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Zustandsgleichungen (III):Zustandsgleichungen (III):dynamische Eigenschaftendynamische Eigenschaften

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Antriebskraft: thermische Konvektion

Rayleigh Zahl „Ra“

Ra =0Tgd3

Thermischer Auftrieb (Energiequelle)

Schicht-dicke

Viskosität(behindernd)

Thermische Dissipation

(behindernd)

- thermischer Ausdehnungskoeffizient

g - Schwerebeschleunigung

0 - Dichte

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Übergang zu einerÜbergang zu einer

kontinuumsmechanischen kontinuumsmechanischen BeschreibungBeschreibung

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zu lösendeszu lösendes

Gl.-System !Gl.-System !

- thermischer Ausdehnungskoeffizient

0 - Dichte - thermische Leitfähigkeit - dynamische Viskositätcp - spezifische Wärmekapazität

g - Schwerebeschleunigung

Erhaltungssatz für die Masse Erhaltungssatz für die Masse **

Erhaltungssatz für die Energie Erhaltungssatz für die Energie **

ErhaltungssatzErhaltungssatzfür den Impuls für den Impuls **

(Kräftebilanz)(Kräftebilanz)

* D/Dt /t + vi/xi (substanzielle Ableitung)

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Zur Bestimmung der Zur Bestimmung der

dynamischen Eigenschaftendynamischen Eigenschaften

im Erdinnern benötigt man die im Erdinnern benötigt man die

Kenntnis der MaterialparameterKenntnis der Materialparameter

, , ccpp

als Funktion der Tiefe !als Funktion der Tiefe !(d.h. insbesondere in Abhängigkeit (d.h. insbesondere in Abhängigkeit

von Druck und Temperatur etc.)von Druck und Temperatur etc.)

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Numerische LösungNumerische Lösung

des denkbar einfachsten Falls:des denkbar einfachsten Falls:

alle Materialparameter alle Materialparameter konstantkonstant

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numerische Lösung für Ra = 104

(finite element solver „citcom“)

Raum-zeitliche Entwicklung der Temperatur TRaum-zeitliche Entwicklung der Temperatur T

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numerische Lösung für Ra = 105

(finite element solver „citcom“)

Raum-zeitliche Entwicklung der Temperatur TRaum-zeitliche Entwicklung der Temperatur T

Modul BWP1216.06.2010

numerische Lösung für Ra = 106

(finite element solver „citcom“)

Raum-zeitliche Entwicklung der Temperatur TRaum-zeitliche Entwicklung der Temperatur T

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Stabilitätsanalyse für denStabilitätsanalyse für den

WärmetransportWärmetransport

in diesem Fallin diesem Fall

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kritischeRayleigh-Zahl

Racr= 658

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Thermal modeling gives a driving force for subduction due to the integrated negative

buoyancy (sinking) of cold dense slab from density contrast between it and the warmer and

less dense material at same depth outside. Negative buoyancy is associated with the cold downgoing limb of mantle convection pattern.

Since the driving force depends on thermal density contrast, it increases for

(i) Higher v, faster subducting & hence colder plate

(ii) Higher L, thicker and older & hence colder plate

Expression is similar to that for “ridge push” since both forces are thermal buoyancy forces

““SLAB PULL” plate driving forceSLAB PULL” plate driving force

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Abschätzung von tektonischen Kräften

„ridge-push“ vs. „slab-pull“

~ 1012 Nm-1 ~ 1013 Nm-1

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Coldest portion reaches only ~ half mantle temperature in about 10 Myr, about the time required for the slab to reach 660 km.

Thus restriction of seismicity to depths < 660 km does not indicate that the slab is no longer a discrete thermal and mechanical entity.

From thermal standpoint, there is no reason for slabs not to penetrate into lower mantle.

When a slab descends through lower mantle at the same rate (it probably slows due to the more viscous lower mantle), it retains a significant thermal anomaly at the core-mantle boundary, consistent with models of that region

Slabs are not thermally equilibrated with mantleSlabs are not thermally equilibrated with mantle

Stein & Stein, 1996

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Was passiert,wenn die Viskosität

in der Erdenicht konstant ist,d.h. mit der Tiefe

abnimmt ?

