Notes on Continental Lithosphere

Transcription

1 Notes on Continental Lithosphere Definitions: First, crust. This is easier than most, as in most places it is the same. Seismologists define Moho from steep P wavespeed gradient near 7.5 km/s (this produces a refracted arrival, termed P n, that was basis for Moho in the first place). Petrologists usually like crust to be material derived from the mantle by melting, where the residuum is in the mantle. Occasionally you can have such material be eclogitic (due to intracrustal differentiation), which is seismologically mantle but petrologically crust. Otherwise both definitions work well together. In contrast, lithosphere is really vague. Petrologically: Upper mantle isolated from convective processes (so having distinct trace element and isotope signatures). Seismologically: Material above low-velocity zone (depth where velocities become lower with depth). LVZ is well defined in oceans, but difficult (often) in continents. Usually best developed as an S-wave LVZ. This is often best measured with surface waves. Thermally: Thermal boundary layer is sometimes called lithosphere. Because the thermal boundary layer is not convecting, this often comes close to the petrologic definition if maintained over a long time. It is defined as where you go from a conductive to a convective (adiabatic) geotherm. Mechanically. This is actually where the definition started, but even it is somewhat vague. Lithosphere has higher stresses than asthenosphere below; it is in this manner a viscosity contrast. One definition is that the lithosphere can transmit (shear) stresses over geologic time. Thus the depth of compensation for isostasy is the asthenosphere through this definition. An extreme version of this is the thickness of an elastic plate, which by definition retains stresses over an infinitely long time. Such thicknesses are typically tens to maybe just over 100 km. Most workers do not take such thicknesses literally. Basic observations: Lang will talk about petrologic constraints. Seismologically, we rely mostly on surface waves. This is largely because surface waves have a strong dependence on depth, as was illustrated in class. It is also because boday waves generated at the surface will not yield a refraction from a low-velocity zone; they might yield a reflection (depending on how abrupt the velocity reversal is) and they will show a shadow zone suh as we discussed from the outer core. Neither of these observations yields much information on the velocities within the low velocity zone, particularly for the continents, where lateral variations in structure can complicate the interpretation of these phenomena. Generally, the low-velocity zone is found about km down under oceanic crust and 200 km or more under cratons, with continental margins having a variety of possible thicknesses. Thermal lithosphere A key here is, what is the thermal structure of the lithosphere? We d like to know where the geotherm switches from the more conductive shallow geotherm to the mantle adiabatic geotherm. As we have seen, the changes in the thermal structure of the lithosphere are

2 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.2 responsible for the topography of most of the ocean basins. Is there something equivalent in continents? To address these, we consider heat in continental lithosphere Our observations are directly made only at the surface. The conductive heat flow q is q = k dt, where T is the temperature with depth and k is the coefficient of thermal conductivity. dz For typical crustal rocks, k is 2-3 W m -1 C -1. In most areas dt/dz = C/km, yielding a heat flow of mw m -2. Another unit currently falling out of favor is the heat flow unit (HFU): 1 HFU = mw m -2. The continental average heat flow is about 65±1.6 mw m 2 ; for oceans it is 101±2.2 mw m -2 (from Turcotte and Schubert s 2 nd edition). Actual measurements of the heat flow are usually made in boreholes. Difficulties that have to be overcome include thermal transients associated with drilling the hole, hydrological effects (hot water bringing heat in or cold water taking heat out), and topographic effects. Most of the

