Douce A.P. Thermodynamics of the Earth and Planets

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168 Energy transfer processes in planetary bodies

2000 2100 2200 2300

100

200

300

400

Heat flux - mW m

–2

2000 2100 2200 2300

Lithospheric thickness - km

1900 2000 2100 2200 2300

1300

1400

1500

1600

1700

Temperature at

base of lithosphere -

100

Convection velocity - cm yr

–1

Characteristic mantle temperature - K

Reference viscosity - Pa s

Present Archaean?

Fig. 3.15 Parametrization of terrestrial mantle convection, as a function of characteristic mantle temperature (calculated with

equations (3.89)–(3.92)). The model assumes a reference mantle viscosity of 10

Pa s at 1950 K and an activation

energy for mantle viscous ﬂow of 300 kJ mol

−1

(see Box 3.3). Other parameters are: α = 10

−5

−1

, ρ =3500 kg

−1

, c

=1kJK

−1

, κ = 10

−6

−1

, k =3Wm

−1

, D =3000 km, T

surface

=300 K, g =9.8ms

−2

Conditions for the present-day mantle correspond to a characteristic mantle temperature of 1950 K. The range in

possible Archaean conditions arises from assumed mantle temperatures of 2100–2300 K.

169 3.8 Parametrization of convection in planetary interiors

today and cycles of continental assembly and break up, which since the late Proterozoic

have had a characteristic time scale of a few hundred million years, may have operated on

time scales of the order of 10

years. The fast rate of crustal growth that characterizes the

Archaean may reflect, at least in part, the higher rates of mantle convection and greater

lengths of plate margins.

3.8.2 Convection with a stagnant lid

An implicit assumption in the derivation of the scaling laws for convection with moving

plates is that the viscosity of the fluid does not vary strongly with temperature. This is what

allows the thermal boundary layer to move with the same velocity as the underlying fluid.

Of course this is not true for mantle rocks nor for any other material, as viscosity is always

a strong function of temperature (Box 3.3). This temperature dependency is what makes

the rigid lithosphere physically distinct from the asthenosphere. Convection with moving

plates is possible on Earth because the lithosphere is broken into fragments that can move

relative to one another, and bend to some extent, by a combination of localized brittle and

ductile deformation mechanisms. The result of this is that the apparent global viscosity of

the thermal boundary layer (the lithosphere) is comparable to that of the asthenosphere.

Locally, however, the lithosphere preserves its rigid behavior that makes diffusion the

only possible heat transport mechanism. Convection with moving plates is a rather special

case, as one might surmise from the uniqueness of plate tectonics in the present-day Solar

System. A more common situation in solid planets is that of convection with a stagnant lid

(Fig. 3.13d).

In order to parametrize this style of convection it is necessary to include the viscosity–

temperature relationship of the fluid (this does not show up in the equations for moving

plate convection because we implicitly assume that viscosity is constant). Let T

be the

temperature of the convecting fluid, T

the temperature at the top of the rheological boundary

layer (Fig. 3.16) and T

the temperature at the top of the thermal boundary layer (see also

Fig. 3.13d). Convection with moving plates is driven by the full temperature difference

T =T

−T

, but if a stagnant lid forms then convection is driven only by the temperature

difference across the rheological boundary layer, T

= T

−T

. This is so because this

is the only volume of cool rock that is entrained in the convective flow, and the negative

buoyancy of the rheological boundary layer is what drives convection. We can then define

the rheological boundary layer as that portion of the thermal boundary layer in which the

viscosity, µ

, is of the same order of magnitude as the viscosity of the convecting fluid, µ

From equation (3.3.3):

=µ

−γ

(

−T

)

=µ

γT

. (3.94)

The condition µ

∼ µ

is satisfied if γT

= 1, where γ is defined by equation (3.3.4),

Box 3.3. The temperature difference that drives convection is therefore given by:

T

. (3.95)

The effective Rayleigh number of the fluid corresponds to this temperature difference. We

will symbolize it by Ra

, to distinguish it from Ra (equation (3.62)), which is calculated

170 Energy transfer processes in planetary bodies

=uy =0

Fig. 3.16

Thermal boundary layer in stagnant lid convection. The diffusive lid has thickness δ, but only a portion of it, the

rheological boundary layer of thickness δ

, participates in convective ﬂow, the remainder is immobile. Vertical heat

transport across the rheological boundary layer is by diffusion (u

=0), so that the temperature gradient is constant

across the entire boundary layer (assuming no heat generation). Convection is driven by the temperature drop across

the rheological boundary layer, T

–T

with the full temperature difference, T. Thus:

gαρ

T

µκ

gαρ

µκγ

=Ra

(

γT

)

