Heat Flow [mW/m2]
100
50
0
150
100
50
0
150
100
50
0
150
100
50
0
0
50
100
150
0
50
100
Age [Ma]
150 0
50
100
150
Figure 4.5. Median heat flow versus age in 2 m.y. bins divided into groups with
50 m increments of sediment thickness. Error bars represent 1-σ. Data points without
error bars represent a single measurement. Cooling model (grey) is heat flow from
GDH1.
50
age and sediment thickness, many bin averages are very noisy within a given sediment
interval. Sediment intervals between 50–100 and 100–150 m exhibit a very low heat
flow when compared to GDH1. In fact, median values at young ages are rarely
elevated much above the asymptotic limit. At 150–200 m there is increased scatter,
but little trend is evident in heat flow at young ages.
As the sediment interval is increased, the heat flow at younger ages generally
increases and in some bins closely approaches or exceeds values of GDH1 (sediment
thickness >450 m). At higher sediment intervals, >550 m, fewer and fewer heat flow
medians meet or exceed GDH1.
To determine the effectiveness of these filters I use two metrics, each highlighting
different aspects of binned heat flow results.
4.6.1
4.6.1.1
Filter Metrics
Statistical Correlation
The heat flow–age pattern resulting from cooling of a half-space can be described
to high accuracy as C/age−1/2 , where C is empirically determined (∼500 Ma1/2 mW/m2 ).
This mathematical model also approximates the plate cooling model
reasonably well. Without biasing filtered heat flow patterns towards any particular
cooling model, I estimate the improvement of various filter combinations by computing the correlation coefficient of a linearized heat flow–age relationship. While the
correlation between heat flow and age−1/2 is a linear space, the distance between 2 m.y.
age bins is large at young ages and decreases rapidly, thereby causing any distribution
to be strongly weighted by the youngest ages as older ages are progressively more
closely spaced. By inverting the equation and testing the linear correlation between
age and heat flow−2 , the time spacing is uniform, which gives equal weight to the
entire range of ages.
I apply the correlation
r=
PN
i=1 (ti γi )
− N t̄γ̄
(N − 1)σt σγ
(4.1)
to filtered data, where ti and γi are the age and heat flow−2 of 2 m.y. binned data,
respectively. The bar denotes mean values of all bins, σ denotes standard deviation,
51
and N is the number of bins. To avoid numerical instability, 1000 is added to each
heat flow datum and zero age is set to 0.1 Ma, neither of which affect the results
significantly.
4.6.1.2
Variability
To complement the correlations, I seek a single value to quantify improvement in
the standard deviations of age bins. The standard deviation is often higher at high
heat flow. Therefore, standard deviation is normalized by the median before making
a direct comparison of binned results. I define the relative variability in heat flow for
a given filter as,
variability =
N
σi
1 X
,
N i=1 q̄i
(4.2)
where q̄i and σi are the average and standard deviation of each age bin, respectively.
Because hydrothermal circulation causes an increase in the spatial standard deviation
in addition to a reduction in the overall average heat flow, I expect filtering to reduce
the standard deviation within each bin.
4.6.2
Filtered Results
Correlation coefficients between binned age and heat flow−2 range from 0.33 to
0.93 (Figure 4.6). Unfiltered data are poorly correlated with age (0.5). The highest
correlation, 0.93, occurs using filters with a minimum sediment thickness of 475 m
and a minimum distance to seamounts of 85 km (‘+’ in top panel of Figure 4.6). The
lowest correlations occur for sediment filters below ∼100 m. The poor correlation may
result from a reduction in hydraulic conductivity as sediments compact. Correlations,
including sites near seamounts, remain low until the sediment thickness filter reaches
∼400 m.
An increase in correlation coefficient occurs very rapidly as the minimum sediment
thickness increases from 300 to 400 m when filtering distance to seamounts is small
(Figure 4.5). A high plateau in correlation coefficient occurs between ∼400–700 m
minimum sediment thickness suggesting all filter results within this range are relatively similar. Seamounts also appear to have very little influence on the median
1.0
Correlation 0-180 Ma
80
0.8
60
40
0.6
20
0
0
200 400 600 800 1000
Minimum Sediment Thickness [m]
Correlation Coeffiient
100
0.4
0.8
100
Variablity 0-56 Ma
80
0.7
60
0.6
40
0.5
20
0
0
200 400 600 800 1000
Minimum Sediment Thickness [m]
Normalized Variability
Minimum Distance from Seamount [km]
Minimum Distance from Seamount [km]
52
0.4
Figure 4.6. Metrics for improvement in globally filtered datasets as a function of
minimum sediment thickness and minimum distance to seamounts. (top) Correlation
of binned heat flow−2 with age. The ‘+’ indicates the filter combination with the
maximum correlation coefficient. (bottom) Normalized variability.
53
values for sediment thicknesses >400 m. Even with coarse sediment thickness resolution, above 400 m it seems likely that local highs in the igneous crust are well
sedimented, preventing fluid flow and contributing to the high correlations.
Seamount distance also shows a significant influence on correlation coefficient.
A slight decrease in correlation coefficient occurs as the distance from seamounts
increases out to ∼30 km. By ∼40–50 km, a large increase in the correlation coefficient occurs, possibly resulting from the maximum lateral extent of most oceanic
hydrothermal systems, well beyond the 18 km suggested by Sclater et al. [1976]. This
40–50 km scale is consistent with observations of the hydrothermal systems southwest
of the Nicoya Peninsula on the Cocos plate [Hutnak et al., 2008]. While distance from
seamounts continues to influence the correlation coefficient beyond 85 km minimum
distance, the proximity to a seamount may also imply rough sediment–basement
morphology and/or a higher likelihood that smaller basement penetrators below the
detection limits of Wessel [2001] are near the heat flow site. Basement highs that
do not penetrate the sediment cover have a smaller effect on the median heat flow.
Therefore, far from seamounts, thinner sediment cover is necessary to retard flow
below noise levels contributing to an increase in the correlations.
Correlations begin to fall off beyond a minimum sediment thickness >700 m as a
result of few data left to compute values of individual bins, making the binned results
more susceptible to individual outliers.
In contrast to the correlation coefficient, variability does not exhibit clearly defined
peaks or troughs that can be used to pick a preferred filter. Variability is highest
for the unfiltered dataset and decreases as both minimum distance to seamounts and
minimum sediment thickness are increased. For 0–180 Ma ages the overall decrease
in variability is from 0.54 to 0.44. A similar pattern in variability exists when the age
range is restricted, but exhibits a much larger range. The variability when restricted
to 0–56 Ma ranges from 0.42–0.78 (Figure 4.6 bottom panel). The higher variabilities
for age restricted data result from the exclusion of bins at older ages with relatively
constant and ‘low’ standard deviation.
54
4.6.3
Preferred Filters and GDH1 Misfit
The preferred set of filters is chosen to minimize the effect of vigorous hydrothermal
circulation on heat flow and maximize the number of data points used in the analysis.
I choose filter values of 325 m for minimum sediment thickness and 85 km as the
distance to the nearest seamount. The filtered dataset has a correlation coefficient of
0.89 and retains 4929 sites, or a little less than 40% of the initial pre-filtered dataset
(Figure 4.7c).
The preferred filter constraints improve agreement between modeled and measured
heat flow as a function of seafloor age (Figures 4.7a. The large increase in heat flow
from thin to thick sediment cover confirms that the effects of hydrothermal circulation
can be eliminated by thick and extensive sediment cover. This assessment is further
confirmed by the reduction in variability (Figure 4.7e and f ). By filtering heat flow
using this simple scheme, the effect of hydrothermal circulation on average heat flow
is substantially reduced. However, a heat flow deficit compared to lithospheric cooling
models persists at ages <55 Ma. I explore some of the causes of this persistent deficit
in Section 4.8.
4.6.4
Cautions Using Global Binned Data
A site by site analysis similar to Sclater et al. [1976] is preferable to global models
like the one performed here. However, a site by site analysis is impractical because
of a lack of the data required to assess many, if not most, sites. A global analysis
can be performed with global models of sediment thickness with some success, but
several issues with global sampling and resolution must be acknowledged.
Many heat flow sites collected in the previous version of the global heat flow
database [Pollack et al., 1993] are isolated and individual determinations. These
provide a good spatial distribution for a global analysis but a single site can be
easily affected by an anomalous measurement. Most data collected since the previous
database update are located in targeted surveys with tens to hundreds of measurements in clusters, lines, and grids. While providing better regional heat flow, they
provide poor spatial coverage globally. The recent data are also commonly located
55
300
sed. thickness ≥ 325 m
seamounts ≥ 85 km
Heat Flow [mW/m2]
250
sed. thickness < 325 m
seamounts - no filter
(a)
(b)
200
150
100
50
No. observations
0
400
unfiltered
filtered, N = 4929
unfiltered
excluded, N = 7370
(d)
200
0
(e)
1.2
SD/HF
(c)
(f )
0.8
0.4
0
0
50
100
Age [Ma]
150
0
50
100
Age [Ma]
150
Figure 4.7. Global filtering results. (left) Heat flow locations filtered for sites with
≥325 m sediment cover and ≥85 km to the nearest seamount. (right) Sites with
<325 m of sediment cover and no seamount distance constraint. (a) and (b) heat flow
versus age in 2 m.y. bins (open circles contain <10 sites). One standard deviation in
grey. (grey squares) Environmentally analyzed data in (a) and (g) (Table 4.1). (c)
and (d) number of observations within each age bin, initial dataset in grey. (e) and
(f ) normalized standard deviation (variability). (g) and (h) ratio of binned heat flow
to GDH1. Black lines on (e − h) represent a 5 m.y. moving average. Grey lines from
plot opposite for direct comparison.
56
1.4
Data/GDH1
(g)
(h)
1.0
0.6
0.2
0
50
100
Age [Ma]
150
0
50
100
Age [Ma]
150
Figure 4.7 continued. (g) and (h) ratio of binned heat flow to GDH1. Black
lines on represent a 5 m.y. moving average. Grey lines from plot opposite for direct
comparison.
57
in regions which exhibit ‘interesting’ heat flow variations/anomalies, rather than
conductive background values. When few data are located in a given age bin, the
spatially clustered data may dominate the bin average, especially if the filtered
sediment thickness is great.
Uncertainties in sediment thickness are also a concern when using global datasets.
At 5 arc minute resolution, a pixel covers (9.3 km)2 at the equator. For the 1◦
dataset, equatorial resolution is (111.2 km)2 . Because the sediment thickness represents an average for the region covered by the pixel, basement highs in either
case may short-circuit otherwise impermeable sediment cover allowing hydrothermal
circulation except where sediment cover is very thick. Numerical models of hydrothermal circulation using reasonable parameters for fluid driving forces and sediment
permeability suggest that 150–200 m of sediment cover is typically required to reduce
fluid flow to a level with negligible influence on heat flow [Grevemeyer and Bartetzko,
2004], although the thickness required varies based on sediment type, permeability,
and the magnitude of the driving force. However, as the mean sediment thickness
increases, fewer volcanic highs approach the surface and their influence on heat flow
is diminished. This is consistent with improvements in filtered results with sediment
thicknesses greater than 150 m.
4.7
Environmental Analysis
To overcome some of the limitations in a global analysis, I choose four regions with
adequate site environment data to reveal hydrothermal circulation patterns and hence
to estimate the background lithospheric thermal regime.
The Juan de Fuca ridge flank is by far the most extensively studied ridge flank
with 1068 heat flow measurements co-located with seismic data [Davis et al., 1992,
1997]. Very few of these data are completely free of hydrothermal circulation but,
properly selected, the data can still be used to estimate conductive heat flow (Table
4.1). An example of heat flow data from the Juan de Fuca flank is shown in Figure
4.8. Temperatures at the sediment-basement interface measured from nine Deep
Sea Drilling Project (DSDP) boreholes with >100 m sediment cover are relatively
58
Table 4.1. Heat flow from environmental analysis.
Locality
Age
Ma
1.4
3.5
6.0
12.9
16.7
18.2
19.8
21.4
22.8
26.6
Juan de Fuca
Juan de Fuca
Costa Rica rift
Gulf of Aden
Gulf of Aden
Cocos plate
Cocos plate
Gulf of Aden
Gulf of Aden
Gulf of Aden
qs ± σq a
mW-m−2
361 ± 92
253 ± 61
216 ± 20
139 ± 9
118 ± 6
117 ± 20
114 ± 14
125 ± 8
110 ± 10
104 ± 8
N
98
118
33
6
3
37
24
6
17
3
qobs /qs b
0.976
0.934
0.965
0.85
0.85
0.967
0.972
0.85
0.85
0.85
a
mean and standard deviation of sedimentation corrected heat flow. b sedimentation
correction using a thermal diffusivity of 0.3×10−6 mm2 /s. Sediment corrections for
Gulf of Aden are average of Lucazeau et al. [2008] and Lucazeau et al. [2009].
0
1
Age [Ma]
2
3
4
5
Observed
GDH1
Heat Flow
[mW/m2]
600
400
200
(a)
0
80
40
(b)
Sediment
4.0
4.5
(c)
1026
1027
3.5
1032
TWTT
[sec]
3.0
1029
0
1023
1024
1025
1030/31
1028
Basement
Temperature
[oC]
120
Basement
0
40
80
120
Distance from ridge axis [km]
Figure 4.8. Heat flow case study from Juan de Fuca flank. (a) heat flow, (b)
basement temperature, (c) sediment and basement interfaces as determined from
seismic reflection. Figure after Harris and Chapman [2004].
59
constant despite significant variability in basement topography and sediment thickness. The heat flow pattern over the well-sedimented seafloor with variable basement
topography is related to vigorous hydrothermal circulation within the basement and,
while variable, the average should be close to conductive background [Davis et al.,
1999]. Additional profiles from the Juan de Fuca flank above smooth basement
topography exhibit a sinusoidal heat flow pattern, suggesting cellular convection
within the basement beneath these sites. Because several cycles are captured, the
average should represent the background heat flow [Davis et al., 1996]. A total 153
sites used from the Juan de Fuca flank allow us to estimate the conductive lithospheric
heat flow in seafloor between 1 and 4 Ma.
There are 48 detailed heat flow sites on the Costa Rica rift flank for which there
are high resolution seismic data [Davis et al., 2004]. The sites are filtered to exclude
high heat flow anomalies associated with basement highs that do not penetrate the
sediment cover. The resulting heat flow is near GDH1 estimates for 6 Ma seafloor.
Modeled isotherms in the region strongly up-warp as a result of variations in the
basement topography and vigorous hydrothermal circulation in the upper part of the
igneous crust, not hydrothermal circulation through the sediments. Thus the high
anomaly is excluded from this analysis.
The Gulf of Aden is a well sedimented rift and relatively young oceanic spreading
center (<30 Ma). I analyzed 35 sites on the ridge flank [Cochran, 1981; Lucazeau
et al., 2008, 2009]. All 35 sites have >100 m of sediment and are located above
relatively smooth basement.
Heat flow values are well behaved and have small
standard deviations (Table 4.1).
A survey of 327 sites co-located with seismic estimates of crustal thickness are used
to estimate heat flow on the Cocos plate [Hutnak et al., 2008]. The authors identified
regionally low basement temperatures due to fluid extraction of heat discharged
through several seamounts. The affected hydrothermal zone extends several tens of
kilometers beyond each exposed seamount. On the south end of the study region
where there are no basement exposures, heat flow is higher and much closer to
background predictions. By excluding data within the identified hydrothermally
60
affected zone and restricting to sites >100 m, median heat flow fits the GDH1 plate
cooling model to within error (Figure 4.9).
Data from these site-specific analyses result in heat flow values similar to the
estimated conductive heat flow. Standard deviations for these data are <25% of the
observed mean, significantly better than variability in global filtered results (Figure
4.7). The Juan de Fuca flank, Costa Rica Rift, Gulf of Aden, and Cocos data all show
heat flow values above the globally filtered data, suggesting hydrothermally disturbed
data are not completely filtered from the global analysis and/or additional corrections
are necessary to account for the lower heat flow.
300
Environmental Analysis
Global Filters:
≥ 10 determinations
< 10 determinations
Heat Flow [mW/m2]
250
200
150
100
50
(a)
0
1.4
Data/GDH1
(b)
1.0
0.6
0.2
0
50
100
Age [Ma]
150
Figure 4.9. Filtered heat flow adjusted for sedimentation and thermal rebound. (a)
adjusted and filtered heat flow (black circles). An environmental analysis is performed
on the grey squares (Table 4.1). (b) fraction of heat flow to GDH1. Lines represent
a moving average of 5 m.y. (black) adjusted and filtered, (grey) filtered only.
61
4.8
Corrections to Global Data
The systematic low at young ages between filtered heat flow and plate cooling
models raises the question of whether a heat flow deficit still exists, or if plate models
should be revised. The heat flow at 10 Ma is 60% of GDH1, and increases to 80% by
20 Ma but does not reach the predicted background heat flow until ∼55 Ma (Figure
4.7g).
