GOME THiXEAL T0N$MJB:s" OF A THEORY OF 00NTINElT G OuTH by JAMS P. DOWNI 3.58, GOREIA INSTIUTE Ot fEOWOGT (196) S7MITElD IN PARTIAL FOIJIIMNT OF M, axnuusm"ne ?OR onI mmiR N' o7 I0T-OR0 O PHILOSOPYT st the MfASACOUS2TTS INSTITUTE OF TSOHNQLOGT June, 1960 ,/1 S Aooptt tur Of AntbOtlP by ..s...-e e . -* * .* * * .* * * * *tant of 6o0otg aM Gophyso May 11, 1960 ,,........ ,,......,,,..,,,,.. ?hairman, Departxental Committee on Graduae stadents TABLIE OF QONTHNTS Title Page................ 6**. **#**d****a**940*4 Table of Otate......... . Table of # 2 we***Cee*****b*...*.** Tables............... ............. 3 ...... 4 Table of Figures..e,.......c...*..e..........eee Abs tract.......... ............................. Aokneawedgementa.. ...... *, *......... 5 . *................. 6 Chapter I - Intreduction......................... 7 12 Chapter II - Formalation of tMdl......,,....,..,,,. 12 Mathesatioal Model....,,.,....,................. 15 Pressure and Temperature Dependoene on the Yieosity.. ........... ** Iu.merical Approximation..............,... ............ ..... 22 24 Parameters Used in Theas,..........i*.,,.,.. Chapter III - General Features of Model and arly Differentiaton of Earth,.,,... Ras to Study EarlyX iffehntiation..........,. 29 29 Possible Yarly Hitory of arth......,......... 36 Initial Distribution of Aoid Frsstion for Near surface Differentiation..................... 40 Time Scale of Initial Separation................, 41 Qomparison of Model to Other Modele,........... 43 45 45 oaputer run*s... ..... * ...... *... * C *... **... *, Disassion of Results..................,.... ,.e 47 49 Chapter V - 0onalusions and fStgestioas for er Work,..............*............ Sa 54 estions for ?urther Work...............,..... Appeanit A - Visosity,*,..*. .. ,, ,....*,*.*..**. Pro ears and Taemprat re D pendonoe.....'e The Effelt of Volatiles on tb* ises .... ..... Appendix s *- er Po.sible ......... 67 w ,a..... 70 83 SQea1S.*...,,.,................ ..... iour*oes................. e...a* Appendix a - Nmerlal ApproxiAto...... ... ... .. genegrl.. **** ............... #... 4,...... **.,.. $tability and oaergeneoe onXitioaes.,........ Aocuracy..9.9............. ... Differonoe Eouations... ,,a..,.a 4 Ciphioal ... 83 92 92 93 #.#............,.96 q.. a a e••.. 96 ....................... *tde. 956 56 e,.. Value of Vieesity Parasters Used in * OSalulations.............. 51 1... 100 TABLE OF TABL"S 3.1 Desription of \sna Usd in Study of 3.2 IRune to Ztudy Effect of 1ha"des in Uo,***...... 3.3 Goera Ieatures MIeR to Illstrat# of MOeLl.............,..,.e Low 4asity layer Pfeattare... 31 32 35 3.4 Euns to Determine Distribrtion of I for Near Surface Differentiation Study*............ 42 4.1 R=us to Stdy lear surface Differentiationa..... 48 5.1 Dpth to Top of Low D*nstty Laer for Several ases....................,......... 52 A.1 Viseosit Glases...,* A.2 Parameters for Some Silioate + *#4 e *6*4*9*0*4*06*4* ... 44 A.5 A.6 * Least S~uares Fit of Vie siVty . **........** .. .,..eoo 73 74 omparison of lit of Viscosity vs 1/T for Two Natural Avaes by Two Yorms of the Visoosity as a unotion of Tmp eraSure......... 75 Differeance Between Activation Energy for Wiet and Dry elt........,.....,,,,,...... 69 Data from Stewart (Unpublished) Used in Study of Effot of Water on Visaosity in Silicate elts............................. .i 62 1/T of Earth-Like Materials.o........,..... A,4 4** Least Squaresoo Fit of Visoosity vs 1/T by Three Cures. .*.*......... A,3 * 69 Omparison of Two i'una to Indicate Aocuray of Numerioal Solution....................0* *6 97 TABLIE O1 FfIGURES 2,1 Temperatue vs Depth Used as Initial Temperature 211DisribPion............ 3.1 Melting Point 38 Ad Energy of ACtivation for Visaos flaw in A.2 A43 A.4 ve Depth for Iron and Diopside... Binary Liquid Silioates..................... 76 Relation Between the Preexponential Constant and Composition for iLinary Liquid Sfilioates... 77 Visoosity vs 1/T, 1/T-273, 1/T--473 for a Silioate GlaSe s..o... ... .. ,....,........ 78 Visoosity va 1/2 of Silioate Glass Fitted by Three Anotion Forms of the Visosity....... 79 A.5 Viscosityvs I/T -for System Diopside Albite-Anorthite,........ . 4 6644 44 90 4 4 ...,,..,... 80 A.6 Viscosity vs /IT for System OrthoolaseAlbite...............*.....,,............,.. 81 Viscosity vse 1/T for Two Natural Laas......... 82 A.7 ABSTRACT Titles Author: Some Thermal Consequences of a Theory of Continent Growth James P. Downs Submitted to the Department of Geolo and Geophysics on Mayi11 1960 in partil fulfillaent of the requirements for the degree of Dotor of Philosophy at Xassanusetts Institute of Teohnology. A nthaMatoial model is formlated whioh is thought to contain the essential features of the process whtich would be active in an earth with *ontimusly growing continents. The model is based on partial melting leading to the formatio of a "glassy" or molten "acid fration" with a visoosity given by q qo ezp AI + XZ)A/k(T-C). The form of the viscosity ahd the value of the parameters used are based on experimental information reported in the literature for rooks and artificial elts. Once te the acid fraction melts, it is suggested that flcw will result due to the action of the buoyanoy fore resulting from the density difference between the solid and glass if the melted sones are eoneoted over a saffioient depth. The flow regime is approximated by a steady state oiseuile flow through tubes. The equations of the mathematical model were solved numerically on a omuuter for several values of the parameters involved in the formalation of the model. The oelolations indicate that the hypothesis of slowly growing sontinents requires an uIalted and comparastively unbroken near surface region in the early history of he earth. Unless these oonditions are met, differentiation will be completed early in the earth's history. If these conditions are met, a "low density layer" seems to be a consequaene of this hypothesis. The depth to the top of this layer is between 30 and 90 kms. In the model, the lowest energy density neoessfy for later differentiation to oqTw in avolume 10' km (down to 100 kas) is 7.7 x i0 " ol/omv**a This is equivalent to dissipation of the amount of energy carried in seiamio waves for the whole earth in this volume. If this energy is provided, it seems that differentiation in this volme would ocour with a time seale between one to fifty million years after the onset of the energy release. Thesis upervieor: H. Hughes Assistant Professor of Geophysics ACKNO L1I4METS The author wishes to express his sincerest thanks to Irofessor Harry judhes who asuggested the problm and offered valuable suggestions in the formulation and in the writing of the thesis. The many conversations with the staff and students of the Department of Geoloy and Geophysics at M.I.T., in particular with Professors S. U. Simpson, T. R. Madden, and . F. 1raee were extremely helpful at critical moments. Professor 1iadden, among other things, read the final draft. Professor Simpson helped in the earlier stages of the programming. Professor Brace made valuable suggestlons of source material. The assistance of the staff at the M.I.T. Computational Center was invaluable in getting the program to work. The oaloulations were made on the IBM 704 at the M.I.T. Computational Center at no cost to the author. The author is indebted to M.I.T. for awarding him a tuition scholarship for his first two years of study at M.I.T. (1956 and 1957). The National Jcienee Founda- tion awarded the Fauthor a fellowship for his last two years in the Graduate .ohool at '.I.T. (1958 and 1959). Credit for the spelling in this thesis being conventional is due to the author's wife, Cecole H. Downs, who typed, proofread and d helped organize the thesis. Ui1.i1 I INRODUJTION Theories have been advanced sugkgestii that the earth was formed by a process of cold aoretion during lthe eaxth has ntver been much, if which the temperature of any, in excess of the temperature that is thought to pervail in the earth today. These theories have gained considerable acceptanoe in spite of the many unsolved problems associated with them. One of these unsolved problems is how the conti- nents could have segregated out of an earth that was never molten, at least not near the surface, Wilson (1954) has suggested that the continents are now iyowing and have grown throughout geological time by a process of adding aoidia material at the continental margins. The continents are thought to be growing at the expense of the ocean basins. Wilson states that "Iach oontinent oan be divided into structures o three tyes- ranges, and continental shelves old atuble shields, mountain and it is becoming apparent that there is a mountain building prooess which oonvets shelves into mountains while mountains, by a process of erosion, are in time converted into provinces of shields. There seems to be a process by which the continents expand and encroach upon the ocean basins. "There is a suggestion that the ocean floors represent those parts of the original crust of tthe earth that has been least altered and that they owe their eeneral laevel to that cause. They will be oonsidered first and then shelves, mountains, and shields which seem to represent suocessive stages in the growth o the continents over former ocean floors." The above quotation contains the main sequence of events that seem to be assooiated with the bypothesis of the continuing growth of the oontinents. "irst sediants are laid down ocf the shore of the current continents. These sediments are altered by the addition of acidio material from within the mantle and by heat and pressure into gneisses and elevated into mountain ranges. These mountain ranges are then reduced by erosion to shield areas and provide the sediments for the next stage of mountain building. The inarease in the mass of the conti-- nents is due to the addition o the acidic material from within the mantle. The addition of the acidio material is not continuously occurring in any partioular region, such as the shelves off the coast of any continent, at aall times, but rather, it ocours in a particular region at a particular time. A consequence of this type of a process would be that the mantle under the oceans should be less differentiated than under the continents. The geolocical arguments concerning the above prooess will not be discussed here and the reader is referred to the above paper by Wilson and the references listed there. urrent seismological information oan=ot resolve the question oaf whether the material under the Qoeans is - 11111 11111 YI leose differentiated than the material under the continents for it is a question of being able to determine chemical oomposition at depth (i.e., being able to distinguish between say cologite and forsterite). The use of the "coamic abundances" of the elements in oonjunction with the seismological information can limit speculations but not solve the Problem (Birch 1952). tecent measurements have revealed that the heat flow in oceanio regions is not significantly different from what is observed in continental regions. Due to the laok of a granitic layer under the oceans, the heat souroes needed to give the observed heat flow are thought to be distributed to depth in the mantle. In other words, the radioactive material under the oceans seems to be less differentiated than under the continents where, except for potassium, most of the radioactive materials are thought to be conentrated in the acidio material of the orust. This lack of segrega- tion of the radioaotive materials under the ocean may be taken as an indication thatthe the mantle under the ocean is ntleunder less differentiated than the material under the continents. In the more or less qualitative geological arguments such as the above, it iis desirable to abstract the physical processes that are thought to be important and include them in a mathematical model from which numerioal values of the quantities involved can be obtained and evaluated as to their plausibility. Some arguments can be shown to be -Li implausible for the values of the quantities required to make the model work are in disagreement with the measured values by many orders of magnitude. Other arguments are found to be consistent with the laws of physics and the measured values of the quantities involved. The class of mathematical models which can be treated by numerical techniques using high speed computers include rather involved nonlinear systems. This permits considera- tion of many geological arguments which have a mathematical formulation that is beyond the reach of analysis alone. By solving the mathematical model for several values of the parameters involved, of values if that is it is possible to determine the range any for which the model gives a prediction realistic. These values can be compared with existing experimental values or they can be taken as suggestive of experiments that may be performed to check the argument, Rubey (1951) and Wilson (1954) have argued that the volume and composition of the volatiles and lava that is escaping from the interior of the earth today is if sufficient, taken as typical of what has existed over geological time, to explain the composition and volume of the atmosphere, oceans and continents. All the previous examinations of the hypothesis of the slow srowth of the continents do not show any serious conflict with the observational data available. Consequently, a further examination of the ideas contained in the hypothesis is desirable. In particular, the ideas -U involved in the addition of acidic material from within the mantle to the near asurface regions are uaseeptible to a mathematical model, and it iis with this part of the above theory of continent formation that this thesis is concerned. It is sugiested that any process which is advanoed to explain the upward migration of material from within the mantle mast be based on a process involving the partial melting of material within a energeties of such a meohanism is . cal region. discused, The and the processes that are found to be important are inaluded in a mathematioal model which is treated as des e ibed above. *(eVJxwe As"Al s*1 UOftrTT -P 1 #Ut#4X* Uv OT 4.T 4A114 'oe42op Moxo t' AOTenTf OTMtnSPOIPAI P1 *R14 01? 