−1

. (3.96)

With the Rayleigh number defined in this way the problem of stagnant lid convection

becomes the same as that of convection with moving plates, as long as we ignore the

portion of the thermal boundary layer that lies beyond δ

(Fig. 3.16) – we will return to

this shortly. Convection is driven by the rheological boundary layer becoming negatively

buoyant and sinking. We can therefore think of the rheological boundary layer as being

made up of plates that move under the stagnant upper portion of the thermal boundary

layer, suggesting that we modify equations (3.89), (3.91) and (3.92) as follows:

q ∼ 0.25Ra

1/3



T



=0.25Ra

1/3



γD



(3.97)

∼4DRa

−1/3

(3.98)

u ∼0.25

2/3

. (3.99)

We have to be careful with the Nusselt number, however, as it is defined on the basis of the

full temperature difference, T and not T

. From equation (3.65):

Nu=

T

=0.25Ra

1/3

(

γT

)

−1

(3.100)

171 3.8 Parametrization of convection in planetary interiors

or, using the relationship between Ra

and Ra (equation (3.96)):

Nu= 0.25Ra

1/3

(

γT

)

−4/3

. (3.101)

These equations highlight an important difference with moving plate convection. In that

case the efficiency of convective heat transport scales with the Rayleigh number only

(equation (3.90)), whereas in stagnant lid convection the viscosity–temperature relationship

of the fluid also enters the scaling law for heat transfer. We should expect this on physical

grounds, as the viscosity–temperature law is what determines the thicknessof the rheological

boundary layer and hence the effective temperature difference that drives convection.

We still need to calculate the total thickness of the thermal boundary layer, δ. If we assume

that the stagnant lid is in steady-state thermal equilibrium with the convective fluid and that

there is no heat generation in it then the thermal gradient across the entire thermal boundary

layer is linear. The thickness of the thermal boundary layer is in that case related to that of

the rheological boundary layer as follows (Fig. 3.16):

δ =

T

=γTδ

. (3.102)

Worked Example 3.5 Stagnant lid convection vs. moving plate convection

An instructive way of comparing the two styles of convection is by considering what may

happen during the transition from moving plate to stagnant lid convection, for example, if

plate tectonics on Earth were to seize up. During moving plate convection the mantle is

cooled by subduction of cold lithosphere. If subduction stops then convection becomes less

efficient at transporting heat. We can consider two end-member situations: either the rate

of heating does not change, in which case the internal energy content of the mantle must go

up, or the temperature of the mantle does not change, because there is no active heat source.

In both cases we can expect the lithosphere to thicken because it ceases to be recycled and

is thus no longer heated by being immersed in the hot convecting fluid.

Mantle convection is driven by a combination of radioactive decay and secular cooling,

of which core crystallization may be an important component. Radioactive decay delivers

energy at the same rate independently of temperature and, therefore, of the rate at which

heat is transported away from its source. In contrast, the rate of energy supply by secular

cooling generally varies with temperature. For example, if mantle temperature increases

the thermal gradient relative to the core will decrease, and so will the rate of delivery of

enthalpy of crystallization. It follows that the relative contributions of radioactive decay

and secular cooling will determine the evolution of mantle temperature in our hypothetical

planet in which plate tectonics comes to a sudden stop.

Figure 3.17 is constructed for a planet identical to the present day Earth. The thick curves

labeled “Moving plates” are the same ones for lithospheric thickness and heat flux as in

Fig. 3.15, calculated with equations (3.91) and (3.89), respectively. The thick curves labeled

“Stagnant lid” show lithospheric thickness and heat flux for a planet with the same charac-

teristics as Earth, but calculated with equations (3.102) and (3.97). We seek to understand

how the planet may transition from the moving plate to the stagnant lid regime. If stagnant

lid convection had been taking place on Earth since its formation then its mantle temper-

ature would be higher than it is now, because plate tectonics is a more efficient cooling

mechanism than stagnant lid convection, but the comparison between the two convection

styles would still be valid.

172 Energy transfer processes in planetary bodies

1800 2000 2200 2400 2600 2800 300

200

400

600

800

Lithospheric thickness - km

1800 2000 2200 2400 2600 2800 300

100

Characteristic mantle temperature - K

Heat flux - W m

–2

Diffusive thermal boundary layer

at 4.56 x 10

Stagnant lid

Moving plates

Diffusive

heat

flux

4.56 x 10

Moving plates

(i) Constant temperature

(ii) Constant heating

(ii)

Constant

heatin

Fig. 3.17 Transition from moving plate to stagnant lid convection for a hypothetical planet with the characteristics of the Earth.