4.8.1
Sedimentation
Sediments are deposited at the surface temperature, depressing the thermal gradient
and thus the heat flow. The effect of sedimentation on heat flow is near zero initially,
but grows with time (Figure 4.10a). The estimated heat flow with age assuming
constant sedimentation rate is shown in Figure 4.10b. Sedimentation cannot cause the
observed persistent deficit at young ages unless the sedimentation rates are systematic
and extremely high (>500 m/m.y.). The sediment correction can be estimated by
dividing the observed heat flow by the fraction computed in Figure 4.10a. More
than 80% of the seafloor has a sedimentation rate less than 50 m/m.y. and therefore
at most locations there is a small or negligible sedimentation effect on heat flow.
Sedimentation rates >100 m/m.y. occur only in regions near continents with high
river discharge such as the northeast Pacific and Bay of Bengal.
4.8.2
Thermal Rebound
In regions where sedimentation is not so rapid that hydrothermal circulation is
quickly cut off, a large quantity of heat can be extracted from the lithosphere.
Once hydrothermal circulation has ceased, heat flow will return to background given
sufficient time. A delay in the return to background could cause a persistent low heat
flow to be observed. Hutnak and Fisher [2007] explored the effect of sedimentation
and thermal rebound on oceanic heat flow. Their model results show a rebound of
90% in about 1 Ma for 200 m of sediment cover or less (Figure 4.11). The rebound
fraction shown in Figure 4.11 is estimated from an empirical relationship that fits the
Hutnak and Fisher [2007] models for 500, 1000 and 2000 m very well. The empirical
62
1.0
10
(a)
q/qref
0.8
100
0.6
1000 m/m.y.
0.4
0.2 mm2/s
0.5 mm2/s
0.2
0 -4
10
10-2
100
Time [m.y.]
102
200
Heat Flow [mW/m2]
(b)
0.2 mm2/s
0.5 mm2/s
150
100
0 m/m.y.
50
10
0
100
1000
0
50
100
Age [Ma]
150
Figure 4.10. Estimated heat flow response to sedimentation. (a) ratio of heat flow
with sedimentation to heat flow without. Curves are computed using Von Herzen
and Uyeda [1963] with sedimentation rates of 10, 100, and 1000 m/m.y. with thermal
diffusivities of 0.2 and 0.5 mm2 /s, which approximate effective diffusivity with thin
and thick sediment cover. (b) heat flow as a function of seafloor age with sedimentation
models in (a).
63
0.8
0.6
200
m
500
100 m
0
200 m
0m
Fraction Rebound, f
1.0
0.4
0.2
0
10-4
10-2
100
Recovery Time, tr [m.y.]
102
Figure 4.11. Estimated fraction of thermal rebound as a function of sediment cover
and time since cessation of hydrothermal circulation.
estimate for 250 m is equivalent to the 200 m model reported Hutnak and Fisher
[2007], but for my purposes will be sufficient.
If I assume a constant sedimentation rate, the duration thermal rebound, tr , is
given by
!
hc
,
(4.3)
tr = tp 1 −
hp
where tp is the age of the crust, and hp and hc are the present-day sediment thickness
and thickness at cessation of hydrothermal circulation.
I estimate the fully recovered heat flow, qf r from the partially recovered (observed)
heat flow, qpr , and the depressed heat flow from hydrothermal circulation, qh , by
qf r = qh +
qpr − qh
,
f
(4.4)
where f is the fractional recovery estimated from Hutnak and Fisher [2007]. The
rebound pattern I would expect on well-sedimented seafloor has a larger deficit at
young ages because less time has elapsed before measurement and therefore less time
to return to equilibrium.
4.8.3
Adjustments to Filtered Heat Flow
I estimate the sedimentation rate by dividing the present day sediment thickness
by the age of the igneous crust. While this estimate is simplistic, it is beyond the
scope of this study to model the sedimentation history of the entire ocean. I then
64
compute the sedimentation effect following Von Herzen and Uyeda [1963] using a
thermal diffusivity of 0.3 mm2 /s (Figure 4.10).
To estimate the hydrothermal rebound, I assume that hydrothermal circulation
falls below measurable levels for sites with >325 m of sediment (my preferred filter).
Because it is difficult to know a priori the hydrothermally disturbed heat flow, I
estimate qh from a smoothed estimate of the average deficit with age (Figure 4.7b).
Since the lithosphere continues to cool following cessation of hydrothermal circulation,
I choose the deficit at the present seafloor age for simplicity. Thus qr calculated in
this manner is likely an underestimate of the true rebound.
To avoid issues with which correction to apply first (order matters), I estimate
the adjustments for incomplete rebound and sedimentation separately and add them
to the filtered heat flow (Figure 4.9). The sedimentation adjustment increases the
heat flow for nearly all ages. Heat flow observations within individual age bins at
crust older than 100 Ma show no discernible trend in heat flow as a function of
sediment thickness, implying that my sedimentation adjustment overestimates the
effect of sedimentation. The adjustment for incomplete rebound is negligible for ages
>70 Ma, but significantly reduces the deficit at young ages. The combination of the
two adjustments reduces the underprediction of heat flow at 10 Ma from 0.6 to 0.7
of GDH1 model estimates. By 30 Ma, the deficit is reduced to near zero, indicating
GDH1 heat flow is a good predictor of background conductive heat flow through the
oceanic lithosphere.
To estimate the remaining deficit from incomplete thermal rebound and sedimentation more accurately, one requires a thermal model specific to the history of each
site. The simplistic analysis performed above suggests a persistent deficit may be
small relative to the estimated background conductive heat flow model GDH1.
4.9
Global Heat Loss
The current power output of hydrothermal circulation is estimated by the cumulative heat flow deficit in each age bin multiplied by the respective area over which
hydrothermal circulation occurs,
65
P =
N
X
(qi◦ − qi )Ai ,
(4.5)
i
where qi◦ and qi are the reference and observed heat flow between isochrons ti and
ti−1 with associated area, Ai . A Monte Carlo analysis is performed to determine the
advective heat loss using GDH1 heat flow as the reference. The data excluded by
my preferred filter out to 73 Ma is used as an estimate of the hydrothermal deficit
(Figure 4.7b). For 106 realizations, the estimated total power mined by hydrothermal
circulation is 6.9±1.8 TW (Figure 4.12). Hydrothermal power accounts for ∼17% of
the Earth’s total heat loss (44 TW [Pollack et al., 1993]).
My advective power loss is similar to the estimate of power output obtained from
chemical fluxes [Elderfield and Schultz , 1996], but lower than most previous estimates
based on heat flow [Williams and Von Herzen, 1974; Sclater et al., 1980; Stein and
Stein, 1994] with the exception of Wolery and Sleep [1976]. The previous higher estimates result from the use of total seafloor area between isochrons rather than limiting
to the area over which hydrothermal circulation occurs. This restriction reduces the
estimated area of hydrothermally affected seafloor to regions with sediment thickness
<325 m ultimately resulting in my lower estimated hydrothermal heat transport.
# Realizations
2.0
× 104
1.6
6.9 ± 1.8 TW
1.2
0.8
0.4
0
0
2
4
6
8 10 12
Advective Heat Loss [TW]
14
Figure 4.12. Estimated global advective power loss from Monte Carlo analysis with
106 realizations.
66
4.10
Conclusions
My results provide an updated and improved analysis oceanic heat flow, specifically
estimating the extent and magnitude of hydrothermal circulation within the crust.
My global analysis highlights a systematic bias toward low heat flow by measuring
temperatures predominantly in sedimented seafloor.
Heat flow measurements on the seafloor are rarely used for modeling the thermal
evolution of the oceanic lithosphere because they are frequently influenced by nearsurface hydrothermal circulation. A systematic low bias results from collecting an
overwhelming majority of heat flow estimates in sedimented regions of the seafloor
where hydrothermal recharge commonly occurs. The deficit from hydrothermal circulation is observed up to ∼65 Ma and causes an increase in the measured variability.
I show that by simple filtering and adjustments for additional physical effects, binned
heat flow approaches the plate model estimate from GDH1 [Stein and Stein, 1992].
Thus, heat flow can be used in addition to bathymetry to constrain cooling models
of the oceanic lithosphere for nearly all seafloor ages.
Filtering heat flow for localities with a minimum sediment cover of 325 m and
minimum distance to seamounts of 85 km, excludes most of the data perturbed by
hydrothermal circulation. Filtering also reduces the variability in heat flow from
0.6->1 to typical background variability (0.3-0.5).
A deficit in heat flow at young ages persists despite filtering, but is likely due
to a combination of incomplete thermal rebound following cessation of significant
hydrothermal circulation and depression of the thermal gradient by high sedimentation rates. An adjustment for these effects removes the deficit for ages >25 Ma and
reduces the deficit below 25 Ma. However, these adjustments are crude and may be
improved by incorporating more site specific history to each observation. Adjusted
heat flow approaches the estimated reference model GDH1. Thus carefully filtered
and corrected heat flow data can be used as a constraint on cooling models of the
oceanic lithosphere.
Detailed analysis of heat flow and seismic estimates of sediment thickness from the
Juan de Fuca plate, Costa Rica rift, Gulf of Aden and Cocos Plate, allow for very
67
high confidence estimates of conductive lithospheric heat flow. Heat flow in these
regions are consistent with the GDH1.
Total power loss from hydrothermal circulation is ∼7±2 TW, consistent with an
estimate from chemical fluxes [Elderfield and Schultz , 1996], but considerably lower
than previous estimates derived from heat flow studies (10-11 TW) [Williams and
Von Herzen, 1974; Sclater et al., 1980; Stein and Stein, 1994]. My model is an improvement over previous heat flow estimates of advective heat loss because I obtained
a better estimate of hydrothermally affected heat flow and restricted the integration
to regions where hydrothermal circulation is likely.
CHAPTER 5
PLATE COOLING MODELS FOR THE
OCEANIC LITHOSPHERE: ARE
COMPLEXITIES NECESSARY?
5.1
Abstract
Plate models developed over the past five years have been growing in numerical complexity. It is difficult to compare the importance of each complexity since
each study uses a different set of observational constraints. I develop a standard
sediment-corrected bathymetry–age and heat flow–age dataset cleaned of hydrothermal influence. This standard dataset is used to test plate cooling models. The use of
effective expansivity rather than volumetric expansivity exerts the largest influence
on modeled subsidence. Crustal thickness has the greatest influence on heat flow
and subsidence at young ages. Inclusion of radiogenic heat production estimated
from chemical models of oceanic lithosphere produces a negligible effect on cooling
models. Using a 7 km thick crust consistent with seismological observations, the
minimum misfit plate cooling model has a 90 km maximum plate thickness and
potential temperature of 1425◦ C.
5.2
Introduction
One of the triumphs of plate tectonic theory is the ability to predict oceanic
bathymetry as a function of crustal age. As hot lithosphere created at the mid-ocean
ridge is transported towards the abyssal plain, it cools, contracts and subsides. This
pattern can be easily modeled using simple 1-D transient cooling models with fixed
surface and basal boundary conditions, also know as a plate cooling model [McKenzie,
1967].
69
Early plate cooling models assume constant thermophysical properties [McKenzie,
1967; Sclater and Francheteau, 1970; Parker and Oldenburg, 1973; Davis and Lister ,
1974; Crough, 1975; Parsons and Sclater , 1977; Sclater et al., 1980, 1981; Stein and
Stein, 1992]. More recently, modeling efforts are increasingly complex and incorporate
a number of additional effects including temperature- and/or pressure-dependent
thermophysical properties [McKenzie et al., 2005], polymineralic composition [Afonso
et al., 2005], asthenospheric melt fraction [Afonso et al., 2007], etc. Each of these
models uses a different estimate for average bathymetry. For example, the Stein and
Stein [1992] (GDH1) model uses bathymetry from the north Pacific and northwest
Atlantic, far from seamounts and with low sediment cover, whereas the model by
McKenzie et al. [2005] uses bathymetry only from the north Pacific. Heat flow data
used to constrain for each of these models also vary, but are in general only used to
fix the asymptotic behavior.
Because many of thee models are calibrated against different bathymetry and/or
heat flow datasets, it is difficult to compare models directly. Additionally, the model
sensitivity to variations in parameter complexities are rarely discussed. In this study,
I propose a standardized dataset for global bathymetry and heat flow that I use
to investigate the importance of complexities beyond the simple constant property
models.
5.3
Theoretical Formulation
The equation for one-dimensional transient heat conduction with sources is given
by
"
#
∂T
∂
∂
k
+ A,
[ρCP T ] =
∂t
∂z
∂z
(5.1)
where T is temperature, t is time, and z is depth. Thermophysical properties CP , ρ,
k, and A represent heat capacity, density, thermal conductivity, and heat production,
respectively. If thermophysical properties are temperature-dependent, CP and ρ are
not constant in time and k is not constant in depth and Equation 5.1 expands to
∂T
∂k ∂T
∂2T
∂ [ρCP ]
T + ρCP
=
+ k 2 + A.
∂t
∂t
∂z ∂z
∂z
(5.2)
70
The recent model by McKenzie et al. [2005] includes time-dependent variations of
ρ and CP whereas Afonso et al. [2005, 2007] ignore them. Both studies assume
negligible heat sources. I numerically solve Equation 5.2 using the Crank-Nicholson
method with a central difference formulation. I initially ignore the time derivatives
for CP and ρ and solve the system using tridiagonal elimination.
The solution
is then perturbed by a time dependent term including both CP and ρ [McKenzie
et al., 2005]. Perturbations are iteratively applied until the solution converges. (The
formulation given by Equation 11 of McKenzie et al. appears to be incorrect despite
the correctness of their results. See section 5.7 for a discussion.)
Surface heat flow is estimated from modeled temperatures from the first two nodes
and the harmonic mean of conductivity at the top and bottom of the first layer,
q=
2k0 k1
k0 + k1
!
T1 − T0
.
z1 − z0
(5.3)
Subsidence is computed from the change in density relative to a reference density
column by
s=
N
X
ρobs,i − ρref,i
i=0 ρref,N − ρw
(5.4)
where ρobs,i and ρref,i are the observed and reference density for the i-th node.
5.3.1
5.3.1.1
Thermophysical Properties
Thermal Expansivity, αV and Density, ρ
The temperature-dependent volumetric expansivity, αV , is estimated by
αV (T ) = a0 + a1 T + a2 T −2 ,
(5.5)
where ai are empirical constants [Fei , 1995]. The temperature-dependent density is
then be easily solved by:
◦
"
ρ(T ) = ρ exp −
Z
T
Tref
#
αV (T )dT ,
(5.6)
where ρ◦ is the density at reference temperature and pressure (0 GPa and 298 K),
and Tref is the reference temperature.
71
The pressure- and temperature-dependent thermal expansion is computed by
ρ(P )
αV (P, T ) = αV (T )
ρ0
!−δT (ρ0 /ρ(P ))
,
(5.7)
where ρ0 and ρ(P ) are the densities at 0 GPa and at the pressure P , and δT is the
Anderson-Grünison parameter [Afonso et al., 2005]. Using the logarithmic equation
of state [Poirier and Tarantola, 1998],
ρ(P )
ρ(P )
P = KT
log
ρ0
ρ0
!"
KT′ − 2
ρ(P )
1+
log
2
ρ0
!#
,
(5.8)
where KT and KT′ are the isothermal bulk modulus and first pressure derivative,
respectively, I iteratively solve for αV (P, T ) by the Newton-Raphson method. Average
density and expansivity are estimated as the weighted mean of the individual phases,
which are in turn computed from a weighted mean of the individual mineral endmembers. Constants used to estimate expansivity and density are summarized in
Appendix D.
5.3.1.2
αV vs. αeff
Laboratory studies are used to estimate the volumetric expansivity by allowing full
expansion or contraction. Because the Earth’s surface is a free moving boundary,
contraction can occur completely in the vertical direction. However, the strength of
the lithosphere in the lateral dimensions is enough to overcome some degree of contraction. Thus, using the laboratory expansivity estimates leads to an overestimate of the
total contraction and consequently subsidence as the lithosphere cools. Pollack [1980]
first noted that the effective expansivity required to explain subsidence of the oceanic
lithosphere is between 70–85% of the full volumetric expansivity. The viscoelastic
rheology of the lithosphere depends upon temperature and allows for relaxation
of stress and an increase in contraction toward the volumetric limit.
Recently,
Korenaga [2007a] rigorously modeled this effect, showing the significant time- as
well as depth-dependence of this phenomenon with a similar effect on expansivity as
estimated by Pollack [1980].
Rather than model the effect directly, which is numerically expensive, I use the
results of Korenaga [2007a] to estimate an empirical approximation for the time
72
and temperature dependence of the effective expansivity.
