Conductivity of the Subcontinental Upper Mantle: An Analysis Using Quiet-Day

Transcription

1 Conductivity of the Subcontinental Upper Mantle: An Analysis Using Quiet-Day Geomagnetic Records of North America Wallace H. CAMPBELL* and Robert S. ANDERSSEN** *U.S. Geological Survey, Denver Federal Center, Denver, CO 80225, U.S.A. **Division of Mathematics and Statistics, CSIRO, Canberra A.C.T. 2601, Australia (Received May 4, 1983 ; Revised September 12, 1983) Electrical conductivity properties of the upper mantle for a North American sector of the Earth have been determined using the 24-, 12-, 8-, and 6-hr spectral components of the quiet-day geomagnetic field variations. Spherical harmonic coefficients obtained from an analysis of the three components of the quiet daily variation (Sq) field for the solar-quiet year of 1965 were applied to a modeling procedure that was modified from SCHMUCKER'S (1970) publication. From depths of about 140 km to about 540 km, the conductivity, 6(ohm-meter) -1, may be represented by where d is the depth in kilometers. Small perturbations of conductivity indicating some layering at 140 to 220, 220 to 400, and 400 to 600 km correspond to the similar behavior of the Earth's density in these regions. From temperature-depth models we infer that the multiphase bulk properties of the expected silicates in these regions behave approximately as where T is the temperature in degrees kelvin. The constants of this equation do not seem to be very model dependent. 1. Introduction The atmospheric electric currents that vary in daily, seasonal, and latitudedependent patterns above the Earth's surface act as a source that induces currents to flow in the conducting layers of the Earth. The depth of these induced currents is not only related to the spectral and spatial composition characteristics of the source field but also to the variations of conductivity in the Earth. At the Earth's surface the

2 W. H. CAMPBELL and R. S. ANDERSSEN observed mixture of fields from the source and induced currents can be separated by spherical harmonic analyses and the relationship between the internal and external amplitudes and phases can be used to infer the Earth's conductivity profile at great depths (ScHMUCKER, 1970, 1979). In two recent papers (CAMPBELL, 1982, 1983), the quiet geomagnetic daily variations were studied for the American region from the Equator to the North Pole. Observatory data for 1965 were used because that year contained a suitable distribution of 15 stations and the greatest number of quiet intervals since The quietest daily geomagnetic field variation (Sq) patterns for each orthogonal field component representative of each month of 1965 were established. The 24-, 12-, 8-, and 6-hr Fourier spectral components for the field changes were smoothed to determine latitude and seasonal component patterns. The daily passage of observatories beneath the source current region was taken to represent 360 in longitude change. After constructing an appropriately adjusted image of the northern field coefficients to represent the southern hemisphere fields, a spherical harmonic analysis (SHA) was applied to obtain a separation into internal and external field coefficients up to and including order (m) 4 and degree (n) 12. The SHA coefficients yield 12 separate representations (one for each month) of the magnetic scalar potential, V, in colatitude 0 and longitude 1 described at the Earth's surface by where R is the Earth's radius and Pm(8) denotes the Schmidt-normalized associated cosine and sine SHA coefficients composed of external (ex) and internal (in) parts. It is well known that the external and internal fields can reconstructed from these a and b coefficients (CHAPMAN and BARTELS, 1940 ; MATSUSHITA, 1967). The induction equations for our analysis are best written in terms of the SHA coefficients a and b. Using the usual complex number notation of induction analysis, S n is the ratio of the internal to external components of order m and degree n of the geomagnetic surface field. From Eqs. 20, 23, and 24 of SCHMUCKER (1970), it can be shown that with we obtain and

3 Conductivity of the Subcontinental Upper Mantle Then the complex induction transfer function CrT, in kilometers, can be written as with and where R is the radius of the Earth in kilometers. The ratio of the Z (into the Earth) to the X (northward) component of the field is then and the ratio of the Z to the Y (eastward) coefficient is for a given colatitude location 0. The depth to the equivalent conducting layer determined by the coefficient of the order m and degree n is just and the conductivity of a uniform substitute layer at that depth is given by The validity of Eqs. 5 and 6 is limited to situations for which

