On the effects of a bumpy core-mantle interface

Motivated by the high degree of correlation between the variable parts of the magnetic and gravitational potentials of the Earth discovered by Hide and Malin (using a harmonic analysis approach and utilizing the geomagnetic data) when one field is suitably displaced relative to the other, Moffatt and Dillon (1976) studied a simple planar model in an attempt to find a quantitative explanation for the suggestion that this high degree of correlation may be due to the influences produced by bumps on the core-mantle interface. Moffatt and Dillon assumed that the core-mantle interface was z = η(x) where |?η/?χ| ? 1 and such that in the core [z < η(x)] a uniform flow (U₀, 0, 0) prevails in the presence of a uniform ‘toroidal’ field (B₀, 0, 0); (here z is the vertical coordinate and x is the eastward distance). The whole system rotates uniformly about the vertical with angular velocity Ω. The present work extends the model discussed by Moffatt and Dillon to include a horizontal component of angular velocity Ω_H and a uniform small poloidal field B_p. In addition, the uniform toroidal field is here replaced by one which vanishes everywhere in the mantle and increases linearly, from zero on the interface, with z. It is shown that the presence of Ω_H and B_p, together with the present choice of toroidal magnetic field, has a profound effect both on the correlation between the variable parts of the magnetic and gravitational fields of the Earth, and on how far the disturbances caused by the topography of the interface [which is necessarily three-dimensional i.e. z = η(x, y) here] can penetrate into the liquid core. In particular it is found that the highest value of the correlation function is +0.79 which corresponds to a situation in which the magnetic potential is displaced both latitudinally and longitudinally relative to the gravitational potential.     

Density profile and modern crustal and uppermost mantle geotherm of the Voronezh crystalline massif

A density profile and a modern temperature distribution in the lithosphere of the Voronezh crystalline massif (VCM) are derived through the use of the V_P(z), V_S(z) seismic velocity models, petrological data, measurements of V_P, V_S, density (ρ) and mean atomic weight ($\text{m}$) for several groups of rocks and minerals of different composition and genesis, as well as from pressure and temperature derivatives for different thermodynamic regimes.     

On the topographic coupling at the core-mantle interface

The mean tangential stresses at a corrugated interface between a solid, electrically insulating mantle and a liquid core of magnetic diffusivity λ are calculated for uniform rotation of both mantle and core at an angular velocity Ω in the presence of a corotating magnetic field B. The core and mantle are assumed to extend indefinitely in the horizontal plane. The interface has the form z = η(x, y), where z is the upward vertical distance and x, y are the zonal and latitudinal distances respectively. The function η(x, y) has a planetary horizontal length scale (i.e. of the order of the radius of the Earth) and small amplitude and vertical gradient. The liquid core flows with uniform mean zonal velocity U₀ relative to the mantle. Ω and B possess vertical and horizontal components.The vertical (poloidal) component B_p is uniform and has a value of 5 G while the horizontal (toroidal) field B_T = B_pαz, where α is a constant. When |α| ? 1, the mean horizontal stresses are found to have the same order of magnitude (10^?2 N m^?2) as those inferred from variations in the decade fluctuations in the length of the day, although the exact numerical values depend on the orientation of Ω as well as on the wavenumbers in the zonal and latitudinal directions.The influence of the steepness (as measured by α) of the toroidal field on the stresses is investigated to examine whether the constraint that the mean horizontal stresses at the core-mantle interface be of the order of 10^?2 N m^?2 might provide a selection mechanism for the behaviour of the toroidal field in the upper reaches of the outer core of the Earth. The results indicate that the restriction imposed on α is related to the value assigned to the toroidal field deep into the core. For example, if |α| ? 1 then the tangential stresses are of the right order of magnitude only if the toroidal field is comparable with the poloidal field deep in the core.     