The effective thermal
expansivity is reasonably approximated by,
αeff
= 0.5707 + 4.777 × 10−4u − 5.823 × 10−7 u2 + 5.403 × 10−10 u3 ,
αV
(5.9)
where u is given by,
u = T − 552.1t−0.25 ,
(5.10)
and is a function of both temperature, T in Kelvins, and plate age, t in m.y. Because
this approximation is unstable when t is close to zero, I set u = T −552.1 for ages t < 1
m.y. The ratio of effective to volumetric expansivity is also bounded between 0.55 and
1. I multiply the ratio calculated using Equation 5.9 by the temperature-dependent
expansivity before estimating density.
Figure 5.1 illustrates the accuracy of the Equation 5.9 relative to the model proposed by Korenaga [2007a]. While the approximation is not perfect, it is accurate
enough for my purposes given the uncertainty in estimating the effective expansivity
from rheologic properties.
5.3.1.3
Heat Capacity, CP
The heat capacity can be approximated as
CP = c0 + c1 T −0.5 + c2 T −2 + c3 T −3 +
dCP
P,
dP
(5.11)
with empirically derived constants, ci [Fei and Saxena, 1987]. The pressure contribution is computed using the thermodynamic identity [Osako et al., 2004],
∂CP
∂P
!
T
T
∂α
=−
α2 +
ρ(P, T )
∂T
!
.
(5.12)
P
The bulk heat capacity of the rock matrix is computed as the mean heat capacity
of all phases weighted by their volume fractions. Constants used to estimate heat
capacity are summarized in Appendix D.
73
Temperature [K]
400
800
1200
10 m.y.
50 m.y.
100 m.y.
1600
0.5
0.6
0.7
0.8
αeff/αV
0.9
1.0
Figure 5.1. Effective thermal expansivity model. The solid lines represent the
modeled thermal expansivity by Korenaga [2007a] for 10, 50 and 100 m.y. The
dashed lines are computed using Equation 5.9 at the same times.
5.3.1.4
Thermal Conductivity, k
Thermal conductivity results from the combination of lattice (phonon-phonon) and
radiative (photon-phonon) mechanisms. The total conductivity is written simply as
a sum of these two components [Schatz and Simmons, 1972],
k(P, T ) = kL (P, T ) + kR (P, T ).
(5.13)
For any given phase, the lattice thermal conductivity, kL , can be given by
kL (P, T ) = k ◦
298
T
n
K′
1+ TP ,
KT
!
(5.14)
where k ◦ is the conductivity at 0 GPa and 298 K. The conductivity for a solid-solution
mineral is slightly more complicated, but is easily adjusted by replacing k ◦ with a
quadratic as a function of the mole fraction of an end-member. Hence, the thermal
conductivity of a two-member solid-solution series is written:
2
kL (P, T ) = k0 + k1 χ + k2 χ
298 n
T
K′
1+ TP ,
KT
!
(5.15)
where χ is the mole fraction of a given end-member. This equation is a simplification
of Hofmeister [1999b] Equation 10, as suggested by Beck et al. [2007]. Thermal
74
conductivity of a more complex solid-solution mineral such as amphibole and garnet
is more complicated still.
The radiative contribution to thermal conductivity, kR , is near zero at room temperature and grows in influence with temperature. Hofmeister [1999a] estimated
the radiative contribution for olivine using a third-order empirical equation. However, polynomial formulations grows quickly to unrealistic values or return negative
conductivities beyond the experimentally calibrated bounds.
Therefore, I prefer
a formulation that is always positive and well-behaved when extrapolated beyond
experimental conditions. The estimated radiative conductivity data used to calibrate
my models appear to behave similar enough to that of an arctangent or error function
may be used to fit the data (Appendices A and C). I estimate radiative conductivity
using
kR (T ) = kRmax [1 + erf (ω(T − Tm ))] ,
(5.16)
where kRmax is the maximum radiative conductivity, ω is a scaling factor and Tm is
the temperature at 0.5kRmax .
The radiative contribution has been measured on very few minerals. For olivine,
by far the best studied, the experimental estimates are highly variable. The most
recent estimates suggest that the radiative contribution is small (Hofmeister [2005],
Appendix A) with even smaller values for lherzolite [Gibert et al., 2003]. I use the
estimate from lherzolite for olivine and pyroxenes. Spinel and garnet are opaque
at mantle temperatures and may have little influence on radiative conductivity and
therefore are ignored [Shankland et al., 2005]. No data exist for the radiative effect on
plagioclase, amphibole or chlorite. I assume no radiative contribution for plagioclase,
values similar to mica for chlorite, and similar to lherzolite for amphibole. A pressure
component to radiative conductivity may also exist, but it is even more poorly studied
than the temperature effect and therefore ignored [T. Shankland, pers. comm. 2006]
For a mineral composite with N phases with mole fractions wi , the effective thermal
conductivity, keff , is computed using the geometric mean [Clauser and Huenges, 1995],
keff = exp
" PN
i=1
#
wi log(kLi + kRi )
.
PN
i=1 wi
(5.17)
75
I use this formulation for the geometric mean rather than the more common product form because it is easier to differentiate. Constants used to compute mineral
conductivities are given in Appendix D.
Very few measurements have been made on chlorite, clinopyroxene, or spinel from
which to estimate room-temperature conductivity, and variable and P –T data are
even more limited. For minerals with limited conductivity data, k ◦ is estimated
to be the mean of all available determinations rather than estimating a quadratic
between end-members. When temperature-dependent conductivity data do not exist,
the temperature exponent, n, is estimated as 0.5 as suggested by Xu et al. [2004];
however, this assumption may not be true for all silicates.
Conductivity measurements have been performed on a limited range of clinopyroxene compositions, most Di>90 [Horai , 1971; Diment and Pratt, 1988; Harrell ,
2002; Hofmeister and Pertermann, 2008] and three with slightly higher iron contents (augite) [Horai , 1971; Hofmeister and Pertermann, 2008]. The conductivity
of clinopyroxene measured using contact methods at room temperature ranges from
4.05–5.57 W/m-K. Room-temperature conductivity of diopside estimated by Hofmeister and Pertermann [2008] using laser flash analysis is 8.84 W/m-K, significantly
higher than contact methods. Hofmeister and Pertermann [2008] suggest that laser
flash analysis estimates are typically ∼20% higher than contact methods due to
contact resistance, which appears to explain less than half the difference. However,
a correction is often applied for contact resistance and the only studies reporting
conductivities using the laser method are by Hofmeister et al., making it difficult to
verify their claim. Therefore, I use the average conductivity of seven clinopyroxene
samples measured using contact methods. The temperature coefficient is estimated
using the results by Hofmeister and Pertermann [2008].
Because of extremely complex cation and anion substitution in amphiboles, computing the thermal conductivity from a simple quadratic of two end-members is not
possible. A more complex scheme is necessary. One based on an analysis of major
element chemistry and cation substitution may provide a reasonable conductivity
estimate (Appendix B). I use the following approximation for the conductivity:
76
◦
kamph
= 4.0654χtr + 3.7834χfact + 1.7066χparg,
(5.18)
using the mole fraction of end-member compositions for tremolite (tr), ferro-actinolite
(fact), and pargasite (parg).
5.3.2
Mantle Phase Changes
Two major phase changes occur within the lithospheric mantle modeled in this
study. The plagioclase-spinel transition,
plagioclase + 2 olivine ↔ clinopyroxene + 2 orthopyroxene + spinel.
(5.19)
is roughly independent of temperature, with a transition pressure of ∼0.9 GPa [Herzberg,
1976]. The spinel-garnet transition,
2 orthopyroxene + spinel ↔ garnet + olivine,
(5.20)
varies in pressure as a function of temperature. The pressure of the spinel-garnet
transition is estimated using the empirical relationship
Psg (T ) = 1.4209 + exp(3.9073 × 10−3 T − 6.8041),
(5.21)
where T is in Kelvins. This curve is calibrated by inversion of the data reported
by Robinson and Wood [1998], Walter et al. [2002], Klemme and O’Neill [2000] and
references therein.
5.4
5.4.1
Datasets
Bathymetry
I prefilter ETOPO2 bathymetry, excluding regions defined as large igneous provinces
(LIPs) [Coffin and Eldholm, 1994]. While LIPs are excluded, the flexural effect
due to loading of the lithosphere can extend for several hundred kilometers. By
excluding the high bathymetry associated with LIPs and accepting the associated
sedimentary moats, the average bathymetry–age relationship may be biased towards
greater subsidence [Crosby et al., 2006; Crosby and McKenzie, 2009]. Bathymetry
77
deviations from the average also occur as a result of variations in crustal thickness
and/or mantle temperatures. For example, petrologic models predict higher melt
fractions with higher adiabatic temperatures.
The thickness of oceanic crust is
then related to the quantity of partial melt generated within the mantle reservoir
[Asimow et al., 2001]. Both the increased crustal thickness and temperatures increase
bathymetry. However, depending upon how one defines ‘anomalous’ bathymetry,
these natural variations may or may not be removed.
Attempts to filter out ‘anomalous’ bathymetry in response to loading are made
by restricting the magnitude of gravity anomalies [Crosby et al., 2006] and by using
a spatial correlation and distances from seamounts [Korenaga and Korenaga, 2008].
Using Korenaga and Korenaga’s filters, average bathymetry is ∼20 m shallower with
higher scatter than an unfiltered analysis.
Therefore, I have chosen to use the
unfiltered data. By doing so, I capture the entire flexural signals for most seamounts
but may bias bathymetry to greater average depths, particularly in old ages where
LIPs are more common.
Sediment loading significantly influences bathymetry and is particularly apparent
on passive continental margins where large sedimentary deposits result in shallowing
water depth with age. Assuming negligible flexure, the equilibrated isostatic response
to sediment loading is quite straightforward. It can be shown that the isostatic
correction to bathymetry as a result of sediment loading is computed from the the
average sediment density (ρ̄s ), density of seawater (ρw ), and density of the displaced
mantle (ρm ) as
ρm − ρ̄s
,
(5.22)
ρm − ρw
where h is the sediment thickness, ∆ε, is the change in water depth between the
∆ε = h
adjusted and observed bathymetry [Sykes, 1996].
The effective sediment density, ρ(z), is computed as a function of the porosity, φ(z):
ρs (z) = ρg + (ρw − ρg )φ(z),
(5.23)
where ρg is the matrix (grain) density. I use a two-layer porosity model given by
φ(z) =
(
φ1 e−z/h1
φ2 e−z/h2
if z ≤ H
,
if z > H
(5.24)
78
Table 5.1. Parameters for sediment correction.
Property
grain density, ρg
porosity, φ1
porosity, φ2
characteristic depth, h1
characteristic depth, h2
characteristic thickenss, H
seawater density, ρw
mantle density, ρm
Carbonate Oozea
2700
0.656
0.409
1278
2591
1190
1030
3340
Units
kg-m−3
m
m
m
kg-m−3
kg-m−3
Pelagic Clayb
2500
0.812
0.504
664
2160
457
a
Constants determined by inversion of chalk data reported by Mallon and Swarbrick
[2002]. b Constants from Velde [1996].
where φ1 and φ2 are the extrapolated surface porosity, h1 and h2 are the decay
constants (units of depth), and z is the depth. There is a fundamental change in the
nature of compaction curves for most sediment types requiring a two-layered model at
sediment thicknesses greater than a characteristic thickness, H [Sclater and Christie,
1980; Velde, 1996; Mallon and Swarbrick , 2002]. The average density is computed by
integration from the surface to the base of the sediment column,
ρ̄s = h−1
Z
0
h
ρs (z)dz,
(5.25)
where h is the sediment thickness. The average density for thicknesses h < H is
simply
ρ̄s = ρg + (ρw − ρg )φ1
h1 1 − e−h/h1 ,
h
(5.26)
where hc is the compaction decay length, φ0 is the unloaded porosity, and ρs and ρw
are the maximum rock and water densities, respectively. If h > H,
h
ρ̄s = ρg + (ρw − ρg )h−1 φ1 h1 1 − e−H/h1 + φ2 h2 e−H/h2 − e−h/h2
i
,
(5.27)
All of the constants depend on the sediment type (Table 5.1).
The two dominant oceanic sediment types are pelagic clay and carbonate ooze
which spatially cover most of the oceanic crust although, near continents, terrigenous
sediments are more common. Carbonates dominate shallow bathymetric regions, but
are not stable below ∼4500 m water depth, allowing pelagic clay to dominate. This
79
depth is called the Carbonate Compensation Depth (CCD) and depends on water
temperature, salinity, and pH [Spinelli et al., 2004]. Since the CCD depth is somewhat
uncertain and some near-surface carbonate deposited before the seafloor descends
below the CCD may be re-absorbed, sediment density is computed as carbonate ooze
above 4000 m, pelagic clay below 5000 m and a linear combination of carbonate ooze
and pelagic clay between 4000 and 5000 m. This simplification is similar to those
proposed by Sykes [1996].
5.4.2
Heat Flow
I use the heat flow database from a new global compilation to estimate the heat
flow as a function of seafloor age (Chapter 2). Heat flow at young ages is strongly
affected by hydrothermal circulation. Binned values at young ages exhibit a significant
deficit as a result of a systematic bias caused by hydrothermal circulation and by
measurement constraints.
Filtering heat flow data to retain only sites far from
seamounts (>85 km) and with significant sediment thickness (>325 km) minimizes
the effect of hydrothermal circulation on estimated heat flow (Table 5.2).
The filtered heat flow values in this study are largely uncorrected for sedimentation
and thermal rebound. Chapter 4 presents a detailed analysis of filtering heat flow to
produce the results used in this study.
Even after filtering, heat flow out of young seafloor may still exhibit a hydrothermal
deficit. By examining high-density data from regional surveys with seismic control
on sediment thickness and basement roughness, it is possible to identify sites with
minimal hydrothermal disturbance and obtain more accurate estimates of conductive
heat loss (Davis et al. [1992, 1999], and Chapter 4). I use regional estimates from
several localities to supplement heat flow out of young seafloor (Chapter 4, Table
4.1).
80
Table 5.2. Global bathymetry and heat flow data.
Age
Ma
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
51
53
55
57
59
61
63
65
67
69
71
εadj ± σε
m
2980 ± 552
3159 ± 548
3332 ± 559
3476 ± 580
3579 ± 589
3672 ± 542
3772 ± 564
3883 ± 570
3962 ± 594
4028 ± 561
4115 ± 522
4188 ± 532
4250 ± 576
4327 ± 623
4412 ± 568
4454 ± 566
4534 ± 568
4566 ± 597
4615 ± 618
4672 ± 689
4719 ± 643
4767 ± 657
4801 ± 667
4892 ± 665
4925 ± 688
4986 ± 718
5039 ± 713
5060 ± 671
5044 ± 676
5057 ± 725
5110 ± 757
5169 ± 809
5145 ± 719
5160 ± 745
5261 ± 681
5313 ± 686
Nq
433
622
189
165
152
1044
1082
74
119
172
115
73
52
77
63
46
66
114
45
36
39
39
14
22
38
37
30
21
18
17
28
26
27
27
36
14
q ± σq
mW-m−2
205.0 ± 214.2
214.0 ± 121.3
124.0 ± 38.7
100.0 ± 37.7
101.5 ± 50.4
116.0 ± 80.1
103.2 ± 65.0
100.0 ± 40.3
95.0 ± 45.6
83.9 ± 46.2
86.6 ± 40.4
93.0 ± 37.8
80.0 ± 38.5
87.0 ± 40.8
79.0 ± 30.1
76.0 ± 29.8
69.1 ± 38.2
77.0 ± 97.9
69.9 ± 32.0
59.0 ± 29.8
67.4 ± 56.9
76.6 ± 47.4
63.0 ± 35.0
76.2 ± 31.7
62.2 ± 28.2
64.9 ± 16.7
57.0 ± 59.2
68.2 ± 9.5
69.3 ± 20.2
64.0 ± 31.2
62.8 ± 14.7
59.7 ± 30.4
65.0 ± 95.4
58.1 ± 13.1
61.5 ± 15.7
68.5 ± 24.6
Age
Ma
73
75
77
79
81
83
85
87
89
91
93
95
97
99
101
103
105
107
109
111
113
115
117
119
121
123
125
127
129
131
133
135
137
139
141
143
εadj ± σε
m
5321 ± 651
5297 ± 674
5292 ± 669
5268 ± 682
5283 ± 735
5268 ± 690
5279 ± 694
5291 ± 761
5368 ± 780
5388 ± 704
5385 ± 720
5391 ± 753
5402 ± 739
5419 ± 746
5455 ± 693
5443 ± 723
5424 ± 772
5458 ± 815
5477 ± 951
5498 ± 854
5416 ± 917
5412 ± 955
5399 ± 975
5347 ± 995
5397 ± 808
5439 ± 693
5398 ± 762
5477 ± 712
5506 ± 734
5501 ± 810
5524 ± 873
5565 ± 975
5523 ± 965
5486 ± 925
5567 ± 1056
5579 ± 949
Nq
22
23
15
18
95
40
133
56
35
46
46
62
32
16
9
29
19
28
38
19
11
22
15
105
38
90
77
16
11
18
16
23
34
19
13
25
q ± σq
mW-m−2
57.2 ± 37.3
64.0 ± 36.3
59.3 ± 24.3
66.4 ± 31.6
72.0 ± 24.3
48.3 ± 25.7
57.0 ± 50.1
36.3 ± 42.6
45.0 ± 47.1
53.5 ± 14.4
56.9 ± 15.5
58.7 ± 15.6
43.6 ± 18.6
56.3 ± 81.0
48.0 ± 44.1
55.4 ± 31.0
49.0 ± 10.9
55.9 ± 12.5
53.2 ± 23.8
49.8 ± 25.0
38.4 ± 22.2
52.0 ± 55.5
53.6 ± 15.2
45.6 ± 8.3
49.2 ± 14.3
54.7 ± 13.6
51.1 ± 21.8
50.1 ± 14.1
50.0 ± 28.1
46.3 ± 18.9
51.1 ± 14.3
47.7 ± 16.6
49.8 ± 8.5
49.1 ± 8.4
47.7 ± 45.5
50.7 ± 4.6
81
Table 5.2. continued.
