Lecture 16

Heat

Pollack et al. (1993) Source: Peter Bird

3090 STEIN AND STEIN: CONSTRAINTS ON HYDROTHERMAL HEAT FLOW

250] • • • ] 250] • • •-, 200 t Anderson and Skilbeck [1981] t 200 Pacific Ocean _

• 150 I•••, [] Aflanfic x Galapagos ]J 150 I+\\• PSM o ,n,on . , _

I I + + O/ I ! I I O/ I I I 0 •0 100 1 s0 o so 1 oo 1 so

! , , I , , , "• /•X • GDH1 / I X• • GDH1

___ , ___ -

I•- •,••• - [ I• o o ••o o Oooo o _o o O O O

0/ I I / o I I I 0 50 100 150 0 50 100

AGE (Ma) AGE (Ma) Fig. 6. Observed and predicted heat flow versus age for the ocean basins. (top left) Summary of results from earlier study [Anderson and Skilbeck, 1981]. In their figure, the predicted curve is schematic. Other panels show results for our larger data set, compared to predictions of reference models GDH1 and PSM [Parsons and Sclater, 1977]. We find that the seal- ing age for the Pacific is comparable to that for the other ocean basins, in contrast to the earlier result showing a much younger sealing age.

150

The linear fitting approach characterizes simply the presum- ably complex age and site dependence of the heat flow fraction. The scattered data appear inadequate to characterize the age dependence beyond a linear trend. The heat flow fraction might be expected to increase most rapidly for the younger (0-4 Ma) ages and more slowly near the sealing age, as perhaps suggested by the data.

Our Pacific sealing age differs from that of Anderson and Skilbeck [1981] because their data were compared to the predictions of a thermal model with an 80-km-thick lithosphere, a 1200øC basal temperature, and a thermal conductivity of 6.15 x10 -3 cal *C -I cm -I s -l' This model predicts too low heat flow in older lithosphere, which was not apparent because they had only data for 0-50 Ma lithosphere [Anderson and Hobart, 1976], whereas our data extend to older ages. As a result, their model at 20 Ma predicts heat flow -29 mW in -2 lower than GDH1 and hence implies a younger sealing age. Thus global hydrothermal heat flux estimates using this young sealing age [Wolery and Sleep, 1976, 1988; Anderson eta/., 1977; Sleep and Wolery, 1978] are significantly lower than ours.

It is not clear how much significance to ascribe to the scatter in the heat flow fraction for ages exceeding the nominal sealing

age. Some of the scatter presumably reflects the noise in the data, both from measurement uncertainties and topographic effects [e.g., Lachenbruch, 1968; Noel, 1984]. There also appear to be sites in relatively old lithosphere where heat flow studies indicate the water circulation occurs, such as the 80 Ma site dis- cussed by Ernbley et al. [1983]. The limited number of sufficiently dense heat flow surveys on old lithosphere suggest that such water flow is not common. Considerably more such surveys would be required for any meaningful generalization about water flow. The heat flow data presently available indicate only that any water flow in older lithosphere is not transporting significant amounts of heat through the seafloor.

MECHANISM OF SEALING

The mechanisms which cause the heat flow discrepancy, and hence presumably hydrothermal circulation, to decrease with lithospheric age have long been of interest. Two primary effects, both of which may be present, are thought to contribute. Ander- son and Hobart [ 1976] proposed that the heat flow fraction was low for young, unsedimented, crust and rose to about one once the basement rock was covered by 150-200 m of sediment. In

3090 STEIN AND STEIN: CONSTRAINTS ON HYDROTHERMAL HEAT FLOW

250] • • • ] 250] • • •-, 200 t Anderson and Skilbeck [1981] t 200 Pacific Ocean _