3 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.3 practitioners of heat flow are well aware of these problems and usually will have avoided holes that will not yield reasonably representative values, but it is well to be aware of them as occasionally maps of thermal gradients are presented that include a lot of bad holes. In the ocean basins, where the concentration of radioactive elements is not so great and not very variable, we could ignore it in plotting heat flow vs. age. On continents, which, as we have seen, are global repositories for things like potassium, thorium, and uranium, we cannot. Heat in the Earth comes both from heat being transferred out of the mantle (the end result of both gravitational energy from formation and differentiation of the core and radioactive decay in the Earth's interior) and from radioactive decay in the crust. Locally there can be heat sinks or sources from chemical reactions and shear heating; in a global sense, these are unimportant (see sections and of Stüwe for details) but can be important in the vicinity of magmas and large shear zones. Because most radioisotopes are concentrated in the crust, this term needs to be considered before we can infer the thermal structure of the deeper levels of the lithosphere. It turns out, somewhat surprisingly, that in many areas the surface heat flow is linearly related to the heat production of the surface rocks (measured as a volumetric pruduction in µw m -3 or by mass as µw kg -1 ). The Sierra Nevada turns out to be a classic example: The linear fit allows us to remove the effect of heat production in the crust by seeing where the line hits the axis of 0 heat production; this value is called the reduced heat flow q r. This is frequently interpreted to be the heat coming from the mantle, but this is in fact the very uppermost bound on what can be coming from the mantle. First, a constant contribution from radioactivity in the lower crust is likely present. Second, the lateral variations in radioactivity will cause heat to flow laterally (Jaupart, C., Horizontal heattransfer due to radioactivity contrasts - causes and consequences of the linear heat-flow relation, Geophysical Journal of the Royal Astronomical Society 75 (2): ). We shall return to this point after considering the onedimensional case. The very simplest explanation for the linear relationship of heat flow to heat production is that the radiogenic isotopes are in a layer of some thickness D. In this case, q o = q r + DA, where

4 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.4 A is the volumetric heat production. The slope of the line above then determines the thickness of the slab. An alternative that will produce the same result is if the heat production decreases exponentially with depth (discussed in Saltus and Lachenbruch's Sierra Nevada paper and in Turcotte and Schubert, pp ). While the Sierra and many parts of the eastern U.S. have the linear behavior with heat production, other areas do not, including the Rocky Mtns. and the Basin and Range. The geotherm in this case requires a new term. If we are at steady state, so there is no change in temperature, then the heat flow out of a volume of rock has to equal that in plus the heat produced in the rock. If heat is flowing upward along the z axis and our volume is a slab of thickness dz with a heat production A (expressed as W m -3 ), then we find q( z dz) = q( z) + A dz dq dz = A (1) We have made a mild change in definitions so that q = k dt dz positive up, so for z positive down but heat flow dq dz = d dz k dt = k d 2 T = A (2) 2 dz dz Over a slab where A and k are constant, we can simply integrate this to get k dt dz = Az + c 1 (3) This is the heat flow, and if z= 0 is the surface, then c 1 = q o, the observed surface heat flow. We integrate again to get (within the radioactive body) T = A 2k z2 + q o k z + T s (4) where the second constant of integration falls out easily as the temperature at the surface. This is a very helpful equation, though, as we shall see, it has had some misleading implications. We can rewrite this in terms of the heat flow entering from below the slab, as we noted that q 0 = q r + DA: T = A 2k z ( 2D z ) + q r k z + T s = A 2k D2 + q r k z + T s z < D z > D (4a) Note that the two righthand terms are the linear conductive geotherm in the crust without radioactivity. Thus this would require the temperatures to be hotter beneath the radioactive material than to the sides.

5 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.5 Although useful, the derivation above has a problem encountered in many places, such as the Sierra, is that the depth of erosion of plutons is variable, suggesting that the thickness D should vary, but the heat flows still plot on the nice regression line. Although there are a number of possible solutions, an elegant one suggested by Art Lachenbruch some years ago is an exponential decay of heat, A = A 0 e z / h r, where h r is a length scale of decay of radioactive heat production with depth. We can redo the integral from (2) to (3): q = k dt dz = h r A + c 1 (5) In this case, we might specify the heat flow at great depth be a constraint, which we will designate q m, and so c 1 = q m. Thus we see that q will depend linearly on A independent of the depth z, satisfying our observations in some areas. We also get (once again) that the heat flow at the surface when A equals zero is the heat flow at depth. Another integration yields T = h 2 r k A 0e z / h r + q m k z + c 2 = h 2 r k A 0 1 e z / h ( r ) + q m k z + T s = T s + q m k z + ( q 0 q m )h r k 1 e z / h r ( ) (6) This alternative geotherm isn t very different from what we would get with a slab, but the geotherms will diverge with greater depth such that an exponential decay will have higher temperatures than the slab below the slab thickness. Neither a slab of constant heat production nor an exponential decay of heat production is consistent with what we observe in the field (usually it seems to be somewhere in between). These two models provide us some bounds for the shallow subsurface and for greater depths under certain circumstances. Real heat flow from the mantle We return to what the real deep thermal structure looks like. This can most easily be seen by calculating one-dimensional geotherms from the surface down assuming a constant deep heat flow (q r ) but differing heat productions in a slab using equation (4):

