Heat Transfer in the Earth What are the 3 types of heat transfer ? 1. Conduction 2. Convection 3. Radioactive heating Where are each dominant in the Earth.

Heat Transfer in the Earth

What are the 3 types of heat transfer ?

1. Conduction2. Convection3. Radioactive heating

Where are each dominant in the Earth ?

Heat Transfer in the Earth Conduction: - Oceanic Lithosphere - Some conduction occurs everywhere a temperature gradient exists - Inner core (?)

Convection: - Ocean water - Mantle interior - Outer Core - Inner core (?) Radioactive heating: - Mantle interior - Continental crust

Radioactive Element Abundance in Continental Crust

The continental crust has the highest concentration of radiogenic elements by volume, A ~ 2.5 W/m3 .

Let's consider the time-dependent heat conduction equation

dT/dt = d2T/dx2 + a

If we assume steady state conditions:

dT/dt = d2T/dx2 + a0

d2T/dx2 = a /then

We can obtain a function T(x) which satisfies this equation.

Radioactive Element Abundance in Continental Crust

The major heat producing elements in the crust are 40K , 238U, 235U, 232Th.

These elements have a half-life of about 1-10 Ga.

Heat production from elements in the continental crust is ~0.6 pW/Kg and can account for nearly ½ the observed surface heat flow

For example: A heat production value of 2.5 mW/m3 through a 10 km depth slice produces 25 mW/m2 surface heat flux.

The Mantle Heat Budget Puzzle

The observed surface heat flux is 60-100 mW/m2.

Total crust ~ 10%

Upper mantle ~ 3% (3 nW/m3 to 650 km)

Full mantle ~ 20 -50 % ( extend to 3000 km)

TOTAL = 65% max

What other factors may contribute to surface heat flow ?


The observed surface heat flux is 60-100 mW/m2.

Convecting mantle plumes ~ 10% Lower mantle may have higher radiogenic concentration

- Reservoirs of “primitive” mantle- Accumulation of subducted oceanic crust

This still may leave a discrepancy of at least 15-20%

Heat from the outer core could contribute – can this be calculated ?


What kind of convective behavior will a heat source at the base of a box produce ?

Can the number and wavelength of plumes be calculated ?

We can study convection with a combination of internal heat sources and base heating and study style and even number of plumes produced...

We can compare thesepredictions to what weknow about plumes in theEarth's mantle from surfaceobservations (volcanism,seismic tomography, etc.)

Convective Heat Transport

Convection is fluid flow driven by internal buoyancy and gravity

Buoyancy is driven by horizontal density gradients

Buoyancy can be positive or negative and occurs when a boundary layer becomes unstable.

Mantle convection in the Earthoccurs by solid state deformationand creep mechanisms (the mantle is NOT a fluid) overmillions of years.

Convective Heat Transport

There is an intimate relationshipbetween interior convection and the surface topographythat it produces.

Most convecting systems aredescribed by two thermal boundary layers (at the top andbottom). Some by only one TBL.

Fluid Mechanics and Mantle Flow

The Earth's interior deforms by creep mechanisms over long periods of time – geologic time

We approximate movement of solid rocks as a viscous material

We use fluid mechanical laws to understand mantle flow over geologic time scales

Fluid Mechanics and Mantle Flow

First we consider the governing conservation equations

Conservation of Mass

Conservation of Momentum

Conservation of Energy

Fluid Mechanics

Conservation of Mass

Assume that the mantle behaves as an incompressible fluid

Consider conservation of fluid volume

Then the rate fluid flows into a given volume is equal to the rate fluid flows out.

v1 v

1 + dv

1

Fluid Mechanics

Flow through the sides plus flow from bottom to top has a net balance such that

v1 v

1 + dv

1dx

1

dv1/dx

1 + dv

2/dx

2 = 0

In other words, the divergence is zero v = 0

This is known as the continuity equation.

Fluid Mechanics

If the fluid is compressible, we must allow for small changes in density with position and time,

The time rate of change in mass equals the net flux in and out

v1 v

1 + dv

1dx

1

d/dt (mass in xz) = flux out – flux in

d/dt xz = - xz d/dx(vx ) - xz d/dz(v

z )

dz1

d/dt = d/dx(vx ) + d/dz(v

z )

d/dt = . v

Fluid Mechanics

If density is constant in space, then we get back the continuity equation.

v1 v

1 + dv

1dx

1

dz1

Putting everything on one side gives the Material Derivative:

d/dt + . v

time position

d/dt = . v

See Class notes on development of Navier-Stokes Equation

Buoyancy

Buoyancy arises from gravity acting on density differences.