• 150 I•••, [] Aflanfic x Galapagos ]J 150 I+\\• PSM o ,n,on . , _

I I + + O/ I ! I I O/ I I I 0 •0 100 1 s0 o so 1 oo 1 so

! , , I , , , "• /•X • GDH1 / I X• • GDH1

___ , ___ -

I•- •,••• - [ I• o o ••o o Oooo o _o o O O O

0/ I I / o I I I 0 50 100 150 0 50 100

AGE (Ma) AGE (Ma) Fig. 6. Observed and predicted heat flow versus age for the ocean basins. (top left) Summary of results from earlier study [Anderson and Skilbeck, 1981]. In their figure, the predicted curve is schematic. Other panels show results for our larger data set, compared to predictions of reference models GDH1 and PSM [Parsons and Sclater, 1977]. We find that the seal- ing age for the Pacific is comparable to that for the other ocean basins, in contrast to the earlier result showing a much younger sealing age.

150

The linear fitting approach characterizes simply the presum- ably complex age and site dependence of the heat flow fraction. The scattered data appear inadequate to characterize the age dependence beyond a linear trend. The heat flow fraction might be expected to increase most rapidly for the younger (0-4 Ma) ages and more slowly near the sealing age, as perhaps suggested by the data.

Our Pacific sealing age differs from that of Anderson and Skilbeck [1981] because their data were compared to the predictions of a thermal model with an 80-km-thick lithosphere, a 1200øC basal temperature, and a thermal conductivity of 6.15 x10 -3 cal *C -I cm -I s -l' This model predicts too low heat flow in older lithosphere, which was not apparent because they had only data for 0-50 Ma lithosphere [Anderson and Hobart, 1976], whereas our data extend to older ages. As a result, their model at 20 Ma predicts heat flow -29 mW in -2 lower than GDH1 and hence implies a younger sealing age. Thus global hydrothermal heat flux estimates using this young sealing age [Wolery and Sleep, 1976, 1988; Anderson eta/., 1977; Sleep and Wolery, 1978] are significantly lower than ours.

It is not clear how much significance to ascribe to the scatter in the heat flow fraction for ages exceeding the nominal sealing

age. Some of the scatter presumably reflects the noise in the data, both from measurement uncertainties and topographic effects [e.g., Lachenbruch, 1968; Noel, 1984]. There also appear to be sites in relatively old lithosphere where heat flow studies indicate the water circulation occurs, such as the 80 Ma site dis- cussed by Ernbley et al. [1983]. The limited number of sufficiently dense heat flow surveys on old lithosphere suggest that such water flow is not common. Considerably more such surveys would be required for any meaningful generalization about water flow. The heat flow data presently available indicate only that any water flow in older lithosphere is not transporting significant amounts of heat through the seafloor.

MECHANISM OF SEALING

The mechanisms which cause the heat flow discrepancy, and hence presumably hydrothermal circulation, to decrease with lithospheric age have long been of interest. Two primary effects, both of which may be present, are thought to contribute. Ander- son and Hobart [ 1976] proposed that the heat flow fraction was low for young, unsedimented, crust and rose to about one once the basement rock was covered by 150-200 m of sediment. In

3090 STEIN AND STEIN: CONSTRAINTS ON HYDROTHERMAL HEAT FLOW

250] • • • ] 250] • • •-, 200 t Anderson and Skilbeck [1981] t 200 Pacific Ocean _

• 150 I•••, [] Aflanfic x Galapagos ]J 150 I+\\• PSM o ,n,on . , _

I I + + O/ I ! I I O/ I I I 0 •0 100 1 s0 o so 1 oo 1 so

! , , I , , , "• /•X • GDH1 / I X• • GDH1

___ , ___ -

I•- •,••• - [ I• o o ••o o Oooo o _o o O O O

0/ I I / o I I I 0 50 100 150 0 50 100

AGE (Ma) AGE (Ma) Fig. 6. Observed and predicted heat flow versus age for the ocean basins. (top left) Summary of results from earlier study [Anderson and Skilbeck, 1981]. In their figure, the predicted curve is schematic. Other panels show results for our larger data set, compared to predictions of reference models GDH1 and PSM [Parsons and Sclater, 1977]. We find that the seal- ing age for the Pacific is comparable to that for the other ocean basins, in contrast to the earlier result showing a much younger sealing age.