6 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.6 Sierran geotherms -2 0 A = 0.3 A = 4 A = 2 A = 0.3, q r =2q r (today) A = 4, q r =2q r (today) A = 2, q r =2q r (today) 2 Depth (km) Temperature Above Surface Temp ( C) As you go down from a fixed surface temperature, the geotherms diverge, and while they bend back to being parallel as you emerge from the bottom of the radioactive part of the crust (see eqn 4a), the temperatures would seem to be higher under the more radioactive crust. At first blush, this would seem to require the lithosphere to vary in thickness with variations in heat production, which is really rather ridiculous unless the length scales of radioactive heat production are on the order of the thickness of the lithosphere. For much shorter variations (which is true in the Sierra to a large degree) we suspect that the heat from the mantle under high heat producing regions will instead flow into the colder regions under lower heat producing regions. This means that the actual heat being produced under a given measurement is actually lower for low heat producing areas and higher for high heat producing areas:

7 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.7 Thus the reduced heat flow is no better than the maximum heat flow from the mantle. Mike Sandiford has made some nice plots of this effect ( The summary is illustrated here:

8 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.8 Here the lateral length scale of variations in heat production is h x and the actual heat flow from depth is 20 mw m -3 (the dashed line is the relationship you would expect if there was no lateral conduction of heat). From this analysis, the thickness of the lithosphere calculated from a conductive geotherm assuming a heat flow equal to the reduced heat flow will be too thin. What you really need is the properly averaged heat production over the region (notice that the dashed lines cross the observed curves above at some point likely to represent the average heat production) with a proper (lower) mantle heat flow. Assumptions, assumptions, assumptions The discussion above includes a number of assumptions we should be very careful about. One is that the thermal conductivity k is constant with depth. This is not true in the mantle (there is a depth dependence) and it is not true for crustal rocks. Most notably, this is not true for shales, which tend to have a lower k than most other crustal rocks. This means that a pile of shale will act as a blanket, and temperatures will tend to be higher under them than other rocks. It turns out this is a big issue in interpreting fission track ages in the Rocky Mountains. Another huge assumption is that things are in equilibrium. You can go out of equilibrium in two easy ways: change the temperature or heat flow at the bottom of the lithosphere, or add or remove material from the top by sedimentation or erosion, from the middle through tectonism, or from the bottom through convective processes. The final big assumption is that heat is conducted. In fact, within the crust there are processes like hydrothermal systems, regional aquifers, and magmatic systems that all are

9 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.9 capable of moving heat through advection. We will not explicitly consider this issue here, but you should be aware that these can be largescale effects. For instance, low heat flow over much of the Colorado Plateau has been attributed to water flow transporting heat sideways in aquifers such as the Coconino sandstone; a similar anomaly is along the Snake River Plain. Also, we previously mentioned that heat flow measurements in oceans were highly scattered towards the ridges because of geothermal systems. Thermal disequilibrium The plot below from Saltus and Lachenbruch (1991) helps us understand this: The idea is that if a new fixed temperature is emplaced at some depth H, then the heat flow with time will change as shown by the " T at boundary" curve above. If instead of a fixed temperature we change the heat flux, then the surface heat flow will follow the " q at boundary"