Age
Ma
145
149
151
153
155
157
159
161
163
εadj ± σε
m
5528 ± 1019
5571 ± 1042
5520 ± 1040
5495 ± 1025
5199 ± 977
5006 ± 1138
5281 ± 1526
5660 ± 790
5578 ± 783
Nq
8
54
19
21
20
13
39
59
26
q ± σq
mW-m−2
36.4 ± 10.0
51.8 ± 20.3
50.7 ± 38.6
50.7 ± 11.0
44.2 ± 9.5
40.7 ± 8.4
49.4 ± 29.4
46.9 ± 8.7
48.9 ± 19.9
Age
Ma
147
165
167
169
171
173
175
177
179
εadj ± σε
m
5535 ± 1051
5572 ± 788
5595 ± 772
5538 ± 737
5492 ± 698
5535 ± 815
5558 ± 897
5525 ± 997
5426 ± 1204
Italicized values excluded from misfit computation.
Nq
12
16
36
12
12
15
21
44
16
q ± σq
mW-m−2
50.6 ± 5.0
48.7 ± 28.0
48.4 ± 30.6
49.4 ± 18.1
53.0 ± 15.2
49.0 ± 10.0
49.0 ± 8.4
49.9 ± 10.8
48.1 ± 15.6
82
5.5
Results and Discussion
5.5.1
Plate Thickness
Before fitting models to data, I explore the influence of model parameters on
subsidence and heat flow. A suite of models varying the maximum plate thickness
between 65 and 125 km is shown in Figure 5.2. The effect of maximum plate thickness
variations on subsidence and heat flow in young seafloor is negligible. However, the
models rapidly diverge as diffusion lengths reach and are limited by maximum plate
thickness. As plate thickness grows, the asymptotic limits on water depth and heat
flow converge as the solution approaches half-space cooling within the time interval
investigated.
Because of the large range of heat flow, it can be difficult to see differences between
the various models on a linear scale. I therefore plot the heat flow on a nonlinear scale
where q −2 ∝ t to highlight the differences between the various models at old ages. To
more easily observe model differences at young ages, bathymetry and heat flow are
shown relative to a preferred model (Section 5.5.5). From these plate models, I expect
models with greater maximum plate thicknesses to better fit heat flow observations.
5.5.2
Mantle Adiabat
Subsidence and heat flow of plate cooling models with mantle potential temperatures varying from 1200 to 1600◦ C begin to diverge at very early time (Figure 5.3).
The subsidence patterns differ markedly from the half-space cooling model. Unlike
the subsidence pattern resulting from plate thickness variations, subsidence due to
differences in the mantle potential temperature largely occur at ages <80 m.y. At old
ages, the subsidence difference for 50 K difference in mantle potential temperature
is ∼200 m. At the youngest ages, the 400 K range in adiabatic temperatures tested
produces >60 mW/m2 difference in predicted heat flow. However, this range is rapidly
reduced to ∼20 mW/m2 by 50 m.y. At old ages, a 50 K difference in adiabatic
temperature asymptotically approaches a heat flow difference of 1 mW/m2 . Most
median heat flow bins are captured by the tested range of potential temperatures.
Plate Thickness
0
0
HSC
4
0
50
100
Age [m.y.]
250
(d)
Heat Flow [mW/m2]
200
150
100
65
75
50
0
2
75
85
95
105
115
125
3
4
150
65
0
400
200
125
100
80
70
50
100
Age [m.y.]
150
115
105
95
0
85
-400
75
-800
65
0
50
100
Age [m.y.]
150
30
(e)
65
60
75
85
50
95
105
125
HSC
40
(c)
400
50
100 150
Age [m.y.]
125
0
∆ Subsidence [m]
3
75
85
95
105
115
125
1
0
50
100
Age [m.y.]
150
∆ Heat Flow [mW/m2]
2
65
Subsidence [km]
1
125
800
(b)
Heat Flow [mW/m2]
Subsidence [km]
(a)
(f)
20
65
10
75
85
95
0
125
-10
-20
-30
0
50
100
Age [m.y.]
150
83
Figure 5.2. Influence of plate thickness variations on subsidence and heat flow. The dashed line represents the half-space
cooling model (HSC). Bathymetry data are reduced by the approximate depth to the ridge, 2750 m. Data (dots) and
reference model (solid line) same as Figure 5.6.
Potential Temperature
0
0
2
1200
1300
1400
3
1500
1600
1600
1
1200
2
1300
1400
3
1500
∆ Subsidence [m]
1
100
Age [m.y.]
(d)
Heat Flow [mW/m2]
200
150
100
0
4
150
250
1600
50
0
50
100
Age [m.y.]
1350
1300
-400
50
100 150
Age [m.y.]
400
200
125
100
80
70
150
1200
0
50
100
Age [m.y.]
150
30
(e)
60
1600
1500
50
1300
1200
1200
0
0
1250
∆ Heat Flow [mW/m2]
50
1550
1500
1450
1400
-800
Heat Flow [mW/m2]
0
400
1600
HSC
4
(c)
800
(b)
Subsidence [km]
Subsidence [km]
(a)
(f)
20
1600
10
1500
0
1400
1300
1200
-10
-20
HSC
40
0
50
100
Age [m.y.]
150
-30
0
50
100
Age [m.y.]
150
84
Figure 5.3. Influence of mantle potential temperature variations on subsidence and heat flow. The dashed line represents
the half-space cooling model (HSC). Data and reference model same as Figure 5.2.
85
5.5.3
Crustal Thickness
At early time during the evolution of the plate models computed in this study, the
crust represents a significant fraction of the total lithospheric thickness. As the system
evolves, the fraction of crustal thickness to that of the entire lithosphere rapidly
decreases. As a result, I expect the crustal influence on the thermal evolution to be
greatest at early time. Models computed with crustal thickness varying between 1 and
15 km suggest that the impact of a crust is greatest in the first 10 m.y. with almost
no influence beyond 50 m.y. However, some of this difference is due to the difference
in initial temperatures conditions (Section 5.5.6.2). Variations in crustal thickness
produce patterns similar to those produced by varying adiabatic temperatures except
that the differences between models rapidly (20–30 m.y.) reach a nearly constant
difference (Figure 5.4), whereas differences between adiabatic models increase out to
∼80 m.y. (Figure 5.3). Subsidence and heat flow are less sensitive to crustal thickness
variations than to differences in the mantle potential temperature.
5.5.4
Model Misfits
I run a grid search to identify the minimum misfit of plate cooling models to heat
flow and bathymetric data. Misfit of the models to data are computed by,
M
N
1 X
1 X
(εobs,i − εmod,i)2
(qobs,i − qmod,i )2
misfit =
+
2
2
M i=0
σε,i
N i=0
σq,i
"
#−1/2
,
(5.28)
where εi and qi are the individual water depth and heat flow associated with a total of
M and N ages. The subscripts ‘obs’ and ‘mod’ are the observed and model values, respectively. The misfits are normalized by the standard deviations, σ, of the respective
datasets and are therefore dimensionless. Because models compute subsidence, not
bathymetry, the initial ridge depth must also be computed to determine the model
misfit to the data. The initial ridge depth is determined such that the misfit between
the subsidence curve and the bathymetry is minimized. The initial temperature
condition is 0◦ C at the seafloor and increased linearly to the adiabatic temperature
at the base of the crust. For models with no crust, temperatures are increased to the
adiabat at 1 km to increase numerical stability. The adiabatic gradient is assumed to
Crustal Thickness
0
0
(b)
2
15
11
3
1
5
1
2
15
11
3
1
5
HSC
50
100
Age [m.y.]
150
250
(d)
Heat Flow [mW/m2]
200
150
100
1
50
0
0
400
200
125
100
80
70
50
100
Age [m.y.]
5
9
13
-400
150
(e)
60
1
50
15
HSC
40
0
50
100
Age [m.y.]
150
30
15
0
1
0
50
100 150
Age [m.y.]
0
50
100
Age [m.y.]
150
∆ Heat Flow [mW/m2]
0
400
-800
4
Heat Flow [mW/m2]
4
(c)
800
∆ Subsidence [m]
1
Subsidence [km]
Subsidence [km]
(a)
(f)
20
10
1
3
5
7
9
11
15
0
-10
-20
-30
0
50
100
Age [m.y.]
150
86
Figure 5.4. Influence of crustal thickness variations on subsidence and heat flow. The dashed line represents the half-space
cooling model (HSC). Preferred model in represented by the heavy line. Data used to estimate misfit are shown for reference.
Data and reference model same as Figure 5.2.
87
Table 5.3. Compositional model for the oceanic lithosphere.
Mineral
albite
anorthite
clinochore
daphnite
tremolite
ferro-actinolite
pargasite
diopside
hedenbergite
enstatite
ferrosillite
forsterite
fayalite
spinel
pyrope
almandine
grossular
A [µW/m3 ]
Crust
basalt gabbro
17.85
10.08
42.05
37.92
1.89
0.21
3.08
5.28
0.44
15.70
26.14
4.90
5.86
5.14
5.86
5.59
7.09
3.01
1.91
0.1
0.03
plag-lherzolite
Mantle
sp-lherzolite
gt-lherzolite
0.71
3.82
0.41
17.49
1.94
68.43
7.20
0.003
4.49
0.41
18.77
1.93
66.55
7.15
0.70
0.003
4.49
0.41
17.32
1.78
67.00
7.20
1.19
0.43
0.18
0.003
Values as molar fraction of approximate end-member mineralogy. Crustal compositions from Hacker et al. [2003], mantle composition computed from Niu et al.
[1997].
be 0.3 K/km. The model is discretized at 200 m in depth and 0.01 m.y. in time. Finer
discretizations in depth and time have been tested to ensure models are not improved
within my desired level of precision. A total of 2873 models are computed, with crustal
thickness ranging from 0–15 km, maximum plate thickness ranging from 65–125 km
and mantle potential temperatures from 1200–1600◦C. Layer compositions, including
the heat production, are given in Table 5.3.
The basaltic layer is assumed to be
2 km thick and all changes in crustal thickness are attributed to variations in the
cumulate layer (seismic layer 3), which tends to dominate crustal structure [Mutter
and Mutter , 1993].
Seismic evidence suggests average thickness of the igneous oceanic crust is 7.1±0.8 km
[White et al., 1992]. Crustal thickness varies from 0 km at slow spreading centers dom-
88
inated by structural deformation [Tucholke et al., 1998, 2001] to >10 km thickness at
hot fast spreading ridges, although it can be much greater in regions with anomalous
volcanism such as Galapagos and Iceland [White et al., 1992; Mutter and Mutter ,
1993; Asimow et al., 2001].
In this study, minimum misfit is achieved with 2 km thick crust (Figure The misfit
surface shows more of a trough than a well defined global minima, suggesting that
crustal thicknesses and adiabatic temperatures within this zone yield similarly fitting
models. This misfit trough roughly correlates with chemically calibrated estimates
of oceanic crustal thickness generated by melting MORB source as a function of
potential temperature [Asimow et al., 2001]. However, the theoretical 5.5). function
predicts lower adiabatic temperatures than my misfit would suggest. This may result
from additional physical influences on lithospheric buoyancy, surface heat flow, or
uncertainties in model parameters. Because this trough poorly estimates crustal
thickness, I choose a value of 7 km consistent with seismic estimates in order to
estimate other input parameters. The misfit assuming a 7 km thick crust is 0.43.
Best-fitting potential temperature using a 7 km thick crust yields an estimated
potential temperature of 1425◦ C. The range of potential temperatures with low misfit
is ∼75 K. Adiabatic temperature estimates discussed above are only one of several
proposed models derived from basaltic chemistry. Another competing model suggests potential temperatures are low (∼1300◦ C) with an additional 250 K variation
resulting from heating in proximity to hotspot sources (plumes?). These hypotheses
are evaluated by Herzberg et al. [2007], who find the more consistent with a lower
adiabatic temperature. Below I discuss further the difference between my model and
the geochemical prediction.
My choice of crustal thickness results in a best-fitting plate thickness of 90 km
(Figure 5.5). The misfit surface shows a well defined range of plate thickness (∼10 km)
which reasonably fits the bathymetry and heat flow data. The 90 km maximum
plate thickness is relatively independent of the choice of crustal thickness. Plate
thickness estimates from Parsons and Sclater [1977] and Stein and Stein [1992] that
use temperature and pressure independent physical properties yield 125 and 95 km,
89
Plate Thickness, 90 km
Potential Temperature [°C]
1.2
(a)
1.0
0.8
1500
0.6
w
mo
Asi
l.
et a
]
01
[20
1400
0.6
1300
0.4
0.8
1.0
1200
0
Crustal Thickness, 7 km
1600
Potential Temperature [°C]
1600
1.2
5
10
Crustal Thickness [km]
15
(b)
1.2
1.0
0.8
1500
0.6
1400
0.6
1300
1200
0.8
70
80 90 100 110 120
Plate Thickness [km]
Potential Temperature, 1425°C
0.8
Plate Thickness [km]
120
(c)
110
0.6
100
90
80
0.6
0.8
1.0
1.2
70
0
5
10
Crustal Thickness [km]
15
Figure 5.5. Slices through the misfit surface at (a) a plate thickness of 90 km, (b)
a crustal thickness of 7 km, and (c) a mantle potential temperature of 1425◦ C. ‘×’
denote the minimum misfit for each slice.
90
respectively. McKenzie et al. [2005] showed that models incorporating temperatureand pressure-dependent properties yield ∼85% lower plate thicknesses than their
constant-property counterparts (106 and 83 km). However, McKenzie et al. [2005] use
volumetric rather than effective expansivity estimates. Because the volumetric expansivity plate models have overall greater subsidence than effective expansivity models
for given plate thickness, the best-fitting plate thickness in models incorporating the
effective expansivity is greater, consistent with my results.
5.5.5
Preferred Model
My preferred model with 7 km thick igneous crust, 90 km maximum plate thickness,
and a mantle potential temperature of 1425◦C is shown in Figure 5.6. Modeled
subsidence is shifted by 2760 m to match bathymetric data. This model shows
excellent agreement for median bathymetry. A slight under-prediction of bathymetry
occurs at young ages (<50 Ma), which compensates for the increased variability in
bathymetry at older ages that are over-predicted. Modeled heat flow also agrees well
with the data. When a inverse square transform is applied to heat flow to make
modeled results more linear, differences between the observed heat flow and model
are apparent. Heat flow at young ages fit very well, a region traditionally ignored in
oceanic cooling models because of hydrothermal effects on heat flow. While heat flow
at older ages appears to fit less well in the transformed space, small variations that
are well within uncertainty appear very large and are not a cause for concern.
5.5.6
5.5.6.1
Additional Influence on Plate Models
Volumetric vs. Effective Expansivity
As mentioned previously, ignoring the effects of strength within the lithosphere
affects estimates of expansivity. Korenaga [2007a] showed the importance of incorporating a viscoelastic rheology into estimates of expansivity. Figure 5.7 shows the
difference between model outputs computed using my empirical estimate of effective
expansivity and the volumetric expansivity. The difference in subsidence quickly
grows to reach half the maximum by 30 Ma. The total subsidence difference at
91
2.5
2.5
3.5
(b)
Bathymetry [km]
Bathymetry [km]
(a)
4.5
5.5
50
100
Age [Ma]
150
400
(c)
300
0
200
100
0
50
100
Age [Ma]
150
50
Age [Ma]
400
200
125
100
80
70
Heat Flow [mW/m2]
Heat Flow [mW/m2]
4.5
5.5
0
0
3.5
100 150
(d)
60
50
40
0
50
100
Age [Ma]
150
Figure 5.6. Preferred plate cooling model with a crustal thickness of 7 km, plate
thickness of 90 km and potential temperature of 1425◦C including heat production.
