Three-dimensional Spherical Shell Research Articles

Internal shear layers in librating spherical shells: the case of attractors

Following our previous work on periodic ray paths (He et al., J. Fluid Mech., vol. 939, 2022, A3), we study asymptotically and numerically the structure of internal shear layers for very small Ekman numbers in a three-dimensional spherical shell and in a two-dimensional cylindrical annulus when the rays converge towards an attractor. We first show that the asymptotic solution obtained by propagating the self-similar solution generated at the critical latitude on the librating inner core describes the main features of the numerical solution. The internal shear layer structure and the scaling for its width and velocity amplitude in $E^{1/3}$ and $E^{1/12}$ , respectively, are recovered. The amplitude of the asymptotic solution is shown to decrease to $E^{1/6}$ when it reaches the attractor, as is also observed numerically. However, some discrepancies are observed close to the particular attractors along which the phase of the wave beam remains constant. Another asymptotic solution close to those attractors is then constructed using the model of Ogilvie (J. Fluid Mech., vol. 543, 2005, pp. 19–44). The solution obtained for the velocity has an $O(E^{1/6})$ amplitude, but a self-similar structure different from the critical-latitude solution. It also depends on the Ekman pumping at the contact points of the attractor with the boundaries. We demonstrate that it reproduces correctly the numerical solution. Surprisingly, the numerical solution close to an attractor with phase shift (that is, an attractor touching the axis in three or two dimensions with a symmetric forcing) is also found to be $O(E^{1/6})$ , but its amplitude is much weaker. However, its asymptotic structure remains a mystery.

Linear analysis on the onset of thermal convection of highly compressible fluids with variable viscosity and thermal conductivity in spherical geometry: implications for the mantle convection of super-Earths

In this paper, we carried out a series of linear analyses on the onset of thermal convection of highly compressible fluids whose physical properties strongly vary in space in convecting vessels either of a three-dimensional spherical shell or a two-dimensional spherical annulus geometry. The variations in thermodynamic properties (thermal expansivity and reference density) with depth are taken to be relevant for the super-Earths with ten times the Earth’s mass, while the thermal conductivity and viscosity are assumed to exponentially depend on depth and temperature, respectively. Our analysis showed that, for the cases with strong temperature dependence in viscosity and strong depth dependence in thermal conductivity, the critical Rayleigh number is on the order of 108–109, implying that the mantle convection of massive super-Earths is most likely to fall in the stagnant-lid regime very close to the critical condition, if the properties of their mantle materials are quite similar to the Earth’s. Our analysis also demonstrated that the structures of incipient flows of stagnant-lid convection in the presence of strong adiabatic compression are significantly affected by the depth dependence in thermal conductivity and the geometries of convecting vessels, through the changes in the static stability of thermal stratification of the reference state. When the increase in thermal conductivity with depth is sufficiently large, the thermal stratification can be greatly stabilized at depth, further inducing regions of insignificant fluid motions above the bottom hot boundaries in addition to the stagnant lids along the top cold surfaces. We can therefore speculate that the stagnant-lid convection in the mantles of massive super-Earths is accompanied by another motionless regions at the base of the mantles if the thermal conductivity strongly increases with depth (or pressure), even though their occurrence is hindered by the effects the spherical geometries of convecting vessels.

Radial symmetry on three-dimensional shells in the Landau–de Gennes theory

We study the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau–de Gennes theory for nematic liquid crystals. We prove that the radial-hedgehog solution is the unique minimizer of the Landau–de Gennes energy in two separate regimes: (i) for thin shells when the temperature is below the critical nematic supercooling temperature and (ii) for a fixed shell width at sufficiently low temperatures. In case (i), we provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution.

