Estimation of flow duration of basaltic magma in fissures

Journal of Volcanology and Geothermal Research, 37 (1989) 167-186

167

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

ESTIMATION OF FLOW DURATION OF BASALTIC MAGMA IN FISSURES P. FABRE, Y. KAST and M. GIROD Centre G~ologique et Gdophysique, C.N.R.S., U.S.T.L., Place E. Bataillon, 34060 Montpellier Cddex, France (Received August 6, 1986; revised and accepted September 5, 1988)

Abstract Fabre, P., Kast, Y. and Girod, M., 1989. Estimation of flow duration of basaltic magma in fissures. J. Volcanol. Geotherm. Res., 37: 167-186. We suggest a relationship between the thickness of chilled margins often observed in basaltic dikes and magma flow duration. For this purpose, a new thermal model of cooling basaltic dikes is proposed which takes into account flow duration in a fissure before solidification of the magma. The thermal problem is solved by adding a boundary condition (serving as a substitute for magma flow) to the classical initial conditions for dikes of Fourier's equation. We assume a bound temperature To at the center of the dike as long as magma flows. At the beginning of the cooling process, the new model gives the same results as the classical one and, consequently, the same chilled margin thicknesses. But later on, when the classical model enforces temperature decreases, our temperatures increase due to the heat brought by magma flow. The chilled margin is reheated and partially or fully destroyed. Petrological observations on the chilled margins of dikes from Salagou and B~darieux (South of France) and their country rock support this theoretical study. The chilled margins are always thinner than 10 ram; glass includes rare true phenocrysts (mainly olivine and Fe-Ti oxides) and branching quench crystallites of clinopyroxene. The conditions of formation and evolution of the chilled margins have been specified from experimental data: cooling rate 800 ° C h - 1; glass transition temperature (TM) = 675 ° C and glass deformation temperature (TD) = 705 ° C. The initial chilled margins predicted by the two models are always thicker than those actually observed; there is evidence for reduction due to reheating by devitrification and softening. In our new model, the chilled margin thickness which remains is directly dependent on the flow duration. For example, for a dike of Salagou ( 1-m wide and a chilled margin of 5 ram), a flow duration of six days has been computed. At B~darieux, for a 1-m-wide dike and a thicker chilled margin (8.5 ram), flow duration was 4.6 days. These values must be regarded as upper limits because the effects of the latent heat on the temperature distribution during reheating have not been taken into account. The metamorphism of the country rock (pelites and bauxites) has been investigated; temperatures estimated from classical geothermometers are in agreement with those computed by our model and support its validity.

1 Preliminary remarks and introduction It is well-known that many dikes have a glassy crust, called a chilled margin, that is formed when hot magma is quickly cooled against cold country rock. The purpose of this paper is to suggest a relationship between chilled margin

thickness and the duration of magmatic flow. This relationship has been obtained from a new model for cooling dikes which takes into account the flow duration in a fissure. Chilled margins observed in the outer parts of basaltic dikes are systematically thinner than those calculated from the classical models of

168

cooling dikes (Lovering, 1935; Jaeger, 1967). In order to explain this observation (thinner chilled margins), additional heat sources should be considered, such as the temperature of country rocks and latent heat of crystallization. Temperatures of country rocks can be easily adjusted in order to obtain any required chilled margin thickness. However, in many geological situations such as those investigated in this study, country rock temperatures are low compared to magma temperature and cannot account for the reduced thicknesses of chilled margins. It is more pertinent to look at the contribution of latent heat. Jaeger (1967) took this factor into account for a static magma, by considering a fictitious To. His method is too crude to provide temperatures near the initial instant of emplacement. On the other hand, at each point within a dike, latent heat depends on the rate of crystallization, which, in turn, depends on the rates of nucleation and crystal growth. Unfortunately, up to now, there are insufficient data to evaluate these two parameters. However, we have tried, with the few available data, to perform an iterative method by introducing heat source points in the Fourier equation. At the beginning of cooling, latent heat is, of course, zero in the outer glassy part of the dike; in other parts, it remains at low values, since temperatures are close to the liquidus TL. Using this method, the numerical calculations did not indicate any significant thinning of the chilled margin due to latent heat liberated during flow (section 4). Next, we tried to explore the effect of "preheating" country rock. We assumed that if magma flows along the contact for a certain time, the effect would be to raise the temperature at the contact To. This seems a more reasonable hypothesis than that of a sudden intrusion, but the effect on the chilled margin thickness appears too drastic: a one hour preheating entirely suppresses the chilled margin. Because of these failures, we decided to consider a new model (section 3): we assume that

temperature To is imposed, during the period of flow, in the central part of the dike, or just on the median plane, which is our boundary condition. From this, we get a possible relationship between the thickness of a chilled margin and the duration of flow of magma. Some geologists may be surprised to see a physical reality replaced by a boundary condition. It must be recalled, however, that the advantage of a mathematical model is to put a boundary condition in place of a complex physical process and to isolate one phenomenon from the others, in order to investigate its effects. The relevance of the selected phenomenon can be appreciated from the effects (in our case, temperature in the country rock). A comparison of calculated effects with observations will partially or totally justify the starting assumption. Let us note that our approach is similar to that of Lovering (1935) who put a simple assumption about the initial temperature distribution in a homogeneous, infinite medium, in place of an injection of magma (a rather complex physical process). Our solution indicates that thinning of the chilled margin as a result of flow is a likely process. This does not mean that other processes, such as latent heat, must be ruled out. A study of the contribution of latent heat is in progress and the results will be published later. Basaltic magmas rise along vertical fissures, some of which reach the surface. When fissure eruptions occur, two phases can be recognised: the first occurs along the entire length of the fissure; during the second phase (several hours to several days later), volcanic activity becomes localized in a few specific centers whose number decreases as the eruptive process slows down. Such observations were made, for instance, in Iceland (Williams and McBirney, 1979) and in Hawaii (McDonald, 1972). Even where fissures do not reach the Earth's surface, we can suppose that volcanic cones are connected to dikes at depth, because when the volcanic structures are eroded, dikes are usually observed which locally swell to form necks. In short, fissures which open during eruptions to

169

allow the rise of magma are rapidly blocked except at some locations where magma continues to ascend to the surface. The behaviour of the basaltic magma flowing in fissures at shallow depth is therefore an important topic which may provide useful information about volcanic processes.

lm L

Dol

2 Petrological data The dikes which are used to support our model are part of two different sets in Southern France: the set of dikes at Salagou (50 km west of Montpellier), which cuts Permian pelites (Fig. 1) and the dike of the "Carri~res d'Arboussas" near B~darieux (Fig. 2), which cuts a Cretaceous bauxite and its Eocene conglomeratic roof. These basaltic intrusions (alkali olivine basalts) date from the recent volcanic events of Lod~vois, corresponding to 1.7 _+ 0.05 Ma for B~darieux (Gas~aud, 1981). They have been studied by Bardossy et al. (1970), and by Maury and Mervoyer (1973). The additional observations presented here concern mainly the variations of mineralogical composition inside the dikes and the chilled margin (used for calculation) and the metamorphic reactions observed in the country rock (used for verification of the model).

