Browsing by Author "Wolf, Detlef"

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Open Access
Boussinesq’s problem of viscoelasticity
(1991) Wolf, Detlef
We consider an isochemical, isentropic, incompressible fluid half-space and study quasistatic viscoelastic perturbations, induced by two-dimensional (2D) surface loads, of a hydrostatic initial state. In view of the regional or local scale required for deformations of planets to be amenable to the half-space approximation, the model is assumed to be externally gravitating. We derive analytic solutions for the displacement and incremental stress components and study several approximations to the expressions. Particular emphasis is placed on discriminating between the material and local incremental stresses. Based on this distinction, deeper insight is gained into the physical significance of the solution.
Open Access
The changing role of the lithosphere in models of glacial isostasy : a historical review
(1993) Wolf, Detlef
During the last 125 years, the role of the lithosphere in models of glacial-isostatic adjustment experienced several changes. Following the postulation of glacial isostasy by Jamieson in 1865, the lithosphere was generally regarded as comparable in importance for the adjustment process to the fluid substratum. This changed with the initiation of quantitative modelling by Van Bemmelen and Berlage and by Haskell in 1935, whereupon effects due to the lithosphere were commonly neglected in interpretations of postglacial uplift for 30 years. After the development of a layered viscous earth model with an elastic surface layer by McConnell in 1965, the lithosphere was eventually reintroduced into models of glacial isostasy. Subsequent studies largely confirmed the original ideas regarding the importance of the lithosphere for the adjustment process, although the effects are pronounced only for short-wavelength deformations. Using this response characteristic of the lithosphere, estimates of its thickness have recently become available for several tectonic provinces.
Open Access
Comment on "Geomagnetic depth sounding by induction arrow representation: a review" by G. P. Gregori and L. P. Lanzerotti
(1982) Wolf, Detlef
In conclusion we may safely state that Gregori and Lanzerotti [1980] oversimplified their comparison of real induction vectors considerably and completely disregarded the four points summarized here. However, careful consideration of these differences is not merely a matter of academic interest but is also indispensable when interpreting actual field data.
Open Access
Glacial isostasy and long-term crustal movements in Fennoscandia with respect to lithospheric and asthenospheric processes and properties : comment
(1991) Wolf, Detlef
Recently, Mörner discussed his model of glacio-isostatic compensation in the light of new seismological models of the upper mantle below Fennoscandia. He concluded that the P-wave velocity distribution is in "full agreement" with his estimate of the viscosity distribution as inferred from glacio-isostatic relaxation. Whereas the consistency of seismological and geodynamical earth models is a necessary condition for their soundness, the particular type of agreement sought by Mörner is not required or expected, nor can his method of attaining such agreement be approved. For easier understanding of my criticism, I briefly recall the type of information provided by the seismological and by the glacio-isostatic evidence.
Open Access
Glacio isostatic adjustment in Fennoscandia revisited
(1986) Wolf, Detlef
The theory of load-induced relaxation of the layered Maxwell half-space is applied to the study of glacioisostatic adjustment in Fennoscandia. The interpretation uses emergence data from Angermanland (Sweden) and uplift-rate data from the Gulf of Bothnia. With mantle viscosity fixed at 10²¹ Pa s, the data can be explained by an earth model characterized by (a) an elastic surface layer of conventional thickness (about 100 km) superimposed on a low-viscosity layer or (b) an elastic surface layer of enhanced thickness (about 200 km) and no low-viscosity layer. Reasons for this ambiguity and possible ways to resolve it in future studies are suggested. The investigation also attempts to interpret the negative free-air gravity anomaly observed over the deglaciated region of Fennoscandia. Either (i) enhanced mantle viscosity below 670 km depth or (ii) a transition zone in viscosity between an elastic surface layer and a viscous mantle is required to predict negative anomalies substantially larger than 3 mgal.
