Variations of surface heat flow and lithospheric thermal structure beneath the North American craton

transcript

www.elsevier.com/locate/epsl

Earth and Planetary Science Letters 223 (2004) 65–77

Variations of surface heat flow and lithospheric thermal structure

beneath the North American craton

J.C. Mareschala,*, C. Jaupartb

aGEOTOP-UQAM-McGill, P.O. Box 8888, Sta. ‘‘downtown’’, Montreal, H3C3P8, Canadab Institut de Physique du Globe, 4 pl. Jussieu, 75252 Paris, France

Received 31 October 2003; received in revised form 6 April 2004; accepted 7 April 2004

Abstract

Two end-member models have been proposed to explain the variations in surface heat flow in stable continents, calling

for changes of either crustal heat production or heat flow at the base of the lithosphere. The scale of the surface heat flow

variations controls how these variations affect the thermal structure and thickness of the lithosphere and provides

constraints on these models. Data in the Canadian Shield and the Appalachians are now extensive enough to address

problems of scale and relationship between average heat flow and heat production. We analyze the global data set as well

as data from five compositionally distinctive subprovinces. Within each province, on scales < 500 km, observed heat flow

variations are linked to changes of local crustal structure. For the five subprovinces, the average values of heat flow (Q)

and heat production (A) conform to the simple relationship Q=Qo +HA, where Hc 9 km and Qoc 33 mW m� 2. This

shows that, on scales larger than the dimensions of these provinces (>500 km), variations in crustal heat production

dominate and hence that variations of mantle (Moho) heat flow must be small. The large heat flow step at the Grenville–

Appalachian boundary (c 16 mW m� 2) may be accounted for by a change in crustal heat generation only. In that case,

the lithosphere is c 40 km thinner in the Appalachians than in the Shield. At wavelengths of 500 km or more, mantle

(Moho) heat flow variations are constrained to be smaller than the detection limit of heat flow studies, or about F 2 mW

m� 2, and may not be correlated with surface geology. Downward continued to the base of the lithosphere, the amplitude

of these variations depends on wavelength and must be smaller than F 7 mW m� 2. Such variations imply that

temperature differences must be smaller than 400 K at 150 km depth. These bounds are consistent with seismic shear wave

velocity variations and geothermobarometry studies on mantle xenoliths.

Keywords: lithosphere; heat flow; heat production

1. Introduction from oceanic mantle sampled at mid-ocean ridges and

Interest in continental roots has recently soared as

it has become clear that they are chemically distinct

doi:10.1016/j.epsl.2004.04.002

* Corresponding author.

E-mail addresses: jcm@olympus.geotop.uqam.ca

(J.C. Mareschal), cj@ccr.jussieu.fr (C. Jaupart).

that they affect mantle convection currents [1–4]. The

thermal regime of the roots is one of the key elements

to understand their nature and evolution. Determining

the thermal conditions at the base of the continental

lithosphere is also important because they are used as

a constraint that models of mantle convection must

satisfy [5]. Relevant data on the thermal structure of

J.C. Mareschal, C. Jaupart / Earth and Planetary Science Letters 223 (2004) 65–7766

the roots have come from heat flow measurements [6]

and from geothermobarometry on mantle xenoliths

[7]. Knowledge of the distribution of seismic veloc-

ities in these roots has been steadily improving, but it

is difficult to discriminate between temperature and

composition effects [4,8].

In stable continents, for ages greater than about

500 Ma, thermal transients have decayed and surface

heat flow is the sum of crustal heat production and

of the heat supply at the base of the lithosphere.

Identifying these two components is an essential step

for calculating continental geotherms. Surface heat

flow varies by large amounts because of shallow

heat production contrasts. Thus, as long as sampling

of continental heat flow and heat production was

inadequate, it was necessary to define systematic

trends in the data to supplement insufficient coverage

in one area with measurements from other areas. Age

is an obvious control variable, and hence ‘‘charac-

teristic’’ continental geotherms were calculated for

heat flow values corresponding to worldwide aver-

ages for specific age groups [9,10]. These geotherms

may not apply to any specific province because of

the large variations of bulk crustal heat generation

among provinces of the same age [11].

