Arthur Holmes (1945)
?
Various Conceptual Models (i.e., hypotheses) of Large-Scale Mantle
Convection Inspired by Seismology and/or geochemistry.
Each hypothesis has fundamentally different consequences for
our understanding of mantle convection and how it cools the
Earth and drives plate tectonics.
Dziewonski et al. (2010)
Courtillot et al. (2003)
Kellogg et al. (1999)
Jellinek and Manga (2004)
Garnero (web)
?
Discovering which, if any, of these are occurring in the Earth is critical toward building
the foundation for our understanding of:
 Driving forces of mantle convection and plate tectonics
(e.g., slab-driven versus superplume-driven convection)
 Heat transport and thermal evolution
 Chemical evolution and MORB/OIB chemistry.
 The geodynamo
 Hotspots and the morphology, size, temperature, and the chemistry of plumes.
Dziewonski et al. (2010)
Courtillot et al. (2003)
Kellogg et al. (1999)
Jellinek and Manga (2004)
Garnero (web)
Geodynamics of Mantle Convection
Physics (Conservations of mass, momentum, and energy)
Application/Modeling (Numerical modeling, laboratory experiments)
Science (Observations, hypothesis development, hypothesis testing)
Conservation of Mass
Rate of mass
change in a
volume
=
Rate of mass
entering the volume
Rate of mass
exiting the
volume


   V 
t
This is identical to:



     V   V   
t
Independent Variables:
t = time
x, y, z = position
Dependent Variables:
  x, y , z , t 
Density (scalar field)

V ( x, y , z , t )
Velocity (vector field)
Conservation of Momentum
Rate of momentum
change per volume =
Rate of
momentum entering the
volume
Rate of
momentum +
exiting the
volume
Force acting on
the volume
Conservation of Momentum



V


   V   V      g
t
Derivation includes using conservation of mass.
Independent Variables:
t = time
x, y, z = position
Dependent Variables:
  x, y , z , t 
Density (scalar field)

V ( x, y , z , t )
Velocity (vector field)
  x, y , z , t 
Stress (tensor field)

g  x, y , z , t 
Gravitational acceleration (vector field)
Stress
 ab
“a” is the normal to the plane that the stress acts upon
“b” is the direction of the stress
NOTE: 1, 2, 3 are the same as x, y, z
Image from http://homepage.ufp.pt/biblioteca/GlossarySaltTectonics/Pages/PageS.html
It is often convenient to decompose the stress tensor into 2 parts:
1. Pressure, p (scalar)
2. Deviatoric Stress,

(tensor field)
Pressure is defined to be the average of normal stresses:
1
p    xx   yy   zz 
3
By convention, pressure is in the opposite direction of stress direction.
Pressure acts only in the normal direction.
Pressure is the same in each direction.
Image from http://www.see.ed.ac.uk/~johnc/teaching/fluidmechanics4/2003-04/fluids5/stress.html
By construction:
Note that:
1
p    xx   yy   zz 
3

    pI

1 0 0


I  0 1 0
0 0 1


Therefore, the deviatoric stress is defined as:

    pI

Important note: directions for stress, deviatoric stress, and pressure are not a universal convention, so be careful!
Let’s look at the following term in the momentum equation:

        pI

       pI
 p 0 0

  ˆ  ˆ  ˆ 
      i , j, k    0 p 0 
 x y z   0 0 p 


 p ˆ p ˆ p ˆ 
      i  j  k 
 x y z 
     p
 
Conservation of Momentum (different form, with pressure)



V


   V   V      p  g
t
Independent Variables:
t = time
x, y, z = position
Dependent Variables:
  x, y , z , t 
Density (scalar)

V ( x, y , z , t )
Velocity (vector field)
  x, y , z , t 
Deviatoric stress (tensor field)

g  x, y , z , t 
Gravitational acceleration (vector field)
p  x, y , z , t 
Pressure (scalar)
Conservation of Energy
Rate of internal
energy change in a =
volume
Rate of heat
Rate of heat
transferred to _ exiting the _
the volume
volume
Rate of work
performed by
the volume
This is the time derivative of the first law of thermodynamics.
dU  Q  W
Conservation of Energy

 DS*
T
Dp
c p
  c p V  T     kT  T
  : V 
t
Dt
Dt
Derivation includes using conservation of mass, conservation of momentum, and dU = TdS – PdV (1st and 2nd laws of thermo).
Independent Variables (x,y,z,t)
k  x, y , z , t 
Dependent Variables:
  x, y , z , t 
Density (scalar field)
 S 
CP  
 T
 T  P
T  x, y , z , t 
1  V 