Numerical studies of the effects of phase transitions on Venusian mantle convection

This paper presents a study on the effects of phase transitions on the mantle convection of Venus in a three-dimensional (3D) spherical shell domain. Our model includes strong depth- and temperature-dependent viscosity and exothermic phase change from olivine to spinel as well as endothermic phase change from spinel to perovskite. From extensive numerical simulations of the effects of Rayleigh number (Ra), and the Clapeyron slopes and depths of phase changes, we found the following: (1) The endothermic phase change prevents mass flow through the interface. Increasing the absolute value of the Clapeyron slopes decreases radial mass flux and normalized radial mass flux at the endothermic phase boundary, and decreases the number of mantle plumes. In other words, mass flow through the phase boundary decreases. The inhibition influence of phase changes increases, as do convective wavelengths. (2) Increasing Ra also increases the convective wavelength and decreases the number of mantle plumes, but it has less influence on the mass exchange. As Ra increases, the convective vigor increases along with the radial mass flux and the mass flow through the phase boundary; however, the normalized mass flux through the phase boundary varies little with Ra, which is different from the conclusion that increasing Ra will greatly increase the inhibition of mass flow through the phase boundary based on two-dimensional (2D) modeling. (3) Increasing the depth of endothermic phase change will slightly decrease the number of mantle plumes, but has little effect on the mass flow through the phase boundary. Consistent with previous studies, our results show that the phase change from spinel to perovskite could inhibit the mass flow through the phase boundary, but they also show that the buildup of hot materials under the endothermic phase boundary in the 3D model could not be so large as to cause strong episodic overturns of mantle materials, which is quite different from previous 2D studies. Our results suggest that it is difficult for phase changes to cause significant magmatism on Venus; in other words, phase changes may not be the primary cause of catastrophic resurfacing on Venus.

A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of...

Non-Linear Dynamic Characteristics of Single-Layer Shallow Reticulated Spherical Shells

The nonlinear dynamical equations are established by using the method of quasi-shells for three-dimensional shallow spherical shells with circular bottom. Displacement mode that meets the boundary conditions of fixed edges is given by using the method of the separate variable, A nonlinear forced vibration equation containing the second and the third order is derived by using the method of Galerkin. The stability of the equilibrium point is studied by using the Floquet exponent.

Constraints of the topography, gravity and volcanism on Venusian mantle dynamics and generation of plate tectonics

Venus's mantle convection model was studied in a three-dimensional spherical shell domain with depth- and temperature-dependent viscosity. Numerical results show that key observations of Venus including the number of major “hotspot” volcanic systems, spectral patterns of the surface topography and geoid at long- and intermediate-wavelengths can be explained in models that have a spinel-to-post-spinel endothermic phase change of −3.5MPa/K Clapeyron slope and averaged mantle viscosity of 2×1021Pas (i.e., convective Rayleigh number of 1.8×107). Our models with the endothermic phase change show relatively weak time-dependence, suggesting that the phase change may not be the primary cause for “catastrophic” resurfacing on Venus. Our calculations also show that Venus cannot have a weak asthenosphere that is similar to that on the Earth, in order to match the observations, thus supporting a key role of asthenosphere in producing plate tectonics.

A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test case we consider the evolution of a single black hole in numerical general relativity. A natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates. Instead, we consider the simpler and potentially more efficient possibility of a double Fourier expansion on the sphere for tensors in Cartesian coordinates. As usual for the double Fourier method, we employ a filter to address time-step limitations and certain stability issues. We find that a tensor filter based on spin-weighted spherical harmonics is successful, while two simplified, non-spin-weighted filters do not lead to stable evolutions. The derivatives and the filter are implemented by matrix multiplication for efficiency. A key technical point is the construction of a matrix multiplication method for the spin-weighted spherical harmonic filter. As example for the efficient parallelization of the double Fourier, spin-weighted filter method we discuss an implementation on a GPU, which achieves a speed-up of up to a factor of 20 compared to a single core CPU implementation.

