HR: 0800h
AN: GP11D-0855    [Abstracts]
TI: Taylor Expansion Method for Paleosecular Variation
AU: * Kono, M
EM: mkono@misasa.okayama-u.ac.jp
AF: Okayama University Institute for Study of the Earth's Interior, Yamada 827, Misasa, Tottori-ken, 682-0193 Japan
AB: Constable and Parker (1988) first suggested that auss coefficients can be treated as independent normal random variables with variances depending only on the degree. Kono et al. (2000) and Kono and Roberts (2002) demonstrated that this postulate is well satisfied in the long time behavior of various numerical dynamo models. However, significant deviations from the original model of Constable and Parker were also found by the analysis of the paleomagnetic data of the last 5 Ma; most notably the large amplitude in the fluctuation of the (2,1) harmonic. In order to express these facts better, Kono and Tanaka (1995) developed Taylor expansion method. But its performance was not satisfactory enough because the approximation used in the calculation was only up to the second order. This paper presents the Taylor expansion method extended to arbitrarily high approximation. We assume that the long term behavior of the geomagnetic field can be modeled as the sum of two parts; the mean field and the fluctuation $\boldsymbol{m=\mu+\Delta m}$, where all the elements of $\Delta\boldsymbol{m}$, $m_j$, are zero-mean normal variates with the variance $\sigma_j^2$. The mean value of a nonlinear quantity can be obtained by averaging the Taylor series about the mean model, with summation rules such as E$[\Delta m_j\Delta m_k] = \sigma_j^2\delta_{jk}$, etc. Summation of the series to high orders is made possible because the nonlinear quantities (or their first derivatives with respect to a Gauss coefficient) used in paleomagnetism (inclination $I$, declination $D$, virtual geomagnetic pole (VGP) position $\theta_p$, $\phi_p$, etc.) can be expressed as the product of the linear quantities and nonlinear quantities such as the total intensity $F$. These nonlinear quantities can always be expressed as the square root of the sum of squared linear quantities (e.g., $F = \sqrt{X^2+Y^2+Z^2}$). Because of this property, the general form of the derivatives of nonlinear quantities can be written down in a form that is convenient for calculation using the computers. The assumption of the normal distribution also make it possible to estimate quantities such as E$[\Delta m_j^2\Delta m_k^2\cdots\Delta m_n^2]$. The number of terms increase roughly $O(N^L)$, where $N$ is the number of fluctuating coefficients and $L$ is the truncation level. It is thus not possible to include more than a few parameters in the model. This method was used to obtain the statistics of the mean VGP and angular standard deviation. To make the problem simpler, only the axisymmetric models were studied. It is found that the truncation must be quite high to obtain a good convergence. The main features of the PSV can be surprizingly well represented by fluctuations of $g_2^1$ and $h_2^1$ only. The present results suggest that the exceeding importance of the (2,1) harmonic, rather than the equal partition of the energy as envisioned by Constable and Parker, may be the most important characteristic of the dynamo operating in the Earth.
DE: 1522 Paleomagnetic secular variation
DE: 1560 Time variations--secular and long term
SC: Geomagnetism and Paleomagnetism [GP]
MN: 2004 AGU Fall Meeting