Francis Nimmo

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EART162 PLANETARY INTERIORS

• Francis Nimmo

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Last week - Seismology

• Seismic velocities tell us about interior
  properties

• Adams-Williamson equation allows us to relate
  density directly to seismic velocities

• Travel-time curves can be used to infer seismic
  velocities as a function of depth
• Midterm

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This Week Fluid Flow Convection

• Fluid flow and Navier-Stokes
• Simple examples and scaling arguments
• Post-glacial rebound
• What is convection?
• Rayleigh number and boundary layer thickness
• Adiabatic temperature gradient
• See Turcotte and Schubert ch. 6

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Viscosity

• Youngs modulus gives the stress required to
  cause a given deformation (strain) applies to a
  solid
• Viscosity is the stress required to cause a given
  strain rate applies to a fluid
• Viscosity is basically the fluids resistance to
  flow

viscous
elastic
viscosity
Youngs modulus

• Kinematic viscosity h measured in Pa s
• Dynamic viscosity nh/r measured in m2s-1
• Typical values for viscosity water 10-3 Pa s,
  basaltic lava 104 Pa s, ice near melting 1014 Pa
  s, mantle 1021 Pa s
• Viscosity often temperature-dependent (see Week 3)

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Defining Viscosity

• Recall
• Viscosity is the stress generated for a given
  strain rate
• Example moving plate

u
(Shear) stress s required to generate velocity
gradient u / d ( ) Viscosity hs d / u
h

• Example moving lava flow

Driving shear stress rgd sinq Surface velocity
rgd2 sinq / h
d
e.g. Hawaiian flow h104 Pa s q5o d3m gives u2
ms-1 (walking pace)
q
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Adding in pressure

• In 1D, shear stress (now using t) is

• Lets assume u only varies in the y direction

Fluid velocity u
y
x
Pressure force (x direction, per unit
volume)
dx
Why the minus sign?
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Putting it together

• Total force/volume viscous pressure effects

• We can use Fma to derive the response to this
  force

What does this mean?

• So the 1D equation of motion in the x direction is

What does each term represent?

• In the y-direction, we would also have to add in
  buoyancy forces (due to gravity)

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Navier-Stokes

• We can write the general (3D) formula in a more
  compact form given below the Navier-Stokes
  equation
• The formula is really a mnemonic it contains
  all the physics youre likely to need in a single
  equation
• The vector form given here is general (not just
  Cartesian)

Yuk! Inertial term. Source of turbulence. See
next slide.
Pressure gradient
Buoyancy force (e.g. thermal or electromagnetic)
is a unit vector
Diffusion-like viscosity term. Warning
is complicated, especially in non-Cartesian geom.
Zero for steady- state flows
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Reynolds number

• Is the inertial or viscous term more important?
• We can use a scaling argument to get the ratio Re

Here L is a characteristic lengthscale of the
problem
Re

• Re is the Reynolds number and tells us whether a
  flow is turbulent (inertial forces dominate) or
  not
• Fortunately, many geological situations allow us
  to neglect inertial forces (Reltlt1)
• E.g. what is Re for the convecting mantle?

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Example 1 Channel Flow
L
y
x
u
2d
0
P2
P1
(Here u doesnt vary in x-direction)

• 2D channel, steady state, u0 at yd and y-d

• Max. velocity (at centreline) (DP/L) d2/2h
• Does this result make sense?
• We could have derived a similar answer from a
  scaling argument how?

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Example 2 falling sphere
r

• Steady-state. What are the important terms?

h
u
An order of magnitude argument gives drag force
hur Is this dimensionally correct? The full
answer is 6phur, first derived by George Stokes
in 1851 (apparently under exam conditions) By
balancing the drag force against the excess
weight of the sphere (4pr3Drg/3 ) we can obtain
the terminal velocity (here Dr is the density
contrast between sphere and fluid)
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Postglacial Rebound

• Postglacial rebound problem How long does it
  take for the mantle to rebound?
• Two approaches
• Scaling argument
• Stream function j see TS

mantle

• Scaling argument

• Assume u is constant (steady flow) and that u
  dw/dt
• We end up with

decay constant

• What does this equation mean?

