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Introduction to Tectonophysics Patrice F. Rey CHAPTER 1 The Earth’s Geotherm The geotherm, i.e. the distribution of temperature with depth, is an important characteristics of the Earth's lithosphere because temperature impacts on all physical properties of rocks (e.g. density, viscosity, conductivity, elasticity, magnetism etc). In particular, temperature controls the rheology of rocks and therefore how they deform in response to applied deviatoric stresses, and how the Earth's lithosphere reacts to tectonic forces. In this chapter we derive, from first principles, a simple expression for the geotherm. 1 SECTION 1 Heat Transfer in the Earth’s lithosphere In this section 1. Heat transfer in the Earth’s Lithosphere 2. Heat energy and temperature 3. Heat conduction 4. Heat advection 5. Radiogenic heat production The continental geotherm is a function of the i/ rate at which heat is produced or consumed within the lithosphere, ii/ the rate at which the lithosphere looses heat to the atmosphere/ ocean system, and iii/ the rate at which the lithosphere gains heat from the hot convective mantle. When the heat lost by the lithosphere balances the heat gained by the lithosphere, an equilibrium is reached and the geotherm is said to be steady state 2 (i.e. the temperature at any given depth does not change through time). In contrast, when the lithosphere has a net gain or a net loss of heat, the geotherm is said to be transient (i.e the temperature changes through time) until a new equilibrium is reached. On a billion year time scale, the geotherm is always transient because the primordial accretionary heat and the Earth’s supply in radiogenic isotopes progressively decrease, •Heat conduction (transfer of kinetic energy between molecules or atoms from a hot to a less hot region) •Heat advection (replacement of a volume of rock at temperature T1 with an equivalent volume at temperature T2) •Heat production (heat produced by radioactive isotopes, viscous heating, exothermic metamorphic reactions) • Heat consumption (heat consumed by endothermic metamorphic reactions, in particular partial melting) however, on a scale of 100 myr, and in the absence of tectonic activity, the geotherm can approach an equilibrium which expresses the balance between heat gained and heat lost by the lithosphere. Here, we first review the processes involved in heat generation and heat transfer, and we derive from the rate of these processes a general equation which describes the change in temperature with depth and through time. From this general equation we derive a particular solution for the so called "steady state" continental geotherm (temperature changes with depth but not with time, i.e. zero net heat gain or loss). In a second part, we discuss how the steady state continental geotherm is affected by a number of geological processes including, lithospheric thinning and thickening, burial via sedimentary or volcanic processes, and basal heating via the spreading of mantle plumes at the base of the Earth's lithosphere. The variation of temperature dT over an increment of time dt depends on the sum of heat variations dE due to each process. In what follows, we derive three expressions for i/ the rate of heat conduction, ii/ for the rate of heat advection, and iii/ for the rate of radiogenic heating. From these, we derive the 1D conductionadvection heat transfer equation from which an expression for the steady state geotherm can be derived. Sounds more complicated than it really is. So bear with me ... Temperature and Heat The temperature (degree of hotness or coldest) of a small volume of rock somewhere in the lithosphere varies if heat energy (a form of kinetic energy) is gained or lost. The relationship that gives the variation of temperature dT as a function of a variation of heat dE is: Heat conduction Conduction transports heat from hot to cold regions. The flow of heat (Q) is proportional to the negative temperature gradient (dT/ dz) between the cold and the hot region, with the coefficient of proportionality being the conductivity (k). Mathematically this translates into the Fourier's law where Q is in W.m-2 and k is in W.m-1.K-1. In our reference frame, z increases downward (T(z+dz)>T(z)). Conduction occurs in the direction of decreasing temperature (i.e. dEc is a gain for upward conduction) hence the dT=dE/(Cp.m) with Cp the heat capacity, and m the mass. The main processes able to change the amount of heat energy in the lithosphere are: 3 sign "-" insures that Q is positive upward (dEc is positive when T increases downward). Q =−K E1=Q(z).a.dt z Q(z) dEc = E2 - E1 dEc = (dQ/dz).dz.a.dt a z+dz E2=Q(z+dz).a.dt Radiogenic heat production Radiogenic desintegration of radioactive isotopes (238U, 235U, 232Th, and 40K) releases heat. The increment of radiogenic heat (dEr) produced in a small cylinder of rock of section a and length dz over an increment of time dt is: dT dz Let's consider a small cylinder of rock of section a (area in m2). If the incoming and outgoing heat at both ends of the cylinder are the same, there is no net heat gain or loss, and the temperature remains unchanged. Temperature changes when the total heat E1=Q(z) ⋅ a ⋅ dt leaving the volume over a time interval dt is different to the total heat E2=Q(z + dz)⋅a ⋅dt entering it. A⋅a⋅dz⋅dt = A⋅dV⋅dt z A dEr = A.a.dz.dt z+dz a with A the rate of radiogenic heat production. Radioactive heat is the main internal heat source for the earth as a whole (it is measured in W.m-3). The entering heat flow Q(z + dz) can be Q(z+dz) approximated with a Taylor serie in which only the two first terms are of significance. (nb: mathematically f (xn + d x) can be approximated from f (xn) and the derivatives at location xn : f ′(xn) , f ′′(xn) , etc: Heat Advection Advection of heat implies that an amount of material at temE1 =Cp.ρ.a.uz .dt.T z perature T (in yellow on the sketch) is being pushed out of dEu = E2 - E1 our cylinder and replaced by dEu = C .ρ.a.u .dt.dT an equivalent amount of matez p rial at temperature T+dT. The increment of heat gained or a lost (dEu) over an increment of E ) time dt is proportional to the z+dz 2 =Cp.ρ.a.uz .dt.( T+dT mass of material displaced ( ρ ⋅a⋅uz ⋅dt ), the heat capacity dQ dz 2 d 2Q Q(z + dz) = Q(z) + dz⋅ + ⋅ + ⋅⋅⋅ dz 2 dz 2 Therefore the increment of heat (dEc) gained or lost in an increment of time dt is dEc = E1 − E2 : dQ d 2T a⋅Q(z)⋅dt − a⋅Q(z + dz)⋅dt = − a⋅dz⋅ ⋅dt = K⋅ 2 ⋅dV⋅dt dz dz 4 a.uz . dt a.uz . dt to right, the conduction term, the term of radiogenic heat production, and the advective term. In 3D, this equation becomes: of the material (Cp, J kg-1 K-1: the amount of energy required by 1kg of the material to increase its temperature by 1 K), and the temperature contrast of the two exchanged volumes. (nb: because z increasing with depth, uz is negative for upward convection, however, dEu is a gain for upward convection hence the sign - insures that when uz is negative (i.e. upward motion) the advective heat is positive). We get that the rate of heat advection is: ∂T K ∂2T ∂2T ∂2T A ∂T = ⋅[ 2 + + ] + − u ⋅ z ∂t ρ⋅Cp ∂x ∂y 2 ∂z 2 ρ⋅Cp ∂z −Cp ⋅ρ⋅a⋅uz ⋅dt ⋅dT The geotherm in the continental crust Total heat gained or lost We derive here the equation for the steady state continental geotherm. For a steady state geotherm we get that dT/dt=0 and uz = 0 (no convection in the lithosphere and no erosion or sedimentation); therefore the heat transfer equation simplifies to: Adding up the rate of conductive heat (dEc), the rate of convective heat (dEu), and the rate of radiogenic heat (dEr) gives the total rate of heat gained or lost. This variation of heat triggers a change in temperature dT = dE/(Cp.m) therefore : d 2T A =− dz 2 K Cp.m. dT= dEr + dEu + dEc and: This is a second order differential equation. This kind of equation is solved by integrating twice and by using two boundary conditions. For example we may know the temperature at the surface let's say: T=To at z=0; and we may know the surface heat flow for instance at z=0 the heat flow is -Qo (remember this is positive). Assuming that A is constant with depth, the first integration led to the temperature gradient dT/dz: d 2T Cp ⋅ρ⋅dV⋅dT = A⋅dV⋅dt − Cp ⋅ρ⋅a⋅uz ⋅dt ⋅dT + K⋅ 2 ⋅dV⋅dt dz This leads to the 1D conduction-advection heat transfer equation: dT K d 2T A dT = ⋅ 2 + − uz ⋅ dt ρ⋅Cp dz ρ⋅Cp dz dT A = − ⋅z + C1 dz K This equation describes the variation of temperature with depth and through time due to heat conduction, radiogenic heat and heat advection. This equation assumes no lateral heat flow (hence 1D). On the right end side of the equation the three terms are, from left This gives the slope of the geotherm as a function of depth. From this function we get at the surface: dt/dz=C1 5 relationships, one for the crust (equation in the previous slide), one for the lithospheric mantle. The second boundary condition demands, via the Fourier's law, that at z=0: dT Q0 = dz K If the production of radiogenic heat is zero in the mantle then we get: d 2T =0 dz 2 by combining the last two equations for z=0 we get C1 = Qo/K, we can therefore write that... Q dT A = − ⋅z + 0 dz K K Integrating twice we get: T(z) = C2 ⋅z + C3 Integrating a second time led to: With the two following boundary conditions: T=Tc at z=zc=Moho, and Q=-Qm (the basal heat flow), we get that: A 2 Q0 z + C2 T(z) = − ⋅z + K 2K C3 = Tc − The first boundary condition says that T=To at z=0, therefore we get that: T(z) = − and that A 2 Q0 z + T0 ⋅z + K 2K C2 = Qm ⋅z K c Qm K The geotherm in the lithospheric mantle is therefore: This relationship is the steady state geotherm. It gives the distribution through depth of temperature in a layer with homogeneous radiogenic production A, conductivity K, with a surface temperature of To and a surface heat flow of Qo. T(z) = Qm ⋅ (z − zc) + Tc K Hence, the geotherm in the lithosphere is defined by a two steps function: The geotherm in the continental lithosphere and beyond A 2 Q0 T(z) = − z + T0, for 0 < z < zc ⋅z + K 2K Qm T(z) = ⋅ (z − zc) + Tc, for z > zc K The continental lithosphere consists in two layers with contrasted thermal properties. In particular the radiogenic heat production in the mantle is negligible compared to that of the crust. The geotherm in continental lithosphere is therefore best described by two 6 alternatively, if Qm instead of Qo is known the crustal geotherm (i.e. when 0<z<zc): T(z) = − This is called a differential equation because this equation links todT dT dT d 2T gether the derivatives ( , , 2 ) of a function T(z, t). Here, dt dz dz dt dT is the rate of temperature change at a given depth; is the gradidz d 2T ent of temperature change at a given time; and 2 is the gradient dz of the gradient of temperature change at a given time. Q A 2 A ⋅z + [ m + ⋅zc]⋅z + T0, for 0 < z < zc 2K K K In the asthenospheric mantle, convective motion is such that the temperature shows relatively little variation as