Showing papers in "Geophysical Journal International in 1967"

Linear Models of Dissipation whose Q is almost Frequency Independent-II

Abstract: Summary Laboratory experiments and field observations indicate that the Q of many non-ferromagnetic inorganic solids is almost frequency independent in the range 10-2-107 cis, although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression for a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stressstrain relation. Since the aim of this research is also to contribute to elucidating the dissipating mechanism in the Earth free modes, we shall treat the dissipation in the free, purely torsional, modes of a shell. The dissipation in a plane wave will also be treated. The theory is checked with the new values determined for the Q of spheroidal free modes of the Earth in the range between 10 and 5 min integrated with the Q of Rayleigh waves in the range between 5 and 0.6 min. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an aluminium rod in the range between lo-’ and 10-3s. In both checks the theory represents the observed phenomena very satisfactorily. The time derivative which enters the stress-strain relation in both cases is of order 0.15. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. In a homogeneous isotropic elastic field the elastic properties of the substance are specified by a description of the strains and stresses in a limited portion of the field since the strains and stresses are linearly related by two parameters which describe the elastic properties of the field. If the elastic field is not homogeneous nor isotropic the properties of the field are specified in a similar manner by a larger number of parameters which also depend on the position.

Numerical Applications of a Formalism for Geophysical Inverse Problems

Abstract: Summary A gross datum of the Earth is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We prove that the collection of Earth models which yield the physically observed values of any independent set of gross Earth data is either empty or infinite dimensional. We exploit this very high degree of non-uniqueness in real geophysical inverse problems to generate computer programs which iteratively produce Earth models to fit given gross Earth data and satisfy other criteria. We describe techniques for exploring the collection of all Earth models which fit given gross Earth data. Finally, we apply the theory to the normal modes of elastic-gravitational oscillation of the Earth.

A Seismic Equation of State

Abstract: Birch's hypothesis of a close relationship between seismic velocity and density is extended and modified so as to be in accord with theoretical predictions concerning the form of the equation of state. Although developed as a simple method to assure consistency between the seismic velocities and densities in free oscillation calculations the resulting equation of state is of quite general utility in geophysical studies where the seismic velocities, rather than hydrostatic pressure and temperature, are the directly measured variables. A simplified form of the seismic equation of state is ρ= AMΦ^* where ρ is the density, M is the mean atomic weight, n is a constant of the order of ¼ to ⅓ and is related to the Gruneisen constant γ, and Φ is the seismic parameter V_P^2 - (4/3)V_S^2. The exponent n is slightly different for constant temperature and constant pressure experiments but its magnitude, in both cases, can be estimated from lattice dynamics. On the other hand n is roughly the same number for compositional, structural and pressure effects. Since Φ also is (∂P/∂ρ)_S and KS/ρ, data from static compression and shock wave as well as ultrasonic experiments can be used to determine the parameters in the equation of state and to extend its range beyond that available from ultrasonic data. Static pressure and shock wave data extend to much higher pressures, or compressions, than the ultrasonic data used by Birch and many more materials have been tested. The general tendency of density to increase with Φ can be used to determine the density in the C-region even if this is a region of phase changes. New density models for the Earth are constructed on these considerations.

