Chapter
7
C
onvection
and
complexity
...
if
your
theory
is
found
to
be
against
the
second
law
of
thermodynamics,
I
can
give
you
no
hope;
there
is
nothing
for
it
but
to
collapse
in
deepest
humiliation.
Eddington
Contrary
to
current
textbooks
...
the
observed
world
does
not
proceed
from
lower
to
higher
"degrees
of
disorder",
since
when
all
gravitationally-induced
phenomena
are
taken
into
account
the
emerging
result
indicates
a
net
decrease
in
the
"degrees
of
disorder",
a
greater
"degree
of
structuring"
...
classical
equilibrium
thermod
ynamics
...
has
to
be
completed
by a
theor
y
of
'creation
of
gravitationally-induced
structures'
...
Gal-or
Overview
In
1900
Henri
Benard
heated
whale
oil
in
a
pan
and
noted
a
system
of
hexagonal
convec-
tion
cells.
Lord
Rayleigh
in
1916
analyzed
this
in
terms
of
the
instability
of
a
fluid
heated
from
below.
Since
that
time
Rayleigh-Bemard
convection
has
been
taken
as
the
classic
exam-
ple
of
thermal
convection,
and
the
hexagonal
planform
has
been
considered
to
be
typical
of
convective
patterns
at
the
onset
of
thermal
con-
vection.
Fifty
years
went
by
before
it
was
real-
ized
that
Benard's
patterns
were
actually
driven
from
above,
by
surface
tension.
not
from
below
by
an
unstable
thermal
boundary
layer.
Experiments
showed
the
san1.e
style
of
convection
when
the
fluid
was
heated
from
above,
cooled
from
below
or
when
performed
in
the
absence
of
gravity.
This
confirmed
the
top-down
surface-driven
nature
of
the
convection
which
is
now
called
Marangoni
or
B€mard-Marangoni
convection.
Although
it
is
not
generally
recognized
as
such,
mantle
convection
is
a
branch
of
the
newly
renamed
science
of
complexity.
Plate
tectonics
may
be
a
self-drivenfarjrom-equilibrium
system
that
orga-
nizes
itself
by
dissipation
in
and
between
the
plates,
the
mantle
being
a
passive
provider
of
energy
and
material.
Far-from-equilibrium
sys-
tems,
particularly
those
in
a
gravity
field,
can
locally
evolve
toward
a
high
degree
of
order.
Plate
tectonics
was
once
regarded
as
passive
motion
of
plates
on
top
of
mantle
convection
cells
but
it
now
appears
that
continents
and
plate
tecton-
ics
organize
the
flow
in
the
mantle
.
But
man-
tle
convection
and
plate
tectonics
involve
more
than
geometry
and
space-filling
considerations.
The
mantle
is
a
heat
engine,
controlled
by
the
laws
of
thermodynamics
.
One
can
go
just
so
far
without
physics
.
Conservation
of
mass
and
energy
are
involved
, as
are
balancing
of
forces.
Although
the
mantle
behaves
as
a fluid,
mineral
physics
principles
and
classical
solid-state
physics
74
CONVE
CT
ION
AND
C O
MP
LE
XITY
are
needed
to
understand
this
fluid.
The
effect
of
pressure
suppresses
the
role
of
the
lower
ther-
mal
boundary
layer
at
the
core-mantle-boundary
(CMB)
interface
.
The
slow
uplift
of
the
surface
of
the
Earth
in
response
to
the
removal
of
an
ice
cap
or
drainage
of
a
large
lake
changes
the
shape
of
the
Earth
and
the
geoid;
this
is
not
only
proof
of
the
fluid-like
behavior
of
the
mantle
but
also
pro-
vides
the
data
for
estimating
its
viscosity
.
In
con-
tra
st
to
everyday
experience
mantle
convection
has
some
unusual
characteristics.
The
container
has
spherical
geometry
.
The
'
fluid'
has
stress-
,
pressure-
and
temperature-dependent
properties.