4 W. H. CAMPBELL and R. S. ANDERSSEN The relevance of this condition is explained in Schmucker's 1970 paper, but, as given on his page 31, the equation contains a typographical error. In order to cope with possible outliers in the data, it was necessary to introduce two acceptability criteria. The first concerns arg (S,T), which is the difference in phase between the internal and external SHA coefficients. When this difference is small or negative, quite unrealistic values of conductivity are obtained. We have therefore introduced the requirement that in order to remove such poorly determined data. The second concerns the fact that SHA coefficients of very small amplitudes typically yield poor estimates of the conductivity. We have therefore introduced the requirement that where Km is a different constant for each m and each data set. Our method for determining Km will be discussed shortly. The frequencies available in the daily variations of the field limit the depths to which the conductivity can be determinated to less than 650 km. Lateral lithospheric heterogeneity in the first few tens of kilometers down to the continental Mohorovicic discontinuity (KAY and KAY, 1981) limit the utility of spherical harmonics in that shallow range. By 140 km depth, the conductivity has a regional scale rather than a local structure and the conductivity has reached a value that is higher than that found in shallower layers s (SHANKLAND et al., 1981). In addition, the SCHMUCKER (1970) analysis method that is used here requires the conductivity to increase with depth oceanic and continental plate conductivities may be higher than the adjacent lower layers. For such reasons, the region of applicability of the present study would be expected to be about 140 to 600 km. In the following sections we describe the conductivity analysis applied to our 1965 data, the method for excluding outliers from the data, and the construction of a simple curve fit. Our conductivity values show an exponential change with depth and are shown to be within the range of values estimated by other authors. Our results are compared to conductivities that we derive from published SHA analyses of S, by other researchers. We then explain the irregularities in our conductivity-depth values about the constructed exponental curve in terms of the seismic evidence for density layering. Finally, we evaluate some bulk property constants of the local multiphase silicates from a relationship of conductivity and temperature. 2. Analysis We know that the relative errors inherent in the SHA coefficients increase as the amplitudes of the coefficients decrease. Various values for Km were tested in Eq. 9. Figure 1 shows the conductivities that were obtained when Km was set at 0.4 gamma

5 Conductivity of the Subcontinental Upper Mantle Fig. 1. Conductivity values for 1965 obtained for test with the SHA coefficient amplitude exclusion constant, Km, set at 0.4 gamma for all m. Exponential curve arbitrarily drawn to indicate trend of main sequence of values. for all m. Note the clustering of values near a representative exponential curve. It was discovered that further increases in Km did little to decrease the scatter in the conductivity-depth values but did increase the removal of the higher order m components that are, by nature of the source current, of lower amplitude. An earlier Sq study (CAMPBELL, 1982) indicated that the ratio of the amplitudes of the four harmonic components for the 1965 North American data, averaged over 90 of latitude in 5 increments, is :0.719 :0.355 :0.163 for the 24-,12-, 8-, and 6- hr components. Accordingly, it was assumed that the four Km values should be in a similar ratio. This choice implies that the fractional errors are similar for the harmonic components of Sq. Then a number of conductivity-depth distributions were determined for gradually increased values of K1 (with K2, K3, and K4 determined by the above ratio). We first assumed that the general conductivity profile in the Earth should show a regular change with depth such as that displayed by the smoothed density or temperature profiles. Therefore, the best K1 was determined to be at a value of 1.4 gamma where we obtained the smoothest structure in the data. The data of Fig. 2 have two distinct features : a smooth cluster of data to which the exponential curve

6 W. H. CAMPBELL and R. S. ANDERSSEN Fig. 2. Conductivity values for 1965 obtained for test with the SHA coefficient amplitude exclusion constant, K1 = 1.4 gamma, adjusted to Sq amplitude component ratio for m 2 to 4 (see text). Exponential curve (text Eq. 10) is best fit with seven outliers (large dots) excluded. defines the least square fit (coefficient of determination, r2, is 0.913) with d denoting the depth in kilometers, and seven erratic points (large dots in Fig. 2) which we have excluded as outliers. To determine more precisely what is involved in excluding the 7 outliers from Fig. 2 we plotted a linearized form of Eq. 10 as (In ei) vs. ( d) and determined the distribution of distances from that line for all 46 values (Fig. 3). The seven erratic points lie outside the 88% confidence interval, in the tail of the distribution of Fig. 3. Therefore, they are treated as "outliers" and can be excluded from further consideration. 3. Results The conductivity-depth values computed by applying Eqs. 5 and 6 to the (relevant) 1965 SHA coefficients listed in Table 1 are shown in Table 2 and Fig. 2 (+ points only). Equation 10 defines the exponential curve which is the least squares fit to (10)

