....... On te formation of Caldera s BY Dr. B.G. Escer Publised on te occasion ol' te Fourt Pacific Science Congress Batavia Bandoeng (Java). May 1929. Contents Page. Acknowledgment 184 I. Teory 184 II. Calculation 187 1. Deduction of te general equation 1871 2. Example: Te Tengger-caldera 188. General solution 191 4. Tabl&s for 4r and 194, III. Discussion of te values found and teoretical considerations 214'. 1. On te applicability of te new teory. Te value of x.. 214, 2. On te dept of te cored-out cylinders. Te value of?> 214-. On te degree of force of volcanic eruptions 215! 4. On te most probable combination of x and ;! ;. 216' 5. On te duration of te paroxysms and of te periods of quiescence in caldera-forming volcanoes 217 IV. Synopsis of te teory of te formation of ealdora's 218 Bibliograpy 219
ACKNOWLEDGMENT. In te following study more matematics are used tan is usual in dealing wit problems of dynamical geology. Altoug te matematics tat are applied are simple, it is only by te elp of some colleagues tat tis study could be carried troug. Prof. W. v. d. Woude elped me in te solution of te equations of te tird degree. To Prof. Dr. W. de Sitter I owe te elegant metod for determining te middle root of te 60 equations of te tird degree by a simpler metod. Tese values were calculated in is institute. To bot tese colleagues as well as to Prof. Erenfest I tender my sincere tanks for te elp tey gave me. LEIDEN, Februari 1929. I. THEOY. In two previous publications (bibl. 1 and 2) I ave brougt te formation of calderas into relation wit te gas pase, observed by Perret during te eruption of Vesuvius in 1906 (bibl. ). In tese papers I arrived at te conclusion tat during te gas pase a cylinder is cored out, and tat tis may be te cause of caldera formation. In te first te paper subject was treated geometrically, wile in te second calculations were made of a particular case (te Krakatoa eruption of 188) to see if tey would bear out tis teory. Tis caldera-formation, owever, is not a typical case, as tere must previously ave been an older Krakatoa-caldera, and in Aug. 188 it was not a large portion of te volcanic cone tat disappeared, but only an island wic projected little above sealevel; te nortern part of te ancient island akata, wit te volcanoes Perboewatan and Danan. How a caldera migt be formed from a cored-out cylinder I ave tried to explain in two different ways. In te case of te Tengger-caldera I assumed, in analogy wit wat appened in Vesuvius after 1906 (bibl. and 4) tat te uppermost part of te cylinder was transformed into a funnel-sape by crumbling away of te walls, and tat rising lava, as in Vesuvius 191 1926, formed a flat bottom wic continually reaced iger levels. Tis explanation does not apply to te caldera of Krakatoa, as after te great eruption of Aug. 26 t to 28 t 188 no furter signs of eruption were observed, until in Dec. 1927 a new pase began in tis famous
185 volcano. In te case of Krakatoa in 188, terefore, I tougt it justifiable to apply te penomena, known to occur in coal mining, of recent subsidences wic are caused by te working of coal seams lower down. Goldreic (bibl. 5) gives 45 as te angle of inclination of te slides in loose, detrital formations, and a. larger angle, as great as 85 for more compact formations (fig. 1). Fig. 1. Subsidences due to coalmining,according to A. H. Goldreic (bibl. 5, p. 88, fig. 59). If we imagine a sufficiently large disc-saped mass to be removed from te coal seam, te sliding plane will become funnel-saped. Te applications of tis penomenon to te caldera-formation brougt me to te following ypotesis: if not only te cylindrical disc in te dept is removed, but a complete cylinder from te bottom to te surface, a forteriori a funnel-saped sliding plane 1 ) will form, at least if te radius of te eruption cylinder is not too small. Te collapsed material will ten presumably ave a sallow basin-saped surface. Tis basin sape migt be transformed into a more or less orizontal plane by lava-streams or volcanic ases or bot. In tis article te line of reasoning is resumed, and te last mentioned views on caldera formation from cored-out cylinders are treated as generally as possible. Prom tis, owever, it must not be concluded tat te first point of view is entirely abandoned, as narrow eruption cylinders will not subside tso easily, and te new teory is especially meant for larger eruption cylinders. If tis teory is applied to te normal volcanic cone we arrive at te following results (see fig. 2). As generating line of te external slope of te volcanic cone a straigt line is taken, forming an angle ß wit te orizontal. Te angle of te slide, is also measured wit regard to a orizontal line. a, Wen te cylinder (AA'B'B) is expelled from te volcano, te volume ') Te term "sliding plane" is used ere for te sake of simplicity. It first forms as a funnel saped fault along wic subsequently sliding takes place.
186 witin te surface of revolution (ABC) will collapse into te volume (A A'DP). Te contents of tese two bodies must e equal. Fig. 2. Diagram of te formation of a caldera from an eruption cylinder by a funnelsaped slide. = 1000, d = 50, α = 70, ß = 15, x = 246, = 2021 (See table B 5). Of a given caldera te radius of te upper rim is known as well as d te dept below te rim of te caldera bottom, wile ß and a may vary witin certain limits and can be taken as te known variables. Te dept E K =, of te former eruption cylinder, after te collapse, is a function of a, of te radius of te eruption cylinder x, and in a lesser degree of ß also. In tis paper te general equation will be deduced from wic x can be found and afterwards,. Ten a particular case will be treated, and finally a general solution will be given, to wic teoretical considerations will be attaced. Of course in reality neiter A C not B C will be straigt lines and in te calculations, for te sake of simplification, in bot cases a mean slope as been taken, wile D F as also been taken as straigt.