2.a Chilled margins

Fig. 1. Dike swarm of Salagou. The investigated outcrops are specified (A,B...). The isograd " + diopside" is mentioned for three locations.

The chilled margins of Salagou and B6darieux dikes contain 11-12 % of true phenocrysts (olivines, Fe-Ti oxides, rare clinopyroxenes) set in a predominantly pale yellow matrix (Fig. 3a). The presence of phenocrysts indicates that the magma was injected at a temperature of a few tens of degrees below the liquidus. The liquidus temperature, experimentally estimated by a Leitz heating stage, is 1250°C + 10°C. Classical geothermometers on clinopyroxene/liquid (Nielsen and Drake, 1979) and olivine/liquid (Roeder and Emslie, 1970; Leeman and Scheidegger, 1977) indicate temperatures ranging from 1150 °C to 1200 °C for the rising magma.

In addition to phenocrysts, tiny crystals of clinopyroxene formed during the quench also occur in the glass; they have two main shapes: branching microliths and spherulites (Fig. 3b and 3c). The inner boundary of the chilled margin is identified by a large and abrupt decrease in the glass content. Figure 4 represents glass percentage variations from the outer margin inwards for five dikes of different width (from 2 cm to 1 m). With the exception of very thin portions of dikes (~< 5 cm) which are entirely chilled, the chilled margin thickness never exceeds 10 mm. In the range 0 to 10 mm, thicknesses differ widely, even along one dike. From

170

Herr Boer

~Hall +

/ / I t I i

0 I

Fig. 2. Schematic location map of B~darieux dike (right side) and metamorphic transformations in the country rock (left side); the following isograds are drawn: H e r = hercynite; H e m = hematite; B o e = boehmite; K a o = kaolinite and H a l l = halloysite. Symbols + and - represent mineral appearance and disappearance respectively. 1 -- Jurassic dolomite; 2 = Cretaceous bauxite; 3 = Eocene conglomerate; 4 = Plio-Quaternary basalt; 5 = Plio-Quaternary basaltic dikes.

50 measurements, we can conclude that there is no relationship between the dike width and the chilled margin thickness (lower part of Fig. 5). Beyond the inner boundary of the chilled margin, the vitreous aspect disappears. However, glass is still present, although in much lower amounts (~<25%) and with a distinct brown color for a few millimeters. Thus, we can define a "transition zone" which contains patches of a residual brown glass and numerous interwoven fibrous and sheaf-shaped crystals coexisting with branching pyroxene crystallites and spherulites very similar to those observed in the chilled margin itself. The thickness of the transition zone is not readily appreciated, even with the microscope; it probably does not exceed 6-7 ram.

2.b Conditions of formation of chilled margins Kinetic experimental data on basalts (Lofgren et al., 1974; Donaldson et al., 1977; Walker et al., 1978; Schiffman and Lofgren, 1982) indicate that very high cooling rates (>/1000°C h -1) are generally necessary to prevent any crystallization. The occurrence, in the investigated chilled margins, of branching crystallites and spherulites formed during quench suggests a cooling rate slightly slower than 1000 °C h-1. We have carried out some quench experiments on the Salagou basalt with a Pyrox HM 70 furnace connected with an Eurotherm temperature regulator and programmator in order to obtain a vitro-crystalline material similar to the observed chilled margins. The cooling rate required is about 800 °C h-1. In short, the condi-

171

Fig. 3. Photomicrographs of a chilled margin ofa Salagou dike. a. Clinopyroxene and oxides phenocrysts occurring in a glass displaying branching microliths and very tiny spherulites. × 160. b. Different shapes of clinopyroxene quench crystals (spherulites and branching microliths). × 780. c. Clinopyroxene spherulites observed with a Scanning Electron Microscope (Cambridge). × 3800.

172

100

2c r n ~

80 o\O

~0 60

,o

w %

20

~70 ,

i

J

2

4

6

8

j

,

,

10

12

14

~ 16

L

18

~

I00

,,,

20

mm

Fig. 4. Glass percentage variations versus distance from the margin inwards (mm), for dikes displaying different widths (100 cm, 70 cm .... ). The main slope change represents the inner limit of the chilled margin.

merical applications of our model, we have taken 800 ° C as a lower boundary; this is a likely but rather arbitrary value. However, calculations with other temperatures (700 ° C or 900 ° C, for instance) do not change significantly the conclusions of our study.

6 5

4 la

4 q~ II a

Ib

3

2.c Thickness reduction

~Ic

\\

II b IIc

2 I

t i 10

20

: 30

:

.

,

40

50

.

a (cm)

Fig. 5. Comparison between chilled margin thicknesses (e in cm) of the investigated samples (dots and stars) and those calculated from the Jaeger's model for the following conditions: I: 0=800°C, To=ll00°C, t = l hour; IIa: 0=800°C, To=1200°C, t = l hour; IIb: 0=800°C, To= 1200°C, t=0.5 hour; a (cm) is the half-width of the dike; on the dashed straight line, each star represents an entirely chilled dike. Accuracy on the measurements of the chilled margin thickness: 0.05 mm.

tions for the chilled margin formation are expected to be as follows: magma temperature, when injected, is around 1200°C; chilled margins are formed when magma temperature falls down from 1200°C to 800°C in about half an hour (800 ° C h - 1 cooling rate ), regardless of the following temperature evolution. For the nu-

