首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献，搜索用时 15 毫秒
1.
Planetary topography can either be modelled as a load supported by the lithosphere, or as a dynamic effect due to lithospheric flexure caused by mantle convection. In both cases the response of the lithosphere to external forces can be calculated with the theory of thin elastic plates or shells. On one-plate planets the spherical geometry of the lithospheric shell plays an important role in the flexure mechanism. So far the equations governing the deformations and stresses of a spherical shell have only been derived under the assumption of a shell of constant thickness. However, local studies of gravity and topography data suggest large variations in the thickness of the lithosphere. In this paper, we obtain the scalar flexure equations governing the deformations of a thin spherical shell with variable thickness or variable Young's modulus. The resulting equations can be solved in succession, except for a system of two simultaneous equations, the solutions of which are the transverse deflection and an associated stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include loads with a toroidal tangential component. We further show that toroidal tangential displacement always occurs if the shell thickness varies, even in the absence of toroidal loads. We finally prove that the degree-one harmonic components of the transverse deflection and of the toroidal tangential displacement are independent of the elastic properties of the shell and are associated with translational and rotational freedom. While being constrained by the static assumption, degree-one loads can deform the shell and generate stresses. The flexure equations for a shell of variable thickness are useful not only for the prediction of the gravity signal in local admittance studies, but also for the construction of stress maps in tectonic analysis.  相似文献   

2.
Summary. Numerical convection models are presented in which plates are simulated by imposing piecewise constant horizontal velocities on the upper boundary. A 4 × 1 box of constant viscosity fluid and two-dimensional (2-D) flow is assumed. Four heating modes are compared: the four combinations of internal or bottom heating and prescribed bottom temperature or heat flux. The case with internal heating and an isothermal base is relevant to lower mantle or whole mantle convection, and it yields a lower thermal boundary layer which is laterally variable and can be locally reversed, corresponding to heat flowing back into the core locally. When scaled to the whole mantle, the surface deflections and gravity and geoid perturbations calculated from the models are comparable to those observed at the Earth's surface. For models with migrating ridges and trenches, the flow structure lags well behind the changing surface 'plate'configurations. This may help to explain the poor correlation between the main geoid features and plate boundaries. Trench migration substantially affects the dip of the cool descending fluid because of induced horizontal shear in the vicinity of the trench. Such shear is small for whole mantle convection, but is large for upper mantle convection, and would probably result in the Tonga Benioff zone dipping to the SE, opposite to the observed dip, for the case of upper mantle convection.  相似文献   

3.
Recent studies on the relationship between the Nusselt ( Nu ) and Rayleigh ( Ra ) numbers for base-heated convection in a spherical shell with a constant viscosity show that the power-law index is around 1/4, which is different from the value of 1/3 predicted by a simple boundary layer theory. We show that such a difference may be caused by the flow pattern due to the geometry. The flow pattern of the convection in a spherical shell at relatively low Ra , at least, less than 106, is characterized by narrow upwelling and broad downwelling, which is similar to the opposite flow pattern of internally heated convection. Convection in the internally heated case predicts the power-law index of 1/4. We demonstrate this relationship based on the concept of 'local' Rayleigh ( Ra1 ) and Nusselt ( Nu1 ) numbers  相似文献   

4.
A 2-D time-dependent finite-difference numerical model is used to investigate the thermal character and evolution of a convecting layer which is cooling as it convects. Two basic cooling modes are considered: in the first, both upper and lower boundaries are cooled at the same rate, while maintaining the same temperature difference across the layer; in the second, the lower boundary temperature decreases with time while the upper boundary temperature is fixed at 0°C. The first cooling mode simulates the effects of internal heating while the second simulates planetary cooling as mantle convection extracts heat from, and thereby cools, the Earth's core. The mathematical analogue between the effects of cooling and internal heating is verified for finite-amplitude convection. It is found that after an initial transient period the central core of a steady but vigorous convection cell cools at a constant rate which is governed by the rate of cooling of the boundaries and the viscosity structure of the layer. For upper-mantle models the transient stage lasts for about 30 per cent of the age of the Earth, while for the whole mantle it lasts for longer than the age of the Earth. Consequently, in our models the bulk cooling of the mantle lags behind the cooling of the core-mantle boundary. Models with temperature-dependent viscosity are found to cool in the same manner as models with depth-dependent viscosity; the rate of cooling is controlled primarily by the horizontally averaged variation of viscosity with depth. If the Earth's mantle cools in a similar fashion, secular cooling of the planet may be insensitive to lateral variations of viscosity.  相似文献   