 Thermal expansivity [V is volume] (scalar field)
V  T  P
Specific heat at constant pressure (scalar field)
Temperature (scalar field)

V ( x, y , z , t )

Velocity (vector field)
Thermal conductivity (scalar field)
dS*
  x, y , z , t 
Deviatoric stress (tensor field)
Entropy changes related to processes other
than those for a homogenous material
undergoing changes in temperature and
pressure.
Some Explanation:
During the derivation, entropy changes were split into 2 types:
1. Entropy changes due to single phase, homogeneous material undergoing changes in
temperature and pressure.
2. All other entropy changes are lumped into: S* These include things such as radioactive
heat production, phase changes, chemical reactions, nuclear reactions, etc. These items
are best extracted as needed for the particular problem at hand. Radioactive heat
production of uranium, thorium, and potassium is often extracted as: H
Conservation of Energy (with heat production explicitly defined)


T
Dp
DS*
c p
  c p V  T     kT  T
  : V  H 
t
Dt
Dt
Derivation includes using conservation of mass, conservation of momentum, and dU = TdS – PdV (1st and 2nd laws of thermo).
Independent Variables (x,y,z,t)
Dependent Variables:
  x, y , z , t 
Density (scalar field)
 S 
CP  
 T
 T  P
T  x, y , z , t 
Velocity (vector field)
Thermal conductivity (scalar field)
1  V 
 
 Thermal expansivity [V is volume] (scalar field)
V  T  P
Specific heat at constant pressure (scalar field)
Temperature (scalar field)

V ( x, y , z , t )
k  x, y , z , t 
dS*
  x, y , z , t 
Deviatoric stress (tensor field)
Entropy changes related to “extra” processes
H  x, y , z , t 
Heat production (power per mass)
Some Explanation on Notation:

 : V
This is a scalar, defined by:
3
3
j 1
i 1

Vi
 ij
x j
This term describes the heat produced by friction, and is often called the “viscous dissipation.”
Some Explanation on Notation:
D
Dt
Is called the “material derivative,” and it is defined as:
D  
 V 
Dt t
The material derivative can act on a scalar or a vector. For example:
DT T 

 V  T
Dt
t




DV V

 V   V
Dt
t
Summary: Conservation Equations (No approximations)


   V 
t
Mass

DV


 p      g
Dt
Momentum

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass
 Inertia term. Material remains in constant motion unless acted upon by forces on the R.H.S.
DV

Momentum

 p      g
Dt

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass
Pressure term: material typically flows from high to low pressure.

DV


 p      g
Dt
Momentum

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass
Stress term: viscous and elastic processes that transfer stress.

DV


 p      g
Dt
Momentum

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass

DV


 p      g
Dt
Buoyancy term: the weight of the volume.
Momentum

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass

DV


 p      g
Dt
Momentum
Conduction term: heat diffuses from hot to cold

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass

DV


 p      g
Dt
Momentum
Adiabatic term: rising material expands and cools, vice versa.

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass

DV


 p      g
Dt
Momentum
Viscous dissipation term: viscous friction generates heat.

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass

DV


 p      g
Dt
Momentum
Heat production term: can be prescribed as needed.

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
Summary: Conservation Equations (No approximations)


   V 
t
Mass

DV


 p      g
Dt
Momentum
Additional entropy changes: can be prescribed as needed.

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt
Energy
We haven’t made any approximations thus far.
Now, we’ll work toward transforming these equations into the forms that we
typically use in modeling.
This involves 3 things:
1. Definition of a reference model.
2. Non-dimensionalization of variables
3. Approximations of physics and material parameters.
Reference Model
If the dynamics are largely driven by perturbations to a stable reference state, then
it often becomes convenient to decompose some variables into 2 parts, a
reference part and a perturbation part.
Density:
 x, y, z , t    REF
~
 x, y , z     x, y , z , t 
Pressure:
~
p x, y, z , t   pREF  x, y, z   p  x, y, z , t 
Note: The perturbation for pressure is given the name: dynamic pressure
~
p
Gravitational acceleration:


~
g  x, y, z , t   g REF  x, y, z   g  x, y, z , t 
Temperature:
~
T  x, y, z , t   TREF  x, y, z   T  x, y, z , t 
In our reference model, the reference pressure should be a self-consistent
hydrostatic pressure due to the reference density and reference gravity.
pREF