Formation of Magnetic Anomaly Pattern of Mid-Atlantic Ridge

Abstract: Summary Surveys of the crest of the Mid-Atlantic Ridge and similar portions of the other mid-ocean ridges characteristically reveal a pattern of magnetic anomalies which have an elongation of about 4:1 parallel with the median line of the ridge. The linear pattern can be roughly simulated by a model in which steep-sided blocks within the oceanic crust between the sea floor and the depth of the Curie point isotherm have alternately normal and reversed directions of remanent magnetization. Vine & Matthews (1963) suggested that this structure might be due to the continued injection of basaltic feeder dykes along the median line of the ridge, each dyke shouldering aside its predecessors and being magnetized in the direction of the Earth's magnetic field. As the direction of the Earth's field reverses periodically this mechanism can account in a crude way for the formation of the model. A survey of the crest of the Mid-Atlantic Ridge at 45° N (Loncarevic, Mason & Matthews 1966) showed a less regular pattern of blocks suggesting that the mechanism of dyke injection is less regular than that utilized in the original theory. It seems plausible that dyke injections do not always occur precisely along the median line but are scattered about it. An analogue model experiment involving marbles, and a computer program written for titan are used to investigate the consequence of such scatter. First results indicate that the great majority of dyke injections must occur within the width of the median valley; the standard deviation of the distribution of dykes (assumed normal) must be less than about 5 km (half the width of the median valley).

Inverse Problems of Seismology (Structural Review)

Abstract: Summary The inverse problem of seismology is formulated as the determination of the set of cross-sections fitting all given observations. The schemes of solution of this problem by the trial-and-error method and the inversion method are described. Mathematical and computational problems involved in the solution are outlined. Not all these problems are solved at the present time. Among them optimal parametrization of cross-section and the determination of a probability distribution function for cross-section are the most urgent problems. But with certain limitations the described scheme of a trial-and-error method for joint interpretation of surface and body waves is in routine operation now. Results obtained are described in papers [2, 4, 6, 15, 22, 23]. Usually it is the non-uniqueness that strikes us when we see the solution of the inverse problem. But there is another side to the matter: if we do not look for an excessively detailed cross-section we may obtain quite definite conclusions even from very rough observations which are commonly regarded as useless.

Converting Vector and Tensor Equations to Scalar Equations in Spherical Coordinates

Abstract: Sunzriiary We show how to represent a vector field in a spherical shell in terms of three scalar fields, and a second-order tensor field in terms of nine scalar fields. We derive the scalar representations of the most common algebraic and differential operations on vector and second-order tensor fields, and apply the results to obtain scalar formulations of various tensor problems in the continuum mechanics of the Earth’s mantle. We give an exhaustive catalogue of all possible equilibrium stress fields in the mantle, we deduce the scalar equations of elastic-gravitational oscillation of a transversely isotropic, radially stratified, spherically symmetrical Earth, and we give the scalar convection equations for a viscous, self-gravitating, spherically symmetrical mantle with radially variable viscosity.

Correlation of Petrology and Natural Magnetic Polarity in Columbia Plateau Basalts

Abstract: Summary This paper reports clear penological differences between normally and reversely magnetized lavas from the Columbia Plateau basalts. We conclude that self-reversal of magnetization cannot explain this correlation, but that perhaps a causal connection between polarity of the Earth's magnetic field and oxidation state in lavas is necessary to reconcile all the facts.

The Intensity of the Ancient Geomagnetic Field: A Review and Analysis

Abstract: Summary In this review all palaeomagnetic determinations of the intensity of the ancient geomagnetic field known to the author at 1966 June are brought together and analysed. The lists of data cover determinations from both historic (archaeomagnetic) and geological specimens. The problems encountered in the determination of ancient field intensities from rocks, especially that of rock alteration during the formation of artificial remanence and that of the presence of secondary components of magnetization, are discussed, and methods used to overcome them are described. Experimental techniques used hitherto are briefly summarized. Methods of treating data are reviewed; and suggestions for comparing data from different latitudes are made. The main conclusions drawn from an analysis of both the field intensity data and the problems involved in obtaining this data are: (i) that the problem of rock alteration during the heating necessary to induce an artificial TRM may be overcome by heating selected naturally highly-oxidized specimens in air; (ii) that the effects of secondary components of magnetization may be avoided by utilizing that part of the primary moment which remains after the removal of secondary moments by thermal or a.c. demagnetization; (iii) that the geomagnetic dipole is not constant within any given polarity but fluctuates in strength, possibly with a period of the order of lo4 years; (iv) that because of these fluctuations and non-dipole field variations no conclusions regarding the mean strength of the geomagnetic dipole at any given time may be made if only a few rock samples are measured, and hence that future workers should use sufficient samples to average out these effects; (v) that the limited data from transition zones suggest that during a field reversal the dipole moment reduces to zero but the non-dipole field remains.