( x) x) )* 2 187 II. CALCULATION. 1. Deduction of te general equation. Known:, d, ß and a. Fig.. Te measurements used in te calculation. Te drawing was made wit: = 40, d =, α = 6 26, (tan α = 2), ß = 26 4 (tan ß = ½). Unknown: x, wen: Contents (AB C) = Contents (AKEF)., = (, = tan ß tan x r = tan a. =t H( -ti)- x S w Cont. ( t + 2 ) ( 2 +x + x 2) -jr (4 + 2 ) x 2 = =\ tt (tan ß + tan *) ( x ) ( 2 +x - x 2) (II). Cont. (A KEF) J = % (r 2 +rx + x 2) = ó = I-srtanajlß x! (III). ' ( \ tan a, ) )
x) 1080x - 60x tan =0 2 12.2 11.0 8.4 188 (II) = (III) terefore: (tan ß 4- tan *) ( - ( 2 + x x 2 ) = 8 >. (V tan x > ) = K ir tan x \ ( ) (2 tan js + tan x) x (tan ß + tan x) x 2 + 4- (tan ß + tan x) / d \ x V tan a, i (IV). 2. Example: Te Tengger Caldera. Te Sandsea is ere taken, wereby te straigt boundery, te Tjemora Lawang, is left out of consideration; = 4000 m., d = 00 m. Wat te mean angle of incline ß is for te part of te volcanic cone tat as disappeared, could only e approximated by a special investigation. Te larger te angle is, te larger te volume of te eruption cylinder must be under oterwise equal conditions of te d and. About 25 would be te maximum tat can be assumed for te mean angle of incline. To simplify te calculations tan ß = y 2 is taken ß = 26 4' terefore. For a was taken 45, 66 19', 6 25', 75 58'' and 84 17' successively, sotat tan«becomes successively: 1, / 2, 2, 4, 10. Te general equation (IV) ten gives te following five equations, togeter wit wic te approximated roots are listed, everyting being expressed in ectometers. TABEL A (ectometers). ß tan ß oc tan x Equation. Approximate roots. x(i) *(.) x( ) ( 2 ) 26 4' Vi 45 1 8x 8 -- 2 + 90694 =0-1.9-4- 22.4 + 6.5 14.6» 11 5619'»/. llx - 480x 2 + 9184= 0 - + 18 + 7.8 0.00 Î) 11 6 26' 2 14x 8-600x 2 + 9174=0 - + 15.5 + 8.4 46.00 ÏÎ 11 75 58' 4 26x' - 2 + 9216 = 0 + 10.7 9.2 11.80 + n ii 84 17' 10 62x 8-2520x 2 + 92585 =0 5.7 + 6.6 + 9.7 1.00 Tese five groups of tree roots are grapically represented in fig 4.- Te negative and te smallest positive root approac zero, wic is reaced at a = 90. Te greatest positive root lies close to, te value of 40 being reaced at a=90.
189 Te negative root lias no real significance for our problem, nor as te largest positive root any volcaiiological importance for our problem, Fig. 4. Grap of te tree roots of te equations of table A. as te diameter of te eruption cylinder would ten e almost equal to tat of te caldera and at te same time te dept of te eruption cylinder would be very sligt (fig. 5). It is only te middle root tat is of importance, tat is te smallest of te two positive roots.
190 Fig. 5. epresentation of an application of te two positive roots of te equation. 14x³ 600x ²+ 9174 = 0. = 40, d =, α = 6 26 (tan α = 2), ß = 26 4 ( tan ß = 4). Te two positive values of x are 15.5 and 8.4, te values of belonging to tese are 46 and 0.2. In fig. 6 te equation for te Tengger caldera for 26 4' and = /? a = 6 26': y 14 = x 2 600 x + 9174 is represented grapically. Tis sows te general course of te functions ere dealt wit. Fig. 6. Grap of te equation y = 14 x³ 600 x² + 9174. Tis equation olds for = 40, d =, α = 6 26 (tan α = 2),ß = 26 4 (tan ß = ½). Te tree roots for x lie on te point of section of te curve wit te x-axis (y = 0).