Open Access
An improved estimate of lithospheric thickness based on a reinterpretation of tilt data from Pleistocene Lake Algonquin
(1985) Wolf, Detlef
A reinterpretation of relative-uplift data associated with the deformed strandline of Pleistocene Lake Algonquin is presented. The new analysis is based on a dynamic model that allows for the finite rate of retreat of the Laurentide ice sheet and the viscous relaxation of the Earth's mantle. Attention is also paid to the effects of the shape of the load profile on the interpretation. In the context of this model the thickness of the lithosphere is constrained to be not more than 85 krn. This value is based on the assumption of a model ice sheet with a parabolic cross section. If the load has a sharper edge, a thickness of 110 km is consistent with the data. These revised values slightly exceed Walcoll's original estimate. They are, on the other hand, distinctly lower than the value of 200 km, which has recently been inferred by Peltier for the thickness of the North American lithosphere. The relations between the different estimates are briefly discussed. It is suggested that a reconciliation of the estimates may require that effects caused by geoidal perturbations be included in future analyses of strandline tilt.
Open Access
Inductive coupling between idealized conductors and its significance for the geomagnetic coast effect
(1982) Wolf, Detlef
A problem of current interest is the inductive coupling between an ocean, a solid earth conductor and a conductosphere. The anomaly of this configuration is modelled by (i) the inductive response of a two-dimensional system consisting of two thin half sheets and an underlying thin whole sheet or (ii) the superposition of the responses of two related systems, each consisiting of only one of the two half sheets and the whole sheet. The conductivity of these conducting planes is assumed to be perfect, and rigorous solutions for the erived by conformal mapping. induced magnetic fields are a comparison between the anomalies (i) and (ii) permits us to ascertain the degree of inductive coupling between the idealized conductors. This establishes a reference for estimating the inductive coupling between more realistic conductors and may therefore assist in the interpretation of complicated magnetic variation anomalies in coastal regions. Our substitute configurations can also be used directly for the rapid modelling of the inductive response of the earth in the vicinity of coastlines. This is demonstrated by analyzing some field data from the recent literature.
Open Access
Inductive coupling between idealized conductors and its significance for the geomagnetic coast effect
(1983) Wolf, Detlef
A problem of current interest is the inductive coupling between an ocean, a solid earth conductor and a conductosphere. The anomaly of this configuration is modelled by (i) the inductive response of a system consisting of two thin half planes and an underlying thin whole plane and (ii) the superposition of the responses of two related systems, each consisting of only one of the two half plane and the whole plane. The configuration is two-dimensional, and the planes are perfectly conducting. These two assumptions allow the derivation of rigorous solutions for the induced magnetic fields by conformal mapping methods. A comparison between the anomalies (i) and (ii) permits the determination of the degree of inductive coupling between the idealized conductors. This establishes a reference for estimating the inductive coupling between more realistic conductors and may therefore assist in the interpretation of complicated magnetic variation anomalies in coastal regions. Our substitute configurations can also be used directly for the rapid modelling of the inductive response of the earth in the vicinity of coastlines. This is demonstrated by analyzing some field data from the recent literature.
Open Access
Isostatic compensation and continental lithospheric thickness
(1984) Wolf, Detlef
Lateral density inhomogeneities are a general feature of the continental crust and upper mantle. This suggests the generalization of the interpretation of isostatic response functions by introducing internal loads into the conventional model of an elastic plate flexed under a topographic load. With this modification, analyses of continental response functions favour lithospheric thicknesses larger than the average thickness of the crust. These values also conform with independent estimates of the long-term thickness of the continental lithosphere and are therefore more satisfactory than recent estimates of only a few kilometres based on a pure flexure model.