Within a single craton, such as the North Amer-

ican craton for example, there are significant varia-

tions of surface heat flow over a range of scales

[6,13]. Variability at short spatial scales bears no

consequence for the deep lithosphere. For a study of

lithospheric structure, one must determine an average

heat flow value over a scale which is sufficiently

large for a reliable downward continuation of the

temperature field: The thicker the lithosphere, the

larger-scale the horizontal averaging. Thus, the de-

termination of lithospheric thickness and average

heat flow cannot be treated independently. Ignoring

this may lead to substantial errors, as shown by two

extreme examples. For a global study of mantle

convection, Pari and Peltier [5] used the worldwide

heat flow database and retained only degrees 1–12

in the spherical harmonic expansion of heat flow. On

that scale (c 3000 km), different provinces are

lumped together, and the geologically active Basin

and Range province is associated with a lower heat

flow than the Canadian Shield. On the other hand,

one-dimensional calculations on 1j�1j grid

(c 100� 100 km) in the eastern European platform

yield some very short wavelength heat flow varia-

tions near the base of the lithosphere that are

implausible [12]. Since the Pollack and Chapman

studies [9,10], a large number of heat flow and heat

production data have been obtained in previously

poorly sampled regions, notably in Precambrian

Shield areas of Canada and India. Furthermore, it

is now clear that the Precambrian lithosphere is quite

thick (>200 km), and this must be considered when

averaging heat flow.

In this paper, we analyze data now available for

the Canadian Shield and the Appalachians, including

recent new heat flow and heat production measure-

ments [14–19]. These data are extensive enough to

address problems of scale and relationship between

average heat flow and heat production. Two classes

of continental geotherms are derived from heat flow

data depending on the heat flow at the base of the

lithosphere, which is taken to be either constant or

proportional to the surface heat flow. We use simple

considerations on diffusive heat transport to show

that significant (>10%) variations in mantle (Moho)

heat flow are not consistent with the presence of a

thick lithosphere. We establish a new relationship

between heat flow and crustal heat production at the

province–subprovince scale (c 500 km) in the

Canadian Shield and in the Appalachians. Variations

in crustal heat production and mantle composition

account for a good part of the observed seismic

velocity heterogeneities in the mantle part of the

lithosphere.

2. Two classes of geotherms for the continental

lithosphere

Fig. 1 illustrates the basic variables and parameters

of this problem. The average value of and variations

in surface heat flow data are QsFDqs. Subtracting the

contribution of crustal heat sources, one obtains the

‘‘mantle’’ heat flow and its variations, QmFDqm.

Finally, the supply of heat at the base of the litho-

sphere at depth z = L is called the ‘‘basal’’ heat flow

and is denoted as QbFDqb. The relationships be-

tween heat flow variations at these three levels are

scale-dependent. Here, the ‘‘lithosphere’’ is defined as

the mantle upper boundary layer where heat is trans-

ported by conduction only. There is no reason to

Fig. 1. Diagram showing the three kinds of heat flow values

considered in this paper. Qs is the surface heat flow. Subtracting

from it the contribution of crustal heat sources yields the mantle heat

flow Qm. The supply of heat at the base of the lithosphere is Qb.

Each heat flow value is associated with horizontal variations of

magnitude Dqs, Dqm, Dqb. The relationships between these three

variations depends strongly on horizontal scale.

Fig. 2. Various relationships that have been proposed between

surface and mantle heat flow. Pollack and Chapman’s predicted

mantle heat flow was calculated following [37] with D = 10 km, Pari

and Peltier [5] used Qm= 0.35Qs and Jaupart and Mareschal [6] give

the range 11–15 mW m� 2.

J.C. Mareschal, C. Jaupart / Earth and Planetary Science Letters 223 (2004) 65–77 67

assume that it extends to the same depth everywhere

and that its base is perfectly flat. Thus, when the

lithosphere thickness varies spatially, calculations for

conductive heat transport are carried out down to the

smallest lithosphere thickness.

2.1. The local linear relationship

Early work on continental heat flow focused on a

linear relationship between the local values of heat

flow and heat production within a few geological

provinces [20]:

Q ¼ Qr þ AD ð1Þ

whereQ and A stand for the observed heat flow and the

local heat production of rocks of the crystalline base-

ment, respectively. The slope D has dimension of

length and Qr is called the reduced heat flow. This

relationship documented small-scale (c 10–30 km)

variations of heat flow and heat production.

The accumulation of large data sets of heat flow and

heat production and theoretical considerations have led

to re-evaluate this relationship. It is now clear that the

crust is heterogeneous on all scales in both the hori-

zontal and vertical directions. The correlation between

local values of surface heat flow and heat production

only holds over exposed plutons very enriched in

radioactive elements [20]. Over other rock types, it is

weak at best, as demonstrated by data from large

Precambrian provinces of India [21], Canada [14]

and South Africa [22,23]. Theory shows that, at small

wavelengths, horizontal heat transport smoothes out

deep differences in heat production rates. Thus, surface

heat flow is only sensitive to shallow heat production

contrasts and the depth scale D is related to the

horizontal correlation distance of heat production

[24–26]. In Appendix A, we also show that for

wavelengths larger than crustal thickness, the heat

flow and heat production power spectra become pro-

portional, and therefore variations in integrated crustal

heat production are directly reflected in the surface

heat flow.