10 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.10 curve, where the original heat flow is at the 0 position on the y-axis and the new steady-state is at 100. The two extra axes below are for emplacement of a body at 5 km depth and for the depth of a body emplaced 20 m.y. ago and producing the changes. Using the bottom curve, for instance, if we suspect that the surface heat flow is exactly halfway from old values to new, and we believe this was caused by a change in heat flow, the depth of that change would be about 40 km. Thus we can see that changes to the thermal structure at the base of the lithosphere take tens of millions of years to show up at the surface. Heat flow is in fact the only geophysical observable that is more a measure of past conditions than present ones. Mechanical lithosphere As we shall see, the definition of mechanical lithosphere is fraught with ambiguity, but let us start with a simple conceptualization and move forward from there. A useful approximation to the lithosphere is an elastic plate over a fluid substrate. There is a well developed theory from engineering for the behavior of plates, and Earth scientists have only had to modify it a little to produce something of use. The full derivation can be found in Chapter 3 of Turcotte and Schubert; we will begin with the general equation in two dimensions (x and z) that applies to plates D d4 w dx 4 + P d2 w dx 2 + ( ρ a ρ f )gw = q a ( x) (7) where D is the flexural rigidity, P is the force parallel to the plate, ρ a is the density of the underlying fluid, ρ w is the density of material filling in the top of the plate (water if in the oceans, air or sediments on land, depending on the situation), w is the deflection downwards of the plate, and q a is the load applied to the plate. Basically, when you plop a load on a plate, it doesn t go into local isostatic equilibrium, it distributed the load by bending. The broader it distributed the load, the stronger the plate. If the plate is of a constant Young s modulus E, Poisson s ratio ν, and a thickness T e, then ET e 3 D 12 1 ν 2 ( ) (8) Periodic topography (Turcotte and Schubert, sec. 3-14) The simplest place to start is with a solution that requires little effort and yet gives us our first real insights into the scales where isostasy takes over from rigidity. Consider some region where topography is sinusoidal with x such that the elevation e(x) = e 0 sin (2πx/λ) and does not vary with y. The load on the lithosphere, q a, is then the variation of weight that accompanies this deflection: q a (x) = ρ c ge 0 sin2π x λ (9) where ρ c is the density of the crust associated with the height variation. If we now assume there is no end load pressure P, we find in eqn. (7) that D d4 w dx 4 +(ρ a ρ c )gw = ρ c ge 0 sin2π x λ (10)

11 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.11 Note that we use a fill density of ρ c, which means that our applied load is added on a level upper surface and so is the final (observed) topography of the system. Given the periodic nature of the load, we would expect that the deflection of the lithosphere would also be periodic. Let us guess a solution of the form w(x) = w 0 sin2π x λ Putting this into equation 10 produces 4 2π Dw 0 sin2π x λ λ + (ρ ρ )gw sin2π x a c 0 λ = ρ ge sin2π x c 0 λ w 0 D 2π 4 + (ρ λ a ρ c )g = ρ c ge 0 (11) (12) w 0 = e 0 4 D 2π gρ c λ + ρ a ρ c 1 We notice two endmembers: one where the wavelength is very small, one where it is very large. 1/4 D When λ is small (much smaller than 2π ), the first term in the denominator dominates gρ c and w 0 approaches zero. This is the familiar case of the rigid earth not yielding beneath a landfill; the wavelength of the landfill turns out to be much smaller than the flexural strength of the lithosphere. In the other extreme, as λ goes to infinity (at least much greater than 1/4 D 2π ) then the first term drops out and we are left with w 0 = e 0 ρ c /(ρ a - ρ c ), which is the gρ c equation for local, Airy isostatic support of mountains. (For references, you might note that the factor ρ c /(ρ a - ρ c ) is how much more the Moho is perturbed for a given topographic signal; it varies from about 4 to 8, so 1 km high mountains should have 4-8 km deeper Mohos than adjacent lowlands). For values of the wavelength in between, the topography is partially compensated; this is illustrated in Figure 3-26 of Turcotte and Schubert. The sinusoidal solution can be the basis for a Fourier expansion of the response to a load and is one potential strategy for solving more complex loads. The wavelength where support for topography transitions from rigid to isostatic depends on the flexural rigidity D of the plate, and determining this transition in the basis for so-called admittance analysis of gravity (a proxy for the deflection w of the Moho) against topography that has been very successful in revealing the rigidity of oceanic lithosphere. For the cases of greatest interest here, there are simpler solutions. Linear load on unbroken lithosphere (sec. 3-16) Things are somewhat more complicated when we make our load finite. Let us start with a simple load isolated along a line; let the magnitude of this force be V 0 at x = 0. Elsewhere the load is zero. We once again let the end load be negligible and wish to solve (7) everywhere but x = 0 and thus face