convection acts as an efficient mixing process. For z>zl (zl being the base of the lithosphere) the geotherm follows the adiabatic gradient ~0.3 K per kilometer. For instance, from the top of the convective mantle (i.e. the base of the lithosphere) to the core/mantle boundary (i.e. a distance of ca. 2700-2800km) temperature increases by only ~3000ºC. This contrast with the evolution of the temperature within the lithosphere where temperature increases from ca. 0ºC at the surface to 1300ºC at depth of 100-200 km. From this differential equation, one can derive a solution for the particular case when thermal equilibrium is reached (i.e. there is no dT change of temperature through time: =0 and uz=0). This is the dt case of steady geotherm for which the differential equation simplifies to: d 2T A T’’(z) = 2 = − dz K This tells us that the change of the temperature gradient with d dT ) is equal to a constant (-A/K). This says that the depth ( )( dz dz geotherm in the continental crust is not a straight line. Appendix 1: Solving differential equation Very often the rate of natural processes (i.e. rate of heat conduction, rate of heat advection, the rate of radiogenic heating etc) can easily be calculated or measured. In the context of the Earth’s geotherm, we have seen that from the rate of individual heat transfer mechanisms, we can express the overall rate of temperature change through time (t) at any depth (z) by: Lets plot T’’(z): T''(z) dT K d 2T A dT = ⋅ 2 + − uz ⋅ dt ρ ⋅Cp dz ρ ⋅Cp dz -A/K 7 T''(z) z d 2T A The differential equation 2 = − can be solved to find out the dz K geotherm T(z). For this the differential equation is integrated twice: The curve in blue is the geotherm, however bear in mind that this curve is only relevant for 0 < z < zmoho. T(z) The first integration leads to the temperature gradient : T’(z) = dT A = − ⋅z + C1 dz K T(0) zmoho Graphically: z T'(z) c1 -A/K One may recognizes that the integration constants C1 and C2 have the following meanings: C1 is the temperature gradient (the slope of the geotherm) at the surface (i.e. z=0); and C2 is the temperature at the surface. The temperature at the surface is typically in the range of 0 to 30ºC, and the temperature gradient at the surface can be determined by measuring the temperature at the surface and at the bottom of a well. z -A/K T''(z) The second integration leads to the geotherm: T(z) = − Appendix 2: Steady state geotherm with decreasing RHP A 2 ⋅z + C1 ⋅z + C2 2K The steady state crustal geotherm we derived earlier assumed that the radiogenic heat production is constant with depth. This is usually not the case as cycles of partial melting and crustal differentiation, through upward flow of granitic melt, tend to deplet the lower crust and enrich the upper crust in radiogenic elements. Hence, the decrease with depth of the radiogenic heat production is often described by an exponential law: Graphically: T'(z) c1 -A/K T(z) c2 −z A(z) = A0 . Exp( ) hc z -A/K T''(z) 8 This means that the radiogenic heat production measured at the surface ( A0 ) in divided by e (e=2.71) every hc meters (typically 5000m< hc <25000). Therefore the 1D conduction-advection heat transfer equation is: After substituting C1 and C2 into T(z), expanding and simplifying, one gets: dT K d 2T A(z) dT = ⋅ 2 + − uz ⋅ dt ρ⋅Cp dz ρ⋅Cp dz T(z) = If one knows the surface heat flow instead of the mantle heat flow then: In the case of a steady state geotherm, this equation becomes: d 2T −A(z) = 2 dz K replacing A(z) we get that T(z) = A0 d 2T −z ⋅Exp( = − ) 2 dz K hc Following two successive integrations one gets: A dT −z = 0 ⋅hc ⋅Exp( ) + C1 and dz K hc T(z) = − A0 2 −z ⋅hc ⋅Exp( ) + C1z + C2 K hc To find out the two integration constants C1 and C2 we need to call upon two boundary conditions. Lets say that we know the temperature at the Earth' s surface T0 and the heat flow Qm entering the base of the lithosphere. The first boundary condition leads to: C2 ... and second to: C1 = A0 2 A −z Q −z ⋅hc ⋅(1 − Exp( )) + ( m − 0 ⋅hc ⋅Exp( c ))⋅z + T0 K hc K K hc = T0 + A0 2 ⋅hc K −z Qm A0 − ⋅hc ⋅Exp( c ) K K hc 9 A0 2 Q −z ⋅hc ⋅(1 − Exp( )) + 0 ⋅z + T0 K hc K SECTION 2 Earth’s Geotherm In this section 0 1. Steady state geotherm 2. Transient geotherm 400 800 1200 TºC 1600 20 40 0 400 800 TºC 1200 1600 20 Moho 40 0 400 800 1200 TºC 1600 20 40 Moho to Moho t1 60 60 to Mechanical boundary layer 80 80 60 t5 t100 t10 80 t t 8 100 8 t20 100 t10 t20 t100 t1 Thermal boundary layer t5 100 to 8 t Plume spreads under the lithosphere Steady state geotherm In the previous section we have derive the 1D conductionadvection heat transfer equation and we have derive an equation desbribing the equilibrium geotherm, also called “steady state geotherm” because the temperature at every depth doesn’t change through time. Here we explore how sensitive is the geotherm to the radiogenic heat production when it is not constant 10 120 140 140 Plume spreads at the top of the thermal boundary layer Depth (km) Depth (km) 140 120 Depth (km) 120 Plume spreads at the Moho with depth (as we have assumed in the first section). We then look at how sensitive is the geotherm to chance in mantle heat flow, conducvity and thickness of the radiogenic crust. Distribution of radiogenic heat production We have assumed so far that the volumetric radiogenic heat production A was depth-independent ( A remains constant with * nb: These are elements that tend to concentrate into the melt phase during partial melting. Due to the buoyancy of the melt phase, incompatible elements concentrate over time into the upper part of the crust. Incompatible elements such as U, Th and K have a large radius and therefore do not fit easily into crystals lattice, hence their tendency to move into the melt phase when partial melting occurs. depth). However, because the upper crust is enriched in incompatible elements* the radiogenic heat production decreases with depth. A common model assumes that A is divided by e (e=2.71) every h metres, h being the length scale of the exponential law. This model of distribution is given by: A(z) = A0 ⋅Exp( − z /h) Basal Heat Flow (Qm) The graph illustrates the sensitivity of the geotherm to the mantle heat flow Qm, also called basal heat flow. It shows three geotherms calculated assuming same TºC rate of radiogenic heat pro0 500 1000 1500 0 duction, same conductivity and same crustal thickness. 20 Only the mantle heat flow varies. Moho 40 Depth (km) 100 =24 80 Depth (km) =36 Qm 11 60 Qm=12 A basal heat flow of 12 x 10-3 W.m-2 (yellow geotherm) is characteristic of cold and thick cratonic lithospheres, whereas a basal heat of 36 103 W.m-2 is representative of thinned lithosphere. Increasing the mantle heat flow from 12 to 36 10-3 W.m-2 increases the temperature at the Moho from ~320ºC to 720ºC. Qm Depth (km) The graphs shows the geotherm for A decreasing exponentially with various length scale h (10, 25, 50km). Ao is adjusted so that the total radiogenic heat production (R.H.P which is given by the integration of the radiogenic heat Temperature (TºC) Radiogenic Heat (mW.m-3) profile with 500 1000 6 2 4 8 depth) is the same in all models. The larger is 25 25 h (i.e. the deeper are the radioMoho 50 50 genic elements) TMoho: the hotter is the 724, geotherm. This 935, 10 km 1284 75 75 suggests that at 25 km 50 km an early stage of R.H.P.: 0.078 W.m-2 the Earth evolu100 100 Archaean (3.5-3 Ga) tion (before the extraction of the crust) the Earth had a warmer geotherm. the geotherm increases as warmer temperatures are reached at lesser depths (the slope of the geotherm increases). During thickening the geotherm decreases as cooler conditions are met deeper in the crust (the slope of the geotherm decreases). Second, as the thickness of the crust changes, the steady state geotherm is affected. This is because a thicker crust will produce more radiogenic heat, therefore crustal thickening leads to warmer geotherm whereas crustal thinning leads to cooler geotherm. Thermal conductivity (K) The graph below illustrates the sensitivity of the geotherm to the thermal conductivity K (in W.m-1.K-1). It shows three geotherms calculated assuming same rate of radiogenic heat production, TºC same same crustal thickness, 0 500 1000 1500 0 and same mantle heat flow. The conductivity is known to 20 vary with temperature. Here however, we assume that K is Moho 40 constant through depth. K is proportional to the ability of 60 material to conduct heat away. Therefore the larger the ther80 mal conductivity the lesser 100 heat they can stored and the cooler is the geotherm. Decreasing the thermal conductivity from 2.75 to 1.75 W.m-1.K-1 increases the temperature at the Moho from 500 to 750ºC. Lithospheric thickening 5 .25 k=2 1.7 k= Thickening produces heat advection as a volume of rock, and the heat attached to it, is displaced vertically resulting in a rapid cooling of the geotherm. Thickening also increases the thickness of the radiogenic layer therefore increasing the production of radiogenic heat in the lithosphere. Isostasy (cf. section of Isostasy and Gravitational Forces) produces uplift of the lithosphere leading to erosion which in turn affects the amount and distribution of the radiogenic heat elements in the crust which also impacts on the geotherm. The interplay between the mode of the thickening (heterogeneous via thrusting vs homogeneous), the thickening rate and the erosion rate leads to contrasted thermal histories. The graphs show transient geotherms (at 0, 0.5, 2, 10...Myr) following heterogeneous thickening (thickening is achieved by doubling the thickness of the crust via a single thrust, which explains the temperature discontinuity at t=0), and homogeneous thickening (the thicknesses of the crust and the lithospheric mantle are doubled by pure shear deformation). Erosion is discarded here. The discontinuity in the case of heterogeneous thickening is smoothed out in a few Ma. Transient geotherms in both cases are similar for time > 10 Myr. Depth (km) .75 k=2 Thickness of the radiogenic crust Crustal thickening and thinning, via tectonics, modify the geotherm in two different ways. First, during deformation, heat is advected mainly upward (in the case of thinning) and mainly downward (in the case of thickening) as rocks carry their temperature with them during fast thinning or fast thickening. During thinning 12 0 200 400 600 800 1000 1200 TºC 0 50 600 100 150 150 Heterogeneous thickening via thrusting and doubling the thickness of the crust Depth (km) 100 Homogeneous thickening via pure shear doubling the thickness of the lithosphere Sedimentation of burial of the crust Lithospheric thinning 0 0 200 400 600 TºC 800 1000 1200 Sedimentation and burial of the continental crust under a few kilometre of volcano-sedimentary rocks have a significant long-term effect on the geotherm. The newly deposited layer effectively insulates the heat producing layer. If the conductivity of the upper layer is lower or equal to that of