Convection in the Earthâ€™s mantle - BSL: Berkeley Seismological

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EPS 122: Lecture 20 – Convection

Convection in the Earth’s mantle Reading: Fowler p331-337, 353-367


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

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The adiabatic temperature gradient Need the change of temperature with pressure at constant entropy, S

using reciprocal theory

S

P

T

=V

T

P

Some thermodynamics…

coefficient of thermal expansion

Maxwell’s thermodynamic relation

specific heat

Substitute… …adiabatic gradient as a

function of pressure


The adiabatic temperature gradient

…but we want it as a function of depth

Substitute…

…adiabatic gradient as a function of pressure

For the Earth

…adiabatic gradient as a function of radius

Temperature gradient for the uppermost mantle

0.4 °C km-1 using T = 1700 K = 3 x 10-5 °C-1

g = 9.8 m s-2

cp = 1.25 x 103 J kg °C-1 at greater depth 0.3 °C km-1

due to reduced

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Adiabatic temperature gradients Models agree that gradient is close to adiabatic, particularly in upper mantle

…why would it not be adiabatic?

greater uncertainty for the lowest 500-1000 km of the mantle

big range of estimated T for CMB

2500K to ~4000K

This is the work of

Jeanloz and

Bukowinski in our dept


Melting in the mantle

200 km

100 km

Potential temperature: T of rock at surface if rises along the adiabat

Different adiabatic gradient

for fluids: ~ 1 °C km-1

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1D velocity model for the Earth

Using the arrival times of seismic phases at stations

around the globe we can calculate a 1D average velocity model for the Earth

VP VS

Uppermost mantle low-velocity zone

Transition zone: 410-660 km

Lower mantle

Core-mantle boundary

Outer core: must be fluid as VS = 0

Inner core solid


Density and elastic moduli for whole Earth

P-velocity S-velocity

two equations, three unknowns K, μ and bulk modulus, shear modulus and density

Adams-Williamson equation – the 3rd equation

tells us the density gradient as a function of depth

using our understanding of gravitational attraction and the radial mass distribution of the Earth

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Adams-Williamson equation

Mr is the mass within radius r

…which we don’t know

Start at Earth’s surface and work inwards applying the equation successively to shells of uniform composition

self compression model

1. Choose a density for the top of the mantle and work downward to the CMB

2. Choose a density for the top of the core

3. Once at the center of the Earth, the total mass must equal the known value. If not, pick a new starting density and re-calculate.

final model satisfies seismic data and mass of the Earth

Additional constraints:

1. Moment of inertia – distribution of mass within the Earth

2. Need to account for changes in phase as well as composition


Density and elastic moduli

Density from the Adams-Williamson equation

then

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Mantle convection?

What is mantle convection?

Why do we believe there is convection in the mantle?


Convection

In a fluid:

• Occurs when density distribution deviates from equilibrium

• Fluid may then flow to achieve equilibrium again

In a viscous solid heated from below:

• Initially heat is transported by conduction into the fluid at the base

• Increased temperature reduces the density making the material at the base less dense than fluid above

• Once the buoyancy force due to the density contrast overcomes the

inertia of the fluid convection begins

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Rayleigh-Benard convection

Newtonian viscous fluid:

stress = dynamic viscosity x strain rate

As fluid at the base heats up, initial convection is in 2D

rolls

As heating proceeds, a second set of rolls forms perpendicular to the first

– bimodal

Continued heating – hexagonal pattern

Then a spoke pattern, and finally an irregular pattern forms as vigorous convection takes place (not shown)


Rayleigh number Non-dimensional number which describes the nature of heat transfer

g - gravity

- density - thermal expansion coefficient

T - temperature variation

a - length scale: thickness of fluid layer - thermal diffusivity

- viscosity

c – conductive time constant

a – advective time constant

The critical Rayleigh number:

• the point at which convection initiates

• approximately 103 (dependent on geometry)

• By knowing the material properties and physical geometry we can

determine if there will be convection and the nature of that convection

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Rayleigh number and convective mode

103

104

105

106

Ra

yle

igh

nu

mb

er

Convection plan view

Rayleigh number of the mantle:

Upper mantle (thickness 700

km): 106

Lower mantle (thickness 2000 km):

3x107

Whole mantle (thickness 2700 km):

108


the effect of heating

• Heat from below

• T fixed on upper boundary

aspect ration of 1

…not what we see on Earth

• Heat from below

• Constant heat flow across upper boundary

large aspect ratio

• Internal heating only

no upwelling sheets

Simple convection models

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the effect of heating

• Isotopic ratios of oceanic basalts are very uniform but

different from bulk earth values

the mantle is well mixed

any body smaller than 1000 km is reduced to less than 1

cm thick in 825 Ma!

Simple convection models


the mantle Constant viscosity incompressible mantle internally heated

The many views of

increasing lower mantle viscosity

flow dominated by sheets that extend through the mantle …more Earth like

Factor of 30 difference

Compressible fluid

short wavelength

again

Heating from the core

hot upwellings dominate

short wavelength, numerous

downwellings

Bunge models

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the mantle

Phase changes at 400 and 660 km

increasing Rayleigh number

mantle becomes stratified

avalanches of material into lower mantle

The many views of

Yuen


What about the plates?

One layer or two?

Subduction ? Downwelling

Ridges ? Upwelling

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Downwelling = subduction Low temperature high density slabs observed extending through the entire mantle


Subduction

Farallon slab

• Originates from a time when there was subduction all along the western US.

• We find evidence of this slab extending all the way to the core-mantle

boundary

and mantle convection

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Subduction

Japan trench

Izu slab

Kuril slab

Western pacific

Some evidence for slab

penetration into

the lower mantle


Subduction Lau Basin

3D viewing available at http://ldorman.home.mindspring.com/VRC/VRmjl.html

Lau Basin

Observations:

• High velocity slab, low velocity wedge

• Earthquakes in slab and rift

• Large number of compressional

earthquakes above 660 km

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Lower mantle subduction

Two slabs do extend into the lower mantle

• Farallon and Tethys


Modes of mantle convection

Subduction ? Downwelling

• Slabs clearly represent the downwelling mode in the upper mantle

• Some slabs pass through the transition zone into the lower mantle

…Yes

…and upwelling?

Convection in the Earthâ€™s mantle - BSL: Berkeley Seismological

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