Viscous Mantle Research Articles

Sinker tectonics: An approach to the surface of Miranda

Images of the Uranian moon Miranda show three areas of complex terrain, trapezoidal to ovoid in shape, with no previously known counterpart. These features, called coronae, are characterized by an inner core of intersecting ridges and troughs surrounded by a series of subparallel and concentric bands. Two of the proposed explanations for these tectonic features involve mantle convection driven by density anomalies. The present study determines the surface stresses induced by such density anomalies and their expected surface expression. Miranda is modeled as having a rigid core surrounded by a mantle capable of viscous flow which is overlain by a thin, elastic lithosphere. The density anomaly is modeled as a point mass representing the sinker or riser. The model is developed in a spherical coordinate system having axial symmetry about a θ=0 axis passing through the sinker. The flow field set up by the sinking (or rising) mass is expressed in terms of a stream function using Legendre sums. Streamlines are functions of the position of the mass anomaly and the radii of the core/mantle and mantle/lithosphere boundaries only. Magnitudes of the stream function depend upon the size of the mass anomaly, its local gravitational field, and the mantle/lithosphere radius. The stresses on the base of the lithosphere are then determined from the equations of motion. The viscosity of the mantle material has no effect on the stresses produced at any given time but does control the velocity of the anomaly. Expressions for displacements and stresses in the elastic shell are developed as Legendre sums and coupled to the viscous mantle solution with appropriate boundary conditions. The stresses derived yield a pattern of expected tectonic expression. This tectonic style varies with the radius of the body. For a small body, surface stresses are dominant in the area above the mass anomaly, while farther away stresses at the base of the lithosphere control the tectonics. Directly over a sinker and extending outward ∼15° is a zone of both latitudinal (θ) and longitudinal (Φ) surface compression which produces folding and thrust faults with no preferred orientation. Beyond 15°, longitudinal stresses at the base of the lithosphere are extensional, while latitudinal stresses are compressional, producing a zone of folding and thrusts concentric about the sinker. Stresses induced by a riser are opposite in sign and predict extensional features with the same orientation as the compressional features predicted for a sinker. Stress magnitudes generally decrease with distance from the sinker and also decrease with both increasing sinker depth and lithosphere thickness. The predictions of the sinker model are in good agreement with the tectonic styles of the banded and ridged terrains of Miranda as determined from the topography as imaged by Voyager 2. This agreement supports the idea that the coronae on Miranda were produced by dense masses, originally located near the surface, that sank into Miranda's interior. For larger bodies, the stresses at the surface and base are nearly equal. Longitudinal stresses are greater than latitudinal stresses resulting in compressional features with a preferred orientation radial to the sinker location. The changeover in tectonic styles occurs at a planetary radius of approximately 1500 km.

Small-scale mantle flows and induced lithospheric stress near island arcs

This paper explores the middle ground between complex thermally-coupled viscous flow models and simple corner flow models of island arc environments. The calculation retains the density-driven nature of convection and relaxes the geometrical constraints of corner flow, yet still provides semianalytical solutions for velocity and stress. A novel aspect of the procedure is its allowance for a coupled elastic lithosphere on top of a Newtonian viscous mantle. Initially, simple box-like density drivers illustrate how vertical and horizontal forces are transmitted through the mantle and how the lithosphere responds by trench formation. The flexural strength of the lithosphere spatially broadens the surface topography and gravity anomalies relative to the functional form of the vertical flow stresses applied to the plate base. I find that drivers in the form of inclined subducting slabs cannot induce self-driven parallel flow; however, the necessary flow can be provided by supplying a basal drag of 1–5 MPa to the mantle from the oceanic lithosphere. These basal drag forces create regional lithospheric stress and they should be quantifiable through seismic observations of the neutral surface. The existence of a shallow elevated phase transition is suggested in two slab models of 300 km length where a maximum excess density of 0.2 g cm−3 was needed to generate an acceptable mantle flow. A North New Hebrides subduction model which satisfies flow requirements and reproduces general features of topography and gravity contains a high shear stress zone (75 MPa) around the upper slab surface to a depth of 150 km and a deviatoric tensional stress in the back arc to a depth of 70 km. The lithospheric stress state of this model suggests that slab detachment is possible through whole plate fracture.