'PTnbTT mmnzozetp VePOUW0 VtU S0 tCTTTR ItT hOTS 05 10T47ZT OVA P*TPr49 (956T) -tJJcTP ,UTV '*9T) O1 UT flM*W-TWUX0 To 4.e SJ ST 14101mM w4.os V t 0. PT~tJT £0 PTTOS Vt UT ShZOT 30 UOTvnflJTV pB0Wq sq tmO 880004 t 0141 00O 40f{m uOCfl 8EmflW1 U0T$OflJ PT09 014 IOTMAA SA *oo?)Jns 014 XSOU pOPPtV 9aq hO no;.xo s seeoo44d owos oq 4Sflfl *1014! 'xSW! oxT et ozreoo pun s'1- Or 0! Ot.p0flTJ*TP eq ej. euzeos %q.flq sro;Sox Tv4U0UaT~uO0 UT 4.Stu0 014 UT PflWx4tDUOO *10TtO.fl BT pOUT'gtdxs 04 ;SU PTOv 014 *1T* q*nRa uo-;!.nAsqo wvd~our~d ogu, *t0(J P#OATOAZI 800xJ 04OU OJ *smIC OT#?)q 014 ptw 1ZflTo1w 91 0214 WHIt TT pTO1 P 01 IOT1I#A "0J o*xnvwsdw4 jo ostx 0,10M4 'Otu4,XTw Ofl flflO 10 8T fV-jlW 014 S0otU T #*=wtoA&014 J0 4.90M 3upTdnooo qe 4TSUOP so OSWMS OTtI4 .mgttp I Pup 4e 4Tsuop JO (e4-Vewrp eAT4,OflTPt& P-cn *#OT*yA AitV fUTWPv4u00) *ztmd 0T1PJ4S US P0 ft14,T0* Stq poqszwndnJ eq triO 181A105 *fl twq 30 %4 sq, Tr' vSZ~ Ur *ooa v q41* .1fl0q14 0-; S$Utt$QU Oo 0q4, 04 JrTUITS IZoThTSOdISOO 180 tao OfljtE 01 t 014 IR4vfl T ITXtfl*OWL Tow'T Rorl Ili IMmok ltccr A4WTAY M 0J1X4 O114 *4,v pnnwn81 TYTZ0~SE OUTttt4OflQUtd v 1 0 (4 strttr joBugx OD4T --------~ I ~ -------- P--~L--. "wI I I- -~-P OWN conclude that "Diffusion rates would become appreciable only at temperatures well within the ma tie range, and large scale tratnsformations (e.g., granitination) at temperatures around 400 GC. would require a span of time considerably in excess of reasonable geologic estimatee. This confliots with what some petrologists would consider irrefutable evidence from field observations. The only alternative seems that either granitization effeotively o0curs only at temperatures within the maz atic range (>70000.) or that the phenomenon is metasomatio." onsequently, it seems asaif any process which attempts to explain the formation of the crust must be based on a mechanism involving bydrodynamie flow in a region where at least partial melting has occurred, In the following, the possibility of diffusion in the liquid is neglected. In regions where a prooess of partial melting has occurred, the acidic phase would be molten and oocupy the pore spaes amon4gst the basic phase. aue t tthe density differenoe between the two phases and their presence in the earth's gravitational field, a buoyancy force exists which tends to drive the lihter acid phase towards the surface. This ~ilration is possible if the pore spaces are sufficiently well connected. The solid material in such a region would be ina masses of various sizes and shapes, and would sink to the _ _~ bottom of the melted region. if the basti material had infinite strength, these masses would settle until they touched and supported bhemselves. This would force part of the acid material towards the surface and retain part in the enolose4 pore apace. However, has a finite strength, and if the solid portion the stress differences resulting from the material attempting to support itself exsceeds this strength, the materiul will tail either elastically or plastically forcing the remaining fluid part towards the surface if connected. If the strength of material in of order 108 dynes/m 1/2 3 ya/o the pore spaces are still 2 the matle is and the density difference is , the alidie phase would be squeezed toward the top of any column with connected pores extending over 2 km. factors that govern the rate at which the fluid flows towards the surfaoe are the cross sectional area of the pore spaoes, the viaoosity of the acid phase, and the length of time it takes the oasio phase to adjust to the stresses built up in settling. It sems that the basic solid should adjust as rapidly as the acidio material can flow out of the voids. A simple model is formulated which shows the relation- ships between the important paxameters and allows limits to be set upon the values of the parameters in the earth for reasonable time scales -or the misration of the acidio material. The justification for the simplifying assumptions L^~~^Il---~I"Lllllllllil lili 15 is found in the unoertainty in the values ot the parameters appropriate to the earth. a Ca MO$Idel tthaot The simplest one dimensional model which retains the physics of the problem is to assume that the pore spaces can be approximated by uniorm tubes of length L and ratdius 'a' The eoordinate orientated in a vertical direotion, system is selected so that the Z axis is vertical with inoreasing positive Z into the interior. viscosity Assuming that the of the material oan be represented by some average value over the length L, the pressure gradient P= 6fI where & is the density difference between the At phases and g is the acceleration of gravity, the average stady state Poiseuille velooity of the acidio phase in a tube is 61 2.1 The velocity of the acidio material is toward the surface. on the average always U is negative in the coordinate system selected. GPsersaot f as EQuation Let N be the number of pores per unit area and $ be the fraction of the unit volume oeoupied by the aoidle phase, then S = Nua 2, N is a funotion of depth in order to allow for varying concentration of the acidic phase. By -----~----- --- ----- I -~ IIIY 16 considering volume elements with linear dimenasions large the density of such a two phase model is compared to 'a*, :P The ohanb 2.2 - e in the density of the individual phases with depth are neglected for they are maller than the ohange in the gross density, p, due to ohanges in the ohemical composition. The onservation of massan the system can be written straightaway by reelizing that %he assumption that the density of the phases is independent of depth requires in the model that the volume of the basio phase which moves down be equal to the volume of the acidie phase moved up. condition, it is With this only neaessary to eonaider the oonservation of the aidio phase. The not influx of mass of the sidio phase into a unit volume is whioh oet equal the time rate of change of the mass of the aoidio phase per unit volume a Pq ) f-1 2.4 _I IL-~-(-----l^--- ~-I~1--~ 1_III~-- 111 11 1 -C- 17 Equating equations 2.3 and 2.4 giVr e of onservation ohe massa equation as ) -Q 2.5 where U is given by equation 2.1 The boundary condtione are that there is no mass transport across the surfaces Z = 0 and Z a 1 where R is the greatest depth to which the acidio material was distributed before the segregation prooess started. to be the a H is assaumed af depth at which melting ooeurs. In Appendix B, it is argued that it is only necessary to consider the radioaotive heat production and the strain energy released locally as heat in the energy balance. The oonversion of potential energy into heat energy is negligible in the energy balanoe, but is included for this was not realized when the program was written. in the one dimensional model, the oonoervation of energy is expressed by (heat/unit volume) = rate heat laded by oonduotion + by oveation + by oonversion of potential energy + by heat generation per unit volume. - 2.6 The rate heat is added by oonduction is well known as R 3- T -- where R is the theral conductivity, 2.7 3d C 50 uTOTZAuoo 30 01* 4 4 4ZJ TOTUOBrre 09* *0#TzOtWO A!UTPT&TP Pmr s0,v Iq 04 qsnZOuoe 0q. q 0 =(0)0 MQfl. 1ZT4*J204uT 014 RUTZZU403X# -p'TU00 Ltiptxnoq #14 3u~t~d44 0.412p ss'nssuoo0 sbn 9 TV11 t*0 1*14 BU0lWrWW98 -TJOP #UIT4. 0T46 U0T4*IW&00i pfl4x*TX00 B3 #*Sm~d 094S izeeq en" 91q4 40q *4 R*vtat S *#M- 04311 #14 o4. sup Attn. 0RA,30 T~v4'OT*,;q uemjd o&4 #14 T TMX0T U0T1* V BfLOOTA Sq £ttOU 1*UT*Bfl8 TvtnUnod uT 000# 014VP-O PtIW; t00 Szt# 4*#4R OrsTo4' *4 Onga .#w'd OTpTOw pun o~tsq ?u;X*qwUt pUw WMTOA. wrO 1 *rj j 4-04 toT1qxto Oqn 820T AOtnT oq4. SPu'sspruuoo tq XO1UWV flurif w14 111 PourTq0 ST lb Tstq;:~uoo OAPwPUtIOO oizU 19 U 2.9 The he&t generation per unit volume consiste of two parts. One is the contribution from radtoaotive deay and the release of strain energy as heat. the other is strain energy released locally as heat is treated as a constant. obtained by assumin The called Q and is The radioactivity contribution is that the radloactive souroes have a distribution proportional to the as*i4o material. If A is the rat* of radioaotive heat produotion/ce for a granite, the souree term appropriate to the model under the above sawmption in 2.o qr, An 7 The total heat contained in a volume daV containing a two phase mtixture is cT (euwhere QO is A(dff a constant. A The time rate of change of this quanttty is cT ej-~.cffa A tc(e e N4 a _--~Lsll-II~LII-F IX----.1 L-^.II ~,DI^I 20 The conservation of mas -f equation (equation 2.5) gives a At where as usual p a2.11 t b - 0 wa2 N. Substituting equations 2.7, 2.8, 2.9. 2.10, 2.11 and QE in equation 2.6 gives the oonservation of energy equation as I" ) T/ - c a- -WEa2-2,l - 2.12 The boundary oonditlons are -/ (6) = 7o where the first condition reflects a constant surface temperature, and the second neglects any heat flow at the bottom of the partially melted region. The selection of the initial temperature distributions used in the various models is rather arbitrary due to the uncertainty in the temperature that exists under the oceans. In the runs to study the general features of the model, the nethod used to select the initial temperature distribution was to assume a steady state temperature under the oceans (i.e., a parabola down to H, the bottom of the region -- TEMPERATURE IN Cg 5S& MODEL 4 aooo t MODL 6 1750 t 1500- 1000. Figure 2.1. Plot of steady state temperature vs. depth for the two cases reported by MacDonald (1959) used as initial temperature distributions in the calculations. 750- 500 DEPTH I K(M ?50 I /00oo I /50o /00 Zoo SO2co ?0 ao I 300 30 - I 3 0 .36 400 40 22 containing heal sources, a constant below this depth and no heat flow between these regions). the The specification of uerface temperature, To, and the tnperature at the depth H, TE, in addition to gives the initial 0 temperature T(Z,O) as 2.13 4 Using the suZrface temperature, and the value of the tmapera, The *treat the depth H specifies the initial tea perature. distribution of heat souroes which gives this distribu- tion is 2/ For the cases to study near surf ao the initial temperature was taken fro 2.14 differentiation, a plot (figure 2.1) of either model 4 or 6 in MacDonald's (1959) paper. The particular mdel used to the initial temperatare for 'ive a given oase is ~iven with a escoription of the case. rEsare0d and Te eept rs eedence of the V s1ositi A discussion of the preessuare and temperature dependenoe of the Viscosity of silioate melts given in Appendix A shows that the viscosity of the acid traction which must be a polysomponet allioate can be represented by two forms (l - 7o exp EkT t) 2.15 k-c) 2.16 These two formulas describe the Viscosity with thee necessary in this thesis. ouracy E, E' are activation energies for the procesaes, Q6 and 7 are the visoosity at infinite temperature, k is Boltzman's constant, and X (=1.5 pgdL where K is the bulk modulus) gives the pressure dependence on the proceas. The hydrostatic approximation to the gressure has been made. it was not possible to neglect volume chancre in the formulation of the viscosity (included in X) as in the energy balance equation. The form 2.15 is shown to be applicable only when temperatures near the solidus point are not considered, for in this region, this form predicts values of the viscosity much lower than are observed. The form 2.16 is shown to be valid over a wide range of temperature inaluding temperatures near the solidus point. Both physical argu- ments and experimental evidence requires zhe equation 2.14 to be used for complex melts consisting of cany omponents which melt over a wide rane of temperature when temperatures near the solidues oint are considered. -4119r--~--~~-7 ~-LPI LI - -l~P -a~-- 24 Num ri al arximation The equations 2.1, 2.5, and 2.12 with the appropriate and boundary conditions form of the visoosity, initial This set of nonlinear oonstitute the mathematical model. equations cannot be solved in closed form. Jonsequently, they are approximated by a set of finite difference equations which were solved on the ~lA 704 at the M. I. Computational center. T. The dixLerence equations ares for the velocity equation t-x ~U ( II+ I 2.17 J where d6,=p~~ t.ndj for the conservation of mass equation 4/ A4 ~~A -0;) (/ j + n / 2.18 and for the conservation of energy equation TI I-4 ,I Ui M ,r' -e 0 7jsk 4__ PI r 7 c y V 4 Jr ' 7 +4 47c + -+.l-AJrjL 2.19 ~_I_ ____ 25 where m and A refer to the time and *saceloeation of a point in the differenoe net respetively, The boundary conditions for the energy equation are 'o= To at upper surfaoe X + 1 temperature is The initial - at lower surface t- t---- or taken from Figure 2.1. The boundary oonditions for the oonservation of mass equation are U e, at upper surface = 0Q at lower surface and the h0 must be speciifed. The requirements which the parameters must satisfy for stability are 