The moving plate regime (thick curve labeled “Moving plates”) is the same as in Fig. 3.15. The stagnant lid regime

(thick curve labeled “Stagnant lid”) is calculated with equations 3.97–3.102. If the plates seize up and mantle

temperature cannot increase (because there are no active heat sources) surface heat ﬂux and lithospheric thickness

will vary along path (i). The dashed lines show the values of these two variables at different times after the inception

of stagnant lid convection, as a function of mantle temperature when plate tectonics end (the temperature then

remains constant). The thermal boundary layer cannot be thicker than ∼760 km, its thickness for the age of the Solar

System. If there is heat generation in the mantle, or heat input from the core, and the rate of heat supply remains

constant, then steady state stagnant lid convection will be attained when the mobile-plate heat ﬂux is reestablished,

at the end of path (ii). This is accompanied by a substantial increase in mantle temperature, and modest increase in

lithospheric thickness.

173 3.9 Convection and cooling of solid planetary interiors

Suppose that there is no radioactive heat generation, nor any other active source of internal

energy (e.g. tidal, electromagnetic). In this case mantle temperature cannot increase when

plate tectonics stops and the thermal boundary layer will cool and thicken by diffusion.

Assuming that the moving plates before seizing were thin relative to the fluid layer, the

thickness of the rigid lid capping the fluid mantle a time t after the plates lock up will be

of order 2

√

κt, and the surface heat flux at that time will be given by equation (3.22). The

broken lines in Fig. 3.17 show heat flux and thickness of the thermal boundary layer for

three times, 10

years, 10

years and the age of the Solar System. If the temperature of the

mantle at any given time is such that the convective heat flux is less than the diffusive heat

loss across the lid then the fluid mantle will keep convecting (as long as its Rayleigh number

is above the critical value) and the lid will continue cooling diffusively and thickening. This

is shown by the thin arrows labeled (i) in Fig. 3.17. For example, if the Earth had formed

with a mantle temperature of 1950 K and had never had plate tectonics, then convective

heat flux under a stagnant lid at the age of the Solar System would still be far lower than

diffusive heat loss across the thermal boundary layer (recall that in this model there is no heat

generation – this is essentially Kelvin’s model). If enough time is available then diffusive

heat loss across the lid and convective heat transport may eventually become equal and, if

at this point the mantle is still convecting, then subsequent changes in the thickness of the

stagnant lid will be controlled by the evolution of the fluid mantle, rather than by diffusive

cooling. The maximum possible stagnant lid thickness for this end-member situation is the

diffusive cooling length 2

√

κt (e.g. ∼760 km for the age of the Solar System, Fig. 3.6).

The other end-member situation is one in which there is an active energy source, such

as radioactive decay. In this case mantle temperature must increase, as the rate of energy

supply does not change but the rate of heat loss decreases when subduction stops. The

evolution is shown by the thin arrows labeled (ii) in Fig. 3.17. Steady-state convection is

reestablished once the heat flux in the stagnant lid regime matches the moving plate heat

flux. The continuous lithosphere at this point is thicker than it was when it was broken into

plates. This may sound counterintuitive, given that mantle temperature is now higher, but

the point is that, in order to preserve the same heat flux under a higher temperature gradient,

the diffusive lid must become thicker (by Fourier’s law, equation (3.5)).

Real planetary bodies with unbroken lithospheres must lie somewhere in between these

two end-members. Lithospheric thickness cannot be greater, and mantle heat flux cannot be

lower, than the diffusion values corresponding to the time since inception of the stagnant

lid regime, because diffusion is the least efficient heat transfer mechanism. Minimum litho-

spheric thickness or maximum heat flux are not so easily constrained, but either variable

can be calculated from the other one with the equations of stagnant lid convection.

3.9 Convection and cooling of solid planetary interiors

I have treated convection as being driven exclusively by heat loss across the cold upper

boundary layer. I have ignored the bottom boundary layer, and have implicitly assumed

that it is able to transport heat at whatever rate is demanded by the convective system,

and otherwise behave in a completely passive way. This is not true in nature. What I have

described is only one of the possible modes of heat advection in planetary mantles. The

hot lower boundary layer drives another type of advective heat transport process which is

174 Energy transfer processes in planetary bodies

expressed as mantle plumes. This mode is superimposed on the convection process driven

by the top boundary layer. I will not offer a quantitative description of mantle plumes, but an

accessible and lucid explanation can be found in the book by Davies (1999). On the Earth,

mantle plumes are responsible for perhaps 20% of the total heat loss from the planet’s deep

interior (Box 3.2). The proportion may be different in other planets.