  REF g REF
Conservation of Mass:


   V 
t

 REF ~

   V 
t
t

~
   V 
t
Conservation of Momentum:

DV

~
~
~

  pREF  p        REF    g REF  g
Dt

DV


~
~
~
~

 pREF  p       REF g REF  g REF  g  REF   
Dt

DV



~
~
~
~


   REF g REF  p       REF g REF  g REF  g  REF   
Dt

DV

~
~
~

 p      g REF  g  REF  ~ 
Dt


Conservation of Energy:

DT
Dp
DS*
c p
   kT  T
  : V  H 
Dt
Dt
Dt

DT
D
DS*
~
 pREF  p    : V  H 
c p
   kT  T
Dt
Dt
Dt

DT
D
D ~
DS*
c p
   kT  T
pREF  T
p   : V  H 
Dt
Dt
Dt
Dt

D

pREF  pREF  V  pREF
Dt
t

D

pREF  V   REF g REF
Dt
So
 

DT
D ~
DS*
c p
   kT  T REF V  g REF   T
p   : V  H 
Dt
Dt
Dt
Summary: Conservation Equations, with reference model.
(Still no approximations)

~
   V 
t
Mass

DV

~
~
~

 p      g REF  g  REF  ~ 
Dt
Momentum
Energy
 

DT
D ~
DS*
c p
   kT  T REF V  g REF   T
p   : V  H 
Dt
Dt
Dt
Non-dimensionalization of variables.
Why!?
1. To make the problem scalable. For example, we can model mantle
convection in a fish tank, as long as we scale the parameters appropriately.
2. Leads to non-dimensional collection of variables that can be used to
characterize the system. For example, Rayleigh number, Reynolds number,
Dissipation number, Buoyancy number.
3. Allows for a better understanding of parameter trade-offs. For example, if
you double both density and viscosity, the system remains unchanged. For
example doubling density is equivalent to halving the viscosity.
4. Numbers are closer to unity. Allows better computational applicability.
There are many ways to non-dimensionalize a system. The key is: we make the rules.
Choose transformations that will be useful. For thermal convection problems, we can
transform the following:
Note: primed variables
are non-dimensional
 x, y, z , t    o  x, y, z , t 
  x, y , z 
 REF x, y, z    o  REF
~
~
 x, y, z , t    o  o T  x, y, z , t 

o 
V  x, y , z , t  
h
V  x, y, z , t 
 o o ~
~
p  x, y, z , t   2 p  x, y, z , t 
h
  x, y , z , t  
 o o
2
 x, y, z , t 
h


g REF  x, y, z   g o g REF  x, y, z 
~
~
g  x, y, z , t   g o g  x, y, z , t 

C p  C po C p
T ( x, y, z , t )  T T ( x, y, z , t )
k ( x, y, z , t )   o  oC p o k ( x, y, z , t )
   o 
H ( x, y , z , t ) 
t
h
2
o
C po T o
t
1
  
h

D o  D 
 2 
Dt h  Dt 
h
2
H ( x, y, z , t )
Summary: Non-dimensional Conservation Equations, with
reference model. Primes have been dropped. (Still no
approximations)

~
 o T     V 
t
Mass

3

 ~


g

Th
DV
~
o o o
 g REF 
 p      
Dt
 o o


  o o 

 
 o 
  o g o h 3  ~

 g  REF   o T ~ 
  o o 
Momentum
  o goh 
 
DT
T REF V  g REF  
c p
   kT  
 cp 
Dt
o


 o 

D
~

T
p   : V  H
2
 c p o h 
Dt
 o

Energy
We introduce the following collections of variables:
 o  o g o Th
Ra 
 o o
Di 
 o goh
3
Rayleigh number
Dissipation number
c po
o
Pr 
 o o
Prandtl number
Summary: Non-dimensional Conservation Equations, with
reference model. Primes have been dropped. (Still no
approximations)

~
 o T     V 
t
Mass
Momentum
 1
 
 Pr 


DV

~  1
~
~
~
 p      Ra g REF  Ra g 
 REF   
Dt
  o T

Energy
 

DT
D ~  Di 
 Di  o T 
c p
   kT  Di T REF V  g REF   
p 
T
 : V  H
Dt
Dt
 Ra 
 Ra 
Now, lets examine various approximations appropriate for mantle convection modeling:
Assumption:
~
0
t
Justification: This term is mainly relevant if the system has shock waves. Shock waves
can be important if convection velocities are comparable to the speed of sound.
Mantle convection velocities are much slower! cm/yr versus km/s.