2.2. Two models for the mantle heat flow

Because of the high radioactivity of crustal rocks,

the ‘‘age dependence of heat flow’’ implies that there is

a relatively simple relation between age and bulk

crustal heat production. Pollack and Chapman [9,10]

addressed this question by noting a correlation be-

tween the reduced heat flow and the average heat flow

Q in a few geological ‘‘provinces’’. They added

estimates of lower crustal heat production and

obtained a model in which the mantle heat flow is

nearly proportional to the surface heat flow. This

relationship is not valid at short spatial scales by

construction. It is supposed to hold for sufficiently

Fig. 3. Two models of lithospheric thickness variation with surface

heat flow. We use the intersection of the geotherm with the 1250 jCadiabat as the base of the lithosphere.

Fig. 4. Heat flow map of the Canadian Shield. White dots are heat flow sit

Provinces: Trans-Hudson Orogen (THO), Grenville (Gre); Paleozoic Provin

Bay (H.B.). The Abitibi subprovince is outlined in south eastern part of S

large scales to allow 1-D models of vertical thermal

structure down to the base of the lithosphere, but the

validity of this implicit assumption has never been

examined. A different model was put forward by Pinet

et al. [13] from their analysis of heat flow and heat

production in the Abitibi and Grenville provinces of

the Canadian Shield. They proposed that surface heat

flow varies mostly because of crustal composition and

that mantle heat flow remains in the narrow range of

11–15 mW m� 2.

These studies suggest two convenient end-mem-

bers for the basal heat flow, such that it is either

constant or proportional to the surface heat flow.

These two models are shown in Fig. 2. They lead to

markedly different predictions for lithosphere struc-

ture and thickness because temperatures in the mantle

lithosphere vary much more when the basal heat flow

varies than when crustal heat production varies. For

example, Fig. 3 shows how the depth to the 1250 jCisentropic profile varies with surface heat flow accord-

es. Archean provinces: Superior (Sup), Hearne and Rae; Proterozoic

ce: Appalachians (App); Paleozoic Basins: Williston (Will), Hudson

uperior. Eastern limit of Thompson Belt is also shown.

Table 1

Mean heat flow and heat production in different belts and provinces

of North America

hQia(mW m� 2)

rQb NQ

c hAia(AW m� 3)

rAb NA

Superior

(>2.5 Ga)

42F 2 12 57 0.95F 0.15 1 44

Superior

(excluding

Abitibi)

45F 2.4 12 26 1.4F 0.26 1.2 21

Abitibi 37F 1 7 26 0.41F 0.07 0.33 21

Trans-Hudson

Orogen

(1.8 Ga)

42F 2 11 49 0.73F 0.07 0.50 47

THO (juvenile

crust only)

37F 1.4 7 38 0.6F 0.08 0.48 36

(Thompson

53F 1.6 5 10 1.12F 0.10 0.32 11

Grenville

(1.1 Ga)

41F 2 11 30 0.80e

Appalachians

( < 0.5 Ga)

57F 1.5 13 79 2.6F 0.27 1.9 50

a MeanF one standard error.b Standard deviation on the distribution.c Number of sites.d Number of heat production values. Each value is based on

many samples.e Area-weighted average.

Table 2

Mean heat flow and heat production in four different belts of the

Trans-Hudson Orogen

hQi(mW m� 2)

rQ NQ hAi(AW m� 3)

Northern

volcanic belt

34F 2.3 7 9 0.81F 0.15 0.45 9

Southern

volcanic belt

42F 1.8 8 23 0.45F 0.07 0.32 22

Central gneiss

domain

37F 3.0 6 4 1.02F 0.31 0.63 4

Thompson Belt 52F 1.2 4 9 0.98F 0.07 0.19 8

hQi is the mean heat flowF one standard error, rQ is the standard

deviation on the heat flow distribution, NQ is the number of sites,

hAi is the mean heat productionF one standard error, rA is the

standard deviation on heat production and NA is the number of sites

for which heat production values are available. Each ‘‘site’’ value is

based on many samples.

ing to both models. These two models differ in one

crucial aspect, the magnitude of basal heat flow

variations. The relationship between horizontal varia-

tions of basal heat flow, mantle heat flow and surface

heat flow is the main theme of this study.

2.3. Variations in heat flow and crustal heat

production in the Canadian Shield

There are now more than 300 reliable heat flow

values in the Canadian Shield (including 150 good heat

flow determinations in Lake Superior) with corre-

sponding heat production values for most of the land

sites. (This data set is available upon request fromhttp://

www.geotop.uqam.ca/geophysique/flux/index.htm).