Open Access
Lamé's problem of gravitational viscoelasticity : the isochemical, incompressible planet
(1994) Wolf, Detlef
We consider a spherical, isochemical, incompressible, non-rotating fluid planet and study infinitesimal, quasi-static, gravitational-viscoelastic perturbations, induced by surface loads, of a hydrostatic initial state. The analytic solution to the incremental field equations and interface conditions governing the problem is derived using a formulation in terms of the isopotential incremental pressure measuring the increment of the hydrostatic initial pressure with respect to a particular level surface of the gravitational potential. This admits the decoupling of the incremental equilibrium equation from the incremental potential equation. As result, two mutually independent (4 times 4) and (2 times 2) first-order ordinary differential systems in terms of the mechanical and gravitational quantities, respectively, are obtained, whose integration is algebraically easier than that of the conventional (6 X 6) differential system. In support of various types of application, we provide transfer functions, impulse-response functions and Green's functions for the full range of incremental field quantities of interest in studies of planetary deformations. The functional forms in the different solution domains involve explicit expressions for the Legendre degrees n= 0, n= 1 and n≥ 2, apply to any location in the interior or exterior of the planet and are valid for any type of generalized Maxwell viscoelasticity and for arbitrary surface loads.
Open Access
A method of calculating lithosphere thickness from observations of deglacial land uplift and tilt
(1986) Wolf, Detlef
The study proposes an inversion method applicable to observations of deglacial land uplift and tilt near the margin of a major Pleistocene ice-sheet. This information allows the determination of the cross-section of the ice-sheet and the calculation of the thickness of the lithosphere. The method is applied to the Pleistocene glaciations in Fennoscandia and Laurentia and yields similar lithosphere thicknesses of 110 ± 30 km and 130 ± 35 km, respectively. Since the method is based on a static theoretical model, the thicknesses must be interpreted as upper bounds.
Open Access
The normal modes of a layered, incompressible Maxwell half-space
(1985) Wolf, Detlef
The theory describing the relaxation of an incompressible, layered Maxwell half-space is developed. The approach is based on the analytic solution of the associated elastic model and the subsequent application of the correspondence principle. The viscoelastic theory follows normal-mode theory, which allows the independent and exact determination of the relaxationtime and amplitude spectra for each mode of relaxation. The solution is tested by calculating the response of several models in the wavenumber and spatial domains. The examples are selected with regard to postglacial adjustment in Fennoscandia and analyse effects caused by (a) varying lithospheric thickness, (b) adding an asthenosphere, (c) increasing lower-mantle viscosity, (d) permitting relaxation of the lower lithosphere or (e) introducing density contrasts at 400-km and 670-km depths.
Open Access
The normal modes of a uniform, compressible Maxwell half-space
(1985) Wolf, Detlef
The analytical solution for the load-induced deformation of a uniform, compressible, hydrostatically pre-stressed elastic half-space is derived. The solution is correct to first order in the quantity e, which is inversely proportional to the wave number k of the deformation. Usually e is very small compared with unity for Earth deformations on a scale amenable to the halfspace approximation. Since pre-stress advection is included in the analysis, the correspondence principle allows us to solve the field equations governing the deformation of the associated Maxwell half-space. The viscoelastic solution shows that the relaxation of the Maxwell continuum is characterized by a fundamental mode and a rapidly decaying overtone of much smaller amplitude. In the incompressible limit the overtone is not excited. The significance of the results for the relaxation of the Earth's mantle is briefly discussed.
Open Access
On Boussinesq’s problem for Maxwell continua subject to an external gravity field : research note
(1985) Wolf, Detlef
Boussinesq's problem is solved for a uniform and incompressible Maxwell half-space subject to an external gravity field. The solution is based on momentum equations which account for stress advection in the hydrostatically pre-stressed continuum during its deformation. The analysis shows that disregarding the pre-stress term renders the theoretical stress distribution incorrect and the deformation singular in the inviscid limit of the Maxwell continuum. Our solution is contrasted with a recently published alternative solution of the same problem, where regularity in the inviscid limit was forced by modified boundary conditions.
Open Access
On deglaciation induced perturbations of the geoid
(1986) Wolf, Detlef
The importance of geoid perturbations in interpreting tilts of glacial lake levels is investigated, using a representative theoretical model. It is shown that near the ice margin, the slope of the Earth's perturbed surface is everywhere large compared with the slope of the associated undulation of the geoid. Perturbations of the geoid may therefore be neglected in interpretations of glacial strandline tilt.