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„Lithosphäre“

„stagnant lid“

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Plattentektonik

3 Typen von Plattengrenzen

Ozeane Kontinente

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simple scaling viewL

W

D

FR

FB

vplate T

density after expansion

t)1/2

cooling thickness time t

- bouyancy forceFR - resistance force

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density size gravity

mass acceleration*

„bouyancy force“

stress

area„resistance force“

because of

Plate tectonics: scaling view (I)

FB = DW ) g

FR = v/LDW )

and = = vL

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FB FR

~ Ra 2/3

Plate tectonics: scaling view (II)

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T = 1400 K temperature difference

= 3 ·10-6 m2/s thermal expansion

= 1022 Pa s viscosity

= 10-6 m2/s thermal diffusivity

= 3 ·103 kg/m3 density

g = 10 m/s2 grav. acceleration

L = 3 ·106 m layer thickness

plate velocity ~ 14 cm/yr !

Modul BWP1216.06.2010Stein & Wysession, Blackwell 2003

Different stresses result if weight of column of material supported in

different ways

similar to what seismic focal mechanisms show !

Forces within subducting platesForces within subducting plates

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Clapeyron slope describes how mineral phase Clapeyron slope describes how mineral phase changes occur at different depths in cold slabschanges occur at different depths in cold slabs

use thermal model to find dT, phase

relations to find and thus dP

convert to convert to depth change depth change

dzdz

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Opposite deflection of mineral phase boundariesOpposite deflection of mineral phase boundaries

Upward deflection of the 410 km and downward deflection of the 660 km discontinuities have been observed in travel time studies.

In contrast, the ringwoodite ( spinel phase) to perovoskite plus magnesiowustite transition, thought to

give rise to the 660 km discontinuity, is endothermic (absorbs heat) so H > 0. Because this is a

transformation to denser phases (V < 0), Clapeyron slope is negative, and the 660 km discontinuity should be

deeper in slabs than outside

Because spinel is denser than olivine, V < 0. This reaction is exothermic (gives off heat) so H < 0 is also negative, causing a positive Clapeyron slope. The slab is

colder than the ambient mantle (T<0 ), so this phase change occurs at a lower pressure (P<0), corresponding

to shallower depth

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Kirby et al., Rev. Geophys. 1996

Metastable delay of mineral phase transformationsMetastable delay of mineral phase transformations

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Predicted mineral phase boundaries and resulting buoyancy forces in slab with

and without metastable olivine wedge

For equilibrium mineralogy cold slab has negative thermal buoyancy, negative

compositional buoyancy from elevated 410 km discontinuity, and positive

compositional buoyancy from depressed 660 km discontinuity

Metastable wedge gives positive compositional buoyancy and decreases

force driving subduction

Stein & Rubie, Science 1999

negative buoyancy favours subduction, whereas positive buoyancy opposes it.

Metastable delay of mineral phase transformationsMetastable delay of mineral phase transformations

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Deep earthquakes from metastable olivine ?Deep earthquakes from metastable olivine ?

Kirby et al., Rev. Geophys. 1996

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Intermediate depth earthquakes (I)Intermediate depth earthquakes (I)

Under equilibrium conditions, eclogite

should form by the time slab reaches ~70 km

depth. However, travel time studies in some slabs

find low-velocity waveguide interpreted as

subducting crust extending to deeper

depths. Hence eclogite-forming reaction may be

slowed in cold downgoing slabs,

allowing gabbro to persist metastably.

Oceanic crust should undergo two important mineralogic transitions as it subducts. Hydrous (water-bearing) minerals formed at fractures and faults

warm up and dehydrate. Gabbro transforms to eclogite, rock of same composition composed of denser minerals.

Kirby et al., Rev. Geophys. 1996

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Intermediate depth earthquakes (II)Intermediate depth earthquakes (II)

Support for this model comes from the fact that

the intermediateearthquakes occur below the island arc volcanoes,

which are thought to result when water released from the

subducting slab causes partial melting in the

overlying asthenosphere.

In this model intermediate earthquakes occur by slip on faults, but phase changes favor faulting. The extensional focal mechanisms may also reflect the phase change, which would

produce extension in the subducting crust.

Kirby et al., Rev. Geophys. 1996

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various kineticprocesses during

subduction

P. van Keken, 2004

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Deep subduction process is a chemical reactor that brings cold

shallow minerals into temperature and pressure

conditions of mantle transition zone where these

phases are no longer thermodynamically stable.

Because there is no direct way of studying what is happening and what comes out, one seeks to

understandthe system by studying

earthquakes that somehow reflect what is happening.

Kirby et al., 1996

Complex thermal structure, mineralogy & geometry of Complex thermal structure, mineralogy & geometry of subducted slabs in the mantle transition zonesubducted slabs in the mantle transition zone