Solution of the Linear Inverse Problem in Magnetic Interpretation with Application to Oceanic Magnetic Anomalies

Abstract: Summary The problem of interpreting a magnetic anomaly usually reduces to either (1) determining the distribution of magnetization given the shape of the body and the direction of magnetization, or (2) determining the shape of one interface, given the magnetization and other interfaces. These involve solution of an integral equation, linear in case (1) and non-linear in case (2). The paper gives two solutions of the linear problem applicable either to gravity or magnetic interpretation, which may also be used as a starting point for the iterative solution of the non-linear problem. Method (1): The Fourier convolution theorem has been used to derive a two-dimensional magnetic version of the ‘equivalent layer’ theorem, which is the simplest case of the inverse problem. This enables a given magnetic anomaly to be replaced by a coating of magnetic moment per unit area of specified direction over the horizontal plane of the measured anomalies. The ‘equivalent layer’ can be continued downwards by existing methods. A computer program applicable to the method is available at Durham. Method (2): A more versatile approach is to approximate the linear integral equation by a summation, resulting in a set of linear equations. If the number of observations are equal to the unknown magnetization parameters the equations are solved directly; if there are more observations than unknowns, least squares is used. In either case, the matrix schemes available on most computers provide the main tool for the method. Method (2) has been applied to the interpretation of oceanic magnetic anomalies in terms of two-dimensional rectangular blocks confined between two specified depths, the magnetization varying laterally from block to block. The matrix inversion needs to be done once only for specified depths, block width and direction of magnetization. It provides a weighting function for the observed anomaly and repeated application of the convolution gives the underlying distribution of magnetization. Applications of the method to the Juan de Fuca Ridge, suggests that unacceptably high contrasts in magnetization are required if the main source is below the oceanic layer 2, supporting the view that rocks in layer 2 cause a substantial part of the anomalies.

The Krakatoa Air—Sea Waves: An Example of Pulse Propagation in Coupled Systems

Abstract: The theory of pulse propagation in an atmosphere coupled to an ocean is applied to the air-sea waves excited by the explosion of the volcano Krakatoa. Numerical results for a realistic atmosphere-ocean system show that the principal air pulse corresponds to the fundamental gravity mode GR_0. A small sea wave is associated with the mode GW_0 with phase velocities close to the √(gh) velocity of the ocean. Free waves with this velocity exist in the atmosphere and transfer energy to the ocean in an efficient manner. These air waves ‘jump’ over land barriers and re-excite the sea waves. An explosion of 100–150 megatons is required to produce the equivalent of the Krakatoa pressure disturbance.

The Intensity of the Tertiary Geomagnetic Field

Abstract: Summary Although many paleomagnetic directional data are now available, very little is known of the ancient intensity of the Earth's magnetic field. This paper describes the determination of the Tertiary geomagnetic field intensity using baked contacts (red sandstones, conglomerates and dykes) and dykes from Scotland and lavas and baked laterites from Iceland. The field intensity results obtained correspond to a mean Tertiary dipole moment of 4.94 × 1025 gauss. cm3 in Scotland and a mean Tertiary dipole moment of 5.34 × 1025 gauss. cm3 in Iceland. These values contrast with the present geomagnetic dipole moment which is 8.0 × 1025 gauss. cm3.