Open Access
On the relation between two-dimensional and axisymmetric loads in plate flexure problems
(1984) Wolf, Detlef
This note has attempted to clarify the relation between straight-edged axisymmetric loads and straight-edged two-dimensional loads by emphasizing the critical role of the elastic plate in this relation. It has been shown that loads of widely varying scale can be approximated by two-dImensional edges, provided that the lithosphere is sufficiently thin. As the distance from the edge of the load increases, the displacement w decreases. This results in enhanced relative discrepancies between the two models in the region of the peripheral bulge. Whether this is significant when modelling data will ultimately depend on the magnitude of the errors associated with the measurements.
Open Access
The relaxation of spherical and flat Maxwell earth models and effects due to the presence of the lithosphere
(1984) Wolf, Detlef
The radial surface deflection of a spherical earth model appropriate to time scales characteristic of glacial loading is compared with the associated halfspace response. For that purpose the analytical solution for a Maxwell sphere surrounded by a thick elastic shell is derived under the assumption of incompressibility. The half-space approximation is deduced as a special case directly from the spherical solution. Comparison of the response spectra, for different thicknesses of the elastic shell, reveals only minor differences. In the spatial domain, the half-space approximation may, nevertheless, diverge significantly from the spherical solution. For a disk load radius R = 800 km (Fennoscandia), the half-space approximation is adequate, whereas it is usually inappropriate if R = 1600 km (Laurentia). The sensitivity of the response to the thickness of the elastic shell is also investigated. For R = 800 km, the surface deflection in the central region below the load is fairly diagnostic of the shell thickness. If R = 1600 km, the peripheral region is more sensitive to this parameter.
Open Access
Reply to comments by Robert P. Comer [Letter to the editor]
(1986) Wolf, Detlef
In his letter, Comer (1986) addresses three different points which bear on three related publications on load-induced flexure of elastic plates (Comer 1983; Ward 1984; Wolf 1985a). As his first point is primarily an annotation to his own publication on the subject (Comer 1983), I will restrict my reply to the remaining two points.
Open Access
Singular solutions to Maxwell's equations and their significance for geomagnetic induction
(1983) Wolf, Detlef
In conclusion we may state that the correctness of Summers' (1982) response curves must be questioned. The main reason for this is that a dramatic increase of the anomalous field for decreasing frequency, i.e. towards the resistive limit is required. Intuitively, such response characteristics appear highly unusual. As demonstrated by the present discussion, its theoretical justification is likewise difficult. It is therefore suggested that Summers' numerical results be checked against analytical solutions. Kertz's (1960) cylindrical body seems very appropriate for this and also resembles the configuration discussed by Summers (1982) very closely. As the cylinder is embedded into a non-conducting medium, spurious channelling effects are excluded and induction is strictly local. Consequently, Kertz's response curves are as expected, i.e. frequency independence is confined to the high-frequency limit. An easy test of the accuracy of Summers' numerical code is therefore possible, and the character and frequency range of the discrepancies between the two solutions can be determined.
Open Access
Thick plate flexure re-examined : research note
(1985) Wolf, Detlef
The flexure of an incompressible, thick elastic plate floating on an inviscid substratum and subject to an external gravity field is re-analysed. The solution is derived from momentum equations which account for the advection of hydrostatic pre-stress. This is contrasted with a recently published thick-plate solution derived from momentum equations without a pre-stress term. It is demonstrated that neglecting pre-stress advection renders the solution singular when the model degenerates into an inviscid half-space. If pre-stress advection is included, the solution remains correct in this limit. A numerical comparison of both types of thick-plate solution with results based on conventional thin-plate theory further shows that, for geophysically relevant models, the difference in the momentum balance entails discrepancies between the thick-plate solutions which are comparable to the errors introduced by the thin-plate approximation.