Introduction to Tectonophysics

Home , Geodynamics, Ridge push, Tectonophysics

Patrice F. Rey Preface

i PREFACE This eBook, in a permanent state of unfinishedness, focusses mechanical coupling between the lithosphere and the astheno- on tectonic processes, from the forces that drive them to the pa- sphere. rameters that control the mechanical behavior of the Earth’s How the Earth’s lithosphere deforms depends on the relation- lithosphere. ship between applied forces and the mechanical response We start in Chapter 1 by looking at the Earth’s geotherm be- measured by the accumulated strain and the strain rate, hence cause the mechanical behavior of rocks - and therefore that of Chapter 4 tackles the rheology of rocks. This chapter intro- the lithosphere - is very sensitive to temperature and because duces the concepts of elastic, plastic and viscous strain and ex- tectonic processes can significantly bring the geotherm out of plores the influence of parameters such as temperature, pres- equilibrium forcing the cooling or heating of rocks. sure, and fluid, before presenting the important concept of lithospheric strength envelopes. The second chapter introduces the notion of isostatic equilibrium, critical to understand the topography of the Earth’s sur- Chapter 5 introduces the new discipline of computational tec- face when its lithosphere is at rest. This chapter explains how tonics and geodynamics. Equipped with a reasonable under- isostatic equilibrium differs from the notion of mechanical equi- standing of the thermal and mechanical properties of the litho- librium. Isostasy introduces gravitational stresses because den- sphere, one can tap on the power of high-performance comput- sity interfaces are no longer parallel to gravitational equipoten- ing to explore through numerically experiments tectonic and tial surfaces. The notion of gravitational stress is important to geodynamic processes. understand how the Earth’s lithosphere can deform in the ab- Chapter 6 explores through numerical experiments a range of sence of plate boundary forces. tectonic processes at lithospheric scale, in a variety of tectonic Plate boundary forces are the focus of Chapter 3. Slab-pull, settings from continental collisions, to extensional tectonics ridge push, and drag from asthenospheric flow contribute to and transcurrent tectonics. drive the motion of lithospheric plates at the Earth’s surface, Finally, Chapter 7 focuses on the dynamics of mantle convec- and their deformation. Ridge-push can be evaluated analyti- tion. cally using the concept of the gravitational forces. The evalua- tion of the asthenospheric drag and that of the slab-pull is more difficult because it requires some understanding of the copyright © Patrice F. Rey, 2018

ii CHAPTER 1 The Earth’s Geotherm

The geotherm, i.e. the distribution of temperature with depth, is an important characteristic 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 steady-state geotherm. We then consider the notion of transient geotherm.

3 SECTION 1 Heat Transfer in the Earth’s lithosphere

In this section

1. Heat transfer in the Earth’s Lithosphere 2. Heat energy & temperature 3. Heat conduction 4. Heat advection 5. Radiogenic heat production

The continental geotherm is a function of the i/ rate at which given depth does not change through time). In contrast, when heat is produced or consumed within the lithosphere, ii/ the the lithosphere has a net gain or a net loss of heat, the geotherm rate at which the lithosphere looses heat to the atmosphere/ is said to be transient (i.e the temperature changes through ocean system, and iii/ the rate at which the lithosphere gains time) until a new equilibrium is reached between heat lost and heat from the hot convective mantle. When the heat lost bal- gained. On a billion year time scale, the geotherm is always tran- ances the heat gained, an equilibrium is reached and the geo- sient because the primordial accretionary heat and the Earth’s therm is said to be in steady state (i.e. the temperature at any supply in radiogenic isotopes progressively decrease, however,

4 on a scale of ~100 myr, and in the absence of tectonic or magmatic The main processes able to change the amount of heat energy in activity, the geotherm can approach an equilibrium state. The litho- the lithosphere are: spheric geotherm varies laterally. The base of the lithosphere is de- •Heat conduction (transfer of kinetic energy between molecules fined by the isotherm ~1300ºC. In cratonic regions, this isotherm or atoms from a hot to a less hot region) can stand at a depth between 200 and 300 km. In other continental •Heat advection (replacement of a volume of rock at tempera- regions it can be as shallow as 50 km, and at mid-oceanic ridge it is ture T1 with an equivalent volume at temperature T2) met at a few kilometers underneath the oceanic floor. •Heat production (heat produced by radioactive isotopes, vis- Here, we first review the processes involved in heat generation and cous heating, exothermic metamorphic reactions) heat transfer, and we derive from the rate of these processes a gen- • Heat consumption (heat consumed by endothermic metamor- eral equation which describes the change in temperature with phic reactions, in particular partial melting) depth and through time. From this general equation we derive a particular solution for the so called "steady state" continental geo- The variation of temperature dT over an increment of time dt de- therm (temperature changes with depth but not with time, i.e. zero pends on the sum of heat variations dE due to each process. In net heat gain or loss). In a second part, we discuss how the steady what follows, we derive three expressions for i/ the rate of heat state continental geotherm is affected by a number of geological conduction, ii/ for the rate of heat advection, and iii/ for the rate processes including, lithospheric thinning and thickening, burial of radiogenic heating. From these, we derive the 1D conduction- via sedimentary or volcanic processes, and basal heating via the advection heat transfer equation from which an expression for the spreading of mantle plumes at the base of the Earth's lithosphere. 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 Heat conduction of rock somewhere in the lithosphere varies if heat energy (a form Conduction transports heat from hot to cold regions. The ﬂow of of kinetic energy) is gained or lost. The relationship that gives the heat (Q) is proportional to the negative temperature gradient (dT/ variation of temperature dT as a function of a variation of heat dE dz) between the cold and the hot region, with the coefﬁcient of pro- is: portionality being the conductivity (K). Mathematically this trans- . dT = dE / (Cp m) lates into the Fourier's law where Q is in W.m-2 and K is in W.m-1.K-1. In our geological reference frame, z increases downward with Cp the heat capacity, and m the mass.

5 (T(z+dz)>T(z)). Conduction occurs in the direction of decreasing Radiogenic heat production temperature (i.e. dEc is a gain for upward conduction) hence the z Radiogenic disintegration of radioac- sign "-" insures that Q is positive upward (dEc is positive when T tive isotopes (238U, 235U, 232Th, and increases downward). 40K) releases heat. The increment of dT A Q = − K radiogenic heat (dEr) produced in a E =- .a.dt z 1 Q(z) dz small cylinder of rock of section a Q(z) dE = A.a.dz.dt Let's consider a small cylinder of rock of sec- and length dz over an increment of r dEc = E2 - E1 2 time dt is: dE = .a.dt tion a (area in m ). If the incoming and outgo- c (dQ/dz).dz z+dz a ing heat at both ends of the cylinder are the a same, there is no net heat gain or loss, and A⋅a⋅dz⋅dt = A⋅dV⋅dt

E2=-Q(z+dz).a.dt the temperature remains unchanged. Tem- z+dz where A is the rate of radiogenic heat production. Radioactive heat perature changes when the heat E1 leaving is the main internal heat source for the earth as a whole (it is meas- Q(z+dz) the volume over an increment of time ( ured in W.m-3). E1=Q(z) ⋅ a ⋅ dt) is different to the heat E2 entering it (E2=Q(z + dz)⋅a⋅dt). Heat Advection The entering heat flow Q(z + dz) can be approximated with a Tay- lor series in which only the two first terms are of significance. (n.b.: Advection of heat implies that mathematically f (x + dx) can be approximated from f (x ) and the n n a mass of material at tempera- a.uz . dt derivatives at location : , , etc: E1=C ..a.u .dt.T xn f ′(xn) f ′′(xn) ture T (in yellow on the z p z sketch) is being pushed out of dE = E - E 2 2 our cylinder and replaced by u 2 1 dQ dz d Q dE t.dT Q(z + dz) = Q(z) + dz⋅ + ⋅ + ⋅⋅⋅ an equivalent mass of material u = Cp..a.uz .d 2 dz 2 dz at temperature T+dT. The increment of heat gained or lost ( Therefore the increment of heat (dEc) gained or lost in an increment a of time dt is : dEu) over an increment of time a.u . dt dEc = E2 − E1 z E2=C dT) dt is proportional to the mass z+dz p..a.uz .dt.(T+ dQ d2T of material displaced ( a⋅Q(z)⋅dt − a⋅Q(z + dz)⋅dt = − a⋅dz⋅ ⋅dt = K⋅ ⋅dV⋅dt 2 dz dz ρ⋅a⋅uz ⋅dt ), the heat capacity of

6 -1 -1 the material (Cp in J . kg . K : is the amount of energy required by left to right, the conduction term, the term of radiogenic heat pro- 1 kg of the material to increase its temperature by 1 K), and the tem- duction, and the advective term. In 3D, this equation becomes: perature contrast of the incoming and outgoing masses. (n.b.: be- ∂T K ∂2T ∂2T ∂2T A ∂T cause z increasing with depth, uz is negative for upward convec- = ⋅[ + + ] + − u ⋅ 2 2 2 z tion, however, dEu is a gain for upward convection hence the sign - ∂t ρ⋅Cp ∂x ∂y ∂z ρ⋅Cp ∂z insures that when uz is negative (i.e. upward motion) the advective heat gained is positive). We get that the rate of heat advection is:

−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 (no variation of temperature Adding up the rate of conductive heat (dEc), the rate of convective through time) by definition dT/dt=0 and uz = 0 (no advection or ero- heat (dE ), and the rate of radiogenic heat (dE ) gives the total rate u r sion or sedimentation), therefore the 1D heat transfer equation sim- of heat gained or lost. This variation of heat triggers a change in d2T A temperature dT = dE/(Cp.m) therefore : plifies to: = − dz2 K and Cp.m. dT= dEr + dEu + dEc : This is a second order differential equation. It says that the gradient of the temperature gradient is a constant equal to the negative 2 d T ratio between the radiogenic heat production and conductivity. Cp⋅ρ⋅dV⋅dT = A⋅dV⋅dt − Cp⋅ρ⋅a⋅uz⋅dt ⋅dT + K⋅ ⋅dV⋅dt dz2 This kind of equation is solved by integrating twice and by using two boundary conditions. For example we may know the tempera- This leads to the 1D conduction-advection heat transfer equation: ture 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 dT K d2T A dT = ⋅ + − u ⋅ this is a positive number). Assuming that A is constant with depth, 2 z dt ρ⋅Cp dz ρ⋅Cp dz the first integration led to the temperature gradient dT/dz:

This equation describes the variation of temperature with depth dT A = − ⋅z + C and through time due to heat conduction, radiogenic heat and dz K 1 heat advection. This equation assumes no lateral heat ﬂow (hence This gives the gradient of the geotherm as a function of depth. 1D). On the right end side of the equation the three terms are, from From this function we get that at the surface (i.e. z=0): dt/dz=C1

7 Because of the Fourier's law, the second boundary condition im- relationships, one for the crust (the very last equation), and one for poses that at z=0: the lithospheric mantle. dT Q If the production of radiogenic heat is zero in the mantle then the = 0 dz K 1D conduction-advection heat transfer equation simpliﬁes to: 2 by combining the last two equations for z=0 we get C = Qo/K, we d T 1 = 0 therefore conclude that the thermal gradient at any depth z is ... dz2 dT A Q Integrating twice we get: = − ⋅z + 0 dz K K dT = C2 and T(z) = C2⋅z + C3 Integrating a second time led to: dz

With the two following boundary conditions: at z = zc (zc = Moho A 2 Q0 T(z) = − ⋅z + z + C depth): T(zc ) = Tc and Q(zc ) = -Qm (the basal heat flow), we get that: 2K K 2 Qm Qm The first boundary condition says that T=To at z=0, therefore we C2 = and that C3 = Tc − ⋅zc get that: K K The geotherm in the lithospheric mantle is therefore: A Q T(z) = − ⋅z2 + 0 z + T 2K K 0 Qm T(z) = ⋅(z − zc) + Tc This relationship is the steady state geotherm. It gives the distribu- K tion of temperature with depth in a layer with homogeneous radio- Hence, the geotherm in the lithosphere is defined by a two steps genic production A, conductivity K, with a surface temperature of function: To and a surface heat flow of Qo. A Q T(z) = − ⋅z2 + 0 z + T , for 0 < z ≤ z The geotherm in the continental lithosphere and beyond 2K K 0 c

The continental lithosphere consists in two layers with contrasted Qm T(z) = ⋅(z − z ) + T , for z > z thermal properties. In particular the radiogenic heat production in K c c c the mantle is negligible compared to that of the crust. The geotherm in continental lithosphere is therefore best described by two

8 2 alternatively, if Qm instead of Qo is known the crustal geotherm (i.e. dT dT d T A “differential equation” links together the derivatives ( , , ) when 0 < z < zc): dt dz dz2 dT dT of a function T(z, t). Here, is the rate of temperature change; A Q A dt dz T(z) = − ⋅z2 + [ m + ⋅z ]⋅z + T , for 0 < z ≤ z 2K K K c 0 c is the gradient of temperature (i.e. the slope of the geotherm); and d2T In the asthenospheric mantle, convective motion is such that the is the gradient of the gradient of temperature. dz2 temperature shows relatively little variation as convection acts as From this differential equation, one can derive a solution for the an efﬁcient mixing process. For z > zl (zl being the base of the lithosphere) the geotherm follows the adiabatic gradient ~0.3 K per kilo- particular case when thermal equilibrium is reached (i.e. there is no dT meter. For instance, from the top of the convective mantle (i.e. the change of temperature through time: = 0 and necessarily there base of the lithosphere) to the core/mantle boundary (i.e. a dis- dt is no advection of heat, hence uz = 0). This corresponds to the tance of ca. 2700-2800km) temperature increases by only ~3000ºC. “steady geotherm” for which the differential equation simpliﬁes to: This contrast with the evolution of the temperature within the lithosphere where temperature increases from ca. 0ºC at the surface to d2T A T’’(z) = = − 1300ºC at depth of 100-200km. dz2 K

Appendix 1: Solving differential equations This tells us that the change of the temperature gradient with d dT depth ( )( ) is equal to a constant -A/K (we assume here that Very often, the rate of natural processes (i.e. rate of heat conduc- dz dz tion, rate of heat advection, the rate of radiogenic heating etc) can the radiogenic production A and conductivity K are constant with easily be calculated or measured. In the context of the Earth’s geo- depth). The negative sign implies that the rate of temperature in- therm, we have seen that, from the rate of individual heat transfer crease decreases with depth. mechanisms, we can express the overall rate of temperature change Lets plot T’’(z): through time (t) at any depth (z) via the following differential equa- T''(z) tion:

dT K d2T A dT = ⋅ + − u ⋅ 2 z dt ρ⋅Cp dz ρ⋅Cp dz z -A/K

9 d2T A The curve in blue is the geotherm, bearing in mind that this curve The differential equation = − can be solved to ﬁnd out the dz2 K is only valid for 0 < z < zmoho. geotherm T(z). For this the differential equation is integrated twice: T(z) The ﬁrst integration leads to the temperature gradient :

dT A T(0) T’(z) = = − ⋅z + C dz K 1 zmoho z Graphically we get that: T'(z) C is therefore the temperature 1 c We have recognized that the integration constants C1 and C2 have 1 -A/K gradient at z=0 (Earth’s surface), the following meanings: C1 is the temperature gradient (the slope and -A/K is the slope of the gradient. of the geotherm) at the surface (i.e. at z=0); and C2 is the temperature at the surface. The temperature at the surface is typically in

z the range of 0 to 30ºC, and the temperature gradient at the surface -A/K can be determined by measuring the temperature at the surface T''(z) and at the bottom of a well a few 100s to a few 1000s of meter deep.

The second integration leads to the geotherm: Appendix 2: Taylor Series

A 2 According to the Taylor theorem any smooth function f(x) can be T(z) = − ⋅z + C1 ⋅z + C2 2K approximated as a polynomial involving its derivatives following: Graphically we get that: df(x) d x2 d2f(x) d x3 d3f(x) f (x + d x) = f (x) + d x⋅ + ⋅ + ⋅ + ⋅⋅⋅ 2 3 C2 is therefore the temperature T'(z) d x 2! d x 3! d x at z=0. c 1 -A/K f(x+dx) ≈ f(x)

f(x+dx) ≈ f(x) + f’(x) . dx c f’’(x) 2 T(z) f(x) f(x+dx) ≈ f(x) + f’(x) . dx + . dx2 2! z -A/K T''(z) x x+dx x

10 Appendix 3: Steady state geotherm with decreasing RHP A0 2 −z T(z) = − ⋅hc ⋅Exp( ) + C1z + C2 The steady state crustal geotherm we derived earlier assumed that K hc the radiogenic heat production is constant with depth. This is usu- To find out the two integration constants C1 and C2 we need to call ally not the case as cycles of partial melting and crustal differentia- two boundary conditions. Lets say that we know the temperature tion through upward flow of melt tend to deplete the lower crust at the Earth' s surface T0 and the heat flow Qm entering the base of and enrich the upper crust in radiogenic elements. The decrease the lithosphere. with depth of the radiogenic heat production is often modeled by A an exponential law: The first boundary condition leads to: C = T + 0 ⋅h 2 2 0 K c −z

A(z) = A0 . Exp( ) Qm A0 −zc hc ... and second to: C1 = − ⋅hc⋅Exp( ) K K hc This means that the radiogenic heat production measured at the surface ( A0 ) in divided by e (e=2.71) every hc meters (typically After substituting C1 and C2 into T(z), expanding and simplifying, 5000 m< hc <25000 m). Therefore the 1D conduction-advection one gets: dT K d2T A(z) dT A −z Q A −z heat transfer equation is: = ⋅ + − uz⋅ 0 2 m 0 c 2 T(z) = ⋅hc ⋅(1 − Exp( )) + ( − ⋅hc ⋅Exp( ))⋅z + T0 dt ρ⋅Cp dz ρ⋅Cp dz K hc K K hc In the case of a steady state geotherm, this equation becomes: If one knows the surface heat ﬂow instead of the mantle heat ﬂow then: d2T −A(z) = 2 A0 2 −z Q0 dz K T(z) = ⋅hc ⋅(1 − Exp( )) + ⋅z + T0 K hc K d2T A −z replacing A(z) we get that = − 0 ⋅Exp( ) 2 dz K hc Funbits: What love as to do with differential equations? Following two successive integrations one gets: https://fabiandablander.com/r/Linear-Love.html dT A0 −z = ⋅hc⋅Exp( ) + C1 and dz K hc

11 SECTION 2 Earth’s Geotherm

In this section TºC TºC TºC 1. Steady state geotherm 0 400 800 1200 1600 0 400 800 1200 1600 0 400 800 1200 1600

2. Transient geotherm 20 20 20

40 Moho 40 Moho 40 Moho to

t1 60 60 60 t5 Mechanical boundary layer t o t10 t100 Thermal boundary layer t1 t20

t5 80 80 t100 80 t10 t t20 8

t 100 100 8 100

to 120 120 120

140 t 140 140

Plume spreads at the top of the Plume spreads under the lithosphere thermal boundary layer Plume spreads at the Moho Depth (km) Depth (km) Depth (km)

Steady state geotherm tion). We then look at how sensitive is the geotherm to change In the previous section we have derived the 1D conduction- in mantle heat ﬂow, conductivity and thickness of the radio- advection heat transfer equation, and we have derived an equa- genic crust. tion describing the equilibrium geotherm, also called the Distribution of radiogenic heat production “steady state geotherm” because the temperature at every depth doesn’t change through time. Here we explore how sensi- We have assumed so far that the volumetric radiogenic heat pro- tive is the geotherm to the radiogenic heat production when it is duction A was depth-independent ( A remains constant with not constant with depth (as we have assumed in the ﬁrst sec- depth). However, because the upper crust is enriched in incom-

12 patible elements* the radiogenic heat production decreases with partial melting, they tend to concentrate into the melt phase. Due to the depth. A common model assumes that A is divided by e (e=2.71) buoyancy of the melt, incompatible elements concentrate over time into every h meters, h being the length scale of the exponential law. This the upper part of the crust. Through repeated cycles of melting, the con- model of distribution is given by: centration of radiogenic elements contributes to the cooling of the Earth’s geotherm. A(z) = A0⋅Exp( − z /h) Basal Heat Flow (Qm) ... in which A0 is the radiogenic heat production at z = 0 m. The graph below illustrates the sensitivity of the geotherm to the The graphs shows mantle heat ﬂow (Qm), also -3 the geotherm for Radiogenic Heat (mW.m ) Temperature (TºC) called in some textbook ba- 2 4 6 8 500 1000 TºC A(z) decreasing sal heat ﬂow. This graph 0 500 1000 1500 0 exponentially shows three geotherms cal- with various culated assuming same 25 25 20 length scale h (10, rate of radiogenic heat pro-

25, 50km). Ao is duction, same conductivity Moho Moho 40 adjusted so that 50 50 and same crustal thickness. the total radio- TMoho: Only the mantle heat flow 724, 60 Qm=36 Depth (km) genic heat produc- Depth (km) 935, varies. 10 km 75 75 1284 Qm=24 tion (R.H.P which 25 km is given by the in- 50 km A basal heat flow of 12 x 80 -2 10-3 W.m-2 (yellow geo- tegration of the R.H.P.: 0.078 W.m Qm=12 100 100 therm) is characteristic of 100 radiogenic heat Archaean (3.5-3 Ga) profile with cold and thick cratonic depth) is the same lithospheres, whereas a ba- -3 in all models. The larger is h (i.e. the deeper are the radiogenic ele- sal heat flow of 36 x 10

-2 Depth (km) ments) the hotter is the geotherm. This suggests that at an early W.m is representative of stage of the Earth evolution (before the extraction of the crust) the thinned lithosphere. In- Earth had a warmer geotherm. creasing the mantle heat ﬂow from 12 to 36 x 10-3 W.m-2 increases the temperature at the * n.b.: Incompatible elements such as U, Th and K have a large radius and Moho from ~320 ºC to 720 ºC. therefore do not ﬁt easily into crystals lattice. Hence, during episodes of

13 Thermal conductivity (K) thinning or fast thickening. During thinning the geotherm increases as warmer temperatures are reached at lesser depths (the The graph below illustrates the sensitivity of the geotherm to the slope of the geotherm increases). During thickening the geotherm thermal conductivity K (W.m-1.K-1). It shows three geotherms calcu- decreases as cooler conditions are met deeper in the crust (the lated assuming same rate of slope of the geotherm decreases). Second, as the thickness of the radiogenic heat production, TºC crust changes, the steady state geotherm also changes because a same same crustal thickness, 0 500 1000 1500 0 thicker crust will produce more radiogenic heat, therefore crustal and same mantle heat ﬂow. thickening leads to warmer geotherm, whereas crustal thinning The conductivity is known to 20 leads to cooler geotherm. vary with temperature. Here however, we assume that K is Lithospheric thickening 40 Moho constant through depth. K is Thickening produces heat advection resulting in a rapid cooling of proportional to the ability of 60 k=1.75 the geotherm. Thickening also increases the thickness of the radio- material to conduct heat k=2.25 genic layer therefore increasing the production of radiogenic heat away. Therefore, the larger 80 in the lithosphere. Isostasy (cf. section Isostasy and Gravitational the thermal conductivity of k=2.75 Forces) produces uplift leading to erosion which in turn affects the rocks, the lesser the capacity 100 amount and distribution of the radiogenic heat elements in the of the crust to store heat, and crust which cools geotherm. The interplay between the mode of the the cooler is the geotherm. thickening (heterogeneous via thrusting vs homogeneous), the Increasing the thermal con-

Depth (km) thickening rate, and the erosion rate leads to contrasted thermal his- ductivity from 1.75 to 2.75 tories. W.m-1.K-1 decreases the tem- The graphs on the next page show transient geotherms (at 0, 0.5, 2, perature at the Moho from 10...Myr) following heterogeneous thickening (thickening is 750 to 500ºC. achieved by doubling the thickness of the crust via a single thrust, Thickness of the radiogenic crust which explains the temperature discontinuity at t=0), and homogeneous thickening (the thicknesses of the crust and the lithospheric Crustal thickening or thinning modify the geotherm in two differ- mantle are doubled via pure shear deformation). Erosion is dis- ent ways. First, during deformation, heat is advected mainly up- carded here. The discontinuity in the case of heterogeneous thicken- ward (in the case of thinning) and mainly downward (in the case of ing is smoothed out in a few Ma. Transient geotherms in both cases thickening) as rocks carry their temperature with them during fast are similar for time > 10 Myr.

