Transcript No Slide Title

TOWARD THE FULLY SELF-CONSISTENT MODELING OF CONTINENTAL THERMAL STRUCTURE
Prong 3: We now move to numerical simulations that allow for a lot of Earth
pertinent stuff not present in the simple analog systems. A long-term goal is to
have a heat flow scaling theory that can address situations of the complexity of
the
simulations. The partial lid scalings will help move us toward such a
theory and the simulations will test it. For now, the simulations are presented
on a stand alone basis. An interesting result is that local mantle heat flow into a
continent decreases as internal crustal heat goes up. This shows the non
seperability of the problem. One can not safely solve for continental thermal
structure by considering a conduction solution with a lower boundary condition
prescribed for mantle convection as convective flux depends on the structure of
the conductive portion of the lithosphere. The simulations, & partial lid scalings,
also show that local thermal structure depends on the % of continental to oceanic
lithosphere. Thus, continental growth comes into play. By the way, an extended
theory predicts the results rather well but more later after it’s fully tested ...
upper crust
8
9
10
11
12
108
109
1010
1011
1012
10
10
10
10
10
10 9
50
50
d=0.009
30
30
1010
bulk mantle
20
20
10
10
11
101011
12
101012
Ra
FULL LID SCALINGS AND SIMPLE NUMERICAL SIMULATIONS: The goal here is simple: for a convecting fluid layer covered by a conducting lid, predict the system
heat transfer properties as a function of lid thickness, lid conductivity, fluid properties, and the applied temperature drop across the system. This is a conjugated heat transfer
problem in that the degree of convective vigor in the fluid depends on the lid base temperature which itself depends on the degree of convective heat loss and on the properties of
the conducting lid. This makes things a bit different from the classic Rayleigh-Bernard case in which the temperature drop driving convection is not part of the solution itself. The
few simple assumptions noted above make the problem not as nasty as it may seem. Of course we must convince ourselves that the assumptions, and the scaling relations they lead
to, are valid. To do this we have compared the scaling predictions to the results of numerical simulations that solve the full conservation equations describing thermal convection
below a conducting lid. The results are good enough to convince us that the assumptions and the scaling ideas that follow from them are not at all bad. This system is very
idealized relative to the Earth but the ideas developed can serve as a springboard to more complex situations such as allowance for internal heating and for lids of finite extent.
L
T
~
Lid
d

Ti
V
~
Rd
R1
theory
simulation
100
100
Ra = 10 9
R2
0.4
0.6
0.8
Nu
Nu
0
0.018
10
10
Ra = 10
11
0
0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
11
LidLidExtent
Extent
200
200
250
250
200
200
mean %var 150
no lid lid lid
theory 151.4 27.8 5.0
sim t1 153.6 28.1 5.2 100
sim t2 181.4 27.7 5.2
50
150
100
100
100
50
50
50
Ra = 10 10 L = 0.2 d= 0.018
0
0
0.2
0.4
0.4
0.6
Length c
0.6
Length
1200
1600
moho
70 root base
140
210 mean surface
heat flux = 53 mW/m 2
280
0.018
80
250
375
500
125
250
375
500
51
22
70
140
125
80
root base
210 mean surface
heat flux = 52 mW/m 2
280
34.5
70
140 root base
11
210 mean surface
heat flux = 51 mW/m 2
280
18
moho
70
140 root base
210 mean surface
heat flux = 46 mW/m 2
280
0
400
800
1200
Temperature ( oK )
13.5
9
1600
0
Upper Crust Heat Generation Rate
Relative to that of the Bulk Mantle
Ra = 10 10 L = 0.6 d= 0.018
150
150
0.2
1010
800
0
300
300
250
250
0.076
d=0.076
mean %var
no lid lid lid
theory 125.1 23.0 7.1
sim t1 120.6 23.4 7.7
sim t2 115.3 23.5 7.0
0
The final scaling expressions are a bit cumbersome but
are fully given in the preprint available w/ the poster
10 7
d=0.018
6r1110c 254000
Lid
d=0.076
400
moho
d=0.018
5
0
Mean Craton Heat Fluxes for Increasing
Upper Crustal Heat Concentration
moho
d
7
Ra = 10
300
300
Lid
0.2
d=0.076
The First Main Assumption is that the analogy above is valid. Thus, if
we know something about the thermal resistance in the no-lid zone,
which we can get from classic boundary layer theory, and in the lid
zone, which we can get from our full lid scaling, then we can build a
composite thermal resistance via the circuit analogy. Its not only a
matter of resistance as we must also consider the driving voltage, err
that is to say, temperature drop. This depends on the average internal
temperature of the convecting fluid. The Second Main Assumption is
that this value is the volume weighted avg of the internal temperature
for the lid and no lid case. Pictorially it looks like this:
Lid
10 7
d=0.018
Convecting
Fluid
solve for
internal
temp in
no lid &
full lid
case then
average
accounting
for %
covered
by lid &
% free
Ra L
Geotherm Envelopes for
Increasing Root Thickness
mW/m 2
Ra
10
101010
mW/m 2
1099
10
mW/m2
1088
10
Surface Heat Flux
00
1077
10
The graphs to the left compare scaling predictions to numerical simulation
results for the average surface heat flux (top graph) and surface heat flux
variations (bottom graph) versus Rayleigh number & non-dimensional lid
thickness. Thermal fields from simulations are show in the images above.
