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 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/m 3. Let's consider the time-dependent heat conduction equation dT/dt =  d 2 T/dx 2 + a If we assume steady state conditions: dT/dt =  d 2 T/dx 2 + a 0 d 2 T/dx 2 = 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 40 K, 238 U, 235 U, 232 Th. 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/m 3 through a 10 km depth slice produces 25 mW/m 2 surface heat flux.

The Mantle Heat Budget Puzzle The observed surface heat flux is mW/m 2. Total crust~ 10% Upper mantle ~ 3% (3 nW/m3 to 650 km) Full mantle ~ % ( extend to 3000 km) TOTAL = 65% max What other factors may contribute to surface heat flow ?

The Mantle Heat Budget Puzzle The observed surface heat flux is mW/m 2. 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 ?

The Mantle Heat Budget Puzzle 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 these predictions to what we know about plumes in the Earth's mantle from surface observations (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 Earth occurs by solid state deformation and creep mechanisms (the mantle is NOT a fluid) over millions of years.

Convective Heat Transport There is an intimate relationship between interior convection and the surface topography that it produces. Most convecting systems are described by two thermal boundary layers (at the top and bottom). 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. v1v1 v 1 + dv 1

Fluid Mechanics Flow through the sides plus flow from bottom to top has a net balance such that v1v1 v 1 + dv 1 dx 1 dv 1 /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 v1v1 v 1 + dv 1 dx 1 d/dt (mass in  x  z) = flux out – flux in d  /dt  x  z = -  x  z d/dx(v x  ) -  x  z d/dz(v z  ) dz 1 d  /dt = d/dx(v x  ) + d/dz(v z  ) d  /dt = . v 

Fluid Mechanics If density is constant in space, then we get back the continuity equation. v1v1 v 1 + dv 1 dx 1 dz 1 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 F B = 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. F B = 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/cm 3 ) is about 75% heavier than mantle materials (3.25 g/cm 3 )! 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 ? QuantitySymbolValue mantle Value CMB Unit Thermal expansion  3 x x o C -1 Thermal conductivityk3 9W/m o C Thermal diffusivity  1 x x m 2 /s Heat CapacityC p J/kg o C 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/cm 3 ) 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 N F B for a plume head of 1000 km diameter and  300 o C is a buoyancy of 2 x N. Subducting lithosphere to 600 km depth exerts a negative buoyancy of -40 x 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 Convection ACTIVITY: 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) F B- = -g L     T  is the average Temperature difference between the slab and mantle and is approximated by -T/2 F B- = -g L    T/2

Subduction, Mantle Viscosity, and Plate Velocity Lithospheric thickness (  ) varies with age and can be estimated by T = L / V. F B- = -g L    T/2 We must also consider conductive cooling (previous lecture):  = sqrt (  t)

Subduction, Mantle Viscosity, and Plate Velocity 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

Subduction, Mantle Viscosity, and Plate Velocity Once plate velocity adjusts to the viscous shear in the mantle the forces are balanced, Buoyancy Force = Shear Force F B =  -g L    T/2 =  2V Solve for V to get the resultant plate velocity V = -g L    T/4 

Subduction, Mantle Viscosity, and Plate Velocity 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

Subduction, Mantle Viscosity, and Plate Velocity 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/m 3  = 2 x o C -1 T = 1400 o C  = m 2 /s  = Pas

Subduction, Mantle Viscosity, and Plate Velocity 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 !

Subduction, Mantle Viscosity, and Plate Velocity 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 how various 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 lower then how would the velocity change..... ? the velocity would then increase by 10 2/3 (~ 4.6 times greater).

Scaling Fluid Dynamic Models to Earth Systems Can we really compare experiments in the laboratory or on a computer performed in a small box to the Earth ?

Scaling Fluid Dynamic Models to Earth Systems 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 the original equation for velocity: velocity

Scaling Fluid Dynamic Models to Earth Systems To obtain: (D/  ) 3 =  g  T D 3 / 4  This is written in a general form which is often used to describe a non-dimensional number, the Rayleigh number. Ra =  g  T D 3 /  What is a non-dimensional number ?

Non-Dimensional Numbers Ra =  g  T D 3 /  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

Non-Dimensional Numbers Ra =  g  T D 3 /  The Rayleigh number describes the vigor of convection. (ratio: of diffusion time / advection time) In the Earth, Ra ~ 10 9 A fluid will start to convect when the Ra > 1 x 10 3 What does convect mean ? Convection describes the physical movement (advection) of fluid particles (e.g. convection cells, plumes) -this comes from the material derivative

Non-Dimensional Numbers 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 Ra earth = 1 x 10 9 Ra lab = 1 x 10 9

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 ~ ! Reynolds number: is a property of fluid flow Re =  VL /  (ratio: of inertial forces / viscous forces) In the Earth, Re ~

Non-Dimensional Numbers The Nusselt and Rayleigh numbers give thermal similarity : Nusselt number: describes thermal properties Nu = LF heat /  (ratio: of total heat flux / conductive heat flux) Rayleigh number: describes thermal and dynamic properties

Non-Dimensional Numbers The Nusselt number measures the efficiency of convection and is related to the Rayleigh number in classical theory: Nu = Ra 1/3 Weeraratne and Manga, 1998

Non-Dimensional Numbers Length scale =  / D Velocity scale: V =  / D Characteristic time: t = D 2 /  Other relevant scaling parameters: Can you use any of these non-dimensional parameters in your class projects ?

Boundary Layer Theory Boundary layers are everywhere! Airplane wing: note particles in boundary layer surrounding wing geometry Wind Chill Factor: wind that is strong enough to blow away the warm thermal boundary layer surrounding your skin.

Boundary Layer Theory  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

Boundary Layer Theory  v We can describe this instability using buoyancy forces F B = m a =  g  Where the wavelength ( ) can be measured.

Boundary Layer Theory  v There is also a resistive force from the surrounding fluid F R =  V  fluid F R =  d  / dt

Boundary Layer Theory  v The buoyancy force balances the viscous force so:  fluid F B = F R d  / dt =  g  / 

Boundary Layer Theory  v The wavelength ( ) of instabilities is given by:  fluid =      

Boundary Layer Theory  v The characterisitic time (  ) of growth of the instability:  fluid  =  g 

Boundary Layer Theory  v How do boundary layers react to different modes of heating ? Conductive heating ? Convective heating from top and bottom ? Internal heating ?  fluid