( ) (1) ρ c crustal density 2600 kg m -3 ρ w water density 1000 kg m -3. HEAT FLOW PARADOX (Copyright 2001, David T. Sandwell)

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1 1 HEAT FLOW PARADOX (Cpyright 2001, David T. Sandwell) (See Special Issue f J. Gephys. Res., v.85, 1980: A) Turctte, Tag, and Cper, A Steady-State mdel fr the distributin f stress and temperature n the San Andreas fault, p ; B) Lachenbruch and Sass, Heat flw and energetics f the San Andreas Fault Zne, p ; Schlz, C. H., The Mechanics f earthquakes and Faulting, Cambridge University Press, Cambridge England, 1990.) Paradx - The seismgenic zne extends frm the surface t a depth f abut 10 km. Accrding t Byerlee's law, the shear stress n the fault shuld be sme large fractin f the hydrstatic stress. ( ) (1) τ() z = f ρ ρ gz c w f static cefficient f frictin ~ 0.60 ρ c crustal density 2600 kg m -3 ρ w water density 1000 kg m -3 g acceleratin f gravity 9.8 m s -2 D depth f seismgenic zne 12 km This assumes that water perclates t 12 km depths t lwer frictin n the fault. We can cmpute the average shear stress n the fault. D 1 1 τ = f ( ρc ρw) gzdz = f ( ρc ρw) gd = 56 MPa (2) D 2 The bserved stress drp during an earthquake ranges frm 0.1 t 10 MPa with a typical value f 5 MPa which is abut 10 times smaller than the average stress frm Byerlee's Law. This implies that nly a fractin f the ttal stress is released during an earthquake. The average stress during the earthquake times the earthquake displacement prduces energy bth as seismic radiatin (small fractin) and as heat (large fractin). If this heat energy is averaged ver many earthquake cycles, then this average heat/area generated n the fault plane will appear as a heat flw anmaly n the surface having a similar heat/area as alng the fault. T calculate this heat anmaly fr a variety f frictinal heating mdels, first cnsider a line surce f heat.

2 z 2 y -V/2 V/2 Q(x,z) = Vτ(z)δ(x) x The differential equatin and bundary cnditins fr a unit-amplitude, line surce at depth -a is T = k Qxz (, ) = δ( x) δ( z+ a) (3) k Tx (, 0) = 0 lim Txz (, ) = 0 z lim Txz (, ) = 0 x where T is the temperature anmaly in K, k is the thermal cnductivity (3.3 Wm -1 K -1 ), and Q is the heat generatin in Wm -3. Nte this is the same differential equatin as equatin (5) f the last sectin. The nly difference is the surface bundary cnditin. The surface stress prblem has vanishing shear stress at the surface (i.e., vertical derivative f displacement is v is zer) s we intrduced a psitive image surce t frce the displacement field t be symmetric abut z = 0. In this heat flw case, we have vanishing temperature anmaly at the surface s we intrduce a negative line heat surce at z = a t frm an anti-symmetric temperature functin. The slutin t the full-space prblem is identical t equatin (14) f the previus sectin. [ ( ) ] Txz (, ) = 1 ln x + z+ a 2πk / 1 2 After including the image surce, the result is / / { [ ] [ + ( ) ] } Txz (, ) = 1 ln x + ( z+ a) ln x z a 2πk (4) (5)

3 3 Nte that this is similar t equatin (23) in the Turctte et al., [1980]. The quantity f interest is the surface heat flw versus distance frm the fault. / / { [ ] [ + ( ) ] } (6) δ qxz k T 1 δ (, ) = = ln x + ( z+ a) ln x z a δz 2π δz After a little algebra ne arrives at the heat flw. qxz (, ) = 1 ( z+ a) ( z a) 2π x + ( z+ a) x + z a ( ) (7) Thus the surface heat flw fr a line surce f unit strength at depth a is 1 qx ( )= π x a + a (8) Fr an arbitrary shear stress distributin with depth τ(z) the surface heat flw is z z qx ( ) V τ = () π x + z dz (9) Nw lets assume that the stress fllws equatin (1), Byerlees's law (i.e. high stress and high heat flw). Als allw hydrthermal circulatin t extend frm the surface t sme depth d which effectively remves all the heat prduced between the surface and that depth. The integratin is qx ( )= 2 f( ρc - ρw) gv z π x + z dz D d (10) This integral is dne with help frm the table f integrals. 2 x a bx dx x a 1 = - b b a bx dx + (11) + After sme algebra ne arrives at the fllwing analytic frmula fr the heat flw f( ρc - ρw) gv d D qx ( ) = ( D d) 1 1 xtan x tan + x x (12) π It is interesting t cmpare this heat flux t the heat flux at a mid-cean ridge fr the same ttal pening rate V (see figure n next page). The frmula is