150

The linear fitting approach characterizes simply the presum- ably complex age and site dependence of the heat flow fraction. The scattered data appear inadequate to characterize the age dependence beyond a linear trend. The heat flow fraction might be expected to increase most rapidly for the younger (0-4 Ma) ages and more slowly near the sealing age, as perhaps suggested by the data.

Our Pacific sealing age differs from that of Anderson and Skilbeck [1981] because their data were compared to the predictions of a thermal model with an 80-km-thick lithosphere, a 1200øC basal temperature, and a thermal conductivity of 6.15 x10 -3 cal *C -I cm -I s -l' This model predicts too low heat flow in older lithosphere, which was not apparent because they had only data for 0-50 Ma lithosphere [Anderson and Hobart, 1976], whereas our data extend to older ages. As a result, their model at 20 Ma predicts heat flow -29 mW in -2 lower than GDH1 and hence implies a younger sealing age. Thus global hydrothermal heat flux estimates using this young sealing age [Wolery and Sleep, 1976, 1988; Anderson eta/., 1977; Sleep and Wolery, 1978] are significantly lower than ours.

It is not clear how much significance to ascribe to the scatter in the heat flow fraction for ages exceeding the nominal sealing

age. Some of the scatter presumably reflects the noise in the data, both from measurement uncertainties and topographic effects [e.g., Lachenbruch, 1968; Noel, 1984]. There also appear to be sites in relatively old lithosphere where heat flow studies indicate the water circulation occurs, such as the 80 Ma site dis- cussed by Ernbley et al. [1983]. The limited number of sufficiently dense heat flow surveys on old lithosphere suggest that such water flow is not common. Considerably more such surveys would be required for any meaningful generalization about water flow. The heat flow data presently available indicate only that any water flow in older lithosphere is not transporting significant amounts of heat through the seafloor.

MECHANISM OF SEALING

The mechanisms which cause the heat flow discrepancy, and hence presumably hydrothermal circulation, to decrease with lithospheric age have long been of interest. Two primary effects, both of which may be present, are thought to contribute. Ander- son and Hobart [ 1976] proposed that the heat flow fraction was low for young, unsedimented, crust and rose to about one once the basement rock was covered by 150-200 m of sediment. In

5

EPS 122: Lecture 19 – Geotherms

The ocean basins Depth distribution

Depth distribution is related to age

ie the time available for cooling

Good approximation to observation out to ~70 Ma

squares: North Atlantic circles: North Pacific

EPS 122: Lecture 19 – Geotherms

Depth and heat flow – observations

Stei

n &

Ste

in, 1

994

…works best till ~70 Ma

for greater ages depth decreases more slowly

Depth

Heat flux

for greater ages Q decreases more slowly

initially…

5

EPS 122: Lecture 19 – Geotherms

The ocean basins Depth distribution

Depth distribution is related to age

ie the time available for cooling

Good approximation to observation out to ~70 Ma

squares: North Atlantic circles: North Pacific

EPS 122: Lecture 19 – Geotherms

Depth and heat flow – observations

Stei

n &

Ste

in, 1

994

…works best till ~70 Ma

for greater ages depth decreases more slowly

Depth

Heat flux

for greater ages Q decreases more slowly

initially…

(i)  Depth increases more slowly with increasing age (ii)  Heat flow decreases with increasing age

1.  Half space cooling model

2.  Plate model

∂T

∂t

=k

rC

p

—2T +

A

rC

p

∂T

∂t

=k

rC

p

—2T +

A

rC

p

�u ·—T

3

Half space cooling model:

Assumptions: (i)  temperature is in equilibrium (ii)  horizontal advection >>

horizontal conduction

6

EPS 122: Lecture 19 – Geotherms

A simple half-space model T

= T

a

T = 0

ridge

3D convection and advection equation

Assume: •� temperature field is in equilibrium •� advection of heat horizontally is

greater than conduction

Also, t = x/u i.e. distance and time related by the spreading rate

z

x

We have already seen the solution….