There are different scales of variations in heat flow

and heat production. For this study, we shall not

consider the scale of the individual intrusions which

bears no consequence for the deep lithosphere. The

new heat flow map of the Shield (Fig. 4) shows that

heat flow contours cross province boundaries but are

related to subprovinces with relatively specific geolog-

ical and petrological characteristics. This is confirmed

by the statistics that show that, on the largest scale, the

mean heat flow does not vary significantly between

Provinces (Table 1). However, on the scale of individ-

ual belts or sub provinces, there is much variability.

This is best documented in the Trans-Hudson Orogen

(Table 2).

Data are too sparse and unevenly distributed to

properly estimate the power spectrum of surface heat

flow, but they are sufficient to assess the scales of

heat flow variability. To this aim, we have calculated

the average heat flow over increasingly larger areas in

the Precambrian (i.e., excluding the Appalachians

which forms a rather narrow belt at the edge of the

continent). We have paved the Shield with squares of

given side length and calculated the average heat flow

for each square. We have then determined how the

mean and standard deviation of these averages vary

with scale. The results are presented in Table 3. We

see that there is almost no difference between the

standard deviation of the individual heat flow values

and that of the 50� 50 km averages (8.9 vs. 8.8 mW

m� 2). The standard deviation decreases to 7.3 mW

m� 2 for areas 250� 250 km and to 4.3 for 500� 500

km. The mean is almost constant, which shows that

sampling is adequate. The very slight decrease in the

mean may be explained by the increased weight of

isolated very low heat flow values (Voisey Bay on the

coast of Labrador and Nielsen Island in Hudson Bay).

In these remote areas, sampling remains insufficient.

This analysis confirms that, in the Archean and

Table 3

Variations of the mean and standard deviation of heat flow values

with length scale of averaging window

l(Q)(mW m� 2)

r(Q)(mW m� 2)

– b 40.6 8.9 316c

50 39.8 8.8 126

250 39.5 7.3 36

500 39.9 4.3 15

a Number of windows.b All data set with individual heat flow values.c Number of heat flow measurements in the Canadian Shield

data set.

Proterozoic, most heat flow variations are of short

wavelengths (V 250 km).

3. Scale of heat flow variations and deep thermal

structure

We shall now examine the differences in the deep

thermal structure beneath the Precambrian provinces

which are compatible with the surface observations.

The magnitude of these differences depends on the

basal heat flow.

3.1. Changes in basal heat flow

In the Precambrian Shield of North America, there

are heat flow variations with an amplitude of about 5

mW m� 2 over wavelengths c 500 km, i.e., with

peak-to-peak variations of 10 mW m� 2. Without

knowledge of crustal heterogeneity, such variations

might be attributed to changes of basal heat flow

alone. We shall now show that this interpretation is

untenable. By downward continuation, the amplitude

of heat flow perturbations increases with depth. Clear-

ly, one condition is that such perturbations must not

exceed the mean heat flow value; otherwise, there

would be areas where temperature decreases with

depth and heat is conducted downward into the con-

vecting mantle. For given wavenumber k and wave-

length k (such that k = 2p/k), the amplitude of the heat

flow perturbation relative to the basal heat flow Qb

increases with depth as:

DqðzÞQb

¼ Dqs

coshkz ð2Þ

This ratio becomes >1 below zmax:

zmax ¼k2p

cosh�1 Qb

� �ð3Þ

for the Canadian Shield, where surface heat flow

values of 22 mW m� 2 have been measured, Qb must

be < 20 mW m� 2. For surface heat flow variations

Dqs = 5 mW m� 2 and with wavelength k = 500 km,

we have zmaxc 160 km. According to this model,

therefore, the lithosphere thickness cannot exceed 160

km beneath the Shield, which is not consistent with

present estimates.

Additional constraints can be obtained by a study of

temperature variations at depth. Changes in heat flow at

the base of the lithosphere affect temperatures at depth

when their wavelength is much larger than lithospheric

thickness L. For a given wavenumber, k, variations of

temperature DTb(k) and heat flow Dqb(k) at the base of

the lithosphere are related by:

DTbðkÞ ¼sinhðkLÞ

kLcoshðkLÞDqbðkÞL

Kð4Þ

where K thermal conductivity. This can be rewritten as

a function of surface heat flow:

DTbðkÞ ¼sinhðkLÞ

DqsðkÞLK

This shows that lateral variations in basal heat flow lead

to large variations in basal temperature when kL>1. For

L = 160 km, kc 500 km and Dqs = 5 mW m� 2,

DTb = 500 K. The overall temperature change at 160

km depth would thus be as large as 1000K, which is not

realistic.