12 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.12 There is a general solution to this equation: D d4 w dx 4 +(ρ a ρ w )gw = 0 (13) w = e x /α c 1 cos x α + c 2 sin x α + e x/ α c 3 cos x α + c 4 sin x α (14) where the flexural parameter α is defined as 4D α = g(ρ a ρ w ) 1/4 (15) There is a natural symmetry about x = 0, so we can limit our efforts for x > 0. For distances far from the load, we expect the deflection to go to 0; this causes the positive exponent terms to be dropped (c 1 and c 2 = 0). Because our plate is continuous and symmetric, the slope = dw/dx must be zero at x = 0. This in turn forces c 3 = c 4. The last constant is related to the magnitude of our load V 0. We can show that V = dm/dx, which is Eh 3 V = 12 1 ν 2 ( ) d 3 w dx 3 = D d3 w dx 3 (16) Since our load is pointing down, and since we are solving the half of this for x > 0, we find that We can now return to (14) and find our solution to be V 0 2 = 4Dc 3 α 3 (17) x /α w = w 0 e cos x α + sin x α where our maximum deflection w 0 is at x = 0 and is w 0 = V 0α 3 8D There are several characteristics of this solution worth examination. First is to note that α does not depend on V 0. Thus an increase in the load does not alter the shape of the solution, only its amplitude. This is particularly noteworthy because the solution varies about w = 0: (18) (19)

13 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.13 Here the vertical axis is w/w 0 and the horizontal is x/α. The crossing of w = 0 at x = 0.75πα separates the depression from the forebulge. The position of this crossing is unaffected by changing w 0 (or V 0 ); thus the position of the forebulge is only affected by the position of the load and α, which depends on the physical characteristics of the lithosphere. Note that the forebulge is in general much smaller than the depression itself, only reaching a maximum amplitude just under 5% of the main depression. (Also note that α depends on the density contrast of the infilling material and the asthenosphere--what happens when a sediment starved basin is filled?). Flexure of a broken plate (sec. 3-17) The final instance we are interested in is if the plate is broken. Strictly speaking this simply doesn't happen--there is never an open edge to a lithospheric plate. But once again this kind of approximation gives us insight to situations where the strength of the lithosphere might vary dramatically where the load is applied. In this instance we no longer require that the slope of the plate be flat at x = 0; we replace this with the requirement that the moment be zero on the end of the plate (there is no source of moment in our initial analysis). This means that we require that the second derivative of w with x is 0 at x = 0. Returning to eqn (23) we now find that c 1, c 2, and c 4 must be zero. Repeating the exercise that went into equations (25) and (26), we find that Plugging back in, we find V 0 2 = 2Dc 3 α 3 (29) w = w 0 e x /α cos x α (30) where w 0 = V 0α 3 4D Plotting on the same horizontal scale as above (and with w/w 0 with the new w 0 ), we see several things. (31)

14 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.14 As before, the zero crossings and forebulge positions rely on α exclusively. The zero crossing has moved in considerably closer to the load for a given α, as has the forebulge. The amplitude of the forebulge is now nearly 7% of the maximum depression (which itself is twice the amplitude as for the unbroken plate). When trying to fit a given deflection profile using a broken plate will require about a 50% thicker effective elastic plate thickness than fitting the same profile with a continuous plate. The largest difference in the shapes of the plots is near the load, which is where the assumption of a load only at x = 0 breaks down. Additional variations are possible where a moment is placed on the end of the elastic plate; this is covered in detail in Turcotte and Schubert; it is most applicable when a large subsurface load (usually a subducted plate) is present. Use of this formulation predicts the observed topography at ocean trenches quite well, as well as the deflection of (unfaulted) continental sedimentary basins, as we shall discuss further. Application to lithospheric thickness Ref: Forsyth, D. W., Subsurface loading and estimates of the flexural rigidity of continental lithosphere, J. Geophys. Res., 90, 12,623-12,632, There are two basic classes of approaches to measuring the flexural rigidity of the lithosphere, one a spectral approach and the other a forward modeling approach. In the first, we use the sinusoidal solution from (11) and (12). The first attempt at this used what is generally termed an admittance calculation. This assumed that the topography at the surface was the load on the plate, and that the deflection of the plate could be estimated from the Bouguer gravity anomaly. In this case, if you spectrally divide the Bouguer anomaly by the topography, you will get an admittance Q(ω). As the wavelength gets longer, there should be perfect correlation of the two signals and Q goes to 1. When the topography is supported by the strength of the lithosphere with essentially no deflection, it drops to zero. How the shift occurs determines the elastic plate thickness. This approach worked well in the oceans, where most of the shorter wavelength topography is due to volcanic edifices. However, when applied to continents the result seemed silly: continental lithosphere had values of T e of about 5 to 10 km, much less than the thickness over which earthquakes are found to occur. The problem is that there are loads applied at depth that may not produce much topography. A second spectral approach developed by Forsyth allows loads to be both in the subsurface and the surface. In this case it is the coherence between the topography and the gravity that is exploited; the more coherent these two signals (which is a value of 1), the more the plate acts as though isostasy was local. The less coherent (a value of 0), the closer to a rigid