14 stasy leads the subsidence of the surface of the lithosphere leading 0 200 400 600 800 1000 1200 TºC 0 200 400 600 800 1000 1200 TºC 0 0 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, 10 myr the thinning rate, and the sedimentation rate lead to contrasted 50 50 50 myr 0.5 myr2 myr 10 myr 250 myr 50 myr 250 myr thermal histories.

Moho Moho The graph on the bottom left shows transient geotherms (0, 10, 50, 100, 250... myr) following homogeneous thinning (the thicknesses 100 100 of the crust and the lithospheric mantle are halved by pure shear deformation). Sedimentation is discarded here. Following the increase of the geothermal gradient due to extensional deformation, 150 150 thermal relaxation leads to cooling and therefore the thickening of Homogeneous thickening via pure shear the lithosphere. Slowly, the transient geotherm approaches the Heterogeneous thickening via thrusting doubling the thickness of the lithosphere and doubling the thickness of the crust steady state geotherm. Depth (km) Depth (km) Sedimentation of burial of the crust

Sedimentation and burial of the continental crust under a few kilo- Lithospheric thinning 0 200 400 600 800 1000 1200 TºC meter of volcano-sedimentary rocks have a signiﬁcant long-term 0 Thinning drives heat advec- effect on the geotherm. The newly deposited layers effectively insu- late the buried heat producing layer. If the conductivity of the up- tion as rocks, and the heat at- Moho tached to them, are displaced per layer is lower or equal to that of the buried layer, then heat ac- vertically mainly upward re- cumulates increasing the geotherm. 50 sulting in a rapid warming of This effect could have played a major role in the differentiation of the geotherm (the geothermal 10 myr the continental crust in the Archaean. At that time, radiogenic heat 50 myr 250 myr production in the crust was 2 to 6 time larger that of present day gradient increases). However, 100 myr thinning also decreases the rate of radiogenic heat production. Furthermore it was a time 100 thickness of the radiogenic when 5 to 20 km thick continental ﬂood basalts (the so called green- crust, and therefore reduces stones) where deposited at the surface of the Earth, insulating the 60% homogeneous thinning of heat producing crust. the production of radiogenic the lithosphere via pure shear heat in the lithosphere. Iso- Depth (km) The graphs show transient geotherms following the emplacement

15 thick layer with a temperature of 1700ºC. The three graphs show Archaean continental lithosphere Archaean continental lithosphere TºC TºC Greenstone 200 400 600 800 1000 1200 1400 Greenstone 200 400 600 800 1000 1200 1400 the results for various depth of emplacement.

Crust Crust Crust 25 25 TºC TºC TºC

Geotherm t 8 Geotherm to Geotherm to 0 400 800 1200 1600 0 400 800 1200 1600 0 400 800 1200 1600 t +10Ma to+50Ma o 50 50 to+20Ma

Geotherm t 8 20 20 20 to+100Ma to+30Ma

75 75 Moho to to+200Ma 40 40 Moho 40 Moho Mantle Mantle to+40Ma t1 t +400Ma o 60 60 60 t5 t +50Ma Mechanical boundary layer t 100 100 o o t10 t100 Thermal boundary layer t1 t20 to+100Ma t5 80 80 t100 80 t10 to+200Ma t 125 125 8 t20 to+1Ga t 100 100 8 100 Depth (km) Depth (km) 150 150 to 120 120 120

175 175 140 t 140 140

Plume spreads at the top of the Plume spreads under the lithosphere thermal boundary layer Plume spreads at the Moho Depth (km) of a 6 km (graph on the left) or 12 km thick greenstone covers Depth (km) Depth (km) (graph on the right), with no radiogenic heat production in them. Transient geotherms The temperature increase is large enough to lead to significant partial melting of the crust. Calculation of the steady state geotherm is relatively straightfor- ward. In contrast, the calculation of transient geotherms, such as Emplacement of a mantle plume those displayed above, requires computational tricks as the temperature changes in both space and time. Many analytical solutions Mantle plumes initiate at the core-mantle boundary and rise fast have been proposed for a range of problems. The well-known book through the convective mantle. Upon approaching the more rigid from Carslaw and Jagger, first published in 1946 "Conduction of lithosphere, they spread laterally under the lithosphere. They may Heat in Solids", provides hundreds of analytical solutions to many also thermally erode to base of the lithosphere and spread higher problems. Here, we give a couple of them. up. This is equivalent to put a hot layer, a few tens of kilometer thick, underneath or within the colder lithosphere. In the Ar- Progressive cooling of the oceanic lithosphere chaean, the Earth was warmer and plumes were most likely more numerous. The next graphs document the thermal impact of a man- The formation of oceanic lithosphere at mid-oceanic-ridge is a ther- tle plume assuming that the plume's head spreads into a 50 km mal problem involving the progressive cooling of the asthenosphere. The temperature at the seafloor Ts is maintained constant,

16 so is the temperature in the asthenosphere Tm. The lithosphere be- (he assumed Tm =2000ºC) and that Earth had comes cooler and therefore thicker as cooling proceeds. Neglecting cooled by conduction to its present surface -3 -2 radiogenic heat in the basaltic crust and the mantle, and assuming heat flow (he assumed Q0= -30 10 W.m ). He no sedimentation, the 1D heat transfer equation becomes: assumed that all heat was lost at the surface by conduction (with κ = 10-6.m-2.s-1). Thus the dT d2T = κ⋅ problem is reduced to that of finding the tem- dt dz2 perature within a cooling half-space of infinite extent as a function of time after the half with the thermal diffusivity which can be expressed in terms of κ space is set at a specific temperature, a prob- conductivity (K), density ( ) and heat capacity (Cp): ρ κ = K⋅ρ⋅Cp lem for which the equations above apply. Kelvin found that the Earth was 50 myr old. This 2 order of magnitude off the mark, but The analytical solution of this differential equation is: still much better than the biblical 6000 yr. z T(z, t) = Ts + (Tm − Ts)⋅erf( ) Cooling of a dike 2 κ⋅t The cooling of a dike with half width w is another problem which By differentiating with respect to z one get the temperature gradi- has an analytical solution of the following form: ent ... T w − x w + x s dT T − T z2 T(x, t) = ⋅(erf( ) + erf( )) = m s ⋅Exp( − ) 2 2 κ⋅t 2 κ⋅t dz π⋅κ⋅t 4⋅κ⋅t If the dike has a width of 2 m, i.e. w=1 m, and its initial tempera- -6 2 -1 ...from which the variation of the surface heat flow through time ture was Ts = 1000 °C, and κ = 10 m s , then the temperature at can get extracted as: 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. dT Tm − Ts = Some swarms, such as the giant McKenzie swarm in Canada, ap- dz π⋅κ⋅t pear to radiate from a point, commonly interpreted as a plume source for the magmas. The mid-Proterozoic Coppermine River In the 19th century, Lord Kelvin used this equation to find out the flood basalts were erupted at the same time near the plume head. age of the Earth. Kelvin made the assumption that the Earth had Individual dykes range from 10 to 50 m in width, with some up to formed as a molten body at the temperature at which basalt melts 200 m wide. Some dykes can be traced for up to 2000 km.

17 Numerical solutions and therefore:

Since the arrival of com- ∂T L ∂2T A ∂T puters, which allow for (1) − (1 − ) = κ⋅ + − U ∂t C ⋅(T − T ) ∂z2 ρ⋅C ∂z large numbers of calcula- p liq sol p tion to be performed Boundary conditions: T(z, 0)=0; T(0, t)=T0; dT/dz(zl, t)=-Qm/K very fast, new techniques based on numeri- An approximation of this differential equation can be obtained by cal algorithms have been replacing ∂t and ∂z with ﬁnite differences ∆t and h. The central designed to solve differ- tenet of this computational technique is that for a given time t=n, ential equations. These the temperature at a depth z = i (Tin) can be calculated from knowl- numerical recipes are based on the "discretization" of differential edge of the temperature at depth z = i-h and z = i+h. In a similar equations ... fashion, for a given depth z = i, temperature at time t = n can be calculated from knowledge or the temperature at time t = n-∆t and • Consider the 1D heat conduction-advection equation in a slab zl temperature at time t = n+ ∆t. The level a accuracy depend of the thick. The upper surface of the slab has at temperature To. The tran- size of the time and space ﬁnite differences. There are many ways sient temperature is then described by the conductive-advective to express T(z, t) as a function of T(z-h, t), T(z+h, t), T(z, t- ∆), T(z, t+ equation of heat balance. Here we consider a situation where inter- ∆). Here we present the Crank-Nicholson scheme where, equation nal radiogenic heat is creation at a rate A, heat is lost or gain by ad- (1) is rewritten as: vection at speed U (since z increases downward, if erosion U < 0, if sedimentation U > 0), and heat is lost through phase transition (2) solid-liquid (L < 0 for partial melting, L > 0 for crystallization) X T n+1 − T n L the melt fraction is a function of T and therefore z. i i − 1 − Δt ( Cp ⋅(Tliq − Tsol) ) ∂T ∂2T A ∂T ∂X L = κ⋅ + − U + ⋅ n n n n+1 n+1 n+1 n+1 n 2 1 Ti+1 − 2Ti + Ti−1 Ti+1 − 2Ti + Ti−1 T − T A ∂t ∂z ρ⋅C ∂z ∂t C = κ⋅ ⋅ + − U i i + p p 2 2 2 ( h h ) h ρ⋅Cp re-arranging we get: where: n n+1 Ti = T(zi, tn), T i = T(zi, tn+1), etc ∂T 1 ∂T ∂2T A ∂T − ( ⋅ ) = κ⋅ + − U 2 ∂t Tliq − Tsol ∂t ∂z ρ⋅Cp ∂z

18 The ﬁnite difference equation (2) is known as the Crank-Nicholson This formulation results in N equations and N+4 unknowns, the n n n+1 n+1 scheme. It is based on central differences for the spatial derivatives aver- four additional unknowns being u0 and uN+1 , u0 , uN+1 . How- aged forward in time over time steps n and n+1. ever, these four additional unknowns lie outside the computational grid (time, space). Also, from the boundary conditions the follow- Expanding (2) and re-arranging to express temperature at time n+1 ing conditions must be satisﬁed at the nth and n+1 time steps: as a function of the temperature at time n we get equation (3): T n = 1, T n = 0, T n+1 = 1, T n+1 = 0 −κ ⋅Δt L κ ⋅Δt UΔt −κ ⋅Δt 1 N 1 N ⋅T1+n + 1 − + + ⋅T1+n + ⋅T1+n 2 −1+i 2 i 2 1+i 2h ( Cp ⋅(Tliq − Tsol) h h ) ( 2h ) This gives the following set of N equations (5) for the temperature at the n+1 time step : κ ⋅Δt n L κ ⋅Δt UΔt n κ ⋅Δt n A⋅Δt = ⋅T + 1 − − + ⋅Ti + ⋅T + 2h2 −1+i ( C ⋅(T − T ) h2 h ) ( 2h2 1+i) ρ⋅C p liq sol p A⋅Δt a ⋅T1+n + b ⋅T1+n + c ⋅T1+n = a ⋅T n + b ⋅T n + c ⋅T n + , i = 1 replacing and substituting: 11 0 11 i 11 2 1 0 1 1 1 2 k A⋅Δt κ ⋅Δt U⋅Δt L 1+n 1+n 1+n n n n R = , V = , Lt = a1i ⋅Ti−1 + b1i ⋅Ti + c1i ⋅Ti+1 = a1 ⋅Ti−1 + bi ⋅Ti + c1 ⋅Ti+1 + , i = 2 to N − 1 d 2h2 h C (T − T ) k p liq sol A⋅Δt 1+n 1+n 1+n n n n a1N ⋅TN−1 + b1N ⋅TN + c1N ⋅TN+1 = aN ⋅TN−1 + bN ⋅TN + cN ⋅TN+1 + , i = N we get... k with (6): 1+n 1+n 1+n −Rd ⋅T−1+i + (1 + 2Rd + V − Lt)⋅Ti − Rd ⋅T1+i

A⋅Δt b11 = 1, c11 = 0, a1N = 0, b1N = 1 = R ⋅T n + (1 − 2R − Lt + V)⋅T n + (R ⋅T n ) + d −1+i d i d 1+i ρ⋅C p 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) The N equations above can be rearranged into the following matrix and a = Rd, c = Rd, b = (1 − 2R + V − Lt) i i i d equations: 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 nth time step (4): A⋅Δt a ⋅T1+n + b ⋅T1+n + c ⋅T1+n = a ⋅T n + b ⋅T n + c ⋅T n + = dn 1i −1+i 1i i 1i 1+i i −1+i i i i 1+i k i

19 c .. .. Numerical solutions an example b11 11 0 0 0 n+1 n T1 d .. .. 1 500 1000 TºC a12 b12 c12 0 0 n+1 n T2 d • We illustrate here the problem .. 2 0 a13 b13 c13 0 0 n+1 n where a continental crust with initial T3 d3 0 ...... 0 ⋅ . = . (7) . . thickness zc is thickened by a factor 2 0 ...... 0 n n+1 d via the emplacement of one single zt 50 ...... a c TN−1 N−1 0 1N−1 b1N−1 1N−1 n thick thrust. z = 40 km, z = 40 km. n+1 dN c t ...... TN 0 a1N b1N Moho The figure on the right shows the in- 100 with: stantaneous and steady state geotherm (potential geotherm reached c .. .. n b1 1 0 0 0 after infinite time). The aims is to dis- d1 1 a2 b c2 0 .. .. 0 n play transient geotherms down to a Heterogeneous thickening via thrusting n 2 Depth (km) and doubling the thickness of the crust d2 T2 depth z = zl + zt at various time inter- n 0 a3 b3 c3 0 .. 0 n d3 T3 A⋅Δt . = 0 ...... 0 ⋅ . ⋅ vals, where zl is the thickness of the . . k n 0 ...... 0 n lithosphere before thickening zl = 120 km. d TN−1 N−1 ...... n 0 aN−1 bN−1 cN−1 n dN TN • We choose a spatial finite difference h = 4000 m (spatial grid). 0 ...... aN b N Therefore the number of column in the tridiagonal matrixes will be

zl/h. The Crank Nicholson scheme imposes the maximum time step Thus the RHS of (7) is determined explicitly from the solution at of h2/(2.κ) = 253,000 yr. the nth time step. Note that the boundary conditions are satisfied by ensuring that the coefficients given in (6) are satisfied. Hence in or- • We get that Rd = (κ.∆t)/(2h2) = 0.25. We assume no erosion: v = 0; der to find the solution at the time step, one must solve N linear and we allow for partial melting: equations. Since the coefficient matrixes in (7) and (8) are tridiago- Lt=Latent_heat/(Cp.(Tliquidus-Tsolidus)). With Rd, v and Lt we can nal, one can make use of efficient algorithms (e.g. Thomas algo- determine ai1, bi1, ci1, the coefficient of the tridiagonal matrix at rithm) to find the solution. time n+1, and ai, bi, and ci the coefficient of the tridiagonal matrix at time n.

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.

20 • With this, we construct the tridiagonal coefﬁcient matrixes (7) and (8) , the equation system (7) is solved via matrix inversion (Numpy, R, MatLab or Mathematica are well equipped to do the 0 200 400 600 800 1000 1200 TºC 0 dirty work...). Graph on the right show transient geotherms from 0.5 to 250 myr and to inﬁnite time.

50 0.5 myr2 myr 10 myr 50 myr 250 myr

Moho (7)

100

150

Heterogeneous thickening via thrusting (8) and doubling the thickness of the crust Depth (km)

21 Review 1.1 Heat transfer & the Geotherm Review 1.2 Heat transfer & the Geotherm Review 1.3 Heat transfer & the Geotherm

The relationship between heat and Using the Taylor series and bearing in mind the Fou- List ﬁve processes consuming or re- temperature is: rier’s law: Q = -K . ( dT/dz), which of the following leasing heat into a volume of rock is/are correct: somewhere in the crust.

A. T(z+dz) = T(z) + dz x dT A. A. T = E x Cp x m

B. T(z+dz) = T(z) + dz x (dT/dz) B. B. dE = dT / (Cp x m)

C. T(z+dz) = T(z) x (dT/dz) C. C. dE = dT x (Cp x m)

D. T(z+dz) = T(z) - dz x (Q/K) D. D. dT=dE / (Cp x m) E. T(z+dz) = T(z) + (dT/dz) x (Q/K) E.

Check Answer Check Answer Check Answer

In a mathematical equation involving dimensional parameters, physical dimensions such as meter (m), second (s), kilogram (kg), Joule (J), Kel- vin (K) or degree centigrade (ºC) can be balanced to verify whether or not a proposed relationship between physical quantities is correct. For instance lets propose that the velocity (v) is a relationship between time (t) and distance (l) such that : v = t / l2. Knowing that the unit of velocity is m/s, it is clear that the proposed relationship is wrong since the unit of the right hand side of the proposed equation is s.m-2 . Considering that velocity is in m/s, one can easily ﬁgure out that the correct relationship must be: v = l/t. Review 1.4 Heat transfer and Geotherm Expected learning outcomes

Which of the following is/are correct: At this stage you should be able to:

• Explain why and how temperature increases with depth. This means: 1/ being able to comment on the A. The long term consequence of crustal various terms of the equation describing the steady state geotherm, and the meaning for the various pa- thinning is a hotter geotherm. rameters; and 2/ being able to explain the shape the geotherm in a temperature versus depth graph. This B. The long term consequence of crustal should involve concepts like conductivity, advection, radiogenic heat production etc. thickening is a hotter geotherm. • Explain the relationship between heat flow, the tem- C. The long term consequence of deposi- perature gradient and thermal conductivity. tion of a thick layer of sediments is a • Explain how tectonic and other processes can colder crustal geotherm. change the lithospheric geotherm i/ on the short terms through advective heat transfer, and ii/ on the D. Higher rock conductivity results in long term through heat conduction. hotter geotherm. • At a more advanced level you should be able to derive an expression of the steady state geotherm from E. For the same total radiogenic iso- the 1-D conduction-advection heat transfer equation topes content, a colder syeady steady and knowledge of two boundary conditions. state geotherm is reach when radiogenic isotopes concentrate in the upper crust.