High-temperature anelasticity and elasticity of mantle peridotite

An apparatus has been devised which allows precise creep and relaxation measurements to be made on minerals and rocks at temperatures up to 1600°C and at very low deviatoric stresses (1 < σ < 300 bar). This paper is concerned with measurements on mantle peridotite (lherzolite) from Balmuccia (Zone of Ivrea, Italy).The reaction of the sample to a step-like increase in stress is called its “creep function”. It is shown that the creep function contains all the necessary information to derive the spectra of the quality factor Q(ω) and of Young's modulus E(ω), within the seismic range of frequencies, provided the material behaves as a linear system. This has been proven up to a strain of 5 × 10^?5.The Q^?1-spectra at 1200 and 1300°C, obtained by Fourier inversion from the creep function, show no pronounced peak in the frequency band 0.01 < tf < 1 Hz and exhibit a general tendency to decrease slightly with frequency. The creep function: ?(t) = ?_u · [1 + 3.7 · q · {(1 + 50t)^0.27 ? }], where q is related to Q, satisfactorily describes the data at high temperatures and leads to $\text{Q}{}^{\mathrm{?1}}\text{(ω, T) = 3 × 10}{}^3\text{· ω}{}^{\mathrm{<mtext>?0.27\; ·\; </mtext><mtext>exp</mtext>}}\text{(}\text{?30}\text{RT}\text{)}$E(ω) is related to Q(ω) by the material dispersion equation. Above 1100°C the unrelaxed Young's modulus decreases rapidly with temperature according to an activation energy of about 20 kcal/mole. A lowering of short period S-wave velocity by 40% and P-wave velocity by 10% occurs below the solidus. Therefore, no partial melting is required in the asthenosphere.Steady-state creep at low axial stresses (20 < σ < 100 bar), obtained from the same rock, follows the relation $\text{?}\text{?}\text{= 3 × 10}{}^7\text{· δ}{}^{1.4}\text{·}\text{exp}\text{(}\text{?125}\text{RT}\text{)}$ indicative of grain boundary diffusion or superplasticity. At higher stresses a power law $\text{?}\text{?}\text{= 45 · δ}{}^4\text{·}\text{exp}\text{(}\text{?125}\text{RT}\text{)}$ typical of dislocation creep, is found.The frequency dependence of Q and the ratio of the activation energies of Q and are indicative of so called “high-temperature background absorption”, as the dominant mechanism, and of a diffusion-controlled dislocation mobility common to both absorption and creep. From a, b, and c, relations between the effective viscosity η_f and Q of the form: $\text{log}\text{η}{}_{\mathrm{e??}}\text{=}\text{1}\text{α}\text{·}\text{log}\text{Q ? (n ? 1) ·}\text{log}\text{ω +}\text{log}\text{D}$ are derived, where α ～ 0.25, n is the power of σ, and D is a constant.     

Spatial power spectra of the crustal geomagnetic field and core geomagnetic field

Lowes (1966, 1974) has introduced the function R_n defined by $\text{R}{}^{\mathrm{n}}\text{=(n + 1)}\text{∑}\text{m=0}\text{∞}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}\text{where}\text{g}{}_{\mathrm{n}}{}^{\mathrm{m}}\text{and}\text{h}{}_{\mathrm{n}}{}^{\mathrm{m}}$ are the coefficients of a spherical harmonic expansion of the scalar potential of the geomagnetic field at the Earth's surface. The mean squared value of the magnetic field B = ??V on a sphere of radius r > α is given by $\text{〈}\text{B}\text{·〉 =}\text{∑}\text{n=1}\text{∞}\text{R}{}^{\mathrm{n}}\text{(}\text{a/}\text{r}\text{)}{}^{\mathrm{2n=4}}\text{where}\text{a}$ is the Earth's radius. We refer to R_n as the spherical harmonic spatial power spectrum of the geomagnetic field.In this paper it is shown that Rⁿ = R^M_n = R^C_n where the components R_n^M due to the main (or core) field and R_n^C due to the crustal field are given approximately by $\text{R}{}^{\mathrm{M}}{}_{\mathrm{n}}\text{= [}\text{(n =1)/}\text{(n + 2)}\text{](1.142 × 10}{}^9\text{)(0.288}{}^{\mathrm{n}}\text{Λ}{}^2\text{R}{}^{\mathrm{C}}{}_{\mathrm{n}}\text{= [}\text{(n =1){[1 — exp(-n/290)]/}\text{(n/290)} 0.52 Λ}{}^2\text{where}\text{Iγ = 1 nT}$. The two components are approximately equal for n = 15.Lowes has given equations for the core and crustal field spectra. His equation for the crustal field spectrum is significantly different from the one given here. The equation given in this paper is in better agreement with data obtained on the POGO spacecraft and with data for the crustal field given by Alldredge et al. (1963).The equations for the main and crustal geomagnetic field spectra are consistent with data for the core field given by Peddie and Fabiano (1976) and data for the crustal field given by Alldredge et al. The equations are based on a statistical model that makes use of the principle of equipartition of energy and predicts the shape of both the crustal and core spectra. The model also predicts the core radius accurately. The numerical values given by the equations are not strongly dependent on the model.Equations relating average great circle power spectra of the geomagnetic field components to R_n are derived. The three field components are in the radial direction, along the great circle track, and perpendicular to the first two. These equations can, in principle, be inverted to compute the R_n for celestial bodies from average great circle power spectra of the magnetic field components.     