At present only the Earth undergoes convection with moving plates. But was this true for

all epochs of the Solar System? Differentiation dissipates a large amount of gravitational

potential energy, which may be sufficient to at least partially melt a planet (Section 2.6).

In addition, there may be other important heat sources during the early stages of planetary

evolution, such as tidal heating, decay of short-lived radioisotopes and planet-sized impacts.

Vigorous convection must take place in partially molten planetary mantles. If a magma ocean

forms it may be reasonable to postulate, from analogy with lava lakes, that when the surface

layer first solidifies it will form thin moving plates that subduct. But what if this solid layer

consists of low density material, such as lunar anorthosites or water ice? These materials

remain buoyant relative to ultramafic melts and liquid water, respectively. The Moon and

the icy worlds must have undergone stagnant lid convection from the time of formation of

the first continuous solid outer layer. As the mantle solidifies the thermal boundary layer

thickens in response to declining heat flux and convection of the solid mantle continues

under the stagnant lid as long as the value of Ra

(equation (3.96)) is above the critical

value for convection (∼10

–10

This is not the only way in which stagnant lid convection can be established, however.

The Earth today convects with a healthy moving-plate regime, but at some time in the future,

as heat flux dwindles, it is inevitable that the plates will thicken to the point that they will

be unable to break, bend and subduct. The thermal boundary layer will then lock up and

display its true (high) viscosity. Could this have happened in the other two large terrestrial

planets? Mars may have had something similar to plate tectonics early in its history, as

hinted by magnetic anomalies in its ancient southern hemisphere, that resemble stripes of

alternately magnetized ocean floor on Earth (Acuña et al., 1999). The case of Venus is more

puzzling.

We will attempt to gain some insight about the internal structures of Venus and Mars as an

example of the application of the equations of parametrized convection. We will approach

each planet with two competing working hypotheses. The first one will be to assume that

we know for how long the present-day stagnant lid regime has existed, and that the thermal

boundary layer has attained its maximum possible thickness (case (i) in Fig. 3.17). Based

on their average surface ages, we will take this time to be ∼4 billion years for Mars (Carr,

1999), and 600 million years for Venus (Head & Basilevsky, 1999). Using equation 3.16

we estimate maximum lid thicknesses of ∼250 km for Venus and ∼700 km for Mars. The

other working hypothesis will be that we know the planetary heat flux and that there is no

heat generation in the lithosphere. This last statement is patently untrue, but we will address

that too. Assuming that the full planetary heat flux comes from the deep mantle, and that

convection is in a steady state, we can calculate a minimum thickness for the stagnant lid as

in case (ii) in Fig. 3.17, and a characteristic mantle temperature. As discussed in Box 3.2,

we postulate mantle heat flux values of 30 mW m

−2

for Mars and 63 mW m

−2

for Venus.

What viscosity should we use in our calculations? There is no clear nor unique answer

to this question. The mantles of Venus and Mars must be made up of Mg-rich silicates,

but there may be important differences with the Earth’s mantle that may affect viscosity.

For example, we know that the Martian mantle is richer in Fe than the terrestrial mantle,

and we can surmise that the Venusian mantle is probably drier than the Earth’s mantle. The

175 3.9 Convection and cooling of solid planetary interiors

approach that I follow here is to allow the viscosity to vary inside a feasible range, and

to study the sensitivity of the results to these variations. As an example, I will allow the

reference viscosity µ

(equation (3.3.3)) to vary between 10

and 10

Pa s at 1950 K, i.e.

one order of magnitude in each direction relative to the likely terrestrial value. I assume

that the activation energy is 300 kJ mol

−1

, as in the Earth’s mantle.

Let us look at the results for Venus first (Fig. 3.18). A steady state mantle heat flux of

63 mW m

−2

requires mantle temperatures 600–1000 K higher than on Earth, depending on

the assumed viscosity (shown by the thin dashed line in the center panel of the figure). This

appears unlikely, as at such temperatures there should be a continuous layer of partially

molten asthenosphere and Venus would be expected to be much more volcanically active

than Earth. This is not the case, for although there certainly are young volcanoes on Venus,

they do not appear to cover the planet with a density comparable to Earth’s. In fact, Venus

may be somewhat less volcanically active than Earth. Given the planet’s high surface

temperature (∼800 K), its lithosphere is likely to be less rigid than Earth’s and, moreover, it

would rest on a significantly less viscous and partially molten asthenosphere. Gravitational

potential energy stored in topography should be dissipated faster on Venus than on Earth.