2.20 2.21 26 The lengtL of h is selected by h = B/L where H is the depth to which the acidio phase extends and L is the number of space net points desired. time interva The is selected during the ruaing so that equations 2.20 and 2.21 are satisfied. The compatational marohing type. shame used was a siaple forward In all of the cases run, 100 time inere- ments were used. ?sraneters U94d 4a Tkheis The parasters involved in the formulation of the problem ares k K g a N b pb x qt E',E r 0 S k, 1, , = Boltzmn oonstant = bulk modulus a=oeleration of gravity = radius of tubes = number of tabes per unit area = density difference between the phases = density of basio phase = pressure dependenoe of the visoosity = rseexponential constant = activatlon energy for viseous flow = thermal oonduotivity = speolfic heat = distribution of acidit , phase and a need no special disoussion. The values used whenever they oeour in the salculations ares k =1.38 x i10 1 6 27 erge g 1033 /see S= .003 oal/o/so/ 0 0. o a .25 oal/ga/ 0 . For detailed studies, the aoitatio energy and the preexponential constant should be empiricaly determined for a melt of a given chemical composition. For the order of magnitude studies made in this thesis where the ahemical composition of the melt is uncertain, E and f are estimated from Maexensie (1957) which gives the activation energy and the preexponential constant appropriate to equation 2.15 as a funation of ohemical composition for dry melts. These values are then adjusted for the estimated effect of the presence of volatiles. this is done in Appendix A. The values of the activation energy used in the caluwlations are E = 20, 35 sad 50 koal/mel, the value of the preexponential oonstant is S= 35 koal and = 10 4 poises. The values 0- 4 polses.re thought to be appro- priate to a "wet" plutonio granite, and the other two values are intended to allow for errors in the estimation of E. Less information is available to aid in the seleotion of Ef,q' and O. Viscosity as a function of tempera- ture for some natural rooks (Biroh et al 1942) and glasses (Shartsis and Spinner 1951) was plotted and ourves ~-i_ --------.-11_1~- 1 ~- --- I~-_ L-L-L L1(_-.-~I~P~LI -- 28 were fitted in order to determine typical values of E', Qoand c. The values of TO determined by Dienes (1953) was considered in determining 0. The values used in the thesis are E' = 10, 20 keal/mol, /a 10- 2 poises and 0 = 7000 K. The form of A is based on Frankel (1946) and the value (A = 5 z 10" 9 c -1 ) is based on Hughes' (1955) measurement on the pressure dependence of the activation energy for cations in the ionic semi-conduction mechanism in peridot. Value for A p An estimation of p is obtained by using jadeite phase. (pa = 3.3 ga/or 3 ) as representative of the acidic Hypersthene (b = 3.4 gm/e 3 ) and andradite (Pb = 3.5 g1n/m 3) are used as typical of the basic phase. For these two examples of the basic phase, #p .2 gm/a 3. = .1 and The presence of volatiles with a density of 2 gm/cm 3 would lower the density of the acidic phase to say 3 gm/cm 3 . This givesp = .4 and .5 gm/on3 . The 3.5 m/icn 3 are used in values 6 p = .5 g/cm3 and Pb the calculations. To determine the initial values of a, N and S for the study of the near surface differentiation, it is necessary to obtain a better understanding of the general features of the model. Chapter III. This is the subjeot of 29 (HA1ERT.TU III GERBLAC LiTURtY OV MODEL AND ARLY DIFERETIATION OF v;ARTH AUI$ to £ttyz£srIlytrtierentiation £4sription of une Before an estimation of the distribution and value of a, N and 3 for use in the study of near surface differentiation can be made, it is necessary to study the solution of several cases run on the oomputer using arbitrary values of the parameters in order to make olear certain general features of the model and the nature of some approximations. These cases give some insight into what the concept of the slow growth of the continents implies about the early history of the earth and will allow the initial distribution of the acid fraction to be determined. For the runa, oexp of E/kT . the form of the viscosity used is The form was used beosase the values and B are better known than <and E', and only general features of the model are wanted. S was taken as a constant (S = .1) Z = H = 300 km. This is down to to approximate what would happen in an earth that hat had an initial uniform composition. value A is arbitrary. It The was selected by assuming the continents to be 30 km thick at the end of differentiation. If , .14a ma111 IM 30 If the 30 ka of material was initially distributed throughout 300 km, S m=st be .1. Due to the uncertainty in the gross ohemical composition of the initial earth, no more meaningful selection of 8 *an be made. The computations show that the value of 8 used in the model is only important in the oonveotive transport of heat and the conversion of potential energy terms. The effeot in these terms do not change the general features of the model. The heat procation per unit volume was selected by equation 2.14 to keep the temperature oonstant in time. This is done to make the effect of the temperature distri- bution clear. It negleots the change in the distribution of radioaotive sources due to ohanges in the distribution of the aoldio phase. This is treated later, The cases are desoribed in Table 3.1. Effoot of Changes in Uo Oases 1, 2, and 3 examine the effeoot of chages in Ut on the results. The only difference between the three oases are the radius of the tubes and the initial value of the number of tubes per unit area which aeast be changed to aeooomodate the change in the radius. The extreme values of the radius used differ by a factor of 106. The relative number of tubes per unit area, the temperature, and the elapsed model time after a hundred time steps are given in Table 3.2. The ratio (number of tubes/rit area)final/(number of tubes/unit area)inital is the T BLE 3.1 DESCRIPTION OF RUNS USED IN STUDY OF GEN2RAL FEATURES OF MODEL Run To o H a (cm) 2400 10-6 kcal/ml al #*see 0ooK 2 300 3 300 4 300 6 300 300 7 8 9 28004 300 16-4 2800 300 10- 2800 --- For these cases to 0 3.18 x 106 3.18 x 106 6 3.18 x 10 3.18 x 106 '- 100 million years er 10mlo 300 10 1.4 35 o-" 1.4 x i0 104 35 1.4 x 10-14 10-4 20 1.4 x 10-14 50 1.4 x 10" 20 1.67 x 10- 1 4 300 35 1. 67 x 10 300 50 " 1.67 x 10 300 106 zx x 1$0 3.18 3.18 x 106 L 35 10l les 100 o10 10o-4 -I NOTEt 10 2400 2400 x 10 - 1 H(im) 104 1-4 10- - -I~ ~' - ~-I 300 300 300 10 300 10 10 TiBL 3.2 RINSs TO sToDT a -To OTF MOAVs TN 100s Aol tCTITVA'IOT Inttl 2, t,nI $$ 7, Tas1 IN X m TInn 1 (z, )/ ( o), **str emD ta n1hr0Y, = 35 KcAI/OL Fm ALL THRi? CASn. Case RO4,o) 2 r W7, OaCe 2 t a 10r .18 lx 1 0 0tuobes/ea 2 - c S 10,0)=C 3(8O Case 3 t = 5.1 x 10yr 3o 700o 6 1 oo 1370 16t4 1875 180 300o 69 3 70o 1.035 106 4.*027 r370 4.880 164 1875 0.057 o0.o000 .000 2060 o.00oo 2210 0.000 2311 000 .000 2315 0.000 2379 24o .000oo .000 2380 24~o o.oo o.oo 300 1.000 1.000 1.000 1056 13 1643 1873 1056 1370 1603 187 1.0oo 1.035 4.035 4.872 0.057 .000 2060 1.0oo 2060 1051 2059 210 2210 1.000 2206 1.01 29 20 2315 1.000 230 1.018 270 S3o 2380 Ahoo 1.0oo 1.ooo 2371 2392 .979 .831 6 90 120 150 .0oo 1.00 tab oo/ n 1.000 1.oo0 1.0o0 1.009 1.026 1.0k 30o . o038 aT T 0 =1=5 ts 3.A8o 699 1.000oo 2 yr _~___ ~L _1 111111 1~---- 33 relative number of tubes. There is no difference between case 2 and 3 in the relative number of tubes or temperature at the end of the same number of iterations. How- ever, there is a signifioant difference in the elapsed model time. There is a stradghtforward explanation of the difference in model tine. In all cases, except where the velocity is so slow that the time interval is given by equation 2.20 as in case 1, the change in the time seale is inversely proportional to the change in the velooity. This is expected from equation 2.21. That the numerical solutions keep this relationship is another indication that the program is free of any serious errors. A change in U0 oauses a change in the time scale for the process. Effeet of Initial TeaOr are and Activato EuurA Qasea 2, 4, 6, 7, 8, and 9 are used to show the effect of changes in the initial temperature distribution and the activation energy on the final distribution of the acid fraction. The final distribution is said to occur when there has been no significant change in N for a 100 million years period. To cover a sufficient amount of time, it was necessary to make suooessive computer runs for the same ease. The reason is that there are only a hundred time steps programmed for each run. The length of a step can be anywhere from a fraction of a second to a million years depending on the velocity with a corresponding spread in the elapsed model time. -- - NNM1I'116I1 L -"~I -^- 34 The initial conditions for a later run are the final conditions from the previous run, In order to lengthen the time interval in the later runs, the extent of the region, H, is redefined at the start of each run by dropping all points where the fraction of the acidic material is .01 of its initial value. This removes the points where the temperature is highest thus lowering ( )') giving a longer time interval in succeeding runs. The same boundary oonditions are applied to the new region. Uo was taken to be the same in all six models to eliminate the effects of changes in this quantity. Table 3.3 gives the temperature, number of tubes per unit area, and the elapped model time for each run. A feature of the model revealed by the sixeases described in Table 3.3 is in that there is a marked inarease the concentration of the acid fraction in depth Z1 forming a low density layer. a layer at The depth Z1 is seen to depend on the temperature distribution, and the value of the activation energy used. depends on the selection of Uo , the value of Uo used. Since the time scale this depth also depends on In fact, there are selections of Uo which allow the differentiataion tO be almost instantaneous all the way to the surface, and others which make the separa- tion negligible at all depths If the 1/(T - 0) form of the visoosity had been used in the study, a seleotion of Uo TABLE 3.3 RUNS TO ILLUSTRATE LOW DENSITY LAYER DEPTH, Z, IN KM AND TEMPERATURE IN*K. INITIAL 1 m'y. = 106yr, CASE 2 FEATURE AND TIME SCALE OF MODELS. IN THE TABLE, MODEL TIME IN PARENTHESES AFTER RUN NUMBER. CASE 4 DATA =35kcallmol, a= 10 cm E=20 kcal/mol, a =10" cm N =CONS. N=3.67x 1dtu beslcrri N= 3.67x 10Otubes/ cm conditions at end of run 1 (5.2 my) 2 (100 my) 3 (100 my) 1 (.14 my) 2 (2.6 my) Z T 0 300 30 699 60 105 90 137 120 164 150 1875 180 210 240 270 3001 N() N(0) IT T 1.00 300 1.00 300 1.04 699 2.91 691 4.04 1056 7.09 1021 4.87 1370 .581643 N()T NO) 1.00 309 6.91 300 561 667 conditions 1 T N(t) N (0 0 300 1.04 30 60 90 775 6.19 1200 3.76 1575 .01 120 1900 150 18 210 240 270 N() N (0) 1.01 4.49 5.45 300 700 1056 1.94 9.01 2175 2400 2 570 270( 277 300 280 E = 50 kcal mol, a= 10 cm IN= 3.67x 10" tubes/ cm a 1 (100 my) N(t) N (0) 300 700 20¢ 2210 2315 238 240 CASE 7 E=20kcall/mol, a=10 cm N =3.67x1d* tubes/cm' Z m NT N () CASE 6 1.00 1.00 1. 09 3.84 3.98 .097 300 699 1056 1371 1643 1873 2 ( 100my) N(t) N (0) T 1.00 1. 00 1.18 4.35 3.37 300 691 1021 1258 1348 CASE 8 4 E= 35kcal/mol, a =10 cm N= 3.67 x 10otubes/cm CASE 9 E=5Okca/mol, a=10 cm N=3.67 x10'0 tubes/cm' 1 (1.5myv) 1 t end of run (.O 07 my) T 2 ( 8.1 my ) N(t) -~(0) T 300 9.53 775 1201 1575 300 2.68 772 1120 N(t) 1.00 T 2 (100 my) 3 (100 my N( t ) N(0) 300 1.00 1.12 775 5.66 1200 3.21 575 .01 1900 T N(t ) N 300 1.00 7.15 730 7.69 2.84 1044 2.29 1165 T 300 611 737 ( 34 my) N(t ) N(0) 1.00 1.00 1.71 641 T 2 (100m ) N(t ) FW-) 30011.00 T 3 (100my) N(t) N T 300 1.00 300 775 1.01 765 1.02 765 1200 3.40 11 59 4.74 1159 1575 5.43 1441 4.21 1441 .88 1900 .43 1549 .29 1549 NOTE Thitial temp. used for these runs. 4 was not increased when H was shorted. 3b oould be made that would allow the differentiation to be instantaneous to the depth where the temperature equaled 0. This is the shallowest depth in this model to which differentiation can ocaur. The uncertainty in the value of 0 appropriate to the acid fraction coupled with the unoertainty in the temperature under the oceans rule out an accurate estimation of the depth at which the motion ceases. If 0 is in the range 400-8000 0, and MaoDonald's estimation of the temperatures under the oceans, plotted as figure 2.1 is assumed as being representative, the minimum depth to the top of the low density layer would be between 30-90 km. These oaloulations show that if partial melting occurs in a region, the mobile material must move toward the top of the melted region in a comparatively short time. The shape of the masses in which this fraction would collect should have a larger horizontal dimension than vertical. eor as argued in Chapter II, the vertical dimension of thee masses can be no longer than the maximum height that the basic material can support. msib le Bar& Hiater of rth Now that the formation of the low veloeity layer in the model has been described and some understanding gained as to the parameters which govern its is depth, it possible to further develop the physical description of the model. The next few paragraphs are a description _I_ __~ ~I ~--L-~-~PL---~ 37 of a process which might have occurred in the earlier history of the earth. The disoussion is given in order to fix ideas for a later section and to sug6gest possible extensions of this work. No one is more aware of the incompletenesa of the following discussion than this author. For a complete disoussion, considerably more experimental data on the flow properties of rooks particularly near their melting points is necessary. it is also necessary to extend the work of this thesis to cover a three phase system including the terms involving pressure which were omitted in the model formulated for the upper part of the earth, Assume that the earth was initially of uniform chemIal composition including radioactive elements and that the material eonsisted of three fractionsi basic, and acid. Aarthermore, iron, assume that the tempera- ture at the time of formation was everywhere below the melting point of the material. D4e to the burial of the radioactive material, would be a general warming up of the earth. This process would continue until one of the fraotions melted. (e.g. 4acDonald 1959, Studies p. 1979) of the melting point gradient of basic materials and iron have been made, point of iron is there and the melting 2000-20000 C, less than that of diopside for depths greater than 300 km (see figure 3.1). No studies l litll milYllil IIlI W nI lt01II lI hi, huAYIIY W I' II2Y I10=__mll I I mlI 38 1000 DEPTH IN KM 2000 2900 5000 ,- 4000 5 3000 . 