5.
We present a new numerical method to describe the internal dynamics of planetary mantles through the coupling of a dynamic model with the prediction of geoid and surface topography. Our tool is based on the simulation of thermal convection with variable viscosity in a spherical shell with a finite-volume formulation. The grid mesh is based on the 'cubed sphere' technique that divides the shell into six identical blocks. An investigation of various numerical advection schemes is proposed: we opted for a high-resolution, flux-limiter method. Benchmarks of thermal convection are then presented on steady-state tetrahedral and cubic solutions and time-dependent cases with a good agreement with the few recent programs developed to solve this problem.
A dimensionless framework is proposed for the calculation of geoid and topography introducing two dimensionless numbers: such a formulation provides a good basis for the systematic study of the geoid and surface dynamic topography associated to the convection calculations. The evaluation of geoid and surface dynamic topography from the gridded data is performed in the spectral domain. The flow solver is then tested extensively against a precise spectral program, producing response functions for geoid as well as bottom and surface topographies. For a grid mesh of a reasonable size (6 × 64 × 64 × 64) a very good agreement (to within ∼1 per cent) is found up to spherical harmonic degree 15.  相似文献   

6.
Summary. A finite element method with uniform and variable resolution meshes is used to model very high Rayleigh number Ra thermal convection in a square box of infinite Prandtl number, Boussinesq fluid with constant viscosity and thermodynamic properties. Heating is either entirely from below or mostly from within and the boundaries are stress free. The variable mesh is coarse in the interior of the convection cell and it is fine in the very thin boundary layers and plumes surrounding the core. The highest resolution variable mesh has a dimensionless grid spacing of 0.027 in the core and 0.0017 in the boundary layers. The boundary layers contain about 10 mesh points even at the highest values of Ra considered and are thus highly resolved. The variable mesh approach is shown to yield reliable simulations of convection as long as the aspect ratio of the most elongated boundary layer elements is not too large; values of about 4 to 6 work well. This aspect ratio also measures the increase in resolution in the boundary layers as compared with the central core. Steady single-cell rolls are computed for bottom heating and Ra up to 5 × 105 times the marginal instability value of the Rayleigh number Racr. One and two-cell roll solutions are calculated for f = 1, 0.8 and 0.6, where f is the fraction of the heat escaping through the top of the box that is generated internally. The values of Racr for f = 1, 0.8 and 0.6 are 1296, 1024 and 864, respectively. The largest of Ra/Racr at which unicellular convection is stable (steady) are approximately 390, 610 and 970, for f = 1, 0.8 and 0.6.  相似文献   

7.
Summary. The ascent of a hot spherical body through a fluid with a strongly temperature-dependent viscosity has been studied using an axisymmetric finite element method. Numerical solutions range over Peclet numbers of 10−1– 103 from constant viscosity up to viscosity variations of 105. Both rigid and stress-free boundary conditions were applied at the surface of the sphere. The dependence of drag on viscosity variation was shown to have no dependence on the stress boundary condition except for a Stokes flow scaling factor. A Nusselt number parameterization based on the stress-free constant viscosity functional dependence on the Peclet number scaled by a parameter depending on the viscosity structure fits both stress-free and rigid boundary condition data above viscosity variations of 100. The temperature scale height was determined as a function of sphere radius. For the simple physical model studied in this paper pre-heating is required to reduce the ambient viscosity of the country rock to less than 1022 cm2 s−1 in order for a 10 km diapir to penetrate a distance of several radii.  相似文献   