The lithospheric thickness for this model (∼70–80 km, shown by the thin arrow in the

top panel) is therefore hard to reconcile with the significant topographic relief of Venus,

comparable to that of the Earth. Note that the calculated lithospheric thickness is fairly

insensitive to mantle viscosity (interval between the thin dashed lines in the top panel), and

that the Rayleigh number required to sustain the assumed heat flux varies by a factor of

∼2 for a viscosity contrast of 2 orders of magnitude (bottom panel).

The other end-member possibility is that Venus has a lithosphere ∼250 km thick, that

has been cooling conductively since the planet was resurfaced about 600 million years

ago. The steady-state mantle heat flux in this case would be ∼20 mW m

−2

, varying slightly

depending on the assumed mantle viscosity (center panel).This would necessitate that Venus

has considerably less radioactive heat generation than Earth, that the Venusian core is not an

important heat source, that the planet has undergone a stronger fractionation of incompatible

elements towards the surface than Earth, or a combination of all of the above. Interestingly,

Venus has no intrinsic magnetic field, suggesting that its core is not convecting, and that

it might therefore not be an important energy source for mantle convection. Lack of core

convection could result either from a “cold” core in which crystallization is complete or

near-complete, or from a “hot” core that is mostly above its liquidus and is thus not liberating

enthalpy of crystallization. High mantle temperatures that arise from stagnant lid convection

would argue for the latter explanation. Venus could also be in a steady state if the proportion

of internal energy that is transported to the surface by mantle plumes is larger than in the

Earth, in which case it is required that there be significant heat flux from the core (see

Davies, 1999). In this view the planet’s notable coronae could be active plume heads.

An alternative is that Venus is not in a steady state, but is rather in the process of following

a path such as (ii) in Fig. 3.17. The mantle may be heating up as the rigid lid thickens.

Depending on the mechanical behavior of the lithosphere, it may eventually complete the

transition to a steady state stagnant lid regime (the end point of path (ii) in Fig. 3.17),

or the lithosphere may break up and a regime of convection with moving plates may be

established. In this view (suggested by Turcotte, 1995), Venusian plate tectonics would

consist of catastrophic global subduction events separated by periods of non-steady-state

stagnant lid convection, and the coronae could be incipient subduction zones rather than

plume heads. I do not take sides in this argument regarding steady state vs. episodic Venusian

convection, but I would like to suggest that they lead to tectonic interpretations of coronae

176 Energy transfer processes in planetary bodies

2400 2600 2800

100

200

Lithospheric thickness - km

2400 2600 2800

Heat flux - mW m

–2

2400 2600 2800 300

Average mantle temperature - K

=10

Pa s

=10

Pa s

=10

Pa s

=10

Pa s

=10

Pa s

=10

Pa s

diffusive

q at

Venus

Surface T = 800 K

Fig. 3.18

Possible stagnant lid convection regimes in Venus. Two models are shown, calculated as in Fig. 3.17 for reference

viscosities at 1950 K of 10

and 10

Pa s. It is assumed that the stagnant lid regime has existed for 600 million years.

Lithospheric thickness varies between a maximum of 250 km (corresponding to a constant temperature mantle, case

(i) in Fig. 3.17) and a minimum of 70–80 km constrained by an assumed mantle heat ﬂux of 63 mW m

−2

(thin dashed

line in center panel). The minimum mantle heat ﬂux for Venus would be ∼20 mW m

−2

, corresponding to a 250-km

thick, 600 million year old stagnant lid. Model parameters as in Fig, 3.14, except T

surface

=800 K, D =2700 km and

g = 8.9 m s

−2

177 3.9 Convection and cooling of solid planetary interiors

2200 2400 2600 2800

200

400

600

Lithospheric thickness - km

2200 2400 2600 2800

Heat flux - mW m

–2

2200 2400 2600 2800 300

Average mantle temperature - K

=10

Pa s

=10

Pa s

=10

Pa s

=10

Pa s

=10

Pa s

=10

Pa s

diffusive q at

Mars

Surface T = 250 K

Fig. 3.19

Possible parametrization of stagnant lid convection in Mars. Same as Fig. 3.18, except that the age of the stagnant lid

regime is 4 billion years. Other model parameters as in Fig. 3.15, except T

surface

=250 K, D =1600 km and

g = 3.7ms

−2