Check Answer CHAPTER 2 Isostasy & Gravitational Forces The Tibetan plateau stands 5000 m above sea level. It is the surface expression of a 75 km thick continental crust, which reduces the weight of the continental lithosphere. It imparts a significant horizontal gravitational push force on its surrounding. Isostasy is the physical process that explains the surface elevation of the lithosphere at rest. It is based on the Archimedes’ principle which relates the elevation of a floating body above the surrounding fluid i/ to the density contrast between the floating body and the supporting fluid, and ii/ the vertical length of the floating body.

24 SECTION 1 Isostasy, elevation, gravitational potential energy & gravitational force

Photo: DscobiePhotography.co.uk/creativecommon At rest, a lithosphere with a thick crust stands above equipotential surfaces, introduces gravitational forces adjacent non-thicken regions, and it is said to be in iso- driving the ﬂow of rocks from regions of high pressure static equilibrium. The notion of isostatic equilibrium to regions of low pressure. In this section, we docu- must not be confused with mechanical equilibrium. In- ment the concept of isostasy and its relationship to sur- deed, a horizontal gravitational force (a volume force) face elevation. We then introduce the notion of gravita- appears when density interfaces are no longer parallel tional potential energy and that of horizontal gravita- to gravity equipotential surfaces. Hence, the action of tional force. isostasy, which forces density interfaces away from

25 The sketch below represents a schematic cross-section through a lithospheric plate at rest (i.e. no tectonics). Although the thicknesses of both the crust (in pink) and the sub-continental lithospheric mantle (in green) vary, isostatic equilibrium is maintained (i.e. no vertical motion is observed) which explains the lateral variation of the surface topography. A B C • Isostatic equilibrium implies that at - and below - a particular depth called the compensation depth, the pressure becomes hydrostatic (i.e. the pressure shows no lateral variation). In other terms, at or below the compensation level the weight of columns with the same cross-sectional areas standing on the same gravitational equipotential surface are the same. The minimum depth for the compensation level is that of the base of the thickest lithospheric column, in other terms it is the minimum depth as which there is no lateral change in density. Compensation level

Isostatic equilibrium=>Lithostatic pressure@z = constant for z > Compensation level

• 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. One can easily determine the surface elevation of column A and C with respect to the elevation of the undeformed column B, the surface of which is assumed here to be at sea level. When this heterogeneous lithosphere is at rest then isostatic equilibrium is reached via differential uplift and subsidence, and the pressure at the base of columns A, B and C, all standing on the compensation level, is the same. The pressure at the base of column B is: PB = ρcrust ⋅g⋅zc + ρsclm ⋅g⋅(zl − zc) + ρasth ⋅g⋅( fl ⋅zl + h − zl), and the pressure at the base of column C is: PC = ρcrust ⋅g⋅fc ⋅zc + ρsclm ⋅g⋅( fl ⋅zl − fc ⋅zc) where zc is the thickness of the crust, zl is the thickness of the lithosphere (i.e. crust plus sub-continental lithospheric mantle), fc is the vertical strain factor for the crust (i.e. the ratio of the thickness of the deformed crust over the thickness of the initial undeformed crust), fl is the vertical strain factor for the entire lithosphere. h the elevation of column C, and ρcrust, ρsclm and ρasth are the density of the crust, the sub- continental lithospheric mantle and the asthenosphere respectively. To calculate elevation of the high plateau h one has to solve PB = PC . To calculate the bathymetry of column A one has to solve PA = PB .

26 The sketch on the right illustrates the variation with depth of the lithostatic pressure B C zz = .g.z(MPa) (P(z) = ρ⋅g⋅z) along two lithospheric columns (B and C) in isostatic equilibrium. We assumed here that the density of the crust and that of the mantle are the same in both columns (with ρ ≤ ρ ), and that there is no density difference between the litho- gz (C) c m gz (B) spheric mantle and the asthenosphere. Because it has a thicker crust, column C has a surface elevation higher than that of column B. Consequently, the lithostatic pressure GPE (C) at any depth within the crust of column C is 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 measured at a particular depth is GPE (B) the same in both column. This is because both column are in isostatic equilibrium.

Isostatic equilibrium does not mean mechanical equilibrium. Indeed column C has a larger Gravitational Potential Energy (GPE in Pa) than column B. On the graph the GPE of each column is equal to the area under their respective lithostatic pressure GPE (B) < GPE (C) => Horizontal force acting from C to B proﬁle and down to the compensation level (i.e. orange area for column B, yellow area for column C which extends underneath the orange area). What is the implication of this?

The graph on the lower right shows water jets driven by the hydrostatic pressure in a leaking tank. Water tank The horizontal outreach of each jet is proportional to the hydrostatic pressure at the outlet. If we con- free surface nect two tanks of same height but different water depth, the difference in water level - and therefore Low pressure in pressure - drives an outﬂow has shown in the sketch on the dropping free surface left. This ﬂow stops when the water level, and therefore the pressure, in both tanks equalizes. The same physics applies to our two rising free surface lithospheric columns. Hence, lateral variations of density produce horizontal gravitational forces (body forces) acting inside the

Out!ow pressure Increasing lithosphere and pushing material laterally from high pressure to low pressure regions. The ﬂow stops when all the density inter- High pressure faces are parallel to equipotential gravitational surfaces, or when the gravitational deviatoric stresses are no longer strong enough to overcome the resistance to ﬂow.

27 The gravitational force (Fg in N.m-1) acting between C and B is equal to the difference of their B C zz = .g.z(MPa) respective GPE (difference between the two colored surface areas on the graph, i.e. the thin

-1 gz (C) yellow wedge on the graph). The excess in GPE in column C (Fg > 0 N.m ) will drive exten- gz (B) sion, whereas a deﬁcit (Fg < 0 N.m-1) will drive contraction. Mathematically: GPE (C) Top Top GPE = ρ (z)⋅g⋅z dz and GPE = ρ (z)⋅g⋅z dz C ∫ C B ∫ B GPE (B) Bottom Bottom

Fg = GPEC − GPEB GPE (B) < GPE (C) => Horizontal force acting from C to B

This movie below is the result of a numerical experiment documenting the gravitational collapse of an orogenic plateau. This model shows a 65 km thick crust adjacent to a 40 km thick crust (brown), with the upper mantle in green. The experiment starts at a stage when thermal relaxation has produced 20 to 25% of melt (dark pink) in the plateau’s lower crust. The only force acting is the gravitational force (i.e. the vertical sides are ﬁxed). The ﬂow of the orogenic plateau is powered by the same physics responsible of the spreading of camembert.

Movie 2.1 Collapse of orogenic plateaux

28 zz (MPa) THE CASE OF LITHOSPHERIC THICKENING AND DEBLOBBING Compression 50 100 Tension 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 gravi- 50 tational 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 up- Depth (km) per part of the thickened lithosphere is more or less balanced by a deﬁcit in gravitational potential 100

energy in the deeper part of the lithosphere. Indeed the integration of σzz (i.e. the lithostatic pressure) with depth gives a very small gravitational force (Fg = 0.77 1012 N.m-1). 150

Following convective thinning (the cold and heavy lithospheric keel has been removed), the de- -1 to Fg=0.77xTNm formed lithosphere has a larger excess in gravitational potential energy. This excess in GPE gives a (MPa) 12 -1 zz gravitational force Fg = 6.7 10 N.m . Such an Fg could easily balance or overcome tectonic forces, -50 50 100 150 and in some circumstances could be sufﬁcient to drive extensional collapse of the mountain belt. 20

60 THE CASE OF LITHOSPHERIC THINNING AND THERMAL RELAXATION 80 -1 100 Fg=6.7xTNm t +30Ma The two last sketches show the gravitational force (Fg) due to instantaneous homogeneous stretch- o zz (MPa) Compression -50 50 Tension ing (at time to) and following 200 myr of thermal relaxation. The yellow shaded areas represent the gravitational force acting on the lithosphere. Fg is given by the integration with depth of the differ-

50 ence in lithostatic pressure (σzz) between the thinned and the undeformed lithosphere. Depth (km) Following homogeneous thinning by a factor of 2 (50% reduction of the thickness of the crust and 100 -1 to Fg=0.17xTNm that the whole lithosphere), Fg is rather small (< 1012 N.m-1).

zz (MPa) As thermal relaxation and cooling proceeds, the thickness of the lithospheric mantle increases. This -150 -100 -50 50 100 150 leads to a decrease in the extensional gravitational force acting on the lower part of the lithosphere 20 40

and an increase in the compressional gravitational force acting in its upper part. The integration of 60 12 -1 Fg with depth gives a value > 10 N.m . For contraction structures to develop Fg must be larger 80 Depth (km) -1 100 Fg=1.2xTNm that the integrated strength of the thinned lithosphere. to+200Ma

29 GLOBAL VS LOCAL GRAVITATIONAL STRESS FIELDS

The global plate’s average 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. Mid- oceanic 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 the evolution of the gravity-related stress acting on B during thinning, which will shift the geometry of B toward that of C?

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 lithosphere since C has a GPE C1 greater A B C than that of the global mean potential. Thick Normal Thin 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

Potential Energy Potential Compression Pe<0 B2 then thinning of B will lead to a decrease in GPE (path B2->C2) . With a M.O.R. A M.O.R. B C GPE C2, column C is in compression Crust since it has a deﬁcit of GPE with its im- Oceanic lithosphere Lithospheric mediate surroundings (B2). This re- Mantle 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 Effec- tive Gravitational Stress Field (EGSF) is the superimposition of both gravitational stress fields.

30 SECTION 2 Consequences for deformation of the lithosphere

Horizontal gravitational forces arises when density inter- strength of rocks. Under the action of gravitational force, faces are no longer parallel to equipotential surfaces. hot rocks ﬂow from regions of high-pressure to regions of These volume forces interfere in the development of oro- low pressure until all density interfaces are parallel to gens and possibly in the development of rift basins. In equipotential surfaces. The ﬂow can last for millions of both cases gravitational forces tend to oppose the tectonic years, and occur in the absence of plate tectonic forces. force driving thickening or thinning. When the tectonic force wane, gravitational forces are only resisted by the

31 GRAVITATIONAL FORCE AND THE FORMATION OF CONTINENTAL PLATEAUX

Depending on its sign, GPE may oppose or enhance thickening. This example illustrates the transition from a narrow mountain belt to an broad oro- 0 myr genic plateau due to the build up of GPE. The upper and lower continental crust are in yellow and orange respectively. The lithospheric mantle is in green and the asthenosphere in pale yellow. The dark shading is proportional to stain rate. 5 myr

Following continental extension and rifting a narrow oceanic basin has formed (0 myr). This basin is then inverted (i.e. the tectonic regime changes from extensional to contractional) and the thinned continental margins collide forming a narrow mountain belt via localized crustal scale 10 myr reverse faults (5 to 10 myr).

As the crust becomes thicker, the excess in GPE produces a gravitational force (Fg) that opposes the tectonic driving force (Fd). At one stage the sum of Fg and the viscous strength of the thick crust (also a force to be 15 myr overcomed) balances Fd. As a result deformation migrates into adjacent weaker regions (15 to 25 myr). The mountain grows laterally and a plateau develops.

Should the tectonic driving force Fd diminish or stop, there would be no 20 myr forces to oppose Fg (beside the strength of rocks). This will led to gravitational collapse of the orogenic plateau.

25 myr

32 GRAVITATIONAL FORCE AND CONVECTIVE THINNING Expected learning outcomes

Convective thinning describes the drag into the convective At this stage you should be able to: mantle of the lower part of the lithospheric mantle. This mechanism corresponds to the development of a Rayleigh- • Use your understanding of the Archimedes’ principle Taylor gravitational instability (driven by a density inver- to explain the concept of isostatic equilibrium and derive an expression for the pressure balance between sion): the heavy and weak lower part of the lithospheric man- two lithospheric columns in isostatic equilibrium. tle is gravitationally unstable with respect to its surrounding. • Use your understanding of the the gravitational po- Depending on the rheology of the lithospheric mantle House- tential energy of an object of mass m standing z meter man and Molnar (1997, Geophys. J. Int. ,128,125–150.) esti- above ground to explain the concept of gravitational mated that the part comprised in between the isotherm 900ºC potential energy of an entire lithospheric column. and the isotherm 1300ºC is gravitationally unstable. The detachment of this heavy keel from the rest of the lithosphere • Explain the concept of gravitational force acting between two lithospheric columns of contrasting poten- dramatically changes the balance of forces. Convective thin- tial energies. ning results in a sudden increase in GPE stored in the thickened crust, and therefore promotes extensional collapse. • Comment on the fact that isostatic equilibrium must not be confused with mechanical equilibrium.

• Use the concept of gravitational force to explain the formation of orogenic plateaux, and the concept of gravitational collapse.

33 CHAPTER 3 Tectonic Forces

M.O.R. Fg Frp Oceanic Fo dFf dFr Continental lithosphere Fox 100 KM dFr dF Fss dFaz a Fss Mantle !ow Mantle !ow

dF r Mantle !ow

Mantle ! ow Mantle !ow x dFp

Fsp z

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.

34 FORCES DRIVING PLATES MOTION M.O.R. Fg Based on a simple model of lithosphere and man- Frp Oceanic Fo tle interactions, geophysicists have found that dFf dFr Continental lithosphere Fox three major tectonic forces — ridge push, slab pull 100 KM dFr and asthenospheric drive — can interact to explain dF Fss dFaz a Fss most observed plates motion at the Earth sur- Mantle !ow Mantle !ow face. The gravitational force acting between the dF high-standing mid-oceanic-ridge and the distant, r Mantle !ow Mantle ! ow low-standing oceanic abyssal plain is called the Mantle !ow x dFp ridge push force (Frp). It is of second order impor- Fsp tance when compared to the slab pull force (Fsp), z which is due to the pull exerted by the subducting slab onto the attached oceanic plate. Accord- ing to Don Anderson, “... slabs drive tectonics. There is no need for other driving mechanisms such as plumes or mantle convection that are independent of plate tectonics”. However, which forces keep India moving northward since the detachment of the slab? The viscous drag imposed by the flowing asthenosphere can drive plate tectonics. This force is called the asthenospheric-drive (Fad). The flow field in the asthenosphere is the product of convective motion, as well as the flow induced by subducting slabs (slab suction force Fss a component of Fad), upper mantle avalanches into the deeper mantle, and rising plumes (plume tectonics) pushing or pulling on the surrounding mantle. The mantle flow can also drive the dynamic subsidence or uplift of the lithosphere above.

FORCES DRIVING PLATES DEFORMATION

FRP

The dynamic of continental margins is under the inﬂuence of global plate tectonic Oceanic lithosphere Gravitational Force FG forces, and local gravitational forces (Fg) acting between the high-standing con- Asthenosphere

Continental lithosphere tinental plate and the low-standing adjacent abyssal plain. The buoyant mantle Gravitational Force

FSS wedge can also play an important dynamic role. The interplay between Frp , Fad, Fsp can Fg, FSP can result in contrasting tectonic regimes. FSS

35 In the context of an active continental margin, the tectonic regime expe- M.O.R. Trench Plate rienced by the upper plate (i.e. that standing above the subduction Frp Ocea nic litho sph Fg zone) can vary from extensional tectonics, to contractional tectonics, ere Continental lithosphere and to transcurrent tectonics. The variability of this tectonic regime expresses the very changing balance of forces acting on the continental margin.

Extensional tectonics is the tectonic regime which unfolds when the MOR-Trench Fsp subduction trench moves away from the upper continental plate (a). a/ Trench-Plate The trench retreats from the continent in response to slab rollback M.O.R. Trench Plate driven by the slab pull force, and/or under the push from the gravita- F F g rp Ocea tional force acting at the transition between of the oceanic plate and the nic litho sph ere Continental lithosphere continent. This tectonic regime leads to the detachment of continental fragments (or continental ribbons) from the main continental plate.

When the distance between the trench and a reference location inside the continental plate decreases, the continental margin is shortened (b). Fsp The tectonic regime is contractional , leading to the development of a MOR-Trench b/ Trench-Plate cordillera orogen. Through times, the interplay between ridge-push, slab-pull, slab-suction and gravitational force results in a changing tec- M.O.R. Trench Plate Fg tonic regime with alternating periods of contraction, extension through Frp Oceanic lit hosp he gravitational collapse and transcurrent tectonic regime. These changes re Continental lithosphere can be triggered by small variations in the magnitude of plate velocity, in the direction of plates motion, or asthenospheric ﬂow.

An active yet tectonically stable continental margin indicates that the Fsp resulting effective force acting on the margin is very small and unable MOR-Trench Trench-Plate to deform the margin (c). The margin is in mechanical equilibrium. c/

36 MANTLE FLOW: THE CASE OF STRONG COUPLING

Underneath the lithosphere, the flow of the asthenosphere exerts a dynamic influence on the motion of overriding plates through the v asthenospheric drive. The true extent of this influence is strongly debated as it depends on the level of mechanical coupling between the lithosphere and the convecting asthenosphere.

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 A the subduction zone (this is the slab suction force). 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 drags the nearby asthenospheric mantle into the subduction zone. v

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 drags the slab into the subduction zone.

37 MANTLE FLOW: THE CASE OF WEAK COUPLING

If the strength of the asthenosphere/lithosphere coupling is not that signiﬁcant, the convective asthenosphere can still interfere with plate motion. In particular, 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 re- treat (therefore extensional tectonics in the upper plate) or advance (therefore contractional tectonics in the upper plate), as shown on the sketches on the right.

C/ The asthenospheric wind under the subducting slab forces the slab to bend downward, augmenting the slab-pull. This effect is here enhanced by the downward asthenospheric ﬂow above the C slab forcing the steepening of the slab and promoting extensional tectonics in the upper plate.

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, promoting contractional tectonics in the upper plate.

Overall, the forces Frp , Fg, Fsp and Fad acting on the plates, the sublithospheric flow, and the flow of the mantle in the vicinity of the slab influence plate motion, and the tectonic regime at continental plate margins. D

38 RIDGE PUSH AND OTHER HORIZONTAL GRAVITA- TIONAL FORCES

Gravitational force at mid-ocean ridge Mid-oceanic-ridges (MOR) form a ~55,000 km long mountain belt in the middle of the oceans. This mountain belt stands ~2500 m above the average depth of oceanic abyssal plains (-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 asthenospheric mantle.

The map above shows the bathymetry of the north Atlantic ocean between North Africa and Newfoundland. The bathymetric pro- ﬁle (along the thin red line) across the mid-oceanic ridge reveals the bathymetry across the MOR, with its crest at around 1000 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 F = g⋅e⋅(ρ − ρ )⋅ + rp m w ( 3 2 )

-3 where e is the elevation of the mid-oceanic ridge above the cooling lithosphere, ρm is the density of asthenosphere (3320 kg m ), 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 -2 12 -1 (10 m s ). With these values we get Frp = 2 × 10 Nm . This estimate is an order of magnitude less than the slab-pull force. How- ever, 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 thor- ough formulation must integrate the temperature dependence of rocks density. Using the concept of gravitational force, the total

39 ridge push can be expressed as the integration with depth of the difference in lithostatic pressure between a lithospheric column at the mid-oceanic ridge and a lithospheric 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 that 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 stands in average ca.8000 m above the adjacent abyssal plain of the east Pa- ciﬁc ocean. 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, standing 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.

40 SLAB PULL FORCE M.O.R. Trench Plate A simple formulation for the slab-pull per unit length parallel to Frp z . . . l Ocea F the trench is given by : zm (zl ∆ρ) g. This formulation assumes that nic litho g sph er Continental lithosphere the density of the plate and that of the asthenosphere are e temperature-independent. Assuming an average density contrast of -3 about 60 kg.m , a depth of the subducting slab zm = 660 km, a thick- z -2 m ness of the slab zl = 100 km and g = 10 m s , we can calculate an abso- 13 -1 lute maximum for the slab pull Fsp of ~4 × 10 N m . Fsp

Note: the value would be an order of magnitude greater (Fsp ~ 2 × 14 -1 10 N m ) for a slab going all the way down to the core-mantle Asthenosphere boundary.

A more accurate formulation takes into account the temperature dependence of the density of rocks, 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:

2 2 2 8⋅g⋅α⋅ρm⋅Ta⋅L ⋅Re π ⋅z π ⋅d Fsp = ⋅ Exp( − ) − Exp( − ) π4 ( 2⋅Re⋅L 2⋅Re⋅L )

ρm⋅Cp⋅ν⋅L with: Re = 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 -5 -1 oceanic plate, α is the coefﬁcient of thermal expansion (3 × 10 K ), Ta is the temperature of the asthenosphere (1350ºC), k is the -1 -1 thermal conductivity (2 W m K ), 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 13 -1 above values we get Fsp = 2.5 × 10 N m .

41 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 olivine-spinel phase change, which occurs at around 350 to 420 km, increases the density in the subducting slab, provid- ing an extra pull. Note that this 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 resisted by the friction between the slab and the lithosphere, and the 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.