Data shown with 1-σ bars. Data used to compute misfit are in dark grey, unused in
light grey.
92
1000
30
800
volumetric expansivity
600
ol-only mantle
400
no crust
no crust-2
200
∆ Heat Flow [mW/m2]
∆ Subsidence [m]
(a)
(b)
25
20
15
10
ol-only mantle
5
no crust
0
0
0
50
100
Age [m.y.]
150
-5
volumetric expansivity
0
50
100
Age [m.y.]
150
Figure 5.7. Model sensitivity to selected parameters. See text for description of
models. Reference model (zero lines) is same as preferred (Section 5.5.5).
180 Ma is >700 m. Thus, to achieve a similar fit to bathymetry using the volumetric
expansivity, the plate thickness would have to be decreased by ∼20 km. There
is decrease in heat flow when using the volumetric expansivity that results from
conductivity and heat capacity changes from increased lithospheric pressures.
My analysis does, however, ignore the influence of thermal-cracking and crackfilling in the upper part of the lithosphere, which would reduce the difference between
the effective and volumetric expansivity. While this effect is not large enough to
influence the results strongly, a few percent change in effective expansivity decreases
the estimated plate thickness slightly. An excellent exploration of thermal-cracking
in oceanic lithosphere is made by Korenaga [2007b].
5.5.6.2
Lithospheric Composition
Thus far I have only explored one compositional model. Most plate cooling models
use monomineralic composition (Fo-90) for the entire lithosphere. A few plate cooling models include a polymineralic compositions but still use Fo-90 composition to
compute conductivity. However, conductivity of polymineralic peridotite can differ
by 10-20% from Fo-90. Comparison between the polymineralic models lacking a crust
and a model with Fo-90 olivine composition results in ∼50 m higher subsidence in the
93
Fo-90 model (Figure 5.7). The heat flow difference between the two is <1 mW/m2 .
Thus, the effect of mantle composition on plate cooling models is minor and potentially a second-order effect on subsidence.
While very few models include polymineralic assemblages, even fewer include a
crust. The one exception is the study by Afonso et al. [2008], which again uses
an Fo-90 composition to estimate conductivity. Since conductivity is required to
compute heat flow, predicted heat flow should be initially higher without a crust.
This difference in heat flow affects the efficacy of cooling, and as a result increases
the subsidence rate. One would also expect the density difference from decreased
lithostatic pressure gradients between models and differences in thermal expansion
with and without crustal layers to cause a difference in subsidence, particularly at
young ages where the crust represents a significant fraction of the total lithospheric
thickness.
As predicted, subsidence is much greater in the model without a crust. The initial
depth at which temperatures reach the adiabat differs for varying crustal thicknesses
in my model; therefore, a second model lacking a crust is run with the same initial
temperature condition as my preferred model so they may be compared more fairly
(Figure 5.7, no crust-2). The result indicates that 200 m of additional subsidence
(∼2/3 total subsidence) results from my choice of initial condition as temperatures
rapidly diffuse at young ages (∼10 m.y.).
The inclusion of a crust reduces the
overall subsidence by 100–150 m. The effect on heat flow is even more striking with
an initially higher heat flow (>30 mW/m2 ) in the models lacking a crust with an
asymptotically higher heat flow of ∼2 mW/m2 at ages >90 m.y. Including a crust
results in a ∼20% lower minimum misfit than the minimum misfit without a crust.
5.5.6.3
Heat Production
Data that can be used to estimate heat production within the oceanic lithosphere
are sparse, but sufficient to make a preliminary estimate. Heat production results
from the radiogenic decay of U, Th, and K which can be estimated from geochemical
analyses of mid-ocean ridge basalts (MORB). Average estimates of heat production
94
Table 5.4. Compositional model for the oceanic lithosphere. Values as molar fraction
of approximate end-member mineralogy.
Element
E-MORBa
N-MORBa
N-MORBb
DMb
DMc
K2 O
wt.%
0.48 ± 0.26
0.14 ± 0.07
0.09 ± 0.045
0.072
0.060
Th
ppm
0.97 ± 0.68
0.20 ± 0.14
1.2 ± 0.08
0.0137 0.0079
U
ppm
0.37 ± 0.31 0.083 ± 0.054
0.05 ± 0.04
0.0047 0.0032
d
3
A
µW/m
0.23 ± 0.11 0.055 ± 0.020 0.033 ± 0.014 0.0036 0.0024
a
Niu et al. [2002] from the East Pacific Rise b Salters and Stracke [2004] derived
from modeled chemical ratios c Workman and Hart [2005] estimated from abyssal
peridotites d assuming density of 3000 kg/m3 for basalts and 3340 kg/m3 for peridotites.
range from 0.03 µW/m3 in depleted (normal) N-MORB to 0.2 µW/m3 in (enriched)
E-MORB (Table 5.4). Most of the ocean crust is made of N-MORB, while E-MORB
is more frequently found near seamounts and oceanic island basalts. However, EMORB has been found far from seamounts representing a small but significant fraction
(<∼5%) of analyzed basalts [Donnelly et al., 2004]. Estimated heat production from
a bulk oceanic crust consisting of 95% N-MORB and 5% E-MORB is ∼0.05 µW/m3 ,
representing ∼0.35 mW/m2 of the surface heat flow for a 7 km thick crust.
Hydrothermal alteration of the crust tends to increase the concentration of heat
producing elements substantially [Wheat and Mottl , 2004]. Weight percent K can
be increased by ∼7 times, while U and possibly Th [Hart and Staudigel , 1982] can
be increased even more, bringing estimated upper crustal heat production in altered
N-MORB to values similar to unaltered E-MORB. However, this effect has yet to be
estimated on a large scale and is estimated from three deep-sea drill holes.
Radiogenic heat production within the oceanic mantle lithosphere is at least an
order of magnitude smaller than within the oceanic crust. This depleted mantle
(DM) contributes <0.3 mW/m2 to the surface heat flow at old ages. It is interesting
to note that heat production within the DM is ∼10% of continental mantle estimates
(Chapter 3). The effect of oceanic heat production on bathymetry and heat flow is
very small, ∼5 m and 0.5 mW/m2 , respectively. Heat production within the oceanic
lithosphere is a third-order effect and can be ignored when modeling the thermal
evolution of the oceanic lithosphere.
95
5.5.6.4
Hydrothermal Alteration
Aside from its influence on concentrations of radiogenic elements, hydrothermal
alteration of the seafloor has been ignored. Hydrothermal alteration may affect 5–15%
of the lower crust and is likely more pronounced in the upper crust. Amphiboles
and phylosilicates are the most common alteration products [Carlson and Miller ,
2004]. Within the mantle, serpentinization has been estimated to be as high as ∼20%
[Schmidt and Poli , 1998]. The reactions producing these products are exothermic and
most commonly occur in young seafloor where thermal gradients are high. Some of
the heat is presumably carried out of the system as seawater fluxes through. This is
a complex process and it may be difficult to estimate the total heat imparted to the
lithosphere. Therefore, I choose to ignore this process.
The effect of alteration on thermophysical properties influences both the efficiency of
heat transport and buoyancy within the upper oceanic lithosphere. Except for density,
the required physical properties of many alteration minerals are poorly understood, in
particular their P –T dependence. As laboratory studies improve estimates of physical
properties for alteration products oceanic cooling models may be improved.
5.5.6.5
Adiabatic Gradient
Often quoted adiabatic gradients for the mantle range from 0.2 to 0.5 K/km. A
comparison between my preferred model and a model with no adiabatic gradient
results suggests models that do not include a gradient under-predict heat flow by
∼1 mW/m2 and under-predict bathymetry by ∼25 m. Thus the effect is well within
the uncertainty and variability of the datasets and can be ignored.
5.5.6.6
Constant Melt vs. Variable Melt Fraction
Melt fraction varies with time beneath a cooling plate. The predicted bathymetric
profile is only affected if the melt fraction decreases to a level such that the asthenosphere can support loads for a significant period of time. Afonso et al. [2007] explore
the variability in and effect of melt fraction on plate cooling models. This effect is
beyond the scope of this study.
96
5.6
Conclusions
Recent plate cooling models of the oceanic lithosphere have focused on incorporating selected physical effects. Because each study uses a different set of observational constraints, it is difficult to compare directly the relative importance of each
complexity beyond simple analytical solutions. In this study I develop a standard
bathymetry–age and heat flow–age dataset that can be used to compare plate cooling
models and estimate the average thermal structure of the oceanic lithosphere.
Calculated misfits from >2600 models of bathymetry and heat flow–age data suggest crustal thickness and potential temperatures form a misfit trough roughly parallel
to crustal thicknesses estimated from chemical melting models of MORB source at
different temperatures. By incorporating P –T dependence of polymineralic assemblages, heat production, and assuming a 7 km thick crust, the best-fitting maximum
plate thickness is ∼90 km and has a potential temperature of 1425◦C.
Models using polymineralic rather than monomineralic assemblages yield only minor differences in subsidence and heat flow, ∼50 m and 1 mW/m2 , respectively.
However, the inclusion of a crust is a considerably larger influence on plate models,
particularly at young ages where the crust represents a dominant fraction of the total
lithospheric thickness. Incorporating effective expansivity rather than volumetric
expansivity producing the largest influence on subsidence, reducing subsidence by
several hundred meters but produces a very small change in heat flow estimates.
The title of this chapter poses the question whether or not these complexities are
necessary for plate cooling models. If one only wishes to model subsidence and
bathymetry, the answer is no. Simple one-dimensional cooling models with constant
thermophysical properties like Stein and Stein [1992] fit the data well within the
bounds of heat flow uncertainties and bathymetry variations. However, if one wishes
to model state variables (pressure and temperature) or physical properties (density,
specific heat capacity, thermal conductivity) properly, the answer is yes to most. The
adiabatic gradient and radiogenic heat production produce very small effects and can
easily be ignored.
97
Future improvements to plate cooling models should focus on several additional effects not incorporated into this study that may influence subsidence and/or heat flow.
These processes include metamorphic reactions driven by hydrothermal circulation,
thermal-cracking and crack-filling, and partial melt in the asthenosphere.
5.7
A Note on McKenzie et al. [2005]
The formulation by McKenzie et al. [2005] solves the one-dimensional diffusion
equation without sources,
∂
∂T
∂
k
[ρCP T ] =
∂t
∂z
∂z
!
.
(5.29)
If an analytic solution to
G=
Z
k(T )dT
(5.30)
exists, then Equation 5.29 can be written
T
∂ (ρCP )
∂T
∂2G
+ ρCP
=
.
∂t
∂t
∂z 2
(5.31)
McKenzie et al. [2005] solves Equation 5.31 iteratively using the Crank-Nicholson
method and adds the time derivative of ρ and CP as a perturbation to subsequent
steps. Their finite difference approximation for the initial step (their Equation 11) is
written,
n+1
n+1
n
n
Tj−1
−Hkj+1
Tj+1
+ 1 + 2Hkjn Tjn+1 − Hkj−1
= Tjn + 2H Gnj−1 − 2Gnj + Gnj−1
n
n
n
n
,
−H kj+1
Tj+1
− 2kjn Tjn + kj−1
Tj−1
(5.32)
where H is given by
H=
∆t
2ρ Tjn CP Tjn ∆z 2
,
(5.33)
where ∆t and ∆z are the time step and depth spacing. The indicies n and i refer
to time and to space, respectively. The terms with thermal conductivity, k, appear
98
to be incorrect. The reverse-engineered PDE from Equation 5.32 with the additional
time derivatives of ρ and CP is
T
∂ (ρCP )
∂T
∂ 2 G ∂(kT )
+ ρCP
=
−
.
∂t
∂t
∂z 2
∂z
(5.34)
By comparision between Equations 5.34 and 5.31, I can see that their finite-difference
formulation, Equation 5.32, is incorrect.
CHAPTER 6
CONCLUSIONS
This study addresses the thermal states of the continental and oceanic lithosphere.
An enhanced global heat flow database combined with recent advances in measuring
thermophysical properties of minerals at high pressures and temperatures, new compilations of lithospheric heat production, and a better understanding of convective
heat loss in young sea-floor collectively permit a reexamination of current concepts
regarding the thermal state of the outer 300 km of the Earth.
1. An updated global heat flow database is produced, incorporating ∼40,000 new
records to the existing database, which now includes >60,000 records and
>44,800 heat flow determinations. The database will be shared with the Earth
science community.
I anticipate considerable use of the database, in part
because heat flow serves as the prime constraint in calculating temperatures
through the lithosphere. This database also presents the opportunity to determine heat loss of the solid Earth to improved accuracy.
2. I develop and reassess a number of thermo-geophysical reference models of the
continental lithosphere. An important part of this development is the establishment of a reference lithospheric heat production model for North America.
Heat production in the lower crust estimated from exposed granulite terranes
is ∼0.4 µW/m3 and geochemical estimates of heat production from xenoliths
places median subcrustal lithospheric heat production at ∼0.02 µW/m3 . To estimate upper-crustal heat production I use compositionally normalized elevation
and a field of allowable temperatures from xenolith P –T estimates. Geotherm
models of the continental lithosphere best fit these data when, on average,
26% of the surface heat flow is derived from upper-crustal heat production.
100
Using this geotherm model, estimated surface heat flow in Precambrian regions
37-47 mW/m2 , consistent with most observations.
3. Reference models for cooling of the oceanic lithosphere typically ignore heat
flow as a constraint, particularly for young sea-floor. Heat flow through young
sea-floor is generally considered strongly biased to low values as a result of
hydrothermal circulation.
Global filtering out of heat flow sites with thin
sedimentary cover (<325 m) and in proximity to seamounts (<85 km) brings
heat flow–age relationships much closer to theoretical predictions for cooling lithosphere. However, a heat flow deficit still persists. Adjustments for estimated
thermal rebound and sedimentation effects on heat flow further reduce deficits,
bringing data in line with model estimates for regions >25 Ma. Heat flow–age
estimates at younger ages are improved by examining in detail several localities
with high density sampling and co-located seismic data.
4. The filtered heat flow data are used to construct plate cooling models of the
oceanic lithosphere which predict a maximum plate thickness of 90 km and
mantle potential temperature of 1425◦C. Model results indicate that inclusion of
a crust results in an improvement compare with the models lacking a crust that
are typically computed. These models incorporate several physical processes
acting on the cooling lithosphere, but may be refined further by incorporating
additional chemical (e.g., hydrothermal alteration) and physical processes (e.g.,
melting and thermal-cracking).
5. Data excluded from the heat flow–age analysis that come from areas with relatively thin sediment cover and/or sites within 85 km of an identified seamount
can be used to estimate the heat loss due to hydrothermal circulation of seawater
through the oceanic upper crust. Using a Monte Carlo analysis to incorporate
uncertainties in heat flow, I estimate advective heat loss to be 7±2 TW, or
∼20% of the total oceanic heat loss. This result is smaller than previous heat
flow estimates due to a reduction in the estimated area affected by hydrothermal
101
circulation. This estimate can be used to help constrain total fluid fluxes
through the oceanic lithosphere.
6. One of the eventual goals started during the course of this dissertation is a
revised estimate of global heat loss. Despite the significant improvement in
spatial coverage of this global heat flow database update, large regions of the
planet remain unsampled. To estimate global heat loss, heat flow in unsampled
regions must first be estimated. The thermal models developed within this
dissertation serve as predictors for unsampled regions. Future work will include
updating additional heat flow estimators on continents such as heat flow–age
and –lithology relationships.
By developing a reference model of lithospheric heat production and combining
it with global heat flow, it is possible to estimate the total lithospheric heat
production. By incorporating estimates of radiogenic abundances within the
bulk silicate Earth, and using lithospheric estimates of heat production, it may
be possible to help constrain the Urey ratio. The Urey ratio is the ratio of
radiogenic heat production to the total convective heat flow within the mantle,
which has important implications for the initial thermal state of the Earth. This
implication is one of the exciting avenues for research opened by this study as
well as exploring the myriad of implications of these models.
APPENDIX A
RADIATIVE DIFFUSIVITY AND
CONDUCTIVITY OF OLIVINE
REVISITED
A.1
Abstract
Olivine comprises >60% of the upper mantle, making it the dominant control on
heat transport through the lithosphere. Accurate estimates of the thermal diffusivity
and thermal conductivity are important in modeling geotherms, the thickness of
the lithosphere, depth of melt generation, etc. Diffusivity and conductivity result
from lattice vibration and radiative transport of thermal energy and depend on
temperature. In this study, a new model for radiative diffusivity and conductivity
is based on prior measurements of effective and lattice diffusivity on olivine samples
with Mg# ∼0.9. Diffusivity can be expressed as
T − 1140
Dr = 0.133 1 + erf
370
,
(A.1)
and conductivity as
kr = 0.56 1 + erf
T − 1150
370
,
(A.2)
with misfits of 0.024 mm2 /s and 0.086 W/m-K, respectively. A geotherm estimate
using this model is consistent with geophysical estimates of lithospheric thickness of
the Kalahari craton, but the geotherm may be slightly cool relative to xenolith P –T
conditions. Conductivity estimates using this method appear to be inconsistent with
recent spectral estimates of radiative conductivity, which also depend significantly
on grain size. Additional experimental data are required to resolve the disagreement
between the two models.