Three-Dimensional Spherical Shell Resonator Gyroscope Fabricated Using Wafer-Scale Glassblowing

We report, for the first time, experimental demonstration of a 3-D spherical shell resonator microelectromechanical systems gyroscope fabricated using wafer-scale glassblowing. The gyroscope utilizes a 1-mm-diameter 10-μm-thick spherical isotropic shell surrounded by eight cofabricated satellite spheres serving as actuation and detection electrodes. To demonstrate functionality of the new device, a four-node wineglass mode at a 945-kHz frequency was utilized as the drive mode of a Coriolis vibratory gyroscope. The input rotation causes the transfer of energy from the drive mode to the sense mode, which is a complementary four-node wineglass mode oriented at a 45 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> angle. Sense-mode vibrations were capacitively detected using cofabricated 3-D metal electrodes. Experimental characterization of the spherical shell gyroscope demonstrated a wide linear range of 1000 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> /s, currently limited only by the experimental setup.

Non-Linear Dynamic Stability of Shallow Reticulated Spherical Shells

The nonlinear dynamical equations are established by using the method of quasi-shells for three-dimensional shallow spherical shells with circular bottom. Displacement mode that meets the boundary conditions of fixed edges is given by using the method of the separate variable, A nonlinear forced vibration equation containing the second and the third order is derived by using the method of Galerkin. The stability of the equilibrium point is studied by using the Floquet exponent.

Construction of semi-dynamic model of subduction zone with given plate kinematics in 3D sphere

We present a semi-dynamic subduction zone model in a three-dimensional spherical shell. In this model, velocity is imposed on the top surface and in a small three-dimensional region around the shallow plate boundary while below this region, the slab is able to subduct under its own weight. Surface plate velocities are given by Euler’s theorem of rigid plate rotation on a sphere. The velocity imposed in the region around the plate boundary is determined so that mass conservation inside the region is satisfied. A kinematic trench migration can be easily incorporated in this model. As an application of this model, mantle flow around slab edges is considered, and we find that the effect of Earth curvature is small by comparing our model with a similar one in a rectangular box, at least for the parameters used in this study. As a second application of the model, mantle flow around a plate junction is studied, and we find the existence of mantle return flow perpendicular to the plate boundary. Since this model can naturally incorporate the spherical geometry and plate movement on the sphere, it is useful for studying a specific subduction zone where the plate kinematics is well constrained.

Duration and extent of lunar volcanism: Comparison of 3D convection models to mare basalt ages

It is widely accepted that lunar volcanism started before the emplacement of the mare fills ( ≈ 3.1 – 3.9 Ga b.p.) and lasted for probably more than 3.0 Ga. While the early volcanic activity is relatively easy to understand from a thermal point of view, the late stages of volcanism are harder to explain, because a relatively small body like the Earth's Moon is expected to cool rapidly and any molten layer in the interior should solidify rather quickly. We present several thermal evolution models, in which we varied the boundary conditions at the model surface in order to evaluate the influence on the extent and lifetime of a molten layer in the lunar interior. To investigate the influence of a top insulating layer we used a fully three-dimensional spherical shell convection code for the modelling of the lunar thermal history. In all our models, a partial melt zone formed nearly immediately after the simulation started (early in lunar history), consistent with the identification of lunar cryptomare and early mare basalt volcanism on the Moon. Due to the characteristic thickening of the Moon's lithosphere the melt zone solidified from above. This suggests that the source regions of volcanic rock material proceeded to increasing depth with time. The rapid growth of a massive lithosphere kept the Moon's interior warm and prevented the melt zone from fast freezing. The lifetimes of the melt zones derived from our models are consistent with basalt ages obtained from crater chronology. We conclude that an insulating megaregolith layer is sufficient to prevent the interior from fast cooling, allowing for the thermal regime necessary for the production and eruption of young lava flows in Oceanus Procellarum.