8.
Summary Many accept thermal convection within the mantle of the Earth as the driving mechanism for continental drift. It is also of considerable interest to determine whether thermal convection is occurring within Venus, Mars, and the Moon. In this paper a systematic treatment of the stability of planetary interiors is given. The thermal stability problem for a layer of fluid heated from below is solved when the viscosity of the fluid increases exponentially with depth. For a semi-infinite fluid with exponentially increasing viscosity the critical Rayleigh number based on the surface viscosity and the scale length of the viscosity increase is found to be 30 for a fixed surface boundary condition and 23 for a free surface boundary condition. This stability analysis is also extended to include volume heat release. The thermal stability of the interiors of the Earth, Venus, Mars and the Moon is examined. Using temperature-depth profiles in the literature and a theoretical expression for the viscosity of a crystalline solid based on diffusion creep, viscosity depth profiles for planetary interiors are obtained. Because of the strong pressure effect the viscosity within the Earth and Venus increases greatly from a near surface minimum. For Mars the increase is less pronounced and within the interior of the Moon the viscosity is nearly constant. For all the cases considered the planetary interiors are found to be thermally unstable. Because of the dependence of viscosity on depth the interiors of the Moon and Mars are found to be considerably less stable than Venus and the Earth. It is concluded that thermal convection is occurring within the planetary bodies considered.  相似文献   

9.
We consider the two coupled differential equations of the two radial functions appearing in the displacement components of spheroidal oscillations for a transversely isotropic (TI) medium in spherical coordinates. Elements of the layer matrix have been explicitly written—perhaps for the first time—to extend the use of the Thomson-Haskell matrix method to the derivation of the dispersion function of Rayleigh waves in a transversely isotropic spherical layered earth. Furthermore, an earth-flattening transformation (EFT) is found and effectively used for spheroidal oscillations. The exponential function solutions obtained for each layer give the dispersion function for TI spherical media the same form as that on a flat earth. This has been achieved by assuming that the five elastic parameters involved vary as r p and that the density varies as r p-2, where p is an arbitrary constant and r is the radial distance. A numerical illustration with p = - 2 shows that, in spite of the inhomogeneity assumed within layers, the results for spherical harmonic degree n , versus time period T , obtained here for the Primary Reference Earth Model (PREM), agree well with those obtained earlier by other authors using numerical integration or variational methods. The results for isotropic media derived here are also in agreement with previous results. The effect of transverse isotropy on phase velocity for the first two modes of Rayleigh waves in the period range 20 to 240 s is calculated and discussed for continental and oceanic models.  相似文献   

10.
The state of stress within a bending spherical shell has some special features that are caused by sphericity. While most lithospheres are more like spherical shells than flat plates, our ideas of the state of stress have been dominated by flat-plate models. As a consequence, we might be missing some important aspects of the state of stress within subducting lithospheres. In order to examine this problem, we analyse spherical-shell bending problems from basic equations. We present two approaches to solve spherical-shell bending problems: one by the variational approach, which is suitable for global-scale problems, and the other by the asymptotic equation, which is valid to first order in h/R , where h is the thickness of the lithosphere and R is its curvature radius (i.e. under the assumption of small curvature). The form of the equation for displacement shows that wavelengths of deformation are determined by the spherical (elastic) effect and the gravitational buoyancy effect, for which only the latter effect is included in the usual flat-plate formulations. In the case of the Earth, the buoyancy force is dominant and, consequently, spherical effects are suppressed to a large extent; this explains why flat-plate models have been successful for Earth's lithospheric problems. On the other hand, the state of stress shows interesting spherical effects: while bending (fibre) stress along the subduction zone is always important, bending stress along the trench-strike direction can also be important, in particular when the subduction zone arc is small. Numerical results also indicate that compressive normal stress along the trench-strike direction is important when a subduction zone arc is large. These two stresses, the bending stress and the compressive normal stress, both along the trench-strike direction, may have important implications for intraplate earthquakes at subduction zones.  相似文献   

