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Date Created: 10/22/15
SIG 224 Mantle convection 7 continued 1 Boundary layer theory For the purposes of this section we consider incompressible Boussinesq ow in a 2d geometry The equations then become mass V V 0 1 momentum V T VP apgT 7 TA 2 For an incompressible newtonian uid with visocsity independent of position this becomes nV2V V apgT 7 TA 3 energy 7 this assumes that k is roughly independent of position DT 8T h 77 T 2T 7 4 Dt 8tv V IN 010 H We introduce the vorticity w V gtlt V 5 Note that V2V VV V 7 V gtlt V gtlt V 7V gtlt w Take the curl of the momentum equation to eliminate pressure which is proportional to the gradient of a scalar giving W2 paVT gtlt g 6 Since g effectively acts in the z direction g 792 then 8T V2w Pay 7 7 where w is the y component of 01 At suf ciently high Rayleigh number though provided that the ow is reasonably steady state analytical approximations to the convective solution can be made The ow is characterized by thin boundary layers thin plumes and an isothermal interior Consider the energy equation For no internal heating and steady state this reduces to 78 v VT HV2T 8 In the thin boundary layer V 2 0 and away from the rising and descending plumes the isotherms are nearly horizontal so conduction in the z direction is the most important Thus in boundary layers U1 7 lt67 9 H mm wmbwmmm mama wymuamgmm 1mm ma myxmu 0 manc7z om wg mmmxmj u xm www mm In a sex an 37 3 m mmmmmaawmmauaz 0 mmyxmu an 45 Aumagm swan ymMMWme mammsmmmwmu Cquot n 7 52 v 52 WW mmmmgmymmmmwmgm m z o 12 mum MmMmmmmeommnmmy Mmmm wgcmmm mxmmm muymagwmmm 51mmwa 1 mmswmm uw mamm mu Mummmmm qmmmwmmam mm mum mm W m we Mamb camp ny yw mamammmzm m m m mumth h Mbwufm WWW ng 12 mm my 51 mama 1 m mum gamma m mu m W mhyu mmmammmomomgm Muhammuuh m m h 4 Mum mm mm m mmmwm my mmme bag In Lummmmmmhmmstum mmmddkufdub x m A T m a 5 garazmmmmm m2 mgmmmw mmmuwmm gm W m Mum hyu wank ma 1 Am ns m mm mm mmvihu mm mm bmndn39yhyu m M m E 7 TI 16 m 5 m z E n Smuhdy m Plum m m 15m ma ms m MA am PM Pmuwy mm m 7mm N pagAT 15 3 v 3 i m u an m my mmmmmuwu m m m Plum m armc712 5 m m M 9m w wmwmmaummmmmm WWW N n my a A x 6 w u x w LS5ltSlt WMSWW 39 quot M H mm m m mum a n WWW M m m l w de WNW xnmmmmemawm 24 52 T m whamummymmmy mamme mmy mmmmnmum W m mam m m 2amp d 42 21 where 121 is the mean horizontal velocity at the edge of the boundary layer Thus v1 2 v at least for a square box which isn t surprising as we have to conserve mass let u 2 v1 2 U and we have I Ld and 210049AT6 d 27 22 The Rayleigh number for this system is given by Ra pagAngm and combining equations appropriately gives 6 x dBa l3 23 U x SF an 24 The heat ow out of the surface of the box when we have convection is kAT lql T lt25 Thus the Nusselt number is kAT kAT i d 13 26 d 26 Ba 26 Numerical calculations verify this relationship for Rayleigh numbers at least a factor of 10 above critical and the relation is extremely useful for calculating thermal histories of the earth We also note that this boundary layer analysis can also be developed for internally heated ows where slightly different results are obtained 2 Convective ef ciency A treatment of the global entropy and energy equations leads to the concept of convective ef ciency The energy equation can be manipulated to give the desired result For simplicity we assume a steady state and that the boundaries of the mantle are not moving radially V 2 0 A steady state implies that 8p8t 0 so conservation of mass gives V pv 0 Integrating the energy equation over the whole mantle gives the rather obvious result for steady state QSqdSVpth 27 where Q the net heat ux out of the mantle is just balanced by internal radioactive heat production To look at dissipation we must use the entropy equation Ds DT 8T DP T7C if 7 0 Dt Pplpt 8P SDt ph7VqT 28 Integrating this equation over the mantle gives assuming steady state 4 pCpVVTdV7O TVVPdVqdsiphdvT dVltI 29 V V S V V where I is the global rate of Viscous heating To a good approximation Cp is a constant in the mantle then we can write pvadeV va vadV pTva dS 0 30 V V S where we have assumed the mantle is neither expanding or contracting and we have used V pv 0 If we now use the global conservation of energy we have ltIgt 7 aTV VPdV 31 V This equation implies that the global rate of dissipative heating is exactly cancelled by the work done against the adiabatic gradient It turns out that the pressure gradient is dominated by the hydrostatic background term so that VP 2 7092 where 2 points in the upward vertical direction Then lt1gt Epcprvzdv 32 CP V where V is the vertical velocity If we average over horizontal surfaces we see that lt pCpTVz gt is the horizontally averaged convective heat ux and using the de nition of the Dissipation number assumed constant we nd that lt1gt Dime g D112 33 where Q5 is the total heat ux out of the top surface and me is the convected heat ux When the convection is vigorous ie large Rayleigh number the Nusselt number is large and me Q5 so we can de ne an quotef ciencyquot as 3 g Di 34 Q5 Clearly when the dissipation number is small and the Boussinesq approximation is valid the global rate of viscous dissipation is small and can be neglected