Bernoulli process with 282 ky periodicity is detected in the R-N reversals of the earth s magnetic field

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1 Submed o: Suden Essay Awards n Magnecs Bernoull process wh 8 ky perodcy s deeced n he R-N reversals of he earh s magnec feld Jozsef Gara Deparmen of Earh Scences Florda Inernaonal Unversy Unversy Park, PC-344 Mam, FL 3399 Ph: Fax: E-mal: jozsef.gara@fu.edu

2 ABSTRACT Invesgang he polary me scales for he las 8 My a Bernoull process wh p= overrdng wh a Gaussan nose have been deeced for he R-N magnec reversals. Usng he deeced paern he probably of an upcomng R-N reversal can be calculaed beween any me nervals. The Bernoull rals are separaed by 8 ky. Ths frequency s conssen wh planeary and Mlankovch cycles. The p= probably of he Bernoull process could be he resul of he posve and negave nerferences of hese cycles. Key words: earh's magnec feld; R-N reversals; perodcy; probably

3 . Inroducon Prevous sudes assumed ha he mechansm responsble for he rggerng of he magnec reversal s he same n boh cases. These sudes were lookng for sequences n he me seres of he earh s magnec feld reversals by nvesgang he elapsed me beween wo consecuve reversals (e.g. Cox e al. 98; Marzocch and Mularga, 99; Pelleer, 997; Consoln and De Mchels, 3). The me sequence elapsed beween he same polary reversals has never been nvesgaed. I has been proposed ha he wo knds of magnec reversals, normal o reverse (N-R), and reverse o normal (R-N) mgh be rggered by dfferen mechansms (Gara, ). If he wo dfferen knds of magnec reversals are rggered by dfferen mechansms hen by lookng for a paern n he sequences of he reversals would more accuraely be deeced by focusng analyss on hose reversals, whch are rggered by a common mechansm. In he presen sudy he perodcy of he R-N reversals wll be nvesgaed.. Daa Analyss Usng he polary me scales for he las 8 My (Cande, and Ken, 995) he me elapsed (R N) beween wo consecuve R-N reversals has been deermned frs. ( R N) = (R N) + (R N) () where (R N) s he me before presen of he h R-N magnec reversal. The number of daa (R N) for longer me perods ( >.6 My.) s very few. These long perods were omed because he avalable daa s nsuffcen for sascal analyss. The frequency dsrbuon of R 3 N

4 reveals fve relavely dsnc groups (Fg. ). If one assumes a Gaussan dsrbuon for each of he groups, hen hese dsrbuons overlap and he analyses are sgnfcanly more complcaed. However, snce he means of he varous groups lay several sandard devaons away from each oher, a more racable way o model he groups s o deermne a pon beween hem such ha he probably of a pon beng mscaegorzed s mnmzed. To make hs precse, consder G and G daa se for wo consecuve groups whch have (mean, sandard devaon) ( µ, σ ) and ( µ, σ ) respecvely, and µ < µ. Then we wan o fnd µ < s- < µ such ha p(x s ) + p (X < s ) s mnmzed, where X s a varable equal o R N, p s a probably calculaed from a Gaussan densy funcon wh parameers of µ, σ, and p s a probably calculaed from a Gaussan densy funcon wh parameers of µ, σ. A frs he groups were separaed by vsual nspecon and parameers ( µ, σ ) were calculaed for each group ( G ). Usng he calculaed group parameers he mnmum pons ( ) beween each consecuve group were deermned. If hs mnmum pon concded wh he nal assumpon hen he group separaon was acceped. If he mnmum pon dd no concde wh he nal separaon hen he above procedure was repeaed usng he new mnmum pon for he s + ( ) separaon. Calculaons were repeaed as long as all he calculaed concded wh he nal separaon pon. The calculaed parameers of he separaed groups are gven n able. The means of he overlappng Gaussan dsrbuons are 4 s σ apar, herefore, he overlappng daa on each sde of he dsrbuon s less han.5 percen. Ths small percenage of overlappng does no have sgnfcan affec on he sascal parameers. Addonally, he cu off pars for groups, 3, and 4 are close o symmercal furher reducng he overlappng affec.

5 Analyzng he calculaed sascal parameers, he mean values and sandard devaons of hese separaed groups was found ha he mean values are neger mulples of he smalles mean, revealng a perodc paern. If we le he normalze mean of a group denoe he mean of ha group dvded by s group number (e.g. group 5 s normalzed mean s.8376 My), hen we fnd ha he normalzed means range from.8 o.87 My. The weghed average of he means s.84 My, he correspondng sandard devaon s.73 My The frequency dsrbuon of he reversal me shows ha he R-N reversal s more lkely o have a shorer erm han a longer one. Calculang he relave frequency ha a reversal falls no a group was fnd ha he frequency for he frs group wh mean of.8 My s.5. The frequency of he second group (mean.56 My) s., whle he frequency for he hrd group (mean.85 My) s. ec. [Tab. ]. Ths s jus a Bernoull process wh p= overrdng wh a Gaussan nose. Ths means ha for every ~.8 My, elapsed snce he prevous R-N reversal, here s approxmaely 5% chance ha he earh s magnec feld would make a shf from a reversed o a normal orenaon (Fg. ). 3. Calculang he probably of an R-N Reversal The deeced paern and he sascal parameers allow one o predc he probably of he nex R-N reversal beween any me nervals. The probably of a Bernoull ral s. P (n) = p n ( p) n for n =, () p = where. If zero of n means no R-N reversal, whle one s equvalen wh an R-N reversal hen he probably of he h R-N magnec reversal s: 5