the buried layer, then heat accumulates increasing the geotherm. This effect could have played a major role in the differentiation of the continental crust in the Archaean. At that time, radiogenic heat production in the crust was 2 to 6 time larger that of present day rate of radiogenic heat production in the continental crust. Furthermore it was a time when 5 to 15 km thick continental flood basalts (the so called greenstones) where deposited at the surface of the Moho 50 yr m Depth (km) 0 yr m 100 10 50 my m r yr 10 0 25 Thinning drives heat advection as rock masses, and the heat attached to them, are displaced vertically mainly upward resulting in an instantaneous warming of the geotherm (the geothermal gradient increases). However, it also decreases the thickness of the radiogenic crust therefore reduces the production of radiogenic heat in the litho- r my 250 yr Moho yr 0m 25 yr m 50 m Moho r my 10 yr 2 yr sphere. Isostasy leads the subsidence of the surface of the lithosphere leading to sedimentation which in turn affects the amount and distribution of the radiogenic heat elements in the crust which also impacts on the geotherm. The interplay between the geometry of the thinning, the thinning rate, and the sedimentation rate lead to contrasted thermal histories. The graph on the right shows transient geotherms (0, 10, 50, 100, 250...Myr) following homogeneous thinning (the thicknesses of the crust and the lithospheric mantle are halfed by pure shear deformation). Sedimentation is discarded here. Following the increase of the geothermal gradient due to extensional deformation, thermal relaxation leads to cooling and therefore the thickening of the lithosphere. Slowy, the transient geotherm approaches the steady state geotherm. TºC 800 1000 1200 m 50 yr 0. 5m Depth (km) 200 400 m 10 50 0 0 60% homogeneous thinning of the lithosphere via pure shear 13 Archaean continental lithosphere Crust to+50Ma 50 Geotherm t Depth (km) 100 125 8 to+100Ma 75 Mantle 25 Geotherm t Geotherm to to+10Ma to+20Ma 0 to+30Ma to+200Ma 75 Mantle to+400Ma to+1Ga 100 to+40Ma 400 800 1200 TºC 1600 20 to+50Ma 40 0 400 800 TºC 1200 1600 20 Moho 40 0 400 800 1200 TºC 1600 20 40 Moho to Moho to+100Ma 125 t1 to+200Ma 60 60 to Mechanical boundary layer 150 150 175 175 80 100 100 t5 t100 t10 t20 t t10 t100 t1 Thermal boundary layer 80 t5 60 t20 80 t 8 Geotherm to 8 Crust 100 to 120 Depth (km) Earth, insulating the heat producing Earth's crust. The graphs show transient geotherms following the emplacement of a 6 km (graph on the left) and 12 km thick greenstone covers, with no radiogenic heat production in them. The temperature increase is large enough to lead to profound partial melting in the crust. t 8 140 Plume spreads under the lithosphere 120 120 140 140 Plume spreads at the top of the thermal boundary layer Depth (km) 50 Greenstone 200 400 600 800 1000 1200 1400 Depth (km) Crust 25 mantle plume assuming that the plume's head spreads into a 50km thick layer with a temperature of 1700ºC. The three graphs show the results for various depth of emplacement. TºC 8 TºC Greenstone 200 400 600 800 1000 1200 1400 Depth (km) Archaean continental lithosphere Plume spreads at the Moho Transient geotherms Calculation of the steady state geotherm is relatively straightforward as it is not time dependent. Calculation of transient geotherms, such as those displayed above, requires computational trickery as the temperature changes in both space and time. Many analytical solutions have been proposed for a range of problems. The well-known book from Carslaw and Jagger, first published in 1946 "Conduction of Heat in Solids", provides hundreds of analytical solutions to many problems. Here, we give a couple of them. Emplacement of a mantle plume Mantle plumes initiate at the core-mantle boundary and rise through the convective mantle. Upon approaching the more rigid lithosphere, they spreads laterally under the lithosphere. They may also thermally erode to base of the lithosphere and spread higher up. This is equivalent to put a hot layer, a few tens of kilometre thick, underneath or within the colder lithosphere. In the Archaean, the Earth was warmer and plumes were most likely more numerous. The graphs below document the thermal impact of a 14 Progressive cooling of the oceanic lithosphere In the 19th century, Lord Kelvin used this last relationship to find out the age of the Earth. Kelvin made the assumption that the Earth had formed as a molten body at the temperature at which basalt melts (he assumed Tm=2000ºC) and that it had cooled by conduction to its present surface heat flow (he assumed Q0= -30 10-3 W.m-2). He considered the Earth to be spherically symmetric, and assumed that all heat was lost at the surface by conduction (with κ =10-6.m-2.s-1). Thus the problem is reduced to that of finding the temperature within a cooling half-space of infinite extent as a function of time after the half space is set at a specific temperature, a problem for which the equations above apply. Kelvin found that the Earth was 50 Myr old... The formation of oceanic lithosphere at mid-oceanic-ridge is a thermal problem involving the progressive cooling of the asthenosphere. The temperature at the seafloor Ts is maintained constant, so is the temperature in the asthenosphere Tm. The lithosphere becomes cooler and therefore thickner as cooling proceeds. Assuming no heat production and no sedimentation the 1D advectionconduction heat transfer equation becomes: dT d 2T = κ⋅ 2 dt dz with κ the thermal diffuvity: κ = K⋅ρ⋅Cp The analytical solution of this differential equation is: T(z, t) = Ts + (Tm − Ts)⋅erf( z 2 κ⋅t ) By differentiating with respect to z one get the temperature gradient ... Cooling of a dike Tm − Ts dT z2 = ) ⋅Exp( − dz 4⋅κ⋅t π⋅κ⋅t The cooling of a dike with half width w is another problem which has an analytical solution in the following form: Ts w−x w+x T(x, t) = ⋅(erf( ) + erf( )) 2 2 κ⋅t 2 κ⋅t ...from which the variation of the surface heat flow through time can get extracted as: Tm − Ts dT = dz π⋅κ⋅t If the dike has a width of 2m, i.e. w=1m, and its initial temperature was Ts=1000°C, and κ = 10-6m2s-1, then the temperature at the center of the dike would be about 640°C after one week, 340°C after one month, and only 100°C after one year. 15 solid-liquid (L<0 for partial melting, L>0 for crystallization) X the melt fraction is a function of T and therefore z. Some swarms, such as the giant McKenzie swarm in Canada, appear to radiate from a point, commonly interpreted as a plume source for the magmas. The mid-Proterozoic Coppermine River flood basalts were erupted at the same time near the plume head. Individual dykes range from 10-50 m in width, with some up to 200 m wide. Some dykes can be traced for up to 2000 km. ∂T ∂ 2T A ∂T ∂X L = κ⋅ 2 + −U + ⋅ ∂t ∂z ρ⋅Cp ∂z ∂t Cp re-arranging we get: ∂T 1 ∂T ∂ 2T A ∂T −( ⋅ ) = κ⋅ 2 + −U ∂t Tliq − Tsol ∂t ∂z ρ⋅Cp ∂z and therefore: (1) ∂T L ∂ 2T A ∂T − (1 − ) = κ⋅ 2 + −U ∂t Cp ⋅(Tliq − Tsol) ∂z ρ⋅Cp ∂z Boundary conditions: T(z, 0)=0; T(0, t)=T0; dT/dz(zl, t)=-Qm/K Numerical solutions An approximation of this differential equation can be obtained by replacing ∂t and ∂z with finite differences ∆t and h. The central tenet of this computational technique is that for a given time t=n, the temperature at a depth z=i (Tin) can be calculated from knowledge of the temperature at depth z=i-h and z=i+h. In a similar fashion, for a given depth z=i, temperature at time t=n can be caculated from knowledge or the temperature at time t=n-∆t and temperature at time t=n+ ∆t. The level a accuracy depend of the size of the time and space finite differences. There are many ways to express T(z, t) as a function of T(z-h, t), T(z+h, t), T(z, t- ∆), T(z, t+ ∆). Here we present the Crank-Nicholson scheme where, equation (1) is rewriten as: Since the arrival of computers, which allow for large numbers of operation to be performed in routine, new techniques based on numerical algorithms have been designed to solve differential equations. These numerical recipes are based on "discretization" of the differential equation... •Consider the 1D heat conduction-advection equation in a slab zl thick. The upper surface of the slab has at temperature To. The transient temperature is then described by the conductive-advective equation of heat balance. Here we consider a situation where internal radiogenic heat is creation at a rate A, heat is lost or gain by advection at speed U (since z increases downward, if erosion U<0, if sedimentation U>0), and heat is lost through phase transition (2) 16 Tin+1 − Tin L − 1− Δt Cp ⋅(Tliq − Tsol) ) ( = κ⋅ n n n+1 n+1 Ti+1 Ti+1 − 2Tin + Ti−1 − 2Tin+1 + Ti−1 1 ⋅ + 2 ( h2 h2 ) where: Tin = T(zi, tn), T n+1 i and ai = Rd, ci = Rd, bi = (1 − 2Rd + V − Lt) Rearranging terms, we obtain the following set of implicit equations for the temperature at the n+1 time step in terms of the temperature at the n-th time step (4): Tin+1 − Tin A + −U h ρ⋅Cp A ⋅Δt = din k This formulation results in N equations and N+4 unknowns, the four additional unknowns being u0n and uN+1n, u0n+1, uN+1n+1. However, these four additional unknowns lie outside the computational grid (time, space). Also, from the boundary conditions the following conditions must be satisfied at the n-th and n+1 time steps: 1+n 1+n n n a1i ⋅T−1+i + b1i ⋅Ti1+n + c1i ⋅T1+i = ai ⋅T−1+i + bi ⋅Tin + ci ⋅T1+i + = T(zi, tn+1), etc The finite difference equation (2) is known as the Crank-Nicholson scheme. It is based on central differences for the spatial derivatives averaged forward in time over time steps n and n+1. Expanding (2) and re-arranging to express temperature at time n+1 as a function of the temperatura at time n we get equation (3): T1n = 1, TNn = 0, T1n+1 = 1, TNn+1 = 0 L −κ ⋅ Δt 1+n −κ ⋅ Δt 1+n κ ⋅ Δt UΔt 1+n ⋅ T + 1 − + + ⋅ T1+i + ⋅T i −1+i ) ( 2h 2 2h 2 Cp ⋅(Tliq − Tsol) h2 h ) ( = This gives the following set of N equations (5) for the temperature at the n+1 time step : κ ⋅Δt UΔt A ⋅Δt κ ⋅Δt n L κ ⋅ Δt n n − + ⋅T + 1 − + + ⋅T ⋅T i −1+i 1+i ) ρ ⋅ Cp ( 2h 2 2h 2 Cp ⋅ (Tliq − Tsol) h2 h ) ( a11 ⋅ T01+n + b11 ⋅ Ti1+n + c11 ⋅ T21+n = a1 ⋅ T0n + b1 ⋅ T1n + c1 ⋅ T2n + remplacing and substituing: Rd = A ⋅ Δt , i = 1 k A ⋅ Δt , i = 2 to N − 1 k A ⋅ Δt 1+n 1+n n n a1N ⋅ TN−1 + b1N ⋅ TN1+n + c1N ⋅ TN+1 = aN ⋅ TN−1 + bN ⋅ TNn + cN ⋅ TN+1 + , i = N k 1+n 1+n n n a1i ⋅ Ti−1 + b1i ⋅ Ti1+n + c1i ⋅ Ti+1 = a1 ⋅ Ti−1 + bi ⋅ Tin + c1 ⋅ Ti+1 + κ ⋅Δt U⋅Δt L , V = , Lt = 2h 2 h Cp(Tliq − Tsol) we get... 1+n 1+n −Rd ⋅T−1+i + (1 + 2Rd + V − Lt) ⋅ Ti1+n − Rd ⋅T1+i n n = Rd ⋅T−1+i + (1 − 2Rd − Lt + V ) ⋅Tin + (Rd ⋅T1+i )+ with (6): A⋅Δt ρ ⋅Cp b11 = 1, c11 = 0, a1N = 0, b1N = 1 b1 = 1, c1 = 0, aN = 0, bN = 0 further replacement... These latter conditions ensure that the boundary conditions are always satisfied. a1i = − Rd, c1i = −Rd, b1i = (1 + 2Rd + V − Lt) 17 Numerical solutions an example The N equations above can be rearranged into the following matrix equations: 200 400 0. 