2000 I- 1000 1.O 0.5 PRESSURE X 1 0 BARS 1.5 Figure 3.1 Adapted from MacDonald (1959) to give the extrapolated melting relations for iron and diopside within the earth, Iron I curve is Strong's (1959) estimate. Iron II curve is extrapolated by using the initial slope obtained by Strong and Gruneisen's constant as appropriat high pressureso 31--~~ 1--~*1~ r~rr. 39 of the melting point of acidio silicates with depth have been made but a reasonable assumption is that the melting point of the acidio traction at depths lies between that of the iron and basio phases, Due to its lower melting point, the iron fraction would be the first fraction to melt in the warming earth. Once the iron fraction melted, differentiation would begin with the dense iron phase settling towards the center of the earth, In settling, aravitational potential energy would be released and be converted into heat energy through viscous dissipation. The temperature rise aooam- panying this release of heat would help in the melting of the acidic fraction, Once melted, the acid fraction containing a large fraction of the radioactive elements and volatiles would be forced towards the surface. The upward migration of the acid fraction would continue until an unelted zone (i.e., low temperature zone) was invaded. The upward motion of the acid fraction of this material would stop in this zone forming a low density layer at the bottom of the uzmelted region. The amid fraction would cool and solidify unless the unmelted region was fractured so that the acid fraction could move adiabatically upwards through the channels opened by fracturing or else the near surface material was melted by some prooess. Geophysical evidence indicates that there are no large scale (i.e., of oceanic dimensions) fluid chambers JIQI*-UI-Q-"~-l"1~4LUs*-C-LL -- -* 40 in the upper part of the mantle. The assumption that differentiation was not completed in the early stages of the earth's history implies that the near surtfaoe region was to a very large extent unbroken and unmelted and that the acid fraction has solidified. The existance of a low velocity layer is implicit in theories of alow accretion of the continents. It pro- vides a physical feature that may be found by selmic or gravity methods. This investigation would determine if the ideas of incomplete differentiation are plausible. The teohnique to be used in such a study is straightforward and need not be elaborated on here. Initial Distribution of Acid ?racti n for Near Siarfac To determine the initial distribution of the acid fraction for use in the following study of near surface differentiation, it was necessary to run several cases on the computer using the ftonmof the viscosity given in equation 2.16. The initial temperature distribution was taken from model 6 in figure 2,1. The requirement oif no significant change in N in 100 million years was used as a criterion of when a stable configuration was reached. The initial value of N makes use of the observation that differentiation in the model below 100 km is so swift that the distribution after the initial run can be approximated by a function of the form _ LIIII_--~ -.-111 1~11-~--21_ ?b 0s o o N= 6o0km A S3 60 kam < Z 4 90 km C Z >90 km The strain energy contribution was taken as zero. The runs are described and the results presented in Table 3.4. The ooncentrations for the last run in the cases listed in Table 3.4 are used as initial distributions for the study of near surface differentiation in Ohapter IV. Larger values of the radius were used but the time scale for the separation was so short that it was not msaningful, The time scale for a = 1 cm can be used as a lower limit for larger values of the radius. the time scale of such a.process could be estimated from the equations given if the value of the radius of the tubes appropriate to the melted region could be estimated. However, this is not possible from aailable The time scales given in table 3.3 for information. a= 10 an upper limit to the possible time scale. 4 amS give The ohange in the time scale caused by assuming a larger radius can be estimated as argued previously. The previous runs indi- eate that if the effective radius of the pore spaces in the melted regions are contimaters or larger, the initial differentiation would be very swift, requiring 100,000 years or less once the partial melting of the acidio fraction occurred. & ~ ---~- _--LI~T=-~-- ~-L-slL3~CIIII~~ ~-~-f~-- TAB LE 3.4 RUNS TO DETERMINE DISTRIBUTION OF N FOR ATURE, T, IN OK. n= N(Z,t)/N(O,O). QE= 0 FOR GIVEN IN PARENTHESES AFTER RUN NUMBER. INITIAL CASE 10 DATA E'=10 kca/lmol, a = 10 cm N(O,0)=3.18x10 tubes/cm7 conditions at end of run 1 (16my) 2 (100my) T 0 15 30 45 60 75 90 300 523 693 980 1238 1493 1693 n n n 1. 00 1.00 1. 00 1. 00 1. 00 1.01 1.0 13.99 5.00 1.69 5.00 5.00 NEAR SURFACE DIFFERENTIATION STUDY. DEPTH, Z,IN KM, TEMPER ALL CASES. A = 5.3 x 10 callcmr.sec. ELAPSED MODEL TIME imy =106 yr CASE 12 CASE 14 E'= 20 kcallmd, a =10 cm E'- 10 kcallmol, a = lcm N(O,0)=3.18x10' tubes/cma N(0,0)= .0318 tubes/ cm' 1 T n T n 300 517 742 1000 1237 1.00 1.00 1.02 14.96 .98 300 483 659 799 845 1.00 1.00 1. 00 1.00 5.44 6.80 2.72 CASE 17 E'= 20 kcallmol, a =lcm N(0,0)=.0318 tubes/cm (100 my) T 1 300 525 742 942 1118 1242 1299 CASE 25 E'= 10kcall/mol, N(O,0)=3.18x 10 (1.lyr) n T (7.8yr) n T 3 (.44my) n T 1.00 1.00 1.28 15.68 300 1.00 523 1.00 693 1..9 CASE 28 E'= 20 kcal/mol,_ N(O,O)= 3.18x 10 a = 10 , cm tubeslcmz 1.00 1.00 1.00 12.07 a= 10 cm tubes/crm 300 523 693 980 2 300 523 693 conditions at end of run (2Oyr) 2 1 Z 0 15 30 45 60 75 90 T 300 523 693 980 1238 1493 1693 n n 1.00 1.00 1.00 1. 00 5.00 5.00 O 5.00 1.0 1.00 1.00 1.26 15.13 T 300 523 694 98 1238 (.17my) n 1.00 1.00 1.00 2725 T 3 (100my) n 300 1.00 523 1.00 6941.00 980 27.25 T 300 401 483 528 1 (1.1xlOyr) n T 2 (.7 n 1.00 300 1.00 1.00 523 1.00 1.00 693 1.28 12.02 980 1560 3.86 1238 my) 3 (100my) n 300 1.00 523 1.00 694 6.47 980 10.41 T 300 402 484 529 (. O my) 1 2 (100 my) n T n T 1.00 1.00 1.00 1.26 14.57 300 523 694 980 1238 1.00 1.OC 1.00 4.56 11.32 300 456 594 697 752 43 At the end of this stage of differentiation, earth would have the iron core, the a basic mantle and a mixed near surface region containing excess acid material collected in masses buried at some depth, radioactivity would be in Most of the the low density layer so that the heat production in the deep interior from this source would be negligible. Compris*on oftodel to Other ,odels This model is based on the aCrrent ideas of partial melting as contrasted to the earlier earth models which were based on total melting of the earth. The essential differences between models based on partial melting and those involving total melting are that partial melting permits the retention of a solid outer shell at all times. It is only necessary to supply energy to melt a small fraction of the earth in order that differentiation occur. The completely fluid earth requires differentiation to be more or less completed early in its history. This model requires the existence of an inter- mediate undifferentiated near surface region. The feature of a solid outer shell allows the consideration of the formation of the earth by a process in which the temperature of the outer shell is the equilibrium temperature resulting from a balance between the absorption and radiation of the sun's energy at the surface. only a small fraction of the earth to melt, Lequiring reduces the 44 energy needed to cause differentiation in an earth formed by a cold process. Both of these features are helpful in explaining differentiation in an earth formed by the current cold accretion hypotheses. In the models where the earth is asmed to be fluid at some stage in its history in order that differentiation can occur, it is still necessary to appeal to "fluid filaments" in the final stages of the separation process. The following quotation from Jeffreys (1959) illustrates this. ". . . the fluid filaments left at any stage carried up the latent heat and the new heat by convection, and that as solidification proceeded the residual fluid was forced up by deformation of the crystals, so that solidification and the raisinc of the radioactive material to the upper layers were part of the same process." This mechanism suggested by Jeffreys is the mechanism used in this thesis to explain the differentiation of a partially melted earth. Consequently, the mathematics in this thesis can be used to describe the last stages in the differentiation of an initially molten earth. ------ --rr~---^-I~--~lru~ ~ II- - ------L-Lsl-~... ---- CI'I--- ..~.-------I~~. .-------- 01111011aII I 45 OWIPTER IV NEAR SURFAGE DIFFERlETIATION From the disssion in Ghapter III, it that seeas the hpothesis of an incomplete differentiation of the mantle requires that the said fraotion be more or less onacentrated in masses at depths smebere between 30 and 90 km. The question is how oan this material be brought nearer to the aurface so that it lated into the continents. It can be assimi- was suggested in Chapter I that any process which attempts to bring this material up from these intermediate depths must be based on a process of partial melting, and tbus, another question is this melting occur. how can Melting is used to mean that the acid fraction has been oonverted from the crystalline state into the glassy state. Once glassy, equation 2.16 describes the viseosity of the aaid fraction. The partial melting of material for this near surface differentiation is vi aalized as the result of the conentration of strain energy released as heat in a rather localized region. Some meohanism similar to that described by Orowan (1960) and discussed in Appendix B of this thesis or to Treitel's (1958) sugestion that shook waves might be very important in the region of failure may provide a mechanism for this ooneontration of the release of strain energy. It is not certain that material in the mantle behaves as Orowaen suglests, but after studying his arguments, it is snees possible that some similar mechanism ative in the earth. If the acid material under the oceans is is trated in masses at some deapth, it onn- should be closer to its melting point and oentain a larger percentage of volatile material than basio material at the same depths. It seems plausible that the masses of acid material would be the weakest regions in the mantle. The release of strain energy should occur in these masses when sufficient stress differences have been set up. This leads to their reuelting, Using 300 cal/em3 as a tyical heat, it value of the latent would require 4 x 10 5 or 4 x 10 6 years to melt the acid fraction in 10 Q w 7.7 x l 13 or 7.7 km3 (10,x 1000 x 100) for 10 Ix 14 cal/ca3.sec respectively. The acid fraction occupies a third or less of this volume. This sets a lower limit on the time required for differentia- tion after the onset of intense asismic activity in a region. This time amst be added to the model times obtained in the runs reported in this chapter, for no provision for latent heat was included in the model, The model assumes that the material is in a glassy state at the temperatures existing and that flow will ocur for T> 0. _ _ II~_~_C__C _^__~_j_______ _III___I__ 47 If the shear zones where some melting has oocurred extend from these masses up toward the surface, the buoyance force would cause the upward motion to restart. The shear zones are not visualied as voids, but rather as regions that have become viscous. These fluid ohannels permit upward motion to ooaur. a process of this type that the equations in approximate. It is Chapter I are intended to The radius of the tubes in the mathematical model may be associated with an effective radius of the channels in the shear zones. The cases run to determine the time soale of differentiation after remelting as a function of the visoosity parameters, strain energy is cases reported in radius of the channels, and the rate fed into the system are extensions of the Table 3.4. The description of the cases and the results are reported in Table 3.4 that the initial Table 4.1. The ease in conditions were taken from is indicated by the number in parentheses beside the Cs.se heading. Cases 18, 26, and 29 indioate that QE = 7.7 x 10- 1 4 oal/cm3 osec is the lowest value of this term for which near surface differentiation would oour in this model. This is the value of ig which is given by n = 1 in equation B.l and is equivalent to dissipating an amount of energy in TABLE 4.1 RESULTS OF RUNS TO STUDY SURFACE DIFFERENTIATION. ELAPSED MODEL TIME GIVEN IN PARENTHESES AFTER RUN NUMBER. lmy= 10 YEARS. INITIAL CONDITIONS TAKEN FROM CASE IN TABLE 3.4 WITH THE NUMBER GIVEN T IN OK IN PARENTHESES BY CASE HEADING. ' CASE 16 CASE 15 a =1cm E=10kcal mol a=lcm E=lOkcal mol Qo .77x10'calIcm-sec QG.77 1'QcaIllcm-sec NITIAL DATA conditions at end of 1 (100my) T n 300 1.00 626 1.00 841 28.5 -947 (100my) T 300 746 960 1176 n 1.00 2591 - n" 100 100 1.00 273 27.3 !1 (lOOmy) T n T 3001 1.00 300 523 1.00 626 694 1.00 841 9948 980 conditions at 1 2 (100lmy 1 (3 5my) 2 (3.7 my) n T 'T n n T 300 300 1.00 1.00 300 1.00 703 17.88 887 1.00 627 1.00 1142 16 87 842 16.87 907 - (100my) T T n n Z 300 1.00 O 1.00 300 627 1 5 100 523 1 00 842 30 1.29 694 1A8 45 60 5.5l 98 5.35 CASE 949 of - 1136 CASE - n 0 1.00 15 1.00 301.00 45 1.00 5.00 7 5.00 90 5.00 T CE=.77x1 calicm 3 sec conditions at end of run 1 (7.7my) 2 (12my) T n T n 1.02 300 1300 30C 1 00 523 1.00 697 17.63 1061 .33 1440 693 2.39 934 1689 98 14.53 1206 1455 .01 123E 149 693 (5.1 my) T 300 770 990 1 1201 1331 a = 10 3=1 QC-.77x10 caI/cmsec 1 n 100 8.53 9.43 - T n 100 300 100 523 1.00 694 1.26 980 14.6 1239, 1 n 1.00 1.00 1.00 6.16 9.72 (1 00 my) T 300 629 874 1037 1119 1 (9.2my) 2 (4.4my) T n T n 300 100 300 1.00 100 830 11.38 1006 13.69 1095 649' 1327 316 1340 - 1538 - 13 a:16dcm E = 20kcallmol E=10kcallmol 10kca Z (100 m 1 I n n T 1. 