191. General solution. Te solution for te smallest positive root tat ere follows I owe to Prof. dr. W. de Sitter, wo also ad te accompanying 10 tables calculated for me. As variable knowns besides a and ß we ave It and d. It is more convenient to take S ( = -=r- ) a.s known instead of and d. By te measurements of a few calderas it was found tat 8 will usually lie between 0.01 and 0.10. Te following caldera's were consulted: a. Tengger caldera (Java) (bibl. 1). Te "Zandzeo", leaving out of consideration te straigt boundery of te Tjemoro Lawang. = 4000 M., d = 00 M.. terefore 0.075. 4000. Batoer Caldera I (Bali) (bibl. 6 and 7). Kemmerunc; (bibl. 6, atlas, plates V and VI) assumes tat tere must ave been an earlier large Batoer caldera, ere called I for wic I ave taken = 6000 M. and d = 250 M. = =0^42 6000 ' o. Batoer caldera II (bibl. b' and 7). Te second Batoer caldera is bounded by te terrace of Kintamani. Following Stens map (bibl. 7) for te diameter of tis caldera and te same map for te difference in eigt between caldera rim and caldera bottom, I take = 600, d = 00 terefore = = 0.08. Sflfl 600 d. Idjen caldera 's (Java) (bibl. 8 and 9). Kemmerling as discovered a terrace on te inner slope of te Goenoeng Kendeng (bibl. 8, p. 57) from wic e concludes tat in tis caldera also, at least two collapses ave taken place. Moreover e is of opinion tat te elliptic form of tis caldera, wit a long axis of 20 k.m. from E. to W. and a sort axis of 16 k.m. from N to S., indicates a sifting of te volcanic axis. He considers te Idjen to be a composite caldera. For te two Kendeng caldera 's Kemmerling estimates te amount of te two collapses (bibl. 8, p. 140) at 00 m. and 500 m. Hero I ave taken for te first caldera formation E = 6000 m., d = 00 and d = 500 m. for te second. For te first caldera we find S = = 0.07. 8000 second S = = 0.06. If we divide te general equation (IV) : tan x 2 tan ( tan x + 2 tan ß) x (tan x -+ ß) 4- (tan x + ß) t d \ - tan a, = 0. \ tan x' by tan a, it is canged to:
(1 0.80 - s s) I 192 (»-«S-Äi)-»f 1 + i)'+(> +S5) - ( -^r» If we now put: x = e, K = 8 cot a, y = tan ß cot a and d=8k. ten te equation: ( + 2 7 ) -(l + y)t.2 +(l + y )-'(&IZ- 8^Hf=0. is arrived at, Z ( + 2y)«; -(l+y) *+l + y -(l-k) =0 (V). If we now put: A = 1 K ) = K (1 k -f I K 2). We arrive at te equation: A+ (l_ 2 + 2 )y (e 2 e a )=0 (VI). Tis equation is linear in y and for given s( =-=--1. By letting e vary between 0 te equation in A and y is expressed in straigt lines. Prom te nomogram tus found wit straigt lines, on te oter and e can be read for different y's and A's. By letting a and ß increase by 5 different y's occur, for one particular 8 different k's are ten found, and terefore A's also. After aving found for 10 different 8's (0.01, 0.02, 0.0 0.10), 9 different «'s (45, 50, 55 85 ) and 7 different js's (0, 5, 10-0 ), te 60 corresponding values for -gwere calculated. In (I) we found: -A = tan*j(t 6)-xj. If. we substitute te values found above for d = 8 and x= e te following equation is arrived at:, = tan # ( 6 V tan«1 ' 1)., = tan x (1 5 1 terefore: = tan a (1 e) (VII).
must first be ascertained from te topograpical map, from wic 8 p Values of α. = S for 19 Te ten tables for te different 8's follow ere. (Tables B 1 B 10 ). If we now wis to apply te reasoning of te new teory to any particular caldera, to see wat conclusions we are brougt to, and d follows and so sows wic of te ten tables must be used. Corresponding to te tables graps ave been made (fig. 7 16) x from wic and can be read for all a's between 45 and 85 and all ß's between 0 and 0. Values of ß. From te incline of te remaining foot of a volcano tat as been canged into a caldera by collapsing, a mean angle of incline ß can be deduced by comparison wit te inclination of large volcanic cones tat ave not been attacked by caldera formation. All /S's tat belong to a particular angle of a lie upon a straigt line in te graps. For eac angle a tere is terefore a straigt line of /-points; te /8-lines x for all angles of a bisect eacoter at te point 1 and -~ just above te 0-abscisse terefore, tat lies at te top of te graps. Prof. W. v. d. Woude was so kind as to provide me wit tis important addition. It enables us to read for one interpolated «-value x and ~- te different /-values. Te coice between te different a's seems more difficult. In III tis point will be furter considered. After a a- and a /?-value as been x are read from te graps or te tables. By multiplying by te particular value for wic we ave read from te map, x (= te radius of te cored-out cylinder) and are found.
194 TABLE B1. σ = 0.01. /\ 45 50 55 60 65 70 75 80 85 T 0,10 0,094 0,085 0,078 0,071 0,06 0,055 0,045 0,02 0 0,887 1,074 1.297 1,587 1,98 2,564,51 5.40 11,05 0,214 0,19 0,178 0,160 0,14 0,126 0,111 0,086 0,062 5 8 0,776 0,952 1,164 1,445 1,828 2,91,1 5.17 10,71 0,282 0,258 0,25 0,216 0,191 0,168 0,144 0,114 0,080 10 0,708 0,874 1,082 1,48 1,725 2,276,18 5,01 10,51 0,48 0,09 0,282 0,258 0,22 0,206 0,174 0,142 0,100 15 0,642 0,812 1,015 1,275 1,67 2,171,07 4,85 10,28 0,85 0,52 0,2 0,294 0,268 0,26 0,202 0,160 0,112 20" 0,605 0,762 0,957 1,21 1,560 2,089 2,97 4,75 10,14 0,427 0,9 0,60 0,28 0,297 0,26 0,228 0,182 0,128 25 8 0,56 0,712 0,904 1,154 1,498 2,015 2,87 4,6 9,96 ~ 0,466 0,40 0.97 0,62 0,27 0,290 0,250 0,202 0,142 0 0,524 0,669 0,851 1,095 1,44 1,940 2,79 4,51 9,80
Fig. 7. Grap of table B1.
