Introduction to 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 equilib- rium, 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 enter- ing 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 incre- ment 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 .