00 1.00 300 1. 00 1.00 639 28.3 100 864 976 27.3 run - 11 2 CASE 30 CASE 29 a=1cm a=10 crr E= 20kccllmol E=20kccl/lmol QE=.77x10'2 call cm sec Q =.77x10- calk.m .sec E E 2 ASE 27 CASE 26 -z 2 AE 2 a=10 c emE= 10kcal/mol E=10kcallmol a=10 OE=.77xlO-'callcr sec QE=.77x10 cal/cm sec E E end 19 CASE CASE 18 a =lcm E=2Okcal mol a=lcm E=20kcal mol Q .77xlc'cal/cm. sec Q .77x-1'al/cm.sec run 1 n T Z 0 1.00 300 15 1.00 523 3028.5 693 4 5 977 INITI AL DATA (43 my) T 300 1218 1784 2128 THE WRONG INITIAL CONDITIONS WERE USED IN THESE CASES. 11.6 SHOULD BE SUBTRACTED FROM THE LARGEST VALUE. n = N(Z,t)/N(0,O) where N(O,0)=.0318 x -? ~B~P1------------ -III-.rCIC-L .~----r~....-ql_ _ -- --~--..---II-_----_~I~lli~--XIIII I1II 49 the region of failure equal to the annal amount of energy carried in seismio waves. The elupsed model times indicate that the tie required for the material in the low density layer to reach the near surface region (i.e., near 15 k) is not appreciably different from the time required to melt the acid fraction. -4th million years is acid fraction. = 3 *See, 7.7 x 10"1 3 cal/onM .4 the estimated time required to melt the The time required to mov thehe material to the surface is 12 million years in ease 11 and one million years for case 16. This leads to the conlusion that if the energy sources are available, differentiation would require 1-50 million years after the onset of intense seismic activity in a region. The elapsed model time depends rather strongly on the value of the radius and the viscosity parameters. Very little can be said about the effective radius of the tubes. a = 1C-4 om is probably the aallest the tubes can be, and if the tubes are larger than one centimeter, the time of separation is negligible compared to the time of melting, The visaosity parameters were deter- mined independently and are thought to span the possible values. Oensequently, there is reason to believe that the time scales are representative. 21mmeion of esuts The most important result of the oaloulations is that the minimum rate of strain energy fed into a region _ __II_~ 1_1 ____ _ ~I 50 of failure required for near surface differentiation maust be at least as large as the annual rate of energy release indioated by the energy in seismiO waves for the entire earth. Unless one of the suggested mechanism or some other process can provide such conOentrations of energy, an intermittent near surface differentiation does not seen possible, and the major differentiation of granitio material in the earth muat have occurred at some time in its past history. The existence of molten lavas indicates that the melting of material is possible. This implies that either the energy sources are available or the estimation of the required energy is too high. If the energy source is available and the low density layer was formed, the existence of such differentiation seems possible. The time required for the differentiation of a local region is not unreasonable and the process is straightforward, ORHA TEI1 V 00NCLUSIONS A4D SUGGESTIONS eOR 'YmUETHE This study indicates that there is ORKi a physical process based on partial melting of silicate rocks that could be responsible for the segregation of an initially homogeneous earth formed by "oold accretion". Differentia- tion would not be complete in the early history of the earth if the surface region were unmelted and relatively unbroken. A feature of this process is the formation of a low density layer containing an excess of acid material at depths of 30-90 kas. Table 5.1 gives the depth to the top of the layer for the cases run on the computer. The existence of sach a layer is shown to be a oonse- quence of the hypothesis that the formation of continents is a continuing process and any model which explains the slow growth of continents mast contain a similar feature. The temperature distribution required for differentiation is not in excess of the temperature thought to exist in the ear-th today. Computations indicate that the time soale of this early differentiation is 100,000 years or less after the aold fraction melts. It is suggested that the periodic addition of acid material to the near surface region is a result of the reelting of acid material in local regions of the low density layer. The reaelting is attributed to tho 52 TABLE 5.1 W DETH TO TOP OF LO TO TOP OF L.a DUENITY LAYrl. L = DBETH DENSITY LAL IN KM. S OISLS 2 OR ALL QAZS. DATA IS Al~TRACT D aROe BTABLsi Z3.4, Casn Iumber and Desription II__ I____I_ _ _ Oase 10 Case 12 ~___ON -mPAm 0 i i I W- M 0_-UIII~II Zt a I _I_n-*_~ a, 10 - oC VEa 10 koal/mol 45 a 10 - 4 am Za n 20 kinal/ml 60 E' 10 koal/aol 30 ' = 20 kal/m/el 45 Case 14 ase 17: Oase 25: a = 10 - B' Case 28 I0 koal/mol a* 10E 2 2 45 am = 20 koal/mol 60 53 local release of elastic energy as heat due to failure by some mechanism similar to that suggested by Orowan Once the acid material is (1960) or Treitel (1958). remelted, it would move oloser to the surface through partially melted shear zones above the melted regions of the lw density layer due to the action of the buoyancy force. The acid fraction in this stage of differentiation does not reach the surface (i.e., volcanic phenomena), but rather stops at depths somewhat above 30 kas. No attempt has been made to disuss the difficult problem of the assimilation of this material into the continents. 7.7 x 10 14 oal/cam3ee is the minimum rate energy can be fed into the region in order that remelting and the near surface differentiation can occur. This is equivalent to dissipating the same amount of energy carried by seismio waves in the region of failure. Unless such a source of energy is provided, it is doubtful that near surface differentiation can occur. However, the molten lava in volcanoes indicates that perhaps such sources do exist or that the estimation of the energy required is too high. If the sources required do exist, calculations indicate that the tie se sal for the upward movement to be completed after the onset of intense seismic activity is not impossibly long. Por example, if the effeotive ____I__~ I __ _~__ji_ _~__ I rauias of the shear zones are between 1-10 - 4 ____ ~___~_ om, the time scales are betneen 1-50 million years for the values of the viscosity used, This includes an estation the time required to remlt are reported iUn the material, of Those results able 4.1 the low density layer seem to be a requ4red feature of earth models which attempt to explain nenat formation by a continuine prooess. conti- Therefore, a thorewh inveostigation should be made to determine it such a layer exists. It is well known that seisml teohniques have not been able to determine whether suah a layer does or does not existE is Gutenberg (1959) evidence of the exi tene However, and others show that there of a loww velooity layer, the evidenoe is not conlusive as to the existence of a low density layer, for the low velocity layer ay be interpretated as due to a doorease in reuultiw, the elastic oonstants frm the temperature increase. An aproaoh using the orbital path of satellites may yield an anawer to this ~roblem. underneath the oceans is or if the mantle undif erentiated compared to continntal regions, there should be ant ~smetry in jravity field ot the earth that is of the continents. It the not due to the presence would take the form o a mass ____ ~_ detiotenoy in the oceanio regions. Siune the sateflites are very sensitive to large scale deviations of small magnitude, orbits. this asymmetry would measurably affeet their 56 YVISOOSITY 14r4eV Orature DepeAence anad -T The flow ofailicate melt has been desoarbed very Muoesasfully in the 41ass induetai by espirical relationships based on the assumption that the flow is ewtonian and that Seperature dependence of the, viscoslty has one of the two forms to5 r. Lo £ '/ (T- A61 o +E Wi> A, + and Lo whre ko and E and 16' measured in and I\. =Lo are the viYsosity at infinfinite temperature are activation energiese. 0 K. The temperature is The Kelvin seale of temperature will be used for all tezperaturee unless other scales are specixfically aindiated. The eonsiderations of several workers in the field will be reviewed and extended in order to determine the temperature -anre over which each of the above two expressions are applicable and the values of the constants which are appropriate to earth materials. -s~-- -I-- _~ II_ - :1 Frenkel (1946) has examined the viscosity of liquids as being the result of the movement of discrete ions and obtains Y e.xp P where I, interatomic distance l the , a roelaation time; k, n constant; 1, the activation energy for the flow; the Boltzs oC, /kT oeffiient of thermal expansion; 1i, the bulk modul sa a pataeaster giving the pressure i, the pressure; and Y, It is noted that the equation given by t*renkel dependence. has the temperature dependence of equation A.1 with the additional feature of pressure dependenoe. The following interpretation of the constant r due to Professor Hughes, permits an estimation of the magnitude of the pressure dependenae. is defined in ?reakel by vriting ---- / anduaing I T T 0V v r--: T i s only a 4notion of the intera~-t o a7 which assumes that distaus e &ivee 4 _ _____1_C_____Ym__s__~NLII__________III ti8 Giving k where V is the Spectifio aA/.4 olume. The experimental value given by Hughes (1955) for the activation energy for cations in the ionic semiconduation meohansm for peridot S AA.5 is used in order to evaluate the manitude of d O= 6 10-5/0K, k = 1.38 x 101 Orgs/*K, and a=nx Taking 10 " 1 2 ergse gives rY: ./74 1 Dane and Biroh (1938) examined the effect of pressure on the viscosity of borio anhydride glass for pressuare to 2,000 bare at two temperatures. The best it to the data is given by r' 1 ex pp p withok = 4.6 x 10-4c2/kg at 5160G., 15 x 10-e4 M2rg at at 35900. up There is insufficient data to draw ay firm rl---l~lrl~ 1 --~^-CIILY r~YIIY PII~- conolusions from this except that the pressure dependence does exist. Xaoienzie (1957) studied the effeot of chemioal composition oonviscosity by examining polycomponent systems of silicates. He finds that equation Ad. is universally obeyed for binary alkali and alkaline-earth liquid silicates, and that the aotivation energy E i independent of the is Oationia speoies at all compositions up to 55; metal oxide in alkaline-earth systems and to 35; metal oxide in alkali systems. Figures A. and A.2 reproduced from Mackenzie's paper give the behavior of the activation energy and the preexponential constant, respectively, on the composition, and cationia species. Aaakenzie (1957) demonstrates that the behavior of polycomponent liquid silioates can be deduced by an analysis of the binary systems which are included, If the oations are of the same group, the activation energy is dependent on the mole fraction of the metallio oxides. If the cations are a mixture of different groups and the melt is regarded as an "ideal" mixture of the individual binary melts, the activation energy is given by E= where na (E ) +h b (Eb and nb are the mole fractions of the two metal exilea and (Aa) x and f)x are the activation energies for A.6 CYII~ ~Cly I~~~yy~~ for the two groups at the total metal oxide ooncentration x. Equation A.6 oan be used to estimate the activation energy and the preexponential constant for silicate melts that have not been measured. This is applied to a plutonio granite to illustrate the use of this equation, Barth (1952) gives the composition of a plutonic granite as being 70, and 30 meatallic oxides. i02 The potassium and sodium oxides Treating the other 24.5% of the metal oxides as are 5.5>. alkaline-earth oxides and using figures A.1, A.2, and equation A.6, gives the activation energy tor a melt of this comonpositio ase = 45 koal/mol. The preexponential constant 1o is approximately 10 - 4 poises for a 30 metal oxide melt. Brookris and Lowe (1954) studied the aotivation energy of the system CaO-1i0 2 as a function of temperature and per cent metal oxide in the melt. The interpretation of the results xas given in terms of the flow of a discrete silicate anions. If the melt contained 52, or more metal oxide, the flow unit was a silicate anion chain. the chain was one unit long. At 66A metal oxide, For the ranle 12-5Z4, the flow unit was of different structure than a chain. less metal oxide, molecule. the flow unit was tu ought to be the silica The behavior of logney. some ourvature. For 12/ or 1/T was found to have It is sugested in the above paper that the change in sativation energy at the hi4her temperatures can be explained by assuming that the bonds of the flow unit are weakened by the addition of thermal energy and that the -~ -IIIIICrPIIIIII ICI ~-~LIC--~-~)IIC~*Y IIC~L--~PIIIII 1 shearing stress is more easily relaxed by a breakdown of the complex ion into simpler (and smaller) species than by their flow. energy. The smaller anions have a lower activation This ourvature in the plot of log r vs. 1/2 may be explained by the introduotion of an activation energy for short range order as follows. Dienes (1953) considers the effect of short range order on the viscosity by using an expression given by Frenkel for the temperature dependence of the short range order and obtains p k(T-) A. where B is the activation energy for visass flow in disordered liquid, U is the structure activation energy of Jrenkel and/is the viscosity at infinite temperature. If U = 0 or T is large, equation A.7 reduoes to equation A,1 and if U>E and U/k (T - To)> 1, it reduces to equation A.2 In equation AT7, Dienes finds that To is a constant and is approximately 5000 K. for some silicate glasses. Table A.l which is part of Table I from the above paper by Dienes is given to show some values of the above quantities, and the closeness of fit is obtained by the use of equation A.7. The range of viseosity oovered by the data in Table A.1 is based on 102 - 1014 poises and the temperaturea are between 5400 0.