Alternatively, we may apply the same principles to

determine bounds on the mantle heat flow for a given

lithosphere thickness, for example, L= 200 km.We use

the fact that the basal heat flow cannot be larger than 20

mW m� 2 and hence that its variations must smaller

than this value. ForDqb < 20mWm� 2 andDqs = 5mW

m� 2, we have k>600 km. This shows that observed

heat flow variations, which occur on a characteristic

wavelength of only 500 km, cannot be attributed to

changes of basal heat flow alone. If they were so, the

implied temperature variations would be ridiculously

large. For k= 500 km and Dqs = 5 mW m� 2,

DTbc 800 K (a peak-to-peak change of 1600 K!).

Note that this is a very conservative argument, as the

upper bound of 20 mW m� 2 for basal heat flow

variations cannot be achieved in practice.

3.2. Changes of crustal heat generation

One alternative interpretation is that heat flow

variations are due to changes of crustal heat production.

We have calculated the effect of variations in the

average crustal heat production on temperature at the

base of the lithosphere. For a variation of heat produc-

tion DA(k) with wavenumber k uniformly distributed

with depth, and assuming that the heat flow at the base

is constant, the temperature at depth z in the lithospher-

ic mantle varies as follows:

DTðz; kÞ ¼ DAðkÞk2K

coshðkzmÞ � 1

coshðkLÞ cosh½kðz� LÞ;

where zm is crustal thickness. In the long-wavelength

limit, this reduces to the 1-D calculation, which gives

the maximum temperature variation at the Moho:

DTmax ¼DA z2m2 K

: ð7Þ

In these equations, as shown in Appendix A, for a given

variation of surface heat flow DQs, the associated

variation of average crustal heat production, DA,

depends on wavelength. At small scales, it is larger

than the estimate based on a 1-D calculation, i.e.,

DAczQs/zm.

Fig. 6 shows the temperature perturbation from Eq.

(6), normalized to DTmax, for a ratio zm/L= 0.2. It

shows that, except for the very long wavelengths (k/Lc 5), crustal heat generation has little effect on

temperature at the base of the lithosphere. For in-

stance, the high heat flow anomaly in the Thompson

Belt is narrow ( < 100 km) and does not affect mantle

temperatures. On a scale of c 500 km, heat flow

variations have a peak to peak amplitude of c 10

mW m� 2 (across the Abitibi or from the THO to the

Superior), resulting in 80 K temperature differences at

the base of the crust.

3.3. The Appalachians

The same line of argument leads to more dramatic

results for the Appalachians. In the 500–800-km-wide

Appalachian Belt, the surface heat flow is 16 mWm� 2

higher than in the Shield. If this change is attributed to

a change of basal heat flow alone, the basal heat flow

and the lithosphere thickness must vary by more than

25 mW m� 2 and 150 km, respectively, which is

clearly unrealistic. On the other hand, the heat flow

change at the Grenville–Appalachian boundary may

be accounted for by a change in crustal heat generation

[15]. In that case, temperatures in the shallow Appa-

lachian mantle are c 150 K higher than in the Shield

and the Appalachian lithosphere is c 40 km thinner

than in the Shield. Both estimates must be regarded as

lower bounds because they do not include the possible

contribution of basal heat flow variations of long

wavelength (500 km) and small amplitude (i.e.,

Dqmc 2 mW m� 2, Dqbc 7 mW m� 2) which will

be discussed later.

4. The magnitude of basal heat flow variations

4.1. Long-wavelength trends in heat flow and heat

production

The observations and physical arguments on heat

transport in thick lithosphere demonstrate that surface

heat flow variations record mostly changes of crustal

heat production and hence that changes of mantle heat

flow must be small. To proceed further, one must

evaluate how much can be accounted for by changes

of heat production alone. This can only be done by

studying the relationship between large-scale average

values of heat flow and heat production. This is now

possible with the large amount of data available in the

Canadian Shield and the Appalachians.

Heat flow data cannot resolve deep variations of

thermal structure at small scales. We shall argue later

that the uncertainty on heat flow data is about F 2 mW

m� 2. This is due partly to measurement errors and

partly to insufficient knowledge on the relationship

between heat flow and heat production. Using again

Eq. (2) for L= 200 km and Dqs>2 mW m� 2, the

condition that Dqb < 20 mW m� 2 implies that k>300km. Thus, the minimum scale for detecting changes of

basal heat flow is about 300 km. In North America,

well-defined provinces or subprovinces with different

geological structures have typical dimensions of about

500 km. We therefore use five of these with contrasting

heat flow and heat production characteristics. In the

Archean Superior Province, the Abitibi subprovince is

distinguished from the rest of the Province because it

has large volumes of greenstones (altered basic flows

and intrusives). In the Proterozoic Trans-Hudson Oro-

gen, we exclude the Thompson belt at the edge of the

Superior craton from the rest of the Orogen, because it

is a narrow (60 km) belt made of reworked Archean

sediments and metamorphic rocks. In contrast, the rest

of the Orogen is such that the upper crust is made of

juvenile rocks of true Proterozoic age. The Grenville

province includes a large volume of reworked Archean

crust and is a complex collage of small terranes [27].