7 Conductivity of the Subcontinental Upper Mantle POINT DISTRNCE TO LINE EXPONENTIAL DISTRIBUTION Fig. 3. Distribution of distances for 1965 depth-conductivity points from exponential best-fit curve. Seven values at right tail are outside the indicated 88 percent confidence interval. these conductivity depth values. The possibility of excluding the two isolated values near 600 km could be explored; this would only change the coefficient of d to , which is a trivial difference in the exponential representation for the depths of greatest interest here. We compared our conductivity computations to values obtained from three other sets of SHA coefficients : 1) Chapman's data set (p. 690, CHAPMAN and BARTELS, 1940), a small set of coefficients obtained from a limited group of stations ; 2) Matsushita's 1958 data set (MATSUSHITA and MAEDA, 1965), an active year, three-season analysis dividing the Earth into three sectors ; and 3) Suzuki's 1958 data set (SuzUKI, 1973), a three-season, three-sector analysis of active-year data designed to avoid equatorial electrojet errors. We determined the K1 exclusion levels separately for each of these sets. Using the procedure described above, the exclusion levels were found to be 3.3, 4.1, and 4.0 gammas, respectively. These values for K1 were larger than that used for our 1965 data but they were of a size appropriate to the increased variability in the data that resulted from a small number of stations and enhanced geomagnetic activity level. The use of these Km values improved the distribution of conductivities computed for these sets of data. Figure 4 shows the values of conductivities we obtained from these three sets along with the reference

8 W. H. CAMPBELL and R. S. ANDERSSEN

9 Conductivity of the Subcontinental Upper Mantle

10 W. H. CAMPBELL and R. S. ANDERSSEN Fig. 4. Depth-conductivity values determined from SHA coefficients of Chapman (CHAPMAN and BARTELS, 1940), MATSUSHITA (1967), and Suzuxl (1973). Exponential curve represents our Eq. 10 fit to the 1965 values. curve of Eq. 10. Note the large scatter of values and the concentration near the exponential line. Some published estimates of the Earth conductivity exist for the depth range of our report. The five that summarize such estimates are : LAHIRI and PRICE, 1939; BANKS, 1969 ; LARSEN, 1975 ; ANDERSSEN et al., 1979 ; and SCHMUCKER, They are shown in Fig. 5 along with the curve of equation 10. This curve appears to be a more or less mid value of the other conductivity estimates. Our conductivity-depth values (+ points in Fig. 2) appear to cluster about fixed levels providing some indication of a layering that is superposed upon the exponential increase of conductivity with depth. An excellent representation of upper mantle layering comes from studies of seismic data by DZIEWONSKI and ANDERSON (1981). The line segments shown in Fig. 6 indicate these authors' determinations of Earth density at the depths of present interest. The conductivity values derived in this paper seem to follow this density layering sufficiently well to suggest the Table 3 description of the subcontinental upper mantle in North America. 4. Conclusions The year 1965 was an exceptionally quiet one for geomagnetic variations. The ionospheric sources of the Sq currents behaved in a most predictable manner; intrusions of disturbance currents into the ionosphere were less likely to occur in 1965 than in other more active years ; in such quiet times smoothly varying, spherical harmonic descriptions of the surface field were more representative of the physical

11 Conductivity of the Subcontinental Upper Mantle

12 W. H. CAMPBELL and R. S. ANDERSSEN Table 3. Conductivity layering. *Values from DZIEWONSKI and ANDERSON, conditions. Conductivity models derived from SHA coefficients are expected to be more realistic and less noisy in 1965 than in the more solar-active years. The method of data processing for the 1965 SHA coefficients probably enhanced the utility of these coefficients for conductivity determinations. By restricting the data sample to the Northern Hemisphere of the American continents, and creating a sphere that modeled that region only, the data scatter introduced by regional differences was limited. Latitude smoothing techniques applied to the Fourier coefficients of the observatories' Sq field changes decreased the effects of local crustal anomalies as noise in the subsequent harmonic analyses. The fact that fine scale variations in conductivity as a function of depth are identified in the above analysis may be a consequence of two independent factors. On the one hand, as is clear from the form of Eqs. 5 and 6 above, the use of the equivalentconductivity layer modeling of SCHMUCKER (1970, 1979) has the major advantage of generating conductivity-depth estimates for each m, n component which is decoupled from every other such estimate. In this way, the smoothing which would result if all the components were lumped to produce simultaneously a conductivity-depth profile is avoided. Such smoothing can easily destroy the appearance of fine structure in a profile. On the other hand, 12 sets of separate monthly SHA coefficients are used to derive the conductivity-depth values shown in Fig. 6. In the past the results of conductivity modeling based on the.complex ratio of external to internal geomagnetic variation fields were presented by averaging the results for the same degree and order (PARKINSON, 1974). This procedure was utilized to produce a mean conductivity and depth (for a given pair of m and n indices) when analysis errors were large. A previously overlooked unique feature of the spherical harmonic analysis with associated Legendre polynomials is that the surface spatial wavelength for any m and n changes with latitude. Thus, if a dominant external source current region shifts latitude, the effective wavelength, and thereby the depth of penetration of the induced current, will correspondingly shift. For the spherical harmonic analysis of Sq fields we have m waves evenly spaced in angle around a circle of latitude ; the surface spatial size of one of these waves varies