-1400O C. LII^X~~ IIIYIIIYI II/ _~1^ b2 45s TABti A.d Parameters for some ailicate glasses fitted by This is part of Table 1 in Dienes (1953). Curve A.7. AVg. devia- 0 flo T0 in % 0J POies a/ 2 -2.135 560 18.28 11.77 .073 i 10 -1.446 560 4,57 17.11 .029 # 330 -2.249 562 18.28 10.47 .002 Silicate a eea NoTs Logl E U tion in k±l122l10 L Visoity of gins )szaples are reported by Robinson ad Peterson (1944). Dienes describes a omputational sheme for caiao- lating the paran eters in equation A.7. However, the avail- able data for earth aterials which eonsists of three or four measurements per sample does not Justify the appioation of A.7. in fact, even 0 in equation A.2 annot be determined with any reliability by a oomputational *oeaso A typioal value of C for equation A.2 is determined later which is used in the oalculations. A better understanding of the effect of the parameter C and its importance in the model can be obtained from a study of the measurements of 3bartais and Spinner (1951) on optical silicate glasses. for sample #8198 which is The results inclxded are typoeal of all the samples studied. Three ourves are plotted in Figure A.3 to show the effect of the parameter C. Iog viscosity in poises is plotted vs. 1/T, 1/T-273, and 1/2-473. The aurves were fitted by eye, -Y--~~n~-~-~_-_-r Y P---c- -I-- lpllli -~LIII~ lrm(( As can be seen, there is a seleation of 0 which removes the ourvature. The difference in the predicted vieoasity between a curve including 0 and one omitting it Figure A.4. vs. I/T. is shown in in this figure, log visoosity was plotted A least squares fit of this data by four curves ( was made where(:toy - -I; io-3 an td ,,. In fitting aurve A10O, ct : 4 ad i r Tk 3 the term b/t was found to be so small that this form could not be distinguished from A.9. Therefore, the form A.10 is omitted from the disnussion. The ourves oorresponding to equations A.8, A.9, and A.11 are plotted in Figure A.4, Table A.2 gives the values of the parameters for the three ourves and the differences between the measured values and the values given by these From Pigure A.4, it i s seen that both the linear form and the 1/T-c form give a good fit to the data in the ly ___~ __1 _11__1^_ ~1 1__111^ _ _CI__ _ll___ __~__1.^_ 1___~ __I_(~I I~_~_~_I_~_ interval of temperature where data was taken. The differences between using either of these two forms in the calculation in this interval of temperature away from the melting point are much less than the error associated with the uncertainty in the chemical composition of the material. However, the agreement between equation A.8 and equations A.9 and A.11 is low temperatures. rather poor in regions of The values of viscosity given by the form A.8 are much lower. Birch et al (1942) lists the measurements of the viscosity as a function of temperature for several rocks and artificial natural rocks. materials with a composition similar to The data given for seven of these samples are plotted in Figures A.5, A.6, and A.7. Values of O7 and E determined by a least squares fit of the data are given in Table A.3. 9. and E' was determined for two natural lavas using 0=70 0 °K (the selection of the value is explained later) and they are given in Table A.3. The curves for these two lavas are plotted in Figure A.7. Table A.4 gives the comparison between the fit obtained with 1/T and l/T-C temperature dependence. There is a definite relationship between the chemical composition and the value of the activation energy as expected with the more basic material having the lower activation energy. and ? However, the values obtained for E using a 1/T dependence are higher than those r_------I~ LIII--~ - -l_ I CL__IIIIIIIYIIIIIIII~L __ _/I4__1C__ls_ _^IC~s~_ predicted by use of the curves in mackensie,'s paper for a This glas y melt with the same per cent of metal oxide. discrapancy arises from the ourvature in plot of the Via-The high values of cosity vs. 1/T for the natural roks. the vis osity at the low tenperature *ad inerease the slope of the line and decrease the interoept at infinite temperature. The physieal interpretation of this is not certain. There zre two Poesible explanations that have some merit. is The first is it even if assumed that theequation A.8 is valid down to the solidus point of an individual component, this relationship cannot be valid in a polyoamponent system due to the dependence of E and ?. on the ohemical composition of the melt. melt is lowered, he temperature of the polycomponent When the crystale forn over a wide temperature range. These orystals remove a greater portion of metal oxide than The remaining fluid would contain a silica. cent of metal oxide. aller per This would lead to a higher activa- tion energy and a smaller the activation energy is q. for the remaining fluid since larger, for more acid melts. temperature scale of the measurements reported in V( vs. change in 1/T to be detected if it existed. slope would be continuous. If the litera- the slope of ture would not permit a discrete change in log The apparent care is not taken to insure that equilibrium conditions were reached, experiment would show a discrete change. the lit dependence of the visoosity ie The Hence, no even if correct for a simple ~C1~_ system, It iis necessary to use a 1/T-C dependence in a complex systsm such ae a granite to describe the viscosity as a function of temperature when taperatures in the range containing the melting point of the various components are included. The second explanation is in that there is an increase the short range order as suggested by Dienes as the approached even for a single component solidus point is system, and that the curvature is There is too little due to this phenomena. known about the behavior of the vie- cosity as a material passes throSh its deoide the melting point to question of which of these explanations is the most realistic. One group of authorities look on the transition from a orystalline to a liquid state as being analogous to the liquid to gas transition. While others view the orystalline to liquid transition as being transition. Boreliu of both groups, (1953) esaentially a discontinuous gives a review of the arguments and presents his arguments for accepting a smooth transition from the crystalline to liquid state. The question in an oversimpliied form is, does a critical point exist for the crystal to liquid transition. Experi- mental evidence on this point does not seem to be conolusive. For the purpose of the calculations in this thesis, it does not matter which of these two viewpoints are "correot". -li-x) l--rru I---- -----~- ----* --_1.. .__^- II~- ----- - - -~~~-----~-- ~a~-6~ ~~~-*111~1 ._---~-~PPIICI~II~L 6 It is enough that the viscosity is described by either equation A.1 or A.2 for the temperature ra4ge of interest. In either oase, the forn A.1 is a poor predictor formula when temperatures near the solidus point must be considered. Two forns of the viscosity are used in the thesis. They are ku / / and (/ where the hydrostatic approximationP = pgZ, for the pressure has been used sad X = 1.5 pgAk. The first relationship is used when the melt is at teamperatures relatively far above its solidus point, and the second form is ued at lower temperatures. oulations in the thesia, the first form is In the cal- used in the study of the initial separation ad the second form is used in the near surface differentiation. th ffot of . Vat Jose the Visaoit y Obervers have remarced on the effect of water on the viseosity of silicate melts. It seems that'et melt" are less viscous than "dry melts" of the same chemical composition except for the water. A rather extensive _YI^L~PPL Y~ _L~PL~ 63 literature search did not disclose any experimental work oontrastine the viscosity of a "wet melt" and a "dry melt" of the same cheaical composition from which any aatitap- estimation of the effect of the water could be made. tie Therefore, it is necessary to use what data is available to estimate the effoot of the water. Minakami (1951) reports that the viscosity of lava as measured from the 1951 eruption of the volcano Oo-sxi in the a. Samples of ield was 4 x 103 poises at 11500 the same gross chemical composition when remelted in the laboratory had a viasosity of 5 x 104 poises at the same temperature. The differenae in the viscosity was attributed to the richer volatile content o the natural lava. It it is assumed that the effeoot of the volatiles is to chane* the activation energy for visoous flow, the relationship wr where bi n -dry" ?4ry A12 wet and T1 is the temperature in % at which the measurements were made. equation A.12 oan be used to estimate the change in the aotivation energy due to the presence of the volatiles. several values of is at Ti Table A.5 gives &E for 1423 O. SE = 7 koal/mol indicated by the measurement on the 0o-siza lavas. activation enery for such basaltlc lavas is 30-40 keal/mol. The approximately A tentative estimation of the effoet of - ~ IIIIIIWLII--~LPLllsl~sl\lP ____ _l__pl__lllUI__WI____II___~-*~L3YIIOI~I; the preeenae of the volatiles is that it can cause a chauge of 2< in the activation energy £. TABUE A.5 b kcal/mol t4 ar/ 4 3.9 12 7.0 13.0 100 David B. Stewart of the United States Geological onn the Survey has furnished some unpublished informatio sttling rate of crystals in hydrous silioate liquids from which the viscosity of the liquids can be estimated. The data and the caloulated visosity for four of the measurements are given in Table A.6, The viaeosity was calculated using Stokes approximation for the slow fall of spheres in a fluid (i.e., U = 26pga2 /9z where 'a' is the radius of the aphere and the other symbols have the same meaning as before). T-ABLE A.6 SiSe in Qent Microns jj ?rosjjr# iare 2,000 5,000 5.5 10. 7 5-10 10,000 12.5 4 10,000 12.5 4 NOTE: Di*- }zrtiole bAPt Tp ( 0 ) " 1040 An7OSi30 900 itj Qomposi- Plaoeaent tion 1320 AnOSi20 1190 go Vias- " in Poises .163 .6 .25 .3 .35 .233-.933 .15 .43 .367 .10 .43 .51 An stndo for anorthite and Si for silioa. _~_ The data is not u floicently extenslve to determine the s4atition energy and the preexponential constant for these sibstances. However, the extremely low valaes of the visoOsity found for these wet melts indicates that the preeanoe of water vapor is a very important parameter, and that the estimation of a 20%, ohange in the aotivation due to this ascse is probably not too high. The density differenoes 3 of .4 g/o# between the hydrous silieate liquid and the orystals indicates that the value of .5 Wrm3 used in the thesis as the density difference between the basic solid and the liquid aid fraction is not too high. The data supplied by Stewart also indicates the Iaportanoe of crystals settling out of even suah a relatyiely simple system as given here. In particular, Stewart mentions that the settling of crystals is an unwanted phenomena for equilibrium is diffi lt to obtain when settling oecurs. The cheeical composition of a plutonio granite should be typioal of the omposition of san "acid fraction" which can exist except for the volatile content. Consequently, the value of the aotivation energy of a plutonio granite previously oaleulated from Fig re A.1 reduced by 201 is used as typical of the value of the activation energy for an "aCid fraction". This valne is 35 koal/mol. Figure A.I ~~. .---^^-.