The Appalachians contain enriched granites and meta-

sediments in the upper crust. On the scale of these five

large provinces (c 500 km), average values of heat

flow and surface heat production are statistically cor-

related (Fig. 5). The data are close to a relationship of

the form

Q ¼ Qo þ HA ð8Þ

where Q and A are province wide averaged heat flow

and heat production. That this relationship takes the

same form as the ‘‘local’’ relationship (1) is fortuitous.

In northern America, the latter is only valid for rela-

tively small-scale variations (typically 10–50 km) of

heat flow and heat production over Appalachian plu-

Fig. 5. Mean heat flow and heat production in various provinces and

subprovinces of the Canadian Shield. Bars show the standard error on

the mean. The ThompsonBelt is a small-scale feature (c 60 km) and

was not included for determining the best fitting line to the data.

tons and does not hold in the older provinces (Gren-

ville, Trans-Hudson Orogen, Superior Province). The

new relationship (Eq. (8)) reflects variations of average

heat flow on a much larger scale (>500 km) and relies

on a very large data set.

The new relationship is clearly inconsistent with

the assumption that surface and mantle heat flows are

proportional to one another. Formally, it is not possi-

ble to rule out variations of mantle heat flow between

the five provinces, but the data require that such

variations get cancelled by opposite variations of

lower crustal heat production. It is hard to explain

how this may be achieved in practice, and the simplest

hypothesis is that the mantle heat flow is approxi-

mately the same beneath the five provinces. For the

Abitibi, Grenville and Appalachian provinces, inde-

pendent geophysical and petrological constraints on

crustal structure show indeed that changes of crustal

heat production account for the observed heat flow

variations [13,14]. A search for all models consistent

with all the available data, including gravity data and

bounds on heat production rates for the various rock

types involved, leads to a range of 11–15 mW m� 2

for the mantle heat flow [28]. The same range is also

valid for the Trans-Hudson Orogen [17]. Variations of

the mantle heat flow may not be exactly zero but must

be smaller than departures from the best-fitting rela-

tionship (Fig. 5), or about 2 mW m� 2. This estimate

is close to the intrinsic uncertainty of heat flow

measurements [6].

4.2. Crustal structure in the North American craton

At a scale of 500 km, the average surface heat flow

is not sensitive to the vertical distribution of heat

production and only records the total amount of heat

generated (Appendix A). Relationship (8) indicates

that variations in the upper crust contribute to most (if

not all) heat flow variations between the five sub-

provinces. This does not imply that there are no

variations in heat production in the lower crust but

that they are smoothed out, which implies that they

occur on short wavelengths. For a mantle heat flow

estimate of 13 mW m� 2 [6] and a crustal thickness of

39 km, the contribution of crustal material below the

surface layer of thickness Hc 9 km is about 20 mW

m� 2, corresponding to an average heat production of

0.67 AW m� 3. This is close to the average value for

Archean crust [11]. On a large scale, therefore, conti-

nental crust in these provinces can be schematically

described as a variable upper layer over a uniform

background made of average Archean crustal material.

Within a single geological province, there may be

long-wavelength heat flow variations. For example,

there is a systematic East–West increase of heat flow

across the Archean Abitibi province on a scale of about

800 km. This heat flow variation is not reflected in

surface heat production and is due to changes in mid to

lower crustal heat production within the Abitibi [13].

Relationship (8) shows that, on average, the difference

between the Abitibi and the rest of the Superior is

explained by a layer of greenstones of thickness Hc 9

km in the Abitibi upper crust. (Heat flow provides an

estimate of the thickness of the greenstones, which has

long been an issue.)

The Thompson Belt, which is also shown in Fig. 5

is a relatively small-scale feature (it is a c 60-km-

wide belt separating the THO and the Superior Prov-

ince) and does not affect temperatures at great litho-

spheric depths. It does not fit the relationship because

the whole crust must be more radiogenic to account

for the large difference in heat flow with the surround-

ing provinces (i.e., the slope of the line between the

THO and the Thompson Belt is 45 km) [18].

4.3. Maximum basal heat flow variations

Relationship (8) shows that only small variations

of mantle heat flow are allowed within the Shield and

between the Shield and the Appalachians. Uncertain-

ties in heat production and heat flow data allow for as

much as c 4 mW m� 2 changes of the non-radiogen-

ic heat flow component (i.e., Dqm = 2 mW m� 2)

[6,28]. This is the magnitude of departures from the

best-fitting relationship in Fig. 5. All indications are

that, in Canada, corrections for the last glaciation are

small [29] and that they should not add to the

uncertainty on mantle heat flow.