The Determination of Q, Dynamic Viscosity and Transient Creep Curves from Wave Propagation Measurements*

Abstract: Summary Perhaps the best theoretical approximation to the experimentally observed attenuation and spreading of seismic body waves is due to Futterman. Wuenschel has utilized the Pierre shale field data of McDonal et al. as well as his own two-dimensional model work using plexiglas sheet, to obtain remarkable confirmation of Futterman's approach. However, Futterman's assumption of a linear attenuation-frequency relation requires truncation at both ends of the frequency spectrum. The arbitrariness of this truncation leads to unnecessary parameters which make his approximation awkward to use. We discuss a power-law approximation to the attenuation characteristic which does not have this difficulty. Emphasis will be on computation of mechanical properties such as spatial ‘Q’, magnitude and loss angle of the modulus, dynamic viscosity and transient creep. These computations which are based upon the wave propagation measurements of Jordan will be seen to be in remarkable agreement with independent measurements by Lethersich, who utilized direct experimental procedures.

Density-Velocity Relations for Rocks

Abstract: Summary Experimental data of Birch for rocks at 10 kilobars are found to fit an empirical relationship between scaled velocity and scaled density. The scaling factors are functions of the representative atomic number. An empirical relationship is found connecting the representative atomic number and the mean atomic weight.

The Amplitude‐Distance Curve for Short Period Teleseismic P‐Waves

Abstract: Summary An amplitude-distance curve has been computed for short period P waves in the range A=30"-102" (and h=O) using the amplitudes of explosion signals only. Effects of source asymmetry can thus be discounted and the problem of deciding what amplitude to measure is reduced because the maximum amplitude of explosion signals always occurs in the first few cycles. To increase the consistency of the measurements all the amplitudes have been measured by the same person. Assuming the effect of distance is constant over a series of discrete intervals of distance, the amplitude-distance curve with its confidence limits has been estimated by least squares using data from 65 stations which recorded some or all of the explosions fired at six sites. The most striking features of the curve are: (1) a sharp maximum at about A = 35", (2) a minimum followed by a sharp increase at A = 75", and (3) a sharp minimum at A =93". Over the remainder of its length the curve is similar to the Gutenberg and Richter curve. Station magnitude corrections have also been estimated. An amplitude-distance curve derived from explosion data should be more reliable than curves derived from earthquake data: explosions have radially symmetrical radiation patterns and their seismic signals are almost always the same shape with the maximum amplitude occurring in the first few cycles. Differences in the amplitude of signals recorded from a single explosion at two recording stations are unlikely to be due to source asymmetry (a significant factor if earthquakes are shears), and the difficulties of defining what amplitude to measure for magnitude determinations are reduced. We have collected the available World Wide Standard Seismograph Network (WWSSN) and Canadian records of underground explosions and the amplitudes of the explosion signals have been read by one seismologist (Mr P. D. Marshall). The accumulated data has been used to estimate both an amplitude-distance curve and station magnitude corrections for short period teleseismic P waves.

An Analysis of Annual Sea Level Variations in European Waters

Abstract: Summary Regression analysis has been applied to all long series of annual mean sea level heights for European stations in order to estimate the components due to secular variation, air pressure variations and the nodal tide. The multiple correlation coefficients are in general found to lie between 0.8 and 1.0. Secular variations (rise or fall in sea level relative to the land) are well defined; they range from almost 90 cm/century fall at the head of the Gulf of Bothnia to a rise of 20 cm/century in the southern North Sea, English Channel and Mediterranean. The variations are consistent enough to enable isopleths to be constructed. The results are used to predict the probable value of mean sea level at certain stations in 1970. Expressions are obtained for annual mean sea level anomalies, arising from variations in the air pressure distribution, which enable the computation of ‘normal’ mean sea level (under normal atmospheric conditions) and ‘isobaric’ mean sea level (on the assumption of an isobaric atmosphere). The isobaric estimates represent an approximate to the geoid in European waters, though clear indications exist of a slope due to the spatial distribution of sea water density; the principal isobaric surface slopes are found to be more than 20 cm upwards from the North Sea into the Baltic, 8 cm downward to the north along the Norwegian coast, and 10 cm downward from the English Channel into the Mediterranean. The permanent deformation of the sea surface due to prevailing winds and air pressure gradients is well marked, rising more than 20 cm from the Mediterranean to the Gulf of Bothnia. Estimates of the nodal tide (period 18.6 years) suggest its existence in the equilibrium form, but the degree of scatter in the results is too great to permit a more definite conclusion.