- l-rM.111411, .. .1il indicates that the activation energy omanot radioally or differ from thiis value for my melt containing 20 Therefore, more metal oxide, three activation energies are used in the calaulations where equation A.1 is used as the form of the viscosity to include all possible aoid The They are E = 20, 35, and 50 koal/mol. fractions. range could be narrowed if the quantitative effect of the water vapor were better understood. preexponenial aconstant, 1, val The value of the is taken as 104 potea. This value is not in error by more than an order of magnitude for melts oontaining 10-50Q El'to of metal oxide. and C ointed out, there is As has booeen a soaroity of data on the temperature dependence of the visoosity ih the region near the solidus point. A result of this that the values of E' scaroity of data is are not and known as well as E and fo , but as has been shown, the 1/T-C dependence ast be used <or temperatures in the melting range of a complex silioate. Therefore, it is necessary to estimate as well as possible the values 2', 16 and C for use in the thesis. The values ' = 10 and 20 koal/mol are used as being typical of the activation onergy of an acid fraction. The values are typical of the activation energy of the natural lavas and the glasses of Shartsai I 10 - 2 and Spinner. poises was used for the preexponential constant. r~Y"CL"Y-YI~Cr~rrrraPIIIPslll ~----C-XI--;L---a--- 0 = 500 PI~------- O,. is a typical value for the glasses reported in shartsis and Spinner. For the natural laas studied, 0 fell in the range 7000 K.-1000 0 K. is the valio reported by DiaSes for his glasses. m 5000 K. The value of 0 adopted for use in the calulations is C = 7000 K. The unaertainty in 0 is indicated by the preeeding discussion. ln the calolations where C is used, the viseosity is made infinite for 2T% temperatures. . This ignores flow at very low The main effoot of changing 0 is to change the depth at which flow becomes negligible (i.e., the visoosity is made infinite). The uncertainty in the temperatures that exist under the ooeans would not permit determination of a partioular value for this depth even if 0 were very well known. TAX 1E A.2 LUAEST SQUA3M P ITO DA IM SAM esoon, m cmv. mC . .670o + 10.30 8- x /t .9 9-. AM SPI Mp 1951) . l OmAP x 5t. - 1.409 + 3.710/t - .473 .11 log l 103/ = j198 (swAt9si INzSe A*6 A.9 /A m / A.8 * / .9 /a- &U. .6398 1.961 1.859 1.968 1.995 .102 -.007 -. 034 .6789 2.306 2.256 2.286 2.301 .050 .020 .005 .7287 2.736 2.764 2.719 2.718 -.. 8 .017 .018 .7862 3.241 3.347 3.254 3.234 -. 106 -.O~k .007 .8525 3.9 4. 3.924 3.891 -..120 -. .011 .931 4.771 4.8t21 .78 4.761 ..o5o *.0t7 -. .oo 1.0163 5.833 5.687 5.810 5.851 .146 +.023 -.018 --. IU : ,L_:. - _":.- _:. - :. : ., .. :;,. ::: . _--.- ,--,'- ,: :, -:.... ..... ' " . . -- " : -': :_: :"":: .: ':::: : :-, "" ... :":.:- - -. -: -:"-- : -- - - "-- - - - - 6,L C96* OL 69a 9 OnTAMOA *U ISAV1 50T 9a~e - 9LEO0Th 96-2 Te S VlaW 001-0 a9W 1 9Wv - 0 1F0 9 4TV 0q*90 - 0-'09-0a a,o 90*. 5Qgla4 . miiuw 1 my szrsmd la=WS , asda uv-,rv-1a U~l-o 0-090't 1 (ai6t) V M~ RhIH MW WR~DLVV1~&~Igo - n TqTy YJN ~ESV~~~Y' 1$rtV-t +£C6-- Wit OTt- 9TO~r Z tioo 9oo ov TI TEO* LTD 7 060- ,o WO /nA" OOe 91 aQ*Z MiUt *t azo DSE"L 9,LTV £?66oaC6, Looa a0? 6* C6o 6W 77ou~VV WA3 40 aPil96 I~W~E tNO-;vv M~O M- 198~t SVOV ILU VfO~MWOU S~~Y ZSWYJL M XPJ DS4 X Li aNoa OK * Mg Co A Sr i S60 - i OBa I~ "0. 20- 10 20 30 Mole 5. metal oxide Fe,. A.. 40 50 2 nrgy of activation for viscous flow in binary liquid silicates. S6 7 8 10 20 M ole F A.2. Relation 30 merol 40 50 oxide sand composition for binary between the pre-exponential constant liquid silicates.1t LOG ViscosITY / / / -----V5 I/T /x 3.9 /T-"73 1.,...., //VS - 29. 1/T-473 where T is--- in 5.9 , */ TEMPERATURE SCA LE / i' , i .65 .70 .75 .o80 .75 .0 .85 .90 .90 1.00 .0 .0 .Ao . I i p o .85 .90 .95 I.oo .95 .oo00 1.05 1.to 1.15 ,.20 .30 1.00 .40 1.0 .50 70 I I/T 1.80 I T- 273 1/T-473 I SLOG VISCOSITY / I.. q -.1409 t3.710/x -. 473 7- 6 LOG .= -.554 6.61 X LOG -4.670o -. o.121 5 4 .3 Figure A.4. Plot of log viscosity in poises for a silicate glass (sample #8189, Shartris and Spinner 1951) vs. l/T, 'T in oK. Three curves were fitted by least square technique. x- IO/T X 0o, 3.5 .4 .5 .6 .7 .8 .9 1.0 1.1 /.2 /.3 -- 80 LOG VISCOSITY 5- 4 4 4 SYSTEM DIOPS IDE-ALBITE-ANORTHITE CURVE 1 DI-AB-AN 40-60-0 2 3 4 20-80-0 40-40-20 20-60-20 SYMBOL x 0 C& Figure A.5. Plot of log viscosity in poises for indicated Data system vs. l/T, T in OK. is from Birch (1942). Curves log r = log q + E/kT fitted by method of least squares. 2 107 S1 .536 .567 .. .599 I .630 .662 .693 , .756 I .788 .81/9 "T 1 LoS VISCOSITY SYSTEM ORTHOCLASE-ALBITE CURVE OR-AB SYMBOL 0 60-40 1 2 0-100 7 6- Figure A.6. Plot of log viscosity in poises vs. 1/T, T inoK, for the indicated system. Data is from Birch (1942). Curve, log r = log 1 + E/T, is fitted by least square technique. 5-I .56 *I .57 .58 .s5 .60 I .61 .62. .63 .(o4 .65 103 l" (2 A LOG V15COSI TY 47/ // // CURVE EQUATION 1 log = -5.118+10.426X 3 log = -6.289+10.645X 4 log = -3.076+ 4.439Y 3 2 / 2 log = -0.253+ 1.866Y where X = 10 /T, Y= 10,-700 .700 .750 .o00 .850 .3o0 10/T Figure A.7, Plot of log viscosity in poises vs. 1/% T in oK, for two natural lavas. Curves are fitted by least squares technique. Curves 1 and 2 are for a lava from Mt. Vesuvius, and curves 3 and 4 are for a tachylite lava from Mauna Iki flow, Kilawea. The data is from Birch (1942). ---.--~I^--I------- --- ---- --- I---~^~*---------L-~LLIII~----~"II~ n~__ APPENDIX B ENflRGY SOmiozs The possible sources or sinks of energy which may be important in the model are the release of gravitational potential ential energy as the material sepaates, chemical and phase ohanges, VdY work, radioactive decay, and strain energy which is locally converted to heat by dissipation. An order of magnitude estimation of the relative importanae of the various possible sources is given in order to determine which sources may be neglected and to indicate the error introduced by their neglect. The change in the gravitational energy as a volume V of the basie phase is replaced by acidic material from depths is where A Z is the distance over which the transfer to made, and M is the not mass transferred, that g constant, negligible. 3.1, it is assumed = 0 and compressibility effectse are These last two assumptions are Justified in the next section. rise is o In For the adiabatic oase, /17k~ Ay'A D - "C--~ the temperature where c is the specifio heat. If A Z = 90 km,d p = .5 gm/Em3 , p = O - Sp = 3.5 t/= 3, C =,25 aal/gM O A T = 330 0. The other sources of energy are compared to this source in order to determine their relative importance, The term 4.1 was included in the energy balance equation and was found to be negligible for reasonable values of the velocity of the aid fraction. The .rdV work done by ±nterohan ing equal volumes of acidio and basic material through a distance A Z is (P V di - Pd V)bC V 35 -T b)_ where Vo is the volume exchanged, 1 is the preseure,r i is the coefficient of thermal expansion of the i phase, anad iT is the isothermal compressibility. The relative importance of this term is found by taking the ratio of equations 3.5 and 3.1. Giving -(AT(r) i p -,gz, b dA P 4 3 X 13 '0 10 5 0/m, 2/4, bf -. ,b = 0 3.5 c~/=3 ~ a /*0, d .5 gm/ca 3 , this ratio becomes 7 x 10"4 z/m. The PdV work is seen to be relatively unimportant if the changes occur above Z = 10 3 ks. 86i Obai Cal and Lhaae. Ohat In a phase change, the amount of energy involved in the reversible conversion of mass M = pV is A wn = pA where 3H.2 H is the heat of reaction/gm for the transition. Forming the ratio of equations 3.1 and 3.2 gives p is the density difference between the aoidia and basic phases before the phase oaanres occur, The transitions albite + aepheline a 2 jadeite and quarts = ocesite are typical of the transitions which might oocur in a segregation process. If the ratio in equation B.3 is set equal one, a value of i Z oan be found where the energy released by both prooesses are equal. For the first relationship, Robertson at al (1957) gives A = - 3.6 keal/aml. Ap = .5 g/on3 , and g = 10 3 Using pj 4 3.3 P/Oa/3 , m/sea gives A Z = 49 kn. For the second relationship, MaoDonald (1956) gives AH = -. 22 keal/aol and poes A z= 9 km. = 3.02 g./ca 3 . This gives a aiai aIa...... -... ..... , . The value of A Z found for the first transition is comparable tto the distances material is the earth. transferred in The energy requirements of the phase changes are of the same order of magnitude as the potential energy released by the separation of the phases, and can be neglected in the energy balance equation, Volume changes are found to be negligible in the energy balance consequently, there is no provision for volume changes in the model. This leads to an unrealistic final configuration, for the surface of the granitio material would be the same as the original surface of the ocean bottom. There are large volume changes associated with the phase changes that would ocur in the acid fraotion durin separation. A brief discussion is given to indicate the magnitude of the possible increases in volume in the acid fraction. The changes in the basic he deep depths at which fraction are neglected due to to such transitions would occur. The feldapars at depth would be in a form similar to jadeite (Robertson et al 1957). A 28% volume increase is associated with the transition 2 Jadeite-*albite + nepheline at 1 bar and 250 . The transition coesite-- quartz gives a 14% volume increase (atoDonald under the same eoditions. is little 1956) The percent change in volume affoected by pressure. If 30 kas of aeidic material is converted from the high density phase stable in the mantle to the low density phase stable in the oraust, the resulting increase in elevation would be between 4 and 9 kas. The loss of volatiles to form the oceans ad atmosphere might decrease this by about I km, If A is the rate of radioactive heat produotion/ m3 . see for a granite, the raPioactive source term for the model is obtained by assuming that the distribution of the radioactive material is proportional to tohe distribution of the acidic phase. This gives A8a2N where for a granite isa2N = 1. A = 5.3 x 10 * "13 3.6 Bullard (1954) 4ives cal/ea 3 . seo, and this is the value used in this thesis. The question of how amuh strain energy is released anmually is not settled. However, it seems certain that all of the strain energy released in a local region is not oarried away as seismiaenergy. It is equally certain that the deep and intermediate-depth seismio faulting is not a brittle fracture. Consequently, the energy carried by seismic waves mast be oonsidered as a lower limit to the elastio energy released by failure. I_ _~~__~_L~ _CI_ ___ -1_1 -1-~-1-1111 ~~I_ ^l~l___I~ ~ ~II C8 timations of the amount of energy released each year by earthquakes as determined by relating the measured amplitudes of the seismic waves to the energy of the earth- quake are not at all certain. estimate 10 2 4 ergs/yr. Gatenberg and Piohter (1956) Their previous estimation was 1026 ergs/yr. The potential energy lost in source of energy which is a cooling earth is a stored as elastic energy and To esti- may subsequently be oonverted into heat energy. mate the magnitude of this souroe, a rough treatment of the potential energy lost in a cooling earth is given. The common assumption made in treating the energy released in an earth which undergoes some sort of failure is to assume that the initial and final stress states are hydrostatic. With this assumaption, the potential energy released in a cooling earth can be expressed by oaloulating the difference in potential energy between two spheres in a hydrostatie state of stress that have thehe but a different radius. ae mass in terms of the change of radius, the difference is S(g)' where A R is the difference in (pg 4 5 R radius and 4 R< R. assume that the release of the potential is continuous, It we more or less this can be written as rate of energy release by replacing A W by A W/d t and A R by ARiB t. Hales (1953) gives A/d t/t = 10 cooling earth. 11 O/sec for a a Taking R = 6.4 x 10 8o E = 10 2 dynras/=2a 13 p = 5.5 gm/m3 g= 103 m/seo2 gives A IA t a 2 x 10 2 5 ergs/yr, This is 10 times the observed seismic energy and 10- 2 of the total heat flow out of the earth (8 x 1027 ergs/yr.). Due to the uncertainty in the rate of strain energy released as heat, the value of this term (called Qg) will be assumed onstant with depthdeph sad some muliple of the rate of energy release as measured by seismic nwne energy. The expression is QE a 1 x 10 04/004 where n is a multiple of th 3.7 energy oontained in seismia waves and V is the volume in which this energy is assipated. V is taken a a cube 100 x 1000 x ka3 where 100 and 1000 km are the area of the region and H is the depth to the bottom of the undifferentiated region. The values used in the caloulations are n QE = 8 x 1 3 10 and I and H = 90 km giving d x 10' 14 and8 al/se/cu3 respectively. When the dimension of the region is decreased to iengthen the time scale as described in Chapter III, the amount of elastic energy released in the region is kept constant by increasing the value of E used in the later runs. These Valaue of n were selected because the calculations in Ohapter IV minit ind oate that they are the values of Q which clan differentiation. easily explained. ause the near surface These values are higher than oan be Unless ome meOhaiaM of failure oan concentrate such large amounts of energy locally, It is diffioult to see how the near surface differentiation oould oocur. The following two mechanioss are mentioned in order to indicate possible means of aohieving this concentration, Treitel (1958) suaets that if shock waves ocur in the region of failure, the energy carried in amall amplitude seismic waves is less than .1 of the energy released with the reander dissipated as heat. This would provide a meohanien for conentrating such large amounts of energy. Orowan (1960) discusses the streosses required for brittle failure to ooour even at relatively moderate depths in the earth and finds that the stress differences required for this type of failure are prohibitively large (l bare). 