By definition, such ‘‘hidden’’ heat flow variations

may or may not be correlated with the surface

geology. There is a limit on wavelength, as discussed

above. For illustration purposes, we take L= 150 km,

corresponding to the smallest estimate of lithosphere

thickness in this part of the north American continent.

For k < 300 km, there is no useful constraint on

variations of basal heat flow. For k>500 km, Dqs < 2

mW m� 2 implies that Dqb < 7 mW m� 2. For an

average thermal conductivity of 3 W m� 1 K� 1, such

a variation in basal heat flow would imply DTb < 200

K, i.e., temperature differences at 150 km depth which

may be as large as 400 K. At very large scale (>1200

km), horizontal diffusion can be neglected and hence

we have Dqb < 2 mW m� 2 and DTb < 100 K. The

range of mantle heat flow compatible with our data in

the Canadian Shield and the Appalachians, 11–15

mW m� 2 [13,17] is wide enough to permit such

variations.

5. Discussion

From the above considerations, horizontal differ-

ences in temperature within the lithosphere of the

Canadian Shield, which includes provinces of Ar-

chean and Proterozoic age, may exceed 200 K

depending on horizontal scale. For example, varia-

tions of basal heat flow at a scale of 500 km could

be as large as F 7 mW m� 2, implying basal

temperature differences of up to 400 K. We now

briefly evaluate these results in the light of recent

seismological studies of lithospheric structure in the

Canadian Shield and constraints from geothermobar-

ometry on mantle xenoliths.

5.1. Seismic velocities in the Precambrian lithospher-

ic mantle

Seismic studies have shown different scales of

velocity variations in the Precambrian lithospheric

mantle of North America. P wave velocity can vary

locally by >2% between 150 and 250 km depth on a

scale of 100–200 km in association with hot-spot

tracks and kimberlite fields [30,31]. Such small-scale

features cannot affect surface heat flow. At the scale

of c 500 km, which is compatible with our heat flow

averages, inversion of surface wave data suggests

lateral variations in shear wave velocity between 80

and 150 km depth [32,4]. At 150 km, Vs varies

between about 4.65 and 4.85 km s� 1 at the latitudes

of the heat flow survey [32].

Both composition and temperature affect seismic

wave velocity [4,33]. One difficulty is to properly

account for anelastic effects [34,35]. The average

compositional difference between Archean and Prote-

rozoic lithospheric mantle can account for about 1% of

Vs variations [4]. At 150 km depth, the 4% Vs variations

indicated by the surface wave data can be accounted for

by temperature differences between 100 and 200 K

depending on the magnitude of anelastic effects [4].

This is consistent with the heat flow constraints. At a

depth of 150 km, given their characteristic wavelength

of c 500 km, the seismically determined temperature

differences cannot be due to variations of crustal heat

production only (Fig. 6) and require small variations of

basal heat flow (c 2 mW m� 2).

5.2. Mantle heat flow variations

Our analysis shows that mantle heat flux variations

are small and that heat flow data allow poor resolution

on lateral changes of heat flux at the base of the

lithosphere. This is due partly to the fact that the basal

heat flux is small, implying that even a small absolute

uncertainty represents a significant fraction of the

large-scale average. The analysis, however, leads to

the important observation that changes of basal heat

flux may not be correlated with surface geology.

Using measurements of phase and group velocities

of fundamental mode surface waves and a ‘‘diffraction

tomography’’ method, Shapiro et al. [32] have deter-

mined the ensemble of shear velocity models which

satisfy the data down to a depth of 400 km with a

horizontal resolution of about 500 km. They used heat

Fig. 6. Mantle temperature perturbations due to horizontal variation

in crustal heat production with wavelength k/L. The temperature is

normalized to the one dimensional temperature change beneath the

crust (DAzm2/2K).

flow data with two end models of crustal heat produc-

tion to obtain the range of Moho temperature and

mantle heat flow consistent with the surface observa-

tions. Shear velocity values are converted to tempera-

ture using the method and parameters of [4] and only

models that fall within the permissible range are

retained. The best-fitting temperature gradients in the

mantle part of the lithosphere are calculated and con-

verted to heat flux using an average conductivity value

of 3 W m� 1 K� 1. The standard deviation of heat flow

values increases with increasing heat flow and reaches

a maximum of 2.5 mW m� 2. One problem with the

procedure is that no allowance is made for horizontal

diffusion, implying in particular that there can be no

difference between mantle and basal heat flows.