6 P n n (R N) = p ( p) =.5 (3) The uncerany of he me of he Bernoull ral s very hgh herefore he dscree Bernoull ral should be replaced wh a Gaussan dsrbuon. P n n (R N) = p ( p) P() d =.5 + (4) where P() s he Gaussan densy funcon for he h ral. P() = σ e π (+ 78 ) σ (5) where s he me from presen n ky, s he mean, and σ s he varance. The mos recen R-N magnec reversal, Mauyama-Brunhes, has been used as reference; herefore, 78 ky has been added o he curren me. In order o calculae he probably of he occurrence of an R-N reversal beween me and [ P(R N) ] frs he sequence number of he Bernoull ral () should be denfed. The sequence numbers can be deermned from equaon 6 by keepng only he neger pars of he calculaed values and (6) If he begnnng and he end of he nvesgaed perod falls no he same Bernoull ral ( = ) hen he probably of he even can be calculaed by negrang equaon 5 beween he me and. P(R N) = p n ( p) n σ e π (+ 78 ) σ d (7) 6

7 The parameers, mean 8 ky, and varance 73 ky, deermned by hs nvesgaon should be used for he calculaon. If he me perod covers wo or more consecuve Bernoull rals ( ) hen he probably of he R-N reversal can be calculaed as: P(R N) = { [p n ( p) n ] (z+ ) }.5 σ e π (+ 78 ) σ d + + z + σ e π (+ 78 ) σ d (8) where z =. 4. Correlaon wh Asronomcal Cycles The presence of he planeary cycles n he srengh and he nclnaon of he earh magnec feld for he pas.5 has been deeced by Yamazak, and Oda (). The Bernoull process of he R-N reversals of he earh's magnec feld has been sable a leas for he me perod of he daa, whch s abou My. Wh hs me lengh only asronomcal cycles are known o reman sable. The frequency of he Bernoull rals s a harmonc of he bea frequency (93,48.3 y.) of he gan planes (Shrley, and Farbrdge, 997). The b frequency of he global clmae oscllaon (Mlankovch, 93) s also conssen wh he frequency of he Bernoull rals. Impac naed global coolng has been proposed as a possble rggerng mechansm for he magnec reversals (Muller, and Morrs, 986). Ths model s conssen wh geologcal observaons for he las wo R-N reversals (Glass, 99). If mpac naed global coolng can 7

8 rgger an R-N reversal hen he p= probably of he Bernoull process could be explaned as posve and negave nerferences beween he Mlankovch and planeary cycles. 5. Concluson In hs sudy he me sequence elapsed beween wo consecuve R-N reversals has been analyzed for he las 8 My. The paern conssen wh a Bernoull process wh p= overrdng wh a Gaussan nose. The frequency of he deeced Bernoull rals s 8 ky. Ths frequency s conssen wh planeary and global clmae cycles. The p= probably of he Bernoull process can be explaned as posve and negave nerferences beween hese cycles. References Cande, S. C., and Ken D. J., Revsed calbraon of he geomagnec polary mescale for he Lae Creaceous and Cenozoc, J. Geophs. Res.,, , 995, Consoln G. and De Mchels P., Sochasc Resonance n Geomagnec Polary Reversals, Phys. Rev. Le., hp://lnk.aps.org/absrac/prl/v9/e585, 3 Cox, A. and Cande, S. C. and Ken D., A sochasc approach owards undersandng he frequency and polary bas of magnec reversals, Phys. Earh Plane. Iner., 4, 78-9, 98 Gara, J., The orgn of he Earh magnec feld and he cause for s reversals, Eos. Trans., AGU Fall Mee. Suppl., 8 (47), F345,, Glass, B. P., Tekes and mcro ekes: key facs and nferences, Teconophyscs, 7, , 99, Marzocch, W. and Mularga F., The perodcy of geomagnec reversals, Phys. Earh Plane. Iner., 73, -8, 99, Mlankovch, M., Maemasche Klmalehre and Asronomsche Theore der Klmaologe, W Koppen und R Geger, Gebr Bornrager, Berln, 93 Muller, R.A., and Morrs, D.E., Geomagnec reversals from mpacs on he Earh, Geophys. Res. Le., 3, 77-8, 986 8

9 Pelleer, J.D., Sascal analyses and modelng of varaons of he earh's magnec feld, hp://xxx.lanl.gov/abs/physcs/97534, 997 Shrley, J. H. and Farbrdge R. W., Encyclopeda of Planeary Scences, p. 57, 997 Yamazak, T and Oda H., Orbal Influence on Earh's Magnec Feld:,-Year Perodcy n Inclnaon, Scence, 95, , 9

10 Fgure Capons Fg. The dsrbuon of he me nervals elapsed beween wo consecuve R-N reversals, and he bes fng normal dsrbuons of he separaed groups. Fg. The probably dsrbuon of he R-N reversals. The calculaed sascal parameers conan he relavely hgh uncerany of he geologc me deermnaon (dashed lnes). If he R-N reversals are generaed by asronomcal cycles hen he uncerany of he parameers should be sgnfcanly smaller (connuous lnes).

11 Table. Number Group Normalzed Group Group Tme Range Of Average Average Sandard Number [My.] Frequency Evens [My.] [My.] Devaon Toal: Average:.8.73

12 7 Frequency [Number of Evens] [My.] Elapsed Tme Beween Two Consecuve R-N Reversals Fgure.

13 Las R-N Reversal f(x) P - P - P P + P + P +3 Tme (My.) = 8 ky. Mauyama-Brunhes Presen Tme (4) Legend -.4 P +3 Probably dsrbuon from paleomagnec daa Probably dsrbuon wh smaller uncerany me before presen probably of an R-N reversal (P =.5) Fgure. 3