A comparative study of parameterized and full thermal-convection models in the interpretation of heat flow from cratons and mobile belts

Size: px

Start display at page:

Download "A comparative study of parameterized and full thermal-convection models in the interpretation of heat flow from cratons and mobile belts"

Garey Harris
5 years ago
Views:

Geophys. J. Int. (1993) ll3,747-751 A comprtive study of prmeterized nd full therml-convection models in the interprettion of het flow from crtons nd mobile belts Andrew A. Nyblde" nd Henry N.

1 Geophys. J. Int. (1993) ll3, A comprtive study of prmeterized nd full therml-convection models in the interprettion of het flow from crtons nd mobile belts Andrew A. Nyblde" nd Henry N. Pollck Deprtment of Geologicl Sciences, University of Michign, Ann Arbor, Michign 48109, USA Accepted 1992 November 2. Received 1992 July 3; in originl form 1991 December 2 SUMMARY Het flow from Archen crtons worldwide is typiclly lower thn from younger mobile belts surrounding them. The contrst in het flow between crtons nd mobile belts hs been ttributed in previous studies to the greter therml resistnce of thicker lithosphere beneth the crtons which impedes the flow of mntle het through the crtons nd forces more mntle het to escpe through thinner mobile belt lithosphere. This interprettion is bsed on therml models which employ prmeterized convection lgorithm to clculte het trnsfer in the sublithospheric mntle. e test this interprettion by compring therml models constructed using the prmeterized convection scheme with models developed using n lgorithm for full therml convection. e show tht therml models constructed using the two different convection lgorithms yield similr surfce het flow nd therml structure to moderte depths within the lithosphere. Therefore, we conclude tht the interprettion of the het-flow observtions in terms of thicker lithosphere under Archen crtons thn under mobile belts is robust in the sense tht surfce het flow is not sensitive to the detils of het trnsfer within the convecting mntle nd how deep mntle het is delivered to the bse of the lithosphere. Key words: convection, crtons, het flow, mobile belts. INTRODUCTION A mrked contrst in het flow between Archen crtons nd younger pre-cmbrin nd Phnerozoic mobile belts is observed in mny shield res round the world (e.g. Morgn 1985; Chpmn & Furlong 1977). For exmple, in southern Afric het flow in the Klhri Crton is m mp2 nd increses to m m-' in the mobile belts surrounding the crton (Nyblde et l. 1990; Bllrd, Pollck & Skinner 1987). Over the pst two decdes there hs been growing number of geophysicl nd petrologicl observtions which support the hypothesis tht the lithosphere extends to depths of severl hundred kilometers beneth Archen crtons (see Jordn 1988, for review). Bllrd & Pollck (1987) interpreted the contrst in het flow between the Klhri Crton nd its surrounding mobile belts within the context of this hypothesis, nd by nlogy, their interprettion cn be pplied to other pre-cmbrin shield * Now t: Deprtment of Geosciences, Pennsylvni Stte University, University Prk, PA 16802, USA. res. Through numericl models, they showed tht the lower het flow in the crton could result entirely from the diversion of deep mntle het from beneth 400 km thick crtonic lithosphere lterlly to thinner mobile belt lithosphere, or lterntively tht if some of the difference in het flow between the crton nd mobile belt rises from differences in crustl het production between the crton nd mobile belt, then less het need be diverted by the crtonic root. In the ltter cse the crtonic lithosphere then need extend only to some 200km depth. In constructing their therml models, Bllrd & Pollck used prmeterized convection lgorithm to clculte het trnsfer in the sublithospheric mntle. They ssumed tht this pproch yields, in time-verged sense, resonbly good representtion of the therml conditions in the convecting mntle nd of the delivery of het to the overlying lithosphere; however, the prmeterized convection lgorithm does not embody full description of mntle convection, nd more complete representtion cn be obtined from full therml convection models. In this pper we test the Bllrd & Pollck interprettion by compring therml models using the prmeterized convection 747

748 A. A. Nyblde nd H. N. Pollck lgorithm with models using full therml convection lgorithm to determine if surfce het flow is sensitive to the detils of how mntle het is delivered to the bse of the lithosphere.