It
is
cooled
from
above
and
from
within
(slabs)
and
heated
from
within
(radioactivity)
and
from
below
(cooling
of
the
core
and
crystalliza-
tion
of
the
inner
core)
.
The
boundary
conditions
and
heat
sources
change
with
time
.
Melting
and
ph
a
se
changes
contribute
to
the
buoyancy
and
provide
additional
heat
sources
and
sinks
.
Mantle
convection
is
driven
partly
by
plate
motions
and
partly
by
chemical
buoyancy.
The
boundaries
ar
e
deformable
rather
than
rigid
.
None
of
these
char-
acteristics
are
fully
treated
in
numerical
calcu-
lations
,
and
we
are
therefore
woefully
ignorant
of
the
style
of
convection
to
be
expected
in
the
mantle.
The
cooling
plates
may
well
organize
and
drive
mantle
convection
, as
well
as
themselves
. A
mantle
with
continents
on
top
will
conv
e
ct
dif-
ferently
from
one
with
no
continents
.
Th
e
theory
of
convection
in
the
mantle
can-
not
be
decoupled
from
the
theories
of
solids
and
petrology
.
The
non-Newtonian
rheology
,
the
pressure
and
temperature
sensitivity
of
viscosity
,
thermal
expansion,
and
thermal
conductivity
,
and
the
effects
of
phase
changes
and
compress-
ibility
make
it
dangerous
to
rely
too
much
on
the
intuition
provided
by
oversimplified
fluid-
dynamic
calculations
or
labor
a
tory
experiments
.
There
are,
however,
some
general
characteris-
tic
s
of
c
onvection
that
transcend
these
details.
Technical
details
of
normal
or
classical
thermal
conve
c
tion
can
be
found
in
text
b
ooks
on
man
-
t l e
co
n
vection.
Plate
tectonics
and
mantle
motions,
however,
are
far
from
normal
thermal
convection
.
Geochemists
consider
convection
and
stir-
ring
to
be
equivalent
.
They
use
convect
in
g
man
t l e
as
convenient
sho
r
thand
for
w h
at
t h
ey
consi
d
er
to
be
the
h o mo g e n o u s
upper
man
t l
e.
The
u
nder
l
ying
assumption
is
that
mi
d
ocean
ri
dge
basa
l
ts
,
known
for
their
chemica
l
homo-
geneity,
must
come
from
a
well-stirred
mantle
reservoir
.
Generalities
SO
FF
E
syste
ms
are
ex
t
rao
rdinaril
y
se
n -
s i
tive
to
bo
un
dary
and
initial
conditions.
The
corollary
is
that
small
differences
between
computer
or
laboratory
simulations
,
or
between
t h
em
and
the
mantle,
can
completely
change
the
outcome
.
The
effect
of
pressure
suppresses
the
role
of
the
lower
thermal
boundary
layer
(TBL)
at
the
core-mantle-boundary
(CMB)
inter-
face
.
The
state
of
stress
in
the
lithosphere
defines
the
plates,
plate
boundaries
and
locations
of
mid-
plate
volcanism
.
Fluctuations
in
stress
,
due
to
changing
boundary
conditions,
are
responsible
for
global
plate
reorganizations
and
evolution
of
volcanic
chains
.
In
Rayleig
h
-BEmard
con-
vect
i
on
, by
contrast,
temperature
fluctuations
are
the
important
parameters.
In
plume
theory
,
plates
break
where
heated
or
uplifted
by
hot
buoyant
upwellings.
Ironically
,
the
fluid
flows
in
the
exper
i me
n
ts
by
Benard
,
which
motivated
the
Rayleigh
theory
of
t h
ermal
convec-
tion
,
were
driven
by
surface
tension,
i.e.
stresses
at
the
surface.
Computer
simulations
of
mantle
convection
have
not
yet
included
a
self-consistent
thermo-
dynamic
treatment
of
the
effect
of
temperature,
pressure,
melting
and
volume
on
the
physical
and
thermal
properties
;
understanding
of
the
'
exterior'
problem
(the
surface
boundary
con-
dition)
is
in
its
infancy.
