Paleosecular variation in field directions due to randomly varying Gauss coefficients

Expressions are derived for the magnetic field directions produced by randomly varying Gauss coefficients. This approach was pioneered by Constable and Parker (1988), and followed by Kono and Tanaka (1995) and Kono and Hiroi (1996). The present treatment is a sequel to the latter two. The basic assumption is that the magnitude of the axial dipole component is much larger than all the other terms: equatorial dipoles or nondipole components. It is also assumed that, in a sufficiently long time interval, Gauss coefficients, g(l)(m) and h(l)(m), behave as independent random variables that have the first and second moments defined. General expressions are obtained for the field directions (inclination I and declination D). Paleosecular variation (PSV) of field directions is formulated as latitude dependences of statistical parameters (the means and variances) of inclination and declination. It is possible to derive a family of PSV models by specifying the statistical nature of individual Gauss coefficients, i.e., assigning means and variances to them. In this paper, we describe a ''homogeneous background model'' (HBM), which consists of a large axial dipole component (g(1)(0)) and other harmonics that have zero mean and variances decreasing exponentially with the degree of the harmonic. This model is too homogeneous to reproduce observed latitudinal variation of secular variation, but is quite useful to identify the harmonics that have effect that is larger than produced by the HBM. Comparison with paleomagnetic data obtained from volcanic rocks of 0-5 Ma ages (Quidelleur et al., 1994) shows that it is necessary that the mean of g(2)(0) must be about 5% of the mean of g(1)(0), and that g(2)(1) (or h(2)(1)) should have about three to four times as large an effect as suggested by the HBM. The former conclusion is consistent with the result of Constable and Parker (1988), while the latter conclusion substantiates the similar conclusion obtained by Kono and Tanaka (1995) by the analysis of the scatter of the virtual geomagnetic poles (VGPs).