196 TABLE B. σ = 0.02. 2 0\ 45 50 55 60 65' 70 D 75 80 85 0,15 0,140 0,126 0,112 0,100 0,087 0,07 0,058 0,046 0 e., 0,827 1,00 1,228 1,518 1,910 2,488,44 5,2 10,88 0,244 0.222 0.201 0,184 0,160 0,141 0,120 0,098 0,070 5 0,76 0,906 1,121 1,9 1,782 2,40,27 5,09 10,61 0,07 0,281 0,254 0,22 0,204 0,180 0,152 0,127 0,086 10 0,67 0,87 1,045 1,10 1,687 2,2,15 4,9 10,42 IT 0,59 0,29 0,00 0,277 0,24 0,21 0,181 0,149 0,10 15 K 0,621 0,780 0,980 1,22 1,604 2,142,04 4,81 10,2 0,405 0,71 0,9 0,09 0,275 0,244 0,208 0,168 0,116 20 0,575 0.70 0,924 1,177 1,55 2,057 2,94 4,70 10,08 0,446 0,411 0,74 0,42 0,06 0,271 0,21 0,189 0,11 25 0,54 0,682 0,874 1,120 1,469 1,98 2,86 4,58 9,91 " 0,484 0,446 0,410 0,74 0,5 0,297 0,256 0,209 0,146 0 8 0,496 0,640 0,82 1,064 1,406 1,911 2,77 4,46 9,74
Fig. 8. Grap of table B2.
198 TABLE B. σ = 0.0. \ 45 50 55 60 65" 70 75 80 85 0,191 0,172 0,158 0,140 0,125 0,110 0,091 0,07 0,057 0, 0,779 0,957 1,172 1,460 1,847 2,415,6 5,2 10,75 0,270 0,246 0,225 0,200 0,179 0,158 0,1 0,104 0,078 5 0,700 0,869 1,077 1,56 1,71 2,28,20 5,05 10,51 0,29 0,00 0,272 0,246 0,220 0,194 0,164 0,11 0,09 10 0,641 0,804 1,010 1,276 1,64 2,184,09 4,90 10,4 ~ 0,80 0,46 0,16 0,285 0,255 0,227 0,192 0,152 0,109 15 0,590 0,750 0,947 1,208 1,568 2,09 2,98 4,78 10,16 0,42 0,88 0,56 0,21 0,288 0,254 0,218 0,172 0,121 20 0,547 0,700 0,890 1,146 1,497 2,019 2,89 4,66 10,02 0,46 0,426 0,90 0,5 0,17 0,282 0,240 0,191 0,16 25 8 0,507 0,654 0,841 1,091 1,45 1,942 2,80 4,56 9,85 0,501 0,461 0,422 0,85 0,46 0,06 0,26 0,211 0,150 0 0,469 0,612 0,795 1,05 1,7 1,876 2,72 4,44 9,69
Fig. 9. Grap of table B.
' 200 TABLE B 4. σ = 0.04. 45 50 55 60 65" 70 75 80 85 1/ \ 0,224 0,205 0,186 0,166 0,149 0,11 0,112 0,087 0,066 0 8 0,76 0,908 1,122 1,404 1,785 2,47,27 5,14 10,64 IT 0,294 0,270 0,245 0,220 0,197 0,172 0,148 0,116 0,084 5, 0,666 0,80 1,08 1,11 1,682 2,25,14 4,97 10,4 0,52 0,21 0,291 0,262 0,24 0,206 0,176 0,141 0,098 10 0,608 0,769 0,972 1,28 1,60 2,141,0 4,8 10,27 "" 0,401 0,66 0,2 0,299 0,269 0,26 0,202 0,161 0,114 15 0,559 0,716 0,914 1,174 1,528 2,059 2,94 4,72 10,09 T 0,442 0,407 0,69 0, 0,00 0,26 0,227 0,180 0,127 20 0,518 0,667 0,861 1,115 1,462 1,985 2,84 4,61 9,94 0,482 0,44 0,40. 0,66 0,29 0,290 0,248 0,199 0,141 25 K 0,478 0,624 0,81 1,058 1,99 1,910 2,76 4,50 9,78 0 0,518 0,478 0,47 0,97 0,57 0,15 0,271 0,218 0,15 0,442 0,582 0,764 1,004 1,9 1,842 2,68 4,9 9,64
Fig. 10. Grap of table B 4.
202 TABLE B 5. σ = 0.05. J0\ 45 50 55 c 60 65 70 75 80 85 ~ 0,252 0,229 0,208 0,189 0,166 0,146 0,120 0,099 0,065 0 0,698 0,869 1,081 1,55 1,79 2,296,2 5,06 10,64 0,21 0,291 0,262 0,27 0,209 0,18 0,154 0,127 0,085. 5 0,629 0,795 1,004 1,272 1,647 2,194,11 4,90 10,41 0,74 0,40 0,07 0,279 0,246 0,216 0,181 0,149 0,098 10" 0,576 0,77 0,940 1,199 1,567 2,101,00 4,78 10,26 ~~ 0,421 0,82 0,48 0,14 0,279 0,246 0,207 0,168 0,112 15 0,529 0,687 0,881 1,18 1,497 2,021 2,91 4,67 10,10 0,46 0,422 0,8 0,47 0,09 0,272 0,21 0,187 0,127 20 8 r" 0,487 0,69 0,81 1,018 1,4 1,950 2,82 4,56 9,9 0,501 0,458 0,418 0,78 0,-8 0,298 0,252 0,206 0,140 25 0,449 0,596 0,781 1,027 1,70 1,878 2,74 4,45 9,78 0 ~ 0,57 0,492 0,450 0,409 0,64 0,22 0,272 0,22 0,154 0,41 0,556 0,75 0,974 1,14 1,812 2,67 4,6 9,62
Fig. 11. Grap of table B 5.