Plate
tectonics
itself
is
implicated
in
the
surface
boundary
condi-
tion
.
Sphericity
,
pressure
and
the
distribution
of
radioactivity
break
the
symmetry
of
the
prob-
lem
and
the
top
and
bottom
boundary
condi-
tions
play
quite
different
roles
than
in
the
simple
calculations
and
cartoons
of
mantle
dy
nam-
ics
and
geochemi
c
al
reservoirs.
Conventional
(Rayleigh
-
Benard)
convection
theory
may
have
lit
-
tle
to
do
with
plate
tectonics.
The
research
oppor-
tunities
are
enormous
.
The
history
of
ideas
Convection
can
be
driven
by
bottom
heating,
top
or
side
cooling,
and
by
motions
of
the
boundaries
.
Although
the
role
of
the
surface
boundary
layer
and
slab-pull
are
now
well
under-
stood
and
the
latter
is
generally
accepted
as
the
prime
mover
in
plate
tectonics,
there
is
a
widespread
perception
that
active
hot
upwellings
from
deep
in
the
interior
of
the
planet,
inde-
pendent
of
plate
tectonics,
are
responsible
for
'e
xtraordinary
'
events
such
as
plate
reorganiza-
tion
,
continental
break-up
,
extensive
magmatism.
a
nd
events
far
away
from
current
plate
bound-
a
ries
. Active
upwellings
from
deep
in
the
mantle
are
viewed
as
controlling
some
aspects
of
surface
tectonics
and
volcanism,
including
reorganiza-
tion
,
implying
that
the
mantle
is
not
passive.
This
is
called
the
plume
mode
of
mantle
convec-
tion.
This
has
been
modeled
by
the
injection
of
hot
fluids
into
the
base
of
a
tank
of
motionless
fluid.
Numerical
experiments
show
that
mantle
convection
is
controlled
from
the
top
by
con-
tinents,
cooling
lithosphere,
slabs
and
plate
motions
and
that
plates
not
only
drive
and
break
themselves
but
can
control
and
reverse
convection
in
the
mantle.
Studies
of
the
time
dependence
in
3D
spherical
mantle
convection
with
continental
drift
show
the
extreme
sensitivity
to
changes
of
conditions
and
give
results
quite
different
from
simpler
sim-
ulations
.
Supercontinents
and
other
large
plates
generate
spatial
and
temporal
temperature
vari-
ations
.
The
migration
of
continents,
ridges
and
trenches
cause
a
constantly
changing
surface
boundary
condition,
and
the
underlying
man-
tle
responds
passively.
Plates
break
up
and
move,
and
trenches
roll
back
because
of
forces
on
the
plates
and
interactions
of
the
lithosphere
with
the
mantle
.
Density
variations
in
the
mantle
are,
by
and
large
,
ge
nerated
by
plate
tectonics
itself
by
slab
cooling,
refertilization
of
the
man-
tle,
continental
insulation;
these
also
affect
the
forces
on
the
plates.
Surface
plates
are
constantly
evolving
and
reorganizing
although
major
global
reorganizations
are
infrequent.
Plates
are
mainly
under
lateral
compression
although
loc
al
regions
having
horizontal
least-compressive
axes
may
be
the
locus
of
dikes
and
volcanic
chains.
The
GENERALITIES
75
Aegean
plate
is
an
example
of
a ' rig id'
plate
col-
lapsing,
or
falling
apart
,
because
of
changes
in
stress
conditions.
The
mantle
is
generally
considered
to
convect
as
a
sing
le
layer
(whole
mantle
convection),
or
at
most
two.
However,
the
mantle
is
more
likely
to
convect
in
multiple
layers
as
a
result
of
gravita-
tional
sorting
during
accretion,
and
the
density
difference
between
the
mantle
products
of
differ-
entiation.