15 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.15 plate. Again, the transition determines the elastic plate thickness, but the problem of determining the load location is mitigated. This approach yields numbers more in line with expectations. Forward modeling requires that a load be applied in some known manner and then examining the resulting flexure. These often are of the form of the line load solutions (above). It has the advantage of avoiding some assumptions in all the spectral techniques, but can only be applied in some areas. Foredeeps, subduction zones, and moats around island chains are all good examples where it has been applied. Strength of the lithosphere References: Brace and Kohlstedt, Limits on lithospheric stress imposed by laboratory experiments, J. Geophys. Res., 85, , Turcotte and Schubert, Geodynamics, chapter 7 (and 8 to some degree). Rheological layering of the lithosphere. There appear to be two fundamental modes of deformation of rocks: brittle failure and ductile flow. The former is related to earthquakes, which represent such failure; the latter should not generate earthquakes. The boundary between the two regimes is termed the brittle-ductile transition. Although it is often represented as a major discontinuity within the Earth, it is in fact most probably a broad zone within which a number of different mechanisms of rock deformation occur. Somewhere within this zone should be the base of that part of the lithosphere that generates earthquakes (the seismogenic layer). Our understanding of deformation within the Earth is largely based upon laboratory experiments and, to a lesser degree, observations of naturally deformed rocks. Lab experiments are necessarily over a much shorter time than actual rock deformation and are of a smaller scale; these substantial differences suggest that some caution must be exercised when employing these observations. (One possible example of the lab being too small has been suggested for the frictional behavior of faults during an earthquake. A theoretical mode of motion on a fault would cause the effective frictional resistance of the fault to go to zero during rupture. This mode is observed in experiments with foam rubber blocks but not with rocks. It turns out that to look for this kind of deformation in rocks would require an apparatus with blocks of rock several meters on a side, an experiment not yet conducted). Brittle upper crust An examination of rock properties reveals a large variation in rock strength with lithology. This would suggest that lithology would be the most important factor in determining the strength of rocks. However, experiments have revealed that this variation is almost entirely restricted to the effort to break a rock; once broken, the frictional behavior of a fault remains pretty uniform from lithology to lithology. What is more, this behavior depends almost entirely on the effective pressure on the fault, with little dependence on temperature or strain rate. This observation is termed Byerlee's Law and is expressed as a relationship of the shear stress and effective normal stress on a fault. This can be reformed as a relationship between the principal stresses: σ 1 5σ 3 σ 1 3.1σ σ 3 <110 MPa σ 3 >110 MPa (1)