It is usually admitted that a chilled margin, quickly formed at the beginning of magma injection, is not modified afterwards. However, our investigation indicates that it may be reheated. In this case, if temperature increases sufficiently, two processes of destruction can occur: devitrification and softening. The conditions of these transformations can be specified. For the basaltic glasses of Salagou and Bddarieux, the glassy transition temperature TM (conventional viscosity 1013'3 poise) has been evaluated by micro-DTA and by dilatometry. It is 675°C + 5°C (Lebeau and Girod, 1987), which stands in the range of published values for basaltic glasses (Ryan and Sammis, 1981). If the temperature exceeds TM, glass will undergo a partial recrystallization, as experimentally specified by Beal and Rittler (1976):

173 the higher the temperature, the quicker and coarser the devitrification. On the other hand, the deformation point (or softening dilatometric point) TD (viscosity 1011-1012 poise) has been estimated by dilatometry; it is 707°C + 5°C (Lebeau and Girod, 1987). In order to get more information on the behaviour of the chilled margins, we have carried out some complementary rheological experiments on the chilled margins of Salagou which clearly indicate that they begin to flow, under a weak load, at temperatures around 700 ° C. Experiments have been performed on cylindrical test bars (diameter: 6 mm; height: 4 m m ) , which have been drilled in samples of three Salagou dikes. For each dike, two test bars were cut, one in the chilled margin, the other in the center of the dike. Each test bar is placed in a furnace and submitted to a 2-kg vertical strength. Temperature measurement is made from a thermocouple close by the sample. Strain and temperature are graphically recorded. For the test bars cut in the central part of dikes, strain begins at 950 ° C, and the more important deformation (80% of shortening) occurs between 980 and 1040 ° C. Chilled margins behave quite differently: strain appears at a lower temperature: it starts at about 640°C and is very important in the temperature range 700-750 ° C. At 750 °C, shortening of the test bar exceeds 4O% (Fig. 6). Therefore, if a chilled margin is heated at temperatures higher t h a n TD, it behaves as a viscous liquid and, consequently, can be removed by the magma flowing in the fissure. Because of the small difference between TM and TD, we will assume a unique temperature above which the chilled margin begins to be destroyed. For the following numerical application of our model, we have selected 750°C as the "destructive temperature". Petrological data strongly support the assumption that some parts of the original chilled margin were destroyed: (a) Portions of dikes, near the neck of Salagou (e.g., G on Fig. 1) or the neck itself, have

(°C) 1000-

/ 900_

800 _

700_

600.

5oo tOO

l

I

5O

0

I00 h/ho

Fig. 6. Experimental shortening of three samplesof chilled margin from dikes of Salagou, versus temperature (°C). Total thicknesses for dikes a, b and c are 25, 150 and 100 cm respectively;ho= initial height of the test bar (4 mm); h = height at temperature 0;,explanations in text. no chilled margin. Glass and crystals with a typical chilling morphology are entirely lacking, and at the very contact of basalt, the country rock has been heated to a temperature higher than 700 ° C (see below); it is obvious that the temperature at the inner part of the dike itself was higher still. Clearly, if a chilled margin was present at some stage, it was removed. (b) Some crystals (interwoven fibers, sheafshaped morphologies) observed in the transition zone could result from devitrification. Indeed, very thin dikes ( < 5 cm) which are entirely chilled (therefore not reheated by the flow), have only quench crystals and no devitrification morphologies. Interwoven fibers and sheaf-shaped crystals are thought to be annealing phases related to reheating of the dike margins. Therefore, the transition zone could represent the inner part of the initial chilled margin.

2.d Country rock metamorphism Metamorphic transformations observed in the country rocks have been investigated in or-

174

der to determine whether or not temperatures inferred from metamorphic reactions are similar to those computed from our model.

The Salagou dikes (Fig. 1) Pelites which have been intruded by basaltic magma have been investigated by optic microscope, electronic microprobe and X-ray diffractometry. The sedimentary rock consists of quartz, albite, illite, clinochlore, montmorillonite, calcite, analcite, dolomite and hematite (all these minerals are not always present in the same sample ). Five pelites suites have been investigated; for each suite, rocks were sampled every five centimeters in a direction normal to the dike wall. Among the various metamorphic reactions that occurred in the pelites (Maury and Mervoyer, 1973 ), only the transformation of dolomite into diopside was considered: CaMg (CO3)2 + 2 SiO2 -~ CaMg Si206 + 2 CO2. The temperature corresponding to this reaction does not exceed 450°C for Pco,~ < 1 kbar (Metz, 1970). The isograd " + diopside" is indicated on Figure 1 for three locations; it stands at a distance from the contact ranging from 20 to 50 cm (Table 1 ). The reaction analcite + quartz ~ albite + H20 (about 200°C) cannot be used, since al-

bite is present in unmetamorphosed pelites; moreover, it is a very sluggish reaction (several tens of days, according to Liou, 1971). Thus, the reaction is incomplete, and it is not surprising to find analcite sometimes up to the contact of the dikes. In addition, it is worth noting that cordierite, K-feldspar and magnetite appear in the three following occurrences: (a) At the contact of the neck of Salagou (Fig. 1 ). There, cordierite is observed between 0 and 25 cm, K-feldspar between 0 and 70 cm, and magnetite between 0 and 90 cm. The diopsidic zone is wider (120 cm) than elsewhere. (b) At the contact of the portion of dike situated very close to the neck (A on Fig. 1). (c) At point G on Fig. 1, cordierite, K-feldspar and magnetite are present within 5, 15 and 30 cm, respectively, from the contact. The diopsidic zone is 50 cm wide. The assemblage cordierite + K-feldspar, which is frequent in high-grade pelitic rocks, is expected to record the highest temperatures (more than 700°C) reached in the country rocks. In the occurrences mentioned above, the lack of chilled margin is a striking feature. It is thus reasonable to expect a negative correlation between the chilled margin thickness and the intensity of the country rock metamorphism.

TABLE 1 Comparison between field observations and model data at different locations of the Salagou dike swarm (see Fig. 1 for locations) Location

Field and mineralogical data

Theoretical data

1

2 3

a

b

c

d

e

f

24 200 1300

100 100 50

0.5 0.7 0.6

50 20-30 10-20

149 143 35

50 25 30 20

a. Distance from the neck (m). b. Dike width, 2a (cm). c. Chilled margin thickness (cm). d. Diopside zone width (cm). e. Flow duration (hours). f. Position of the isotherm 450°C in the country rock, from the contact (cm).