Inhomogeneities in the Earth's Mantle

Abstract: Using seismic body and surface waves, the velocity structure of the Earth's mantle is determined with the emphasis on regions of anomalous variations (so-called ‘discontinuities’). In the upper mantle, the interpretation of Rayleigh and Love wave dispersion curves yields shear velocity profiles with discontinuities at depths 350 km and 700 km, and a low-velocity zone extending to 350km. In the lower mantle P-velocity profile is determined from dt/dΔ measurements using large aperture seismic array and travel times from Long Shot nuclear explosion for the Japan-Kuriles-Aleutian-Montana path. The velocity structure shows anomalous gradients or ‘discontinuities’ at depths 700, 1200 and 1900km, indicating that the lower mantle is not homogeneous. Lateral variations of the velocity structures are investigated. For the upper mantle studies the Earth is divided into three regions: oceanic areas, continental shields, and tectonic zones. Pure path phase velocities of Love waves are extracted from the composite dispersion data. The pure path shear velocity profiles obtained from these data are characterized by lower velocities under the oceans in the uppermost portion of the mantle. Shields have the highest velocities. These velocity differences are interpreted in terms of temperature variations. At a depth of 110 km the temperature of the oceanic mantle is higher (by 100–500° C depending on the temperature coefficient of the velocity) than that of the mantle under the shields. The presence of lateral heterogeneities in the mantle is demonstrated qualitatively by the differences of dt/dΔ vs Δ curves for two separate paths. Undulations of the geoid as determined from satellite observations are investigated for determining the sources of the anomalies. It is concluded that the main sources of lateral density variations must be in the mantle at depths greater than about 100km.

P and S Travel Time Anomalies and their Interpretation

Abstract: Summary Study of the deviations of P and S travel times from the J-B tables at teleseismic distances has shown that there are regional differences in travel time. Both P and S are early in the central and eastern United States, late in the western United States. The differences have a range of about three seconds for P and eight seconds for S. It can be deduced from the relation between the travel time residuals (1) that the change in shear velocity is approximately one and one-quarter times the change in P velocity, (2) that the observations imply a difference in Poisson's ratio between the two regions, and (3) that a model in which the shear modulus, μ, alone varies, the compressibility, k, remaining sensibly constant, fits the data best. It can be shown also that the differences between the P travel time residuals and the gravity anomalies in the central and western United States are not consistent with the Birch relation between velocity and density.

A Theory of Microbaroms

Abstract: summary A theory analogous to the Longuet-Higgins theory on the generation of microseisms explains the generation of microbaroms by standing water waves associated with marine storms. The spectral characteristics and the amplitude order-of-magnitude of microbaroms that are predicted by this theory agree well with observations. The theory is based on the oscillations of the centre of gravity of the air above the Ocean surface on which the standing waves appear (or of the water below, to explain microseisms). These oscillations are of twice the ocean wave frequency and thereby explain the observed frequency-doubling common to both microbaroms and microseisms. The theory is expanded by statistical methods to predict the microbarom-generating effect of more realistic ocean waves, whose phases vary randomly over the Ocean surface. In addition, the effect of the widespread source on microbarom coherence and resolvability at the receiving array is discussed.

An Automated Method of Gravity Interpretation

Abstract: Summary The interpretation of an observed gravity anomaly in terms of an anomalous mass with irregular outline and with uniform density contrast requires the solution of a non-linear problem. It is possible to iterate this non-linear problem by means of a linear approximation, provided some assumption is made about one of the surfaces of the anomalous mass. This paper gives such a method. If there are m observations of a gravity anomaly and if the anomalous mass is assumed to be subdivided into n two-dimensional rectangular blocks (n n-to give a system of blocks of variable density contrast which satisfy. or nearly satisfy in the case of the least squares solution, the observed gravity anomaly. These blocks are then transformed to give blocks of uniform density contrast. Because the gravity effect is non-linear the transformed blocks will not usually satisfy the observed anomaly. It is, therefore, necessary to adjust the model using the same general method. Two computer programs applying respectively to structures with, inward dipping contacts and to structures with outward dipping contacts have been developed. The formulae used in the programs apply to two-dimensional structures, but three-dimensional structures are approximated by end corrections.