204 TABLE B6. σ = 0.06. 45 50 55 60 65 70 75 80 85 ~ 0,280 0,252 0,229 0,208 0,185 0,162 0,14 0,112 0,07 0 0,660 0,82 1,041 1,12 1,688 2,242,17 4,97 10,54 0,42 0,11 0,282 0,254 0,224 0,198 0,167 0,17 0,091 5 0,598 0,761 0,965 1,22 1,605 2,14,05 4,8 10, 0,94 0,59 0,24 0,292 0,260 0.229 0,19 0,158 0,104 10 0,546 0,704 0,905 1,166 1,527 2,058 2,95 4,71 10,18 0,441 0,400 0,64 0,28 0,291 0,257 0,217 0,177 0,120 15 8 0,499 0,655 0,848 1,104 1,461 1,891 2,86 4,61 10,00 0,48 0,48 0,99 0,60 0,22 0,282 0,240 0,194 0,12 20 b 0,457 0,610 0,798 1,048 1,94 1,912 2,77 4,51 9,86 0,521 0,474 0,42 0,90 0,48 0,07 0,261 0,212 0,145 25 0,419 0,567 0,751 0,997 1,9 1,844 2,70 4,41 9,71 ~ 0,555 0,507 0,46 0,420 0,77 0,2 0,282 0,20 0,158 0 0,85 0,528 0,707 0,945 1,276 1,775 2,62 4,1 9,56
Fig. 12. Grap of table B6.
206 TABLE B7. σ = 0.07. 45 50 55* 60 65 70 75 80 85 0,07 0,278 0,249 0,222 0,195 0,172 0,149 0,117 0,080 0 0,62 0,791 1,002 1,277 1,665 2,205,10 4,94 10,45 BT 0,68 0, 0,298 0,268 0,26 0,207 0,180 0,140 0,097 5 0,562 0,725 0,92 1,198 1,570 2,108 2,99 4,81 10,25 T 0,418 0,79 0,41 0,0 0,272 0,28 0,20 0,160 0,110 10 0,512 0,670 0,871 1,17 1,492 2,02 2,90 4,69 10,10 0,461 0,420 0,79 0,9 0,0 0,26 0,227 0,180 0,124 15 0,469 0,621 0,817 1,075 1,425 1,955 2,81 4,58 9,94 0,499 0,455 0,412 0,72 0,2 0,289 0,248 0,198 0,17 20 0,41 0,580 0,770 1,018 1,6 1,88 2,7 4,48 9,80 0,540 0,492 0,445 0,401 0,60 0,14 0,270 0,215 0,148 25 0,90 0,516 0,72 0,967 1,0 1,814 2,65 4,8 9,67 0 0,574 0,525 0,477 0,40 0,86 0,8 0,291 0,22 0,161 0,56 0,496 0,677 0,917 1,247 1,749 2,57 4,28 9,52
Fig. 1. Grap of table B7.
208 TABLE B8. σ = 0.08. 45 50 55 60 65 70 75 80 85 0,1 0,00 0,271 0,240 0,21 0,188 0,158 0,126 0,087 0 0,589 0,754 0,961 1,26 1,608 2,151,06 4,88 10,6 0,90 0,52 0,18 0,285 0,251 0,220 0,186 0,148 0,10 5 0,50 0,692 0,894 1,158 1,527 2,06 2,96 4,75 10,17 0,48 0,97 0,57 0,20 0,284 0,249 0,210 0,167 0,116 10 0,482 0,69 0,88 1,098 1,456 1,98 2,87 4,64 10,02 0,48 0,47 0,9 0,54 0,14 0,276 0,28 0,187 0,10 15 0,47 0,591 0,787 1,09 1,91 1,909 2,78 4,5 9,86 0,52 0,47 0,427 0,85 0,42 0,299 0,25 0,20 0,141 20 0,97 0,548 0,78 0,985 1,1 1,846 2,71 4,44 9,74 0,559 0,508 0,459 0,41 0,68 0,25 0,274 0,220 0,152 25 0,61 0,506 0,69 0,97 1,276 1,774 2,6 4,4 9,62 0,59 0,541 0,492 0,442 0,95 0,48 0,296 0,28 0,165 0 0,27 0,467 0,645 0,886 1,218 1,711 2,55 4,24 9,46
Fig. 14. Grap of table B 8.
210 TABLE B 9. σ = 0.09. 1/ \ 45 50 55 60 65 70 75 80 85 0,57 0,21 0,288 0,258 0,229 0,200 0,170 0,15 0,09 0 0,55 0,719 0,927 1,195 1,564 2,108,01 4,81 10,28 0,412 0,72 0, 0,298 0,266 0,22 0,195 0,155 0,108 5 0,498 0,659 0,862 1,126 1,484 2,020 2,91 4,70 10,11 0,460 0,414 0,72 0,4 0,296 0,260 0,219 0,175 0,120 10 0,450 0,609 0,807 1,064 1,420 1,94 2,82 4,59 9,97 0,504 0,454 0,408 0,67 0,25 0,285 0,240 0,192 0,15 15 0,406 0,561 0,755 1,006 1,58 1,874 2,74 4,49 9,8C 0,541 0,492 0,441 0,97 0,5 0,09 0,261 0,209 0,145 20 a 0,69 0,516 0,708 0,954 1,298 1,808 2,67 4,9 9,68 0,580 0,525 0,472 0,424 0,78 0,2 0,282 0,225 0,156 25 0,0 0,476 0,664 0,908 1,244 1,745 2,59 4,0 9,56 "F 0,615 0,558 0,50 0,454 0,405 0,56 0,02 0,242 0,168 0 0,295 0,47 0,620 0,856 1,186 1,679 2,51 4,21 9,42
Fig. 15. Grap of table B9.