Thermochemical plumes and mantle phase transitions

Integral relations based on boundary layer theory are derived to study the motion of an isolated, two‐dimensional thermal plume through a viscous mantle containing polymorphic phase changes. Analytical results are obtained which show that phase transitions alter average mantle convective velocities by less than 50%. In particular we find that the olivine‐spinel transition, approximated as univariant, can enhance the circulation velocity of upper mantle convection by 30–40%, while it can enhance the overall amplitude of whole mantle convection by a few percent only. Our calculations demonstrate that a possible endothermic phase change located at 650 km will not prevent deep mantle convection unless the Clapeyron slope defining the transition exceeds −0.3 kbar/°K. This large value is more than one order of magnitude greater than what has been proposed for the 650‐km discontinuity. We then extend the method to include compositional buoyancy and effects of the divariant nature of the olivine‐spinel transition. Analysis of the motion of a compositionally buoyant plume (one having an anomalous Mg/Fe ratio relative to the ambient mantle) reveals that the chemical plume locally distorts the transition in a way which contributes buoyancy and enhances convective amplitudes by 10% or less. Finally, we combine thermal and compositional buoyancy to investigate the interaction between a thermochemical plume and a compositionally induced density interface. The results are used to determine the stability criterion for two‐layered mantle convection. We find that a compositionally produced density increase of 4% at 650 km is needed to prevent mixing between the upper and lower mantle. Seismological observations indicate that density jump at the 650‐km discontinuity is between 5% and 9%. If all the density increase observed there is due to change in composition, then two‐layered mantle convection is stable. However, if phase transitions account for part of the density increase, so that the chemically produced component is less than 4%, then two‐layered mantle convection may be unstable.

Thermal evolution of the earth

Models to study the thermal evolution of the earth have been developed based on a calculation of finite-amplitude thermal convection in an incompressible and viscous mantle with temperature and pressure-dependent physical parameters. The lateral dependences of the variables are resolved through their spherical harmonic representations, while the radial and time dependences are calculated using numerical procedures. The major emphasis of this paper is to illustrate the effects of different factors on the thermal evolution of the earth rather than to present a definite earth model. The main conclusions achieved in this study are: 1. (1) A significant portion of the present-day temperature in the mantle and surface heat flux is due to initial heat of the earth resulting from accretion and core formation. 2. (2) Since the effective Rayleigh number of the earth's mantle is high, solid-state convection is found to be sufficiently efficient to maintain an almost adiabatic temperature gradient within the convecting region, while large temperature gradients exist across the thermal boundary layers (core—mantle boundary) and below the lithosphere. 3. (3) The convection is oscillatory with avalanche-type properties. This is due to the slow development of instability in the thermal boundary layer near the surface which causes the episodic behavior. The periods of the oscillations are about 50–250 m.y. The oscillation in the mantle also induces periodic features in the surface heat flux and the thickness of the lithosphere. Mantle viscosity structure plays an important role in the cooling rate and nature of oscillations. 4. (4) The mobility of the lithosphere is an important factor controlling the thermal state of the earth's interior. A mobile lithosphere will cool the interior faster.

Viscous mantle flow under moving lithospheric plates and under subduction zones

Summary. Some simple solutions are presented for slow steady viscous flow in fluid layers or half-spaces with moving boundary segments. The results suggest the following conclusions concerning flow in the Earth's mantle. (a) Viscous flow entrained by the larger lithospheric plates would penetrate the whole mantle if its properties are fairly uniform. (b) Secondary circulation cells are generated under stationary plates by the flow entrained by an adjacent moving plate. The secondary cells are of opposite sense to the adjacent cells, and of very low amplitude when there is no heat input into the fluid. Heating of the fluid from within or below would amplify them. (c) Flow under a convergence boundary between a stationary and a moving plate tends to dip obliquely under the stationary plate. A maximum dip of the flow of about 30 degrees occurs vertically under the convergence boundary if the effect of the subducted plate is ignored. This suggests a straightforward explanation for the observed oblique dip of the Benioff seismic zones: the sinking plates are deflected from the vertical by the mantle flow entrained by (or associated with) the moving plates to which they are attached.