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Surface heat flow

…differentiate T-gradient

The observed heat flux was:

� this simple model provides the t1/2 relation

6

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

3D convection and advection equation

Assume: •� temperature field is in equilibrium •� advection of heat horizontally is

greater than conduction

Also, t = x/u i.e. distance and time related by the spreading rate

z

x

We have already seen the solution….

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Surface heat flow

…differentiate T-gradient

The observed heat flux was:

� this simple model provides the t1/2 relation

6

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

3D convection and advection equation

Assume: •� temperature field is in equilibrium •� advection of heat horizontally is

greater than conduction

Also, t = x/u i.e. distance and time related by the spreading rate

z

x

We have already seen the solution….

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Surface heat flow

…differentiate T-gradient

The observed heat flux was:

� this simple model provides the t1/2 relation

6

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

3D convection and advection equation

Assume: •� temperature field is in equilibrium •� advection of heat horizontally is

greater than conduction

Also, t = x/u i.e. distance and time related by the spreading rate

z

x

We have already seen the solution….

EPS 122: Lecture 19 – Geotherms

A simple half-space model T

= T

a

T = 0

ridge

z

x Temperature gradient

Surface heat flow

…differentiate T-gradient

The observed heat flux was:

� this simple model provides the t1/2 relation

7

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Estimate the lithospheric thickness…

T at base of lithosphere: 1100 °C and Ta = 1300 °C

look up inverse error function

if � = 10-6 m2 s-1

L in km, t in Ma � 10 Ma � L = 35 km � 80 Ma � L = 98 km reasonable?

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Column of lithosphere at the ridge

=

Rearrange

Need �(z) …density as a function of T

coefficient of thermal expansion

…and T as a function of age

Substitute…

Ta = 1300oC T = 1100oC

7

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Estimate the lithospheric thickness…

T at base of lithosphere: 1100 °C and Ta = 1300 °C

look up inverse error function

if � = 10-6 m2 s-1

L in km, t in Ma � 10 Ma � L = 35 km � 80 Ma � L = 98 km reasonable?

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Column of lithosphere at the ridge

=

Rearrange

Need �(z) …density as a function of T

coefficient of thermal expansion

…and T as a function of age

Substitute…

7

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Estimate the lithospheric thickness…

T at base of lithosphere: 1100 °C and Ta = 1300 °C

look up inverse error function

if � = 10-6 m2 s-1

L in km, t in Ma � 10 Ma � L = 35 km � 80 Ma � L = 98 km reasonable?

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Column of lithosphere at the ridge

=

Rearrange

Need �(z) …density as a function of T

coefficient of thermal expansion

…and T as a function of age

Substitute…

L in kms and t in My

What about depth? ……Apply principle of isostacy

7

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Estimate the lithospheric thickness…

T at base of lithosphere: 1100 °C and Ta = 1300 °C

look up inverse error function

if � = 10-6 m2 s-1

L in km, t in Ma � 10 Ma � L = 35 km � 80 Ma � L = 98 km reasonable?

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Column of lithosphere at the ridge

=

Rearrange

Need �(z) …density as a function of T

coefficient of thermal expansion

…and T as a function of age

Substitute…

7

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Estimate the lithospheric thickness…

T at base of lithosphere: 1100 °C and Ta = 1300 °C

look up inverse error function

if � = 10-6 m2 s-1

L in km, t in Ma � 10 Ma � L = 35 km � 80 Ma � L = 98 km reasonable?