Hysteresis properties of titanomagnetites: Grain-size and compositional dependence

Sized fractions of x = 0.6, 0.4, 0.2 and 0.0 titanomagnetites were studied with a vibration magnetometer. In the course particles (d > 150 μm), no compositional dependence of hysteresis parameters was found. H_C was less than 50 Oe, H_R/H_C > 4 and J_R/J_S < 10^?2, reflecting multi-domain behaviour. In contrast, fine particles ($\text{d ? 0.1 μ}\text{m}$) revealed systematic grain-size dependence of parameters with coercive force as high as 2,000 Oe in x = 0.6 titanomagnetite. Grain-size dependence studies revealed broad transition sizes for the onset of true multi-domain behaviour depending upon which factor is chosen. In magnetite it varies from 10 to 20 μm. The experimental critical size for single-domain behaviour for magnetite is about 0.1 μm and for x = 0.6 titanomagnetite 1–2 μm.     

General relationships among sound speeds: II. Theory and discussion

The dependence of bulk sound speed V_φ upon mean atomic weight $\text{m}$ and density ρ can be expressed in a single equation:

$\text{V}{}_{\mathrm{\phi }}\text{=Bρ}{}^{\mathrm{\lambda }}\text{(}\text{m}{}_0\text{m}{}^{\mathrm{<mtext>[</mtext><mtext>1</mtext><mtext>2</mtext><mtext>+\lambda (1?c)]</mtext>}}\text{(}\text{km/sec}\text{)}$

Here B is an empirically determined “universal” parameter equal to 1.42, $\text{m}{}_0\text{= 20.2}$, a reference mean atomic weight for which well-determined elastic properties exist, and λ = 1.25 is a semi empirical parameter equal to $\text{γ ?}\text{1}\text{3}$ where γ is a Grüneisen parameter. The constant c = (? ln V_M/? ln $\text{m}\text{)}{}_{\mathrm{X}}$, where V_M is molar volume, is in general different for different crystal structure series and different cation substitutions. However, it is possible to use c_Fe = 0.14 for Fe²⁺Mg²⁺ and GeSi substitutions and c_Ca ? 1.3 for CaMg substitutional series. With these values it is pos to deduce from the above equation Birch's law, its modifications introduced by Simmons to account for Ca-bearing minerals, variations in the seismic equation of state observed by D.L. Anderson, and the apparent proportionality of bulk modulus K to V_M^?4.     

Generation of superDreicer electric fields in the solar chromosphere

The electric field generation at the front of the current pulse, which originates in a coronal magnetic loop owing to the development of the Rayleigh–Taylor magnetic instability at loop footpoints, has been considered. During the τ_A ≈ l/V _A ≈ 5?25 s time (where l is the plasma plume height entering a magnetic loop as a result of the Rayleigh–Taylor instability), a disturbance related to the magnetic field tension B _?(r,t), “escapes” the instability region with the Alfvén velocity in this case. As a result, an electric current pulse I_z(z ? V _A t), at the front of which an induction magnetic field E _z, which is directed along the magnetic tube axis and can therefore accelerate particles, starts propagating along a magnetic loop with a characteristic scale of Δξ ≈ l. In the case of sufficiently large currents, when B _? ²/8π > p, an electric current pulse propagates nonlinearly, and a relatively large longitudinal electric field originates E _z ≈ 2I _z ³ V _A/c ⁴a²B_z ²l, which can be larger than the Dreicer field, depending on the electric current value.     