5m 600 800 1000 1200 TºC yr yr 0m 25 m 50 yr Moho 100 Depth (km) The figure on the right shows the instantantaneous and steady state geotherm 150 (potential geotherm reached after infinite Heterogeneous thickening via thrusting and doubling the thickness of the crust time). The aims is to display transient geotherms down to a depth z=zl+zt at various time intervals, where zl is the thickness of the lithosphere before thickening zl=120km. with: b1 d1n a2 d2n 0 d3n = 0 . . 0 n dN−1 0 dNn 0 50 0 r my 10 yr m .. .. b11 c11 0 0 0 T1n+1 d1n .. .. a12 b12 c12 0 0 T2n+1 d2n .. 0 0 0 a13 b13 c13 d3n T3n+1 .. .. .. .. .. = (7) . 0 0 ⋅ . . . .. .. .. .. .. 0 0 n n+1 dN−1 T N−1 .. .. .. a1N−1 b1N−1 c1N−1 0 dNn n+1 T N .. .. .. .. a1N b1N 0 0 2 • We illustrate here a probleme where a continental crust with initial thickness zc is thickened by a factor 2 via the emplacement of one single zt thick thrust. zc=40km, zt=40km. .. .. 0 0 0 1 .. .. b2 c2 0 0 T2n .. a3 b3 c3 0 0 T3n A⋅Δt .. .. .. .. .. ⋅ ⋅ . 0 . k .. .. .. .. .. 0 n TN−1 .. .. .. aN−1 bN−1 cN−1 TNn .. .. .. .. aN bN c1 • We choose a spatial finite diffence h=4000m (spatial grid). Therefore the number of column in the tridiagonal matrixes will be zl/h. The Crank Nicholson scheme imposes the maximum time step of h2/(2.κ) = 253,000 years. • We get that Rd=(κ.∆t)/(2h2)=0.25. We assume no erosion: v=0; and we allow for partial melting: Lt=Latent_heat/(Cp.(Tliquidus-Tsolidus)). With Rd, v and Lt we can determine ai1, bi1, ci1, the coefficient of the tridiagonal matrix at time n+1, and ai, bi, and ci the coefficient of the tridiagonal matrix at time n. Thus the RHS of (7) is determined explicitly from the solution at the n-th time step. Note that the boundary conditions are satisfied by ensuring that the coefficients given in (6) are satisfied. Hence in order to find the solution at the time step, one must solve N linear equations. Since the coefficient matrixes in (7) and (8) are tridiagonal, one can make use of efficient algorithms (e.g. Thomas algorithm) to find the solution. We choose to get the transient geotherm at 20 Myr interval up to 400 Myr. The depth of the tridiagonal matrixes (number of row) is therefore 400/20=20. 18 • With this, we construct the tridiagonal coefficient matrixes (7) and (8) , the equation system (7) is solved via matrix inversion (here Numpy, R, MatLab or Mathematica are well equiped to do the dirty work...). Graph on the right show transient geotherms from 20 to 400 Ma. (7) (8) 19 CHAPTER 2 Isostasy & Gravitational Forces Isostasy is the physical process that explains the surface elevation of the Earth’s lithosphere at rest. It relates to the Archimedes’ principle showing that the elevation of a floating body above the surrounding fluid depends on the density contrast and vertical length of the floating body. Isostatic equilibrium must not be confused with gravitational equilibrium. A gravitational force (a volume force) appears when density interfaces are no longer parallel to gravity equipotential surfaces. Hence, the action of isostasy, which forces density interfaces away from equipotentials, introduces gravitational forces driving the flow of rocks from regions of high pressure to regions of low pressure. The Tibetan plateau stands 5000 m above sea level. It is the surface expression of a 75 km thick continental crust, which imparts a significant gravitational push on its surrounding. 20 ISOSTATIC EQUILIBRIUM AND MECHANICAL EQUILIBRIUM The sketch below represents a schematic cross-section through a lithospheric plate. Although the thicknesses of both the crust and the lithospheric mantle vary, isostatic equilibrium is maintained. • Isostatic equilibrium implies that at - and below - a parA B C ticular depth, called the compensation depth, the pressure becomes hydrostatic (ie pressure shows no lateral variation). In other terms, at or below the compensation level, the weight of vertical columns with the same crosssectional areas standing on the same gravitational equipotential surface are the same. • Isostasy controls the elevation of the Earth's surface. Lithospheres with thin/thick continental crusts have lower/higher surface elevation compared to the average lithosphere. However, isostatic equilibrium does not mean Compensation level mechanical equilibrium. Lateral variations of density Isostatic equilibrium=>Lithostatic pressure@z = constant for z > Compensation level along a given equipotential surface produce gravitational forces (body forces) acting inside the lithospheric plate. B σzz = ρ.g.z(MPa) C GRAVITATIONAL POTENTIAL ENERGY AND GRAVITATIONAL FORCE (B) C) z( ρg 21 z ρg The sketch bellow illustrates the variation with depth of the lithostatic pressure along two lithospheric columns, B and C, in isostatic equilibrium. It is assumed here that the density of the crust (pink) and that of the mantle (green) are laterally constant (i.e. they are the same in both columns) and that there is no density difference between the lithospheric mantle and the asthenosphere. Because it has a thicker crust, elevation of column C is higher than that of column B. Consequently the lithostatic pressure at any depth within the crust of column C is a higher than the lithostatic pressure at the same depth in column B. However, at or below the compensation depth (here the base of crust in column C) the lithostatic pressure is the same in both column. GPE (C) GPE (B) GPE (B) < GPE (C) => Horizontal force acting from C to B C) z( ∫ ρg GPEC = σzz = ρ.g.z(MPa) C B) z( Top B ρg The magnitude of the Gravitational Potential Energy (GPE, in Pa) for a given lithospheric column is given by the surface area of the region delimited (i.e. bounded) by i) its ρ.g.z function (lithostatic pressure), ii) the vertical axis (z), and iii) the compensation level. For column B this is the orange region, and for column C the yellow region which extends underneath the orange region. The gravitational force (Fg in N.m-1) that C and B applies on each other is equal to the difference of their respective GPE (difference between the two colored surface areas, ie the thin yellow wedge on the graph below). An excess in GPE in column C (Fg>0) will drive extension, whereas a deficit (Fg<0) will drive contraction. GPE (C) GPE (B) Top ρC(z)⋅g⋅z dz and Bottom GPEB = ∫ ρB(z)⋅g⋅z dz Bottom Fg = GPEC − GPEB Movie 2.1 Collapse of orogenic plateaux This movie shows a 65 km thick crust adjacent to a 40 km thick crust (brown), in green the upper mantle. It starts at a stage when thermal relaxation has produced 20 to 25% of melt (dark pink) in the lower crust. The only force acting is the gravitational force. It is the same physics that is responsible of the spreading of camembert, or that of warm butter. 22 GPE (B) < GPE (C) => Horizontal force acting from C to B THE CASE OF LITHOSPHERIC THICKENING AND DEBLOBBING Δσzz (MPa) Compression 50 The figures on the right show the gravitational force (Fg) due to instantaneous homogeneous thickening (at time to), and following convective thinning (to+30 Myr). Convective thinning is the gravitational process upon which the cold and dense lithospheric keel is dragged into the convective mantle. Following homogeneous thickening, the excess in gravitational potential energy in the upper part of the thickened lithosphere is more or less balanced by a deficit in gravitational potential energy in the deeper part of the lithosphere. Indeed the integration of σzz with depth gives a very small graviational force (Fg = 0.77 1012 N.m-1). 100 Tension Depth (km) 50 100 150 Following convective thinning (the cold and heavy lithospheric keel has been removed), the deformed lithosphere has a larger excess in gravitational potential energy. This excess in GPE gives a gravitational force Fg = 6.7 1012 N.m-1. Such an Fg could easily balance or overcome tectonic forces, and in some circumstances could be sufficient to drive extensional collapse of the mountain belt. -1 Fg=0.77xTNm to Δσzz (MPa) 50 -50 100 150 20 40 60 THE CASE OF LITHOSPHERIC THICKENING AND DEBLOBBING 80 -1 100 Gravitational force (Fg) due to instantaneous homogeneous stretching (at time to) and following thermal relaxation (at time to+200Myr). The yellow shaded areas represent the gravitational force acting on the lithosphere. Fg is given by the integratation with depth of the difference in lithostatic pressure (σzz) between the thinned and the undeformed lithosphere. As thermal relaxation and cooling proceeds, the thickness of the lithospheric mantle increases. This leads to a decrease in the extensional gravitational force acting on the lower part of the lithosphere and an increase in the compressional gravitational force acting in its upper part. The integration of Fg with depth gives a value > 1012 N.m-1. For contraction structures to develop Fg must be larger that the integrated strength of the thinned lithosphere. 23 Δσzz (MPa) 50 Compression -50 Tension Depth (km) 50 100 to -150 -100 -1 Fg=0.17xTNm Δσzz (MPa) -50 50 150 100 20 40 Depth (km) Following homogeneous thinning by a factor of 2 (50% reduction of the thickness of the crust and that the whole lithosphere), Fg is rather small (< 1012 N.m-1). to+30Ma Fg=6.7xTNm to+200Ma 60 80 100 -1 Fg=1.2xTNm GLOBAL VS LOCAL GRAVITATIONAL STRESS FIELDS The Earth’s plate's mean gravitational potential energy (GPE) defines a global reference level. A lithospheric column with an excess of GPE with respect to this reference will be under extensional stresses, or under compressional stresses if it has a deficit. Continental and oceanic columns with an elevation of ~70 m and -4.3 km respectively are in mechanical equilibrium with the global reference GPE. Midoceanic ridges (MOR) have an excess of GPE which explains why they are in extension. Let's focus on the column B on the graph below. Depending on its elevation, B has either an excess (B2) or a deficit (B1) of GPE with respect to the GPE of MOR. What would be evolution of the gravity-related stress acting on B during thinning, which will shift the geometry of B toward that of C? Potential Energy We consider first the case where the potential energy of column B is B1< GPE of MOR. As B becomes thinner, its GPE must increases to reach that of MOR (path B1->C1). Therefore, column C is in extension with respect to its surrounding (column B). It is also in extension with respect to the global Earth's lithoB A C sphere since C has a GPE C1 greater Thick Normal Thin than that of the global mean potential. Hence, the regional gravity-related Extension Pe>0 B2 Elevation B2>~70m state of stress enhances the global M.O.R. C2 gravity-related state of stress. -4.3km C1 Plate's mean potential energy In contrast, if the column B has a GPE B1 Elevation B1~70m Compression Pe<0 B2 then thinning of B will lead to a deM.O.R. M.O.R. A crease in GPE (path B2->C2) . With a B C GPE C2, column C is in compression Crust Oceanic lithosphere since it has a deficit of GPE with its imLithospheric mediate surroundings (B2). This reMantle gional compression opposes the global Not to scale extensional state of stress related to the ∆GPE between the thinned lithosphere (C2) and the global mean potential. From this analysis we can define two gravitational stress field components: The Global Gravitational Stress Field (GGSF), and the Regional Gravitational Stress Field (RGSF). The GGSF results from ∆GPE between a lithospheric column and the global mean potential energy, whereas the RGSF results from the contrast in ∆GPE between a deformed lithospheric column and its immediate surrounding. The Effective Gravitational Stress Field (EGSF) is the superimposition of both gravitational stress fields. 