13 Conductivity of the Subcontinental Upper Mantle with latitude (e.g., it is twice as small at 60 as it is at the equator). Also, we have (n m + 1) waves, not quite so evenly spaced, around a great circle of longitude (p and 615 in CHAPMAN and BARTELS, 1940) so that the spatial size of one of these waves also varies with latitude. Because the Sq source-current spectral components vary regularly in latitude location and relative amplitude with an annual and semiannual periodicity (CAMPBELL, 1982, 1983) a corresponding, but small, regular change occurs in the effective spatial wavelength of each m, n SHA component of the surface eld. Such gradual change from month to month means that the sampling depth fi for any m, n pair in the conductivity equation will vary in a corresponding fashion. For example, observe the complete 12 monthly values for the m = 2, n = 3 components given in Table 2 that show the regular annual and semiannual depth changes. If only an average value of the conductivity and depth is derived from the m = 2, n = 3 components, the fine depth resolution of Fig. 6 would be masked. Equation 10 indicates the general trend of the conductivity with depth in the range studied. Although that function is drawn from the surface to 600 km in Fig. 2, the reliable section lies between about 140 to 540 km. For the relevant range of thermodynamic conditions, the electrical conductivity of the multiphase silicates expected at those depths is thought to vary with the temperature T in the form: (11) (TozER, 1970 ; SHANKLAND and WAFF, 1977), where C and Di are constants, j is generally less than 3, and the C and ID represent the constants for the average conduction characteristics of the silicates (WAFF, 1980). Thus, the variation of the bulk electrical conductivity properties with depth, given in Eq. 10, implies that the inverse of temperature decreases linearly with increasing depth in the range from 140 to 540 km. Equating Eqs. 10 and 11 gives us for the temperature in degrees Kelvin, where C and D are constants from Eq. 11, and d is the depth in kilometers. This equation has the unknowns C, 15, T and d. If the first two may be assumed from appropriate laboratory modeling, then the temperature-depth profile is resolvable for the upper mantle region (c.f., SHANKLAND and WAFF, 1977). Similarly, assuming a T- d relation, the C and D could be tested against the expected chemical differentiation (most likely of olivine) deep in the Earth. Two typical temperature estimates (ANDERSON, 1967 ; STACEY, 1969) are shown in Fig. 7 along with the linear representations in the form of Eq. 12. Evaluating C and D from these two straight lines gives C = 1.42 x 103 and 1.46 x 103, as well as D = 1.72 x 104 and 1.92 x 104, respectively. Taking an average of these two values we obtain for the bulk conductivity properties of the multiphase silicates at this depth range (12)

14 W. H. CAMPBELL and R. S. ANDERSSEN Fig. 7. Reciprocal-temperature versus depth for the range 100 to 600 km. Solid curve represents values from STACEY (1969). Dots are values from ANDERSON (1967). The best straight lines through these two data sets are shown to justify the correspondence to linear Eq. 12, and to obtain the bulk silicate coefficients C and D for Eq. 13. (Note that 40 and 60 correspond to temperatures of 2.5 and 1.7 ~103 K, respectively.) for Tin degrees Kelvin. ANDERSON'S (1967) estimates were interpreted from seismic transition data. STACEY'S (1969) estimates were for the average convecting-model Earth. Although slightly different C and D would be obtained with other temperature profiles, these values appear not to be very model dependent and are consistent with the expected properties of olivine (DuBA et al., 1974). The clustering of the 1965 conductivity values at specific levels seems to match the best estimates of density layering that was obtained from the seismic data. It is likely that small phase transitions in the upper mantle materials occur, giving rise to density steps and occurring in response to pressure and temperature changes. These phase transitions vary the C and Di constants of Eq. 11 and thereby produce the small steps in electrical conductivity. For the depth range of our 1965 data, we see that the density varies almost linearly with the conductivity a (ohm-meter) -1 as where a is the density in grams/cm'. The multiphase silicate at this region of the upper mantle is therefore expected to follow this relationship of density and conductivity. Equation 14 was determined for the range 140 to 540 km. If this relationship continues to greater depths one would expect a value of conductivity near 0.35 (ohmmeter) -1 at 650 to 700 km, placing into doubt either the two conductivity values (13) (14)