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Column of lithosphere at the ridge

=

Rearrange

Need �(z) …density as a function of T

coefficient of thermal expansion

…and T as a function of age

Substitute…

7

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Estimate the lithospheric thickness…

T at base of lithosphere: 1100 °C and Ta = 1300 °C

look up inverse error function

if � = 10-6 m2 s-1

L in km, t in Ma � 10 Ma � L = 35 km � 80 Ma � L = 98 km reasonable?

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Column of lithosphere at the ridge

=

Rearrange

Need �(z) …density as a function of T

coefficient of thermal expansion

…and T as a function of age

Substitute…

8

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Approximate L � �

Rearrange…

Appropriate values: �w = 1.0 x 103 km m-3

�a = 3.3 x 103 km m-3

� = 3 x 10-5 °C-1

� = 10-6 m2 s-1

Ta = 1300 °C t in Ma and d in km

Observed…

The simple half-space cooling model matches ocean depths out to ~70 Ma i.e. lithosphere cools, contracts and subsides

EPS 122: Lecture 19 – Geotherms

The “plate” model

T =

Ta

T = 0

ridge

z

x

Simple half-space model

T =

Ta

T = 0

ridge

z

x

T = Ta

The lithosphere has a fixed thickness at the ridge and cools with time

The asthenosphere below is constant temperature

…asymptotic values of Q, depth etc. …cools and thickens for ever

ρa = 3300 kg/m3

ρw = 1000 kg/m3

α = 3×10-5 oC-1

κ = 10-6 m2s-1

Ta = 1300 oC dr = 2.5 km

8

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Approximate L � �

Rearrange…

Appropriate values: �w = 1.0 x 103 km m-3

�a = 3.3 x 103 km m-3

� = 3 x 10-5 °C-1

� = 10-6 m2 s-1

Ta = 1300 °C t in Ma and d in km

Observed…

The simple half-space cooling model matches ocean depths out to ~70 Ma i.e. lithosphere cools, contracts and subsides

EPS 122: Lecture 19 – Geotherms

The “plate” model

T =

Ta

T = 0

ridge

z

x

Simple half-space model

T =

Ta

T = 0

ridge

z

x

T = Ta

The lithosphere has a fixed thickness at the ridge and cools with time

The asthenosphere below is constant temperature

…asymptotic values of Q, depth etc. …cools and thickens for ever

10

EPS 122: Lecture 19 – Geotherms

A hybrid?

“Plate” model fits depth and Q best

but there is other geophysical evidence for a thickening lithosphere

•� increasing elastic thickness •� increasing depth to low velocity asthenosphere

� thermal boundary layer with small-scale convection

crust

litho

sphe

re

crust

litho

sphe

re

Continents

•� thicker crust

•� similar lithosphere

(cratons?)

EPS 122: Lecture 19 – Geotherms

The mantle geotherm convection rather than conduction

� more rapid heat transfer

Adiabatic temperature gradient

Raise a parcel of rock…

If constant entropy: lower P � expands larger volume � reduced T

This is an adiabatic gradient

Convecting system � close to adiabatic

7

EPS 122: Lecture 19 – Geotherms

A simple half-space model

T =

Ta

T = 0

ridge

z

x Temperature gradient

Estimate the lithospheric thickness…

T at base of lithosphere: 1100 °C and Ta = 1300 °C

look up inverse error function

if � = 10-6 m2 s-1

L in km, t in Ma � 10 Ma � L = 35 km � 80 Ma � L = 98 km reasonable?

