Jozsef Garai
Astronomical cycles are detected in the R-N reversals of the earth’s magnetic field
Abstract
Previous studies assumed that the two kinds of magnetic reversal N-R and R-N are triggered by the same mechanism, and never investigated the time intervals elapsed between the same kinds of reversal. Using the paleomagnetic record of the past 118 My the time length distribution of the R-N reversals was investigated. The distribution reveals to a Bernoulli process with parameter p = 0.5 and arrival time of 282 kyr. The arrival time is harmonics of astronomical cycles.
CONTENTS
1.Introduction
4. Probability of future R-N reversals
5. Correlation to astronomical cycles
6. Conclusions
----Appendix
----References
1. Introduction
Previous studies assumed that the mechanism responsible for the triggering of the magnetic reversal is the same for both R-N and N-R reversals. Researchers did not make distinction between the different polarity shifts and the sequences in the time series of the earth’s magnetic field were investigated between two consecutive reversals (e.g. Cox et al., 1981; Marzocchi and Mulargia , 1992).
There is a consensus that the periodicity of the reversals has progressively increased over the last 100 My. from zero during the quiet Cretaceous interval (85-110 My.) to about four reversals per million years during the last few million years. The increase of the frequency has been modeled with a linear trend on which fluctuation might be superimposed (Lowrie and Kent, 1983; Lowrie, 1982). It has been suggested the superimposed fluctuation has periodicity of 15-32 My (Negi and Tirwari, 1983; Mazud and Laj, 1991; Lutz, 1985). Other investigations have not found any indication of periodicity (McFadden 1984; Raup 1985). The statistical dissimilarity of the normal and reverse polarity regimes has been investigated with negative result (Marzocchi and Mulargia, 1992; Lowerie and Kent, 1983).
Based on laboratory high pressure and temperature experiments it has been suggested that carbonate melt could be produced in the mantle and be trapped at the 660 km boundary globally (Garai, Gasparik, 2000; Garai, 2001a; Garai, Gasparik, 2003). This prediction is consistent with seismic wave attenuation (Cadek and Berg, 1998; Montagner, 1998). Carbonate melt has the lowest viscosity of any known minerals and a globally existing layer would allow developing differential rotation which would generate magnetic field. This simple dynamo model is able to reproduce all of the features of the contemporary field and, within reasonable uncertainty, the paleomagnetic field (Garai, 2001b). The magnetic field changes its orientation when the sign of the differential rotation changes. The angular velocity of the 660 km outer shell can be modified by mass redistribution and/or tidal affect. Global cooling could result an R-N reversal while the tidal affect and/or global warming could lead to an N-R reversal (Garai, 2001b).
Using different model for the reversals impact initiated global cooling has been proposed as a possible triggering mechanism for both kinds of magnetic reversals (Muller and Morris, 1986). Geological records support an impact initiated global cooling process for the R-N reversals since the last two R-N reversals were coincided with global cooling and major impacts (Glass, 1990).
Hypothetically accepting that the two different kinds of magnetic reversals can be triggered by different mechanisms then by looking for a pattern in the sequences of the reversals would more accurately be detected by focusing analysis on those reversals that are triggered by common mechanism. In the present study the periodicity of the R-N reversals will be investigated since these reversals are governed by a single mechanism, global cooling.
2. Bernoulli process
The Bernoulli process is briefly described for uninitiated readers. The probability of a Bernoulli trial is
for n= 0, 1 (1)P(n) = 0.5 for n= 0, 1 (2)
for n= 0, 1 (1)The Bernoulli process is a sequence of independent identically distributed Bernoulli trials (Fig. 1-a). Since the independent Bernoulli trials are separated by a constant time interval, the arrival time [to ], the probability of the superimposed Bernoulli trials can be determined as:
i ∈ N (3)
If there is uncertainty in the arrival time then the discrete probability should be replaced with a continuous function as:
(4)Assuming that the uncertainty can be represented by a Gaussian density function then P(t)i for the ith trial is
(5)where t is the time, to is the mean of the arrival time, and σ is the variance. Assuming that the Gaussian distributions of the individual trials are not overlapping gives the probability of the superimposed trials
(6)
3. Data analysis
Using the polarity time scales for the last 118 My (Cande, and Kent, 1995) the time elapsed between two consecutive R-N reversals Δ t (R-N) has been determined first.
(7)
(7)
where t(R-N) is the time before present of the i th R-N magnetic reversal. The number of data for longer time periods (Δt(R-N)>1.6 My.) is very few. These long periods were omitted because the available data are insufficient for statistical analysis. Plotting the data in bins of 0.04 My. the frequency distribution of Δt(R-N) reveals five relatively distinct groups (Fig. 2). Assuming Gaussian distribution for each of the groups the groups were separated following the mathematics described in the appendix. The calculated parameters of the separated groups are given in Tab. 1. The means of the overlapping Gaussian distributions are 3.86 σ apart, therefore, the overlapping data on each side of the distribution is less than 1.5 percent. This small percentage of overlapping does not have significant affect on the statistical parameters. Additionally, the cut off parts for groups 2, 3, and 4 are close to symmetrical further reducing the overlapping affect.
Analyzing the calculated statistical parameters, the mean values and standard deviations of these separated groups it was found that the mean values are integer multiples of the smallest mean, revealing a periodic pattern. If we let the normalize mean of a group denote the mean of that group divided by its group number (e.g. group 5’s normalized mean is 0.28376 My), then we find that the normalized means range from 0.280 to 0.287 My. The weighted average of the means is 0.28214 My, the corresponding standard deviation is 0.073 My.