212 TABLE B 10. σ = 0.10. / \ 45 50 55 60 65 70 75 80 85 0,82 0,40 0,07 0,272 0,242 0,210 0,181 0,145 0,100 0 0,518 0,687 0,890 1,161 1,526 2,070 2,95 4,75 10,19 0,47 0,91 0,51 0,1 0,279 0,240 0,206 0,164 0,114 5 0,46 0,626 0,827 1,090 1,447 1,988 2,86 4,64 10,0 ~~ 0,485 0,42 0,88 0,47 0,07 0,267 0,228 0,18 0,125 10 0,415 0,577 0,774 1,01 1,86 1,914 2,78 4,5 9,90 0,527 0,472 0,42 0,80 0,8 0,292 0,250 0,200 0,140 15 0,7 0,529 0,724 0,974 1,20 1,845 2,70 4,44 9,7 0,567 0,507 0,456 0,409 0,62 0,15 0,270 0,216 0,149 20 0, 0,488 0,677 0,924 1,269 1,782 2,62 4,5 9,6 0,604 0,541 0,488 0,48 0,89 0,9 0,289 0,22 0,160 25 0,296 0,447 0,6 0,87 1,211 1,716 2,55 4,25 9,50 0,640 0,57 0,519 0,465 0,415 0,62 0,09 0,249 0,17 0 s 0,260 0,409 0,587 0,827 1,155 1,65 2,48 4,16 9,5
Fig. 16. Grap of table B 10.
200 214 III. DISCUSSION OF THE VALUES FOUND, AND THEOETICAL CONSIDEATIONS. 1. On te applicability of te new teory. Te values of x. Te teory of te formation of ealdera's discussed ere is based upon te formation of a cored-out cylinder caused by te gas pase of Perret (bibl. ). Secondarily from tis cylinder, a caldera is formed by te walls subsiding along a funnel-saped surface. Te formation of a caldera (a negative form), tat is to say a large flat crater bottom wit steep sides, is a logical consequence of te fact tat te angle ß is always smaller tan te angle a. Te walls of te cored-out cylinder form a vault wit a vertical axis wic can support radial It is clear tat wit uniform pressure. material te radius of te cored-out cylinder will decide te amount of pressure tat can be supported. Te smaller te radius, te greater te pressure tat can be born by te vertical arc. It sould be possible to make estimations of te relation between tese two, but ere we will do no more tan point out tat te radius of te cored-out cylinder of Vesuvius, according to MaltiAdra's profile (bibl. 4) is about 100 m., wile from Frank Perret 's work (bibl., p. 67 and fig. 1) te conclusion migt be drawn tat x lay between 100 m. and 200 m. After te Vesuvius eruption in 1906 tere did not follow any caldera formation, in te manner discussed. Te "collapse" of wic Perret (bibl., p. 94) speaks and wic e sows in poto's 7 and 56, migt be considered te as beginning of a sliding in of te walls. Te '' external collapse" took place on te sout east flank of te cone. It is tougt to ave been due to a less resistant part of te cored-out cylinder, but I may point out tat Perret explains it in anoter It way. seems, in fact, tat wit te Vesuvius-rock te radius x = 100 m. was too small to occasion a caldera formation from collapse. Notwitstanding, te Vesuvius crater gradually acquired a caldera-like sape, caused by te walls of te cylinder crumbling away from above, were te material was loose, and later on by rising lava filling up te vent (bibl. 1, and 4). Large caldera's, wic migt also be called true caldera's are formed terefore, only wen x is large, as it is only ten, tat te vent can be pressed in. We must not terefore be afraid of assuming a somewat large radius of te cylinder. Tis is one of te necessary deductions of te teory developed ere: it applies only to spacious cored-out cylinders. If we take te angle ß 15, = x for S = 0.5, will be between 0.1 and 0.4 tat is for = 5000 m. between 500 and 2000 m. 2. On te dept of te cored-out cylinder. Te value of. Te dept of te part of te eored-out cylinder tat is ere called, as also tat of te entire original cored-out cylinder, depends primarily
te dept of 50 km te initial pressure would terefore be 15000 kg/cm 2. 215 upon te angle a. As soon as a becomes more acute tan 70, rapidly increases. If we take ß = 15, s for 8= 0.5 varies between 0.5 li and 10. As in comparatively sallow coal mines slides of 85 ave been registered in ard rock, it seems to me probable tat in our case te mean angle of sliding will be 70 or larger. For ß = 15 and 8 = 0.5, tis means tat, as a value between 2 and 10. For =5000 m., would lie between 10 km and 50 km. At tese depts we approac te magma camber. For some geologists tese depts may seem outrageous, and I must confess tat at Bret sigt I found difficulty in accepting suc a conclusion, but we sall see tat furter considerations lead us in te same direction. To make tis clear, we must first examine te degree of force of a volcanic eruption.. On te degree of force of a volcanic eruption. Te motive force of a volcanic eruption is te magmatic gas. As in a violent eruption magma is always trown out, it cannot e asserted in general tat in te magma camber before te eruption tere as l>een a separation between gas and magma. Before te-eruption te gas is in solution under pressure in te magma. According to te present state of penological knowledge, I believe, we must suppose tat by crystallization of te magma in te dept te gas is eld in solution in te residuary magma under a constantly increasing pressure. Te increase of pressure can only go as far as te counter-pressure allows. In oter words te maximum tension of te gasses dissolved in te residual magma is equal to te resistance of te rock above te camber. Te pressure of gas at te moment of eruption, tat is te initial pressure of te eruption, just exceeds te counter-pressure wic is exerted upon "te magma camber. It may be taken tat up to depts of 100 km te pressure will increase wit a constant ratio to te dept. Assuming to tese depts a mean specific gravity of te rock of, te increase of pressure per 10 km will be 000 kg/cm 2. At a dept of 50 km te pressure would be 15000 kg/om 2. Wit an eruption from Tere is anoter factor besides te gas pressure wic determines te force of te eruption. Tis is te lengt of time during wic large quantities of gas per time unite escape. A certain pressure being given by te dept, te force of an eruption will depend upon te total amount of dissolved gas and terefore upon te volume of te magma in te camber, from wic te gas escapes. It is terefore logical to assume, tat violent eruptions can only take place, were te magma camber is large. Te force of a volcanic eruption, terefore, can be pysically defined as te product of gas pressure and te amount of gas, and geologically expressed as te function of te dept of te camber and of its volume. In 1927 I was able to prove by experiments (bibl. 1), tat te radius of a cored-out cylinder is a function of te velocity of te eroding gas.