24 GRAVITATIONAL FORCE AND THE FORMATION OF CONTINENTAL PLATEAUX Depending on its sign, GPE may oppose or enhance thickening. In this example (Vanderhaeghe et al., 2003) a mountain belt develops due to the subduction of the continental lithospheric mantle. a) A localised mountain belt has formed. Crustal thickening produces an extensional gravitational force (Fg) that opposes the tectonic driving force (Fd). b) and c) When Fg balances Fd, thickening migrates into adjacent areas of the foreland region: the mountain belt grows laterally and a plateau develops. The mountain belt is supported by the basal traction exerced by the subduction at the base of the overriding plate. Should the subduction stop, there would be no force to oppose Fg, which will then drive extensional collapse. 25 GRAVITATIONAL FORCE AND CONVECTIVE THINNING Convective thinning describes the drag, into the convective mantle, of the lower part of the lithospheric mantle. This mechanism corresponds to the development of a Rayleigh-Taylor gravitational instability (driven by a density inversion): The heavy yet weak lower part of the lithospheric mantle is gravitationaly instable with respect to its surrounding. Depending on the rheology of the lithospheric mantle Houseman and Molnar (1997) estimated that the part comprised in between the isotherm 900±100º and the isotherm 1300º is gravitationally unstable. The detachment of this heavy keel from the rest of the lithosphere dramatically changes the balance of forces. Convective thinning results in a sudden increase in GPE stored in the thickened crust, and therefore promotes extensional collapse. ioo 26 CHAPTER 3 Tectonic Forces M.O.R. Fg Frp Oceanic dFr Fo dFf Continental lithosphere Fox 100 KM dFr dFaz Mantle flow Mantle flow Fss dFa Fss Mantle flow dFr e ntl Ma dFp w flo z w flo e ntl Ma x Fsp Ever since plate tectonic theory took hold in the late 1960s, geoscientists have argued over what drives plates: mantle upwelling at ridges that pushes plates apart, mantle circulation that drags plates along, or slab pull have been, independently or in combination, proposed as driving forces for plate motion. In this chapter we explore the notion of tectonic forces. 27 FORCES DRIVING PLATES MOTION Mantle flow Ma w flo w flo e ntl e ntl Ma FORCES DRIVING PLATES DEFORMATION A fourth force is involved in the dynamic of continental margins. This force is called the gravitational force (Fg). It is a volume force that acts between the high-standing continental plate and the low-standing adjacent abyssal plain. The interplay between Frp , Fg, Fss and Fsp can result in contrasting tectonic regimes. FG ental n Conti phere Asthenos here lithosp rce nal Fo atio Gravit FSS FSP FSS 28 Ocea ati Gravit phere thos nic li Force onal FRP 100 KM Based on a simple model of lithosphere and mantle M.O.R. Fg interactions, geophysicists have found that three Frp Oceanic Fo dFf major tectonic forces — slab pull, slab suction and Continental lithosphere dFr Fox ridge push — can interact to explain most observed dFr plates motion at the Earth surface. A subducting dFa Fss Fss dFaz slab “pulls” the rest of the oceanic plate behind it. Mantle flow Mantle flow This is refers to as the slab pull force (Fsp). According to Don Anderson, “... slabs drive tectonics. There is no dFr need for other driving mechanisms such as plumes or dFp mantle convection that are independent of plate tectonx ics”. Others consider that the viscous drag imFsp posed by the convective mantle, and/or the asz thenospheric flow forced by the nearby subducting slab, can also contribute to drive the subduction of the oceanic lithosphere. This is the slab suction force (Fss). When slabs detach and sink into the asthenosphere the suction force does disappear. The mantle flow associated to the sinking slab can drive the dynamic subsidence of the lithosphere above. Finally, the gravitational force acting between the high-standing mid-oceanicridge and the distant, low-standing oceanic abyssal plain is called the ridge push force (Frp). It is of second order importance when compared to the slab pull. Hence, plates motion results from the interplay between mantle flow and forces acting at plates boundary. In the context of an active continental margin, the tectonic regime experi- M.O.R. Trench enced by the upper plate above the subduction zone can vary from no tec- Frp tonics, extensional tectonics, contractional tectonics, or transcurrent tecton- Plate Oceanic lith osph ere Fg Continental lithosphere ics. The tectonic regime expresses the balance of forces acting on the conti- Extensional tectonics is the tectonic regime when the subduction trench moves away from the upper continental plate. The trench retreats from the continent in response to slab rollback driven by the slab pull force, or under the push from the gravitational force. This tectonic regime leads to the detachment of continental fragments from the main continental plate. MOR-Trench nental margin. Fsp Trench-Plate M.O.R. Trench Plate Fg Frp Oceanic lith osph ere When the distance between the trench and a reference location inside the Continental lithosphere continental plate decreases, the continental margin is shortened. The tecorogen. Through times, the interplay between, ridge-push, slab-pull, slabsuction and gravitational force results in a changing tectonic regime with alternating periods of contraction, gravitational collapse and extension. MOR-Trench tonic regime is contractional and leads to the development of a cordillera Fsp Trench-Plate These changes can be triggered by small variations in the velocity oo the direction of plates motion. M.O.R. Trench Fg Frp An active yet tectonically stable continental margin indicates that the result- Plate Oceanic lith osph ere ing effective force acting on the margin is very small and unable to deform Continental lithosphere MOR-Trench the margin. The margin is in mechanical equilibrium. Fsp Trench-Plate 29 MANTLE FLOW: THE CASE OF STRONG COUPLING Underneath the lithosphere, the flow of the asthenosphere most likely exerts an influence on the overriding plate motion. The extent of this influence is strongly debated as it depends on the level of mechanical coupling between the lithosphere and the convecting asthenosphere. v The sketches on the right illustrate two cases where the mantle exerts a strong but opposite influence on the plates. The red arrows show the velocity profile across the subducting lithosphere and into the adjacent asthenospheric mantle. A/ The subducting slab drags the surrounding asthenosphere into the subduction zone. The asthenosphere resists and slow down the subducting slab. In this case, the sub-lithospheric mantle flow is the result of plate tectonic processes, as the slab sucks the nearby asthenospheric mantle into the subduction zone. A B/ The flow in the asthenosphere drives plate motion by dragging the oceanic lithosphere into a downwelling zone (suction force through shear traction). Here, it is the flow in the sub-lithospheric mantle that powers plate-tectonic, as mantle convection sucks the slab into the subduction zone. If the strength of the asthenosphere/lithosphere coupling is not that significant, the convective asthenosphere can still interfer with plate motion. In particular, when the flow in the asthenosphere is at an angle to the subducting slab it can forces the steepening or shallowing of the slab. See next slide. 30 v B MANTLE FLOW: THE CASE OF WEAK COUPLING When the flow lines in the asthenosphere are at an angle to the subducting slab, the “mantle wind” can force the steepening or shallowing of the slab, hence promoting trench retreat or advance. C/ The asthenospheric wind under the subducting slab forces the slab to bend downward, enhancing the slab-pull. This effect is here enhanced by the downward asthenospheric flow above the slab forcing the steepening of the slab. D/ In this case, the asthenospheric wind supports the subducting slab, opposing the slab-pull. The slab is pushed upward forcing the shallowing of the slab, and perhaps squeezing the mantle wedge. C Overall, the forces Frp , Fg, Fss and Fsp acting on the plates, the sublithospheric flow and the flow of the mantle in the vicinity of the slab all influence plate motion, and the tectonic regime at continental plate margins. D 31 RIDGE PUSH AND OTHER HORIZONTAL GRAVITATIONAL FORCES Gravitational force at mid-ocean ridge Mid-oceanic-ridges (MOR) form a ~55,000 km long mountain belt in the middle of the ocean. This mountain belt stands ~2500 m above the average depth of oceanic abyssal plain (-5000 m), and the crest of the MOR stands in average 2500 m below sea level. MOR, including the region of hot mantle underneath, applies an horizontal force directed from the MOR to the adjacent abyssal plains and adjacent continents. MORs are in isostatic equilibrium, but not in mechanical equilibrium since they are the locus of important and sustained extensional deformation, which accommodates the continuous production of new oceanic crust through decompression partial melting of the asthenopheric mantle. The map on the right shows the bathymetry of the north Atlantic ocean beween North Africa and Newfoundland. The bathymetric profile (along the red line) across the midoceanic ridge reveals the bathymetry across the MOR, with its crest at around 2000 m depth, and the abyssal plains between -4500 and -5000 m. Assuming temperature independent densities, an estimate of the total ridge-push per unit length parallel to the ridge axis is: & L e Frp = g⋅e⋅ (ρm − ρw)⋅ + (3 2) where e is the elevation of the mid-oceanic ridge above the cooling lithosphere, ρm is the density of asthenosphere (3320 kg m-3), ρw is the density of sea water (1030 kg m-3), L is the thickness of the oceanic plate (85 km), and g is the acceleration of the gravity field (10 m s-2). With these values we get Frp = 2×1012 Nm-1. This estimate, we will see, is an order of magnitude less than the slab-pull force. However, the value of the ridge-push force may increase up to 6.2×1012 Nm-1 when the ridge is underlain by a hot spot. A more thorough formulation must integrate the temperature dependence of the densities. Using the concept of gravitational force, the total ridge push can be expressed 32 as the integration with depth of the difference in lithostatic pressure between the isostatic column at the mid-oceanic ridge and an oceanic column far away from the mid-oceanic ridge. In other terms, the ridge push corresponds to the difference between the gravitational potential energy of the MOR and than of the adjacent abyssal plain. Fg = GPEMOR − GPEPLAIN Continental margins are also regions with large contrasts in elevation, in particular when these margins support mountain belts. For instance, the South American Andes stand in average ~8000 m above the adjacent abyssal plain of the east Pacific. The large contrast in gravitational potential energy produces a gravitational force directed from the continent towards the oceanic basin. The Andes are also in dynamic equilibrium supported by the convergence of the South American and Nazca plates. Gravitational force at continent-ocean transition Even in the absence of cordillera, the transition between continents, which stand in average a few hundred meters above sea level, and adjacent abyssal plains are regions of significant gravitational stresses. Not surprisingly, earthquakes tend to concentrate at plate boundaries even when they involve passive margins. For instance, earthquakes tend to concentrate in a broad zone at the edge of the Australian continent. This zone shows a significant gradient in gravitational potential energy and therefore a strong gravitational force that is at the origin of these earthquakes. 