The Bddarieux dike (Fig. 2) This 1-m-wide dike is particularly interesting, as it intrudes bauxite, a rock very sensitive to temperature variations. Therefore, several markers can be selected in the metamorphic zone. Bauxite mineralogy and mineralogical transformations near the dike have previously been investigated by Bardossy et al. (1970). New investigations by X-ray diffractometry have been carried out in order to specify the exact location of the metamorphosed samples with respect to the dike. On the other hand, thermal experiments on the Bddarieux bauxite have been performed, at atmospheric pressure, in a Pyrox HM 70 furnace. All the phases observed in the metamorphosed bauxite have been syn-

175

thesized and then identified by X-ray diffractometry. The unmetamorphosed bauxite contains boehmite, hematite, and, in lesser amounts, anatase, rutile, kaolinite and halloysite. Figure 9 specifies the main mineralogical variations. Kaolinite breaks down at 60 cm from the contact. The disappearance of this phase is correlated with an increase in halloysite. As suggested by Bardossy et al. (1970), kaolinite is first eliminated in favor of metakaolinite (a well-known reaction occurring between 300 and 500 ° C ), which is further transformed into halloysite by a rehydration process. Boehmite diminishes in quantity at 20 cm from the contact and fully disappears at 15 cm. The dehydration reaction gives way to several A1203 compounds: y-A1203 (between 0 and 20 cm), 0-A1203 (between 5 and 15 cm), J-A12Q (between 5 and 20 cm). Corundum (a-A1203) is present within 5 cm of the contact. Our heating experiments indicate that boehmite breaks down in the range 500-600 ° C. y-A1203 appears around 550 ° C (within few hours), J and 0-A1203 at a slightly higher temperature (600-620 ° C); a-A1203 is observed at 700-720°C (after one day). Hematite is found up to 10 cm from the contact. Nearer the contact, it gives rise to hercynite according to a complex reaction involving breakdown of boehmite. In the FeO-A1203 system, hercynite is a stable phase, at a temperature higher than 800 °C (Baldwin, 1955). However, our experimental data indicate that hercynite crystallizes at a temperature of 700720 °C after a minimal time of two days. In their paper, Bardossy et al. (1970) report the occurrence of Jurassic dolomite lying under the B~darieux bauxite formation. These authors observed that at a few centimeters from the dike, dolomite is totally replaced by calcite. Although periclase (MgO) has not been detected, calcite is expected to be the product of the classical reaction: dolomite -. calcite + MgO + CO2 (Harker and Tuttle, 1955), which is thought to occur at temperatures higher than

700°C (under low pressure). So, this line of evidence indicates that temperatures at the contact occasionally exceeded 700 ° C. 3 Thermal models

3.a The classical model The classical model of cooling dikes (Lovering, 1935; Jaeger, 1967) does not take into account the time necessary for the magma to be emplaced before complete immobilization. The whole period is squeezed into some "initial time". It is a one-dimensional model (the influence of the surface is not significant at depths greater than 50 m, as indicated by Kast and Girod, 1976). The dike is represented by an infinite sheet of width 2a, at an initial temperature To, in sudden contact with two semi-infinite media at 0°C temperature representing the country rock. The thermal diffusivities ~x of magma and of country rock are similar. Each point is defined by its distance x to the midplane of the dike. The temperature O(x,t) is estimated by solving the Fourier equation (00/ Ot=aO20/Ox 2) for a one-dimensional infinite and homogeneous medium, with the initial conditions: 0(x,0) =To at t=0, if Ix[ ~ a 0(x,0)--0

if [x[ > a

The temperature of each point, at time t, is (Jaeger, 1967):

O(x,t) = To/2{erf[ (x + a)/2 (at)1/2] -erf[(x-a)/2(olt)l/2]}

(1)

where the error function, erf u, is 2/n 1/2 fo exp (-v2)dv. It is convenient to introduce the travel time ~, associated with the distance a,

~a-=a2/ 4o~. For example, ra-- 25 hours - 1 day, for a basalt where diffusivity is a = 0.007 cm 2 s-1, and 2a-- 1 m. In terms of %, solution (1) becomes:

176 1.5

0.5

±

J

m

1

1

0.8

0,8

k~ 0.6

0.6

(6.25) -0.4

0.4

(72) 0.2

126) (50)

0,2

T

0,5

1.5

x/a Fig. 7. Temperature distribution across a dike (x/a < 1 ) and the neighbouring country rock (x/a > 1 ) during flow {solution 2 ) for t < Za, the temperature distribution is the same for the two models a n d is represented by the solid lines. Dashed lines are used for the new model and for t / > za. Dot-dashed lines represent results of the classical model for t >/ Ta. T h e curves are n u m b e r e d in multiples of z,.

O(x,t) = To/2{erf[ (x/a) + 1 ] (Za/t) 1/2 -- err[ (x/a) -- 1] ( Za/t) 1/2} The numerical solution of this equation is represented by the curves of Fig. 7 (O/To versus x~ a), for some values of t/Za (solid lines for t < z~, dot-dashed lines for t > za). We will criticize this model on some points, but note, however, that its temperature predictions for large times are excellent.

Computed chilled margin thickness: the classical model first appears to explain well the formation of a glassy selvage at the contact: for x = a, the temperature falls suddenly from To ( 1200 ° C ) to To~2 (600 ° C ). The chilled margin thickness, e, can be defined from the curves of Figure 7 and corresponds to the zone of intrusion where temperature falls from 1200 to 800 ° C in less than half an hour (a condition for glass formation). This thickness seems to be a function of the

half width, a, of the dike (through Za). But, in reality, this is not true. For the short times implied by the glass formation, everything happens as if the dike had an infinite width, provided that a > 10 cm: the defect of a thermal event at one of the dike walls does not have time to reach the center. To calculate the limiting value "e", we can thus use the asymptotic formula for "a" infinite:

O(x,t)= T o / 2 [ 1 - e r f ( x / 2 ( a t ) l / 2 ) ] where x=O is the partition plane, x < 0, the intrusion at initial temperature To, x > 0, the country rock. Indeed, Fig. 5, which gives chilled region thickness, e, versus half-width of the dike, is computed from the complete solution (1) and for different conditions of chilling; curves stop on the straight line e-- a (below this value, about 5 cm, the dike is entirely chilled). Curve IIb corresponds to the chilling condition mentioned above: if a > 10 cm, e = 2.5 cm and is independent ofa. We have plotted on the same