The dissipation number depends on the depth scale of convection and for wholemantle convection Di 2 05 so that it is possible that viscous dissipation is important on a global scale For internal heating the ef ciency must be modi ed a bit as the convective heat ux now changes as a function of depth and we nd that viscous dissipation can be reduced This global analysis says nothing about the local r of quot p heating Q m are presented by Tackley and are shown in the accompanying ppt 11 3 Thermal history The Nusselt number Rayleigh number relationship derived above and veri ed in numerical experi ments on isoviscous uids can be used in thermal history calculations 5 Nu aRa 35 with 2 03 One might think that assuming a constant Viscosity is not a particularly good approxima tion for the Earth and it is certainly true that including a temperature dependent Viscosity can dramatically change the exponent Typically what happens is that the cold upper boundary layer becomes highly Viscous and forms a quotstagnant lidquot which strongly reduces the ef ciency of conVectiVe heat ow and B can become 01 or smaller Such a relationship would probably be appropriate to use on Venus but Earth has plate tectonics and the upper boundary layer gets recycled in a fashion more similar to the is0Viscous calculations To do thermal history calculations we use the energy equation integrated 0Ver the Volume of the mantle The mantle is allowed to cool and in most cases the adiabatic heating term and Viscous heating terms are neglected As shown ab0Ve these terms cancel globally if the mantle neither expands or contracts though this is unlikely to be a good approximation 0Ver the age of the Earth We write the simpli ed energy equation as 8T pCpEdV 76201 Qm 0th 36 V V We now write dTdt as the mean cooling rate of the mantle giVing dT M0103 Qaut Qm E 37 where E is the radioactiVe heat generation in the mantle and M is the mass of the mantle Note that E is a function of time and is roughly exponentially decreasing We are going to use the form for the Nusselt number de ned ab0Ve but we need to be a little careful about our de nition of Rayleigh number The usual form is angdf M where we haVe normalized the temperature such that the surface temperature is zero and T is the mean interior temperature The Nusselt number can be written as Nu Qaut 7 Qcand where and T is the hypothetical heat ow that would emerge with the giVen aVerage temperature T and when only conduction operates Generally we can write andT CT so that Ra 38 39 Qaut CTRGIB Incorporating this with the de nition of the Rayleigh number and treating eVerything as a constant except for 7 which may be a function of temperature and so will change with time we haVe dT T1 MCi39t Eti 41 p d QM gt ltgt a WW lt gt We now haVe to specify a Viscosity law eg gT 7 770 exp Tm 42 Alternatively we can specify the Viscosity relative to a reference temperature and write an approximate form n no in 43 The reference temperature might be the present mean temperature and provided T doesn t change dramatically with time this equation represents the exponential behavior reasonably well Note that n is probably in the range 30 to 40 The energy equation can now be integrated backward in time using the present conditions as initial conditions or forward in time using some guess of the initial conditions The constant a can be estimated from numerical calculations or it can be fudged out by normalizing Qaut to be some speci c value at a particular value of T A natural choice is at t to at present time set Qaut to Q0 and T To Q0 is the current day heat loss from the mantle and is thought to be about 80 of the total surface heat lossafter correction for continental heat production Combining these results together gives 7 1 n MC g owes Ems 7 Q0 p dt This form is convenient as it allows the thermal response of the Earth to be analytically investigated for some simple cases Consider the case when we have no heat sources Mop dT 7620 T gtm 45 44 dt To where m 1 3 713 For n 30 a 40 and 2 3 we nd that m 12 If convective heat transport is ignored and conduction dominates 0 then m 1 Actually m might be larger than 1 for conduction because of contributions of radiative heat transfer which would lead to a temperature dependent thermal conductivity and could give an m of 2 to 4 When m l the solution to the above equation is 7 Q0 T T 7 t 7 t 46 oexp jb lcqlt o lt gt which gives a conductive time scale of cooling of TOMCpQO of approximate 8By If m is greater than 1 then the solution looks like T mil E 1i iaiwxmin MW It is interesting that this equation can lead to in nite temperatures in the past 7 this happens in the last 4By if m is greater than 2 Both of these results suggest that the assumption of no internal heat sources is inconsistent with the present day heat ow When we have internal heat sources it is possible to ask what the thermal response time is if at some time we increase stepwise the amount of heating The solution has a decay constant 7 where 7 MCPTO T on which is the conductive time constant divided by m For m l2 7 700my so there is time for thermal impulses to decay For smaller values of 3 associated with stagnant lid convection 1 so 7 48 7