33 SLAB PULL FORCE M.O.R. A simple formulation for the slab-pull per unit length parallel to the trench is given by : zm. (zl. ∆ρ).g. This formulation assumes that the density of the plate and that of the asthenosphere are temperatureindependent. Assuming an average density contrast of about 60 kg.m-3, a depth of the subducting slab zm = 660 km, a thickness of the slab zl = 100 km and g = 10 m s-2, we get an absolute maximum for the slab pull Fsp ~ 4×1013 N m-1. Trench Frp zl Oceanic lith osph ere Plate Fg Continental lithosphere zm Fsp Note: the value would be an order of magnitude greater (Fsp ~ N m-1.) for a slab going all the way down to the core-mantle boundary. 2×1014 Asthenosphere A more accurate formulation takes into account the temperature dependence of density, the diffusion of heat, and the velocity of the subducting slab. An estimate of the slab-pull force per unit length of subduction zone, Fsp(z), acting at depth Fsp and caused by the density contrast between the cold oceanic plate and the mantle is given by: 8⋅g⋅α⋅ρm ⋅Ta ⋅L 2 ⋅Re π 2 ⋅z π 2 ⋅d Fsp = ⋅ Exp( − ) − Exp( − ) π4 2⋅Re⋅L 2⋅Re⋅L ) ( with: Re = ρm ⋅Cp ⋅ν⋅L 2⋅k where Re is the thermal Reynolds number (the ratio of heat convection to heat conduction), z is the depth beneath the base of the oceanic plate, α is the coefficient of thermal expansion (3×10-5 K-1), Ta is the temperature of the asthenosphere (1350ºC), k is the thermal conductivity (2 W m-1 K-1), L is the thickness of the plate (85 km) and d is the depth of the upper mantle (d+L = 660 km), Cp is the specific heat (1.17×103 J kg-1 K-1), and v is the rate at which the oceanic slab sinks into the mantle (~ 10 cm yr-1). Using the above values we get Fsp = 2.5×1013 N m-1. However, it is worth noting that Fsp linearly depends on the ill-defined coefficient of thermal expansion. With a coefficient of thermal expansion varying in the range of 2×10-5 to 4×10-5 K-1 the slab-pull force varies between 1.7×1013 and 3.4×1013 N m-1. In addition, the olivinespinel phase change which occurs at around 350 to 420 km increases the density in the subducting slab, providing an extra pull. Note: This 34 extra pull depends on the thermodynamical characteristics of the phase transition, notably the slope of its Clapeyron curve (dP/dT). The value of the slope of the Clapeyron curve ranges between 3 and 4 MPa K-1. The resulting extra pull ranges between 1.2×1013 and 1.6×1013 N m-1. The slab-pull force is opposed by the friction force between the slab and the lithosphere viscosity of the asthenospheric mantle. This force is proportional to the velocity of the subducting plate and to the viscosity of the asthenosphere, that is also poorly constrained. Numerical calculations based on the differential equations for the flow of a viscous fluid suggest that the resistive force is of the order of 1013 N m-1. MAXIMUM EFFECTIVE TECTONIC FORCE Overall, the maximum effective tectonic force available for lithospheric deformation is of the order of a few 1013 Nm-1. This force must be able to double the thickness of the continental crust, to account for Tibet, the highest and largest plateau on Earth, and to sustain the gravitational force associated with this thickening and with mountain belts at the edge of continents (e.g. the Andes). Averaged over a lithospheric thickness of 100 to 200 km, the maximum differential stress available to deform the Earth lithospheres is in the range 100-500 MPa. This places some constraints on the rheology of Earth’s material. a/ re sphe ic litho Ocean FRP here hosp nic lit e l Forc Ocea tationa Gravi ental Contin FG sphere Astheno here hosp Cont l lit inenta rce nal Fo tatio Gravi b2/ b1/ FSS FSP here FRP hosp nic lit FSS Ocea FG FG sphere here en Contin l lit inenta e l Forc tationa Gravi sphere Astheno FSS FSS FSS FSS 35 here hosp nic lit Ocea Cont e l Forc tationa Gravi here hosp Astheno hosp tal lit FRP here lithosp here Asthenosp APPENDIX: This R script grabs bathymetry and topography data from the US National Oceanographic and Atmospheric Administration (NOAA), creates a topographic map, and plot a topographic profile (see p.31). # Load the marmap library require(marmap) #setwd stands for set working directory setwd('/Users/Patrice/Documents/GIS/') # Grab data from National Oceanographic and Atmospheric Administration (National Geophysical Data Center) # More data available here: https://www.bodc.ac.uk/data/online_delivery/gebco/ map_Altiplano<-getNOAA.bathy(lon1=-60, lon2=-80, lat1=-30.00, lat2=-10, resolution=2) # Hypsometry of the region # zbreaks = quantile(map_Altiplano, seq(0, 1, length.out=256)) zbreaks = zbreaks <- seq(-8000, 8000, by=100) plot(zbreaks) # Make colour palette ocean.pal <- colorRampPalette(c("#000000", "#000413", "#000728", "#002650", "#005E8C", "#0096C8", "#45BCBB", "#8AE2AE", "#BCF8B9", "#DBFBDC")) land.pal <- colorRampPalette(c("#336600", "#F3CA89", "#D9A627", "#A49019", "#9F7B0D", "#996600", "#B27676", "#C2B0B0", "#E5E5E5", "#FFFFFF")) #land.pal <- colorRampPalette(c("#467832", "#887438", "#B19D48", "#DBC758", "#FAE769", "#FAEB7E", "#FCED93", "#FCF1A7", "#FCF6C1", "#FDFAE0")) mypalette <-c(ocean.pal(sum(zbreaks<=0)-1), land.pal(sum(zbreaks>0))) # Plot the map plot(map_Altiplano, image=T, bpal=mypalette, land=T, deep=c(-6000, -3000, 0), shallow=c(-3000, 500, 0), step=c(1000, 500, 0), col=c("grey10", "grey40", "black"), lty=c(1, 1, 1), drawlabel=c(T, T, T), asp=0, ylim=c(-30,-10), xlim=c(-80,-60)) # Topographic profile points(c(-78, -61),c(-20, -20), type="o", col=2) profile <- get.transect(map_Altiplano, x1 = -78, y1 = -20, x2 = -61, y2 = -20, locator=F, distance = TRUE) plotProfile(profile, ylim=c(-9000,5000), xlim=c(0,1800)) # The following organize map, legend and profile into one single image def.par <- par(no.readonly = TRUE) nf<-layout(matrix(1:4,nc=2), height=c(3,1), width=c(5,1)) par(mar=c(4,4,1,1)) layout.show(nf) plot(map_Altiplano, image=T, bpal=mypalette, land=T, deep=c(-6000, -3000, 0), shallow=c(-3000, 500, 0), step=c(1000, 500, 0), col=c("grey10", "grey40", "black"), lty=c(1, 1, 1), drawlabel=c(T, T, T), asp=1, ylim=c(-30,-10), xlim=c(-80,-60)) points(c(-78, -61),c(-20, -20), type="o", col=2) plotProfile(profile, ylim=c(-9000,5000), xlim=c(0,1800)) image(x=0, y=zbreaks, z=matrix(zbreaks, 1, length(zbreaks)), col=mypalette, breaks=zbreaks, useRaster=FALSE, xlab="", ylab="", axes=FALSE) 36 CHAPTER 4 Introduction to Rheology Rheology is the study of flow, the mechanical response of material to applied deviatoric stresses. “Flow” is used here in its broader meaning, which includes both viscous flow and frictional flow. The relationships between, on one hand applied deviatoric stress and resulting strain, and on the other hand deviatoric stress and strain rate characterise the macroscopic mechanical behavior of rocks. These relationships lead to the constitutive equations linking deviatoric stress and strain rate; "constitutive" as they depend on the constitution of the material. 37 SECTION 1 Elastic, Plastic and Viscous Flows In this section 1. Brittle vs ductile deformation 2. Elastic, plastic, and viscous flow curves Elastic Viscoelastic σ ε = σ (1 - e-t/τ) ε= E E Viscoplastic 3. Rheology of polycrystalline rocks 4. Steady-state flow laws 5. Sensitiviy of flow laws Primary Creep Secondary Creep Rocks display a very large range of response when submitted to deviatoric stresses. Over millions of years, hot rocks can flow under small deviatoric stress like a very low viscosity fluid. Rocks can break under a sudden deviatoric stress load (e.g. hammer). Rocks can resist moderate deviatoric stress, until it reaches a threshold from 38 which permanent strain accumulates. Rocks can deform elastically since they transmit mechanical waves (sound wave, seismic wave). The rheology of rocks is therefore complex and covers a broad range of mechanical behavior from elastic, plastic and viscous. BRITTLE VS DUCTILE DEFORMATION: The role of temperature, composition and strain rate... Objects deform as a continuum (continuous deformation via elastic deformation or via viscous flow) when the flow unit are atoms moving in the lattice of crystal. In contrast, when the flow unit are fragments of crystals or fragments of rocks moving relatively to each other because of fractures or faults, deformationis said to be brittle (discontinuous deformation via frictional flow). Discontinuous deformation: • Case 1: The media is pre-fractured, its strength depends on the frictional strength of pre-existing fractures and faults. • Case 2: The media has no pre-existing fractures, its strength depends on its constitution in particular on the cohesion of its grains. This cohesion must be overcome before fractures and faults can accommodate frictional sliding. Continuous deformation: • Elastic deformation • Plastic deformation • Viscous deformation Ductile deformation. Photograph P. Rey Brittle deformation. Photograph Pui-Leng, flickr 39 CONTINUUM MODELS OF ROCK’S MECHANICAL BEHAVIOR Each type of flow has a characteristic flow curve in the deviatoric stress vs strain space. The graph on the right shows the flow curves for three elementary deformation mechanisms: σ σ Ideal Plastic Elastic σ* Deviatoric stress • Linear elastic flow (in blue): For small stresses, most material are elastic. The characteristics of elastic flow are: 1/ Elastic flow occurs as soon as stress is applied. 2/ Strain increases as long as stress keeps increasing. The elastic flow curve is linear, its slope is 1/E with E the Young modulus, a physical properties characteristics of elastic flow. 3/ Strain does not accumulate if the stress is maintained constant. 