216 Te velocity itself is a function of te amount per time unite of escaping gas. Furter te experiments seemed to sow, tat at a given pressure a certain amount of time is needed for coring out te cylinder. Te widening begins below and works upwards by eddies to te top, provided tere is a sufficient amount of available. gas Caldera-formation as ere described will only take place wen te camber lies deep down, as it is only ten tat te tension of te gas can be great, and wen te volume of te camber is great, as only ten te quantity of gas is sufficient to core out a cylinder. It sould be possible to find te dept of te magma camber from te initial pressure. So far tere are no data known on tis subject, but is ougt to be possible to determine te initial pressure approximately from te gas column of wic Perket gives a representation (bibl., p. 45, fig. 1), as it rose, up during te intermediary gas pase in Vesuvius and wic only spread out orizontally at a eigt of 10 km above te vent. '' Let te reader tink of globular masses of compressed vapour almost exploding in te air at more tan 10 kilometers above te vent, and ten try to imagine te original tension of te. gas and te degree of its acceleration 'witin te saft of te volcano" (bibl., p. 45 46). If it prove leasable to arrive at an idea of te initial pressure in tis way and tat tis is found to be very ig, wc must not forget tat te force of te eruption of Vesuvius in 1906 during te intermediate gas pase was small compared to te force tat is needed to form a true caldera via a cored-out cylinder. In te Krakatoa eruption in 188 te gas column was probably 50 km ig, and terefore te minimuim dept of te magma camber muc greater in 1906. tan in te Vesuvius eruption 4. On te most probable combination of x and. eturning to wat we said on p. 19 and 215 about te angle of a, we must point out in te first place tat a given size of te caldera can lie arrived at matematically by various combinations of x wit tat differ greatly. For S = 0.5 and ß = 15 te combinations vary between x=0.4 wit = 0.5 wit a = 45 and x= 0.1 wit = 10 wit a = 85. Te point now is to ascertain geologically wic combination is te most probable, tat wit a large or wit a small. Te experiments already described sowed tat te greater te pressure is in te duct of te gas, te greater te radius of te cored-out saft becomes. Te pressure in te gas duct was measured in a reduction valve. A similar reduction valve will be present in nature, te opening of te magma camber. But in te natural state te opening will be constantly enlarged by te continuous erosion of te gas-stream. Only wen te magma camber is very large te initial pressure of te escaping gas will be comparatively slowly reduced. Only in tat case a spacious cored-out vent can be formed. Wit a low pressure it is a priori impossible for te vent to be cored-out. It seems to me, terefore, tat a combination of great dept and great volume of te magma camber
217 are necessary for te formation of a caldera in te way suggested. It follows tus tat a must be large. A steep funnel is te most probable, not only for te reasons given above, but also for te following. Te deeper te cause of te sliding down is, te greater must be rigidity of te rock, terefore te greater te angle of slide «must be. Te combination of a large wit a small x seems to me, terefore, te most probable. For a caldera wit = 5000 m. and S= 0.5 and ß t= 15 te most probable combinations would lie between: x= 0.1 wit 10, 500 c= tereforex= m wit = 50 km (<* 85 ). = and x= 0.2 wit =, terefore 1000 x= m wit = 15 km (a 75 ). = 5. On te duration of te paroxysms and te periods of quiescence in caldera forming volcanoes. In bibl. 2 I ave endeavoured to sow tat te cored-out cylinder of Krakatoa in 188 wic led to te caldera being formed, did not lie at te point of te preliminary eruptions wic took place from May 20t to Aug. 26t from te Perboewatan volcano amongst oters, but about 2 km to te sout.. D. M. Verbeek ad already ascertained (bibl. 10) tat te caldera formation of 188 was not te first, but ad been preceeded by a larger one. After te first caldera formation.according to te teory ere developed, te cored-out cylinder must ave been coked up by te sliding down material. It is probable tat a funnel-saped vent closed in tis way by loose material, will sut off te magma camber less effectively tan te former narrow vent filled wit congealed magma. It is also natural tat wit te increasing gas pressure in te magma camber te gas will sometimes find an outlet along a weak line in te sliding plane. I regard te formation of te Perboewatan and Danan volcanoes as due to te escape of gas along a sliding plane, just as I regard te present activity of Krakatoa as escape of gas along te sliding plane of 188. Wit te leaking out of magmatic gas, magma is carried along and tus a volcano is built up. Tis stage may continue for a long time. In te case of Krakatoa it probably lasted for centuries. Later on te leakage was coked (1680), so tat te pressure in te magma camber increased more and more during 20 years. On May 20 188 te weak place was once more opened, te activity gradually increased until on Aug. 26 t 188 te paroxysm took place. Te actual eruption from te central vent wic formed te cored-out cylinder, began on Aug. 26t 188, reaced its eigt on Aug. 27* and died out on Aug. 28 t - at 6 o 'el. in te morning. Te duration of' te actual eruption, wic caused te caldera formation, was tus very sort. Tis is quite natural, because as soon as a wide cannel as been bored by te eroding current of gas, te tension of te gasses dissolved in te magma, quickly decreases. Ten te wide eruption cannel is coked up and te eruption is over. Tere was no activity at Krakatoa after Aug. 28* 188.