Magnetic properties of ulvöspinel

Magnetisation measurements on ulvöspinel have shown that there is a transition from the weakly ferromagnetic state to an essentially antiferromagnetic one at T ～ 60–100 K when moderate measuring fields (24 kOe) are used. Cooling from above 100 K in the presence of a magnetic field of several kilooersteds produces a reversed remanence for $\text{T ? 40}\text{K}$ and the resulting thermomagnetic curve is Néel N-type. Magnetisation in 80 kOe produces a spontaneous moment extrapolated to 0 K of 0.015 μ_B, although this may not be completely saturated. An explanation for the magnetic transition is suggested in terms of an increased anisotropy possibly associated with a crystal transition.     

Patterns of thermal convection in slowly rotating,weakly magnetic fluid shells

The effects of rotation and a toroidal magnetic field on the preferred pattern of small amplitude convection in spherical fluid shells are considered. The convective motions are described in terms of associated Legendre functions P_l^|m| (cos θ). For a given pair of Prandtl number P and magnetic Prandtl number $\text{P}$_m the physically realized solution is represented either by m = 0 or |m| = l depending on the ratio of the rotation rate Λ to the magnetic field amplitude H. The case of m = 0 is preferred if this ratio ranges below a critical value, which is a function of the shell thickness, and |m| = l otherwise.     

Elasticity of ilmenites

Ultrasonic data for the velocities of the ilmenites MgTiO₃ and CoTiO₃ have been determined as a function of pressure to 7.5 kbar at room temperature for polycrystalline specimens hot-pressed in a piston-cylinder apparatus at pressures up to 30 kbar. Titanate and germanate ilmenites define divergent isostructural trends on a Birch diagram of bulk sound velocity (υ_φ) vs. density (ρ). On a υ_φ vs. mean atomic weight ($\text{M}$) diagram, however, all of the ilmenite consistent with a single $\text{υ}{}_{\mathrm{\phi }}\text{M}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{=}\text{constant trend}$. Elasticity systematics for isostructural sequences are used to e the bulk modulus (2.09 Mbar) and bulk sound velocity (7.4 km/sec) of MgSiO₃-elmenite.     

Orthogonality of spherical harmonic coefficients

An essentially arbitrary function V(θ, λ) defined on the surface of a sphere can be expressed in terms of spherical harmonics $\text{V(θ, Λ) = a}\text{∑}\text{n=1}\text{∞}\text{∑}\text{m=0}\text{n}\text{p}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) (g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{cos mΛ + h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{sin mΛ)}$ where the P_n^m are the seminormalized associated Legendre polynomials used in geomagnetism, normalized so that $\text{〈[P}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) cos mΛ]〉}{}^2\text{=1/}\text{(2n+1)}$ The angular brackets denote an average over the sphere. The class of functions V(θ, λ) under consideration is that normally of interest in physics and engineering. If we consider an ensemble of all possible orientations of our coordinate system relative to the sphere, then the coefficients g_n^m and h_n^m will be functions of the particular coordinate system orientation, but $\text{〈:(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{〉) = 〈(h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{=}\text{S}{}_{\mathrm{n}}\text{/}\text{(2n=1)}$ where $\text{S}{}_{\mathrm{n}}\text{=}\text{∑}\text{m=0}\text{n}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}$ for any orientation of the coordinate system (S_n is invariant under rotation of the coordinate system). The averages are over all orientations of the system relative to the sphere. It is also shown that 〈g^m_ng^l_p〉 = 〈h^m_nh^l_p〉 = 0 for l ≠ m or p ≠ n and 〈g^m_nh^l_p〉 = 0 fro all n, m, p, l.     