Oceanic lithosphere

b2/ Asthenosphere a/ Continental lithosphere

FRP b1/ FG Oceanic lithosphere FRP Gravitational Force FG FRP Oceanic lithosphere Continental lithosphere Asthenosphere Oceanic lithosphere Gravitational Force Asthenosphere Continental lithosphere FG Gravitational Force Asthenosphere FSS Continental lithosphere FSP

Gravitational Force FSS FSS

FSS FSS

FSS

42 # 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/ APPENDIX: This R map_Altiplano<-getNOAA.bathy(lon1=-60, lon2=-80, lat1=-30.00, lat2=-10, resolution=2) script grabs bathymetry # Hypsometry of the region and topography data # zbreaks = quantile(map_Altiplano, seq(0, 1, length.out=256)) zbreaks = zbreaks <- seq(-8000, 8000, by=100) from the US National plot(zbreaks) Oceanographic and At- # Make colour palette mospheric Administra- ocean.pal <- colorRampPalette(c("#000000", "#000413", "#000728", "#002650", "#005E8C", "#0096C8", "#45BCBB", "#8AE2AE", "#BCF8B9", "#DBFBDC")) tion (NOAA), creates a land.pal <- colorRampPalette(c("#336600", "#F3CA89", "#D9A627", "#A49019", "#9F7B0D", "#996600", "#B27676", "#C2B0B0", "#E5E5E5", "#FFFFFF")) topographic map, and #land.pal <- colorRampPalette(c("#467832", "#887438", "#B19D48", "#DBC758", "#FAE769", "#FAEB7E", "#FCED93", "#FCF1A7", plot a topographic proﬁle "#FCF6C1", "#FDFAE0")) mypalette <-c(ocean.pal(sum(zbreaks<=0)-1), land.pal(sum(zbreaks>0))) (see p.39 and 40).

# 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) axis(2, at=seq(-9000, 7000, 1000), las=2) mtext("[meters]", side=2, line=3) box()

43 Expected learning outcomes

At this stage you should be able to:

• List and describe the three types of tectonic forces acting on the Earth’s lithosphere, and tell which are plate-boundary forces?

• Calculate the value of the ridge-push force from knowledge of the principle of isostasy and the concept of gravitational force.

• Explain the role of the strength of the mechanical coupling between the convective mantle and the lithosphere on the dynamics of active continental margin.

• Comment on the changing dynamics (i.e. tectonic regime) of active continental margins.

• Comment on the maximum value of the effective tectonic force, and that of individual tectonic forces.

44 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 characterize 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.

45 SECTION 1 Elastic, Plastic and Viscous Flows

In this section 1. Brittle vs ductile Elastic Viscoelastic Viscoplastic deformation 2. Elastic, plastic, and = = (1 - e-t/) viscous flow curves E E 3. Rheology of polycrystalline rocks 4. Steady-state flow laws 5. Sensitivity of flow laws

Primary Creep Secondary Creep

Rocks display a very large range of behavior from which permanent strain accumulates. when submitted to deviatoric stresses. Over mil- Rocks can deform elastically since they transmit lions of years, hot rocks can ﬂow under small de- mechanical waves (sound waves, seismic viatoric stress, like very low viscosity ﬂuids do. waves). The rheology of rocks is therefore com- Rocks can break under a sudden deviatoric stress plex and covers a broad range of mechanical be- load (e.g. hammer strike). Rocks can resist mod- havior from elastic, plastic and viscous. erate deviatoric stress, until it reaches a threshold

46 BRITTLE VS DUCTILE DEFORMATION: The role of temperature, composition and strain rate...

Objects deform as a continuum (i.e. continuous deformation) via elastic deformation or via viscous or plastic 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, deformation is said to be brittle (i.e 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

47 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 below shows the flow curves for three ele- mentary deformation mechanisms, with for each their mechanical analogue:

• Linear elastic flow (in blue): For small stresses, most material are elastic. The characteristics of elastic flow are: 1/ Elas- tic flow occurs as soon as stress is applied. 2/ Strain increases Ideal Plastic as long as stress keeps increasing. The elastic flow curve is linear, its slope is 1/E with E the Young modulus, a physical * = . f(t) properties characteristics of elastic flow. 3/ Strain does not Elastic E accumulate if the stress is maintained constant. 4/The mate- A Plastic B * rial recovers its original shape when stress is relaxed. The mechanical analogue for elastic deformation is a spring. = E Viscous • Plastic flow (in green): In most material elastic flow is lim- = . t 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 Deviatoric stress point A onwards). 2/ Plastic strain accumulates at a level of 1/E C stress which is constant and equal to the yield stress. Hence, Plastic Elastic the amount of strain depends on the duration t over which Cumulative strain ( ) 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.

48 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 characterized by three contrasting relationships.

Elastic strain: When the deviatoric stress is imposed, the material records a ﬁ- nite amount of elastic strain. This amount of elastic strain remains constant = Constant σ through time (and equal to ) as long as the deviatoric stress is maintained. Ideal Plastic E -1 *E + t = . t

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 Viscous E onset of accumulation of plastic strain. Plastic strain accumulates as long as Elastic the deviatoric stress is maintained. For ideal plastic ﬂow, the rate of accumula- =

Cumulative strain ( ) strain Cumulative tion (α) is constant. E Viscous strain: Viscous strain accumulates even for a very small deviatoric σ Time 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.

Elastic Plastic RHEOLOGY OF POLYCRYSTALLINE ROCKS Hardening Creep C B Crystalline rocks display a mechanical behavior that incorporates the three ele- *' C'' Failure mentary flows; rocks are elasto-visco-plastic material. This graph shows a characteristic flow curve for a polycrystalline material. At stresses below the yield * A 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 be- Deviatoric stress E yond B) the material flows under a constant stress. Under the hardening plastic C' Cumulative strain ( )

49 regime, the removal of the driving stress leads to the removal of the elastic component of the deformation (curve CC'). If the sample is re- stressed, 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 perturbed 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 below show how dislocations move about:

Dislocation wall Motion of dislocations is driven by internal elastic energy. Dislocations organize themselves into ar- rays 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 Movie 4.3 Dislocation glide and climb

50 The viscous flow of polycrystalline material can be illustrated on a space strain vs time for experiment performed at constant stress. In Tertiary creep this graph, the plastic flow curve can be divided into three regimes = Constant Secondary creep called: Primary, Secondary and Tertiary creep. A' Primary Failure creep Primary creep corresponds to a reversible flow for which elastic defor- Constant Strain Rate A mation is instantaneously removed following unloading of the sample (at time t1), whereas another component of strain called viscoelas- Elastic

tic deformation is also recovered but over of a time window t1 - t2. ( ) strain Cumulative

V is * c Secondary creep is characterized by a linear relationship between o Elastic e la sti strain and time implying that the material is deforming at constant V c i sc strain rate. Upon unloading elastic deformation is instantaneously re- oe Elastic las covered whereas the viscoelastic deformation is recovered over a pe- tic Plastic ' ' riod of time (t'1 - t'2). The sample, however, records a permanent plas- t1 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 com- Elastic Viscoelastic Viscoplastic ponent), and dashpots (viscous component) connected in series or in parallel to fit a = = (1 - e-t/) real flow curve. E E 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 characterized by slow steady- state 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 characterize the large-strain high-temperature Primary Creep ductile deformation that occurs in the Earth. Secondary Creep

51 SECTION 2 Strength envelopes In this section

1. Low-ModerateYield stress stress (MPa) Yield stress (MPa) Yield stress (MPa) Yield stress (MPa) 2. High-stress3000 regime2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 3. Brittle regime

4. Sensitivity of flow curves 20 20 20 20 5. Rheological profiles Moho 6. Sensitivity of profiles 40 40 40 40

60 60 60 60

80 80 80 80 789MPa 255MPa 598MPa 193MPa 584MPa 213MPa 582MPa 212MPa [12] 100 [11] 100 [16] 100 [1] 100

The notion of strength envelope is relative to sure, the accumulated strain and the tectonic the evolution with depth of the deviatoric regime. At higher temperature where viscous stress necessary either to exceed the yield creep dominates, the strength envelope is sensi- stress, or to achieved a pre-defined strain rate tive to rock types, strain rate, and the magni- through viscous flow. At low temperatures, tude of the deviatoric stress. Here we start by where plastic strain dominates, the strength en- giving the relationship deviatoric stress vs tem- velope is mainly a function of pore fluid pres- perature (i.e. depth).

52 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 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 function of the differential stress, and the differential stress as a function of the strain rate:

• 1/n • −E ϵ E ϵ = A⋅σn⋅Exp rearranging we get >> σ = ⋅Exp ( R . T ) ( A ) ( n⋅R⋅T ) where A is a constant (Pa-n s-1), n is the stress exponent that characterizes 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 This effect is accounted for by the rapid increase, with increasing temperature, of the expo- n=3 3 nential term in the equations above. The graph shows that as n increases from 1 to large n= 8 1

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

FLOW LAWS FOR STEADY-STATE CREEP: HIGH-STRESS REGIME • Power-law creep implies that at 500ºC olivine would only deform at unrealistically high R⋅T ϵ σ = σ ⋅ 1 − ⋅ln d stresses. A better description of the behavior of olivine at high-stress regime (> 200MPa) is d • Ed ( ϵ ) given by the Dorn's law, a relationship that is not as temperature dependent as the power law, in which Ed is an activation energy, σd a critical stress that must be exceeded, and εd is the critical strain rate.

53 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 modeled as frictional sliding through the Byerlee’s law:

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:

−2 R − 1 β = with: R = 1 + μ2 − μ σzz − σ3 ( ) 1 + ⋅(R − 1) ( σ1 − σ3 ) and μ the coefﬁcient of internal friction that characterizes 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 modeled 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):

σ1 + σ3 σ1 − σ3 σ1 − σ3 σ = − ⋅cos(2α) and τ = ⋅sin(2α) with α the angle between the fault plane and σ . n 2 2 2 1

54 SENSITIVITY OF FLOW CURVES TO THERMODYNAMIC PARAMETERS AND CHEMICAL ENVIRONMENT

Confining Pressure and Temperature Confining pressure prevents rocks from falling apart. It is therefore not surpris- 250 Wombeyan Marble ing that increasing confining pressure increases the amount of deformation a =100 MPa sample can accumulate before failure. For instance, in the example of the Wom- 200 beyan marble brittle failure is either reached for a larger amount of accumu- 150 lated strain or even impeded when a higher confining pressure is applied. This =35 MPa higher confining pressure allows the sample to sustain a larger differential 100 stress. Temperature enhances the ductility of material. For instance, in the exam- =10 MPa ple of the Solenhofen limestone, raising temperature decreases the magnitude (MPa) 50 3 Failure

of the deviatoric stress that can be supported, and increases the amount of accu-

1 Failure mulated strain before brittle failure . Under high temperature material can there- 1 2 3 fore accumulate more strain but they can do so for a smaller amount of differen- Cumulative strain ( ) tial stress.

Solenhofen Limestone 400 300ºC

Failure 300 500ºC Failure 200 600ºC

Failure 100 (MPa) 3

Pc~40MPa 1 2 4 6 8 10 Cumulative strain ( )

Paterson and Wong, 2006. Experimental Rock Deformation – The Brittle Field. Surveys in Geophysics, 27, 4, pp 487-488.

55 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.

Dry 300 Yule Marble 600 Solenhofen Limestone 0.5% H20 60 MPa 200 400

2.9% H20 Failure 100 200 Failure 65 MPa

(MPa) Failure

3 78 MPa

(MPa) 93 MPa T=300ºC 3 1 Pc~500MPa 1 4 8 12 18 3 6 9 12 Cumulative strain ( ) Cumulative strain ( )

1 @ Mervin Paterson, ANU

3 3

1 Fluid assisted diffusive mass-transfer through the lattice (Nabarro- Herring creep) or along the grain boundaries (Cobble creep)

56 RHEOLOGICAL PROFILE OF THE CONTINENTAL LITHOSPHERE

It is time now to synthesize what we have learn about the rheology of crystalline rocks to determine the strength proﬁle (i.e. en- velop) 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:

• 1/nc ϵ Ec σ1 − σ3 = β⋅(1 − λ)⋅(ρc⋅g⋅z) or σ1 − σ3 = ⋅Exp ( Ac ) ( 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:

σ1 − σ3 = β⋅(1 − λ)⋅(ρm⋅g⋅z)

• 1/n ϵ Em or σ1 − σ3 = ⋅Exp for (σ1 − σ3) < 200MPa ( Am ) ( nm⋅R⋅T(z) )

• R⋅T(z) ϵ or d σ1 − σ3 = σd⋅ 1 − ⋅ln • for (σ1 − σ3) > 200MPa Ed ( ϵ )

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 Strength in compression 1-3 (MPa) 1000 0 1000 (peridotite with > 80% olivine) Crust respectively. The -6 -n -1 Compression Extension power law and Dorn =3 Ac = 5x10 MPa s creep law are de- 20 pendent on both the nc = 3.0 temperature T and • the strain rate ϵ . Strength in extension -1 Therefore, these Ec = 190 kJ.mol Moho parameters must =0.75 40 be defined first. The geotherm here is such that the temperature at 60 the Moho is 460ºC and the cho- Mantle sen strain rate is 4 -n -1 10-15 s-1. A = 7x10 MPa s Ed=540 kJ.mol-1 Dorn Law + m Power law 80 The straight nm = 3.0 Stress Treshold=8500 MPa parts of the pro- -1 11 -1 100 e =3.05x10 s Depth (km) files (in the Em = 520 kJ.mol d TMoho=460ºC 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.

58 1-3 (MPa) 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 1000 0 1000

20 20 20 20 20 Moho 40 40 40 40 40

60 60 60 60 60 Power law 737 MPa 237 MPa 570 MPa 195 MPa 410 MPa 151 MPa 236 MPa 108 MPa 106 MPa 61 MPa 80 80 80 80 80 Dorn law+ 438 MPa 221 MPa 333 MPa 183 MPa 239 MPa 142 MPa 158 MPa 99 MPa 101 MPa 57 MPa Power law -40% 100 -7% -41% 100 -6% -41% 100 -6% -33% 100 -8% -5% 100 -7%

TMoho=400ºC TMoho=460ºC TMoho=540ºC TMoho=620ºC TMoho=700ºC

500ºC 700ºC 40

-1 30

N.m 20 12 10 TMoho (ºC) 0 400 600 800 1000 -10 A few things to note: -20 Strength • With the Dorn law, brittle failure does not occur in compression, which signiﬁcantly re- -30

Integrated Strength x 10 -40 duces the strength of the upper mantle. 700ºC -50 Depth • The brittle part of the crust is thicker in extension that it is in compression. 500ºC • 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

59 comparatively much weaker. • Because quartz deforms by ductile ﬂow at lower temperature (~300ºC) Yield stress (MPa) Yield stress (MPa) Yield stress (MPa) Yield stress (MPa) than olivine (ductile at ~600ºC) the up- 3000 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 Upper crust

per mantle is much stronger than the 20 20 20 20 Lower crust lower crust for a normal geotherm Moho 40 40 40 40 (TMoho <650ºC). 60 60 60 60 Moho SENSITIVITY OF STRENGTH ENVEL- 80 80 80 80 Mantle OPS TO TEMPERATURE 789MPa 255MPa 598MPa 193MPa 584MPa 213MPa 582MPa 212MPa The dependence of the strength envel- [12] 100 [11] 100 [16] 100 [1] 100 3000 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 Strength in ops on temperature is documented be- compression low. From left to right the temperature 20 20 20 20 at the Moho increases from 400ºC to Moho Strength in 40 40 40 40 extension 700ºC. The strength of the lithosphere, in both compression and extension (the sur- 60 60 60 60 face area of the dark and pale blue re- 80 80 80 80 TMoho=540ºC 460MPa 178MPa 433MPa 166 MPa 425MPa 155 MPa 406MPa 163MPa gions respectively) has been averaged [6] 100 [4] 100 [14-15] 100 [13] 100 TMoho=600ºC over the lithospheric thickness assuming 3000 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 2000 1000 0 1000 for the mantle either a power law creep or a combination of power law and 20 20 20 20 20 Moho Dorn law. As temperature increases, the 40 40 40 40 40 averaged strength of the lithosphere de- 60 60 60 60 60 creases. 80 80 80 80 80 The integrated strength of the continen- 404MPa 160 MPa 379MPa 149MPa 365MPa 153MPa 355MPa 147MPa 78MPa 49MPa [5] 100 [7] 100 [9] 100 [10] 100 [2wet] 100 tal lithosphere (in Nm-1) is deﬁned the integration over depth of the rheological proﬁle. Since the frictional sliding de-

60 pends on the tectonics regime, three "strengths" can be defined down to 78 MPa. In extension the integrated strength ranges for compressional (Fedt), transcurrent (Feds), and extensional from 255 MPa down to 50 MPa. (Fedc) tectonic regime: The large variability of the rheological properties of common The figure on the right shows the integrated strength as a func- crustal and mantellic tion of temperature at the Moho. For TMoho ~500ºC the upper rocks, prevents the mantle is the stronger layer of the lithosphere (in blue in the in- definition of a “stan- set). Because the power law creep is exponentially dependent dard” rheology for on temperature a small increase in temperature strongly re- the continental litho- duces the integrated strength. Indeed, when TMoho is close to sphere. 700ºC the strength of the upper mantle drastically decreases. Over 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. SENSITIVITY OF STRENGTH ENVELOPS TO LITHOLO- GIES

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%

61 #======# This R script calculates and plot the continental geotherm and rheological proﬁles #======# ======Geotherm ======library(mosaic) library(mosaicCalc) # SI unit: m, sec, kg

# Radiogenic heat production (W.m-3) H<-0.7*10^-6 # Thermal conductivity (diffusivity x density x heat capacity), (W.m-1.K-1) K<-(10^-6)*2650*1000 #Mantle heat ﬂow (W.m-2) Qm<-0.020 #Surface Temperature (K) To<-293 #Thickness of the continental crust (m) zc<-42000 #Temperature deﬁning the base of the lithosphere (K) Tl<-1603

#Crustal geotherm Temp_crust<-function(z){(-H/(2*K))*z^2+(((Qm+H*zc)/K))*z+To} #Geotherm in the lithospheric mantle Temp_lithos_mantle<-function(z){(Qm/K)*(z-zc)+Temp_crust(zc)} #Depth of the base of the lithosphere zl<-ﬁndZeros(Temp_lithos_mantle(z) - Tl ~ z, z.lim=range(100000, 200000)) #Geotherm in the asthenosphere (called the adiabat because it follows the adiabatic gradient) Temp_asthenos<-function(z){Tl+0.0003*(z-zl)} #Geotherm in the ﬁrst top 200 km … Geotherm<-function(z){ifelse(z<0, To, ifelse(z<=zc, Temp_crust(z), ifelse(z<=zl, Temp_l- ithos_mantle(z), Tl)))} #Plot of the geotherm above plotFun(Geotherm(z) ~ z, z.lim=range(0, 200000))

# ======Rheology ======

62 CHAPTER 5 Computational Tectonics & Geodynamics

Computational tectonics/geodynamics provides a robust platform for hypotheses testing and exploring coupled tectonic and geodynamic processes. It delivers the unexpected by revealing previously unknown behaviors.

63 SECTION 1 The rise of computer science

During the mid-19th century, the laws of thermodynam- tonic processes which also involves the flow - brittle or ics - which describe the relation between heat and forces ductile - of rocks. 21st century computers have now the between contiguous bodies - and fluid dynamics - which capability to compute the flow of fluid with complex, describes the flow of fluids in relation to pressure gradi- temperature-dependent rheologies, and contrasting physi- ents - reached maturity. Both theories underpin our un- cal properties we typically encounter on Earth. For geosci- derstanding of many natural processes from atmospheric ences, this is a significant shift. and oceanic circulation, mantle convection, and plate tec-

64 Computational tectonics and geodynamics: There is a revolution Navier-Stokes equations: Written in Georges G. Stokes (1819-1903) unfolding at the moment in all corners of science, engineering, and the 1850’s, the Navier-Stokes equations other disciplines like economy and social sciences. This revolution are the set of differential equations that is driven by the growing availability of increasingly powerful describe the motion of incompressible high-performance computers (HPC), and the growing availability fluid and associated heat exchanges. of free global datasets and open source data processing tools (Py- They relate velocity gradients to pres- thon, R, Paraview, QGIS-GRASS, etc). Here in Australia, supercom- sure gradients and can be analytically puters such as Magnus (36,000 cores) at the Pawsey Supercomput- solved only when considering steady ing Center in Perth, and Raijin (58,000 cores) at NCI (National Com- laminar flows. putational Infrastructure) in Canberra, have enabled and democra- Conservation of momemtum: tized the art of numerical modeling. ∂u ρ⋅ + u⋅∇u = −∇p + μ⋅∇2u + ρg Broadly speaking, numerical modeling is a discipline which en- ( ∂t ) ( ) ables the exploration of the behavior of complex systems. It also m.a = internal pressure gradient + viscosity force + gravity force gives scientists, engineers and economists the capacity to extract new knowledge and understanding from big data. In the context of geology and geophysics for instance, numerical modeling allows Conservation of mass (continuity equation incompressible): us to build a model of lithosphere and explore how this lithosphere ∇ u = 0 behaves when submitted to tectonic forces. It enables geoscientists to build spherical models of the Earth to explore mantle convec- Conservation of energy: tion. These models can account for a very broad range of petro- ρ⋅∂C T physical properties including radiogenic heat, heat diffusivity, den- p + ρ⋅ u ⋅∇CpT = ∇⋅ K⋅∇T + A sity, heat capacity, rheology (brittle and ductile), solidus and liqui- ∂t ( ) dus, etc. The behavior of the lithosphere and that of the convective Equation of state: ρ = ρ ⋅ 1 − α T − T mantle is governed by the laws of thermodynamics which de- 0 ( ( 0)) scribes exchange of energies within the system, and fluid dynamics ∂u ∂uj Constitutive equation: i which relates deformation (i.e. flow) to pressure gradients. Both σij = η⋅ + − pδij (∂xj ∂xi ) thermodynamics and fluid dynamics are fully developed theories from 19th century physics. Yet, it is only over the past decade that https://www.youtube.com/watch?v=JH3l-NliCkM HPC became powerful enough to efficiently solve in 3D the rele- https://www.youtube.com/watch?v=ERBVFcutl3M vant equations at a reasonable spatial and temporal resolution. https://www.youtube.com/watch?v=wtIhVwPruwY

65 Let’s try to make sense of all this. Newton’s second law of motion one component must be gained by another component. Hence, states that the sum of all forces applied to an object must be equal heat exchanges are described via the equation of conservation of to its mass (m) times its acceleration (a): m⋅a = ∑F energy.