177 figure the points that give the real thickness of chilled margins versus half-width of observed dikes. No point falls on the theoretical curves, regardless of the conditions of chilling selected. Thicknesses as small as those observed would require a chilling/cooling rate faster than 5,000 ° C h - 1, a quite unlikely value. 3.b Insufficiencies of the classical model The classical model is not applicable for our purpose because: (a) it assumes an instantaneous emplacement which does not permit the description of phenomena that take place soon after the emplacement of magma. Chilling is a very rapid process; therefore, chilled margins cannot be described by a model whose initial time is not well-defined. (b) it implies no heat source except for that which is brought by the hot magma body. With this model, no point of country rock can reach a temperature higher than 600 ° C. Many authors have tried to improve the model by making it more realistic. Their efforts focused mainly on the latent heat of crystallization and never on the emplacement time. Jaeger (1967) considers the latent heat in the model itself. Kirkpatrick (1981), with the same aim, adds a term of heat production rate per unit volume to the heat flow equation: O0/ Ot = o~(0 20/0X2) + (1~pc) ( OQ/ Ot ) This model, perfect in theory, cannot be applied readily to actual eases (see below, section 4), since it assumes that the magma melts congruently from a single crystalline phase. Another method takes latent heat into account more accurately: source-points are distributed in space and time in order to represent latent heat of crystallization where and when crystals appear. Without boundary conditions, the temperatures due to the source-points can be added to O(x,t). A step-by-step method in time defines the effect of the source-points by considering the nature and number of crystals (from

temperature and temperature change calculated in the preceding step with approximate laws for crystallization). We tested this method and found that chilled margin thicknesses were very close to those predicted from the classical model. This does not imply that the latent heat is negligible. After the chilled margin formation, the latent heat may be able to reheat it in a manner similar to that proposed below. Other improvements and heat sources can be added to the classical scheme, but they do not solve its basic inadequacy in defining the initial time. The only satisfactory method would require a thermomechanical solution including all heat sources and describing the magma ascent from a chamber. Delaney and Pollard (1982) are the only authors who have established a thermomechanical solution, but without latent heat; their solution entails important simplifications and, for that reason, is not entirely satisfactory. For the study of a chilled margin, per se, a purely thermal solution seems to be sufficient and is thus the approach that we follow. 3.c Formulation of a new thermal model We hope to replace physical conditions of the magmatic flow by a mathematical one which may be a boundary condition in Fourier's equation in order to take into account the time necessary for emplacement of the magma. Magmatic flow is an advective heat source that might be thermally represented by a continuous source in the middle part of the dike. But the magnitude of the source depends on flow processes that we ignore. The existence of flow in the central part of the fissure means that the magma temperature is high enough to ensure its fluidity. Owing to the rapid variation of viscosity with temperature, in the range TL (Liquidus) - T~ (Solidus), this temperature cannot be much lower than that of the liquidus. We will consider that it is almost constant within a certain zone of the dike ( T = To). First, we use the plane x--0 alone. To is the magma temperature when it arrives at the considered level H,

178 as determined by mineralogical evidence (section 2a). More exactly, it is the difference between the magma temperature and mean initial temperature Ti of the country rock. So that if Ti ~ 0 ° C, we must change To, but values of Ti are in all cases far lower than To ~ 1200°C. This correction is, by the way, of little importance. This assumed knowledge of To is the main reason why we can ignore the mechanics of the problem. Thus, in our model, the flow of magma acts as a thermostat, adjusted to To and active in the middle plane of the dike. We do not need to specify the heat source or its power. Qualitatively, the heat source is primarily the magmatic heat that comes from lower levels (but also other sources), the major part of these being given off by the flowing liquid.

3.d Solution of Fourier equation considering flow duration An energy equation balancing conduction, accumulation, and convection of heat in the fissure is replaced by an equation balancing conduction and accumulation; a boundary condition at the center of the fissure specifies the continuous release of heat as a source that mimics the effect of magma advection. In making this approximation, an explicit dependence of temperature on downstream distance of magma flow is abandoned; it is replaced by a source of heat with a magnitude that can be adjusted as is necessary to fit the data. We are, at present, unable to test the validity of these assumptions; instead, we find that application of the model to a suit of dikes yields reasonable results. A solution 81 (x,t) of Fourier equation is explored, with the same initial conditions as those of the classical model. But we limit its validity to x > 0 and introduce the boundary condition: for x = 0 , 81 (0,t) = To, at every time t As indicated in the appendix, this solution is, for x > 0 :

81 (x,t) = To~2 [ 2 - {erf[ (x+a)/2 ( a t ) '/2] + e r f [ (x-a)/2(at)l/2]}]

(2)

It is one solution of the Fourier equation (sum of elementary solutions), and it satisfies the conditions that: - for x----0, 8, (0,t) = To, at every time t - for t----0, if 0 < x < a , 8,(x,0) = T o (since the two erf are + 1 and - 1 ) if x > a, 81(x,0)=0. By symmetry, it can be applied to all the space by taking 8(x,t)=Ol(x,t) for x > 0 and 8(x,t) = 8 , ( - x , t ) for x < 0. This solution represents the temperature distribution during flow, t < to; to is the time when flow stops. At t = to, equation (2) gives 8 (x, to ). It is the initial distribution of free cooling, which can be represented by the Laplace solution (Carslaw and Jaeger, 1959, p. 53): 8(x,t) =

+~ e x p [ - (x-~)2/4a(t-to)]8(~,to)

f_~

d~

2[7ca(t-to) ] 1/2

t>to

(2')

This integral is estimated, and temperatures are calculated in the appendix. Equation (2) for t < to and (2') for t > to gives the complete solution. One important conclusion can be drawn immediately. For t, sufficiently small with respect to travel time ra (for instance t < ~a/2, or less according to the accuracy selected), the new solution (2) is almost identical to the classical solution (1) since the expression:

a:,,t:

_J

does not differ greatly from 1 (erf ~ = 1, erf 2 - 0.995, erf 2 '/2 = 0.95 ), even if x is very small. Since quenching is a very rapid process (less than an hour), the new model produces chilled