11.
A new class of stratified viscoelastic models by analytical techniques   总被引：2，自引：0，他引：2  
Multilayer, spherically stratified, self-gravitating relaxation models with a large number of layers (more than 100) can be dealt with analytically. Relaxation processes are studied for both Heaviside surface loads and tidal forcings. Simulations of the relaxation process of a realistic earth model with an incompressible Maxwell rheology show that models containing about 30 to 40 layers have reached continuum limits on all timescales and for all harmonic degrees up to at least 150 whenever an elastic lithosphere is present, irrespective of the viscosity profile in the mantle. In particular, fine-graded stratification of the shallow layers proves to be important for high harmonic degrees in these models. The models produce correct long-time (fluid) limits. It is shown that differences in the transient behaviour of the various models are due to the applied volume-averaging procedure of the rheological parameters. Our earlier proposed hypothesis that purported shortcomings in the fundamental physics of (discrete) normal-mode theory are artificial consequences of numerical inaccuracies, theoretical misinterpretations and the use of incomplete sets of normal modes is reinforced by the results presented. We show explicitly that the models produce both continuous behaviour resulting from continuous rheological stratifications and discrete behaviour resulting from sharp density contrasts, as at the outer surface and the core-mantle boundary. The differences between volume-averaged models and fixed-boundary contrast models are outlined. Reducing many-layer models with a volume-averaging procedure before employing a normal mode analysis is both economical and highly accurate on all timescales and for all spherical harmonic degrees. The procedure minimizes the chances of missing contributing modes, while using models with more layers will not result in any substantial increase of accuracy.  相似文献   

12.
Observations of ice movements across the British Isles and of sea-level changes around the shorelines during Late Devensian time (after about 25 000 yr BP) have been used to establish a high spatial and temporal resolution model for the rebound of Great Britain and associated sea-level change. The sea-level observations include sites within the margins of the former ice sheet as well as observations outside the glaciated regions such that it has been possible to separate unknown earth model parameters from some ice-sheet model parameters in the inversion of the glacio-hydro-isostatic equations. The mantle viscosity profile is approximated by a number of radially symmetric layers representing the lithosphere, the upper mantle as two layers from the base of the lithosphere to the phase transition boundary at 400 km, the transition zone down to 670 km depth, and the lower mantle. No evidence is found to support a strong layering in viscosity above 670 km other than the high-viscosity lithospheric layer. Models with a low-viscosity zone in the upper mantle or models with a marked higher viscosity in the transition zone are less satisfactory than models in which the viscosity is constant from the base of the lithosphere to the 670 km boundary. In contrast, a marked increase in viscosity is required across this latter boundary. The optimum effective parameters for the mantle beneath Great Britain are: a lithospheric thickness of about 65 km, a mantle viscosity above 670 km of about (4-5) 1020 Pa s, and a viscosity below 670 km greater than 4 × 1021 Pa s.  相似文献   

13.
We present simulations of large-scale landscape evolution on tectonic time scales obtained from a new numerical model which allows for arbitrary spatial discretization. The new method makes use of efficient algorithms from the field of computational geometry to compute the set of natural neighbours of any irregular distribution of points in a plane. The natural neighbours are used to solve geomorphic equations that include erosion/deposition by channelled flow and diffusion. The algorithm has great geometrical flexibility, which makes it possible to solve problems involving complex boundaries, radially symmetrical uplift functions and horizontal tectonic transport across strike-slip faults. The algorithm is also ideally suited for problems which require large variations in spatial discretization and/or self-adaptive meshing. We present a number of examples to illustrate the power of the new approach and its advantages over more 'classical' models based on regular (rectangular) discretization. We also demonstrate that the synthetic river networks and landscapes generated by the model obey the laws of network composition and have scaling properties similar to those of natural landscapes. Finally we explain how orographically controlled precipitation and flexural isostasy may be easily incorporated in the model without sacrificing efficiency.
  相似文献   