2 748 A. A. Nyblde nd H. N. Pollck lgorithm with models using full therml convection lgorithm to determine if surfce het flow is sensitive to the detils of how mntle het is delivered to the bse of the lithosphere. Our comprtive study hs three prts. e first describe the models, then compre them, nd finlly discuss implictions of the comprison for the Bllrd & Pollck interprettion of het flow from Archen crtons nd mobile belts. MODELS Finite element nlysis is used to solve the governing equtions in both modelling pproches on 64x50 element mesh. The models re heted internlly, nd het trnsfer in the lithosphere is by conduction only. The two pproches differ in how het trnsfer is clculted in the sublithospheric mntle. In the prmeterized convection models, the stedy-stte het conduction eqution is solved with n enhnced therml conductivity, djusted to yield n dibtic temperture grdient beneth the lithosphere (Shrpe & Peltier 1979). For the full convection models, the coupled thermo-mechnicl equtions which govern therml convection in n incompressible fluid with n infinite Prndtl number re solved. These equtions nd their numericl solutions re described by King, Refsky & Hger (1990). In the full convection models, the lithosphere is rigid, the mntle viscosity is constnt, free-slip boundry conditions exist on the sides nd bse of the model, nd there is no het or mss trnsfer cross the sides or bse of the model (i.e. reflecting boundry conditions). Becuse of computtionl limittions we re restricted to models with Ryleigh numbers no higher thn bout lo', somewht lower thn pproprite for the whole mntle. To ccommodte this constrint, one or more of the prmeters tht comprise the Ryleigh number cn be djusted, nd for this study we hve elected to limit the depth extent of our models to 1000 km (lithosphere plus convecting mntle). The therml nd rheologicl prmeters used in the models re given in Tble 1. In both the Prmeterized nd full convection models the crton is locted in the centre of the model to insure tht edge effects do not perturb the therml regime within it. Tble 1. Therml nd rheologicl properties PropeflY coefficient of therml expnsion grvittionl ccelertion sub-lithospheric mntle het production lithospheric mntle het production crustl het production therml conductivity therml diffusivity kinemtic viscosity Vlue 3.0 x 10-5 ~ m s-* 3.5 x 10-8 w m x 10-8 w m x 10-7 m w m-1 K-1 1.O x 10-6 m2 s x 1017 m2 s-l However, becuse the models re symmetric bout their mid-points, we disply only the left hlf of ech model [Figs l() nd (d)]. The models lso include uniform 40km thick crust. In the comprisons to follow, we use simplified version of the models constructed by Bllrd & Pollck. In the first comprison, the crton is 1000 km wide, the crtonic lithosphere is 400 km thick, nd the mobile belt lithosphere is 100km thick. In the second comprison the crtonic lithosphere extends only to 200 km, letting us exmine how vritions in the thickness of crtonic lithosphere ffect results. COMPARISON OF RESULTS Isotherms from the prmeterized nd full therml convection models re shown in Figs l() nd (d). Shllow lithospheric isotherms in the models re generlly similr, but t greter depths they become noticebly different. This is evident from the more closely spced isotherms in the lower hlf of the crtonic lithosphere in the full convection model (Fig. Id) s compred to the prmeterized convection model (Fig. l). In the sublithospheric mntle, the two models show greter differences. Isotherms in the full convection model show n re of cold downwelling, nd ner the bse of the mobile belt lithosphere the convecting mntle hs super-dibtic temperture grdients. 'Stedy-stte' mntle isotherms in the prmeterized convection model re chrcterized by bowing-up under the crton nd pproximtely dibtic temperture grdients beneth the bse of the mobile belt lithosphere. Surfce het flow for ech model is given in Figs l(b) nd (e). The full convection model is time-dependent, nd thus smpling of het flow t different times is shown in Fig. l(e). e verged the curves in Fig. l(e) to produce single time-verged het-flow curve for this model which, together with het flow for the prmeterized convection model, is shown in Fig. 2(). The results re in good greement; both show pronounced difference in het flow between the crton nd mobile belts. Het flow is bout m m-* over the crton nd increses to bout m m-* over the mobile belts. Minor differences include (1) slightly steeper grdient in het flow over the crton-mobile belt contct in the prmeterized convection model, nd (2) slightly lrger contrst in het flow between the crton nd mobile belt in the prmeterized convection model. Figs l(c) nd (f) show tempertures t the bse of the lithosphere. There is good greement between tempertures t the bse of the mobile belt lithosphere in the two models, but not t the bse of the crtonic lithosphere. Tempertures t the bse of the mobile belt lithosphere in both models re bout 1150 C, nd for the crtonic lithosphere, bout 1400 "C on verge in the prmeterized convection model nd bout 1600 C in the full convection model. Additionlly, in the full convection model tempertures vry both temporlly nd sptilly by K t the bse of the mobile belt lithosphere, in contrst to the bse of the crtonic lithosphere where temperture vritions re very smll. The models in Fig. 1 hve lithospheric root extending to depth of 400km beneth the crton. But, Bllrd & Pollck suggested tht if some of the difference in het flow