Buoyancy is a force

FB = m a = -V g

Where is the density difference between the object and its surroundings.

The minus sign assumes buoyancy is positive upwards (and negative downwards, as is gravity).

Will a small and large iron drop have the same buoyancy in the Earth's mantle ?

Buoyancy

In convection, the total buoyancy (not just density differences) determine fluid behavior.

FB = m a = -V g

Will an object with a large density difference but small volume have a large buoyancy force (F

B) ?

The density of a stainless steel ball bearing (6.9 g/cm3) is about 75% heavier than mantle materials (3.25 g/cm3)!

If you drop a ball bearing on the ground, will it sink to the core ?

What if it was 1500 km in diameter ?

Buoyancy and Thermal Expansion

Density differences are caused by thermal expansion () of a material when it is heated.

= o–

When heated material expands and becomes less dense (T

o = reference temp)

Buoyancy the Thermal Expansion

Is thermal expansion constant everywhere in the Earth ?

Quantity Symbol Valuemantle

ValueCMB

Unit

Thermal expansion 3 x 10-5 0.9 x 10-5 oC-1

Thermal conductivity k 3 9 W/moCThermal diffusivity 1 x 10-6 1.5 x 10-6 m2/sHeat Capacity C

p 900 1200 J/kgoC

Deep lower mantle (CMB)

In the lower mantle thermal properties may be pressure-dependent

Buoyancy the Thermal Expansion

In the lower mantle thermal properties may be pressure-dependent

The density contrast in the upper mantle for a of 1000 is about 3%.

In the lower mantle with thermal expansion reduced by only a factor of 3, the density contrast is only 1%.

Buoyancy in the Earth

What other areas of the Earth has density differences ?

Oceanic crust (due to mineralogy composition The contrast between oceanic crust (2.9 g/cm3) and the mantle is ~12%!

The density contrast across the Mantle Transition Zone is 15%. (Due to phase changes, so not a buoyancy source).

The density contrast between the upper and lower mantle is small.

Buoyancy in the Earth The buoyancy force (F

B) of a ball bearing is -0.02 N

FB

for a plume head of 1000 km diameter and 300 oC is a buoyancy of 2 x 1020 N.

Subducting lithosphere to 600 km depth exerts a negative buoyancy of -40 x 1012 N per meter of trench.

Are plumes more dominant ? - Consider the length of oceanic trenches...over 30,000 km!

Buoyancy in the Earth

Oceanic crust undergoes different phase transformations than the lithospheric mantle during subduction, so may be more or less dense than surrounding mantle at different times...

Crustal weight will be more important in young lithosphere which is thinner (or earlier in the Earth's history...).

The large range of magnitudes (10-20 orders of magnitude!) in buoyancy for Earth processes emphasize that fact that we must consider the structural volumes and not just density anomalies alone.

Analytical Calculations of ConvectionACTIVITY:

Consider the force of a subducting plate entering into the mantle

The oceanic plate has a negative buoyancy and sinks of its own weight because it is more dense.

As it sinks it is surrounded by viscous mantle which resists the plate motion by viscous shear.

The viscous stresses influence the plate velocity, slowing it down.

The plate velocity adjusts until an equilibrium (force balance) is reached between the opposing forces of buoyancy and viscous stress.

Subduction, Mantle Viscosity, and Plate Velocity

The buoyancy of the descending lithosphere is given by (see handout for diagram)

FB-

= -g L T

is the average Temperature difference between theslab and mantle and is approximated by -T/2

FB-

= -g L T/2


Lithospheric thickness () varies with age and can be estimated by T = L / V.

FB-

= -g L T/2

We must also consider conductive cooling (previous lecture):

= sqrt (t)


Now consider the viscous resistance of the mantle giving force per unit area

= 2V / L

If we consider force per unit length, multiply by L:

= 2V


Once plate velocity adjusts to the viscous shear in the mantle the forces are balanced,

Buoyancy Force = Shear Force

FB =

-g L T/2 = 2V

Solve for V to get the resultant plate velocity

V = -g L T/4


V = -g L T/4

We must get lithospheric thickness, = sqrt (t)

Two equations, 2 unknowns ( and V)

V = L [g T (sqrt()) /4 ] 2/3


V = L [g T (sqrt()) /4 ] 2/3

Estimate plate velocity using the above equation(which is derived from buoyancy and viscous shear theory)

Use these assumptions for mantle properties: D (mantle thickness) = 3000 km = 4000 kg/m3

= 2 x 10-5 oC-1

T = 1400 oC

= 10-6 m2/s = 1022 Pas


V = L [g T (sqrt()) /4 ] 2/3

How close is your estimate of plate velocity to real velocities that we measure today ?