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Prediction and Observations

• Scaling argument gives

Hudsons Bay deglaciation L1000 km, t2.6
ka So h2x1021 Pa s So we can infer the
viscosity of the mantle A longer wavelength load
would sample the mantle to greater depths
higher viscosity
http//www.geo.ucalgary.ca/wu/TUDelft/Introductio
n.pdf
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Convection
Cold - dense

• Convection arises because fluids expand and
  decrease in density when heated
• The situation on the right is gravitationally
  unstable hot fluid will tend to rise
• But viscous forces oppose fluid motion, so there
  is a competition between viscous and (thermal)
  buoyancy forces

Fluid
Hot - less dense

• So convection will only initiate if the buoyancy
  forces are big enough
• What is the expression for thermal buoyancy
  forces?

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Conductive heat transfer

• Diffusion equation (1D, Cartesian)

Heat production
Advected component
Conductive component

• Thermal diffusivity kk/rCp (m2s-1)
• Diffusion timescale

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Convection equations

• There are two one controlling the evolution of
  temperature, the other the evolution of velocity
• They are coupled because temperature affects flow
  (via buoyancy force) and flow affects temperature
  (via the advective term)

Navier- Stokes
Buoyancy force
Note that here the N-S equation is neglecting the
inertial term
Thermal Evolution
Advective term

• It is this coupling that makes solving convection
  problems hard

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Initiation of Convection
Top temperature T0

• Recall buoyancy forces favour motion, viscous
  forces oppose it
• Another way of looking at the problem is there
  are two competing timescales what are they?

d
Incipient upwelling
d
Hot layer
Bottom temp. T1

• Whether or not convection occurs is governed by
  the dimensionless (Rayleigh) number Ra

• Convection only occurs if Ra is greater than the
  critical Rayleigh number, 1000 (depends a bit
  on geometry)

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Constant viscosity convection
T0
(T0T1)/2
cold
T0

• Convection results in hot and cold boundary
  layers and an isothermal interior
• In constant-viscosity convection, top and bottom
  b.l. have same thickness

d
Isothermal interior
d
d
T1
hot
T1

• Heat is conducted across boundary layers
• In the absence of convection, heat flux
• So convection gives higher heat fluxes than
  conduction
• The Nusselt number defines the convective
  efficiency

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Boundary layer thickness

• We can balance the timescale for conductive
  thickening of the cold boundary layer against the
  timescale for the cold blob to descend to obtain
  an expression for the b.l. thickness d

• So the boundary layer gets thinner as convection
  becomes more vigorous
• Also note that d is independent of d. Why?
• We can therefore calculate the convective heat
  flux

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Example - Earth

• Does this equation make sense?

• Plug in some parameters for the terrestrial
  mantle
• r3000 kg m-3, g10 ms-2, a3x10-5 K-1, k10-6
  m2s-1, h3x1021 Pa s, k3 W m-1K-1, (T1-T0) 1500
  K
• We get a convective heat flux of 170 mWm-2
• This is about a factor of 2 larger than the
  actual terrestrial heat flux (80 mWm-2) not
  bad!
• NB for other planets (lacking plate tectonics), d
  tends to be bigger than these simple calculations
  would predict, and the convective heat flux
  smaller
• Given the heat flux, we can calculate thermal
  evolution

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Thermodynamics Adiabat

• A packet of convecting material is often moving
  fast enough that it exchanges no energy with its
  surroundings
• What factors control whether this is true?
• As the convecting material rises, it will expand
  (due to reduced pressure) and thus do work (W P
  dV)
• This work must come from the internal energy of
  the material, so it cools
• The resulting change in temperature as a function
  of pressure (dT/dP) is called an adiabat
• Adiabats explain e.g. why mountains are cooler
  than valleys

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Adiabatic Gradient (1)

• If no energy is added or taken away, the entropy
  of the system stays constant
• Entropy S is defined by

Here dQ is the amount of energy added to the
system (so if dQ0, then dS0 also and the system
is adiabatic)

• What we want is at constant S. How do we
  get it?
• We need some definitions

Maxwells identity
Specific heat capacity (at constant P)
Thermal expansivity
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Adiabatic Gradient (2)
T
z

• We can assemble these pieces to get the adiabatic
  temperature gradient

adiabat

• NB Youre not going to be expected to reproduce
  the derivation, but you do need to learn the
  final result
• An often more useful expression can be obtained
  by converting pressure to depth (how?)

• What are typical values for terrestrial planets?