3 PARAMETERIZED CONVECTION Prmeterized versus full therml convection 749 FULL CONVECTION - Id) L c U 'III1111II1I1II F1 " I n 3+7 E j LL + 45j U - i 35 I I I I I I I I I 1 I I I I I I I I I P = t z 1200 l h I I i E l I- ~ o o ~ l l l i ~ l l l l ~ l l i l ~ l l l l 3 E LL A :OOO C I I 1400 L -I e 1200 Q s 1000 L I I I 1 1 I I I I I I I I Figure 1. Therml models. (-c) nd (d-f) show isotherms in verticl section, surfce het flow, nd tempertures t the bse of the lithosphere for the prmeterized convection model nd the full therml convection model, respectively. In () nd (d), the surfce of the models re t OT, the contour intervl is 2, nd the shded region is the lithosphere. Het flow nd tempertures in (e) nd (f) re displyed t 100 Myr intervls. between the crton nd mobile belt rises from difference in crustl het production between the crton nd mobile belt, then less het need be diverted by the crtonic root, nd the crtonic lithosphere need not extend s deep s 400 km. e hve therefore compred prmeterized nd full convection models tht hve 200 km thick crtonic root s well. As with the models with 400 km thick crtonic lithosphere, there is good greement between surfce het flow in the prmeterized nd full convection models with 200 km thick crtonic lithosphere (Fig. 2b). However, the contrst in het flow between the crton nd mobile belts is only bout 15 m m-*, illustrting the reduced therml resistnce of the less pronounced crtonic root. For the model with 200km thick crtonic lithosphere to be fully consistent with the het flow observtions from southern Afric, t lest n dditionl 5 m mp2 contrst would hve to come from differences in crustl het production between the crton nd mobile belts, s Bllrd & Pollck suggested previously. DISCUSSION The principl observtion from the comprison of prmeterized nd full therml convection models is tht surfce het flow in both models is very similr. The difference in het flow between the two models is less thn 1mm-2 on verge over the mobile belt nd only 1-2 m m-' over the crton. This result clerly indictes

750 A. A. Nyblde nd H. N. Pollck 45 Prmeterized Convection i, (,, = 1 1, 1 1 1 1, 1 1 1, ( 35 N 75 p) E45 l / Prmeterized Convection = 3 Figure 2.