The frequency distribution of the reversal intervals shows that the R-N reversal is more likely to have a shorter term than a longer one. Calculating the relative frequency that a reversal interval falls into a group it was found that the frequency for the first group with mean of 0.28 My is 0.5. The frequency of the second group (mean 0.56 My) is 0.21, while the frequency for the third group (mean 0.85 My) is 0.12 etc. (Tab. 1). This is just a Bernoulli process with p = 0.5 overridden with a Gaussian noise (Fig. 1). The ‘noise’ might be the result of the uncertainty in the age determination of the reversal. Assuming Gaussian distribution for the uncertainty justifies the method used for the separation.
The presence of the arrival time in the data set can be also tested by calculating the residuals. The residuals of the arrival time of the Bernoulli trials were calculated as:
(8)
where n is an integer. The value of n was selected by applying the criteria:
(9)
residuals = 0 perfect correlation or function
residuals < 0.25 indicates correlation
residuals = 0.25 theoretical value for random distribution
residuals > 0.25 indicates no correlation
The accuracy of the geological records decreases with age; therefore, the data of the last 40 My data was used for calculating the residuals. The period contains 71 R-N reversals. The relative frequency distribution of the residuals is shown on Figure 3. The average of the residuals is 0.187. The distribution and the average of the residuals indicate that the detected 282 kyr. arrival time is characteristic of the data.
4. Probability of future R-N reversals
Based on the detected patter in every ~0.28 My elapsed since the previous R-N reversal, there is approximately 0.5 probability that the earth’s magnetic field would shift from a reversed to a normal orientation. Using this pattern for any given time period the probability of an upcoming R-N reversal can be calculated. Calculating the probability of the occurrence of an R-N reversal between time t1 and t2 [P(R-N)t1-t2 ] requires determining the sequence number of the Bernoulli trial (i). The sequence numbers can be determined from equation (10) by keeping only the integer parts of the calculated values.
and 
(10)
and (10)
The most recent R-N magnetic reversal, Matuyama-Brunhes, has been used as reference; therefore, 780 kyr. has been added to the current time.
If the beginning and the end of the investigated period falls into the same Bernoulli trial (i1 = i2) then the probability of the event can be calculated by integrating equation (4) between the time (t1 and t2).
(11)
The determined parameter, mean 282 kyr., and variance 73 kyr., can be used for the calculation. If the time period covers two or more consecutive Bernoulli trials (i1 ≠ i 2) then the probability of an R-N reversal is:
(12)
where z = i 2 - i 1. This probability shows the probability of the occurrence of an R-N reversal during the time period investigated and contains the probabilities both the single and multiplied R-N reversals.
5. Correlation to astronomical cycles
The identified Bernoulli process had to be stable for the time period covered by the data. For 100 My time length only astronomical cycles are known to remain stable; therefore, it is reasonable to assume that the R-N reversals of the earth’s magnetic field are governed by a process or processes related to astronomical cycles. The presence of planetary cycles in the strength and the inclination of the earth's magnetic field for the past 2.25 My has been demonstrated (Yamazaki and Oda, 2002).
Impact can be triggered by planetary perturbations. The detected arrival time of the Bernoulli trials is the harmonic of the beat frequency (93,418.3 y.) of the giant planets (Shirley and Fairbridge, 1997), which supports an impact initiated global cooling scenario.
The current global climate oscillation, known as Milankovitch cycles (Milankovich, 1930; Olson 1986; Muller, MacDonald, 1997; Lascar, 2003) are the harmonics of the arrival time of the Bernoulli trials. The length of these cycles slightly changed in past 500My (Berger et al., 1992). The detected arrival time seems be present in the sea level oscillation cycles reported for the past 30 My (Tiwari et al., 1997). The presence of the harmonics of the climate and sea level oscillation in the arrival time indicates that global cooling coincides with the R-N magnetic reversal.
5. Conclusions
The time sequence elapsed between two consecutive R-N reversals has been analyzed for the last 118 My. The pattern is consistent with a Bernoulli process with p= overridden with a Gaussian noise. The Bernoulli trials are separated by 282 kyr. from each other. The presence of planetary, climate and sea level cycles in the detected pattern of the R-N reversals and the long term stability of the pattern is consistent with an impact generated global cooling model proposed for the R-N magnetic reversals.
Appendix
If one assumes a Gaussian distribution for each of the groups, then these distributions overlap and the analyses are significantly more complicated. However, since the means of the various groups lie several standard deviations away from each other, a more tractable way to model the groups is to determine a point between them such that the probability of a point being miscategorized is minimized. To make this precise, consider the G1 and G2 data set for two consecutive groups which have (mean, standard deviation) (μ1, σ 1 and (μ2, σ 2 ) respectively, and μ1 < μ2. Then we want to find μ1 < s1-2 < μ2 such that P1(X≥ s1-2) + p2(X < s1-2) is minimized, where is a variable equal to ΔtR-N, p1 is a probability calculated from a Gaussian density function with parameters of μ1, σ1, and p2 is a probability calculated from a Gaussian density function with parameters of μ2, σ2. At first the groups were separated by visual inspection and parameters (μi, σi) were calculated for each group (Gi). Using the calculated group parameters the minimum points si-(i+1) between each consecutive group were determined. If this minimum point coincided with the initial assumption then the group separation was accepted. If the minimum point did not coincide with the initial separation then the above procedure was repeated using the new minimum point for the separation. Calculations were repeated as long as all the calculated si-(i+1) coincided with the initial separation point.
Acknowledgment
I would like to thank Dr. Csaba Gabor for his assistance with the mathematical analysis.
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