Instabilities
Rayleigh-Taylor
(RT)
instabilities
form
when
a
dense,
heavy
fluid
occurs
above
a
low-den
sity
fluid
,
such
as
a
layer
of
dense
oil
placed,
care-
fully,
on
top
of
a
layer
of
water.
Two
plane-
parallel
layers
of
immiscible
fluid
are
stable,
but
the
slightest
perturbation
leads
to
release
of
potential
energy,
as
the
heavier
material
moves
down
under
the
(effective)
gravitational
field,
and
the
light
er
material
is
displaced
upwards
.
As
the
instability
develops,
downward-moving
dim-
ples
are
quickly
magnified
into
sets
of
inter-
penetrating
RT
fingers
or
plumes.
This
process
is
evident
not
only
in
many
examples,
from
boil-
ing
water
to
weather
inversions.
In
mantle
geo-
physics,
plumes
are
often
modeled
by
inserting
a
light
fluid
into
a
tank
of
a
static
higher
density
fluid.
This
is
meant
to
mimic
the
instability
of
a
hot
basal
layer.
In
the
later
situation,
the
insta-
bility
develops
naturally
and
the
density
con-
trast
is
limited.
In
the
injection
experiment,
the
density
contrast
is
impo
sed
by
the
experimenter,
as
is
the
scale
of
the
upwelling.
There
is
a dif-
ference
between
upwellings
of
intrinsically
hot
basal
layers
and
intrinsically
light
chemical
lay-
ers.
The
former
case
sets
up
the
lateral
temper-
ature
gradients
that
are
the
essence
of
thermal
convection.
Rise
of
deep
diapirs
Delamination
and
sinking
of
garnet
pyroxenite
cumulates,
sinking
of
slabs,
and
upwelling
of
mantle
at
ridges
are
important
geodynamic
pro-
cesses
.
Diapiric
ascent,
melt
extraction
and
crys-
tal
settling
are
important
processes
in
i
gneous
petrology
. Basic
melts
apparently
separate
from
magma
chambers,
or
rising
diapirs,
at
depths
as
great
as
90
km
and
possibly
greater.
Eclogite
76
CONVECTION
AND
COMPLEXITY
sinkers
may
equilibrate
at
upper
mantle
or
tran-
sition
zone
depths;
they
then
warm
up
and
rise.
The
basic
law
governing
ascent
and
settling,
Stokes'
Law,
expresses
a
balance
between
grav-
itational
and
viscous
forces,
V
=
2b.pgR
2
/
91J
where
R
is
the
radius
of
a
spherical
particle
or
diapir,
b.p
is
the
density
contrast,
T/
is
the
dynamic
viscosity
and
V
is
the
terminal
veloc-
ity.
This
equation
can
be
applied
to
the
rising
or
sinking
of
blobs
through
a
mantle
or
a
magma
chamber,
with
modifications
to
take
into
account
non-spherical
objects
and
non-Newtonian
viscos-
ity
.
Additional
complications
are
introduced
by
turbulence
in
the
magma
chamber
and
finite
yield
strengths.
Diapirs
are
usually
treated
as
isolated
spheres
or
cylinders
rising
adiabatically
through
a
static
mantle
. Because
of
the
relative
slopes
of
the
geotherm,
and
the
melting
curve,
diapirs
b
eco
me
more
molten
as
they
rise.
At
some
point,
because
of
the
increased
viscosity
or
decreased
density
contrast,
ascent
is
slowed
and
cooling,
crystal-
lization
and
crystal
settling
can
occur.
The
litho-
sphere
serves
as
a
viscosity
or
strength
barrier
,
and
the
crust
serves
as
a
density
barrier
.
Melt
separation
can
therefore
be
expected
to
occur
in
magma
chambers
at
shallow
depths.
In
a
convecting
mantle
the
actual
temperatures
(adi-
abatic
or
subadiabatic)
diverge
from
the
melt-
ing
point
as
depth
increases
.