4 750 A. A. Nyblde nd H. N. Pollck 45 Prmeterized Convection i, (,, = 1 1, , 1 1 1, ( 35 N 75 p) E45 l / Prmeterized Convection = 3 Figure 2. () Surfce het flow for the models in Figs l() (dshed curve) nd (d) (solid curve). (b) Surfce het flow for models with 200 km thick crtonic lithosphere. tht surfce het flow is not sensitive to the detils of how het is delivered to the bse of the lithosphere. The similr contrst in het flow between the crton nd mobile belts in both models demonstrtes the effect of the incresed therml resistnce of thick lithosphere beneth the crton postulted by Bllrd & Pollck: flow of het from the mntle through the thicker crtonic lithosphere is impeded nd therefore more mntle het must escpe through the thinner lithosphere of the mobile belts. It is instructive to clculte how much mntle het is diverted in the models from beneth the crtonic lithosphere to the mobile belt lithosphere. Het flow from the crton nd mobile belt in the bsence of ny het diversion cn be determined by summing the het produced within rock columns under the crton nd mobile belts. This clcultion yields het flow of 64 m m-' from the mobile belts nd 57mm-2 from the crton (see Tble 1 for het production vlues). The models in Fig. 1 show het flow from the crton (bout 47 m m-') nd mobile belts (bout 67 m m-') when mntle het is diverted from beneth the crtonic lithosphere to the mobile belt lithosphere. Thus, het flow in the crton is lowered by bout lomm-' by het diversion, nd het flow from the mobile belts is incresed by bout 3mm-*. Conservtion of energy is mintined becuse of the greter re of the mobile belts. There re other observtions tht lso wrrnt discussion. Although we ttribute the contrst in het flow between the I crton nd mobile belts to thicker lithosphere beneth the crton, our results (nd those of Bllrd & Pollck) lso show tht surfce het flow is not sensitive indictor of bsolute lithospheric thickness. This is evident in Fig. 2, by compring het flow for models with 2km nd 400km thick Archen lithosphere; the thickness in Archen lithosphere vries by 200 km between the models, while the difference in het flow between the mobile belts nd the crton chnges by only bout 5-6 m mp2. As discussed bove, tempertures within nd t the bse of the mobile belt lithosphere re similr in both models, but noticeble temperture differences exist between the lower prts of the crtonic lithosphere nd t its bse. This is consistent with the smll difference in crtonic het flow between the two models of bout 2mm-' (Fig. 2). Given therml conductivity of 3.5 m-l K-' for the lithosphere, n increse in bsl (nd surfce) het flow of 2mm-' will crete 230K difference in temperture t the bse of the 400 km thick lithosphere, nd will led to noticeble vritions in isotherms ner the bse of the lithosphere. For the mobile belts, the smller differences in surfce het flow (<1 m m-') nd the lesser thickness of the lithosphere give rise to much smller temperture vritions within the lithosphere nd t its bse. Finlly, there re two results from the full convection model tht my be ttributed, t lest in prt, to the nture of the boundry conditions imposed on the clcultions. The first result is tht temporl nd sptil vritions in tempertures t the bse of the crtonic lithosphere re negligible nd re only between 100-2K t the bse of the mobile belt lithosphere (Fig. If). Becuse of the reflecting boundry conditions used in the model which preclude horizontl mobility of the lithosphere, tempertures t ny point long the bse of the lithosphere my not vry s much s they would in convecting system with dynmic feedbck between moving pltes nd convecting mntle (Gurnis 1988). If there re temperture vritions t the bse of either the mobile belt or crtonic lithosphere which hve lrger mplitudes or persist for longer periods of time thn do the temperture vritions in our model, then surfce het flow could be ffected to somewht greter extent. However, thick lithosphere is strong low-pss filter, nd it is likely tht mny temperture fluctutions would be dmped out within the lithosphere before reching the surfce. Even in convecting system with lrger sptil nd temporl temperture vritions t the bse of the lithosphere, it seems unlikely tht the generl pttern of surfce het flow would be significntly different from the pttern obtined in this study from the full convection model. The second result from the full convection model tht my be ttributed to the boundry conditions is the persistently lrge difference between tempertures t the bse of the mobile belt nd crtonic lithosphere (bout 450 K). Models with dynmiclly intercting pltes nd convection show tht pltes trnslte lterlly s het builds up beneth them, nd this lterl plte motion could diminish the time-verged mntle temperture t the bse of thick crtonic lithosphere by moving the plte from wrmer to colder regions (Gurnis 1988; Gurnis & Zhong 1991). Our model with sttionry lithosphere my therefore represent n end-member model which provides n estimte