103
A.2
Introduction
Olivine is the most common mineral in the upper mantle, composing >60% by
weight. Therefore, understanding the thermal transport properties of olivine is important to accurate models of lithospheric conduction and upper mantle convection.
Diffusivity and conductivity affect heat production within the lithosphere, the thickness of the mantle lithosphere, sublithospheric heat flow, estimates of heat transport
during convection, depth of melt generation, and thermal isostatic effects [Dubuffet
et al., 1999; Michaut et al., 2007; Hasterok and Chapman, 2007a].
Thermal diffusivity and conductivity result from lattice and radiative components,
the latter of which is poorly known and difficult to measure accurately [Schatz and
Simmons, 1972]. Several studies have estimated radiative conductivity using spectral methods (Shankland et al. [1979] and references therein), but may suffer from
resonance related to similarity between laser frequency and Si-O stretching modes in
silicates [Hofmeister , 1999a]. Conductivity values from these studies range from near
zero at room temperature to ∼2.8 W/m-K at 1100–1700 K.
Radiative conductivity may also be estimated by subtracting the lattice conductivity from measurements of effective conductivity [Hofmeister , 1999a], hereafter referred
to as a difference estimate. The difference estimate reported by Hofmeister [1999a]
is considerably lower and reaches little more than 11% of the spectral estimates at
1700 K. The estimate by Hofmeister [1999a], in widespread use among modelers (e.g.,
Dubuffet et al. [1999], Anderson [2000], van den Berg et al. [2001], McKenzie et al.
[2005], Michaut et al. [2007] and Korenaga and Korenaga [2008]), was not converted
from diffusivity to conductivity as reported.
In this study, I develop a radiative diffusivity and conductivity model using the
difference method and propose a new functional form to fit radiative conductivity that
has several advantages over standard polynomial regression. Geotherms computed
using this and other radiative conductivity models are assessed by comparison to with
xenolith P –T conditions and geophysical estimates of lithospheric thickness beneath
the Kalahari craton.
104
A.3
Estimating Radiative Diffusivity
Heat transfer through minerals occurs by two physical mechanisms, phonon-phonon
interaction (lattice transfer) and black body radiation as photons (radiative transfer)
[Shankland et al., 1979]. Numerous experimental studies have measured the effects
of pressure and/or temperature on conductivity and diffusivity of olivine [Schatz and
Simmons, 1972; Schärmeli , 1982; Katsura, 1995; Chai et al., 1996; Harrell , 2002;
Xu et al., 2004]. The results of these studies show similar temperature derivatives
at low temperatures with scatter in absolute values resulting from individual sample
characteristics and from inclusion or exclusion of radiative conductivity. I focus on
two diffusivity studies, Katsura [1995], which includes both radiative and lattice components, and Xu et al. [2004], which is controlled to measure the lattice contribution
only. Both studies measure diffusivity on olivine of approximately Fo-90 composition.
The diffusivity data from Katsura [1995] are shown in Figure A.1 as a function
of temperature at four pressures (0.1, 3, 6 and 9 GPa). To estimate the radiative
component of diffusivity, the estimated lattice-only contribution is subtracted from
the effective diffusivity. The lattice-only model is estimated using the model by [Xu
et al., 2004] with an additional multiplicative factor tailored to fit the low-temperature
effective diffusivity. Differences between the effective conductivity data and latticeonly data at low temperatures most likely result from sample specific characteristics.
This adjustment does not violate the mathematical model presented by Xu et al.
[2004] to fit lattice diffusivity contributions, whereas an additive adjustment would.
The constants used to reproduce the lattice contributions in Figure A.1 are 0.90, 1.11,
1.01 and 1.09, from low to high pressures respectively.
Radiative diffusivity estimates derived from the difference of the Katsura [1995]
data and model lattice contribution are small near room temperature and become
observable above 800 K. By 1500 K the radiative contribution represents >40% of
effective diffusivity at low pressures and ∼30% at high pressures. Radiative estimates
for 6 and 9 GPa reach a plateau in the conductivity at high temperatures with a
maximum radiative diffusivity of 0.22-0.24 mm2 /s. There appears to be a slight
105
Diffusivity [mm2/s]
1.5
0.1 GPa
3 GPa
6 GPa
9 GPa
1.0
0.5
0.0
Diffusivity [mm2/s]
1.5
1.0
0.5
0.0
500
1000
1500
Temperature [K]
500
1000
1500
Temperature [K]
Figure A.1. Thermal diffusivity of olivine. Effective thermal diffusivity from Katsura
[1995] (squares) measured at (a) 0.1, (b) 3, (c) 6, and (d) 9 GPa. Lattice diffusivity
estimates from Xu et al. [2004] scaled to fit effective diffusivity at 500–600 K (heavy
grey line). Radiative conductivity estimates at each pressure (circles) are shown with
the preferred model (black line).
106
decrease in the overall radiative diffusivity with pressure but uncertainties in the
experimental data a pressure effect is inconclusive.
A.4
Model for Radiative Diffusivity/
Conductivity
The radiative contribution is negligible at room temperature but depends strong-ly
on temperature (∝ T 3 ). However, the absorption spectra show that this process
need not be linear. Hofmeister [1999a] uses a third-order polynomial to fit radiative
conductivity data. There are several reasons that a third-order polynomial is an
undesirable fitting function. First, it can be difficult to fit data at low temperatures
where the radiative contribution remains near zero. Polynomial fits may dip below
zero or may be strongly influenced by noise such as the anomalously high conductivity
at low temperatures. Second, extrapolation beyond the measured bounds can lead to
unrealistic increases or decreases in estimated radiative component.
I suggest a fitting function that can be tailored to be positive for all temperatures,
near zero at low temperatures, and be constant at extrapolated temperatures. An
error function formulation is used to approximate the temperature effect on radiative
diffusivity by,
1
T − Tm
Dr = Dmax 1 + erf
2
a
,
(A.3)
where a is an empirically derived constant, Dmax is the maximum diffusivity and Tm
is the temperature at 0.5Dmax . The resultant fit for the average radiative diffusivity
is shown in Figure A.2. The best fitting diffusivity model is Dmax = 0.266 mm2 /s, a
= 0.0027 K−1 and Tm = 1140 K, with an overall RMS misfit 0.024 mm2 /s.
Conductivity is related to diffusivity by k = DρCP , where ρ is the density and CP
is the heat capacity. The average radiative conductivity in this study ranges from
near 0 below 700 K to a little greater than 1.2 W/m-K at 1600 K. The conductivity
can also be fit using an empirical model identical to that used to fit diffusivity. Best
fitting conductivity parameters are kmax = 1.12 W/m-K, a = 0.0027 K−1 and Tm =
1150 K with a 0.086 W/m-K RMS misfit.
107
Radiative Diffusivity [mm2/s]
0.8
0.6
0.4
1 cm
0.2
0.1 cm
0.01 cm
0
500
1000
Temperature [K]
1500
Figure A.2. Radiative diffusivity models and data for olivine. Radiative diffusivity
(circles) estimated from the difference of Katsura [1995] effective diffusivity measurements and Xu et al. [2004] lattice diffusivity model. Data from Shankland et al. [1979]
and references therein (squares). Difference estimates of radiative conductivity for
lherzolite [Gibert et al., 2003] (diamonds). Preferred model using error function fit
to circles (heavy black line). Model by Hofmeister [1999a] (grey line). Radiative
conductivity estimates for 0.01, 0.1 and 1 cm grain size from Hofmeister [2005]
(dashed lines).
A.5
A.5.1
Discussion
Comparison With Other Estimates
Radiative diffusivity and conductivity estimates using the lattice models by Chai
et al. [1996]; Harrell [2002] yield similar results to mine. The estimated radiative
conductivity by Hofmeister [1999a] is >50% lower than the model developed in
this study. Upon further inspection, I determined that the widely used result by
Hofmeister [1999a] represents the radiative diffusivity, not radiative conductivity.
Figure A.2 compares of my result with other radiative diffusivity data. A difference
estimate of radiative diffusivity can also be made from lherzolite sample BALM4 using
effective diffusivity measurements by Gibert et al. [2003]. The BALM4 lherzolite
sample has ∼76 modal percent olivine with a whole rock Mg# of 0.9. Radiative
diffusivity estimates for the lherzolite sample show a measurable radiative effect above
∼700 K. Above ∼900 K, the radiative diffusivity reaches a nearly constant value of
0.7–0.9 mm2 /s (Figure A.2). The estimated radiative component, while lower for
108
lherzolite, has a functional form similar to that of olivine within measured uncertainty.
The differences could be due to the behavior of additional minerals in the lherzolite
system, partitioning of Fe between phases, and/or differences in grain size.
The earliest diffusivity values based on spectral estimates suggest significantly
higher radiative diffusivity [Shankland et al. [1979] and references therein], but are
subject to resonance in the Si-O bonds as a result of the similarity between the
laser frequency and fundamental oscillation modes [Hofmeister , 1999a]. More recent
estimates based on spectral measurements show a strong dependence on radiative
diffusivity and grain size as well as temperature [Hofmeister , 2005].
Grain sizes in the Katsura [1995] samples used for effective diffusivity measurements
are ∼0.005 cm. While Hofmeister [2005] did not measure grain sizes smaller than
0.01 cm, the extrapolated trend between grain the sizes of 0.01 cm to 0.1 cm suggests
that the radiative conductivity is negligible within the temperature range of the
Katsura [1995] experiment.
The grain sizes measured in the Gibert et al. [2003] sample range from ∼0.01–0.3 cm
with most of the grain diameters near 0.1 cm. Hofmeister’s spectral model for 0.1 cm
is well below difference estimates for the lherzolite sample below ∼1200 K. However,
the spectral model grows rapidly above 1200 K, while the radiative diffusivity for the
lherzolite sample appears relatively constant above 900 K. Recent studies have also
suggested an overestimation of the Hofmeister [2005] radiative conductivity model
[Goncharov et al., 2008, 2009]. Clearly the results from difference estimates and
spectral methods are incompatible at present.
A.5.2
Implications of Radiative Conductivity Models
The thermal gradient within the lithosphere is sensitive to variations in thermal
conductivity. Sensitivity of geotherms to the different radiative conductivity models
discussed above is tested using approximate Kalahari Craton lithospheric parameters
and present day surface heat flow of 47 mW/m2 [Nyblade et al., 1990]).
Figure A.3 shows geotherms computed using thermal conductivities estimated using
Xu et al. [2004] for the lattice component and radiative components from this study
109
0
(a)
(b)
Moho
Adiabat
Depth [km]
50
100
1 cm
150
0.1 cm
0.01 cm
200
This
Study
SS72
Kalahari Craton
250
0
500
1000
Temperature [K]
1500 2
3
4
Thermal Conductivity [W/m-K]
5
Figure A.3. Comparison of geotherms (a) computed from several radiative conductivity models (b). Effective conductivity model by Schatz and Simmons [1972] (SS72)
incorporated into the commonly used Chapman and Pollack [1977] geotherms (heavy
grey line). Models incorporating the radiative model from this study (heavy black
line). Grain size models of 0.01, 0.1 and 1 cm from Hofmeister [2005] (dashed lines).
Xenolith P –T estimates (circles) for peridotite xenoliths from the Kalahari Craton
[Stiefenhofer et al., 1997; Rudnick and Nyblade, 1999; Saltzer et al., 2001; Bell et al.,
2003b; Grégoire et al., 2003; James et al., 2004; Katayama et al., 2008].
110
and the 0.01, 0.1, and 1 cm grain size models by Hofmeister [2005]. The commonly
used Chapman and Pollack [1977] geotherm using the effective conductivity model
by Schatz and Simmons [1972] is also shown for comparison.
All computed geotherms have the same temperature at the Moho and diverge below
the Moho as a result of conductivity differences. Temperatures at 150 km vary by
256◦ C between the coldest (1 cm grain size) and hottest geotherms (0.01 cm grain size)
with a corresponding conductivity range of ∼1.75 W/m-K. Depths of intersection to
the 1300◦C adiabat range from 162 to 236 km. Lithospheric thickness estimates based
on various seismic tomography techniques range from 150–300 km with several around
200 km (Begg et al. [2009] and references therein). Recent receiver function estimates
for the Kaapvaal Craton are on the shallow end at 160 km. Magnetotelluric estimates
of lithospheric thickness in the Kimberley region of the Kaapvaal Craton range from
190–220 km [Muller et al., 2009]. These geophysical estimates are consistent with the
depth to the adiabat obtained using the radiative conductivity model in this study
(197 km) and <1 cm grain size models.
While seismic and electrical methods can be used to estimate lithospheric thickness,
temperatures estimated using these techniques are uncertain. Xenoliths from kimberlite eruptions on the Kalahari Craton provide estimates of P –T conditions. These
equilibrium conditions are believed to represent a near steady-state geotherm because
of a lack of significant tectonic activity in the region for more than a billion years
preceding the eruption (∼100 Ma). The geotherm for 1 cm grain size is significantly
cooler than xenolith estimates, while the 0.01 cm model is on the hot end of xenolith
conditions. The Schatz and Simmons [1972] estimate bounds the cool end, but is more
curved than indicated by the bulk of the xenolith estimates. The geotherm using the
radiative conductivity model from this study also appears cool, but is significantly
less curved than the Schatz and Simmons [1972] model. The geotherm estimated for
a 0.1 cm grain size appears to fit best the bulk of the xenolith data.
Measured grain sizes of south African xenoliths range from 0.4–1.4 cm [Ave Lallemant et al., 1980] and appear inconsistent with the radiative models based on grain
size. The fits of each of these geotherms can be improved by tailoring upper crustal
111
conductivity and heat production, but geotherms with higher grain sizes remain
difficult to fit. A polymineralic model for radiative conductivity derived from the
lherzolite data is likely to provide a better fit to the data because of the more modest
radiative contribution.
A.6
Conclusions
Thermal conductivity and diffusivity are important thermophysical parameters
required for calculating geothermal temperatures and geodynamic modeling.
At
high temperatures, thermal conductivity/diffusivity combine lattice and radiative
mechanisms of heat transfer. A new radiative conductivity estimate is derived and
compared with other radiative estimates. Radiative diffusivity can be estimated by,
T − 1140
Dr = 0.133 1 + erf
370
,
(A.4)
with an RMS misfit of 0.024 mm2 /s, and a radiative conductivity estimated by,
kr = 0.56 1 + erf
T − 1150
370
,
(A.5)
with an RMS of 0.086 W/m-K.
Radiative diffusivity estimates based on spectral methods show a significant grain
size dependence as well as temperature, but appear to be inconsistent with difference
estimates. Geotherms computed using grain sizes consistent with southern African
xenoliths are too cool compared to estimated xenolith P –T conditions for the Kalahari
craton. Temperatures estimated using the radiative conductivity model from this
study are consistent with geophysical estimates of lithospheric thickness but may
also be too cool and be too curved to fit xenolith data accurately.
An estimate of radiative diffusivity within a single lherzolite sample suggests a
smaller contribution to effective diffusivity than from equivalent grain size diffusivity
estimates or monomineralic olivine aggregate results from this study. Additional
experimental results are necessary to resolve the discrepancies between spectral and
difference estimates and grain size contributions to diffusivity/conductivity.
APPENDIX B
THERMAL CONDUCTIVITY OF
AMPHIBOLES: INFLUENCE OF
COMPOSITION
B.1
Abstract
Amphiboles are a major rock forming mineral group and a reservoir of H2 O in
the lithosphere that constitutes >15% of some igneous rocks and >50% of some
metamorphic rocks, as well as small concentrations in hydrous mantle xenoliths.
Hence it is important to understand its physical properties in order to properly model
evolution of the lithosphere. In particular, this study examines the thermal conductivity dependence on amphibole composition. A linear-least squares approach is used
to develop an empirical relationship between cation concentrations and thermal conductivity of amphiboles. Overall misfit between the empirically derived relationship
and measured conductivities is 0.32 W/m-K or better with a maximum misfit of
<0.8 W/m-K. Coefficients in this empirical relationship suggest that incorporation
of Na and Al into the amphibole structure tends to reduce conductivity, whereas
Mg and Ca tend to enhance the conductivity. While this method is promising for
estimating thermal conductivity in amphiboles, it would greatly benefit from the
measurements of additional compositions. This formulation should be applied with
caution to non-end-member compositions.