In
a
homo
ge
nous
mantle,
melting
can
therefore
only
occur
in
the
upper
parts
of
the
rising
limbs
of
convection
cells
or
in
thermal
boundary
layers.
The
addi-
tional
buoyancy
provided
by
melting
contributes
to
the
buoyancy
of
the
ascending
limbs.
Although
the
melts
will
attempt
to
rise
relative
to
the
adjacent
solid
matrix
,
they
are
embedded
in
a
system
that
is
itself
rising
and
melting
further.
If
broad-scale
vertical
convection
is
fast
enough,
diapirs
can
melt
extensively
without
fractionat-
ing.
Fertile,
low
-
melting-point
patches
,
such
as
eclogite,
can
melt
extensively
if
surrounded
by
subsolidus
peridotite
.
The
stresses
and
temperatures
in
the
vicinity
of
rising
plumes
or
diapirs
are
hi
gh,
and
these
serve
to
decrease
the
mantle
viscosity;
thus
rapid
ascent
is
possible.
In
order
to
achieve
observed
magma
temperatures
and
large
degrees
of
partial
melting,
allowing
for
specific
and
latent
heats,
melting
probably
initiates
at
depths
of
order
200
km
under
oceanic
ridges
and
large
volcanic
provinces,
assuming
that
the
mantle
is
mainly
peridotite
.
The
solidus
temperature
of
dry
peri-
dotite
at
this
depth
is
at
least
2100
o
c.
One
ques-
tion
is,
how
fast
can
material
rise
between
about
200
and
90
km,
and
is
the
material
at
90
km
rep-
resentative
of
the
deeper
mantle
source
region
or
has
it
been
fractionated
upon
ascent?
Viscosities
in
silicates
are
very
stress-
and
temperature-
dependent
,
and
diapirs
occur
in
regions
of
the
mantle
that
have
higher
than
normal
stresses
and
temperatures.
Diapiric
emplacement
itself
is
a
high-stress
process
and
occurs
in
regions
where
mantle
convection
may
have
oriented
crys-
tals
along
flow
lines.
Diapirs
may
rise
rapidly
through
such
low-viscosity
material.
A
50
km
par-
tially
molten
diapir
at
a
depth
of
200
km
can
rise
at
a
rate
of
about
40
cmjs
.
Kimberlites
travel
an
order
of
magnitude
faster
still.
Crystal
set-
tling
velocities
in
magmas
are
of
the
order
of
cmjs.
It
appears
therefore
that
deep
diapirs
can
rise
rapidly
enough
to
entrain
both
melt
and
crystals.
At
depth
the
melt
content
and
the
per-
meability
are
low
,
and
melt
segregation
may
be
very
slow
.
The
fertile
low-melting
point
blobs
may
be
encased
in
relatively
impermeable
subsolidus
peridotite
and
can
therefore
melt
extensively
as
they
rise.
In
a
chemically
stratified
mantle
,
for
example
residual
peridotite
over
eclogite
or
fertile
peri-
dotit
e,
there
is
a
conductive
thermal
boundary
between
the
convecting
layers
.
In
such
a
region
the
thermal
gradient
is
in
excess
of
the
meltin
g
gradient,
and
melting
is
likely
to
initiate
at
this
depth
. Eclogite
has
a
melting
temperature
about
200
oc
below
that
of
dry
peridotite
and
meltin
g
of
fertile
blobs
is
also
likely
between
TBLs
.
Partial
melting
causes
a
reduction
in
density,
and
a
Rayleigh-Taylor
instability
can
develop.
Material
can
be
lifted
out
of
the
eclogite
or
piclogite
-
eclogite
plus
peridotite-
layer
by
such
a
mecha-
nism
and
extensive
melting
occurs
during
ascent
to
the
shallower
mantle
. At
shallow
depths
pe
ri-
dotite
elevated
adiabatically
from
greater
depths
can
also
melt
and
magma
mixing
is
likely,
par
-
ticularly
if
the
diapir
is
trapped
beneath
thick
GENERALITIES
77
BUOYANT
ASCENT
AIDED
BY
MELTING
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Methods
of
removing
mate
ria
l
from
a
deep
.
dense
layer
.