4/The material recovers its original shape when stress is relaxed. The mechanical analogue for elastic deformation is a spring. ε= σ E σ* ε= . f(t) E A Plastic ε B Viscous • Plastic flow (in green): In most material elastic flow is limε= σ . t η ited to a certain level of stress beyond which the flow switches from elastic to plastic. This limit is called the yield stress (σ ∗). The characteristics of plastic flow are: 1/ Plastic D flow occurs as soon as the yield stress is reached (i.e. from point A onwards). 2/ Plastic strain accumulates at a level of C 1/E stress which is constant and equal to the yield stress. Hence, Plastic Elastic Cumulative strain (ε ) the amount of strain depends on the duration t over which stress is applied. 3/ The plastic component of strain is permanent. When the imposed deviatoric stress is removed (B>C), the material recovers the component of elastic deformation only. 4/ For and ideal plastic material there is no elastic component. The mechanical analogue of plastic deformation is a rigid block sliding on a rough planar surface. • Linear viscous behavior (in purple): Linear viscous material (newtonian viscosity) have no yield stress. The flow curve is characterized by a linear relationship between stress and strain. Strain accumulates at varying and also constant stress (dashed purple line from D) and when the stress is removed the flow stops but the material does not return to its undeformed state. The mechanical analogue of viscous deformation is a dashpot. 40 In a graph strain vs time, in experiments where the deviatoric stress is applied at t = 0 and maintained constant for t > 0, elastic, plastic and viscous flows are characterised by three contrasting relationships. Elastic strain: When the deviatoric stress is imposed, the material records a finite amount of elastic strain. This amount of elastic strain remains constant σ through time (and equal to ) as long as the deviatoric stress is maintained. E Ideal Plastic ε = σ*E-1 + α . t ε= Cumulative strain (ε ) Plastic strain: The deviatoric stress is constant and equal to the yield stress. At σ∗ t = 0, a component of elastic strain ( ) is instantaneously recorded before the E onset of accumulation of plastic strain. Plastic strain accumulates as long as the deviatoric stress is maintained. For ideal plastic flow, the rate of accumulation (α) is constant. σ = Constant Viscous Elastic ε= σ E Viscous strain: Viscous strain accumumates even for a very small deviatoric σ stress. This strain accumulate at a constant rate ( ), which depends of the η shear viscosity (η) of the material. Viscous strain accumulates as long as the deviatoric stress is maintained. Time σ RHEOLOGY OF POLYCRYSTALLINE ROCKS Elastic Plastic Hardening C 41 C'' σ*' Creep B Failure σ* Deviatoric stress Crystalline rocks display a mechanical behavior that incorporates the three elementary flows; rocks are elasto-visco-plastic material. This graph shows a characteristic flow curve for a polycrystalline material. At stresses below the yield stress (σ ∗), polycrystalline material behave elastically (blue curve). Above the yield stress, the material behave plastically. At low level of strain (green curve from A to B) the material becomes stronger (hardening) as the applied stress must increases in order to keep the material deforming (i.e. the yield stress must be exceeded for strain to accumulate). At high level of strain (red line beyond B) the material flows under a constant stress. Under the hardening plastic σ .t η A E C' Cumulative strain (ε ) regime, the removal of the driving stress leads to the removal of the elastic component of the deformation (curve CC'). If the sample is restressed, elastic deformation occurs under an extended domain as the yield stress has increased (curve C'C''). This means that the material has become stronger. To understand how hardening occurs one must understand how strain is achieved at the microscopic scale. Plastic strain affects the arrangement of atoms in the lattice of minerals. During strain, this arrangement is perturbated by the introduction of defaults or gaps in the lattice called dislocations. Each dislocation introduces a local elastic strain. Under stress, these defaults move around leading to permanent deformation. In the hardening plastic regime, the density of defaults increases but low stress levels impede their displacement, hence the elastic energy in the crystal lattice increases making the material stronger. The animations show how dislocation moves about: Dislocation wall Motion of dislocations is driven by internal elastic energy. Dislocations organize themselves into arrays such as dislocation walls to minimize internal elastic energy. This results in the multiplication of sub-grains and therefore permanent plastic strain. sub-grain Movie 4.1 Dislocation glide Movie 4.2 Dislocation climb 42 Movie 4.3 Dislocation glide and climb σ = Constant Secondary creep A stic Plastic covered whereas the viscoelastic deformation is recovered over a period of time (t'1 - t'2). The sample, however, records a permanent plas- la oe Elastic V isc oe c strain rate. Upon unloading elastic deformation is instantaneously re- Failure ta Cons Vis strain and time implying that the material is deformating at constant A' Rate n i a r t nt S Elastic Primary creep σ* Secondary creep is characterised by a linear relationship between Tertiary creep Elastic Primary creep corresponds to a reversible flow for which elastic deformation is instantaneously removed following unloading of the sample (at time t1), whereas another component of strain called viscoelastic deformation is also recovered but over of a time window t1 - t2. Cumulative strain (ε ) The viscous flow of polycrystalline material can be illustrated on a space strain vs time for experiment performed at constant stress. In this graph, the plastic flow curve can be divided into three regimes called: Primary, Secondary and Tertiary creep. t1 la s tic t2 t'1 t'2 Time tic deformation. It is within the secondary creep regime that the creep parameters that govern the rheology of rocks in the ductile regime are determined. The tertiary creep corresponds to the development of a mechanical instability in which an increase in strain rate leads to the mechanical failure of the stressed sample. Real rocks display a complex elasto-visco-plastic behavior. This mechanical behavior can be represented by a combination of springs (elastic component), blocks on a rough surface (plastic component), and dashpots (viscous component) connected in series or in parallel to fit a real flow curve. The rock analogue on the right has the same flow curve than than presented above. At depth > 10 to 20 km, the deformation of rocks is characterised by slow steadystate creep at constant strain rate, that can accommodates large amounts of ductile deformation. It is generally assumed that steady-state constitutive equations of rocks, or flow laws, can be used to characterise the large-strain high-temperature ductile deformation that occurs in the Earth. 43 Elastic Viscoelastic σ ε = σ (1 - e-t/τ) ε= E E Viscoplastic Primary Creep Secondary Creep SECTION 2 Strength envelopes In this section 1. Low-Moderate stress Yield stress (MPa) 2. High-stress3000 regime 2000 1000 Yield stress (MPa) 1000 0 2000 1000 Yield stress (MPa) 1000 0 2000 1000 Yield stress (MPa) 1000 0 2000 1000 1000 0 3. Brittle regime 4. Sensitivity of flow curves 20 20 20 20 5. Rheological profiles Moho 6. Sensitivity of profiles 40 40 60 80 789MPa 255MPa [12] 100 40 60 40 60 80 80 598MPa 193MPa [11] 100 584MPa 213MPa [16] 100 The notion of strength envelope is relative to the evolution with depth of the deviatoric stress necessary either to exceed the yield stress, or to achieved a pre-defined strain rate through viscous flow. At low temperatures, where plastic strain dominates, the strength envelope is mainly a function of pore fluid pressure and the tectonic re44 60 80 582MPa 212MPa [1] 100 gime. At higher temperature where viscous creep dominates, the stength envelope is sensitive to rock types, strain rate, and the magnitude of the deviatoric stress. Here we start by giving the relationship deviatoric stress vs temperature (i.e. depth). FLOW LAWS FOR STEADY-STATE CREEP: LOW TO MODERATE-STRESS REGIME The constitutive equation that accounts for most low to moderate-stress, steady-state strain, is the power-law creep equation, so called because the absolute value of the steady-state strain rate is proportional to the differential stress raised to a power n. The following equations • give the strain-rate ( ϵ ) as a funtion of the differential stress, and the differential stress as a function of the strain rate: • 1/n ϵ E −E ⋅Exp ϵ = A⋅σ ⋅Exp re-aranging σ = ( n⋅R⋅T ) (R.T) (A) • n where A is a constant (Pa-n s-1), n is the stress exponent that characterises the sensitivity of strain rate on the differential stress (n is dimensionless), E is the activation energy per mole for the creep process (J mol-1), it is the energy barrier that inhibits the creep mechanism, R is the Boltzmann constant (8.3144 J mol-1 K-1), and T is the temperature (K). The constants A, E and n are characteristic of material. The power law creep shows that both temperature and differential stress have a large effect on the strain rate. Thus an increase in temperature increases the strain rate for a constant stress, or lowers the stress required to produce a given strain rate. n=1 n=3 n= 8 σ1−σ3 This effect is accounted for by the rapid increase, with increasing temperature, of the exponential term in the equations above. The graph shows that as n increases from 1 to large values, power-law material evolves from viscous material (for n=1 the power law becomes a linear relationship between stress and strain rate) to near ideal plastic material. For most rocks at moderate stress level 2<n <5, whereas at low level stress, 1<n<2. ε FLOW LAWS FOR STEADY-STATE CREEP: HIGH-STRESS REGIME Power-law creep implies that at 500ºC olivine would only deform at unrealistically high stresses. A better description of the behavior of olivine at high-stress regime (> 200MPa) is given by the Dorn's law, a relationship that is not as temperature dependent as the power law, in which Qd is an activiation energy, σd a critical stress that must be exceeded, and εd is the critical strain rate. 45 σ = σd ⋅ 1 − • ϵ R⋅T ⋅ln •d Ed (ϵ) FLOW CURVES IN THE BRITTLE REGIME With pre-existing fractures: At low temperature or at high strain rate or under high pore-pressure, but mainly in the upper crust and the upper mantle, the failure mechanism is modelled as frictional sliding: σ = β⋅(1 − λ)⋅(ρ⋅g⋅z) where g is the gravitational acceleration, λ is the ratio of fluid pore pressure to the normal stress, ρ the density. β is a parameter dependent on the type of faulting given by: β= R−1 1 + ( σzz − σ 3 ) ⋅(R − 1) σ −σ 1 with: R= ( ) 1 + μ2 − μ −2 3 and μ the coefficient of internal friction that characterises the roughness of the fracture plane. Because the coefficient of internal friction varies little around 0.75 (pretty much independent on the rock composition), we get that R=4, and therefore β varies continuously from 0.75 for normal dip slip faults, to 1.2 for strike-slip fault, to 3 for reverse dip slip faults depending of the exact value of the principal stresses ratio. The frictional sliding equation shows that it takes less differential stress to achieve brittle failure under extensional stress regime (maximum principal stress vertical) than under a compressional stress regime (maximum principal stress horizontal). This makes sense since in a compressional stress regime gravity acts against reverse faulting but enhances the principal stress in extension. Without pre-existing fractures: In this case, the strength of rocks includes the cohesion (C) holding the grains together at atmospheric pressure. The failure mechanism is modelled by the Coulomb's criteria linking the shear stress ( τ ) acting on a fault plane and the normal stress ( σn ) acting perpendicular to it: τ = C + μ⋅σn The normal stress and the shear stress are related to the orientation of the fault plane with respect to the principal stress axes ( σ1, σ2 and σ3): σn = σ1 + σ3 σ1 − σ3 − ⋅cos(2α) 2 2 and τ= σ1 − σ3 ⋅sin(2α) 2 with α the angle between the fault plane and σ1. 