   V   0
Conservation of mass
This is called the anelastic liquid approximation (ALA)
The Prandtl number is a measure of the viscous resistance to inertia.
o
Pr 
 o o
Viscous stresses act to resist continued motion. They diffuse momentum.
Imagine stirring a pot of honey and a pot of water. When you stop stirring, the water
will continue to flow, but the honey will stop.
For Earth’s mantle:
 o ~ 10 20 Pa s
2
m
 o ~ 10 6
s
kg
 o ~ 4000 3
m
Pr ~ 10 22
Therefore 1/Pr is close to zero!
Assumption:
1
0
Pr
This is called the infinite Prandtl number approximation
Justification: The Earth’s mantle has such a high viscosity that it requires constant
forcing to continue to flow.
 1
 
 Pr 


DV

~  1
~
~
~
 p      Ra g REF  Ra g 
 REF   
Dt
  o T



~  1
~
~
~
 p      Ra g REF  Ra g 
 REF     0
  o T

Conservation
of momentum
Assumption: There are no perturbations to the reference gravitational acceleration. This
term is usually only included if one wishes to include self-gravitation due to internal density
heterogeneities and dynamic topography.
~
g 0


~  1
~
~
~
 p      Ra g REF  Ra g 
 REF     0
  o T


~
~
 p      Ra g REF  0
Conservation
of momentum
Summary: Non-dimensional Conservation Equations, with reference
model. Primes have been dropped. Approximations: anelastic liquid
(ALA), infinite Prandtl number, static gravitational field.

   V   0
Mass

~
~
 p      Ra g REF  0
Momentum
Energy
 

DT
D ~  Di 
 Di  o T 
c p
   kT  Di T REF V  g REF   
p 
T
 : V  H
Dt
Dt
 Ra 
 Ra 
Assumption: Truncated Liquid Anelastic Liquid Approximation (TALA)
Dynamic pressure does not contribute to density perturbations:
~
~
 T x, y, z, t , C ( x, y, z, t ), p x, y, z, t 
The material derivative of dynamic pressure is negligible.
D~
p0
Dt
 

DT
D ~  Di 
 Di  o T 
c p
   kT  Di T REF V  g REF   
p 
T
 : V  H
Dt
Dt
 Ra 
 Ra 
Summary: Non-dimensional Conservation Equations, with reference
model. Primes have been dropped. Approximations: truncated
anelastic liquid (TALA), infinite Prandtl number, static gravitational
field.

  V   0 Mass

~
~
 p      Ra g REF  0
Momentum
 

DT
 Di 
c p
   kT  Di T REF V  g REF   
 : V  H
Dt
 Ra 
Energy
Approximation: Extended Boussinesq Approximation (EBA)
Density is constant, ρo, except for density perturbations in the momentum equation.
 REF x, y, z   1
~
 x, y, z   0 Except for in the momentum equation
The main consequence of this approximation is that makes the fluid incompressible:

 V  0
Summary: Non-dimensional Conservation Equations, with reference
model. Primes have been dropped. Approximations: Extended
Boussinesq Approximation (EBA), infinite Prandtl number, static
gravitational field.

 V  0
Mass

~
~
 p      Ra g REF  0
Momentum
 

DT
 Di 
cp
   kT  Di T V  g REF   
 : V  H
Dt
 Ra 
Energy
Approximation: Boussinesq Approximation (BA)
Neglect all terms that include the Dissipation number (viscous dissipation, adiabatic
heating/cooling). Specific heat, thermal expansivity, and thermal conductivity is constant.
Summary: Non-dimensional Conservation Equations, with reference
model. Primes have been dropped. Approximations: Boussinesq
Approximation (BA), infinite Prandtl number, static gravitational
field.

 V  0
Mass

~
~
 p      Ra g REF  0
DT
  2T  H
Dt
Energy
Momentum
The stress and density perturbation terms are prescribed for the particular problem at
hand.
For mantle convection, the stress is usually assumed to be viscous and isotropic.
 
 
1
 1
  bulk    V I   2      V I 
3
3



 
 1
  2      V I  (ALA and TALA)
3


  2 (EBA and BA)
Dimensional density perturbation
~
p
~

~
~
 T x, y, z , t , C x, y, z , t , p x, y, z , t   C   REF  T  
K

~ T x, y, z, t , C x, y, z , t   C  
~
REF T (TALA)
~
~
 T x, y, z , t , C x, y, z , t   C   oT
(EBA and BA)
(ALA)