179

margins as thick as those calculated with the classical model. In Figure 7, the same curves, in solid lines, represent solution (1) or solution (2) for t < ~J2. But when t > ~a/2, the two models have opposite behaviors: temperatures decrease with the classical model (dot-dashed lines on Fig. 7), while they increase with our model (dashed lines ), owing to the heat brought in by flowing magma and implicitly included in the assumption 0-- To constant at x-- 0. Suppose, for instance, a dike of width 2a-- 1 m (so % -~ 1 day) and To=1200°C. A 2.5-cm chilled margin is formed on each wall during the half-hour following magma injection at the investigated location. The corresponding zone (0.95 < x/a < 1), (Fig. 7), cools down up to t = ~ -~ 1 day; at this time, the inner limit of the chilled margin (x/a=0.95) reaches the temperature 0 = 6 3 0 ° C (0/To=0.525). T h e n the temperature increases, and for t= 4.5 ~a, the inner boundary of the chilled margin reaches 750 ° C (0/To = 0.625 ). At this temperature, the chilled margin then begins to thin. At t=6.25 ~a, the 750 ° C isotherm reaches the country rock and removes the chilled margin entirely. Thus, a chilled margin whose thickness is less than 2.5 cm, whatever the dike width 2a, corresponds to a magma flow duration between 4.5 and 6.25 ~a and, therefore, allows us to estimate to. Let us consider a dike of Salagou (Location 1, Fig. 1 ). It is 1 m wide and its chilled margin is 5 m m thick. The chilled margin corresponds to the zone of intrusion where 0.99 < x/a < 1. The 750°C isotherm (represented by O/To= 0.625), cuts the straight line x/a = 0.99 at a point located between the curves 4 and 6.25 ~. By interpolation, a value of about 6 ~a is found. Note that a chilled margin entirely destroyed would only imply flow duration longer t h a n 6.25 ~. The reliability of our model will be described in the next paragraph, by comparing the temperatures inferred from metamorphic minerals observed near the dike, and those predicted by the model in the country rock. But we can already make the following qualitative remarks:

(a) Our model proposes a mechanism allowing the annealing or thinning of a chilled margin. It is consistent with mineralogical studies of chilled margins and explains why they end by a real discontinuity. No continuous cooling process can explain the sudden drop in the percentage of glass of Figure 4. The temperature curves O(x) (from model (1) for instance) do not display, anywhere and at any instant, a variation A0/Ax high enough to account for a discontinuity. (b) According to the model, the thinner the chilled margin, the longer the flow duration. Indeed, field observations show that, when a dike swarm is connected to a neck, where flow duration is more sustained, the chilled margin thickness decreases until it disappears at the vicinity of the neck (it must be recalled, however, that our model is not quantitatively valid for a neck). (c) Consequently, the more important metamorphism is, the thinner chilled margins will be, as observed in the field (Table 1 and section 2.d).

3.e Temperatures in country rock - metamorphism Once the value of To is selected, the temperature evolution for t > to is obtained by solution of (2') or equivalent equations in the appendix. Results are presented on the curves of Figure 8, for to=4 ~a and to--6.25 % [curve 0 is O(x, to)] at the end of flow, curve n is 0 (x, to + nv~). For intermediate values, an interpolation between the two sets of curves is generally sufficient. Turning now to the country rock metamorphism, if the temperature distribution is known at all times, the metamorphic reactions can be predicted: if a solid reaction occurs at temperature TR and gives a new mineral M, the zone where M appears after a time ~R (often poorly known) can be defined. Theoretically, all val-

180

to = 4 Ta 0.8 2

0.6

~,

5 70

0.4 2O 5O

0.2

100

2

3

x/a

to= 6.25 Ta 0.8

,2

0,6

5

~

10

0.4

20 50

0.2

100

2

3

x/a

Fig. 8. Temperature distribution (from the solution 2' ) after a flow duration to, for two values (4 Ta and 6.25 Ta). T h e curve n is at time to + nza.

ues of TR would be uniquely associated with a time ~R. Experimental data are quite insufficient; in most cases, we only know one couple (TR,~R), which is quite unfortunate, because it would be useful to verify the form of the temperature curves. Indeed, if we suppose another mechanism of reheating (e.g., latent heat), the repartition of heat sources will be different and so will the repartition of temperatures in space. Thus, it is not sufficient to specify some temperatures alone, but one must also define its law of variation. Despite its inaccuracy, we think

that the pattern of temperature curves is consistent with the model. Against the dike of B~darieux (2a= 100 cm, e-- 8.5 mm, to = 4.6 days), several reactions were observed which correspond to the appearance of six minerals. Figure 9 summarizes the petrological data and compares them with the maximum temperature reached in each point of the country rock under the hypothesis that to= 4.6 days (the curve derived from the classical model is drawn for comparison). The six points corresponding to hercynite (720 ° C ), c~-

181

T(°c) ~-AI2 03 + Hercyn/te

800

-/

H e m at/te

60C

e-A/z.23

_

400

,

t

t

I

20

@

I

I

40

60

I

-~

80

d (crn) Fig. 9. Mineralogical evolution of the bauxite near the B~darieux dike and comparison between maximaltheoretical temperatures, for to = 4.6 r, (curve 1 ), in the country rock, and distribution of the main appearing and disappearing phases. The bar height is the estimated error. Curve 2, drawn from the Jaeger's model, is indicated for comparison.

A1203 (720 ° C ), 9- and J-A1203 (620 ° C), 7-A1203 (600 ° C ) and halloysite (500 ° C ), fit the curve, though they plot a little above it {this small discrepancy can be readily explained by the uncertainties of both temperatures and times of solid reactions and by the fact that the value to--4.6 days is perhaps a little too low). As indicated earlier, only one reaction was considered at Salagou. Consequently, only one temperature (450 ° C ) can be checked. But samples were collected at different places where chilled margins have different thicknesses and, consequently, where flow duration was different. Thus, the verification is obtained at three points though with a single temperature. (This is another way of verifying the results of the model. ) As indicated in Table 1 {columns d and f), there is a good fit between calculated and observed values. Consider location 1 of the Salagou dike (Fig. 1). Here 2 a = 1 0 0 cm, ~,=1 day, e=5 mm. We found "to"=6 days. Diopside is observed up to 50 cm from the contact (x/a= (0.5 + 0.5)/ 0.5=2). TR=450°C, TR/To=0.375. On Figure 8, for to=6.25 ~a (or, better, on a temperaturetime evolution curve, at x/a--2 ), we note that 9/To reaches 0.375 two days after the end of the

flow and that the corresponding point remains at a temperature higher than 450 ° C for 5 days. This time is sufficient for the appearance of diopside. Moreover, for x/a > 2, or, for a shorter flow duration (e.g., 4 ~,) temperature never reaches 450 ° C, nor does it if we use the classical model (to = 0). 4 Discussion