B.2
Introduction
Amphiboles are a major mineral group found in crustal igneous and metamorphic
rocks and are occasionally found in mantle xenoliths. Abundances of amphiboles in
intermediate composition igneous rocks can be >15% and range up to nearly pure
113
hornblende (hornblendite). Amphiboles are alteration products in the oceanic crust
that can reach ∼8% (locally in fracture zones >20%) of the crust [Carlson and Miller ,
2004] and also are found in concentrations of >50% of some crustal metamorphic
rocks. Crustal cross-sections derived from surface exposures suggest that amphibolitefacies represent approximately 10 km of the mid-crust in the Pikwitonei and Sachigo
region of the Superior Craton [Fountain et al., 1987]. In magmatic arcs amphiboles
may be found in concentrations greater than several tens of percent and have a large
influence on the arc H2 O budget, capturing potentially 20% of water ascending in
mantle-derived melts [Davidson et al., 2007]. In addition to the crust, amphibole is
a major storage mineral for H2 O within the mantle [Lamb and Popp, 2009]. Because
H2 O can have a disproportionately large influence on thermophysical properties, a
few percent amphibole may likewise have a large influence on lithospheric dynamics.
It is therefore important to understand the dependence of thermophysical properties
on amphibole compositions when modeling the current state and evolution of the
crust and upper mantle.
Particularly important in lithospheric evolution is the thermal regime and, hence,
thermal conductivity. Measurements of thermal conductivity of single amphibole
crystals and monomineralic aggregates of ∼12 distinct compositions were made by
Horai and Simmons [1969]; Horai [1971]; Diment and Pratt [1988]. Conductivity of
amphiboles in these studies ranges from 2.2 W/m-K in glaucophane to 5.2 W/m-K in
tremolite. While several compositions have been measured, given the large array of
possible compositions, a relatively small fraction of compositions have been examined.
Therefore, it is desirable to develop a method to predict the thermal conductivity of
an amphibole by its composition.
In this study, I develop a relationship between thermal conductivity to amphiboles
of known composition that may be useful in predicting a wider range of compositions
than has been currently studied. Unfortunately the compositions of amphiboles
analyzed by Horai and Simmons [1969]; Horai [1971]; Diment and Pratt [1988] are
not precise enough to develop a reliable model of thermal conductivity. Therefore, the
114
results presented in this work are preliminary and represent a proposal for a possible
method of estimating the conductivity of amphiboles based on composition.
B.3
Samples
Amphiboles have a wide array of compositions. The general formula for amphibole
is AB2 C5 T8 O22 (OH)2 [Leake and 21 others, 1997]. The A site can generally be filled
with Na or K, or can be vacant. The B and C sites can have a wide variety of
substitutions. In this paper, I focus on amphiboles with A, B and C sites filled with
Na (A or B), Ca (B only), Mg, Fe2+ , Fe3+ (C only) and Al3+ (C or T).
Conductivity measurements were performed on 20 individual amphiboles with 11
distinct compositions by Horai and Simmons [1969]; Horai [1971] using a needle
probe on powdered samples. The compositions were analyzed by x-ray techniques,
but were unfortunately not precisely reported so composition may deviate somewhat
from reported values. Another 6 samples cut from monomineralic aggregates were
measured on a divided bar [Diment and Pratt, 1988]. The compositions of these
samples are somewhat less certain than those measured in the studies by Horai and
Simmons [1969]; Horai [1971]. The thermal conductivity of 12 amphiboles and their
general compositions used in this study given in Table B.1 . Conductivities range from
2.17 W/m-K for glaucophane to 4.66 W/m-K for tremolite with sodic and aluminous
amphiboles having lower conductivities than those without.
Average values for samples of similar composition were used rather than individual measurements to prevent bias from any particular composition. For tremolite,
I excluded the low measurement of 2.76 W/m-K when computing the average of
4.66±0.51 W/m-K, which has a significantly lower standard deviation than when
included (4.38±0.86 W/m-K). Actinolite also displays a large range in conductivity
2.15-4.52 W/m-K, which may result from a deviations from reported compositions
and/or as a result of uncertainties arising from the method used to measure conductivity.
Table B.1. Amphibole compositions and conductivity.
Mineral
1. Anthrophyllite
2. Cummingtonite
3. Grunerite
4. Tremolite
5. Actinolite
6. Ferro-actinolite
7. Hornblende
8. Barkevikite
9. Glaucophane
10. Magnesio-riebeckite
11. Riebeckite
12. Richterite
Chemical Formula
Mg7 (Si8 O22 )(OH)2
Mg3.5 Fe3.5 (Si8 O22 )(OH)2
Fe7 (Si8 O22 )(OH)2
Ca2 Mg5 (Si8 O22 )(OH)2
Ca2 Mg2.5 Fe2.5 (Si8 O22 )(OH)2
Ca2 Fe5 (Si8 O22 )(OH)2
Ca2 Mg3 FeAl(Si7 AlO22 )(OH)2
3+
NaCa2 Fe2+
2 MgFe Al0.5 (Si6.5 Al1.5 O22 )(OH)2
Na2 Mg3 Al2 (Si8 O22 )(OH)2
Na2 Mg3 Fe3+
2 (Si8 O22 )(OH)2
2+
Na2 Fe3 Fe3+
2 (Si8 O22 )(OH)2
Na2 CaMg5 (Si8 O22 )(OH)2
Nb
1
1
2
7 (6)c
3
1
5
1
1
1
2
1
Conductivitya
[W/m-K]
3.96
3.60
3.36
4.66
3.48
3.99
2.74
2.40
2.17
3.63
3.03
3.03
a
Data from Horai and Simmons [1969], Horai [1971], and Diment and Pratt [1988]. b N is the number of conductivity
samples. c Only 6 samples were used to compute the average for tremolite, excluding a low value of 2.76 W/m-K.
115
116
B.4
Estimating Conductivity
Amphibole conductivity, λest , is estimated using the following equation:
λest =
X
ni ki ,
(B.1)
where ni and ki are the number of moles and conductivity contribution from cation
i, respectively.
The conductivity contributions, ki , are given by
Nk = λ,
(B.2)
where λ is a vector of measured conductivity and N is a matrix defined by ni,j , with
columns representing cations from each individual amphibole composition in rows.
The linear-least-squares solution is given by,
k = (NT N)−1 NT λ.
B.5
(B.3)
Results and Discussion
The results of estimated conductivity coefficients for each cation are summarized
in Table B.2 and the compositionally estimated conductivities are shown in Figure
B.1. Conductivity estimated using Na, Ca, Mg, Fe, Al, and Si yields a misfit of
0.32 W/m-K. Vacant sites are not treated as a separate element in this analysis as they
have a negligible effect on the result. Unlike the other cations used in the analysis,
Fe can be partitioned between Fe2+ and Fe3+ and hence may behave differently. By
partitioning Fe into different oxidation states, the results yield a decrease of the RMS
to 0.26 W/m-K. In both cases mentioned above, the misfits of all samples are within
0.8 W/m-K, with the misfits of 67% of the samples within 0.4 W/m-K (approximately
1σ of a typical thermal conductivity measurement).
Some cations in many minerals can fit into different sites, amphiboles included,
and as a result have differing influences on physical properties. For example, Mg in
anthrophyllite sits in both the B and C sites (Table B.1). I decided to focus only on
Al3+ substitution between the C and T sites because Al has a significantly larger ionic
radius than Si4+ . Adding Al substitution further lowers the RMS to 0.21 W/m-K
Table B.2. Fitting constants for estimating conductivity coefficients
Misfit
0.32a,c
0.26a,d
0.21a,e
0.29b,c
0.22b,d
0.07b,e
Na
-0.1769
-0.4640
-0.3626
-0.2366
-0.5256
-0.4095
Ca
0.2014
0.1394
0.5698
0.2296
0.1675
0.7576
Mg
0.1064
0.0431
0.4245
0.0428
-0.0213
0.4714
Fe2+
Fe3+
-0.0554
0.3118
0.3144
0.9439
Fetotal
0.0496
C
Al
0.2794
T
Al
-0.9150
-0.0101
-0.1161
0.3588
0.2555
1.0847
Altotal
-0.4323
-0.3966
-0.5030
-0.4677
0.4203
-1.1908
Si
0.3918
0.4603
0.1328
0.4458
0.5150
0.0918
All table values have units of W/m-K. a Fit using all amphibole compositions. b Excluding actinolite. c No partitioning or
substitution. d Fe partitioned between Fe2+ and Fe3+ . e Fe partitioned between Fe2+ and Fe3+ and Al partitioned between
C and T cation sites.
117
118
Conductivity [W/m/K]
6
(a)
Na
4
3
2
1
Misfit
w/ actinolite
Ca
5
RMS
Al
0.32 W/m/K
0.22 W/m/K, partitioned
0
−1
Conductivity [W/m/K]
6
(b)
5
Na
4
3
2
1
Misfit
w/o actinolite
Ca
RMS
Al
0.29 W/m/K
0.07 W/m/K, partitioned
0
−1
1
2
3
4
5
6
7
8
9 10 11 12
Figure B.1. Thermal conductivity of amphiboles. Amphiboles containing Ca, Na
and Al are illustrate by the labeled horizontal bars. Numbers correlate to samples
in Table B.1. Circles represent natural samples with error bars representing 10%
of the measured thermal conductivity representing typical measurement uncertainty,
rather than absolute error. Misfits are computed by difference of the experimental
and estimated conductivities. (a) estimated thermal conductivity using all samples:
squares-no partitioning or substitution; diamonds-partitioning of Fe between Fe2+
and Fe3+ and substitution of Al in C and T sites (see text). (b) same as (a) excluding
actinolite.
119
(Figure B.1a). Conductivity estimates using partitioning of Fe and substitution of Al
fit nearly all amphiboles well within approximated uncertainty, with the majority of
misfit in the tremolite–ferroactinolite solid-solution series.
If actinolite is excluded from the analysis (see reasoning below), the RMS is slightly
improved in the unpartitioned case (RMS = 0.29 W/m-K) and significantly when Fe
and Al are partitioned (RMS = 0.07 W/m-K, see Figure B.1b). However, when
fitting 11 independent compositions with 8 parameters, there is a fear of over-fitting
the data, especially when Al substitution alone does not improve the fit over the
unpartitioned formulation. More conductivity measurements on an increased number
of end-member compositions are clearly needed to resolve this issue.
Actinolite has a lower thermal conductivity than either tremolite or ferro-actinolite.
This is unsurprising as a number of other minerals with solid solution series, such
as plagioclase and olivine show a quadratic relationship between end-members with
a minimum occurring at an intermediate composition (Figure B.2). The pyropealmandine and andradite-grossular series in garnets may also exhibit a quadratic
behavior between end-member compositions [Giesting and Hofmeister , 2002]. Given
the uncertainties in currently measured actinolite series compositions and conductivities it is difficult to reliably model intermediate compositions. Additional data are
needed to model conductivities in the tremolite-ferroactinolite as well as within other
amphibole solid solution series where no measurements exist.
While the potential nonlinear intermediate low in conductivity reduces the usefulness of this method in predicting conductivity of intermediate compositions, endmembers may still be reliably estimated using this method. And despite this potential
shortcoming, it can still yield a reasonable conductivity estimates when additional
data are not available.
Fitting parameters are given in Table B.2 show the effect of each cation on estimated
amphibole conductivity. While the values of each constant change significantly in
each case, some patterns emerge. Ca and Si are a positive influence on conductivity.
However, when estimated effect of Si on conductivity is large, Ca is small and vice
versa. Mg likewise has a similar relationship with the Si coefficient, but is not strictly
120
Conductivity [W/m/K]
6
5
olivine
actinolite series
4
3
plagioclase
2
1
0
20
40
60
Fo/An/Tr
80
100
Figure B.2. Thermal conductivity of solid solution minerals. Olivine as a function
of forsterite (Fo) content and plagioclase as a function of anorthite (An) content.
The amphibole tremolite-ferroactinolite series is shown for comparison although
intermediate composition is approximate. Error bars are the standard deviation from
multiple samples. Olivine data from Horai [1971], Chai et al. [1996], and Harrell
[2002]. Plagioclase data from Petrunin et al. [2004] and references therein.
positive. Estimated conductivity is always reduced by Na. And while Al in the C
site has a positive influence on conductivity, the overall effect of Al likely reduces the
overall conductivity as the T site coefficient is strongly negative. Fe is more complex
with no clear pattern relative to other ions.
B.6
Implications
To examine the influence of thermal conductivity variations I compute 1-D-steadystate temperature differences through a 10 km thick rock layer as a function of mole
fraction amphibole with compositions ranging from hornblende to tremolite (Figure
B.3). Heat production in the layer is assumed to be a constant at 0.6 µW/m3 , a typical
value for amphibolites. Heat production has a relatively small effect on the results
except when values are high, but unlikely on regional scales. Heat flow, however,
strongly affects the results.
At a heat flow of 20 mW/m2 , the temperature difference arising from amphibole
composition are quite small at the base of the layer, even with high amphibole content.
At 40 mW/m2 the temperature difference computed for tremolite to hornblende is
slightly greater than 10◦ C at mole fractions similar to tonalitic composition (∼10%).
121
0.8
20
40
Fraction Amphibole
20
1
0.6
0.4
0.2
Heat Flow = 20 mW/m2
40 mW/m2
80
40
60
60
0.8
20
0.6
20
40
Fraction Amphibole
0
1
0.4
0.2
60 mW/m2
0
0
0.2
0.4
0.6
0.8
1 0
0.2
0.4
Fraction Tremolite,
1 - Fraction Hornblende
80 mW/m2
0.6
0.8
1
Figure B.3. Influence of amphibole composition on lithospheric temperatures.
Temperature differences for a 10 km thick layer with varying amphibole content
and composition. Radiogenic heat production is assumed constant at 0.6 µW/m3 .
All temperature differences are computed relative to amphibole composition of 100%
tremolite. Contour interval is 10◦ C. Each panel represents a different heat flow at the
top of the layer.
122
At a mole fraction of 50% (amphibolite composition), the temperature difference is
34◦ C for pure hornblende relative to tremolite, but still small for a mole fraction of
50% tremolite (15◦ C). When values of heat flow at the top of the layer are high,
temperature differences are significantly higher for mole fractions less than tremolite
∼0.4 with modest quantities of amphibole (>20%). At high heat flow, temperature
differences at the base of a 10 km thick layer of 15% amphibole could reach as high
as 25◦ C; high enough to possibly effect thermochronology estimates of closure depth
and hence exhumation rate.
If the layer thickness is greater than 10 km, the temperature differences shown in
Figure B.3 will be larger. However, because heat production is included, temperature
differences do not scale exactly 1:1, but are close enough that the temperature
difference at 20 km is ∼5% less than double the 10 km estimate. In general, significant
quantities of amphibole at modest to high heat flow are required to observe significant
temperature differences due to from amphibole composition at the base of a 10 km
thick layer. Thermal conductivity variations due to amphibole composition may
observably change temperatures in arc terranes where heat flow is high and rocks
contain a high fraction of amphibole, but are unlikely to be much of an influence in
shield regions where heat flow is typically low.
B.7
Conclusions
Amphiboles are a major rock forming mineral within the crust and a major reservoir
of H2 O in the lithosphere. Thus understanding the thermophysical properties of
amphibole are important to modeling lithospheric evolution. This study examines the
compositional dependence on the thermal conductivity of amphibole. Conductivity
ranges from 2.2–4.7 W/m-K resulting from compositional variations and possibly
from site-specific substitutions of chemical species within the amphibole structure.
Si and Ca have positive influences on conductivity while incorporation of Na and Al
typically reduce the conductivity.
Conductivity of amphiboles can be estimated by
λest = −0.237nNa + 0.230nCa + 0.043nMg
(B.4)
123
−0.010nFe − 0.503nAl + 0.446nSi .
to with a maximum misfit of 0.5 W/m-K and RMS of 0.3 W/m-K. By partitioning Fe
between Fe2+ and Fe3+ as well as treating Al in the C and T cation sites separately
misfit can be reduced to 0.1 W/m-K with a maximum misfit of 0.1 W/m-K using the
resulting equation:
λest = −0.410nNa + 0.758nCa + 0.471nMg
(B.5)
+0.359nFe2+ + 1.085nFe2+ + 0.420nC Al
−1.191nT Al + 0.092nSi.
Complexities in the conductivity pattern of intermediate composition amphiboles
may be better predicted by a quadratic with a minimum inside the end-member
compositions, thereby reducing the accuracy of this method.
Temperature differences modeled using variations in amphibole composition show
sensitivity to surface heat flow. Arc terranes are likely to have the greatest variation
in temperature estimates due to amphibole conductivity because of their high heat
flow and high amphibole content. Amphibole composition may have little effect on
temperature variations in shields where heat flow is low.
Additional and compositionally more accurate measurements are needed to test for
nonlinear behavior at intermediate compositions, and to further refine and expand
this empirical method to a greater range of compositions. This method may also hold
promise for other complex minerals groups with complex compositions such as micas
and clays.