Intrinsic Q and its frequency dependence

Various techniques for estimating $\text{t1}$ (travel time/quality factor Q) from short-period seismic-array records of body waves have been investigated. Spectral analysis in the frequency domain seems to be more appropriate for this purpose than time domain methods, because of the relative ease with which source and instrument effects can be removed. Of the techniques available, those based on maximum likelihood and homomorphic deconvolution give estimates of relative power versus frequency which best represent the power contained in a time-domain wavelet of short duration. The latter technique seemed to have better noise-eliminating properties than the former. Therefore, homomorphic deconvolution was used to obtain estimates of $\text{t1}$ values from P, PcP, ScP and S phases recorded at the Warramunga array in the Northern Territory of Australia. The source regions for the event studied were the Sunda, Mariana, New Hebrides, Kermadec and Tonga trench zones.The short-period $\text{t1}$ estimates obtained using the above method were much smaller than estimates from published free-oscillation Q models, indicating that the values of Q for compressional and shear waves are frequently-dependent. It was found that short-period $\text{t1}$ values and free-oscillation Q models could be made consistent with one another by assuming Q = Q₀(1+τω) where Q₀ and τ are constants. The results of this investigation suggest another approach to how the Q structure of the mantle can be investigated.     

Interionic repulsive force and compressibility of ions

The compressibility of an individual ion is examined, in comparison with a known set of data for the alkali halides. A simple extrapolation of ionic radius to high pressure is not acceptable, because the pressure derivative of ionic radius changes for different salts. According to the classical concept of an elastic ion, the repulsive potential energy between the ions i and j is specified by the nature of each ion as:

$\text{(ρ}{}_{\mathrm{i}}\text{+ ρ}{}_{\mathrm{j}}\text{) exp[}\text{(ρ}{}_{\mathrm{i}}\text{+ ρ}{}_{\mathrm{j}}\text{? r)}\text{(ρ}{}_{\mathrm{i}}\text{+ ρ}{}_{\mathrm{j}}\text{)}\text{]}$

as a function of the interionic distance r. In this expression, q_i and ρ_i are the ionic radius and ionic compressibility, respectively, in a suitably modified meaning. Such a form of the repulsive potential fits well to the data of lattice constants and bulk moduli. The parameters q_i and ρ_i are evaluated for alkali and halogen ions, and an anion turns out to be much more compressible than a cation. The present treatment may be usefully applied to the minerals in the Earth's mantle, which contain only a few major ions.     

Melting temperature distribution and fractionation in the lower mantle

The melting curve of perovskite MgSiO₃ and the liquidus and solidus curves of the lower mantle were estimated from thermodynamic data and the results of experiments on phase changes and melting in silicates.The initial slope of the melting curve of perovskite MgSiO₃ was obtained as $\text{d}\text{T}{}_{\mathrm{<mtext>m</mtext>}}\text{/}\text{d}\text{P?77}\text{K}\text{GPa}{}^{\mathrm{?1}}$ at 23 GPa. The melting curve of perovskite was expressed by the Kraut-Kennedy equation as $\text{T}{}_{\mathrm{<mtext>m</mtext>}}\text{(}\text{K}\text{)=917(}\text{1+29.6ΔV}\text{V}{}_0\text{)}$, where T_m?2900 K and P?23 GPa; and by the Simon equation, $\text{P(}\text{GPa}\text{)?23=21.2[}\text{(T}{}_{\mathrm{<mtext>m</mtext>}}\text{(}\text{K}\text{)}\text{2900)}{}^{1.75}\text{?1}\text{]}$.The liquidus curve of the lower mantle was estimated as $\text{T}{}_{\mathrm{<mtext>liq</mtext>}}\text{? 0.9 T}{}_{\mathrm{<mtext>m</mtext>}}$ (perovskite) and this gives the liquidus temperature T_liq=7000 ±500 K at the mantle-core boundary. The solidus curve of the lower mantle was also estimated by extrapolating the solidus curve of dry peridotite using the slope of the solidus curve of magnesiowüstite at high pressures. The solidus temperature is ～ 5000 K at the base of the lower mantle. If the temperature distribution of the mantle was 1.5 times higher than that given by the present geotherm in the early stage of the Earth's history, partial melting would have proceeded into the deep interior of the lower mantle.Estimation of the density of melts in the MgOFeOSiO₂ system for lower mantle conditions indicates that the initial melt formed by partial fusion of the lower mantle would be denser than the residual solid because of high concentration of iron into the melt. Thus, the melt generated in the lower mantle would tend to move downward toward the mantle-core boundary. This downward transportation of the melt in the lower mantle might have affected the chemistry of the lower mantle, such as in the D″ layer, and the distribution of the radioactive elements between mantle and core.     