The agreement between calculations and observations is certainly not due to chance. However, the limited number of measurements and the lack of kinetic data (such as those reported in the Time-Temperature-Transformation TT - T diagrams) for the metamorphic phases prevent an accurate check of the results given by our model. It is also important to note that if we had taken a shorter flow duration, latent heat, with its proper distribution, may have acted as a secondary source, leading to temperatures perhaps close enough to those computed here and possibly satisfying the mineralogical conditions. Of course, if one admits that the predicted shape of curves is verified, this objection can be ruled out, but the only way to answer the question is

182

to build a model dealing with both heat sources, magma flow and latent heat of crystallization, and is thus quite complicated. In order to take into account the latent heat of crystallization, it seems possible, as suggested by Kirkpatrick (1981), to add a term of power production OQ/Ot to the heat equation. But the explicit evaluation of this term is not easy. For each point, it depends on the temperature reached at time t, as well as on the temperature variation rate through nucleation and growth rates (I and Y, respectively). The solution requires a step-by-step numerical method, with the major inconvenience that I and Y are badly known in actual cases. Though numerous studies of the subject are available (Dowty, 1980; Lofgren, 1980; Kirkpatrick, 1981; etc.), none is very satisfactory, and a new approach is necessary. Kirkpatrick himself (1976) studied the only case of an eutectic liquid and indicated that this assumption does not really represent any geological condition. We have tried to calculate approximately the latent heat released by unit time and unit volume in order to compare it with the heat flow (F = K. aO/Ox) given by the model, at each time t and for a fixed location (e.g., at the contact). In that way, we could at least evaluate the error made by neglecting the latent heat. We have encountered the same difficulties. All approximate evaluations (without using I and Y and, by the way, the results of the solution itself) are obviously too large, for they oversimplify real conditions. For lack of anything better, we have finally selected a qualitative approach. Two periods can be distinguished: (a) During flow, latent energy is small with respect to the heat flow from the model. This is certain in the chilled margin (zero); it is very low also in the region of flow whose temperature is near that of the liquidus and, if it exists, its major part is released towards the top. In the remaining volume, the large cooling rate during this period prevents complete crystallization. (b) After the flow stops (i.e., during the con-

solidation period), the latent heat may no longer be negligible. The volume is larger, and cooling rates are smaller. All the latent heat released is then conducted towards the chilled margin and prolongs the effect of magma flow. As we are not able to appraise the error introduced by neglecting latent heat, we can use the scale laws of Brandeis and Jaupart (1987). Their study concerns mainly large magmatic chambers, but they focus on the effect of the "Stephan number" a=L/CvAT. If L = 0 (no latent heat), a=O and if a << 1, the effect of latent heat in the heat budget is negligible. Here, the initial contrast of temperature ~ T = To = 1200 ° C. Let us take L = 100 cal g - 1 and Cp = 0.26 cal g - 1 oC (isobaric heat capacity). So, a=0.3. In figure 4b of Brandeis and Jaupart (1987), the curves of undercooling of the front of "solidification interval" are given in dimensionless variable for three values of a (a= 0, a= 0.55, a= 1.65 ). The maximum discrepancy between temperatures for a = 0 and for a=0.3 (interpolated) is certainly never larger t h a n 2.5X 10 -3 (that is to say 3 °C); this is quite negligible in our problem. It is not possible to infer the effect of this small value of a on the chilled margin temperature, but it is likelyto have the same scale. The computed flow duration is probably too long, but the error is unlikely to be larger than 0.06 ~a (in our examples about 1.5 hours). The petrological verification of the theoretical distribution of temperature confirms that the error is less than a few percent. A second point must be discussed: flow is represented by the assumption 0= To for the plane x = 0 alone. This may seem too restrictive. It is easy to extend the model to cases where To is imposed on an interval "b", but a new parameter is introduced, and the interpretation becomes more difficult. After some attempts, we have decided that the extension was not worth while, at least for the cases studied here. Moreover, the hypothesis x = 0 is not as restrictive as it seems. If To = TL, the curves of Figure 7 show that one-third of the volume is at temperatures higher t h a n the solidus temperature Ts and is

183 chilled margin. Eventually, To can be taken as a parameter and adjusted, if necessary. It iraplies that the heat is advected by the flow, which explains the reheating of the chilled margin and its gradual destruction. It seems that this thinning is the only process to explain the abrupt edges of the chilled margins. Our simple model is also the first to explain, at the beginning of cooling, the high temperatures in country rock necessary to produce certain metamorphic minerals, such as corundum or hercynite. The implications of our model are supported by the mineralogy of the dike as well as of the country rock. The heat supplied by the flow does not exclude other heat sources (particularly latent heat of crystallization.) But we think that magma flow is the main reheating source and must be first taken into account. If the power of other sources is not very high relative to heat flow (which is very likely the case) or, if it is known, the chilled margin thickness is a direct means of evaluating the duration of flow or, at least, its upper limit.

affected to some extent by flow. The petrological verification of temperatures definitely indicates that the simplification x = 0 is sufficient. In places where a chilled margin was entirely destroyed, metamorphic reactions must be used to estimate to. For example, families of curves have to be drawn for "to" values higher than 6.25 za, and a family must be examined so that, up to the distance d where a mineral M is found, the temperature reaches the corresponding value TR. In fact, if an accurate determination of to is desired, in order to compensate for the poor definition of TR, several metamorphic minerals must be considered and a more elaborate approach may finally be necessary. We also compare our results with those of Delaney and Pollard (1982), who found that for a l-m dike, only a few hours were required to block the fissure by solidification (the chamber is supposed to be located at a depth of 5 kin), while we found the flow duration to be several days. The two methods, which apply in different ranges, cannot be directly compared, but their discrepancy should be explained. First, note that volcanological observations (Williams and McBirney, 1979) are consistent with a flow of several days. Furthermore, at the investigated sites, the erosion level is about 100 m. Here the magma keeps on flowing when the top of the fissure is blocked (from the orientation of the vesicles, we know that the flow is rarely vertical).

The original manuscript was improved by criticisms and suggestions from several reviewers and from A.R. McBirney. The authors wish to thank A. Bonneville who was most helpful in discussion of the work.