Late Quaternary glacio- and hydro-isostasy, on a layered earth

Isostatic response of the Earth to changes in Quaternary Times of ice and water loads is partly elastic, and partly involves viscous mantle flow. The relaxation spectrum of the Earth, critical for estimation of the mantle flow component, is estimated from published determinations of Fennoscandian and Laurentide rebound, and of the nontidal acceleration of the Earth's rotation. The spectrum is consistent with an asthenosphere viscosity around 10 21P, and a viscosity around 10 23P below 400 km depth. Calculation of relaxation effects is done by convoluting the load history with the response function in spherical harmonics for global effects, and in rectangular or cylindrical transforms for smaller regional effects. Broad-scale deformation of the globe, resulting from the last deglaciation and sea level rise, is calculated to have involved an average depression of ocean basins of about 8 m, and mean upward movement of continents of about 16 m, relative to the center of the Earth, in the last 7000 yr. Deflection in the ocean margin “hinge zone” varies with continental shelf geometry and rigidity of the underlying lithosphere: predictions are made for different model cases. The computational methods is checked by predicting Fennoscandian and Laurentide postglacial warping, from published estimates of icecap histories, with good results. The depth variations of shorelines formed around 17,000 BP (e.g., North America, 90–130 m; Australia, 130–170 m), are largely explainable in terms of combined elastic and relaxation isostasy. Differences between Holocene eustatic records from oceanic islands (Micronesia, Bermuda), and continental coasts (eastern North America, Australia), are largely but not entirely explained in the same terms.

The dependence of convection in planetary mantles upon the magnitude of the Rayleigh number

The conservation equations of mass, energy and momentum when applied to the problem of convection produced by isothermal compression as well as by thermal expansion in a self-gravitating sphere of uniform density, consisting of a core extending to a fractionŋ of the radius of the sphere and with a viscous mantle overlying the core, lead to a relationship between two characteristic numbers. In the present paper these characteristic numbers are evaluated, for the case when both boundaries of the mantle are free, for spherical harmonic disturbances of orders 1 → 6. It is found that the ease with which different patterns of convection can be excited changes as the characteristic numbers are varied.

Permissible modes of non-thermal convection in planetary mantles

Boundary conditions are imposed upon the solutions of the conservation equations for non-thermal convective motion in a self-gravitating, homogeneous, non-rotating sphere of radiusR, consisting of a core extending to a fraction η of the radius of the sphere and with a viscous mantle overlying the core. It is shown that convective modes are permissible in the mantle only for certain values ofn and η.

The combined effect of thermal and nonthermal convection in planetary mantles

In the present paper we consider a self-gravitating sphere, of radius R and of uniform density, consisting of a core extending to a fraction η of the radius of the sphere and with a viscous mantle overlying the core. In the mantle departures from the state of equilibrium have given rise to internal motions, which are time-independent and both thermal and nonthermal in origin. Solutions are found for the variation in density and radial component of velocity throughout the sphere, under various boundary conditions at both the surface of the sphere and the interface between the mantle and the core, which lead to the appearance of a certain relationship between two characteristic numbers.

Convection in planetary mantles

In the present paper we consider a self-gravitating sphere, of radius R and of uniform density, consisting of a core extending to a fraction η of the radius of the sphere and with a viscous mantle overlying the core. In the mantle departures from the state of equilibrium have given rise to internal motions, which are time-independent and nonthermal in origin. Solutions are found for the variations in density and radial component of velocity throughout the sphere, under various boundary conditions at both the surface of the sphere and the interface between the mantle and the core, which lead to the appearance of certain characteristic numbers.

XXV. The onset of convection by thermal instability in spherical shells

Summary In this paper the problem of the thermal instability of an incompressible sphere consisting of an inviscid core and a viscous mantle is considered, and it is shown that the pattern of convection which sets in, at marginal stability, in the mantle shifts to harmonics of the higher orders as the thickness of the mantle decreases. Thus, when the mantle extends to a depth of half the radius of the sphere, the harmonics of orders three and four set in about simultaneously, while the harmonic of order five follows very soon afterwards. The bearing of this result on the problem of convection in the earth's mantle and of the interpretation of the earth's topographic features is indicated.