46 SENSITIVITY OF FLOW CURVES TO THERMODYNAMIC PARAMETERS AND CHEMICAL ENVIRONMENT Confining Pressure and Temperature 250 Wombeyan Marble σ3=100 MPa 200 150 σ3=35 MPa 100 σ1−σ3 (MPa) Confining pressure prevents rocks from falling apart. It is therefore not surprising that increasing confining pressure increases the amount of deformation a sample can accumulate before failure. For instance, in the example of the Wombeyan marble brittle failure is either reached for a larger amount of accumulated strain or even impeded when a higher confining pressure is applied. This higher confining pressure allows the sample to sustain a larger differential stress. Temperature enhances the ductility of materia. For instance, in the example of the Solenhofen limestone, raising temperature decreases the magnitude of the deviatoric stress that can be supported, and increases the amount of accumulated strain before brittle failure . Under high temperature material can therefore accumulate more strain but they can do so for a smaller amount of differential stress. σ3=10 MPa 50 Failure Failure 1 400 2 Cumulative strain (ε ) ε Solenhofen Limestone 300ºC Failure 300 500ºC Failure 600ºC 200 σ1−σ3 (MPa) 3 Failure 100 Pc~40MPa 2 Paterson and Wong, 2006. Experimental Rock Deformation – The Brittle Field. Surveys in Geophysics, 27, 4, pp 487-488. 47 4 6 8 Cumulative strain (ε ) 10 ε H2O and Pore Fluid Pressure Water in rocks acts as a softening agent. Indeed, compared to a dry sample, a wet sample deforms at lower deviatoric stress (bottom left figure). The presence of fluids enhance mechanism of deformation controlled by coupled dissolution (at high stress region) and precipitation (in low stress regions). Pore fluid pressure acts against the confining pressure and therefore promotes failure at lower differential stress and lower strain (figure on the right). This process is called “hydraulic fracturing”, this is the process explaining micro-seismicity following dam water loading. 300 Yule Marble Dry 600 Solenhofen Limestone 0.5% H20 60 MPa 200 400 100 T=300ºC Pc~500MPa 4 σ σ ε 200 Failure 93 MPa 3 Failure 65 MPa Failure 78 MPa 6 9 Cumulative strain (ε ) 1 σ 3 σ 8 12 18 Cumulative strain (ε ) σ1−σ3 (MPa) σ1−σ3 (MPa) 2.9% H20 1 3 Fluid assisted diffusive mass-transfer through the lattice (Nabarro-Herring creep) or along the grain boundaries (Cobble creep) @ Mervin Paterson, ANU 48 12 ε RHEOLOGICAL PROFILE OF THE CONTINENTAL LITHOSPHERE It is time now to synthesise what we have learn about the rheology of crystalline rocks to determine the strength profile (i.e. envelop) of the continental lithosphere. The strength of the lithosphere can be defined as the vertical integration, from the top to the bottom of the lithosphere, of the differential stress required to trigger either brittle failure or the flow failure of rocks. Failure can occur by power-law creep or Dorn law creep at high temperature and low strain-rate, or by frictional sliding at low temperature and high strain rate, in which case the differential stress depends of the tectonic regime. However at any given depth, the failure mechanism is the one that requires the minimum differential stress to operate. The strength of rocks at any depth in the crust is the lowest differential stress to reach either brittle or flow failure: σ1 − σ3 = β⋅(1 − λ)⋅(ρc ⋅g⋅z) • or 1/nc ϵ σ1 − σ3 = ( Ac ) Ec ⋅Exp ( nc ⋅R⋅T(z) ) The strength of rocks at any depth in the lithospheric mantle is the lowest differential stress to reach either brittle, or power law, or Dorn law failure: • or σ1 − σ3 = β⋅(1 − λ)⋅(ρm ⋅g⋅z) 1/n Em ϵ for (σ1 − σ3) < 200MPa σ1 − σ3 = ⋅Exp ) ( A n ⋅R⋅T(z) ( m) m or σ1 − σ3 = σd ⋅ 1 − • ϵd R⋅T(z) for (σ1 − σ3) > 200MPa ⋅ln • Ed (ϵ) 49 The figure below shows the rheological profiles for a compressional and an extensional tectonic regime. The rheological parameters for the crust and the mantle are that of a granite (quartz >40%) and a dunite (peridotite with >80% olivine) respectively. The power law and Dorn creep law are dependent on both the temperature T and the strain rate. Therefore, these parameters must be defined first. The geotherm here is such that the temperature at the Moho is 460ºC and the choosen strain rate is 10-15 s-1. The straight parts of the profiles (in the upper crust and the upper mantle) represent brittle failures. The red lines are the Dorn law creep curves, whereas the black curved lines are the power law flow curves. One can see that the differential stress for both the power law and Dorn law creep strongly decrease with T. β=3 Strength in extension β=0.75 Crust Ac = 5x10-6 MPa-ns-1 nc = 3.0 Ec = 190 kJ.mol-1 Mantle Am = 7x104 MPa-ns-1 Ed=540 kJ.mol-1 nm = 3.0 Stress Treshold=8500 MPa ed=3.05x1011s-1 Em = 520 kJ.mol-1 σ1-σ3 (MPa) 1000 0 1000 Compression Extension 20 Moho 40 60 Dorn Law + Power law TMoho=460ºC Depth (km) Strength in compression 80 100 A few things to note: • With the Dorn law, brittle failure does not occur in compression. Dorn law creep significantly reduces the strength of the upper mantle. • The brittle part of the crust is thicker in extension that it is in compression. • For a normal geotherm (TMoho <650ºC), the upper crust and the upper mantle are the strongest layers of the lithosphere, the lower crust and the lower lithospheric mantle are comparatively much weaker. • Because quartz deforms by ductile flow at lower temperature (~300ºC) than olivine (ductile at ~600ºC) the upper mantle is much stronger than the lower crust for a normal geotherm (TMoho <650ºC). 50 SENSITIVITY OF STRENGTH ENVELOPS TO TEMPERATURE The figure illustrates the dependence of the strength envelops on temperature. From left to right the temperature at the Moho increases from 400ºC up to 700ºC. The strength of the lithosphere, in both compression and extension (the surface area of the dark and pale blue regions respectively) has been averaged over the lithospheric thickness assuming for the mantle either a power law creep or a combination of power law and Dorn law. As temperature increases, the averaged strength of the lithosphere decreases significantly. 0 1000 2000 1000 0 1000 80 80 Dorn law+ 438 MPa 221 MPa Power law -40% 100 -7% TMoho=400ºC 410 MPa 151 MPa -41% 239 MPa 142 MPa 100 -6% -41% TMoho=460ºC 100 -6% TMoho=540ºC 40 60 60 236 MPa 108 MPa 106 MPa 158 MPa 99 MPa 100 -8% -33% 61 MPa 80 80 101 MPa -5% TMoho=620ºC The integrated strength of the continental lithosphere (in Nm-1) is defined by the vertical integration (i.e. over depth) of the rheological profile. Since the frictional sliding depends on the tectonics regime, three "strengths" can be defined for compressional (Fedt), transcurrent (Feds), and extensional (Fedc) tectonic regime: The figure on the right shows the integrated strength as a function of temperature at the Moho. At TMoho about 500ºC the upper mantle is the stronger layer of the lithosphere (in blue in the inset). Because the power law creep is exponentially dependent on temperature, a small increase in temperature significantly reduces the integrated strength. Indeed, when TMoho is close to 700ºC the strength of the upper mantle drastically decreases. Past 700ºC, the stronger layer of the lithosphere is the brittle upper crust the rheology of which has no dependence on temperature. The red and the blue curves are the integrated strength in extension and compression respectively. 51 20 40 80 333 MPa 183 MPa 0 1000 20 60 570 MPa 195 MPa 1000 0 1000 40 60 60 2000 1000 20 40 40 Power law 737 MPa 237 MPa 0 1000 20 20 Moho 2000 1000 57 MPa 100 -7% TMoho=700ºC 500ºC 40 700ºC 30 20 10 TMoho (ºC) 0 -10 400 600 800 1000 Strength -20 -30 -40 -50 Depth 1000 Integrated Strength x 1012 N.m-1 σ1-σ3 (MPa) 2000 700ºC 500ºC SENSITIVITY OF STRENGTH ENVELOPS TO LITHOLOGIE The continental crust displays a wide range of compositional variation from mafic rocks (gabbros, amphibolites) to quartz dominated composition (granites). This contrast with the relative homogeneity the mantle (peridotite with 60%<olivine<95%). The figure on the right illustrates the dependence of the rheology of the lithosphere on its lithological composition. All rheological profiles involve the same geotherm with a temperature at the Moho of 540ºC (thin dashed line TMoho = 600ºC). Depending of the composition and the rheological parameters the integrated strength in contraction of the continental lithosphere varies over one order of magnitude from 790 MPa down to 78 MPa. In extension the integrated strength ranges from 255 MPa down to 50 MPa. The large variability of the rheological properties of common crustal and mantellic rocks, prevents the definition of a “standard” rheology for the continental lithosphere. Yield stress (MPa) 3000 2000 1000 Yield stress (MPa) 1000 0 2000 1000 20 Moho Yield stress (MPa) 1000 0 [12] 100 [11] 100 1000 2000 1000 20 584MPa 213MPa 2000 1000 582MPa 212MPa 20 [1] 100 1000 0 2000 1000 1000 0 20 40 40 60 60 60 80 80 80 460MPa 178MPa 433MPa 166 MPa [6] 100 [4] 100 3000 2000 1000 1000 0 2000 1000 20 1000 0 20 2000 1000 60 20 Moho 40 40 60 60 60 80 80 80 406MPa 163MPa [13] 100 1000 40 TMoho=540ºC 80 100 0 Strength in extension 40 425MPa 155 MPa [14-15] 2000 1000 TMoho=600ºC 1000 0 20 20 40 40 60 80 404MPa 160 MPa 379MPa 149MPa 365MPa 153MPa 355MPa 147MPa [5] 100 [7] 100 [9] 100 [10] 100 52 Strength in compression 20 Moho 40 Moho Mantle 80 [16] 100 1000 0 Lower crust 60 80 598MPa 193MPa Upper crust 40 60 80 789MPa 255MPa 1000 0 20 40 60 80 2000 1000 20 40 60 0 2000 1000 20 40 3000 2000 1000 Yield stress (MPa) 1000 0 60 80 78MPa 49MPa [2wet] 100