Crustal Heat Flux
d=0.018
This allows us to determine the effects of lid thickness and thermal
conductivity on heat flow for a given Rayleigh number. The degree
of lateral heat flux variations can also be solved for (see preprint).
TL
“subducting” lithosphere
viscosity = 10 25 Pa s
cratonic root
theory
simulation
40
40
Rayleigh # based
on the full system
temperature drop
lower crust
Mantle Heat Flux
non-dimensional
avg lid base
temperature
non-dimensional
lid thickness
7
107
10
depth (km)
non-dimensional
lid conductivity
6
106
10
Ra
3
5
K
(1 - TL )
L
= C
3
TL3
(d - d 2 ) Ra
d
1055
10
warm mantle
21
viscosity = 10 Pa s
Ra
scaling constant
and
10 6
d=0.076
10
10
11
1044
10
% %variation
variation
L
d=0.018
local
geotherm
10 8
Express the Assumptions Symbolically, Do Some Algebra & We Get:
q=K
failed region - compression
d=0.140
MAIN ASSUMPTIONS:
1) System is bottom heated & tends toward thermal equilibrium
so that a) heat flow into lid base equals heat flow out
b) heat flow into fluid equals heat out of fluid
2) Nearly linear thermal gradient holds across lid and across active
thermal boundary layer of fluid.
3) The local boundary layer Rayleigh number remains near a
constant value [Howard, 1966]
TL - T s
failed region - extension
d=0.036
TL = lid base temperature *
Ti = avg internal temperature *
 = active thermal boundary
layer thickness *
key unknowns to be solved for
non-dimensional avg surface heat flux
0.076
depth (km)
*
Tb
0.036
depth (km)
Convecting
Fluid
0.018
depth (km)
Ti
0.000
convecting mantle
cold hot
d=0.009
theory
simulation
Nu
KF
Ra d
100
100
Nu
KL
L.-N. Moresi & H. Muhlhaus - CSIRO
0.8
0.8
0
1 0
1 0
0.2
0.2
0.4
0.4
0.6
Length c
0.6
Length
0.8
0.8
1
1
Surface Heat Flux
Prong
Prong 2:
2: The full lid stuff is overly idealized in many ways. One key way relates
to the fact that continents have finite lateral extent. So the second specific aspect
of the larger problem that we consider is how does the situation explored in the
full lid section change when the continental analogs have a finite extent. Any
issue tied to mantle heat partitioning between oceans at continents at present or
over time is related, to some degree, to this question. From a pure fluid dynamics
point of view, few theories exist for the case of thermal convection with imposed
lateral asymmetry. Thus exploring the simple analog system of the
section is
of use in moving toward our Earth oriented goal and also from a fluid dynamic
viewpoint (it may also relate to insulating your house properly).
TL
6r1110c 254000
Prong 1: The first aspect relates to answering the question of how the thickness
of long-lived continental lithosphere effects the amount of heat that flows into its
base from the mantle. If continental lithosphere is chemically distinct, stable, and
preserved at the Earth’s surface for times exceeding the lifetime of oceanic
lithosphere then, in effect, it forms a conducting lid that bounds the convecting
mantle and to explore its equilibrium thermal structure one must attack the
coupled problem involving the interaction of convection and conduction. An
idealized system that is useful for building some insight into this coupling is that
of a convecting fluid layer overlain by a conducting lid. The
section explores
such a system. Although idealized, keep in mind that theories of mantle heat loss
have not considered conjugate heat transfer issues directly which must be done to
get at continental thermal structure. So best to start simple to build needed insight.
Lid
Surface Heat Flux
THREE PRONGED APPROACH: The key missing ingredient in thermal
models of continental lithosphere to date is the convecting mantle. Few models,
geared specifically at constraining the equilibrium thermal structure of continental
lithosphere, have treated mantle convection and continental chemical structure in
a self-consistent way. This means that the majority of continental thermal models
have had to make some type of assumption about the dynamics of mantle
convection. The degree to which any of the assumptions are dynamically selfconsistent with the derived thermal structure of the lithosphere will remain
difficult to evaluate until the full range of dynamic interaction between mantle
convection and continents is physically understood. This is a big task so if you are
expecting a complete answer from this poster prepare to be disappointed. To
break the task up a bit and to allow for the development of physical insight we
have focussed on certain specific aspects of the full problem.