Shapiro et al. [32] find that mantle heat flow

variations are not well correlated with the distribution

of geological provinces and can be as large as 5 mW

m� 2 within a single province, such as the Superior

for example. The average heat flow values vary

significantly across the Canadian Shield and the

Appalachians, ranging from 11 mW m� 2 in the center

of the Shield to 24 mW m� 2 beneath part of the

Appalachians. These results are sensitive to the start-

ing assumption of a constant temperature gradient in

the mantle part of the lithosphere. Nevertheless, they

are consistent with the constraints from heat flow

data: at the wavelength of 500 km of relevance here,

our analysis shows that the range of basal heat flow

values may be as large as 14 mW m� 2 (i.e., Dqbc 7

mW m� 2).

5.3. Geothermobarometry from mantle xenoliths

The resolution of surface heat flow data decreases

with depth in the lithosphere. Xenoliths studies com-

plement the heat flow and are useful to reduce the

uncertainties on the thermal state of the deep litho-

sphere. In the Canadian Shield, temperatures deduced

from mineral assemblages in mantle xenoliths are

systematically lower in the Slave than in the Superior

Province [7]. Temperature gradients are comparable,

which is consistent with weak variations of heat

supply at the base of the lithosphere. Only two heat

flow values are available in the Slave province, which

prevents a quantitative analysis. We note, however,

that, on average, the temperature difference between

the Superior and Slave xenoliths is 100 K, which is

consistent with the variations documented elsewhere

in the Shield.

6. Conclusions

Two end models for the basal heat flow (Qb

proportional to surface heat flow, Qb constant) yield

distinctly different geotherms. Variations in litho-

spheric thickness are less when basal heat flow stays

within a small range than when it is proportional to

surface heat flow.

The spatial scale of the variations cannot be ig-

nored. Our analysis shows that, for reliable models of

deep lithospheric structure, heat flow averages must

be made on a scale of at least 500 km. Variations of

local crustal heat production occur on smaller scales

and must be properly accounted for, which requires a

large number of heat flow and heat production data.

Such dense data coverage is available in very few

areas. In the Canadian Shield, large surface heat flow

variations appear at the scale of the subprovince

( < 500 km). Variations in mantle heat flow can not

account for these variations if the lithosphere is thick

(c 250 km).

Changes in crustal heat production are sufficient to

account for most of the variability in surface heat

flow, with mantle heat flow ranging 11–15 mW m� 2.

The magnitude of deep temperature variations is

poorly constrained by heat flow data and strongly

depends on horizontal scale. At 150 km depth, tem-

perature may vary by as much as 400 K.

Acknowledgements

John Sass and an anonymous reviewer provided

critical reviews and comments which improved the

manuscript. This research was supported by NSERC

(Canada) and INSU (CNRS) (France). [VC]

Appendix A . Scale of heat flow and heat

generation variations

The scale of the crustal component of heat flow

variations is related to that of heat production. The

theory shows that surface heat flow is a much smoother

field than heat production. This is easily demonstrated

by considering the power spectra of both fields. For

heat sources restricted to the crust, the Fourier spectra

of heat flow and heat production are related by [36]:

Qð!kÞ ¼Z zm

Að!k; zÞexpð�kzÞdz ðA1Þ

where!k ¼ ðkx; kyÞ and k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2y :

qFor uniform vertical distribution of heat sources in

the crust, we find the relation for the power spectra of

heat production PA and heat flow PQ:

PQð!kÞ ¼ PAð

k2ð1� expð�kzmÞÞ2 ðA2Þ

which for kzm < 1 (k/zm>2p) yields PQ(k)~PA(k)zm2.

This implies that the change in crustal composition in

the Abitibi subprovince is well recorded in the heat

flow data.

The assumption that heat source variations are

coherent over the entire crustal column is unrealistic

but maybe useful to model variations in heat produc-

tion at the scale of a subprovince. At a smaller scale,

we must also consider the vertical variations in heat

production. Studies of variations of heat production

with depth in the KTB borehole suggest that the 3-D

power spectrum of heat sources follows a power law

PAðkx; ky; kzÞ ¼ Cðk2x þ k2y þ k2z Þ�b=2 ðA3Þ

with bc 3.7 [11]. Using

Að!k; zÞ ¼ 1

�lAð!k; kzÞexpð�ikzzÞdkz ðA4Þ

we find:

Qð!kÞ ¼ 1

Að!k ; kzÞk þ ikz

ð1� expðkzmÞ

� expð�ikzzmÞÞdkz: ðA5Þ

Straightforward but tedious calculations show that for

k/zm>1:

PQð!kÞ~PAð

!k; 0Þz2m; ðA6Þ

which generalizes the results above when the heat

sources vary with depth. At very long wavelengths,

the surface heat flow variations do not depend on how

the heat sources are distributed vertically but only on

the vertically averaged of heat production. For short

wavelengths, the power spectrum of heat flow depends

on the power spectrum of the heat sources. For a power

law spectrum of heat sources, the distribution of heat

flow is smoother than heat production [11].

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Variations of surface heat flow and lithospheric thermal structure beneath the North American craton

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