4 4 qx ( ) kt ( T) 2πκx/ V (13) = ( ) m -1/2 Τ m mantle temperature 1600 K Τ surface temperature 273 K k thermal cnductivity Wm -1 K κ thermal diffusivity 8. x 10-7 m 2 s -1 Matlab Example The fllwing is a Matlab prgram simulates a high-stress fault (i.e., Byerlee's Law) extending t a depth f 12 km and sliding at a rate f 30 mm/yr. Tw cases are cnsidered; the first case (slid curve n next page) has hydrthermal heat remval extending t a depth f 1 km while the secnd (dtted curve) has heat remval t a depth f 5 km. These mdels are cmpared with the heat flw measurements acrss the San Andreas Fault [Lachenbruch and Sass, 1980]. It is clear that the shallw heat remval mdel is incnsistent with the data. Hwever, the deep heat remval mdel is nt precluded by the bservatins, especially if the backgrund level f the mdel heat flw is allwed t vary frm the spatial average. One argument against hydrthermal remval f heat is the absence f ht springs alng the fault with sufficient vigr t remve this heat. Hydrthermal circulatin is the dminant heat remval mechanism at the mid-cean ridges and hydrthermal vents are cmmn. Hwever, as shwn in the fllwing figure, heat generatin alng a strike-slip fault is 2-3 rders f magnitude less than a mid-cean ridge s it is nt clear that the same mechanism shuld perate at a fault. Even if heat lss is cncentrated in small areas it may be difficult t detect at the surface. % % prgram t calculate the surface heat flux due t frictinal heating n a % strike-slip fault D=12; d1=1; d5=5; rc=2600; rw=1000; g=9.8; V=.03/3.15e7; f=.60; q0=1.e6*f*(rc-rw)*g*v/pi; % % calculate the heat flw fr the tw mdels f shallw and deep heat remval % x=-60:.1:60; q1=q0*((d-d1)+x.*atan(d1./x)-x.*atan(d./x)); q5=q0*((d-d5)+x.*atan(d5./x)-x.*atan(d./x)); % plt the results plt(x,q1+73,x,q5+73,':');xlabel('distance (km)'); ylabel('heat flw (mwm-2)') axis([-120,120,0,167]);

5 heat flw (mwm ) distance (km) -2 heat flw (mwm ) mid-cean ridge strike-slip fault distance (km)

6 6 MOMENT PARADOX: Seismic Mment versus Tectnic Saturatin Mment The Mment Paradx described next is really part f the heat-flw paradx except it is expressed in a different way. As discussed in a previus lecture, and in Brace and Khlstedt [1980], measurements f stress difference in the uppermst crust t depths f several kilmeters are cnsistent with a yield strength mdel fllwing Byerlee's law. The static frictinal resistance t sliding is related t a cefficient f frictin f f abut 0.60 times the verburden pressure f ρgz. This leads t differential stress difference f 140 MPa at a depth f nly 10 km. We als fund that these high stresses are required t supprt the 5000 m elevatin f Tibet relative t India. This isstatic mdel is the minimum stress needed t supprt tpgraphy s it is clear that high stresses exist at shallw depths in the crust. Similarly, ne can calculate the bending mment needed t supprt the trench and uter rise tpgraphy at a subductin zne. The mment calculatin is mdel-independent [ McNutt and Menard, Cnstraints n yield strength in the ceanic lithsphere derived frm bservatins f flexure, Gephs. J. R. astr. Sc., 71, p , 1982;] Mx ( )/ L= ρ wx ( ) x x dx (1) x ( ) where M is the mment per unit length alng the trench, ρ is the mantle-t-seawater density cntrast, is the height f the uter rise abve the nrmal depth and x-x is the distance between the first zer crssing f the trench flexure prfile and sme pint ut n the uter rise. The integral cnverges because w(x) ges t zer expnentially with distance. Observed bending mments at uter rises vary frm 5 x N fr yung lithsphere (10 Ma) t 3 x at ld ceanic lithsphere (140 Ma) [Levitt and Sandwell, Lithspheric bending at subductin znes based n depth sundings and satellite gravity, J. Gephys. Res., 100, p , 1995]. Next we'll cmpare these numbers t typical seismic mments f large earthquakes using the Alaska 1964 and Landers 1992 event as examples. The Landers rupture was abut 70 km lng s we'll divide its mment by length fr the cmparisn with mdels. The results are prvided in the table belw.