218 It was 44 years, up to Dec. 1927, before te gasses ad again acquired sufficient tension to leak out along weak lines in te sliding plane. Presumably, terefore, we are now witnessing te formation of a new series of small secondai-y Krakatoa volcanoes, again lying excentric wit respect to te principle axis. (In oter caldera's, suc as in Batoer on Bali (bibl. 7) te secondary volcanoes are formed centrally). To sum up, I tink I may conclude, tat te paroxysmal eruption wic primarily creates a cored-out cylinder and secondarily a caldera, lasts only a very sort time, some days, but tat te period of rest between two caldera forming eruptions is very long (centuries), tat moreover between two paroxysms of te first order a lengty period of subdued constructive volcanic activity intervenes wic may last for centuries. Te great dept of te volcanic magma camber tat is ere postulated, demands a great lengt of time to bring about te ig gas-pressure necessary to overcome te counter-pressure of te overlying rock. A small leakage as little influence upon te increasing gaspressure. Several volcanoes are known in wic a caldera formation must ave taken place more tan once, fe. g. Krakat.oa, Batoer and Idjen. Te raritywit wic caldera forming eruptions ave taken place in istorical time is in itself owever a proof tat te period of rest between two paroxysms of te first order must e very great. IV. SYNOPSIS OF THE THEOY OF THE FOMATION OF CALDEAS. Hypotetical premises. 1. It is possible for Pkrret's gas pase to be more violent tan in te eruption of Vesuvius in 1906. (Perret type). 2. Te primary result of te gas pase is te formation of a cored-out cylinder and secondarily te collapse of te cylinder along a funnelsaped sliding plane, formes a caldera. Conclusions. 1. For tis caldera formation a large quantity of gas under ig pressure is necessary. 2. A matematical treatment of te problem leads to te conclusion tat te dept of te camber magma must be great (probably of te order of 15 50 km).. Only violent eruptions (paroxysms of te first order) can form a large cored-out cylinder. Te degree of force of a volcanic eruption is a function of te dept and te volume of te magma camber. 4. For collapses along sliding planes a wide eruption cylinder is necessary probably of te order of 1000 2000 m).
219 5. Te duration of a paroxysm of te first order (destructive activity) is very sort (some days) as compared to te period of quiescence before te next paroxysm of te first order (some centuries). 6. During tese centuries te gas pressure must rise to equal te weigt of te overlying rocks. 7. During te quiescent period secondary volcanoes may be formed by leakage of gas (constructive activity). BIBLIOGAPHY. 1. B. G. Escer, Vesuvius, te Tenggcr Mountains and te Problem of Caldera 's. Leidsce Geolog. Mededeolingen, II, pp. 51 88, 1927. (Wit topograpical map of te Tengger-caldera 1: 100.000). 2. B. G. Escer, Krakatau in 188 en in 1928. Tijdscr. Kon. Nederlandsc Aardrijkskundig Gen., 2e Ser., dl. LV, pp. 715 74.. Frank A. Perret, Te Vesuvius Eruption of 1906. Study of a Volcanic Cycle. Carnegie Institution of Wasington, Publication No. 9, Juli 1924. 4. A. Malladra, Sul graduale riempimento del cratere del Vesuvio. Communicazione tenuta all' VIII Congresso Geografico Italiano, Vol. II, degl "Atti", 1922. deren Einfluss 5. A. H. Goldkeicii, Die Bodenbewegungen im Kolenrevier und auf die Tagesoberfläcc. Berlin, 1926. b". G. L. L. Kemmerlino, De Vulkanen Goenoeng Batoer en Goenoeng Agoeng op Bali. Jaarboek v.. Mijnwezen in Ned. Oost-Indië. 48e Jaarg., 1927, Verandelingen Ie Gedeelte, Batavia 1918, pp. 50 77. (Wit Atlas). 7. C. E. Steun, De Batoer op Bali en zijn eruptie in 1926. Vulkanologisce en Seismologisee Mededeelingen, No. 9. Dienst v. d. Mijnbouw in Ned. Indië, Bandoeng 1928. (Wit map of te Batoer-caldera 1: 50.000). 8. G. L. L. Kemmerlino, Het Idjen-Hoogland, Monografie II. De Geologie en Geomorpologie van den Idjen. Batavia (Witout date). (Wit geological map 1: 100.000 of te Idjen-caldera). Topografisce kaart 1: 20.000. Idjen-Hoogland. Toopgrafisce Inricting, Batavia 1920. 10. B. D. M. Verbeek, Krakatau, Batavia 1885.