Effects on the long-wavelength geoid anomaly of lateral viscosity variations caused by stiff subducting slabs, weak plate margins and lower mantle rheology

Instantaneous flow numerical calculations in a three-dimensional spherical shell are employed to investigate the effects of lateral viscosity variations (LVVs) in the lithosphere and mantle on the long-wavelength geoid anomaly. The density anomaly model employed is a combination of seismic tomography and subducting slab models based on seismicity. The global strain-rate model is used to represent weak (low-viscosity) plate margins in the lithosphere. LVVs in the mantle are represented on the basis of the relation between seismic velocity and temperature (i.e., temperature-dependent rheology). When highly viscous slabs in the upper mantle are considered, the observed positive geoid anomaly over subduction zones can be accounted for only when the viscosity contrast between the reference upper mantle and the lower mantle is approximately 1 0 3 or lower, and weak plate margins are imposed on the lithosphere. LVVs in the lower mantle exert a large influence on the geoid pattern. The calculated geoid anomalies over subduction zones exhibit generally positive patterns with quite high amplitudes compared with observations, even when the low activation enthalpy of perovskite in the lower mantle is employed. Inferred weak slabs in the lower mantle may be explained in terms of recent mineral physics results, highlighting the possibility of grain-size reduction due to the postspinel phase transition.

Modelling compressible mantle convection with large viscosity contrasts in a three-dimensional spherical shell using the yin-yang grid

Here it is documented how an existing code for modelling mantle convection in a cartesian domain, Stag3D, has been converted to model a 3D spherical shell by using the recently introduced yin-yang grid. StagYY is thus the latest evolution of a code that has been in continuous use and development for about 15 years so incorporates much physics and several features including compressibility, phase transitions, compositional variations, non-linear rheology, parallelisation, tracers to track composition, partial melting and melt migration, and the ability to also model spherical patches, cartesian boxes, and various 2D geometries by changing one input switch. StagYY uses a multigrid solver to obtain a velocity–pressure solution at each timestep on a staggered grid, a finite-volume scheme for advection of temperature and tracers to track composition. Convergence of multigrid solvers in the presence of realistically large viscosity variations has always been a problem; here a new pressure interpolation scheme is presented that can dramatically improve the robustness of the iterations to large viscosity variations, with up to 19 orders of magnitude variation in presented tests. Benchmark tests show that StagYY produces results that are consistent with those produced by other codes. Performance tests show reasonable scaling on a parallel Beowulf cluster up to 64 CPUs, with up to 1.2 billion unknowns solved for in a few minutes. StagYY is designed to be a stand-alone application with no libraries required and if MPI is installed it can be run in parallel. Technical issues and goals for the future are discussed.

Modeling mantle convection in the spherical annulus

Most methods for modeling mantle convection in a two-dimensional (2D) circular annular domain suffer from innate shortcomings in their ability to capture several characteristics of the spherical shell geometry of planetary mantles. While methods such as rescaling the inner and outer radius to reduce anomalous effects in a 2D polar cylindrical coordinate system have been introduced and widely implemented, such fixes may have other drawbacks that adversely affect the outcome of some kinds of mantle convection studies. Here we propose a new approach that we term the “spherical annulus,” which is a 2D slice that bisects the spherical shell and is quantitatively formulated at the equator of a spherical polar coordinate system after neglecting terms in the governing equations related to variations in latitude. Spherical scaling is retained in this approximation since the Jacobian function remains proportional to the square of the radius. We present example calculations to show that the behavior of convection in the spherical annulus compares favorably against calculations performed in other 2D annular domains when measured relative to those in a fully three-dimensional (3D) spherical shell.

Multigrid-based simulation code for mantle convection in spherical shell using Yin–Yang grid

A new simulation code of mantle convection in a three-dimensional spherical shell is presented. Major innovation of the code comes from an combination of two numerical techniques, namely Yin–Yang grid and ACuTE algorithm, which we had developed for large-scale simulations of solid earth sciences. Benchmark comparisons for the steady convection for low Rayleigh numbers ( Ra) with previous calculations revealed that accurate results are successfully reproduced not only for isoviscous cases but also for the cases where the mild temperature-dependence of viscosity is included. We also demonstrated that our code can reproduce the change in convective flow patterns into the “sluggish-lid” regime with increasing the viscosity variation r η up to 1 0 4 .