16 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.16 These stresses are effective stresses, meaning that the pore pressure has been subtracted from them (σ n = σ n P p, where P p is the pore pressure and n is between 1 and 3). Eqn. 1 can be rewritten for compression (where σ 3 is the vertical stress, ρgz) as σ 1 σ 3 4 ( ρ λρ w )gz ( ρ λρ w )gz <110 MPa σ 1 σ 3 2.1( ρ λρ w )gz ( ρ λρ w )gz >110 MPa where λ is the fraction of hydrostatic pore pressure and ρ w is the density of the pore fluid. A similar manipulation can be done for the extensional case by setting σ 1 to the weight of the rock above. This makes pore pressure one large uncertainty in determining the strength of the brittle part of the lithosphere. Observations down to a couple of kilometers depth in continents suggest that a pore pressure between 0 (dry) and hydrostatic (P p = ρ w gz) are probably about correct for much continental crust. If you recall, there is some evidence in accretionary wedges and possibly foldand-thrust belts for higher pore pressures in these environments. Ductile lower lithosphere At higher pressures, rocks begin to flow ductilely. In this environment, there is no longer a maximum strength per se. Instead, there is a certain stress that will cause flow at a certain rate, usually expressed as a strain rate ε. Drop below that stress, and deformation is at a slower rate. Increase the stress, and deformation will go faster. Unlike frictional slip, the relations here depend on the rock type involved, the temperature, and the stress difference σ 1 - σ 3. The general form of this is a power-law flow: ε = C( σ 1 σ 3 ) n e E a RT (4) For olivine (from Goetze, 1978), C is about 7 x 10 4, n is 3, E a (the activation energy) is 0.52 MJ/mol, and R (the ideal (or universal) gas constant) is J K -1 mole -1. T is in Kelvin. (For dry quartzite, E a is 0.19 MJ/mol, n is 3, and C is about 5 x 10-6 (not 10 6 as shown in Brace and Kohlstedt). For purposes of strength of the lithosphere, we may rewrite (4) in terms of a stress difference as a function of temperature: σ 1 σ 3 = ε ee a C If we assume a linear change of temperature with depth, T = A + bz, then we find a fairly rapid decrease in strength with depth for a given strain rate. (Goetze's work indicates a lower differential stress than this equation at high differential stresses; we will use this form for illustrative purposes, but you'd want to use proper values for any research). We can combine (2) and (5), taking the minimum of the two, to get a feel for the strength of the lithosphere: RT 1/n (2) (5)

17 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.17 0 Compressional strength 0 Compressional strength, varying strain rate depth, km /km, total strength N/m 20 /km, total strength N/m 25 /km, total strength N/m depth, km /s / / σ 1 - σ 3, MPa σ 1 - σ 3, MPa Note that these curves are for a pore pressure of zero (dry rock) in compression. The upper part of the lithosphere fails as a brittle/elastic medium, the lower part by creep, which fails aseismically. Thus the boundary between the two is often termed the brittle-ductile transition and is often considered the base of the seismogenic part of the crust. Note that this transition moves upward in response to higher temperatures. This variation in the brittle-ductile transition has been observed, at least crudely, around the globe in continental areas. The figure on the left shows the variation in strength for a strain rate of s -1 ; that on the right shows a variation for several strain rates with a fixed temperature increase of 20 C/km. Note that the change in strain rate over two orders of magnitude is fairly small compared to the change caused by temperature variations. The area under each curve is the total strength of the lithosphere, meaning that deviatoric horizontal forces equal to that area would have to be applied to get the entire lithosphere to strain at the rate assumed. The strengths for the three curves on the left are listed in the caption. The curves above are for a Compression, strain rate s -1, quartzite over olivene monolithologic lithosphere made 0 entirely of olivine, perhaps a good first approximation for the oceanic 20 lithosphere. We know that the continental lithosphere is stratified, 40 with more quartz-rich rocks above the Moho and ultramafic rocks below. 60 Thus we would like to see the effect of this stratification, if any. Using the 25 /km, total strength N/m 80 flow laws for quartzite above a depth 20 /km, total strength N/m 15 /km, total strength of 35 km and that for olivine below, 13 N/m we can redo the left figure above for a 100 stratified lithosphere (at right). Because quartzite is quite weak at much lower temperatures than olivine, the net effect is that lithosphere σ - σ, MPa 1 3 depth, km

18 Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics p.18 should be considerably weaker with quartz present in the crust than when it is not. Once again we can integrate the curves to get the gross strength of the lithosphere and we find it has decreased by 50-80% for the case above. It is important to remember that these are extrapolations of laboratory measurements; the constants used, not to mention the stress exponent n, are subject to considerable uncertainty. Additionally, the brittle-ductile transition itself probably deforms by neither of the mechanisms assumed above but instead by other modes of failure, making the lithosphere somewhat weaker at these depths. The general effects of temperature in weakening the lithosphere, in strain rate mildly strengthening the lithosphere, and in quartz weakening the lithosphere are probably solid results. Understanding these variations of strength allows us to consider the role of strength and force in continental deformation.