In
a
chemically
layered
mantle
,
the
interface
between
layers
is
highly
deformed
. This
may
cause
phase
changes
and
melting.
and
a
reduction
in
density
.
The
deeper
l
ayer
may
also
be
entrained.
If
the
deep
lay
e r
is
eclogite,
from
subduction
or
delamination,
it
will
have
a
lower
melting
temperature
than
ambient
mantle
an
d may
rise
because
of
melt-induc
ed
buoyancy.
Adiabatic
decomp
ressio
n
leads
to
further
melting.
lithosphere.
In
mid
-
plate
environments,
such
as
Hawaii
and
other
midplate
hotspots
,
melts
will
cool,
fractionate
and
mix
with
other
melts
prior
to
eruption.
Such
a
mechanism
seems
capable
of
explaining
the
diversity
of
hotspot
(ocean
island,
continental
flood)
basalts
and
midoc
ean
ridge
magmas.
Material
can
leave
a
deep,
dense
source
region
by
several
mechanisms
.
(1)
Melting
in
the
thermal
boundary
because
of
the
high
thermal
gradient
compared
with
melting
point
grad
ients.
(2)
Melting,
or
phas
e
changes,
due
to
adiabatic
ascent
of
hotter
regions
of
a
layer
and
crossing
of
phase
boundar
y.
(3)
Entra
inment
of
material
by
adjacent
convect-
ing
layers
.
Some
of
these
me
c
hanism
s
are
illustrated
in
Figure
7.1.
The
potential
temperature
of
the
mantle
is
the
temperature
of
the
mantle
adiabat
if
it
were
to
ascend
directly
to
the
surface.
The
potential
tem-
perature
of
the
mantle
is
usually
between
1300
and
1400
°
C,
averaged
over
large
areas.
The
adi-
abatic
temperature
gradient
for
the
so
lid
upper
mantle
is
app
roximately
0 .3
-0.5
°Cfkm.
The
adia-
batic
g
radient
becomes
smaller
at
very
high
pres-
Slues
because
the
thermal
expansivities
of
solids
are
smaller
at
high
pressures
.
The
adiabatic
gra-
dient
for
liquids
(
~
1
°C/km)
is
higher
than
that
for
solids
because
the
thermal
expansivities
of
liquids
are
grea
ter
than
solids.
Meltin
g
occurs
where
the
geotherms
inter
-
sect
the
mantle
so
lidus
. For
a
dry
peridotite
mantle,
the
geo
therm
(conduction
gra
dient)
in
a
region
of
high
surface
heatflow,
100
mW
/m
2
,
intersects
the
solidus
a t
~
30
km
depth.
Surface
heat
flows
of
100
mWjm
2
or
greater
occur
only
at
or
near
midocean
ridges.
A
geotherm
with
40
mWjm
2
surface
heatflow,
c
haracteri
stic
of
the
old
interior
s
of
continents,
never
intersects
the
dry
peridotite
solidus
,
implying
that
partial
m e lt-
ing
does
not
occur
beneath
continental
interiors
.
In
suture
belts
and
a
long
arcs
there
may
be
low-
melting
point
constituents
s
uch
as
eclogite
in
the
shallow
mantle
.
High
heatflow
,
however
,
in
part,
represents
intrusion
into
the
plate
or
thinning
of
the
thermal
boundar
y
la
yer,
rather
than
intrinsi-
cally
high
mantle
temperatures
.
We
can
induce
melting
at
low
temperature
s
if
we
flux
the
mantle
with
basalt
,
eclogite,
C0
2
or
water,
and
plate
tectonics
do
es
all
of
these
things.
If
portions
of
the
mantle
upwell
rapidly
,
either
passively
in
response
to
spreading
at
ridges
or
displacement
by
sinking
slabs,
decompressional
partial
melting
occurs
.
Decompressional
meltin
g