5 Prmeterized versus full therml convection 751 of the mximum likely difference between tempertures t the bse of the crtonic nd mobile belt lithosphere. However, decresing the difference in temperture between the bse of the mobile belt nd crtonic lithosphere somewht is lso unlikely to significntly lter the surfce het-flow pttern; s we showed bove, modifying tempertures t the bse of thick crtonic lithosphere by few hundred degrees lters surfce het flow by only few milliwtts per squre metre. CONCLUSION Surfce het flow from therml models employing either prmeterized or full therml convection lgorithms is very similr. The contrst in het flow between the crton nd mobile belts in both models is lso similr, nd demonstrtes the effect of the incresed therml resistnce of thick lithosphere beneth the crton postulted erlier by Bllrd & Pollck (1987). e conclude tht the Bllrd & Pollck interprettion of het flow from crtons nd mobile belts is robust in the sense tht surfce het flow is not sensitive to the detils of het trnsfer within the convecting mntle nd how mntle het is delivered to the bse of the lithosphere. ACKNOLEDGMENTS e thnk Michel Gurnis nd Geoff Dvies for helpful discussions nd Shijie Zhong for computtionl ssistnce with the finite element nlysis. This work hs. been supported in prt by the US Ntionl Science Foundtion grnt EAR REFERENCES Bllrd, S. & Pollck, H. N., Diversion of het by Archen crtons: model for southern Afric, Erth plner. Sci. Lert., 85, Bllrd, S., Pollck, H. N. & Skinner, N. J., Terrestril het flow in Botswn nd Nmibi, J. geophys. Res., 92, Chpmn, D. S. & Furlong, K. P., Continentl het flow-ge reltionships (bstrct), EOS, Trns. Am. geophys. Union, 58, Gurnis, M., Lrge-scle mntle convection nd the ggregtion nd dispersl of supercontinents, Nture, 332, Gurnis, M. & Zhong, S., Genertion of long wvelength heterogeneity in the mntle by the dynmic interction between pltes nd convection, Geophys. Res. Leu., 18, Jordn, T. H., Structure nd formtion of the continentl tectosphere, J. Petrology, Spec. Lithosphere Issue, King, S. D., Refsky, A. & Hger, B. H., ConMn: vectorizing finite element code for incompressible twodimensionl convection in the Erth s mntle, Phys. Erth plnet. Inrer., 59, Morgn, P., Crustl rdiogenic het production nd the selective survivl of ncient continentl crust, 1. geophys. Res., 90, C561-C570. Nyblde, A. A., Pollck, H. N., Jones, D. L., Podmore, F. & Mushyndebvu, M., Terrestril het flow in est nd southern Afric, J. geophys. Res., 95, Shpe, H. N. & Peltier,. R., A therml history model for the Erth with prmeterized convection, Geophys. J. R. str. SOC., 59,

Entropy ISSN

Entropy 006, 8[], 50-6 50 Entropy ISSN 099-4300 www.mdpi.org/entropy/ ENTROPY GENERATION IN PRESSURE GRADIENT ASSISTED COUETTE FLOW WITH DIFFERENT THERMAL BOUNDARY CONDITIONS Abdul Aziz Deprtment of Mechnicl