14.
The C -response connects the magnetic vertical component and the horizontal gradient of the horizontal components of electromagnetic variations and forms the basis for deriving the conductivitydepth profile of the Earth. Time-series of daily mean values at 42 observatories typically with 50 years of data are used to estimate C -responses for periods between 1 month and 1  yr. The Z : Y method is applied, which means that the vertical component is taken locally whereas the horizontal components are used globally by expansion in a series of spherical harmonics.
In combination with results from previous analyses, the method yields consistent results for European observatories in the entire period range from a few hours to 1  yr, corresponding to penetration depths between 300 and 1800  km.
1-D conductivity models derived from these results show an increase in conductivity with depth z to about 2  S  m-1 at z =800  km, and almost constant conductivity between z =800 and z =2000  km with values of 310  S  m-1, in good agreement with laboratory measurements of mantle material. Below 2000  km the conductivity is poorly resolved. However, the best-fitting models indicate a further increase in conductivity to values between 50 and 150  S  m-1.  相似文献   

15.
Summary. Amplitude spectra of Rayleigh and Love waves in a layered non-gravitating spherical earth have been obtained using as a source, displacement and stress discontinuities. In each layer elastic parameters and density follow specified functions of radial distance and the solutions of the equations of motion are obtained in terms of exponential functions. The Thomson—Haskell method is extended to this case. The problem reduces to simple calculations as in a plane-layered medium. Numerical results of phase and group velocities up to periods of 300 s in various earth models when compared with earlier results (obtained by numerical integration) show that the present method can be used with sufficient accuracy. The differences in phase velocity, group velocity and amplitude (also surface ellipticity in the case of Rayleigh waves) between spherical- and flat-earth models have been investigated in the range 20–300–s period and expressed in polynomials in the period.  相似文献   

16.
Summary. An approximate analytical solution for flow in a mantle plume of constant radius, viscosity, and density contrast is obtained in cylindrical coordinates. the differential equations for vertical velocity of the mantle surrounding the plume and for topography are homologous to the equation for flexure of an elastic plate. Although the model is too simple to be fully applicable to the Earth, one can conclude that the vertical velocity in the mantle changes significantly away from plumes, that the viscosity of the plume is important for controlling flow rate, and that the long-wavelength geoid anomalies are sensitive to the viscosity of the surrounding mantle. the first induced upwelling away from a plume is quite weak and unlikely to control the spacing of plumes.  相似文献   

17.
Summary. This paper describes the attainment of sphericity, or equivalent isopotential forms, for the surface of small Solar System objects (linear dimensions ∼ 300 km) as a consequence of viscous relaxation (creep).
The viscous relaxation time is shown to be directly proportional to the viscosity and proportional to the inverse square of the linear dimensions of the object. Since the effective viscosity varies very rapidly with temperature the detailed internal temperature distribution will in most cases significantly affect the rate of relaxation. If this temperature is sufficiently high during any phase of the object's history, virtually complete relaxation will have occurred.
Because the thermal relaxation time is usually less than the age of the objects considered, and because differentiation and convection are unlikely to occur, a simple model for the thermal regime within the object is easy to develop from which the viscosity is determined at various depths. It is shown that a body is more likely to relax if it is composed of a mixture of ice and silicate rock rather than either pure substance. The relaxed shape of Phoebe and the irregular shape of Hyperion may be accounted for by a lower radiogenic content in Hyperion, a hypothesis supported by albedo observations. The assumption that the object can be treated throughout as having a viscosity corresponding to the mean internal temperature is of course far from exact. It is shown that a plastic core may be formed which can eventually cause fracture of the outer harder shell and consequent sudden relaxation to an approximately spherical shape.  相似文献   