APPENDIX C
MODELS OF THERMAL CONDUCTIVITY
FOR INDIVIDUAL MINERALS
This appendix includes figures of conductivity models for most of the minerals
used in this dissertation. Conductivity between end-members for plagioclase, olivine,
and garnet are computed using a second order polynomial by linear least squares
inversion. Temperature dependence is computed using Newton’s inverse method for
lattice conductivity or a joint inversion for lattice and radiative conductivity.
Conductivity [W/m-K]
10
8
α-quartz β-quartz
6
keff
4
kL
2
kR
0
300
500
700
900
Temperature [K]
1100
Figure C.1. Effective thermal conductivity of quartz. Solid circles are data from
Höfer and Schilling [2002] and open circles are the difference between the observed
conductivity and lattice model. Radiative model is assumed the same for α- and
β-quartz. Grey points are estimated from difference of observed conductivity and
radiative model. Heavy, light, and dashed lines represent effective conductivity,
keff , estimated lattice conductivity, kL , and estimated radiative conductivity, kR ,
respectively.
125
0.5
Conductivity [W/m-K]
muscovite
0.4
keff
0.3
0.2
kL
0.1
kR
0
300
400
500
600
Temperature [K]
700
800
Figure C.2. Effective thermal conductivity of muscovite. Solid circles are data from
Roy et al. [1981] and open circles are the difference between the observed conductivity
and lattice model. Heavy, light, and dashed lines represent effective conductivity,
keff , estimated lattice conductivity, kL , and estimated radiative conductivity, kR ,
respectively.
3
Conductivity [W/m-K]
orthoclase
keff
kL
2
1
kR
0
300
500
700
Temperature [K]
900
1100
Figure C.3. Effective thermal conductivity of orthoclase. Solid circles are data
from Höfer and Schilling [2002] and open circles are the difference between the
observed conductivity and lattice model. Heavy, light, and dashed lines represent
effective conductivity, keff , estimated lattice conductivity, kL , and estimated radiative
conductivity, kR , respectively.
126
3.0
plagioclase
Conductivity [W/m-K]
(a)
2.5
2.0
1.5
1.0
0
20
40
60
% Anorthite (An)
80
100
3.0
Conductivity [W/m-K]
(b)
0
2.5
20
100
2.0
55
1.5
An0
1.0
300
An20
An55
An100
400
500
Temperature [K]
600
700
Figure C.4. Lattice thermal conductivity of plagioclase. Composition data from
Birch and Clark [1940], Sass [1965], and Horai [1971]. (a) conductivity as a function
of anorthite at 0 GPa and 298 K. (b) conductivity as a function of temperature for
four plagioclase compositions. Temperature data from Petrunin et al. [2004].
127
4.0
Conductivity [W/m-K]
orthopyroxene
3.5
3.0
2.5
2.0
200
600
1000
Temperature [K]
1400
Figure C.5. Lattice thermal conductivity of orthopyroxene. Data from Harrell
[2002].
Conductivity [W/m-K]
7
(a)
6
olivine
5
4
3
2
1
0
20
40
60
% Forsterite (Fo)
80
100
Figure C.6. Lattice thermal conductivity of olivine. Composition data from Horai
and Simmons [1969], Horai [1971], and Harrell [2002]. (a) conductivity at 0 GPa
and 298 K as a function of the end-member forsterite.
128
Conductivity [W/m-K]
7
(b)
6
10 GPa
5 7
4
4 0
3
2
Conductivity [W/m-K]
1
7
6
(c)
Fo0
Fo78
Fo91
5
4
3
2
1
200
400
600
800 1000
Temperature [K]
1200
1400
Figure C.6 continued. (b) pressure and temperature dependence of Fo90 . (c)
conductivity estimates for three compositions using model established in (b) and data
from Harrell [2002].
129
Conductivity [W/m-K]
7
75 ≤ Py + Alm < 90
Py + Alm ≥ 90
Py + Alm ≥ 90 used in fit
6
(a)
5
4
3
2
1
0
20
40
60
% Pyrope (Py)
80
100
Conductivity [W/m-K]
7
(b)
6
5
4
garnet
8
4
0
3
2
1
300
500
700
900
Temperature [K]
1100
Figure C.7. Lattice thermal conductivity of garnet. (a) conductivity at 0 GPa and
298 K as a function of the end-member pyrope. Data from Giesting and Hofmeister
[2002]. (b) pressure and temperature dependence of conductivity for garnet with
composition Py25 Alm73 Sp1 Gr1 from Osako et al. [2004].
130
Conductivity [W/m-K]
14
spinel
12
10
8
6
4
2
200
400
600
800 1000
Temperature [K]
1200
1400
Figure C.8. Lattice thermal conductivity of spinel. Data from Clauser and Huenges
[1995].
APPENDIX D
EMPIRICAL CONSTANTS USED TO
MODEL PHYSICAL PROPERTIES
Constants used for computing density, expansivity are given in Table D.1.
Con-
stants used in computing thermal conductivity are given in Table D.2. Amphibole
composition is assumed to be hornblende in continental rocks (Chapter 3) and a
mixture of pargasite, tremolite and ferroactinolite within the oceans (Chapter 5).
Additional constants are needed for oceanic cooling models to compute heat capacity.
The constants necessary to compute heat capacity are given in Table D.3.
Table D.1. Physical properties and empirical constants for mineral end-members.
Mineral
alpha-quartz
beta-quartz
orthoclase
albite
anorthite
phlogopite
clinochlore
daphnite
tremolite
ferro-actinolite
pargasite
hornblende
diopside
hedenbergite
enstatite
ferrosillite
forsterite
fayalite
spinel
hercynite
pyrope
almandine
grossular
w
g mol−1
60.08
60.08
278.34
262.22
278.21
417.29
555.83
713.51
812.41
970.08
835.86
864.70
216.56
248.10
100.40
131.93
140.71
203.78
142.27
173.81
403.15
497.75
450.45
ρ298
kg m−3
2648
2530
2555
2620
2760
2788
2635
3343
2979
3430
3074
3248
3272
3651
3206
4003
3222
4400
3575
4264
3565
4324
3593
a0 × 105
K−1
1.4170
-0.4400
3.4000
1.9801
1.2491
5.8000
2.5
2.5
3.1310
3.1310
3.1310
2.0750
3.3300
2.9800
2.9720
2.8750
2.8540
2.3860
1.9600
0.9700
2.3110
1.7760
1.9510
a1 × 108
K−2
9.6581
a2
K
-1.6973
1.0065
-0.0162
-0.9760
0.0161
1.0270
0.5711
1.0080
1.1530
1.6400
1.9392
0.5956
1.2140
0.8089
-0.3842
-0.0518
-0.4538
-0.5071
-0.4972
KT
GPa
37.1
57
58.3
53.8
82.5
54
85
86.9
85
76
91.2
94
113
119
105.8
100
128
135
205
209
173
174
168
KT′
5.99
4
4
6
3.2
7.8
3.3
4
4
4
4
4
4.8
4
8.5
8.8
4.2
4.2
4.1
4
5
6
5.5
γth
0.7
0.1
0.4
0.6
0.6
0.6
0.3
0.3
0.74
0.73
0.84
1.1
1
1.5
1.01
0.88
0.99
1.06
1.73
1.2
1.29
1.29
1.38
δT
8.42
4.11
4.44
6.57
3.47
4.55
4.3
4.3
4.74
4.73
4.84
5.1
6.04
5.21
9.39
9.05
5.19
5.26
6.5
5.19
5.3
5.52
4.57
Notes
1,2
1,2
1,2
1,2,8
1,2,8
1,2,3,4
1,12
1,12
1,2
1,2
1,2
1,2,3
1,2
1,2,3
1,5,9
1,6,10
1,2,11
1,2,11
1,7
1,2,3
1,2
1,2
1,2
Italicized values are estimated. 1 Densities and several bulk moduli from Hacker et al. [2003] and references therein.
Thermal expansivity from Fei [1995] and references therein. 3 Reformulated expansivity from Holland and Powell [1998]
and references therein. 4 Comodi and Zanazzi [1995]. 5 Jackson et al. [2003]. 6 Average of Yang and Ghose [1994] and
Hugh-Jones [1997]. 7 Sueda et al. [2008]. 8 Average of Downs et al. [1994] and Angel [2004]. 9 Angel and Jackson [2002].
10
Yang and Ghose [1994]. 11 Stixrude and Lithgow-Bertelloni [2005]. 12 Average of results from Pawley et al. [2002] and
Welch and Crichton [2002].
2
132
Table D.2. Empirical constants for estimating conductivity.
mineral
a-quartz
b-quartz
orthoclase
plagioclase (an)
biotite
chlorite
amphibole
hornblende
clinopyroxene
orthopyroxene
olivine (fo)
spinel (sp)
garnet (py)
λ0
W/m-K
8.79
0.99
1.79
2.2
2.27
4.35
tr: 4.0654
2.65
4.25
3.37
3.09
11.94
4.97
λ1
W/m-K
-2.18
λ2
W/m-K
1.9
fact: 3.7834
parg: 1.7066
-1.17
5.47
-7.42
3.35
8.71
8.45
n
1.48
-1.00
-0.27
-0.21
1.54
1.54
0.5
0.5
0.54
0.3
0.49
1.24
0.37
λRmax
W/m-K
0.8107
0.8107
0.1949
0
0.1166
0.1166
0.1725
0.1725
0.1725
0.1725
0.1725
0
0
ω
K−1
6.553e-3
6.553e-3
3.782e-3
TR
K
558
558
864
2.749e-3
2.749e-3
3.900e-3
3.900e-3
3.900e-3
3.900e-3
3.900e-3
520
520
762
762
762
762
762
Notes
1,2
1,2
2
3,4,5,6
7,8
10
5,9,10,11,12
5,9,10,11,12
5,13,12,14
5,13,12
5,9,13,12
15,16
16,17,18
133
Lattice conductivity is computed as, λ◦ = λ0 + λ1 χ+ λ2 χ2 , where χ is the mole fraction of the end-member (an) anorthite,
(fo) fosterite, (sp) spinel, and (py) pyrope. Amphibole conductivities are computed as a function of the end-members (tr)
tremolite, (fact) ferro-actinolite, and (parg) pargasite. All other minerals do not have enough data from end-member
compositions to estimated mixing behavior. Values in italics are estimated. 1 Höfer and Schilling [2002]. 2 Branlund and
Hofmeister [2007]. 3 Birch and Clark [1940]. 4 Sass [1965]. 5 Horai [1971]. 6 Petrunin et al. [2004]. 7 Room temperature values
[Clauser and Huenges, 1995] estimated as λeff = (2λk + λ⊥ )/3. 8 Radiative conductivity assumed same as muscovite. 9 Horai
and Simmons [1969]. 10 Diment and Pratt [1988]. 11 Appendix B. 12 Radiative conductivity estimated from lherzolite Gibert
et al. [2003]. See Hasterok [in prep.] for explanation. 13 Harrell [2002]. 14 Value for n estimated from augite rather than
diopside sample from Hofmeister and Pertermann [2008]. 15 Spinel estimated from Clauser and Huenges [1995], hyrcenite
estimated from magnetite. 16 Radiative contribution is assumed to be small from spectral measurements Shankland et al.
[2005] and Shankland personal communication. 17 Giesting and Hofmeister [2002]. 18 Osako et al. [2004].
134
Table D.3. Empirical constants for computing heat capacity of mineral end-members.
c0
c1
c2 × 10−3
c3 × 10−6
Mineral
kJ-K−1 kg−1
kJ-K−0.5 kg−1
kJ-K-kg−1
kJ-K2 kg−1
Notes
albite
1.5012
-9.2116
-30.0990
4.0829
2
anorthite
1.5793
-4.7252
0
-1.1395
2
clinochlore
2.1726
-19.7390
-2.9739
-1.7267
1
daphnite
1.8204
-18.1750
6.0436
-1.8045
1
tremolite
1.5725
-10.7350
-11.3070
-0.4465
1
ferro-actinolite
1.4695
-12.6660
9.6112
-2.9282
1
pargasite
1.6559
-13.1260
1.5163
-2.6059
1
diopside
1.4103
-7.4109
-33.0900
4.2567
3
hedenbergite
1.3712
-10.6680
-32.7290
-0.0042
1
enstatite
1.6592
-11.9590
-22.6160
2.7805
4
ferrosillite
1.3204
-10.5590
-3.4440
-0.2858
4
forsterite
1.6572
-12.8040
0
-1.9042
4
fayalite
1.2366
-9.8820
0
-0.3052
4
spinel
1.7197
-14.0860
0
0
4
pyrope
1.4657
-7.0120
-33.0420
3.1262
4
almandine
1.2485
-6.6050
-30.2980
4.4437
4
grossular
1.1531
-0.1401
-62.1740
7.7938
4
1
Reformulated expansivity and/or reformulated heat capaciies from Holland and
Powell [1998] and references therein. 2 Average of Downs et al. [1994] and Angel
[2004]. 3 Berman and Brown [1985]. 4 Berman and Aranovich [1996].
APPENDIX E
SIMPLIFIED CONTINENTAL
GEOTHERMS
Computing geotherms using my method is cumbersome, so I offer several empirical
approximations to simplify computations. Thermal conductivity is approximated
by the multidimensional approximation to P –T -dependent conductivity (Chapter 3,
Table 3.1),
λ(z, T ) = (k0 + k1 T −1 + k2 T 2 )(1 + k3 P ),
(E.1)
where pressure is in GPa and temperature in Kelvins. Empirical constants for the
thermal conductivity of each layer are given in Table E.1.
For this empirical
approximation a constant density of 2850 kg/m3 and 3340 kg/m3 are used for the
crust and mantle, respectively. Note that crustal rocks have two temperature ranges
for each layer, which result from the α–β transtion in quartz [Höfer and Schilling,
2002; Branlund and Hofmeister , 2007]. The approximation varies by 5–10% from
the mineral mixing modeled conductivity I use to compute geotherms. Geotherms is
approximated using a boot-strapping solution for temperature,
Ti+1 = Ti +
Ai
qi
∆zi −
∆z 2 ,
λi
2λi i
(E.2)
where ∆zi is the layer thickness, and Ai is the heat produced within the layer [Turcotte
and Schubert, 2002]. The intra-layer conductivity, ki is computed using Equation
E.1 with the average temperature of the upper and lower layer boundries as input.
Heat production is estimated using my model from Chapter 3. Both Ti+1 and ki are
unknowns and solved iteratively using Newton-Raphson method. Upper boundary
conditions for T0 and q0 are the surface temperature and surface heat flow respectively.
The resulting difference in geotherms increases from 2–21 K from 40–100 mW/m2 with
an associated RMS misfit between 2 and 4 K.
136
Table E.1. Empirical conductivity constants.
Upper Crust
T ≤ 844 K
T > 844 K
Middle Crust
T ≤ 844 K
T > 844 K
Lower Crust
T ≤ 844 K
T > 844 K
Sp-peridotite
Gt-peridotite
k0
W/m-K
k1
W/m
k2 × 107
W/m-K3
k3
GPa−1
1.496
2.964
398.84
-495.29
4.573
0.866
0.0950
0.0692
1.733
2.717
194.59
-398.93
2.906
0.032
0.0788
0.0652
1.723
2.320
2.271
2.371
219.88
-96.98
681.12
669.40
1.705
-0.981
-1.259
-1.288
0.0520
0.0463
0.0399
0.0384
Volumetric thermal expansivity is be approximated by,
αV = (a0 + a1T + a2 T −2 )(1 + a3 P ),
(E.3)
where temperature is in Kelvins and pressure in GPa. Coefficients for this approximation are listed in Table E.2.
Using this approximation, geotherms estimated
above, and Equation 3.8 the thermal isostatic relationship is easily esimated. The
difference between this approximation and the numerical methods employed in this
study are ∼70 m low at 40 mW/m2 and ∼70 m high at 100 mW/m2 .
Additional approximations for lithospheric thickness and heat flow into the base of
the lithosphere may be useful for geodynamic modeling. Lithospheric thickness using
geotherms computed in Chapter 3 is approximated by,
hL =
3100
,
qs − 23.5
(E.4)
with a misfit of 1.9 km. Using this lithospheric thickness, the heat flow into the base
of the lithosphere is estimated by
qL = 0.74qs −
62.0
− 8.4.
qs − 23.5
(E.5)
137
Table E.2. Empirical expansivity constants.
Upper Crust
T ≤ 844 K
T > 844 K
Middle Crust
T ≤ 844 K
T > 844 K
Lower Crust
T ≤ 844 K
T > 844 K
Sp-peridotite
Gt-peridotite
a0 × 105
K−1
a1 × 108
K−2
2.355
1.741
3.208
0.500
-0.7938 -0.1193
-0.3094 -0.0778
2.020
1.663
2.149
0.602
-0.6315 -0.1059
-0.3364 -0.0745
2.198
2.134
3.036
3.026
0.921
0.711
0.925
0.906
-0.1820
-0.1177
-0.2730
-0.3116
a2
K
a3
GPa−1
-0.0626
-0.0563
-0.0421
-0.0408
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