Table for Fisher's Test of Significance in Harmonic Analysis

Abstract: Summary A table is calculated which facilitates the estimation of significant spectral peaks obtained by harmonic analysis which are above the 99, 98, 95, and 90% confidence level. The application of the test is also described. In the scheme of harmonic analysis for revealing the periodic structure of a time series, the inferences are based on the magnitude of the amplitude corresponding to each period. Yet, any set of random numbers subject to the harmonic analysis will yield harmonic amplitudes such that some are bigger than others by chance alone. The decision of reliability of these amplitudes, therefore, depends on comparison with the magnitude of the amplitudes which can be produced by harmonic analysis on the assumption that the time series are from a random fluctuation and do not have any physical meaning. 2. Fisher’s test and tables Fisher (1929) developed a test of significance in harmonic analysis from a series x(i), i= 1, 2, . .. , n constituting a random sample from a normally distributed population. The decomposition of x(i) into its harmonic constituents is

Scattered Surface Waves from a Surface Obstacle

Abstract: Summary This paper is concerned with the interpretation of a surface wave arrival at the Eskdalemuir seismological array which is apparently generated by the scattering of the initial P wave. The theory of scattering of a plane wave by a surface obstacle is developed in order to apply to a three-dimensional surface inclusion of any shape. Results are approximate, depending on the depth of the obstacle and the slope of all boundaries being small. Comparison of theory with data from four events leads to an estimate of the breadth of the scatterer. A studyof the relief in the area of the neighbourhood of the array strongly suggests that scattering is due to irregularities of surface topography.

On the Characteristic Properties of Travel-Time Curves

Abstract: Summary The uniqueness of the determination of a velocity cross-section from the travel-time curves for surface and deep sources was investigated by Gerver & Markushevich (1966,1967). The Earth was assumed spherically symmetrical with a finite number of waveguides. The present report states the conditions when a solution of this inverse problem exists.

A New Palaeomagnetic Result from the Lower Cretaceous of East-Central Africa

Abstract: Summary The mean direction of natural remanent magnetization (NRM) in eight samples of syenite of Lower Cretaceous age from the Mlanje Massif, Malawi, has declination 333" and inclination - 54" (upward) after magnetic cleaning. The stability of remanence in alternating magnetic fields suggests that the NRM is primary. On the assumption of a geocentric dipole field, the palaeomagnetic pole is calculated at 60" N, 98" W. This is in agreement with previous results from volcanics of the Lupata Series nearby in Mozambique, which are of similar age. Combination of results from the Mlanje Massif and the Lupata volcanics yields an estimate of the palaeomagnetic pole for the lower Cretaceous of this part of Africa at 609 N, 993" W, with circular standard deviation (related to palaeosecular variation) of 10".0. Directions of natural remanent magnetization in eight samples from the Lauderdale erosion crater (16" 1' S, 35" 329 E) on the southern side of the Mlanje Massif in south-eastern Malawi are reported. The massif is a syenite-granite complex and has been mapped by Garson & Walshaw (in preparation). The samples were collected from the outer syenite ring of the complex, along about 150 metres of a stream running at right-angles to its contact with the Palaeozoic basement. The boundary of the complex is transitional, the adjacent basement having been syenitized. Only the intrusive body has been sampled. It is several kilometres in diameter, and it is likely that the section which has been sampled represents a long period in the cooling history of the body. Therefore, in spite of the small collection there is reason to believe that palaeosecular variation may have been effectively averaged out in calculating the mean direction, and that the dispersion may be a fair estimate of the palaeosecular variation. Two recent K-Ar age determinations by Snelling (1966) indicate the Cretaceous age of the complex : biotite from perthosite (sample WllOS) from the outer ring at Chambe indicated 1 16T 6 m.y., and biotite-hornblende mixture from quartz-syenite (sample W1112) from the Nanchidwa Hills, some 19 km east of the Lauderdale Crater gave 128 T 6 m.y. Gough & Opdyke (1963) have reported the palaeomagnetism of the Lupata alkaline volcanics and also of one site in redbeds of the Lupata Series, some 160 km south-west of Mlanje, in Mozambique. McDougall made two determinations on anorthoclases from the Lupata volcanics at 110.5 and 106.7m.y. (Gough et al. 1964). Because Gough & Opdyke sampled seven sites in single lava flows, and hence sampled no more