18.
Most previous earth models used to calculate viscoelastic relaxation after the removal of the Late Pleistocene ice loads implicitly assume that there is no exchange of mass across the mantle density discontinuities on periods of tens of thousands of years (the material boundary formulation). In the present study, simple incompressible models are used to determine the Earth's behaviour in the case where the density discontinuity remains at a constant pressure rather than deforming with the material (the isobaric boundary formulation). The calculation of the movement of the boundary is more rigorous than in earlier studies and uses the local incremental pressure calculated at the depth of the boundary and allows for the vertical deformation caused by the change in volume as material changes phase. It is shown that the buoyancy modes associated with the density discontinuities decrease in strength and increase in relaxation time analogous to what results when the density contrast is reduced. Also, two viscoelastic modes arise from an isobaric boundary, which is also predicted when there is a contrast in rigidity or viscosity across a material boundary. The difference in predicted radial deformation between the isobaric boundary model and the material boundary model is largest for long-wavelength loads for which the material incremental pressure at depth is largest. If the isobaric boundary model is appropriate for the treatment of the mineral phase changes in the mantle on glacial rebound timescales, then previous inferences of the deep-mantle to shallow-mantle viscosity ratio based on large-scale deformation (spherical harmonic degree < 10) of the Earth and including data from the early part of the glacio-isostatic uplift are too small.  相似文献   

19.
Summary. We determine the variation of effective viscosity η across the lower mantle from models of the Gibb's free energy of activation G * and the adiabatic temperature profile. The variation of G * with depth is calculated using both an elastic strain energy model, in which G * is related to the seismic velocities, and a model which assumes G * is proportional to the melting temperature. The melting temperature is assumed to follow Lindemann's equation. The adiabatic temperature profile is calculated from a model for the density dependence of the Grüneisen parameter. Estimates of η depend on whether the lower mantle is a Newtonian or power law fluid. In the latter case separate estimates of η are obtained for flow with constant stress, constant strain rate, and constant strain energy dissipation rate. For G * based on the melting temperature, increases in η with depth range from a factor of about 100 for Newtonian deformation or power-law flow with constant stress to about 5 for non-Newtonian deformation with constant strain rate. For G * based on elastic defect energy, increases in η with depth range from a factor of about 1500 for Newtonian deformation or power-law flow with constant stress to about 10 for non-Newtonian deformation with constant strain rate. Among these models, only a non-Newtonian lower mantle convecting with constant strain rate or constant strain energy dissipation rate is consistent with recent estimates of mantle viscosity from post-glacial rebound and true polar wander data.  相似文献   

20.
A time-varying spherical harmonic model of the palaeomagnetic field for 0–7 ka is used to investigate large-scale global geomagnetic secular variation on centennial to millennial scales. We study dipole moment evolution over the past 7 kyr, and estimate its rate of change using the Gauss coefficients of degree 1 (dipole coefficients) from the CALS7K.2 field model and by two alternative methods that confirm the robustness of the predicted variations. All methods show substantial dipole moment variation on timescales ranging from centennial to millennial. The dipole moment from CALS7K.2 has the best resolution and is able to resolve the general decrease in dipole moment seen in historical observations since about 1830. The currently observed rate of dipole decay is underestimated by CALS7K.2, but is still not extraordinarily strong in comparison to the rates of change shown by the model over the whole 7 kyr interval. Truly continuous phases of dipole decrease or increase are decadal to centennial in length rather than longer-term features. The general large-scale secular variation shows substantial changes in power in higher spherical harmonic degrees on similar timescales to the dipole. Comparisons are made between statistical variations calculated directly from CALS7K.2 and longer-term palaeosecular variation models: CALS7K.2 has lower overall variance in the dipole and quadrupole terms, but exhibits an imbalance between dispersion in   g 12  and   h 12  , suggestive of long-term non-zonal structure in the secular variations.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号