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Summary

• Fluid dynamics can be applied to a wide variety
  of geophysical problems
• Navier-Stokes equation describes fluid flow

• Post-glacial rebound timescale
• Behaviour of fluid during convection is
  determined by a single dimensionless number, the
  Rayleigh number Ra

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End of lecture

• Supplementary material follows

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Incompressibility Stream Function

• In many fluids the total volume doesnt change

dx
Incompressibility condition

• We can set up a stream function j which
  automatically satisfies incompressibility and
  describes both the horizontal and the vertical
  velocities

Note that these satisfy incompressibility
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Stream Function j

• Only works in 2 dimensions
• Its usefulness is we replace u,v with one
  variable j

Check signs here!
Differentiate LH eqn. w.r.t. z and RH w.r.t x
The velocity field of any 2D viscous flow
satisfies this equation
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Postglacial rebound and j (1)

• Biharmonic equation for viscous fluid flow

• Assume (why?) j is a periodic function jsin kx
  Y(y) Here k is the wavenumber 2p/l
• After a bit of algebra, we get

• All that is left (!) is to determine the
  constants which are set by the boundary
  conditions in real problems, this is often the
  hardest bit
• What are the boundary conditions?
• u0 at z0, vdw/dt at z0, uv0 at large z

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Postglacial rebound and j (2)

• Applying the boundary conditions we get

• We have dw/dt

• Vert. viscous stress at surface (z0) balances
  deformation

Why can we ignore this term?

• For steady flow, we can derive P from
  Navier-Stokes

• Finally, eliminating A from 1 and 2 we get
  (at last!)

This ought to look familiar . . .
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Postglacial rebound (concluded)

• So we get exponential decay of topography, with a
  time constant depending on wavenumber (k) and
  viscosity (h)
• Same result as we got with the scaling argument!
• Relaxation time depends on wavelength of load
• Relaxation time depends on viscosity of fluid

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Rayleigh-Taylor Instability

• This situation is gravitationally unstable if r2
  lt r1 any infinitesimal perturbation will grow
• What wavelength perturbation grows most rapidly?

b
r1
m
r2
m
b

• The full solution is v. complicated (see TS
  6-12) so lets try and think about it
  physically . . .

L
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R-T Instability (contd)

• Recall from Week 5 dissipation per unit volume
• We have two contributions to total dissipation (
  )
• By adding the two contributions, we get

term
term

• What wavelength minimizes the dissipation?

• We end up with dissipation minimized at lmin1.26
  b
• This compares pretty well with the full answer
  (2.57b) and saves us about six pages of maths

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R-T instability (contd)

• The layer thickness determines which wavelength
  minimizes viscous dissipation
• This wavelength is the one that will grow fastest
• So surface features (wavelength) tell us
  something about the interior structure (layer
  thickness)

Salt domes in S Iran. Dome spacing of 15 km
suggests salt layer thickness of 5 km, in
agreement with seismic observations
50km
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Convection
Cold - dense

• Convection arises because fluids expand and
  decrease in density when heated
• The situation on the right is gravitationally
  unstable hot fluid will tend to rise
• But viscous forces oppose fluid motion, so there
  is a competition between viscous and (thermal)
  buoyancy forces

Fluid
Hot - less dense

• So convection will only initiate if the buoyancy
  forces are big enough
• Note that this is different to the
  Rayleigh-Taylor case thermal buoyancy forces
  decay with time (diffusion), compositional ones
  dont
• What is the expression for thermal buoyancy
  forces?

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Two Dimensions . . .
THIS SECTION PROBABLY A WASTE OF TIME

• In 1D, shear stress (now using t) is

x

• In 2D, there are three different stresses

Shear stress
Normal stresses

• Where do the factors of 2 come from?

p(y)dx

• Force due to pressure (x direction, per unit
  cross-sectional area)

p(x)dy
dy
p(xdx)dy
dx
p(ydy)dx
EXPLAIN WHERE VE COMES FROM
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Viscous forces on an element (1)
x
txy

• Viscous force (x direction, per unit
  cross-sectional area)

v
txy
y
txx
u
dy
tyy

• Total force balance given by viscous pressure
  forces

dx

• After some algebra, we get total force in
  x-direction

Note that force in x-direction only depends on
velocity in x-direction and the x-gradient of
pressure
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Viscous forces on an element (2)

• In the y-direction, body forces can also be
  important

• Otherwise, the analysis is the same as before

• We can use Fma to derive the response to this
  force

What does this mean?

• So the equations of motion in x and y directions
  are

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Putting it together

• x-direction
• y-direction

Pressure gradient
Body force
Viscous terms

• Special cases
• Steady-state Du/Dt0
• One-dimension (e.g. v0, u only varies in y
  direction)