Palaeomagnetism of the Exeter Lavas, Devonshire

Abstract: Summary The directions of magnetization and the magnetic properties of the Exeter lavas and the associated red sediments have been measured. The lavas have been placed in the Stephanian on the basis of recent age determinations (280 m. y.) but they are interbedded with the desert type sediments typical of New Red Sandstone times in southwest England. The directions of the lavas generally show little scatter and are very resistant to both thermal and a. c. demagnetization. This is probably connected with the fact that their remanence is to a large extent due to haematite and a crystallization remanence (CRM). It is likely that the CRM was formed when the lavas were exposed to weathering soon after they were extruded in desert conditions. The palaeomagnetic pole position agrees with that established by an earlier investigation of the lavas. The associated sediments are not altogether satisfactory for rock magnetic studies owing to their friable nature but yield a pole position similar to that of the lavas.

Nearly Symmetric Dynamos

Abstract: Summary Braginskii (1964) initiated the study of homogeneous dynamos in a cylindrical symmetric volume in which the conducting fluid moves almost symmetrically, and in which the magnetic Reynolds number of the symmetric components of this motion is large. He showed that, in a first approximation, the effects of the small non-axisymmetric components of flow could be simply represented in the equation for the symmetric part of the magnetic field by a generating term, provided new ‘effective’ components of symmetric flow and field were defined. We show that this remarkable result is also true in the second approximation.

Truncation Errors in the Stokes and Vening Meinesz Formulae for Different Order Spherical Harmonic Gravity Terms

Abstract: Summary A detailed analysis of truncation errors in the Stokes formula integration, using Molodenskii's method, shows the mode of dependence of the errors on the spherical harmonic components of Δg of different order. The results indicate that significant reduction in the truncation errors can be achieved by adopting a reference model for normal gravity of higher order than that based on the International Ellipsoid. Particularly, the use of a seventh order reference model combined with truncation at the first zero crossing of the Stokes kernel function appears very promising. The treatment of truncation errors for deflection of the vertical as given by Molodenskii et al. and Hirvonen & Moritz yields results for deflections as derivatives of geoidal heights explicitly obtained from Stokes formula. These errors can be reduced by the same techniques as suggested for the Stokes integration. The truncation error behaviour of the Vening Meinesz formulae is shown to be different. An adaptation of Molodenskii's approach using an expansion of the truncated deflection of the vertical kernel function in terms of associated Legendre polynomials of first order provides a means for spherical harmonic analysis of the truncation errors in the Vening Meinesz formulae.

The Computation of the Transient Gravity Wave

Abstract: Summary The two-dimensional Boussinesq equations are integrated with given conditions on the velocity and static stability on the left-hand boundary and with a deformed lower boundary, to give the transient gravity wave, or mountain wave. The results are compared with linear theory, and it is found that although in general the overall flow pattern agrees with the linear analysis, in the region near the obstacle the non-linear boundary condition produces several important departures. Good qualitative agreement with observations is attained. 1. List of symbols