This law applies to any object including a parcel of fluid. The right Remembering the 1D conduction-advection heat transfer equation hand side of the equation includes the force of gravity, the viscous (see Chapter 1)... forces due to friction with the surrounding fluid, pressure gradient dT K d2T A dT from the surrounding fluid, surface tension, etc. The left hand side = ⋅ + − u ⋅ 2 z has by necessity the dimension of a force, it is the inertia force. The dt ρ⋅Cp dz ρ⋅Cp dz acceleration a can be splitted into a local acceleration (also called ... re-arrange so it looks similar to that given on page 65 ... the unsteady acceleration because the velocity changes with re- 2 spect to time), and the convective acceleration when a fluid parti- ρ⋅Cp ⋅dT dT d T + ρ⋅Cp ⋅uz ⋅ = K⋅ + A cle moves through a non-uniform flow field (i.e. field with spatial dt dz dz2 gradients of velocity that may not vary with time), hence: This equation assumes that heat capacity Cp and conductivity K do ∂u not vary with temperature, which is not the case in nature. Hence m⋅a = m⋅ + u⋅∇u we re-write the equation above for a more general case in which C ( ∂t ) p and K depend on T: The conservation of mass m is described by stating that the diver- 2 2 gence of the velocity field is zero. The divergence is a scalar field ρ⋅dCp ⋅dT dCp dT d d T + ρ⋅ ⋅uz ⋅ = ⋅ K⋅ + A which measures how much the flow is expanding (positive diver- dt dz dz dz2 ( dz2 ) gence) or contracting (negative divergence) at any given point. Zero divergence implies that the fluid is incompressible, a reason- The 3-D generalization of this equation is the equivalent per cubic nable approximation for rocks (known as the Boussinesq approxi- meter of the equation on page 65: mation). ρ⋅∂Cp ⋅∂T + ρ⋅uz ⋅∇Cp ⋅∇T = ∇⋅ K⋅∇T + A As a fluid parcel flows through a surrounding fluid, it gains or ∂t ( ) looses energy via conduction, advection, shear heating, radiogenic The Reynolds’ number is the ratio inertia to viscous force. Laminar heat production, partial melting, phase changes etc. As its tempera- flow occurs at low Reynolds numbers, where viscous forces domi- ture varies its density changes, and therefore its inertia as well as nantes. Flow at low Reynolds numbers is smooth and uniform (no the gravitational force acting on it. At the scale of an entire system, turbulence). This mode of flow best describes geological flows. the variation of energy is a zero sum game as the energy lost by

66 Some examples of simple laminar flow for which analytical solu- In short, in the second half of the 20th century, computers were not tions exist include the free fall of a spherical object into a newto- powerful enough to take advantage of these new numerical meth- nian fluid (e.g. the settling of crystals into a magma), the flow in- ods. As computers grew in duced by the motion of a rigid plate power, so did the complexity above a newtonian fluid (Couette flow of fluid flow problems that in relation to the motion of tectonic plate one could tackle. However, above the asthenosphere), and the flow for a long time, computational of newtonian fluid between two static tectonics involving a layered plates (Poiseuille flow, for instance the lithosphere made of stronger 1950’s mainframe computers flow of the lower crust in orogenic pla- layers (upper crust and upper teaux). mantle) and weaker layers (lower crust and lower lithospheric mantle) was limited to 2 dimensional models in which the node of the computational grid had to conform to the deformation of the model (Eulerian grid, see image below). In contrast, computational mantle geodynamics could afford 3 dimensional models because models of mantle convection could use a more efficient fixed grid (Lagrangian grid) as long as the convective mantle was made of one single newtonian fluid.

In the mid 1950’s, with the advent of mainframe computers, the Navier-Stokes equations could be discretized and solved at the nodes of a numerical grid to explore the type of complex, time- dependent ﬂuid ﬂow we encounter in nature. At the same time, nu- @ Yuen, Minneapolis merical methods and computer algorithms were progressing so @ Beaumont, Halifax fast that they quickly overtook the computer capability of the time.

67 Particle-in-cell numerical methods. In the late 1990’s, computers Over the past decade, the growing availability of powerful high- become powerful enough to allow the implementation of a 1950’s performance computers has unleashed the power of computer- numerical method called particle-in-cell (PIC, developed at Los Ala- based modeling in all branches of science. Geoscientists are now mos National Laboratory) in which individual fluid elements, car- able to simulate in four dimensions, using realistic coupled ther- rying material properties and flow history, are advected through a mal and mechanical properties, lithospheric-scale deformation and fixed computational grid. For geologists, this progress meant that mantle geodynamics. 21st century HPC have finally caught-up with 19th century physics, and numerical recipes from the mid-50s.

Numerical experiments vs numerical modeling vs numerical simulation: Before going further we need to clarify the difference between experiment, modeling and simulation. For most people these concepts are interchangeable, but not for the experts.

Numerical experiments PIC = Eulerian mesh + Lagrangian particles like physical experi- one could simulate the deformation of mechanically layered sys- ments do not pretend to tems such as the Earth’s lithosphere; albeit in two dimensions. Ellip- reproduce a natural sis developed by Moresi, first at Caltech then at CSIRO, is one of process in a realistic the earliest and most robust codes (along with Citcom from which manner. The aim is to Ellipsis has emerged) implementing an efficient PIC method on the illustrate a concept, or back of a robust multigrid solver. to try to understand the few most important parameters involved in a particular process. The famous analogue experiment from Paul Tapponnier, performed for the first time in the very late 70‘s, per- Rey, Coltice, Flament, fectly illustrates the concept of escape tectonics, a process which Nature 2014 accommodates convergence via the lateral escape of continental blocks in front of a rigid indenter. Clearly, this experiment bears Ellipsis has been used to show that early proto-continents and thick oceanic plateaux very little resemblance with tectonics in Asia and South East Asia, had enough gravitational power to slowly force adjacent oceanic lithospheres to sub- (no orogenic plateau ...) but it does a nice job in illustrating a conduct. The model above includes the continental crust (red), lithospheric mantle (pink), partially molten mantle (bright blue), the rest is the mantle asthenosphere and litho- cept which has changed our understanding of collisional tectonics. sphere).

68 Numerical modeling aims at under- in the context of geosciences, assumes that we are able to imple- standing a particular process within ment a very broad range of coupled natural mechanical, petrologi- a particular tectonic context. The ini- cal, geochemical processes including the migration of partial melt tial and boundary conditions are and associated advection of heat, the release and consumption of carefully thought through to describe water and other volatiles, evolving anisotropic rheologies due to a geologically realistic setting. Re- grain reduction and crystallographic fabrics, dilatancy due to sults of the modeling are detailed micro-cracking and other softening mechanisms. It will take dec- enough to be able to be compared - to ades before we can properly engage with numerical simulation in a the first order - to natural meaningful manner. geological examples, without FEM and Multigrid solver: The Finite Element Method trying to (FEM) consists in discretizing of a set of partial differential match accu- equations across a 2D or 3D domain (i.e. ∂# are transformed rately any- into ∆ # ). This allows solving efficiently the Navier-Stokes one of them. equations at the nodes of a grid covering the model, but at the price of a small error. To minimize this error, users can use a very fine grid. The finer the grid the higher the resolution but the longer the compute time. Hence, the need to balance accuracy and compute time. To solve the set of equations efficiently, computer scientists Numerical simulations aim at reproducing with the greatest level of design methodologies taking advantage of parallel comput- details and the greatest level of realism a particular process in a par- ing, as well as other techniques. One technique uses a multri- ticular region. For most people this is what modeling is all about, grid solver. It involves the use of a stack of computational and therefore they are quick to point to the many shortcomings of grids of increasing resolution. Instead of solving the set of Paul Tapponnier’s experiments and dismiss its relevance since it equations directly at the node of one single high-resolution neither account for the presence of the highest mountain chain nor grid, a long wavelength solution is computed on a coarser the highest plateau on Earth. Numerical simulation assumes that grid, and this solution is iteratively refined at the nodes of our models are able to include parameters with realistic value dis- grids of increasing resolution taking into account short tributions and time-dependencies, as well as nested heterogenei- wavelength pressure gradients. ties on a broad range of scales. In addition, numerical simulations,

69 SECTION 2 Ellipsis: a FEM-PIC code

Ellipsis and Underworld apps solve the equations of Stokes rate, temperature etc). Lithologies are represented by particles flow (Navier-Stokes equations in the limit of small Reynolds distributed inside the grid’s cells (hence the acronym PIC for number) at the nodes of a computational grid using a Finite Particle In Cell). In this section, you will learn to compile Ellip- Element Methods (FEM). Solving the Stokes equations pro- sis on your own Mac or Linux computer, and you will learn vides the velocity and pressure fields from which a range of the command to launch and stop Ellipsis from a terminal win- other field properties can be computed (viscosity, stress, strain dow.

70 Introduction: Ellipsis has a long history of development starting in the late 90‘s at Caltech by Prof. Moresi (now Melbourne Univer- sity). It is derived from Citcom, a code dedicated to mantle convection modeling still under development today at Caltech. Ellipsis was designed to explore 2-D coupled thermal and mechanical tectonic processes on a single CPU. It can simulate both brittle and ductile deformation where brittle rheologies evolve with accumulated strain, and ductile rheologies evolve with temperature, stress, strain rate and melt fraction. Underworld is a parallel version of Ellipsis with much expanded capacity and ﬂexibility. It can handle 2-D models on a laptop and 3-D models on high-performance computers. It is developed by a team led by Prof. Moresi, with support from the Australian government through its Auscope NCRIS program. Its installation from its source code requires an appropriate level of expertise. However, recent developments on virtual machine allow to deploy both Ellipsis and Underworld in a Docker container on a local computer for the purpose of simple 2-D modeling.

Installation: Both codes run on unix-based OS (including Linux and Mac OSX), and it can also be installed on a PC running Win- dows OS. Their source codes can be downloaded from geodynamics.org and underworldcode.org where detailed instructions can be found to compile Underworld. To compile Ellipsis source one needs access to an appropriate compiler and a terminal window. On a Mac computer, both come with the Xcode.app (free from the Mac App store). If you need to install a compiler go to http://hpc.sourceforge.net/ and download the gcc compiler for your OS, then open a terminal window and execute: sudo tar -xvf /path_to/gcc-5.1-bin.tar.gz -C /. On Linux, simply download a gcc compiler for your version of Linux and the 32 bit library (e.g., apt-get install -y gcc-multilib lib32gcc-5.1). Then in a terminal navigate to the Ellipsis source and execute the three lines on the right (you may have to add the ./configure CFLAGS='-m32' path to your compiler by adding CC=/path_to/cc_compiler after CFLAGS=’-m32’ ): make clean An executable (only a few 100s bytes) called ellipsis3d will be created in the source directory. make Copy this executable into the folder hosting your input files. A good practice is to keep a copy of the Ellipsis executable with each model, so you can keep track of which version of Ellipsis you have been running. You are reading to go ... To run (i.e. execute) an input file, simply open a terminal window, navigate to the folder holding ellipsis3d and the input file (here mycoolmodel.input) and enter the following command: ./ellipsis3d mycoolmodel.input To stop the model, click the sequence control-C. More often than not, the reference model described in the input file requires access to a file laying out the temperature at each node of the computational grid. This file is defined in the input file (previous_temperature_file=”mytempfile.node_data”). We will see later how this file can be generated. This temperature file must be

71 present in the folder in which you are running the experiment, as shown on the ﬁgure on the right.

As Ellipsis runs, it dumps at each time step a bunch of files into the folder. These files, specified by the users in the input script, store grid and particles information (temperature, pressure, velocity, stress, strain rate, viscosity, ...). #.ppm0: These files are graphics ouput showing various lithologies, melt fraction, stress, viscosity etc. #.node_data: These files store parameters such as temperature and pressure at the node of the computational grid. The information in node_data can be used for data mapping (mapping of isotherms, super-solidus temperature, melt fraction, ...). For this you will need to process the data stored in .node_data using Matlab, R, Python or any other appropriate tools. #.profiles: These files give access to profiles (horizontal or vertical) passing through "sampling tracers". In the input file one can specify the initial position of up to 50 sampling tracers (this position can remain fixed to the computational grid, or can move with the rocks through the grid), the direction of the profile (horizontal or vertical), and the type of data to be profiled (temperature, melt, depletion, pressure, velocity, strain rate, stress ... ) #.sample : These files are updated at each time step to add the new value of parameters attached to one of the sampling tracers. Hence, there is one .sample file per sampling tracer. These files record the evolution through time of data attached to a tracer (x, z, Data), where data can be temperature, pressure, etc. These can be used to collect data to plot PTt paths.

Resolution of the computational grid and particle density: In Ellipsis, solving the Stokes equations is handled via a multigrid solver which uses a stack of increasingly finer computational grids. The pressure at a particular location is influenced by local/short wavelength and distant/long wavelength sources. A coarse grid can effectively take care of distant sources, the solution is then interpolated onto a finer grid to account for local sources, the new solution is then averaged back onto a coarser grid. The larger the number of grids the better the resolution, but the longer the running time. The pressure field is calculated using a subset of the multigrid stack (pmg_levels), the maximum depth of which is levels-1.

72 The finer grid also contains Lagrangian tracers carrying information about material properties. The more tracers there are, the better defined are the density interfaces. There is no simple rule for how many grid levels and tracers one should use. Consequently, a model may take any time between a few minutes to several months. In most cases, levels=4 is sufficient when testing lithospheric-scale models. Once the model has been properly tested at levels=4 to ensure that geometry and boundary conditions are properly set, it is a good practice to let it run at levels≥4.

The figure on the right shows a grid stack (Levels=3: grid 0, grid 1 and grid 2) for the multigrid solver. The number of cells in the finest grid is determined by the Levels (here 3) and the parameters mgunitx (number of cells along the x direction in grid 0, here 5) and mgunitz (number of cells along the z direction in grid 0, here 3). The tracer_density is defined on the finest grid level. In this example trac- er_density=4, meaning there are 4x4=16 tracers per element.

Initial temperature field: The temperature field is computed by running the input script without any kinematic or dynamic boundary conditions (i.e. the model is not extended or shortened). For maximum computational efficiency one can increase all viscosities and turn plastic rheologies off so the model doesn’t deform while the geotherm evolves. This is achieved using the following overriding toggles (at the end of the Ellipsis input script): TDEPV=off visc_max=5e31 visc_min=1e31 YIELD=off

… and increasing the fixed_timestep to 1 million years. Since we use second for unit of time ...: fixed_timestep=3.1536e13 Then, any #.node_data output can be used as a previous_temperature file by setting up in the Steady-States and Boundary Conditions section of the input file: previous_temperature_file="A_My_Previous_Temperature_File.node_data"

73 Time step: Unless when running thermal equilibration, time step should be kept below 6.3072e12 sec (that’s 200,000 years). If weird things happen (i.e. unexpected short-term ﬂuctuation of some parameters like viscosity) try to reduce the time step.

Ellipsis can also be delivered to a local machine via a Docker container (a kind of virtual machine) in which an Ellipsis executable is ready to run. The next section explains how it works.

Using Ellipsis on Jupyter environment within a Docker container: “Docker is an open platform for developing, shipping, and running applications.” A Docker container contains a lean Linux operating system with all necessary libraries to compile codes and run apps. A Docker container can run on any machine and any operating system as long as it as the Docker app installed (https://www.docker.com). We have created a Docker container for Ellipsis which is available for download through Docker Store and/or Kitematic both web-based community libraries for Docker containers, available from the Docker drop-down menu. nb1: A Docker image is the container in which your code run. The command docker push publishes a Docker image. nb2: A Dockerﬁle is the ﬁle that is used to build (i.e. compile) the Docker image via the command docker build.

The Ellipsis Docker contains the Ellipsis app, and a series of input ﬁles. It also includes Jupyter a browser-based integrated development environment from which Ellipsis input ﬁles can be created and edited, Ellipsis executable can be launched, and Ellipsis outputs can be processed.

To run Docker containers users must ﬁrst download the Docker app onto their machine. This can be done via: https://www.docker.com/community-edition

Ellipsis can be run from the jupyter environment. Jupyter offers a collaborative environment in which one can run codes, scripts and calculations, and process data using the python ecosystem, as well as other computer languages (>50 of them), to produce new data and visual outputs.

74 SECTION 3

Growing access to high-performance computers and Underworld their relentless increase in power are changing the way we do Science. We have now the numerical capability to explore complex non-linear geological processes in a well constrained physical framework. Underworld is to tectonicists what a laboratory is to physicists, a space where hypotheses can be tested not through physical experiments, but through numerical experiments.

Underworld is an open source, modular, hierarchical, computational framework, designed to perform parallel 2D or 3D time-dependent coupled thermal and mechanical tectonic and geodynamic experiments. It uses an expandable range of rheologies covering elastic, plastic and viscous behaviors. It accommodates radiogenic (i.e. internal) heating, partial melting - and other metamorphic reactions - with feedback on rheologies, densities and temperature.

Underworld development is led by Prof. L. Moresi (ANU) and a team of developers led by John Mansour (at Monash University) and Julian Giordani (Melbourne Uni- versity). Underworld started in 2005 with funding from the Victorian Partnership of Advanced Computing, and carried on with support from the Australian NCRIS/ Auscope program.

75 What does Underworld do?

Underworld solves the equations governing Stokes flow (after George Gabriel Stokes 1819-1903) a class of flow characterized by a small Reynolds number (after Osborne Reynolds, 1842-1912) a dimensionless number which expresses the ratio between inertial forces to viscous forces (cf. page 80). Stokes flow are characterized by smooth laminar fluid motion, in contrast to turbu- lent and chaotic flow occurring at high Reynolds number, for which the full Navier-Stokes equations are necessary. The sketch on the right shows an example of Stokes flow (source: wikipedia).

Underworld solves the Stokes equations on a cartesian 2-D or 3-D grid assuming the ﬂowing material is incompressible, which imposes a zero divergence condition on the velocity ﬁeld, and therefore the mass continuity equation ui,i = 0 which ensures conservation of mass. The Stokes’ equation in a tensorial form (see page 84 to learn how to unpack this form) is:

σij = τij − pδij = fi

This equation expresses the conservation of momentum. It states that the total stress tensor σij (the sum of the deviatoric stress tensor τij and the pressure tensor pδij ) balances the buoyancy force fi (fi = ραTg) due to density variations (also known as the gravitational body force) .

The constitutive equation expresses the relationship between the deviatoric stress tensor, the viscosity tensor (or constitutive tensor) and the strain rate tensor: τij = 2ηijkl . ϵ˙kl

Hence, the resulting Stokes’ equation can be writen as: 2 ηijkl . ϵ˙kl − pδij = fi or in a vectorized form the momentum equation is: η∇2u − ∇p = ρg

In which ∇2u and ∇p are the velocity and pressure gradient respectively, η is the viscosity, ρg the driving force and η∇2u is the stress gradient. The left term is the divergence of stress, the right term is the body force.

76 By solving the Stokes’ equation Underworld retrieves the pressure and the velocity at the nodes of a 2-D or 3-D mesh. Material properties including density, viscosity, strain-rate, etc are updated and stored into particles which are advected through the grid.

Keeping track of the temperature is critical as it impacts on both densities and viscosities. The conduction-advection heat transfer equation takes into account the latent heat of fusion consumed during partial melting, a metamorphic reaction of particular importance because it consumes vast amount of latent heat and because a small fraction of melt can drastically reduces the viscosity of partially melted rocks. The following equation describes the rate of change of temperature as a function of heat diffusion ( κ ), radiogenic heat production (H), heat advection ( uz ), and partial melting (DF is the variation of melt fraction):

∂T 2 H ∆ S DF = κ∇ T + − uz ∙ ∇ T − T ∂t Cp ( ) Cp Dt

The equation above ensures conservation of energy and allows the coupling of densities and viscosities to temperature through the following equations:

ρ(T) = gρ0(1 − α( ∆ T ) − βF)

1 E η T = A−1/n ⋅ Exp ⋅ ε˙(1−n)/n ( ) 2 ( n ⋅ R ⋅ T )

Where A is the pre-stress factor, n is the stress exponent, E is the activation energy, and ϵ˙ is the strain rate.

With this the momentum equation becomes: η(T)∇2u − ∇p = gρ(T)

Plastic deformation is introduced by considering that above a yield stress ( τ ) plastic materials become viscous. The yield stress is:

τ = C + μ ⋅ ρgz ⋅ f (ϵ) ( 0 eff ( ))

Where f (ϵ) is strain weakening function (fault zones become weaker as strain accumulate), Co is the cohesion, μeff is the effective co- efﬁcient of friction (i.e. it takes into account the effect of pore pressure), ρgz is the conﬁning pressure.