7 7 tectnic example mment per length (N) uter rise flexure (10 Ma) 5 x N uter rise flexure (140 Ma) 3 x N Alaska flexure 1.2 x N Alaska 1964 earthquake, M x N = 8.2 x /750 km Landers 1992 earthquake, M x N gedetic/length 2.8 x N seismic/length Byerlee's criterin (0-12 km nly) 1.3 x N m This cmparisn highlights tw issues: first, the mment f the Alaska 1964 earthquake was sufficient t cause a cllapse f the uter rise(??); secnd, the seismic/gedetic mment f the Landers 1992 earthquake is times smaller than the mment estimated next using a the simple elastic dislcatin mdel where stress is limited nly by Byerlee's law. Seismic Mment Released During an Earthquake L y x D z The mment released during an earthquake can be estimated in tw ways, either by analysis f the seismic radiatin pattern r by the gedetic analysis f the gedetic grund mtin. They usually prvide similar values; althugh in the case f the Landers 1992 rupture, the seismic mment estimate is abut 2 times the gedetic mment estimate. The mment is defined as M = µ s LD y (2) f static cefficient f frictin ~ 0.60 ρ c crustal density 2600 kg m -3 ρ w water density 1000 kg m -3 g acceleratin f gravity 9.8 m s -2 µ shear mdulus 2.6 x Pa L length f rupture 70 km D depth f rupture 12 km

8 8 y rupture ffset 4.5 m V plate velcity m/yr t earthquake recurrence interval Nw we can d sme simple calculatins. First, a check n the gedetic mment 1.4 x N prvides a match t the published value. The recurrence interval f y /V = 300 years seems OK fr a fault ut in the Mjave Desert away frm the San Andreas Fault. S everything seems cnsistent. Next lets assume that the stress n the fault, as a functin f depth, matches Byerlee's law fr the case f hydrstatic pre pressure. We'll cmpare this saturatin mment and recurrence interval with the bservatins frm earthquakes. Tectnic Saturatin Mment Assume that the simple half-space slutin (develped abve) prvides the stress and strain field fr a fault lcked frm the surface t a depth D. Further, assume that the maximum stress that can be maintained n a fault is given by Byerlee's law [ ( c w) + n] (3) τ() z = f ρ ρ gz τ where τ n is the additinal tectnic nrmal stress applied t the fault plane. The tectnic mment per unit length is given by 0 M / L = y µ ( z) dz T D What is µ(z)? This is the effective shear mdulus needed t keep the stress belw the upper bund prvided by Byerlee's law s τ() z vxz (, ) µ () z = where ε() z = (5) ε() z x Nw assume that v(x,z) is prvided by the interseismic strain slutin develped abve (equatin 19). It is left as an exercise t finish the prblem. Yu will find that ε(z) is prprtinal t y s this factr cancels in equatin (4). The final result is (4) M L ( ρ ρ ) gd 2 = fπd + τ n 4 3 T 2 c w (6) Using the values in the table abve and fr zer nrmal stress, we find the saturatin mment per unit length is 1.3 x N. Again, this is 10 times larger than the bserved mment. Given the fault parameters abve, this mment implies an ptential seismic ffset f 45 m and a recurrence time f 3000 years; a giant earthquake indeed!

9 There are nly tw ways t understand this delemma. A) Faults are smehw lubricated (f~.05) s the average stress n the fault is times smaller than predicted by Byerlee's law. In this case ne has the difficulty f maintaining the elevatin f the tpgraphy in Califrnia. Fr example, San Jacint Muntain, which is less than 25 km frm the San Andreas Fault, has a relief f abut 3000 m which implies stresses f 80 MPa (16 times the stress drp in an earthquake). B) Faults are strng as predicted by Byerlee's law. In this case, faults are always very clse t failure and each earthquake relieves nly a small fractin (~10%) f the tectnic stress. As we saw in the last sectin, this mdel implies a large amunt f energy dissipatin alng the fault; frictin frm bth aseismic creep and seismic rupture will generate heat. It has been prpsed that perhaps during the earthquake, the cefficient f frictin drps frm 0.60 t say 0.05 t temprarily disable the heat generatin. Hwever, is seems that such a slippery fault wuld release all f the elastic energy during an earthquake (~60 m f ffset). Anther pssibility is that heat is generated but a large fractin f the heat is advected t the surface by circulatin f water in the upper cuple km f crust. The unfrtunate implicatin f this high-stress mdel is that since faults are always clse t failure, it will be almst impssible t predict earthquakes. 9