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Column of lithosphere at the ridge

=

Rearrange

Need �(z) …density as a function of T

coefficient of thermal expansion

…and T as a function of age

Substitute…

Plate cooling model:

8

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Approximate L � �

Rearrange…

Appropriate values: �w = 1.0 x 103 km m-3

�a = 3.3 x 103 km m-3

� = 3 x 10-5 °C-1

� = 10-6 m2 s-1

Ta = 1300 °C t in Ma and d in km

Observed…

The simple half-space cooling model matches ocean depths out to ~70 Ma i.e. lithosphere cools, contracts and subsides

EPS 122: Lecture 19 – Geotherms

The “plate” model

T =

Ta

T = 0

ridge

z

x

Simple half-space model

T =

Ta

T = 0

ridge

z

x

T = Ta

The lithosphere has a fixed thickness at the ridge and cools with time

The asthenosphere below is constant temperature

…asymptotic values of Q, depth etc. …cools and thickens for ever

8

EPS 122: Lecture 19 – Geotherms

A simple half-space model Ocean depth …apply isostasy

Approximate L � �

Rearrange…

Appropriate values: �w = 1.0 x 103 km m-3

�a = 3.3 x 103 km m-3

� = 3 x 10-5 °C-1

� = 10-6 m2 s-1

Ta = 1300 °C t in Ma and d in km

Observed…

The simple half-space cooling model matches ocean depths out to ~70 Ma i.e. lithosphere cools, contracts and subsides

EPS 122: Lecture 19 – Geotherms

The “plate” model

T =

Ta

T = 0

ridge

z

x

Simple half-space model

T =

Ta

T = 0

ridge

z

x

T = Ta

The lithosphere has a fixed thickness at the ridge and cools with time

The asthenosphere below is constant temperature

…asymptotic values of Q, depth etc. …cools and thickens for ever ...cools and thickens for ever ...asymptotic values of Q, depth

∂T

∂z

=∂∂z

T0er f

✓z

2p

kt

◆�=

T0ppkt

e

�z

2/4kt

∂T

∂t

=k

rC

p

1r

2∂∂r

✓r

2 ∂T

∂r

◆+

A

rC

p

T =A

6k

(a2 � r

2)

�k

dT

dr

=Ar

3

T =�Q

b

b

2

k

✓1a

� 1r

◆

0 =k

r

2∂∂r

✓r

2 ∂T

∂r

◆+A

T = T

a

✓z

L

+•

Ân=1

2np

sin

✓npz

L

◆exp

✓�n

2p2kt

L

2

◆◆

4

9

EPS 122: Lecture 19 – Geotherms

Depth and heat flow – observations

Stei

n &

Ste

in, 1

994

Which model(s) fit the data? HS – Half-space model GDH1 – plate model PSM – plate model

All fit the heat flow data (within error)

The GDH1 “plate” model does a better job of fitting the depth data (which is better constrained)

EPS 122: Lecture 19 – Geotherms

The ocean basins Depth distribution

Depth distribution is related to age

ie the time available for cooling

Good approximation to observation out to ~70 Ma

squares: North Atlantic circles: North Pacific

Plate model:

There is a limit to the lithospheric thickness available for cooling

10

EPS 122: Lecture 19 – Geotherms

A hybrid?

“Plate” model fits depth and Q best

but there is other geophysical evidence for a thickening lithosphere

•� increasing elastic thickness •� increasing depth to low velocity asthenosphere

� thermal boundary layer with small-scale convection

crust

litho

sphe

re

crust

litho

sphe

re

Continents

•� thicker crust

•� similar lithosphere

(cratons?)

EPS 122: Lecture 19 – Geotherms

The mantle geotherm convection rather than conduction

� more rapid heat transfer

Adiabatic temperature gradient

Raise a parcel of rock…

If constant entropy: lower P � expands larger volume � reduced T

This is an adiabatic gradient

Convecting system � close to adiabatic

Small scale convection

Source: Peter Bird

globally and also are corrected for convective effects.

Fig 3.6 Two different ways of averaging continental heat low data. The first (a) is by the age ofthe last major tectonic event and (b) is by the radiometric date of crustal age. The width of eachbox represents one standard deviation

Fig 3.7 Locations of heat flow measurements in both oceans and continents

87

Continental Heat Flow

Sclater et al., 1981