D = system depth
d = lid thickness
d Ts = surface temperature
Tb = base temperature

K L = lid thermal conductivity
K F = fluid thermal conductivity
Ts
D
OVERVIEW: The equilibrium thermal structure of continents depends on the
distribution of heat producing elements within the continental lithosphere and on
its thermal conductivity structure. It also depends on the amount of heat coming
from the convecting mantle. The majority of continental thermal models to date
have not treated convective mantle heat flux directly. Rather, they have been
based on solving the one-dimensional heat conduction equation in a layered
medium meant to represent the lithosphere. This approach treats mantle heat flow
as a free lower thermal boundary condition that is applied to a lithospheric
column. Although this approach has proved useful, particularly when additional
data constraints on deep thermal structure such as xenoliths exist, it can often hit a
fundamental impasse in that a variety of models with different proportions of
internal heat sources and mantle heat flow can be consistent with heat flow
observations [Rudnick et al., 1998]. The reason the standard thermal modeling
approach hits this impasse is because it inherently assumes that the component
of heat coming from the convecting mantle is decoupled from the local chemical
structure of the continental lithosphere and the strength and distribution of heat
sources within it. That is, within this type of modeling approach mantle heat
flow can take on a wide range of values for the same distribution and strength of
crustal heat sources and vice versa. This is why a variety of models can be made
to match heat flow data. Local mantle heat flux must, however, depend on the
chemical structure and the amount of heat produced within a continent in some,
as yet, unknown way, as these factors influences the local surface conditions that
the convecting mantle experiences. If the dependence of local mantle heat flux on
the local structure of continental lithosphere and on the amount of heat produced
within it could be determined, then the range of allowable thermal models for any
continental region could be more tightly constrained. In the broadest sense, this is
the goal of the work presented here.
A. Lenardic & C.M. Cooper - Rice University
1010
0.076
The upper graph to the left compares scaling predictions to numerical
simulation results for the average surface heat flux versus Rayleigh number,
lid thickness, & lid extent. The lower graphs to the left compare scaling
predictions and simulation results for heat flux profiles across the full
system. Thermal fields from simulations are show in the images above.
PARTIAL LID SCALINGS AND SIMPLE NUMERICAL SIMULATIONS: The goal is much as it was for the full lid except that there is an added parameter, lid extent. This
introduces a lateral symmetry breaking to the system which adds a bit of complexity. Again, a few simple assumptions can make things tractable. The basic one is that the lid and
no lid regions of the system run in parallel. The full lid scaling already made an indirect circuit analogy in that it considered heat transfer through the lid and through the active
thermal boundary layer of the fluid to run in series. We now take that series circuit and connect it in parallel to the free lid zone, whose resistance can be gotten at through classic
boundary layer theory. A final assumption is needed to determine the average internal temperature which, following the circuit analogy, is needed to determine the voltage
(temperature) drop driving electrical current (heat) out of the system. We make the simplest possible assumption to get at this. As with the full lid case we have tested the scaling
relations that result. The partial lid theory allows for more stringent testing as it makes predictions not only about system averaged heat transfer but also about the heat transfer
properties in the lid and no lid zone, i.e., it makes global and local predictions. The top & bottom center graphs show that global & local predictions match simulation results
rather well. The theory is easily extended to cover any number of lids with internal thickness variations above a convecting fluid - just add a more resistors to the circuit. This can
move us further from the idealized and toward the Earth-like case where heat transfer depends on multiple continents, with internal structure, interacting with a convecting mantle.
Geotherms from the center of a continent at several times
from models with varied root thickness. Mean heat fluxes and
variations can be extracted from the envelopes so as to map
the effects of root thickness on continental thermal structure.
Mean heat fluxes from the center of a continent from models
with varied crustal heat. Notice that local mantle heat flux
goes down as crustal heat goes up showing that mantle heat
flow is not decoupled from continental heat generation rate.
REASONABLY COMPLEX NUMERICAL SIMULATIONS: The model shown above has a chemically distinct
continent residing within the upper thermal boundary layer of a convecting mantle. The modeling formulation allows
for a layered continental crust with each layer having distinct heat production & thermal conductivity (lateral variations
can also incorporated although this is not employed above). Chemically distinct continental roots of variable shape and
heat production are also allowed for. The amount of heat that flows into a section of continental lithosphere within the
model is determined by the dynamics of mantle convection and its interaction with a continent. A version of the
CITCOM finite element code is used to solve model equation. Both continental and oceanic plates are incorporated
within the simulation. The incorporation of plate-like behavior relies on allowing for the formation of localized shear
zones that represent plate margins. This is accomplished through the use of a visco-plastic rheology akin to that used
by Moresi & Solomatov [1998] but also allowing for strain dependent weakening along the plastic rheologic branch
[Tackley, 1998]. Continental lithosphere is assumed to be stable by giving it a very high viscosity, which prevents
distributed ductile deformation, and a very high yield stress, which prevents localized shear zone formation. This
assumption allows the equilibrium thermal profile within a presently stable continental section to be determined. It is
not meant to isolate the factors that lead to stability. The preliminary results above explore how variations in crustal
heat generation and root thickness affect global system dynamics. The coupled nature of the system explored means
that local continental thermal structure is inseparably linked to global dynamics and must be considered in this context.