15 Conductivity of the Subcontinental Upper Mantle obtained near 600 km or the continuation of the relationship (Eq. 14) to depths greater than 540 km. Studies by ANDERSSEN et al.(1979) seem to indicate that values of conductivity at depths much greater than about 600 km, are unlikely to be obtained from Sq fields although our two extreme values at such depth seem to follow the lower depth exponential trend (Eq. 10). The region between 600 and 670 km is a major transition zone according to seismic studies (DzIEwoNsKI and ANDERSON, 1981) so the density-conductivity relationship of Eq. 14 may fail in that region where a relatively major phase transition may be occurring. We wish to thank Dr. Ulrich Schmucker of Gottingen University for clarifying the application of his induction equations to geomagnetic, harmonic analysis data. The staff of World Data Center A for Solar Terrestrial Physics supplied the original magnetic observatory data tapes, some analysis facilities, and stimulating discussions. The work was supported in part by the U.S. Office of Naval Research. REFERENCES ANDERSON, D. L., Phase changes in the upper mantle, Science, 157, , ANDERSSEN, R. S., J. F. DEVANE, S. A. GUSTAFSON, and D. E. WINCH, The qualitative character of the global electrical conductivity of the earth, Phys. Earth Planet. Inter., 20, 15-21, BANKS, R. J., Geomagnetic variations and the electrical conductivity of the upper mantle, Geophys. J. R. Astr. Soc., 17, , CAMPBELL, W. H., Annual and semiannual changes of the quiet daily variations (Sq) in the geomagnetic eld at North American locations, J. Geophys. Res., 87, , fi CAMPBELL, W. H., A description of the external and internal quiet daily variation currents at North American locations for a quiet-sun year, Geophys. J. R. Astr. Soc., 73, 51-64, CHAPMAN, S. and J. BARTELS, Geomagnetism, Clarendon Press, 1049 pp. Oxford, DUBA, A., H. C. HEARD, and R. N. ScHocK, Electrical conductivity of olivine at high pressure and under controlled oxygen fugacity, J. Geophys. Res., 79, , DZIEWONSKI, A. M. and D. L. ANDERSON, Preliminary reference earth model, Phys. Earth Planet Inter., 20, , KAY, R. W. and S. M. KAY, The nature of the lower continental crust: Inferences from geophysics, surface geology, and crustal xenoliths, Rev. Geophys. Space Phys., 19, , LAHIRI, B. N. and A. T. PRICE, Electromagnetic induction in non uniform conductors and the determination of the conductivity of the earth from terrestrial magnetic variations, Philos. Trans. R. Soc. Lond., A237, , LARSEN, J. C., Low frequency ( cpd) electromagnetic study of deep mantle electrical conductivity beneath the Hawaiian Islands, Geophys. J. R. Astr. Soc., 43, 17-46, MATSUSHITA, S., Solar quiet and lunar daily variation fields, Chapter III-1, pp , in Physics of Geomagnetic Phenomena, edited by S. Matsushita and W. Campbell, 1398 pp., Academic Press, New York, MATSUSHITA, S. and H. MAEDA, On the geomagnetic solar quiet daily variation field during the IGY, J. Geophys. Res., 70, , PARKINSON, W. D., The reliability of conductivity derived from diurnal variations, J. Geomag. Geoelectr., 26, , SCHMUCKER, U., An introduction to induction anomalies, J. Geomag. Geoelectr., 22, 9-33, SCHMUCKER, U., Erdmagnetische Variationen and die elecktrische Leitfahigkeit in tieferen Schichten der Erde, Sitzungsberichte u. Mitteilungen der Braunschweigischen Wissenschaftlichen Gesellschaft, 4, , 1979.

16 W. H. CAMPBELL and R. S. ANDERSSEN SHANKLAND, T. J., R. J. O'CONNELL, and H. S. WAFF, Geophysical constraints on, partial melt in the upper mantle, Rev. Geophys. Space Phys., 19, , SHANKLAND, T. J. and H. S. WAFF, Partial melting and electrical conductivity anomalies in the upper mantle, J. Geophys. Res., 82, , STACEY, F. D., Physics of the Earth, 324 pp., John Wiley and Sons, Inc., New York, SUZUKI, A., A new analysis of the geomagnetic Sq field, J. Geomag. Geoelectr., 25, , TOZER, D. C., Temperature, conductivity, composition and heat flow, J. Geomag. Geoelectr., 22, 35-51, WAFF, H. S., Relations of electrical conductivity to physical conditions within the asthenosphere, Geophys. Surveys, 4, 31-41, 1980.