Conclusion

Appendix

A simple thermal model of cooling dikes is presented, for consideration of the duration of their emplacement. The initial instant being better defined than in classical models, this new approach better represents the cooling behavior during the initial moments, and represents particularly well the time during which chilled margins are formed. It applies at a specific level (deep enough to neglect the effect of the ground) characterized by the magma temperature, as deduced from mineralogical study of the

Research of the solution (2)

Acknowledgements

A solution O(x,t) of the equation (E), 00/ Ot=a(O20/Ox2), is sought, defined for x > 0 and which is verified for t=0: if x > a, O(x,O)=O; if0 < x < a, O(x,O)= To. Moreover, if x=0, O(O,t) = To. To is a given constant. The Laplace Transform:

O(x,p) = [O(x,t) ] ----~o exp(--tp)O(x,t) dt

184

verifies the transformed equation of (E) which takes two different forms according to the initial conditions. Let us put O(x,p)=02(x,p) if 0 < x < a and O(x,p)=O3(x,p) i f x > a ; (t~2): p 02 - T o = a

With the tables (Carslaw and Jaeger, 1959, p. 494) one can easily do the inversion.

02 = To/2{erfc[ ( x + a ) / 2 ( a t ) 1/2] -erfc[ ( - x + a ) / 2 ( c ~ t ) 1/2] +2} 03 = To/2{erfc [ ( x + a ) / 2 ( ~ t ) 1/2 ] +erfc[ ( x - a ) / 2 ( a t ) l / e ] }

( 0 2 0 2 / 0 x 2) for 0 < x < a

(I~3): p 03 =0~ (020JOx 2) for x > a

or

With q2 =p/o~,

Oe= To~2 [ 2 - {erf[ (x +a) /2 (c~t)1/2]

(t~2): 0202/Ox2-q202 = - To/O~

03 = To~2 [ 2 - {erf[ (x+ a)/2 (at)1/2]

(I~3): 0203/Ox2--q203=0,

- e r f [ ( x - a ) / 2 (~t)1/2] } ].

then 02 ---A exp (qx) +B exp(-qx) + (To/p) 03 = C exp ( - qx ) (forx = + ~ , 03 is finite ) A, B, C are constants determined by the boundary and continuity conditions,

02(O,p) = To/p~A +B + To/p= To~p, whence A = - B ,

02(a,p) = 03(a,p) =>Aexp (qa) +B exp(-qa)

(0020X)a: (O03/Ox)a~qA

The two expressions are the same and form the solution O(x,t) for x > 0. This solution has a physical meaning only during the flow duration, for t < to.

Numerical calculation of the Laplace solution (2') The later temperature evolution (t > to, free cooling) is given by the Laplace solution, with the initial distribution f(x ) = 0 (x, to ) ( Carslaw and Jaeger, 1959, p. 53):

O(x,t) =

+ To/p=C exp(-qa) exp (qa)

-qB exp(-qa) = -qC exp(-qa). One finds

A = - To/Z[exp(-qa) /p] = - B and

C= To~2 [exp(qa) /p+exp( - q a ) /p ] 02 = To/2 [exp( - q (x+ a) )/p -exp( -q(-x

+erf[ ( - x + a ) / 2 ( a t ) l / 2 ] } ]

+a) ) /p+ Z/p ]

-~ f(~). e x p [ - (x-~)2/4o~(t-to)] 217co~(t_to) ]l/2 d~

f_

t>to Numerical calculation of this integral is not immediate but it is made easy by the regularity of the error function (erfc u = l - e r f u) and its rapid decrease. Furthermore, f(~) can be replaced by straight line segments, and one can put [(~) = 0 for ~ > xj. The range [ (xi, f(xi) ), (xi+l, f(xi+l))] is given by the following equation: f(~)=OLi~'JF~i

03 = To~2 [ e x p ( - q ( x + a ) ) /p +exp(-q(x-a)

)/p].

X i < ~ < Xi+ 1

w i t h O Q = ( Y i + l - - Y i ) / ( X i + l - - X i ) , fli=Yi--oLixi,

yi=f(xi) = To~2 [2-erf(xi/a+ 1) (n -1/2) -erf(1-xi/a) (n-l/2) ] if to=nra.

and

185

By symmetry, f(¢i)=f(-~i), = - ai and fl_ ( i + 1 ) = f l i " The integral becomes:

then,

OL (i+1)

N--1

O(x,t)= (To/2) ~, {fli(erfU~-erfU~ 0

N--1 fXi+l O(x,t) = To i-_~-N ~,

- erfU~ + erfU/4) +xai(erfU1 i - e r f U 2i + e r f U ~ - e r f U ~ ) + (aai/n 1/2) [ e x p ( - U~2)

e x p [ - ( x - ¢ ) 2 / 4 a ( t - t o ) ] (olin+ill) d~ 2[TrOZ(t--to) ] 1/2

- e x p ( - U~2) + e x p ( - U~2) - e x p ( - U~2) ] } These calculations can be performed on a minicomputer.

t> to One must calculate such integrals:

(x)

fxi+, e x p [ - ( x - ~ ) e / 4 a

J:xi

(t- to)]

2[TCO~(t--to) ] 1/2

d~

or

II (X ) = 1/2{erf[ ( x - x~) /a ] [~a/ ( t-to) ]1/2 -eft[ ( x - xi+ l ) /a ] [ ~J ( t - t o ) ]1/2}; and,

I~(x) = I x :~÷1e x p [ - ( x - ~ ) 2 / 4 a ( t - t o ) ] d~ 2[TCOl(t--to) ] 1/2

Ii2x ) =x/2{erf[ ( x - xi) /a ] [z J ( t - to) ]1/2 - e f t [ ( x - xi+ l ) /a ] [ za/ ( t-to) ]1/2}

+ (a/2n 1/2) [(t-to)/Z~] 1/2 [exp{ - [(x - x i ) / a ] 2 [~cJ (t-to) ]}-exp{ - [(x -Xi+l) ]/a)2[~a/ (t-to) ]} ]. T h e n for t > to: N--1

O(x,t)=To

~ [fliIl +aiI2] i=--N

If only positive indexes are preferred, one groups the terms with the same fl~ and opposite ai and one can put:

U~= [ ( x - x i ) /a] [r~/ (t-to)11/2 Vi2 : [ (x--xi+l) /a] ['Q/ (t--to) ]1/2

Ui3= [ (x + x~) /a ] [~J ( t-to) ]1/2 vi4 = [ (x + xi+ l ) /a ] [%~(t-to)]1/2 Finally, the solution becomes for t > to:

References

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