Lateral variations in CMB heat flux and deep mantle seismic velocity caused by a thermal–chemical-phase boundary layer in 3D spherical convection

Numerical simulations of thermo-chemical, multi-phase, compressible mantle convection in a three-dimensional spherical shell are used to investigate the relationship between lateral variations in seismic shear-wave velocity V s above the core–mantle boundary (CMB) and lateral variations in heat flux across the CMB ( q CMB), when compositional variations and the post-perovskite phase transition are included. For simple thermal convection, the V s– q CMB relationship is reasonably but not perfectly linear. The post-perovskite transition introduces a non-linearity that amplifies fast V s anomalies in cold regions, but there is still a unique mapping between δ V s and q CMB. Lateral variations in composition such as piles of dense material introduce another non-linearity that affects hot upwelling regions, and introduces a non-uniqueness in δ V s– q CMB if the dense material (e.g., MORB) is seismically fast compared to the surrounding material. In this case, dense piles are ringed by sharp, low- V s anomalies. If the CMB is covered by a global dense layer than variations in δ V s and q CMB are reduced but so is the mean value of q CMB. In all cases, the peak-to-peak lateral variation in q CMB is similar to or larger than twice the mean value, which might create problems for generating a dynamo according to existing numerical dynamo simulations. Analytical scalings are developed to explain the observed trends.

The Three‐dimensional Evolution of Buoyant Magnetic Flux Tubes in a Model Solar Convective Envelope

We present a set of three-dimensional spherical shell anelastic MHD simulations of the buoyant rise of magnetic flux tubes from the base of the convection zone to a depth of 16 Mm below the photosphere. It is found that when a twisted flux tube arches upward due to buoyancy, it rotates out of the plane and thus produces a tilt at the apex. Our simulations show that for tubes with the twist rate that is necessary for a cohesive rise, the twist-induced tilt dominates that caused by the Coriolis force, and furthermore, the twist-induced tilt is of the wrong direction (opposite to the observational Joy's law) if the twist is left-handed (right-handed) in the northern (southern) hemisphere, following the observed hemispheric preference of the sign of the active region twist. It is found that in order for the emerging tube to show the correct tilt direction (consistent with observations), the initial twist rate of the flux tube needs to be less than half of that needed for a cohesive rise. Under such conditions, severe flux loss is found during the rise. We also found that due to the asymmetric stretching of the rising tube by the Coriolis force, a field strength asymmetry develops, with the field in the leading leg (leading in the direction of rotation) of the Ω-shaped emerging tube being stronger than the field in the following leg, which results in a more compact morphology in the leading polarity of the emerging active region.

Supercontinent cycles, true polar wander, and very long-wavelength mantle convection

We show in this paper that mobile-lid mantle convection in a three-dimensional spherical shell with observationally constrained mantle viscosity structure, and realistic convective vigor and internal heating rate is characterized by either a spherical harmonic degree-1 planform with a major upwelling in one hemisphere and a major downwelling in the other hemisphere when continents are absent, or a degree-2 planform with two antipodal major upwellings when a supercontinent is present. We propose that due to modulation of continents, these two modes of mantle convection alternate within the Earth's mantle, causing the cyclic processes of assembly and breakup of supercontinents including Rodinia and Pangea in the last 1 Ga. Our model suggests that the largely degree-2 structure for the present-day mantle with the Africa and Pacific antipodal superplumes, is a natural consequence of this dynamic process of very long-wavelength mantle convection interacting with supercontinent Pangea. Our model explains the basic features of true polar wander (TPW) events for Rodinia and Pangea including their equatorial locations and large variability of TPW inferred from paleomagnetic studies. Our model also suggests that TPW is expected to be more variable and large during supercontinent assembly, but small after a supercontinent acquires its equatorial location and during its subsequent dispersal.

Assembly of Nanoparticle–Protein Binding Complexes: From Monomers to Ordered Arrays

Tag it: Well-defined monomers, dimers, trimers, three-dimensional spherical shells, and two-dimensional ordered arrays of genetically engineered proteins were constructed by using functionalized gold nanoparticles. Nanoparticle size, functionality, and protein genetic motif contribute to the unique geometry, precise stoichiometry, given orientation, and high specificity of these nanoparticle–protein hybrid structures.