77 Once the material passes its yield stress, the material behave as a viscous ﬂuid with a post-yielding viscosity described by:

τ ηyield = 1/2 1 2 ⋅ ( 2 ∑ij ϵ˙ijϵ˙ij)

Where the denominator is twice the second invariant of the strain tensor (also called the effective strain rate).

1 ∂ui ∂uj ϵ˙ij = + 2 [ ∂xj ∂xi ]

78 How does Underworld do it?

Underworld is one component in a modular numerical framework made of a stack of specialized modules (numerical libraries) working together to solve efﬁciently the Stokes’ equations in the context of a tectonics and/geodynamics model. This framework also provides gLucifer a visualization platform developed by the Underworld team.

•Underworld - Geophysical formulations, e.g. rheologies. Underworld •gLucifer - Visualisation package. Constitutive equations: Rheology, Radiogenic heating

•PICellerator - Particle in Cell methods (Lagrangian / Eule- History Particle-in-cellerator gLucifer Handle Particles Parallel visualization rian framework). sensitive materials •StgFEM - Finite Element Method (FEM) framework (Eule- StGFEM Solve system of linear equation rian). Includes input ﬁles and plugins stored in StgFEM_Components or StgFEM_plugins directory. StGDomain Math, geometry, mesh, particles •StgDomain - Basic mathematical formulations, model ge- StGermain ometry, mesh, shape and basic swarm framework. Parallel scientific applications

•StGermain - Core parallel programming architecture, mem- ory management and Object Oriented frameworks.

Many of these components are built themselves upon other libraries (PETsc, OpenMPI or MPICH, HDF5 ...) which explains that the deployment of the Underworld framework can be challenging.

Users access Underworld through “input ﬁles” the purpose of which is to describe the architecture of the model, the physical properties of materials that compose it, the time-dependent boundary conditions under which the model exists, and the optional and user speciﬁed passive markers necessary to capture the thermal and mechanical evolution of the model.

79 Underworld 1.X - Up to the version 1.8 Underworld input ﬁle is distributed across a few xml ﬁles (xml: extensible markup lan- guage). xml allows the description, and transport of data in a simple text format with a trees-like hierarchical structures of elements (elements, sub-elements, sub-sub-elements ...) that can be stich together using the tag “include”: mycrustRheology.xml

Tags () in xml are not pre-deﬁned like in HTML. xml allows for customizable case sensitive tags with quoted attrib- utes:

Because tags are deﬁned by users, xml is self descriptive and more information can be added using : Example1: Example2: Here is a shape: Here is a shape involving the Intersection (!) function: Sphere Intersection 0.5 0.5 myrectangle minZ ! my_diapir 0.1

Example3: Example4: Here we deﬁne a simple viscous rheology: Here we deﬁne a viscosity limiter function: FrankKamenetskii ViscosityLimiter TemperatureField 1e23 1.0e26 1e18 13.815510558

80 Here we define a plastic rheology: We put everything together adding a few bits to define the thermal and Example 5 mechanical properties of materialOne VonMises Example 7 StrainRateField 50.0 my_diapir_host strainWeakening 2800 3e-5 1e-6 Here we define a weakening function in which: initialDamageFraction is the fraction of particle with initial strain. myVonMises initialDamageFactor is the maximum initial strain of a particle strainWeakening initialSofteningStrain is the strain at which material starts to yield my_diapirViscosity viscosityLimiter finalSofteningStrain is the strain at which the material has fully yielded initialDamageWaveNumberCosI (or ...SinI) is the inverse of the horizontal length of the damaged box. 1.8e-12 1000 Example 6 StrainWeakening timeIntegrator_aux materialSwarm 0.1 0.0 0.5 0.5 0.0

81 Underworld 2.0 - Since the version 2.0, Underworld input files are written in python, and therefore can be assembled within a Ju- pyter notebook. The Underworld2 code itself (UW2) can be either deployed and compiled as an executable application stored in the user’s local machine, or deployed and compiled into a remote high-performance computer. This process demands a robust expertise, which involves the installation of numerical libraries (MPI, PETSC, HDF5, etc). Once the input Jupyter notebook has been created, it is saved as a python script via the terminal command jupyter nbconvert --to python myinputfile.ipynb or via File > Download as .py. To execute this UW2 input script, open a terminal window and execute: mpirun -np 4 python myinputfile.py, where -np 4 specifies to run UW2 in parallel on 4 cpu (alternatively mpiexec 4 python myinputfile.py).

To simplify things a little, UW2 executable is made available inside a Docker container. After installing Docker, go to Kitematic (in the Docker’s drop-down menu) and search for Underworld2. There is a number of versions including the robust Underworld2_latest. nb: The installation of Docker and Jupyter are detailed at: https://geoslearn.github.io/LabDeploy.html.

UW2 jupyter notebooks can be run inside Docker using the host machines cpus.

Readers who want to experiment with Underworld2 should consult the following website: http://www.underworldcode.org https://github.com/underworldcode https://github.com/underworldcode/UWGeodynamics.

82 Interactive 5.1 An example of UW-Geodynamics Jupyter input script (double click on the image below to scroll down)

83 INERTIA, VISCOUS FORCE AND THE REYNOLDS’ NUMBER Tensors describe linear relationships between scalars, vectors and other tensors. The dimensionality of tensor, its rank, informs about the Inertia is the resistance to a change of motion it is expressed by New- nature of the tensor, e.g.: ton’s second law of motion F = m.a. Earth geodynamics (mantle flow a tensor of rank 0 is a scalar; processes) and tectonics (plate tectonic processes) involve very small a tensor of rank 1 is a vector (1- dimen- acceleration, hence very small inertia. sional array); a tensor of rank 2 is a matrix (2- dimensional array) ... The viscosity of rocks is a measure of their capacity to resist flow. Tensors provide a concise way for for- Hence, the viscous force is the resistive force to flow. The viscosity of mulating relationship between quanti- solid rocks is very large typically >1e18 Pa.s. ties, the drawback is that they can be hard to read. The Reynolds’ number is the ratio between inertia force and the viscous force. Since on Earth the inertia force is very small and the vis- The stress tensor is a second order tensor which collects into col- cous force is very large, then the Reynolds’ number for tectonic and umns the coordinates of 3 traction vectors (Tei) acting on 3 perpendicu- geodynamic flow is very small. lar faces of a cube. The value of these coordinates depends on the choice of the reference framework. Nevertheless, these coordinates Quiz: Let’s consider a tectonic plate 4000 km x 2500 km x 150 km ac- fully describe the state of stress on the central point in the cube. By convention the component refers to coordinates along axis i of the celerating from 0 to 5 cm.yr-1 in 100,000 years. The density of the σij plate is 3000 kg.m-3, and the viscosity of the asthenosphere is 1e20 traction acting on face j. The number of indices relates to the order of Pa.s. What is the Reynolds number of the mantle flow induced by this the tensor (order, degree, rank are synonymous). acceleration, assuming that the plate motion causes shearing distrib- Einstein’s summation convention implies summation when an index uted over 10 km of asthenosphere? variable appears twice in a single term: n.b.: shear stress = ~ strain rate * viscosity 3

ci xi ⟺ ∑ ci xi ⟺ c1x1 + c2x2 + c3x3 i=1 3 3 hence: τij = 2ηijkl ⋅ ϵ˙kl = ∑ ∑ 2ηijkl ⋅ ϵ˙kl k=1 l=1

84 CHAPTER 6 Lithospheric Deformation

The mechanical behavior of the Earth’s lithosphere under dynamic or kinematic boundary conditions is strongly non-linear, and therefore difﬁcult to predict. This is due to the exponential dependency of the rheology of rocks to temperature, as well as weakening mechanisms including strain (which reduces the grain size of rocks and helps ﬂuids circulation) and partial melting. Hence, small variation in temperature, strain or melt fraction can result in strong variations in mechanical properties and mechanical behaviors. In this chapter we turn to computational tectonics to explore lithospheric deformation in a variety of tectonic settings.

85 SECTION 1 Collisional tectonics Content

1. A generic 2-D thermo- mechanical experiment of collisional tectonics 2. The European Alps

This section shows that numerical experiments, involving realistic mechanical and thermal properties, can produce collisional structures observed in the Euro- pean Alps.

Tectonic convergence between plates is accommodated continental collision. In what follows we explore colli- by a range of processes including subduction (oceanic or sional processes through numerical experiments. In our continental), coupled shortening and thickening, and coupled thermal-mechanical numerical experiments the transcurrent tectonics involving one or more lithospheric- computational grid is 768 km x 1536 km, with a cell reso- scale strike-slip faults. Following the closure of oceanic lution of 2 km x 2 km. The rheology of all rocks in the domains, continents may subduct. However, continental model is visco-plastic, excepted for the asthenospheric subduction is limited by the buoyancy of the continental mantle which is assumed iso-viscous (1e20 Pa.s). crust, and continental subduction is rapidly followed by

86 Subduction and continental collision -

(a) - The converging continents are 160 km thick, and are adjacent to oceanic lithospheres of thicknesses 110 km and 40 a km. The continental crust consists in 20 km of upper crust (lavender) above a basement 20 km thick (orange). The oceanic crust is 6 km thick (pink). A fault dipping left decouples the thick and thin oceanic lithospheres. Convergence is achieved by pushing the right continental lithosphere to the left, whereas the left wall of the continental lithosphere to the left is ﬁxed. Once a ~300 km slab has formed, slab pull becomes the only driving b force. Erosion removes rocks above 5 km elevation, and sedimentation ﬁlls all spaces below -2 km depth. Sediments are color coded yellow to blue from older to younger sediments.

(b-c) - Initiation of subduction leads to extension and thinning of c the margin of the continent on the left, and deposition of sediments above the subducting oceanic plate. The ﬂexure of the subducting plate provides the space for extension.

(d-e) - The thin oceanic plate is forced to subduct, whereas the d sedimentary basin is transferred from the original downgoing plate onto the upper plate. As the continental crust from the upper plate enters the subduction zone its motion is blocked by the colliding lithospheric mantles. A crustal-scale overturned fold develops, and the structural vergence switch to the left. e

87 (f) - The system has reached the stage of continental collision. The thinned margin of the left continent is being subducted underneath a thick wedge made mainly of crustal material from the right continent. Crustal rocks reach a depth > 100 km. Its only at this stage that high topography is being generated.

(g-h) - The lithospheric mantle of the continent on f the right is peeling-off as the crust is being transferred onto the left continental plate, the margin of which is been completely smeared. The upper crust of the left continent and the sedimentary basin are being pushed to the left. A set of left verg- ing folds develop. The upper part of the hinge of the overturned fold is sheared by a normal fault, and translated to the left above the sedimentary g basin. This leads to the exhumation of deep crust.

88 The European Alps is an example of collisional orogen which follows the subduction of oceanic lithosphere. It shows, at various scale, a range of structures resembling those observed in our numerical experiment. When considering similarities between numerical experiments and natural examples, one needs to exert caution and keep in mind that nature is always far more complicated that our numerical representations of it.

The Alps results from the convergence and collision of the Adriatic micro plate, an African promontory sometimes called Apulia (in yellow on the map) and Europe (in blue), in between which the Briançonnais microcontinent (in green) was squeezed. Conver- gence started at around 70 myr ago with the south dipping subduction of the Ligurian-Piemont ocean that separated the Briançon- nais terrane and the African margin. By 35 myr ago this ocean was closed and the Briançonnais, with the narrow Valais ocean that separated the Briançonnais and Europe, were squeezed in between the colliding European and African margins. The deeper, internal part of the Briançonnais terrane was buried to great depth (>100 km), under the African margin (in orange in the cross-section on the next page), whereas the external part (zone houillère, pale green) escaped high- pressure metamorphism. High-temperature metamorphism (up to 800ºC) was limited in time (~30-15 myr) and space (subpen- ninic nappes of the central Alps).

89 The ﬂysch sediments (dotted regions on the cross-sections) that derived from erosion of the African plate and deposited on the subducting Valais margin where translated onto the European margin, as the Briançonnais was exhumed, translated over the European margin, and folded into recumbent folds. A combination of folding and gravitational spreading exhumed the Helvectic nappes (pale blue) in a tectonic window. These thin-skinned nappes are made of sedimentary rocks deposited over the European rifted platform. They were sheared-off from their pre-Mesozoic basement to form a fold-and-thrust belt. The European pre-Mesozoic (i.e. Varis- can) basement was also deformed into the so-called Subpen- ninic nappes (in pink). Ongoing collision resulted on back- thrusting of the Briançonnais over the Adriatic plate.

A the plate-scale, the Alps involves a complex and segmented subduction zone system wrapping around Adria (see ﬁgure next page). The Alps now stands above a suture Transect NFP-20 West, Schmidt, et al., The TRANSMED Atlas, 2004 zone involving: to the west an east to south dipping subduction, and to the east a north to northeast dipping subduction. At the surface, the Alps presents itself as a curvilinear belt with sharp bends: the Cantabrian arc, the arc of the western Alps, the Carpathian arc, and the Helenic arc. The orogen itself is relatively symmetric and bivergent with opposite structural vergences.

Our simple 2-D numerical models cannot account for this sort of complexity. Nevertheless, it is interesting to note that to the ﬁrst order our numerical experiment reproduces: • The burial of basement rocks down to 100 km. • Their deformation into crustal-scale recumbent folds. • Formation and denudation into a tectonic window of thin-skin Helvetic-style nappes. • The emplacement of nappes of external (i.e. upper crustal) basement rocks above younger allochthonous ﬂysch.

90 # R script to produce the map on the next page from gebco data.

# Load the libraries library(marmap) library(raster) library(RColorBrewer) library(ggplot2) library(devtools)

#setwd stands for set working directory setwd('/path_to_your_directory/')

# Etopo color scheme, modiﬁed from https://github.com/cran/marmap/blob/master/R/palette.etopo.R

etopoAlps <- read.csv(textConnection( "altitudes,colours Handy et al., Int. J. Earth Sci., 2014. 4500,#FBFBFB 2000,#864747 1950,#7E4B11 1000,#9B8411 950,#BD8D15 300,#F0CF9F 0,#307424 -1,#AFDCF4 -3000,#090B6A" ), stringsAsFactors=FALSE)

etopoAlps$altitudes01 <- scales::rescale(etopoAlps$altitudes)

etopoAlps.colors <- function(n) { colorRampPalette(etopoAlps$colours)(n) }

scale_ﬁll_etopo <- function(...) { ggplot2::scale_ﬁll_gradientn(colours=etopoAlps$colours, values=etopoAlps$alt- itudes01, limits=range(etopoAlps$altitudes), ...) }

scale_colour_etopo <- function(...) { Moho depth and suture zone around the Adria, Luth ggplot2::scale_colour_gradientn(colours=etopoAlps$colours, values=etopoAlps$alt- et al., Tectonophysics, 2013. itudes01, limits=range(etopoAlps$altitudes), ...) }

91 Topographic map of the Alps (see R script), with geology of the western Alps.

Zhao et al., Geology, 2015

92 SECTION 2 Transcurrent tectonics Lorem Ipsum

1. A generic 3-D thermo- mechanical experiment of pull-apart basin 2. The Dead Sea basin

Here we document crustal ﬂow in an orogenic pull-apart velop in the deep crust. A block of lithosphere where localized extension and crustal thinning focus the 384x256x128 km is mapped over a computational mesh exhumation of deep crust. Using numerical experiments consisting of 240x160x80 elements. It includes an 8 km we show that an extensional strain regime, collinear with thick layer of air, a layer of continental crust 40 km thick the direction of plate motion, is partitioned into the shal- (temperature at the Moho is 610ºC), and a layer of mantle. low crust, whereas contractional structures and fabrics - Two non-overlapping, parallel and vertical faults extend- at a high angle to the direction of imposed transport - de- ing from the surface to 20 km depth, and laterally over 64

93 km from the two opposite vertical walls. Kinematic bound- View from the top showing both ary conditions drive divergence at a total velocity of 3.84 cm extensional and shear fractures. yr-1 promoting the formation of a dextral pull-apart.

After 1 myr of displacement (i.e. total displacement of 38.4 km), a fractures network invades the brittle upper crust in the region in between the two strike slip faults. This network consists of i/short extensional fractures oriented per- pendicularly to the direction of extension imposed by the velocity boundary conditions, and ii/ longer shear fractures that merge into the two faults zones. Damages from this fracture network explain the localization of further deformation into this region.

Brittle strain damage after 1 myr of deformation 256 km 1.28 cm/yr

384 km Fault 1.28 cm/yr Step angle 1.28 cm/yr 48 km 20 km Fault 16 km

1.28 cm/yr 40 km Upper crust

1200 Lower crust 800 400 0 Lithospheric 1330ºCmantle

Asthenosphere 120 km 120

94 Meanwhile in the deeper crust ... As displacement increases the upper crust gets thinner, and pressure gradients in the weak deeper crust drives the ﬂow of the deep crust towards the pull-apart region, resulting in the formation of a dome.

The long axis of the dome joins the tips of the strike-slip faults. Strain markers show that the direction of stretching is co- linear with the imposed velocity boundary condition. As a result, the stretching lineation plunges to the west at the western end of the dome, and to the east at the eastern end of the dome. Deeper in the dome the strain deﬁnes two sub-domes separated by a median high-strain zone.

Gallery 6.1 Architecture of a dome in a pull-apart. N Strain marker 0 km 16 km

60 km

Depth (km) -40 -30 -20 -10

-43.8 -4.43

Shallow crust

Deep crust

Strain markers Solidus initially at Moho 42 km depth

View to the NW showing the dome underneath the upper brittle crust (dark and pale blue layers). The red surface is the solidus.

95 Movie 6.1 Faults network in a pull-apart basin between two non-overlapping vertical faults.

96 Movie 6.2 Faults network in a pull-apart basin between two overlapping vertical faults.

97 The Dead Sea: an example of pull-apart basin

The Dead Sea transform fault system runs along the margin of the Arabian shield, and mechanically links the opening of the Red Sea to the south and to the Taurus- Zagros collision to the north. Over the past 15 to 20 myr, it has accommodated ca. 100 km of sinistral motion between the African plate and the Arabian plate, motion slowing down to 6 mm per year over the past 5 myr. The Dead Sea basin forms in a narrow corridor no large than 15 km in between two left lateral strike-slip fault zones: the Jericho (Jordan) fault to the west and the Arava fault to the east. The Dead Sea itself is a 350 m deep hyper- saline lake (dark blue in far right map), its water level stands 400 m below the Mediterranean sea level.

The sedimentary basin is 8.5 km deep with very steep subsidence Shalev et al., Tectonophysics, 2012 gradient across closely spaced faults, and an unusual flat basin floor. It is segmented into sub-basins by transverse normal faults highly oblique to the strike slip faults (Smit et al., Tectonophysics, 2008). The flanks of the basin show different characteristics. The east flank is higher and dissected by river gorges highly oblique to the basin. The west flank shows NNE trending topographic ridges that conform with the left-lateral sense of shear, and suggests that the lithosphere on the west flank is relatively weaker compared to the Ara- bian shield. Heat flow data reveals a negative thermal anomaly that may corresponds to the downward advection of cold sediments.

98 Interestingly, seismic data shows that the lower crust from ca. 18 km down to the Moho is not affected by the basin development.

In summary: The Dead Sea basin is characterized by a set on particular features including: • Two closely spaced bounding faults • A flat basin floor • No heat flow anomaly • A deep crust that shows no thinning • A Moho that shows no offset

To explains these features one can hypothesize that when at the Earth’s surface the spacing between the two bounding faults (here the Jericho fault and the Arava fault) is smaller than that thickness of the brittle crust, the faults merge into a ductile strike slip shear zone upon entering the deep crust. Hence: extension and thinning is strongly partitioned into the upper brittle crust; sedimentation keeps up with Bottom panel: E-W seismic profile across the Dead Sea basin showing P-wave velocity contours. Interpreted geology: Basin subsidence of the basin floor; no deep isostatic flow is re- fill (dark yellow); pre-basin fill sediments (light yellow); upper quired; and as a consequence the Moho remains flat and crust (green); lower crust (orange). Top panel: comparison be- there is no enhanced heat flow. To test this hypothesis a 3- tween observed and calculated free-air gravity anomaly using a D numerical experimentation can be set-up. density model derived from the P-wave velocity model.

99 SECTION 3 Extensional tectonics Lorem Ipsum

1. Extensional collapse 2. Continental extension 3. Continental rifting 4. Mid-oceanic ridges

Luke Mondy, 2016

Continental extension occurs in a limited number of modes. Because continents have insulating properties, the temperature excess that buildup in the mantle underneath supercontinents lead to dynamic topography which generate extensional stresses that drive continental breakup, deformation focussing on pre-existing zones of weakest such as ancient suture zones. In a process similar to the calving of iceberg from the margin of polar ice caps, the recurrent detachment of continental blocks at the margins of continents is another mode of continental extension. In this case, slab rock back and/or the upward ﬂow of the mantle wedge drive extension. Extension can also be driven by gravitational stresses inside the lithosphere. These stresses are the result of lateral variation of gravitational potential energy due to lateral

100 change in density. On a regional scale, this process explains extensional tectonics in orogenic systems. The north-south trending normal faults in the Himalayas and in the Tibetan plateau, and the north-south trending normal fault array in the Basin and Range (western USA) are the product of gravitational collapse. At a local scale, gravitational collapse can also explain the normal faults in a pull-apart basin (see the section on the Dead Sea Basin).