400,000 He then suggests that earthquakes (i.e., faulting) may be due to a graual conoentration and local aooeleration of creep arising from an inherent physical instability. In any mechanism of failure similar to that suggested by Orowan, a large part of the elastic energy released locally is converted into heat. An equally interesting feature of this type of mechanism of failure is that it 111. 1~r _^11~1--MOMMOMMOOMMOMWA provides a means of concentrating the strain energy from a much larger region into the region of failure. It is difficult to visualize any meehanism of failure under high hydrostatic confining pressure which takes into account the observed plastio flow at high temperatures under long period stress differences that would not convert a large part of the released elastic energy into local heat energy. -II ~-_IIIIYIUlm~-IIX_-___ _ I^IYII~-QII~C-PI^ 8' ~ol ~f A PENDIL 0 NUIMEICAL iEAPROXIMATION General The set of simultaneous nonlinear equations (2.1, 2.5, and 2.12) which form the mathematical model cannot be solved in closed form. Consequently, numerioal techniques are used to obtain a solution. The partial differential equations were approximated by a set of finite difference equations using central differences to replace the differential operators. This set of difference equations was solved on the IBM 704 at the M. I. T. Computational Center. The method of solution used is a simple forward marching scheme in the time variable. There is a danger that answers obtained from such approximations will bear little or no relationship to the solution of the differential equations. This can oocur due to the amplification of round off error, to excessive trunoation error, or to blunders. The net spacing and the order of the differences used determine the relationu ships that must exist between the parameters in the problem in order that the solution to the difference equations approximates the solution of the differential equations. In certain linear and quasilinear eases, an error analysis can determine the required relationships between the parameters for a given net and order of differences. For most nonlinear systems, it is not -rr--~l_-~_--~- ----- -D --^^--1----^III -'~X-II-L'I- Ga possible to perform an error analysis. is In such cases, it necessary to experiment numerically in order to estab- lish the required relationships between the parameters. The set of equations that form the mathematical model in this thesis is such nonlinear system. Stability and Co verqne COnditiones Prom analysis of finite difference approxiations to a quasi-linear parabolic partial differential equation, John (1952) derived requirements that the space and time net spacings and the oefficient of the highest order derivative must satisfy it the solution of the differene approximation is to be stable and convergent to the solu- tion of the differential equation. In the difference net, these requirements take the forn < z< In addition to this requireant, the oonservation of energy equationin the model requires an additional constraint. This iss due to the term giving the rate of conversion of the gravitational potential energy into heat energy. Since in finite dAfference schemes, the value of a function over an interval in mated by its value at a point. 8p4oe and time is approxi- There are selections of the size of the time and space increments that satisfy for which the estimation the requirement in equation 0.3 of the concentration of the acidic phase bears little no relation to its or mean value over the inIteaval selected. This leads to dif ficulty in the potential oner conversion term which involves a product of the velocity and the conThe volume selected by centration of the acidic phase. the sLace interval may be swept clean of the acidic phase in a very small fraction of the time interval selected by equation C.1, The conversion of potential energy tera uses the non-zero concentration for the whole time interval. The amount of energy that is apparently converted to heat energy by viscous dissipation can be several orders of the potential magnitude greater than the decrease in energy, In extreme cases, this can lead to apparent tempera- order 10 5 OK. ture rises of of To insure that the amount of energy oonverted given by the approximation in equation 0.1 does not exceed the amount of energy available, it is necessary to apply a condition which permits only one space interval to be swept clean in a time interval. The required condition is For eaoh iteration, the velocity at all points of the net is caloulated and the spaoe increment is velocities. The smallest quotient is divided by these selected. This is _ ______I~ _L _ ^ the longest time increment that oan be used if, at most, one apae interval is to be swept olean in This value of the time increment is selected by the requixzment in a time interval. compared to the value 0.1, and the smaller of these two values is used as the length of the time incre- ment for that iteration. This makes the time increment a variable which must be calculated at each step, but it insures that both equations ;:.1 and 0.2 are satisfied while using the longest possible time interval consistent with the two conditions. The above is the condition 0.2. the orioinal thinking which lead to Oalculations made using the condition 0.2 indicates that the conversion of potential energy term is negligible in the energy balance and on these grounds could be omitted from the equation. However, the conser- vation of mass equation also requires a condition similar to equation 0.2 for it involves the product NU and since the conversion of potential energy term is very sensitive to the violation of the constraint, it is retained in the calculations. An observation based on a study of the literature conoerning finite difference approximations to linear and quasi-linear equations is that the higher the order of the differences used in an approximation, the smaller is the range of the variables for which the solution given by the approximation represents the solution of the differontial _~ ___1~X l~^~~llil L~_~~ 111_ equation. It is work of this type to use preferable in as low order differences as possible to approximate the differential operators and a fine mesh spaoing in order to cover the widest possible range of variables. A check on the accuracy of the approximation was to caloulate the solution for two mesh spacings and conpare the solutions, these runs. Table C. jives the solution for The parameters used were the same except for the number of mesh points and the length of a space increment. Column 1 is Case 1 described in The difference between the two runs is Table 3.1. insignificant. The physical arguments that were applied to check for blanders in the formulation of the problem or the approximations used are (1) the velocity of the acidic phase must always be toward the surface, tare cannot be negative when and (3) (2) the tempera- ta4sured on the Kelvin scale, the possible range of the temperature variations can be estimated. The subscript / refers to the spaoial location and the superscript a refers to the temporal location in difference net. The space increment is the selected by h =H/L where R is the length of the space and L is the number of 97 tABLZ 0.1 __ __"_"A " _",hl"' " i I_ 1". < / ; "''" L _? 1MERATUMI AU2 ,-0 ....... ::: " " . , .,'''_ ' f. RlATTV , : ' : -010111 .. . " ^:: . . . XWtiTE. : m_ 1110 0. im O'"_ ggOR, . . . . . 01 . . - 110-4, . . . L : . . .. . .. UB*S ........ 141 m W.. . . . for spaae izur4mln't and number of space not are tsed. ratis Sm 15 kam, L = 20) N( )/(o0) N(*t)/N(O) P300 300m~s 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 s0r~i~a 699 1000 1.000 1056 1371 1.009 1.026 1873 1.044 300 1.000 505 1.00 699 883 1056 1219 1371 1513 1644 1765 1874 1.000 1.000 1.002 1.012 1.021 1.031 1,041 1.048 1973 2060 2206 1.51 1.041 2309 2371 2392 B($)~(Qf . see parafaers exAcet ROemit for first run of ~a 1 desribed in Table 3.1. (Resulnts are a ~iven in Table 3.2.) L = 1Q) SOk 3 (b Depyt " ""... .979 .831 ~ ReIatIve 1.052 2062 2140 2207 2264 2310 2346 2372 2387 2393 mber of Tabes/aem. 1.051 1.046 1.037 1.025 1.011 *.991 .935 .690 2 IIII~I~ I~- L~ t~l) The time increment I is selected as indicated increments. above. The approximations to the differential operators used are 2C- where f in is the value of the function at the point (m, Z) the net. The velocity equation in / where - /I U6 when the form used, Tm is 0 this approximation is - a (- =t e for the viscosity is replaced in equation 0.3 by T~ - C and the calculations proceed as before unless t the velocity, U = 0, is A 0, then used. The conservation of energy equation becomes '(T - ),-T slE- ll~ ll^_ l~ and the conservation of mass equation is N - N;+,' N " ,. The boundary oonditions on the energy equation are at 0 a0 O and to express the aondition of no heat flow at Z = H, it is neoessry to extend the domain of definition of the temperature to include the point N + 1, and define The initial temperature distribution is The boundary condition on the ooneervation of mos equa- tion at Z = 0 is The use of the equations are slapler if the oon4ition of no mass transport &cross Z = H is satisfied by extendtg the domain of definition as in the temperature case and defining N+mL + 1 :0 which insures that no mass is transported across the lower surface. Equations C.3, 0.4, and 0.5 are the equations programed and run on the computer. _IIX__ BIOGRAPHIAL NOTE J. Downs was born and attended school in Savsanh, Georgia. He graduated from the Georgia Institute of teohnology with highest honor in 1956, receiving a B, S. in Phylrsis Institute of Teohnologyj At the Masaehusetts ho had a two year tuition scholarship (1956 and 1957). He was a National Solence Foundation Fellow for two years (1958 and 1959). -r~ ~-~-~--- ------- --- -L-r~L~ _~ . ~II_ ^-I II_. II-~l-X~IJ/ BIBLIOGRAPHY Barth, T.F.W., 1952, Theoretical Petrology, John Wiley and Sons, ino., 369 pp. Birch, F., Sohairer, J.F, and Spioer, H.0.,,eds. 1942 Handbook of baysioal Constants, Geologioal Society of Americal Spooial Papers, Number 36, 325 pp. Birch, f,, 1952, 2lastioity and constitution of the Earth's Interior, Journal of Geophysical riesearch, 2, 2 p. 227. Borelius, G., 1953, On the Solid-Liquid transformation in Metals, Arkiv ?or Pysik, Band 6, hr. 20, p. 191. Brockris J. O'M., Lowe, DG., 1954, Viscosity and ths structure of Molten Silicates, Proceedings of hoyal Society of London, A *gh P. 423. Ballard, L. . , 1954, The Interior of the Earth, Kuiper, G., ed, The Barth as a Planet, Chicago University Pres, P'F Jr., and Birch, F., 1938, The Effect Of Dane, iE., Eressre on the Visaosity of Boric Anhydride, Journal of Applied iPhysics, ., p. 669. Dienes, G. J., 1953, Activation Energy for Visous Flow and Short Range Order, Journal of Applied Physics, 24, p. 779. Frankel, J., 1946, Kinetic Theory of Liquide, Oxford University Press, 488pp. Fyfe, W., Turner, 3. J., and Verhoogen, J., 1958, Metamorphic Reaotions and Metamorphia Foles, Geologioal Society of Amerioa, Memoir 73, 259 pp. Gutenberg, 4., 1959, Wave Veloities Below the .ohorovicio Discontinuity, Geophysioal Journal, 2,, 4, p. 348. Gutenberg, B., and 1ichter, 0. '., 1956, Earthquake Magnitude, Intensity, Energy and Aoetleration, Bulletin Seiaaological Sooiety of America, 4f, p. 105. -----rrr-----r~-._ ----~-- l- I-r~-~---l- --- -~-----~..----- I 'PI-C--~ _I. 1IYI Rales, A. L., 1953, The Thermal Contraction Theory of ountain Building, AiAS GS, 6., 7. Hughes, H., 1955, the Presuare on the Eleotrioal Ownduotivity of Peridot. Journal of Geophysical esearoh, §i, 2, p.187. Jeffreys, H., 1959, The Earth, 4th Edition, Cambridge University Press, 420 pp. John, F., 1952, Advanoed Numerical Analysis, New York University institute of athematioal Seienoes, 194 pp. MaDonald G. J. if., 1956, Quartz-Goesite Stability Relations at High temperatures and Pressures, American Journal of Soience, 2~5, p.7 1 3. MaoDonald, G. J. F., 1959, alulations on the Thermr History of the Earth, Journal of Geophysical Research, $4, 11, p. 1967. Makensie, J. D., 1957, Discrete ion Theory and Visoous flow in Liquid Silieates, Transations Faday Sooiety, a, p. 1488. Minakami, Takeshi, 1951, On the Temperature and Visaosity of the ?resh Lava Extru4ed in the 1951 Oo.4imn iraption, Tokyo University Earthquake esearoh 3, 487 pp. t , Part Institute Balletin, Orowan, ., 1960, Mechanism of Seiomi Grigg, Do and Handia J., d*s., FaPulting, p.323, reek Deffo Geological Sooiety of America, Mfeair ti j, pp. 382. Robertson, E. C., Biroh, F., MacDonald, G. J. F.t 1957, rperimental Determination of Jadeite Stability Relations to 25,000 Bars, American Journal of , p. 115. Science, Robinson, H. A. and Peterson, 0. A., 1944, Visoosity of IRecent Container Glass, Journal of American Geramio Society, 2t, ube, 5, p. 129. . ., 1951, Geologic History of Sea Wster Blletin of the Geological Society of America, 62, . 1111. Shartais, L. and Spimer, S., 1951, Visosity and Density of Molten Optical Glasses, Journal of Researc the National BuIreaa of Standards, , 3, p. 176 of IC-'1lllr i ^II -I I - ------ ..I--~ ^1IIIIIYII"'LI~-I --~-~-r--*r~--~I-- ---- 1 Stewart0,D., 1960, ersonal 0 .ation. Trettel, ., 1958, on the Dissipation of Seimi Energy fro Searee to Suarfae, ft. . Thesis, Masashusetts Institute of Teehnology. Wlson, J. T., 1954, The Devml rast, ohicag UnieraG.,I t a Ar,*.. siversity Press, pp. Structure of the "