This general agreement suggests that convection, and plate buoyancy in the mantle is a viable theory to explain why plates move ! THINK ABOUT IT !


V = L [g T (sqrt()) /4 ] 2/3

In the past the Earth may have been hotter (more like Jupiter's moon Io today).

If hotter in the past, would Earth's plates have moved faster or slower ? Why ? (Hint: look at your equation)

Io: showing volcanoes and eruptions

Scaling Fluid Dynamic Models to Earth Systems

The theory we just developed from assumptions of buoyancy forces and shear forces also tell us howvarious physical properties scale with each other.

For example in the equation for fluid velocity:

V = L [g T (sqrt()) /4 ] 2/3

If viscosity was 10 times lowerthen how would the velocity change..... ?

the velocity would then increase by 10 2/3 (~ 4.6 times greater).


Can we really compare experiments in the laboratory or on a computer performed in a small box to the Earth ?


V = L [g T (sqrt()) /4 ] 2/3

Earlier we showed that diffusion across a characteristic distance is given by:

= sqrt (t)

or = sqrt (D /v)

velocity

We can solve for velocity, and set this equal to theoriginal equation for velocity:

velocity


To obtain:

(D/ )3 = g T D3 / 4

This is written in a general form which is often used to describe a non-dimensional number, the Rayleigh number.

Ra = g T D3 /

What is a non-dimensional number ?

Non-Dimensional Numbers

Ra = g T D3 /

What is a non-dimensional number ?

This is a number with no dimensions...how is this possible ?

The units on the RHS (right hand side) will ALL cancel – try it!

Even though units cancel, we still have values for buoyancy on the top and viscous shear & thermal diffusivity on the bottom

So if the number is greater than 1, buoyancy forces are stronger But if the number is less than 1, viscou shear is stronger


Ra = g T D3 /

The Rayleigh number describes the vigor of convection . (ratio: of diffusion time / advection time) In the Earth, Ra ~ 109

A fluid will start to convect when the Ra > 1 x 103

What does convect mean ?

Convection describes the physical movement (advection) of fluid particles (e.g. convection cells, plumes) -this comes from the material derivative


If the Rayleigh number or any non-dimensional number is the same in your experiment and in the Earth

Then we consider the physical behavior to be comparable

Raearth

= 1 x 109Ra

lab = 1 x 109

Non-Dimensional Numbers True compatability requires both dynamic and thermal similarity :

Prandlt number: is a property of the fluid

Pr = / (ratio: diffusion of momentum and vorticity / diffusion of heat)

In the Earth where viscosities are high, Pr ~ 1026 !

Reynolds number: is a property of fluid flow

Re = VL / (ratio: of inertial forces / viscous forces)

In the Earth, Re ~ 10-12

Non-Dimensional Numbers The Nusselt and Rayleigh numbers give thermal similarity :

Nusselt number: describes thermal properties

Nu = LFheat

/ (ratio: of total heat flux / conductive heat flux)

Rayleigh number: describes thermal and dynamic properties


The Nusselt number measures the efficiency of convection andis related to the Rayleigh number in classical theory:

Nu = Ra 1/3

Weeraratne and Manga, 1998


Length scale = / D

Velocity scale: V = / D

Characteristic time: t = D2/

Other relevant scaling parameters:

Can you use any of these non-dimensional parametersin your class projects ?

Boundary Layer Theory

Boundary layers are everywhere!

Airplane wing: note particlesin boundary layer surroundingwing geometry

Wind Chill Factor: windthat is strong enough to blowaway the warm thermal boundarylayer surrounding your skin.


v

Thermal or material behavior at margins indicates that thin layers form which insulate or act to protect the material

These boundary layers may be stable or if heat is increased may grow and go unstable

The perterbation shown above describes a boundary layer instability


v

We can describe this instability using buoyancy forces

F

B = m a = g

Where the wavelength () can be measured.


v

There is also a resistive force from the surrounding fluid

F

R = V

fluid

FR = d/ dt


v

The buoyancy force balances the viscous force so:

fluid

FB = F

R

d / dt = g /


v

The wavelength () of instabilities is given by:

fluid

=


v

The characterisitic time () of growth of the instability:

fluid

= g


v

How do boundary layers react to different modes of heating ?

Conductive heating ?

Convective heating from top and bottom ?

Internal heating ?

fluid

Heat Transfer in the Earth What are the 3 types of heat transfer ? 1. Conduction 2. Convection 3. Radioactive heating Where are each dominant in the Earth.

Documents

major heat

mantle convection

mantle heat budget puzzle

earths mantle

observed surface heat

observed surface heat

lower mantle

upper mantle