The collapse of orogenic plateaux- During their build-up phase, orogenic plateaux store significant excess of gravitational potential energy. As the plateau region grow weaker through thermal relaxation and heating, this excess of energy drives their gravitational collapse, and - in the case of fixed boundary collapse - contractional shortening of foreland regions. One can use 2-D numerical experiments to explore the process of plateau collapse. The Underworld model below is 150 km deep and 900 km long; and it is mapped over a grid of 2 km resolution. It uses appropriate thermo-mechanical properties (radiogenic heat production, diffusivity, rheology, density etc), fixed lateral boundary conditions, constant temperature on top, and an isostatic condition maintain the pressure constant at the base of the model. In this experiment, the plateau lower crust flows horizontally at a velocity of a few cm per year into the foreland which becomes thicker while the orogenic crust becomes thinner. In this model, the strength of the upper crust is such that there is no measurable deformation at the surface. Passive tracers

40 km Continental crust Orogenic plateau 65 km

Mantle

101 Extension of orogenic lithosphere and the formation of metamorphic core complexes- Extension of thick and hot crust leads to the development of metamorphic core complex (MCC). The Underworld experiment below represents a 120 km x 360 km lithosphere mapped over a computational grid with a resolution of 2 km. The crust is 60 km thick and the temperature at the Moho has reached ~800ºC. A ~2 km thick fault-shaped prism made of weak material is embedded Upper crust Fault into the upper crust. Its role is to localize deformation 60 km Lower crust Passive tracers away from the vertical walls in order to mitigate boundary effects. This experiment uses appropriate thermo- Mantle mechanical properties. The boundary conditions involve a kinematic condition pulling of both vertical walls at a velocity of 1.8 cm per year, a constant temperature on top, a constant heat ﬂow and isostatic condition at the base of Partially melted lower crust the model. To assess the deformation of the lower crust, and array of circular passive tracers is embedded into the deep crust.

Extension leads to the thinning of the continental crust, and isothermal decompression of the hot deep crust which results in decompression melting. The presence of melt in the lower crust reduces the viscosity, and increases the mobility of the deep crust which ﬂows easily in response to lateral pressure gradient. Extension and thinning in the upper crust is localized around zones of brittle deformation. The necking of the brittle upper crust is ac- companied by the formation of gneiss dome. The structure in the deeper part of the dome involves a sub-vertical high- strain zone separating two colliding recumbent folds, which evolves into two sub-domes.

102 The “calving” of continental ribbons at passive margins - To the east of Aus- tralia’s east coast, the Tasman sea separates Australia and a girdle of continental rafts with include the north and south islands of New Zealand. Most of these rafts consist of submerged ribbons of thinned continental crust. They were detached at around 85 myr ago from Gondwana’s Paciﬁc margin at a time when Antarctica and Australia were still attached.

A number of hypothesis have been proposed to explain the detachment of continental blocks. These include: • The slab pull force acting on a passive continental margin attached to a oceanic lithosphere being subducted. • The pull on a continental margin from a subducting slab retreating toward the ocean (i.e. slab rollback). • Extensional gravitational stresses due to the buoyant rise of the mantle wedge above a subducting slab. • The collapse of a cordillera due to a reduction of plate velocity or a change in the direction of plate motion.

b1/

FRP

Oceanic lithosphere FSP FG Oceanic lithosphere

Asthenosphere Continental lithosphere Asthenosphere

Gravitational Force

Continental lithosphere Gravitational Force FSS

FSS

FG FRP

Oceanic lithosphere Continental lithosphere

Gravitational Force Asthenosphere

FSS

103 Continental blocks and continental ribbons are a common occurrence in particular along the west margin of the Paciﬁc ocean. Numerical experiments can help understand the range of conditions amenable to their formation. The model below is a 2-D representation of a passive continental margin. Its dimensions are 768 km x 192 km, and it involves an oceanic lithosphere, a continental lithosphere, and a transitional domain in between which represents the continental margin. Each domain is 256 km long. The model is extended through the imposing of a velocity condition. We explore the role of the extensional velocity and the age of the oceanic lithosphere. Ocean Margin Continent When pulling at 3 cm/yr at both sides, our experiment suggests that a 10 myr old oceanic Mantle lithosphere is weaker than the continental margins, in which case rifting occurs at the junction between the ocean and the continental margin. A 20 myr old oceanic lithosphere is stronger than the continental margin. In this case extension and 10 myr old oceanic lithosphere thinning occurs at the junction between the continent and the margin. Interestingly, extension gets Velocity: 3 cm/yr at each vertical wall localized in two rifts.

Reducing the total extensional velocity from 6 cm/ yr down to 2 cm/yr leads to one single rift. One can 20 myr old oceanic lithosphere concludes that a single rift can accommodate exten- Velocity: 1 cm/yr at each vertical wall sion up to a given velocity. For faster extension, the system evolves multiple rift zones to keep up with the imposed velocity, in which case multiple 20 myr old oceanic lithosphere ribbons can develop. Nikita Golkin, Sydney Uni. Honours thesis.2015

104 cont. crust The “calving” of continental ribbons at active margins - In this experi- cordillera lithospheric mantle ments we simulate an active continental margin supporting a cordillera fault orogen (crustal thickness 60 km). The model is 1536 km long and 768 km deep. The initial reference model at time t = to is produced by push- convective mantle ing an oceanic lithosphere toward the continent at 5 cm/yr. A fault- like zone of weakness insures that the oceanic plate slides underneath the continent. To maintain a constant mass into the model and insure isostasy, an equivalent outﬂow is imposed to the right wall (0.791 cm/ yr). A zero horizontal velocity condition (ux = 0) is imposed on the left wall. We turn off the push and the outﬂow once the slab reaches ~300 self-consistent experiment km (time = 5.30 myr). From this time, that we take as our t = to, the de- starts here .... formation is only driven by the slab-pull force (i.e. ux = 0 on both vertical walls).

The oceanic is pulled towards the continent underneath which it sub- ducts. The orogenic crust spreads towards the ocean, and a block of orogenic crust gets almost separated from the continent. However extension in the cordillera is short lived from ~5.3 to ~7.8 myr, and the oro- Arrows show genic block fails to fully detach. The trench retreats towards the ocean, velocity ﬁeld however the slab itself is not retreating. From ~7.80 myr onwards the thinned cordillera is in transient mechanical equilibrium.

An experiment including a buoyant mantle wedge shows that the detachment of the orogenic block is enhanced, but still not completed. This may be explained by the absence of diking in our numerical experiment, which in nature could be a strong weakening mechanism.

105 Movie 6.3 Stretching of a cordillera orogen

106 CHAPTER 7 Mantle Geodynamics

Hassan et al. 2016. Nature, 533, 239-242.

The volume of rocks expands when they become warmer, and contracts when they cool down. This leads to a decrease and a increase in density, respectively. The coefﬁcient of thermal expansion of most rocks, as a volume fraction per degree, is about 3e-5 ºC-1. This means that close to the core-mantle boundary, where temperature reaches ~4000ºC, rocks are less dense by about 12% (3e-5x4000=0.12) compared to rocks closer to the surface. This drives mantle convection, i.e. the buoyant rise of deep hot mantle rocks, and the sinking of an equivalent volume of colder, denser rocks.

107 SECTION 1 The Rayleigh number Summary Figure 7.1 Example of mantle ﬂow underneath an extending lithosphere.

1. Why mantle convection

2. Rayleigh number

3. Mantle heating and the Rayleigh number

The purpose of this section is to explore the nal conditions. We will also illustrate what dynamic of the convective mantle, to under- role lithospheric plates play on the pattern of stand the contribution of this process to cool- mantle convection. We start here by introduc- ing the interior of planets, and - through nu- ing the Rayleigh number, a dimensionless sca- merical experiments - gain insights into how lar which represents the ratio of the forces that mantle convection and plate tectonics relate to drive convection over the forces that resist con- each other. We will test different modes of convective motion. In short, the Rayleigh num- vection driven by various boundary and inter- ber is a measure of the vigor of convection.

108 from the necessity to minimize internal energy. Convection Temperature increase with depth, but why? forces deep, hot and less dense material up, and pushes shallow, cold and denser material down. Because of the re- In the Earth’s interior, temperature increases with depth for lationship between density and temperature the Earth's 3 main reasons: mantle is maintained in a state of permanent gravitational 1/The radioactive decay of radiogenic elements (mainly K40, unstability, as hot material rising toward the surface pro- U235, U238, U232 and Th) involves exothermic reactions that lib- gressively cool down and become more dense. Heat (a erate thermal energy. form of energy) is consumed by mantle convection which 2/ During its formation, the Earth has trapped a ﬁnite acts as a giant heat exchanger, to cool down very efﬁciently amount of accretional energy (gravitational energy transformed into heat). the deep Earth’s interior. 3/ Last but not least, the Earth is cooled at the surface since it is suspended into the chilling ~0 K Universe. Rayleigh Number And The Strength Of Convection

The Rayleigh number measures the vigor of the convection. Mantle Convection It is a dimensionless number that expresses the ratio of the Upon heating, rocks expand and consequently their den- forces driving mantle convection, and the forces opposing sity decreases. Since temperature increases with depth one it. concludes that the density of rock decreases with depth as Activity: Try to ﬁgure out which physical parameters are rele- well. However, this is not ideal, as a system in which den- vant to the process of convection? Which of these parameters pro- sity decreases with depth has a higher internal gravita- mote convection, and which resist convection? tional potential energy compared to that of a system in which density increases with depth. Indeed, a fundamental The buoyancy force - that arises from volumetric expan- principle of physics states that a system naturally evolves sion (thermal expansion) - drives mantle convection. In to minimize its internal energy. Mantle convection emerges the other hand, the effective viscous force (due to the viscosity of the mantle) and the thermal diffusivity (which

109 smooth out thermal gradient) oppose mantle convection. where K is the thermal conductivity. The ratio between the buoyancy force to the viscous force RabT and Rabhf are identical as long as Qm.zm/K = T1 hence is the Rayleigh number. For convection to occur, the Ray- Qm = (T1.K)/zm. leigh number must be larger than a critical value, which depends on whether the surface is free of ﬁxed (i.e. zero velocity at the surface). The Rayleigh Number driven by internal heating: 3 2 ρ0.g.α.zm H.zm Raih = .( −T ) The Rayleigh number takes slightly different forms depend- η.κ K 0 ing on how heat is delivered to the mantle (e.g. internal where H is the radiogenic heat production per cubic meter. heating, basal heating or a mix of both). In details, things Rabhf and Raih are identical as long as H = Qm/zm and there- 2 are complicated but here we simplify and pose that: fore as long as: H = (T1.K)/zm .

The Rayleigh number driven by a hot base (at temperature The Rayleigh number driven by both basal heat ﬂow and in- T1) and a cold surface (at temperature T0 at the top of the ternal heating: 3 2 convective layer): ρ0.g.α.zm Qm.zm+H.zm 3 Ramh = .( −T ) ρ0.g.α.zm 0 RabT = .(T −T ) η.κ K η.κ 1 0 The second term in the Rayleigh numbers describes the heat-

Where ρ0 is the density are room temperature, g is the gravi- ing mode. As long as the ﬁrst term of the heating mode (T1, 2 tational acceleration, α is the thermal expansivity, zm is the Qm.zm/K, H.zm /K, ...) are equals, the Rayleigh numbers are thickness of the convective layer, η is the average viscosity the same. of the mantle, and κ is the thermal diffusivity. For convection to occur, the Rayleigh number must be The Rayleigh number driven by a hot base (at constant basal larger than a critical value, which varies from 600 to 3000 heat ﬂow Qm) and a cold surface (at temperature T0 at the depending on the boundary conditions (e.g. stagnant lid, top of the convective layer): plate tectonics, etc). 3 ρ0.g.α.zm Qm.zm Rabhf = .( −T ) η.κ K 0

110 SECTION 2 Exercise: Keeping it cool ...

Summary

1. Learning outcomes: To http://topography.geotheory.co.uk/ develop a deeper understanding of mantle dynamic and its dependence to the mode of heating of the convective mantle 2. Generic skills: Problem solving ability, computational skills, analytical skills 3. Assumed knowledge: high- school math, some notions of tectonophysics (heat and mass transfer). 4. Tools: R, Python, Matlab 5. Reading: Turcotte & Schubert: Geodynamics Exercise 1: Basal heating vs internal heat- To ensure we compare apple with apple we ing vs mixed heating need to compare models that are at once thermally and convectively equivalent. We will Here we compare mantle convection pow- consider that two models are: 1/ thermally ered by various modes of mantle heating, equivalent if in their non-convective state i.e. basal heating (either constant basal tem- (inﬁnite viscosity) they have the same perature or constant heat ﬂow), internal ra- steady-state average temperature, and 2/ diogenic heating, or a combination of both.

111 convectively equivalent if in their normal convective state spectively; H is the volumetric rate of radiogenic heat pro- they have the same Rayleigh number. duction.

Find the relationships that sets the conditions for thermal You may need to know that: equivalence and convective equivalence and comment on α β a.xn+1 your results. We give the following clues: axn+b dz = +b.x+c ∫ ( n+1 ) • The steady state conductive geotherm for constant basal β α temperature is:

(T1−T0) TbT(z) = .z+T0 Exercise 2: What’s the Earth’s Rayleigh number? zm • The steady state conductive geotherm for constant basal The Earth is currently releasing heat into space at the rate heat flow is: of ~44 TW. About 20% of this amount (~8 TW) originates Q from radiogenic isotopes trapped in the plates. This plate T (z) = m .z+T bhf K 0 component of heat doesn’t contribute to power mantle con- • The steady state conductive geotherm for internal heat- vection. Therefore, Earth is left with a convective heat flow ing is: of 36 TW to drive mantle convection. H.z2 H.z T (z) = − + m .z+T If mantle convection is driven by basal heating alone, then ih 2K K 0 the 36 TW must come from the Earth’s core in the form of a • The steady state conductive geotherm for mixed basal basal heat flow. In the other hand if mantle convection is heat flow and internal heating is: powered by internal heating alone then the 36 TW must be H.z2 Q H.z T (z) = − +( m + m ).z+T partitioned into the convective mantle as radiogenic heat mh 2K K K 0 production. In nature, we expect the convective heat flow where Qm, K and T0 are the basal heat flow entering the to be split into both internal and basal heating. base of the convective mantle at the core-mantle boundary, the thermal conductivity and the surface temperature re-

112 -3 There is however a debate about the partitioning of the tion 2” W.m ; Qm = “your value from question 1” W.m-2; η heat flow between the core and the mantle. This partition- = 5e21 Pa.s; T0 = 273 K ing is described by the Urey ratio, the ratio between internal nb1: For the mixed heating mode, use 24 TW for the internal heat- heating in the convective mantle to total convective heat ing and 12 TW for basal heating. flow (36 TW). Estimates varies from 0.10 to 0.90. nb2: For the case involving a constant basal temperature one can 1/ Calculate the basal heat flow (in W.m-2) in the case calculate the temperature at the core-mantle boundary by extrapo- where the convective heat flow is delivered to the convec- lating the linear conductive geotherm in the lithospheric plate tive mantle in the form of basal heating at the core-mantle (273 K at the surface and 1603K at 150km depth). boundary. 4/ Assuming no mantle convection, estimate the tempera- 2/ Calculate the rate in radiogenic heat production (in ture at the core-mantle boundary in both basal heating (con- W.m-3) in the case where the convective heat flow is in the stant heat flow) and internal heating models using the conform of internal heating. ductive geotherm equations given earlier. Compare these Please note that: The Earth’s radius is: 6371 km, the litho- values with the estimated range of temperature of the core- sphere average thickness is 150 km, the depth of the core- mantle boundary in the convective Earth (3000 to 4000 K), mantle boundary is at 2890 km. Volume of a sphere: and comment. 4 .π.R3 ; surface of a sphere: 4.π.R2 (R is the sphere radius). 3 3/ Calculate the Rayleigh Number for each heating modes (internal heating, basal heating, and mixed heating) using the following values:

-3 -2 ρ0 = 4000 kg.m ; g = 10 m.s ; α = 3e-5; zm = (2890-150) km; K = 4 Wm-1 K-1; κ = 1e-6 m2 s-1; H = “your value from ques-

113 SECTION 3 Exercise: Convective patterns

Summary

1. Learning outcomes: To develop a deeper understanding of mantle convection and its relationship to plate tectonics 2. Learn to use non- dimensional approach 3. Generic skills: Problem solving ability, computational skills, analytical skills 4. Assumed knowledge: high- school math. 5. Tools: Ellipsis, R, Python 6. Reading: Turcotter & Schubert: Geodynamics Ellipsis is a finite-elements code that solves multigrid solver (solutions are iteratively complex 2D fluid dynamic problems. It refined by using an increasingly finer grid). solves the relevant equations for incom- Particles, representing different materials, pressible fluids (Navier Stokes equations, move through the grid carrying relevant in- equations of conservation of mass and ener- formation to solve the equations. The com- gies and constitutive equations) at the bination of a fixed grid (Eulerian grid) and nodes of a fixed computational grid using a free particles (Lagrangian particles) allows

114 numerical experiments involving large deformation of # Initial conditions gravacc=1 rheologically complex models. toptbcval=0 bottbcval=1

The Ellipsis input script (a simple text file) describes the # Embedded Temperature model including the boundary and internal conditions. El- Temp_rect=3 # Nb of rectangle Temp_rect_x1=0.00,0.0,0.00 # Coordinates lipsis then executes the script. Since mantle convection de- Temp_rect_x2=6.00,6.0,6.00 pends only on the Rayleigh number, the values of all pa- Temp_rect_z1=0.00,0.1,0.90 Temp_rect_z2=0.10,0.9,1.00 rameters entering its definition are of little importance. Temp_rect_mag=0.25,0.5,0.75 # magnitude Hence, in Ellipsis we will set gravity, diffusivity, viscosity, Temp_rect_ovl=R,R,R # A/M/R = add/multiply/replace density, mantle thickness, radiogenic heat production to 1, We fill our box with one material representing the mantle. excepted for the coefficient of thermal expansion whose This box is defined by the coordinates of its top-left corner value becomes the Raleigh number. (x1, z1) and bottom-right corner (x2, z2). With this in mind, let’s build our first model in the case of nb: (0,0) is the top-left corner. constant basal temperature. We first define the dimension # Material distributions of the model. The model horizontal length must be a few Material_rect=1 # number of rectangle regions Material_rect_property=0 # Material id times longer than it is depth. Since the depth of the con- Material_rect_x1=0.0 # Coordinates vecting mantle is 1, we choose 6 for the length for the x di- Material_rect_x2=6.0 # Use columns to add rect. Material_rect_z1=0.0 mension. Material_rect_z2=1.0

# Dimensions of the model: Then we deﬁne the boundary conditions. We use a con- dimenx=6.0 dimenz=1.0 stant temperature of 0 at the top of the model, and a constant temperature of 1 at its base. For the initial conditions, we have gravitational acceleration set to 1 and the initial geotherm involves three layers # Boundary conditions Temp_bc_rect=2 # number of bc at temperature 0.25 (from 0 to 0.1), 0.5 (from 0.1 to 0.9) and Temp_bc_rect_norm=Z,Z # Z,X for horiz./vert. BC. 0.75 (from 0.9 to 1.0). Temp_bc_rect_icpt=0,1 # Coordinate of boundary plane

115 Temp_bc_rect_aa1=0,0 # lateral coordinate extent # OUTPUT FILES Temp_bc_rect_aa2=6,6 # lateral coordinate extent datafile="ConvMod1" # root name of output files Temp_bc_rect_hw=0,0 # smoothing half-width datatypes="Temp,Pres" # output variables at nodes Temp_bc_rect_mag=0,1 # magnitude of bc averages="Temp" # horizontally averaged values, Heat_flux_z_bc_rect=0 # no basal heat flow storage_timesteps=1 # writing interval checkpt_timesteps=1 # PPM writing interval In what follow, we define the thermal and mechanical prop- storage_timing=1.e-8 # timestep writing interval checkpt_timing=1.0 # PPM writing interval (time) erties of the mantle. Again, because all relevant parameters are included in the Rayleigh number, density, viscosity, # Specifications for graphical output files PPM_files=1 # number of PPM files at each output diffusivity and heat capacity are equal to 1. Because we con- step sider a basally heated model we set the internal heating to PPM_height=256 # vertical size of output PPM file PPM_coloring=1 # variable upon which to base colouring 0, and the coefficient of thermal expansion is the Rayleigh # 1=temperature, 2=viscosity, 3=stress, ... Number. Finally, we may want to extract some temperature (or other # MATERIAL PROPERTIES variables) profiles across our model: Different_materials=1 # Nb of materials

# for extracting profiles/histories ## Mantle Sampling_tracers=10 # number of sampling tracers Material_0_density=1 # Rheological model Material_0_rheol_T_type=1 # visc(T)=N0*exp(-T1*T) Tracers (fixed to the grid or to the material) can record in- Material_0_viscN0=1 # N0 in viscosity model formation such as Temperature, Pressure, etc. They can also Material_0_viscT1=0 # T1 in viscosity model # Thermal parameters record parameter profiles (vertical or horizontal). Material_0_therm_exp=7.34e7 # coeff. thermal exp # Rayleigh Nb One last thing, to run this model over 1600 time steps. # Material_0_therm_diff=1 # thermal diffusivity Material_0_Cp=1.0 # isobaric heat capacity Material_0_Qt=0.0 # Rad. heating rate / kg ADVECTION-DIFFUSION PARAMETERS # Colouring (Try to experiment with this.) maxstep=1200 # maximum number time steps We now consider the output files we would like to get from Ellipsis:

116 In a terminal navigate to the folder holding the input scritp 3/ Run the models 1 and 2 using a temperature dependent an executable and run the following command: viscosity: ./ellipis3d my_ellipsis_model.input with: Exercise 3: 2D Experiments of Mantle Convection using. Material_0_viscN0=5 # N0 in viscosity models 1/ Run over 1200 steps the previous script (convection Material_0_viscT1=5 # T1 in viscosity models with basal heating, constant temperature). Verify that the average viscosity of the convective mantle is 2/ Build and run a model with internal heating still 1. (Ra=2.62e8). For this model, you will need to 1/ remove 4/ Analyze and comment your results. Give particular at- the constant temperature at the base of the model, 2/ set to tention to the shape of the averaged geotherm for both in- zero the magnitude of the basal heat ﬂow (insulation condi- ternally heated and basal heated models. How do the geo- tion, no heat is lost at the base of the model), and 3/ set the therms compare? internal heating rate to 1. 5/ Design a model with both internal heating and basal

# Boundary conditions heating using constant basal heat ﬂow instead of constant Temp_bc_rect=1 # nb of bc. basal temperature (use the Rayleigh number from exercise Temp_bc_rect_norm=Z,Z # Z (horiz), X (vert.) BC Temp_bc_rect_icpt=0,1 # Coordinate of boundary plane 2). Run this model and compare it with a model in which a Temp_bc_rect_aa1=0,0 # lateral coordinate extent lithospheric plate is added. The plate has a thickness of 0.1 Temp_bc_rect_aa2=6,6 # lateral coordinate extent Temp_bc_rect_hw=0,0 # half-width of bc smoothing edge and extends over 50% of the model. Its density must be Temp_bc_rect_mag=0,1 # magnitude of bc 20% lower than the mantle and its viscosity (constant with

Heat_flux_z_bc_rect=1 # nb bc ranges (surfaces) temperature) must be 1000 times larger than that of the Heat_flux_z_bc_rect_norm=Z # normal to boundary mantle. The plate can be maintained ﬁxed or let to move Heat_flux_z_bc_rect_icpt=1 # normal-axis intercept Heat_flux_z_bc_rect_aa1=0 # lateral coord. extent around. You may want to experiment with both options. Heat_flux_z_bc_rect_aa2=6 Heat_flux_z_bc_rect_hw=0 # smoothing half-width Heat_flux_z_bc_rect_mag=0 # magnitude of bc

117 For a ﬁxed plate, use the following (change X1 and X2 by the horizontal coordinates for your plate):

Velocity_z_bc_rect=1 # nb of bc (surfaces) Velocity_z_bc_rect_norm=X # normal to bc pla Velocity_z_bc_rect_icpt=0 # coord. of the bc plane Velocity_z_bc_rect_aa1=X1 # lateral coordinate Velocity_z_bc_rect_aa2=X2 # lateral coordinate Velocity_z_bc_rect_hw=0 # smoothing half-width Velocity_z_bc_rect_mag=0 # magnitude of bc

6/ How plates inﬂuence the convection, the average temperature and the temperature distribution in the mantle?

118 SECTION 4 The geodyamics of mantle melting

Basaltic volcanism is the result of partial melting of the Earth’s compression melting in small-scale convection cells along thick mantle (Fig. 1A). Volcanic activity in arcs and along mid-ocean lithospheric keels has been proposed to explain minor intraplate ridges and continental rifts is related to plate tectonics, through volcanism (Davies and Rawlinson, 2014). decompression melting of the shallow sublithospheric mantle in One of the main differences between plate-tectonics–related regions of lithospheric extension and thinning, and/or hydra- melting of the sublithospheric mantle and partial melting in hot tion of the mantle wedge above subducting slabs. In contrast, upwelling mantle material is the pressure (P) and temperature intraplate volcanism, responsible for thick continental ﬂood ba- (T) at which the adiabat (i.e., geotherm in the convective man- salts, has usually been linked to upwelling and decompression tle) intersects the mantle solidus (Fig. 1B). These cannot be ob- melting of large volumes of deep, hot mantle (e.g., mantle tained directly from the composition of basalts because they are plume; Morgan, 1983; Richards et al., 1989). More recently, de- affected by fractional crystallization and interaction with sur-

119 rounding rocks, as well as the progressive evolution of the composi- A B Temperature (ºC) 1 2 1200 1400 1600 1800 2000 0 0 tion of the source. The geochemistry of the ﬁrst-formed batch of Continental crust Liquidus 50

Lithospheric mantle a’ 15 Solidus MgO % melt as fertile mantle crosses its solidus (i.e., the primary melt) is a Melt adiabat 100 50

function of the mantle potential temperature, Tp , as shown by for- Depth (km) 150 Conductive a Tectonic thinning and b’ ward thermodynamic modeling (Asimow, 1999; Asimow et al., 200 exhumation of 100 geotherms sublithospheric mantle Deep mantle 2

250 Adiabat upwelling 2001; Herzberg and O’Hara, 2002; Herzberg, 2004). Tp is the tem- 1 Asthenosphere 150 3

perature measured at the Earth’s surface from extrapolation of the 25 er contine p n 30 Su t b Depth (km) mantle adiabat (McKenzie and Bickle, 1988), a convenient way to 3 200 Sublithospheric Mantle 35 mantle warming Mantle exhumation express the intersection temperature of adiabat and solidus. The warming ive mantle ect Adiabat Deep nv 250 o composition of the elusive primary melt of a volcanic province de- C mantle upwelling rives from processing the most magnesian liquids (i.e., parental 300 melts) inferred from a suite of basalts (Herzberg et al., 2007). Com- parison of the calculated and modelled primary melt constrains the Fig. 1. A: Geodynamic settings of mantle melting. B: Phase diagram, contoured for MgO, of mantle melting. 1—lithospheric thinning; 2— mantle upwelling; 3—mantle Tp from which the geodynamic setting of the partial melting can be warming. During (1) and (2), the convective mantle is exhumed adiabatically (blue inferred. By targeting primary melts, petrologists mitigate geo- and red dashed lines in B) intersecting the solidus at ~70 km and 180 km, respec- chemical alteration of parental magmas through fractional crystalli- tively, with Tp (1330 ºC and 1620 ºC) shown by the blue and red dots at Earth’s sur- zation and interaction with surrounding rocks during melt ascent face. Following mantle warming (3), the adiabat (black dashed line) intersects the (e.g., Herzberg et al., 2007). Keeping in mind the many pitfalls solidus at ~120 km depth, with Tp (~1470 °C) shown by the gray dot. Primary melts (Herzberg et al., 2007; Herzberg, 2011), data from present continen- are produced at solidus temperatures (blue, gray, and red stars). Should these melts be extracted as soon as they are produced, they would follow their respective melt tal rifting and spreading ridges derive a Tp of less than 1400 °C (Fal- adiabat (solid blue, gray, and red arrows). In the case of continental rifting leading to loon et al., 2007; Herzberg et al., 2007). Hole (2015) shows that data a mid-ocean ridge, melt is usually extracted at melt fraction of <1% (i.e., fractional from plume-related ﬂood basalts point to a Tp of more than 1500 melting). As latent heat escapes with the melt, the source rock cools and follows the °C, in agreement with, for example, Thompson and Gibson (2000), solidus during exhumation (path a-a’). As the source becomes more residual, it pro- Herzberg and O’Hara (2002), and Herzberg et al. (2007). The range duces parental melts with a range of compositions (empty blue arrows). In plumes, melt continuously reacts with the residue, with which it remains in equilibrium. As of adiabats during tectonic extension and exhumation of the shal- latent heat remains with the melt in the source, the upwelling mantle keeps follow- low mantle (blue region in Fig. 1B) and during deep mantle upwel- ing the adiabat (path b-b’). As the plume head slows down, melt segregates toward ling (red region in Fig. 1B), do not overlap. The concept of Tp can the top of the partially melted column, before escaping to the surface (i.e., batch or thus be used to constrain the geodynamic setting of mantle melt- equilibrium melting) following the parental melt adiabat (empty red arrow). The ing. Let’s now consider the origin of the ca. 200 Ma Central Atlan- pressure at which melt is extracted is the ﬁnal melting pressure. Both fractional melt- tic magmatic province (CAMP). ing and equilibrium melting can occur in succession depending on strain rate, perme- ability, and the relative upwelling velocities of the residue and melt.

120 The CAMP extends over Africa, South America, Europe, and North pends on plate tectonics: continents amalgamate to create supercon- America, covering ~107 km2 (Marzoli et al., 1999). It post-dates by tinents through the Wilson cycle, then break into smaller continen- 50 m.y. the ﬁnal stages of Pangea’s assembly. The peak igneous ac- tal plates. Two- and three-dimensional numerical studies show that tivity at 199–200 Ma pre-dates by ~10 m.y. the initial opening of the the Tp of the convective mantle can increase 10% to 15% (~130–200 central Atlantic Ocean (Marzoli et al., 1999; Sahabi et al., 2004). The °C) as plates aggregate into a supercontinent covering 20% to 35% attribution of the CAMP to a mantle plume-head impinging the of Earth’s surface (Coltice et al., 2009), thus the Tp of partial melt- base of the lithosphere (Hill, 1991; Wilson, 1997; Leitch et al., 1998; ing following the formation of supercontinents is between that of Courtillot et al., 1999; Janney and Castillo, 2001; Ernst and Buchan, plate tectonic processes (<1400 ºC) and deep mantle upwelling 2002) has been challenged on the basis that chemical and isotopic (>1500 ºC). The potential temperature determined by Hole from compositions of basalts point to shallow-mantle sources, including the CAMP’s primary magmas using the most advanced model lithospheric ones, enriched by earlier subduction (Bertrand et al., (PRIMELT2), combined with the observation that magmatism fol- 1982; Bertrand, 1991; Puffer, 2001; Deckart et al., 2005; Verati et al., lowed supercontinent aggregation, conﬁrms that the CAMP was 2005). However, the origin of the CAMP is still strongly debated be- the result of large-scale mantle melting following the aggregation cause basaltic magmas change on their way to the surface. of the last supercontinent (Coltice et al., 2009).

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123 SECTION 5 Modelling mantle melting Summary

1. Solidus/liquidus 2. Basalts MgO content 3. Melting during fast decompression 4. Melting during slow decompression 5. Exercise: Melting in a mantle plume

The mantle solidus and liquidus are relatively well constrained. For instance, according to Katz et al., (2003): Tsol(z): P<15GPa = 1358.7 + 132.899 (ρ.g.z)/(1e9)- 5.104 ((ρ.g.z)/(1e9))2 Tsol(z): P≥15GPa = 1510.76 + 46.27 (ρ.g.z)/(1e9)- 0.8036 ((ρ.g.z)/(1e9))2 Tliq(z): P<15GPa = 2053 + 45 (ρ.g.z)/(1e9)- 2.00 ((ρ.g.z)/(1e9))2 Tliq(z): P≥15GPa = 1470.3025 + 55.53 (ρ.g.z)/(1e9)- 0.9084 ((ρ.g.z)/(1e9))2

Melt fraction F is calculated using the concept of super-solidus (Tss) and eq. 21 of McKenzie and Bickle (1988): 2 3 F(Tss) = 0.3936 + 0.8936 Tss + 0.4256 Tss + 2.988 Tss Basalts can be classiﬁed based on their MgO content: Komatiite (MgO>18%), komatiitic basalts (12%

124 Mantle melting during fast exhumation: The graph below shows a phase diagram with melt fraction mapped between the solidus and the liquidus. This graph describes what happened during a phase of fast homogeneous continental rifting back in the Archean when the mantle was ~200ºC hotter than present. The continental lithosphere (pale green) is thinned rapidly (225 km to 110 km), which triggers the decompression of the mantle below the continent. When extension and thinning is rapid each point on the geotherm follows the adiabatic gradient (i.e. no conductive cooling). As the geotherm becomes warmer, it intersects the solidus at two locations Temperature (K) (A0’ and A3’) which deﬁne the top and base of a par- 1400 1600 1800 2000 2200 0

Liquidus tially molten mantle column. The base of the column re- Solidus Tpot = 1820 K 0 Melt adiabat mains close to 150 km throughout the volcanic history. Geotherm (t 20 40 60 80 Because extension is rapid, the partially molten column 50 1) A ’ (i.e. solid matrix together with the melt) follows also 3 A2’ Geotherm (t 2 the adiabat since there is no melt extraction, and there-

100 0) Pressure (GPa) fore no loss of heat. After ~100% extension, points on LAB at t1 the initial geotherm located at A0 to A3 have moved up- A1’ A3 melt % ward to location A0’ to A3’. The partially molten man- A0’ 4 150 tle column consists of partially molten fertile sub- A2 Exhumation lithospheric mantle (from ~150 to ~110 km), and par- SCLM of the LAB Depth (km) tially molten SCLM (from ~110 to ~55 km). Because the 200 6 melt accumulates in the column, the buoyancy of the A LAB at t molten column increases and its viscosity decreases, un- 1 Adiabatic path 0 After extension til melt escapes to the surface to form a basaltic crust. 250 Fertile The maximum thickness of this crust is given by the Sub-Lithospheric mantle 8 area between the solidus and the geotherm (purple tri- (SLM) 300 Before extension angle), here it is ~ 16 km (i.e. ~ 0.5 x (depth(A0’) - A0 depth(A3’)) * maximum melt fraction).

125 Temperature (K) The phase diagram on the right shows the MgO content between the solidus and 1400 1600 1800 2000 2200 the liquidus, according to Herzberg and Gazel (2009). The MgO content of melts ex- Solidus Liquidus Melt adiabat tracted at various locations in the partially molten column ranges from 9 to 21 %. Geotherm (t 10 20 30 1)

A ’ Mantle melting during slow exhumation: When extension is slow (graphs below), A3’ 2

MgO % the solid matrix ﬂows upwards very slowly, and the melt - whose buoyancy in- 100 A1’ creases as the pressure at which it is produced decreases - ﬂows faster through the porous source and ponds at the top of the partially molten column, before escap- 150 A0’ ing to the surface via diking. In the molten column, melt forms an interconnected

network for a critical melt fraction as low as 0.1 %, limiting the amount of melt in Depth (km) 200 the column to less than 1%. While the ﬂowing melt follows the melt adiabat, the depleted residue follows the solidus as latent heat escapes with the melt. The total 250 Temperature (K) Temperature (K) amount of 1400 1600 1800 2000 2200 1400 1600 1800 2000 2200

Melt adiabat Liquidus Solidus Liquidus Solidus latent heat 300 Geotherm (t 40 60 80 Geotherm (t 10 20 30 consumed 1) 20 1) 50 Melt adiabat A3’ A3’ depends only on the volume of rock advected A2’ MgO % above the solidus, not on the history of melt extrac- 100 100 LAB at t1 A1’ tion. Hence, the melt productivity, and therefore melt % A0’ the depletion of the source, is not signiﬁcantly af- 150 150 A0’ A2 Exhumation of the LAB fected by the residence time of the melt in the SCLM Depth (km) Depth (km) 200 200 source before its extraction. Assuming 100% extension, the thickness of the basaltic crust also A LAB at t 1 Adiabatic path 0 250 250 reaches ~16 km. The MgO content of the melt var- Fertile Sub-Lithospheric mantle (SLM) ies with the pressure at which it is produced and 300 300 varies between 9 and 17%. A0

126 The small fraction of melt remaining in the source has only a modest impact on the buoyancy of the partially molten region. Its impact on the rheology is, however, more signiﬁcant. On one hand, the viscosity of the partially molten mantle strongly decreases because melt lubricates grain boundaries. On the other hand, partial melting drains water out of the solid matrix and increases the olivine content. Although the impact of dehydration on the viscosity may not be as signiﬁ- cant as previously thought (e.g., Fei et al., 2013, Nature 498), both processes should contribute to increase the viscosity of the depleted residue once its temperature drops below the solidus.

Exercise - Melting in a mantle plume: We model the rise of a mantle plume in a box 700 km deep and 4200 km wide (i.e. simulating the upper mantle). The plume is modelled using a positive thermal anomaly 300ºC higher than the surrounding mantle (1347ºC). We take advantage of the symmetry of the problem and embed the anomaly on the lower left corner of the box. The mantle is modelled as a visco-plastic material with temperature, stress, strain-rate and melt dependent viscosity. The viscous rheology of the mantle is based on dry olivine (Brace & Kohlstedt, 1980). Its plastic rheology is approximated with a Coulomb failure criterion with cohesion 40 MPa, a coefﬁcient of friction (μref ) 0.268, and a weakening factor dependent on the accumulated plastic strain (εp). The coefﬁcient of friction (μ) evolves as a function of the accumulated plastic strain as:

0.25 for εp < 0.15: μ = μref ×(1-(1-0.0373)×( εp /0.15) ) for εp > 0.15: μ = μref × 0.0373

This weakening leads to a strong strain localisation simulating faults with nominal viscosity ~25 times weaker than that of the surrounding rocks when plastic strain reaches 15%. This weakening, which is a key factor for the operation of plate tectonics on present Earth, simulates the formation of phyllosilicates (serpentine, talc, micas) during the strain-induced hydra- tion of mantle rocks and basalts. For semi-brittle deformation independent of pressure, we impose an upper limiting yield stress of 300 MPa (Ord & Hobbs, 1980; Escartin & Hirth, 1997).

The top 15 km of the model consists of a thick layer of basalts, which simulates the oceanic crust. Because these basalts are emplaced below sea level, they are strongly hydrothermally altered. For simplicity, we assume that this layer of basalt has

127 a nominal viscosity 1000 times weaker that the underlying mantle, a weak cohesion of 1 MPa, a reference coefﬁcient of friction of 0.134 and the same weakening properties as the rest of the mantle.

In our modelling of a rising plume, we assume that melt in the partially molten column is extracted when the melt fraction reaches 1%. In this case, latent heat escapes with the melt, and the ascending depleted residue follows the solidus closely. We assume the fusion entropy to be 400 J kg-1 K-1.

The small fraction of melt (<1%) has only a modest impact on the buoyancy of the partially molten region. However, the density of the residue decreases as it becomes more depleted. We assume a maximum density decrease of 1.5% upon full depletion. Because even a small fraction of melt lubricates grain boundaries, it affects the viscosity of the partially molten column. Hence, we impose a linear viscosity drop to a maximum of one order of magnitude when the melt fraction reaches 1%. On the other hand, partial melting drains water out of the solid matrix and reduces the number of phases. Al- though the impact of dehydration on the viscosity may not be as signiﬁcant as previously thought (Fei et al., 2013), both processes should contribute to increase the viscosity of the depleted residue once its temperature drops below the solidus. In our experiment we impose an increase in viscosity proportional to depletion, assuming a viscosity increase of two or- ders of magnitude for 100% depletion.

The geotherm is linear in the lithosphere (T(0km)=293 K, T(150km)=1620K). In the convective mantle, the geotherm follows the adiabatic gradient (~0.3 ºC per km).

Run the model (A_Plume100.input), analyze the result, and summarize your findings into a 4-page report (less than 3-4 Mbyte). For this you may want to consider the information stored in the following files (all readable with a simple text edi- tor): .node_data, .profiles, .samples and .timelogs.

128 References Brace, W. F., Kohlstedt, D. L. Limits on lithospheric stress imposed by laboratory experiments. Journal of Geophysical Research 85, 6248-6252 (1980).

Escartin, J. & Hirth, G. Nondilatant brittle deformation of serpen- tinites: Implications for Mohr-Coulomb theory and the strength of faults. Journal of Geophysical Research 102, 2897-2913 (1997).

Fei, H., Wiedenbeck, M., Yamazaki, D. & Katsura, T. Small effect of water on upper-mantle rheology based on silicon self-diffusion co- efﬁcients. Nature 498, 213-216 (2013).

Herzberg, C. & Gazel, E. Petrological evidence for secular cooling in mantle plumes. Nature 458, 619-623 (2009).

Katz, R. F., Spiegelman, M. & Langmuir, C. H. A new parameteriza- tion of hydrous mantle melting. Geochemistry, Geophysics Geosys- tems 4 , doi:10.1029/2002GC000433 (2003).

McKenzie, D. & Bickle, M. J. The volume and composition of melt generated by extension on the lithosphere. Journal of Petrology 29, 625-629 (1988).

Ord, A., Hobbs, B. E. The strength of the continental crust, detachment zones and the development of plastic instabilities. Tectono- physics 158, 269- 289 (1989).

129 Temperature (K) Temperature (K) 1400 1600 1800 2000 2200 1400 1600 1800 2000 2200 0 0

Solidus Liquidus Solidus Liquidus

20 40 60 80 10 20 30 50 2 2 Pressure (GPa) melt % Pressure (GPa) MgO % 100 100 4 4

150 150 Depth (km) 6 Depth (km) 200 6 200

250 8 250 8

Solidus and liquidus according to Katz et al., Solidus and liquidus according to Katz et al., 2003. Melt fraction from McKenzie & Bickle 2003. MgO content from Herzberg & Gazel 300 (1988) (2009) 300

130