Global survey of aftershock area expansion patterns

We developed an objective method to define the aftershock areas of large earthquakes as a function of time after the main shock. The definition is based upon the amount of energy released by aftershocks, the spatial distribution of the energy release is first determined and is contoured. The 1-day aftershock area is defined by a contour line corresponding to the energy release level of 10^15.6 ergs/(100 km² · day). The 10-day, 100-day and 1-y aftershock areas are similarly defined by contour lines corresponding to 10^14.8, 10^14.0, and 10^13.5 ergs/(100 km² · day), respectively. We also define the expansion ratios at time t by the ratio of the aftershock area at t to that at 1 day.Using this method we study the aftershock area expansion patterns of 44 large (M_s ? 7.5) and five moderate shallow earthquakes which occurred from 1963 to 1980. Each aftershock sequence is examined at four different times, i.e., 1 day, 10 days, 100 days, and 1 y after the main event. We define the aftershock area expansion ratios η and η_e by $\text{S(100)/S(}\text{1}\text{)}\text{and}\text{L(100)/L(1)}$, respectively: here S(t) and L(t) are the area and the length of the aftershock area, respectively, at time t. Our study suggests that a distinct regional variation of aftershock area expansion patterns is present; it is strongly correlated with the tectonic environment. In general, the subduction zones of the “Mariana” type have large expansion ratios, and those of the “Chilean” type have small expansion ratios. Some earthquakes that occurred in the areas of complex bathymetry such as aseismic ridges tend to have large expansion ratios.These results can be explained in terms of an asperity model of fault zones in which a fault plane is represented by a distribution of strong spots, called the asperities, and weak zones surrounding the asperities. The rupture immediately after the main shock mostly involves asperities. After the main rupture is completed, the stress change caused by the main shock gradually propagates outward into the surrounding weak zones. This stress propagation manifests itself as expansion of aftershock activity. In this simple picture, if the fault zone is represented by relatively large asperities separated by small weak zones (“Chilean” type), then little expansion of aftershock activity would be expected. On the other hand, if relatively small asperities are sparsely distributed (“Mariana” type), significant expansion occurs. The actual distribution of asperities is likely to be more complex than the two cases described above. However, we would expect that the expansion ratio is in general proportional to the spatial ratio of the total asperity area to the fault area.     

Theoretical grain size limits for single-domain,pseudo-single-domain and multi-domain behavior in titanomagnetite (x = 0.6) as a function of low-temperature oxidation

A theoretical model of grain size variation of domain transitions in titanomagnetite (x = 0.6) as a function of oxidation (z) is presented. The superparamagnetic (SP) to single-domain (SD) transition d_s, the SD to two-domain (TD) transition d₀, the TD to three-domain (3D) transition and the pseudo-single domain (PSD) to multi-domain (MD) transition are calculated as a function of z. It is shown that all the transition grain sizes increase with z, except for the PSD-MD transition for z > 0.6. The calculations predict that d_s increases from 0.044 to 0.197 μm, d₀ increases from 0.54 to 13 μm, the TD-3D transition increases from 1.6 to 49 μm as z varies from 0 to 0.8. The PSD-MD transition increases from 42 μm at z = 0 to 150 μm at z = 0.6, whereas between z = 0.6 to z = 0.8, the PSD-MD transition decreases to 49 μm. Qualitatively, the model explains some of the trends in magnetic properties of submarine basalts with low-temperature oxidation. Quantitatively, the model does give reasonable estimates of the PSD-MD boundary and d₀, which are close to the experimental values for x = 0.6 and z = 0. Furthermore, the model predicts that psarks or two-domain grains could be the major contributors to the remanence of oxidized submarine pillow basalts.