Harmonic Degree Research Articles

Asteroseismic Signature of a Large Active Region

Axisymmetric magnetic activity on the Sun and Sun-like stars increases the frequencies of the modes of acoustic oscillation. However, it is unclear how a corotating patch of activity affects the oscillations, since such a perturbation is unsteady in the frame of the observer. In this paper we qualitatively describe the asteroseismic signature of a large active region in the power spectrum of the dipole and quadrupole p modes. In the corotating frame of the active region, the perturbations due to (differential) rotation and the active region completely lift the $(2\ell + 1)$-fold azimuthal degeneracy of the frequency spectrum of modes with harmonic degree $\ell$. In the frame of the observer, the unsteady nature of the perturbation leads to the appearance of $(2\ell+1)^2$ peaks in the power spectrum of a multiplet. These peaks blend into each other to form asymmetric line profiles. In the limit of a small active region, we approximate the power spectrum of a multiplet in terms of $2\times(2\ell+1)$ peaks, whose amplitudes and frequencies depend on the latitude of the active region and the inclination angle of the star's rotation axis. In order to check the results and to explore the nonlinear regime, we also perform numerical simulations using the 3D time-domain pseudo-spectral linear pulsation code GLASS.

Finite-frequency imaging of the global 410- and 660-km discontinuities using SS precursors

SUMMARY We report finite-frequency imaging of the global 410- and 660-km discontinuities using boundary sensitivity kernels for traveltime measurements made on SS precursors. The application of finite-frequency sensitivity kernels overcomes resolution limits in previous studies associated with large Fresnel zones of SS precursors and their interferences with other seismic phases. In this study, we calculate the finite-frequency sensitivities of SS waves and their precursors based on a single-scattering (Born) approximation in the framework of travelling-wave mode summation. The global discontinuity surface is parametrized using a set of triangular gridpoints with a lateral spacing of about 4°, and we solve the linear finite-frequency inverse problem (2-D tomography) based on singular value decomposition (SVD). The new global models start to show a number of features that were absent (or weak) in ray-theoretical back-projection models at spherical harmonic degree l > 6. The thickness of the mantle transition zone correlates well with wave speed perturbations at a global scale, suggesting dominantly thermal origins for the lateral variations in the mantle transition zone. However, an anticorrelation between the topography of the 410-km discontinuity and wave speed variations is not observed at a global scale. Overall, the mantle transition zone is about 2–3 km thicker beneath the continents than in oceanic regions. The new models of the 410- and 660-km discontinuities show better agreement with the finite-frequency study by Lawrence & Shearer than other global models obtained using SS precursors. However, significant discrepancies between the two models exist in the Pacific Ocean and major subduction zones at spherical harmonic degree >6. This indicates the importance of accounting for wave interactions in the calculations of sensitivity kernels as well as the use of finite-frequency sensitivities in data quality control.

On Some Average Properties of Convex Mosaics

In a convex mosaic in we denote the average number of vertices of a cell by and the average number of cells meeting at a node by Except for the d = 2 planar case, there is no known formula prohibiting points in any range of the plane (except for the unphysical strips). Nevertheless, in d = 3 dimensions if we plot the 28 points corresponding to convex uniform honeycombs, the 28 points corresponding to their duals and the 3 points corresponding to Poisson-Voronoi, Poisson-Delaunay and random hyperplane mosaics, then these points appear to accumulate on a narrow strip of the plane. To explore this phenomenon we introduce the harmonic degree of a d-dimensional mosaic. We show that the observed narrow strip on the plane corresponds to a narrow range of We prove that for every there exists a convex mosaic with harmonic degree and we conjecture that there exist no d-dimensional mosaic outside this range. We also show that the harmonic degree has deeper geometric interpretations. In particular, in case of Euclidean mosaics it is related to the average of the sum of vertex angles and their polars, and in case of 2 D mosaics, it is related to the average excess angle.

Tidal Response of Mars Constrained From Laboratory‐Based Viscoelastic Dissipation Models and Geophysical Data

AbstractWe employ laboratory‐based grain size‐ and temperature‐sensitive rheological models to describe the viscoelastic behavior and tidal response of terrestrial bodies with focus on Mars. We consider five rheological models: Maxwell, extended Burgers, Andrade, Sundberg‐Cooper, and a power law approximation. The question of which model provides the most appropriate description of dissipation in planetary bodies, remains an open issue. To examine this, we build crust and mantle models of Mars (density and elasticity) that are computed self‐consistently through phase equilibrium calculations as a function of pressure, temperature, and bulk composition, whereas core properties are based on an Fe‐S parameterization. We assess the compatibility of the viscoelastic models by inverting tidal response, mean density, and moment of inertia of Mars for its thermal, elastic, and attenuation structure. Our results show that although all viscoelastic models fit the data, (1) their predictions of the tidal response at other periods and harmonic degrees are distinct, implying that our approach can be used to distinguish between the various models from seismic and/or tidal observations (e.g., with InSight), and (2) Maxwell is only capable of fitting data for unrealistically low viscosities. All viscoelastic models converge upon similar interior structure models: large liquid cores (1,750–1,890 km in radius) that contain 17–20.5 wt% S and, consequently, no silicate perovskite‐dominated lower mantle. Finally, the methodology proposed here is generally formulated and applicable to other solar and extrasolar system bodies where the study of tidal dissipation presents an important means for determining interior structure.

A Global Two-scale Helicity Proxy from π-ambiguous Solar Magnetic Fields

Abstract If the α effect plays a role in the generation of the Sun’s magnetic field, the field should show evidence of magnetic helicity of opposite signs at large and small length scales. Measuring this faces two challenges: (i) in weak-field regions, horizontal field measurements are unreliable because of the π ambiguity, and (ii) one needs a truly global approach to computing helicity spectra in the case where one expects a sign reversal across the equator at all wavenumbers. Here we develop such a method using spin-2 spherical harmonics to decompose the linear polarization in terms of the parity-even and parity-odd E and B polarizations, respectively. Using simple one- and two-dimensional models, we show that the product of the spectral decompositions of E and B, taken at spherical harmonic degrees that are shifted by one, can act as a proxy of the global magnetic helicity with a sign that represents that in the northern hemisphere. We then apply this method to the analysis of solar synoptic vector magnetograms, from which we extract a pseudo-polarization corresponding to a “π-ambiguated” magnetic field, i.e., a magnetic field vector that has no arrow. We find a negative sign of the global EB helicity proxy at spherical harmonic degrees of around 6. This could indicate a positive magnetic helicity at large length scales, but the spectrum fails to capture clear evidence of the well-known negative magnetic helicity at smaller scales. This method might also be applicable to stellar and Galactic polarization data.

Unified Linear Stability Analysis for Thermal Convections in Spherical Shells Under Different Boundary Conditions and Heating Modes

AbstractLayered interior structure of most planets may be well approximated by spherical shells. An important subject for analyzing planet thermal structure and evolution is to determine the transitional condition from heat conduction dominated mechanism to thermal convection dominated mechanism inside a spherical shell, which can be accomplished through linear stability analysis. In this paper, we propose an analytical solution in terms of Frobenius series to the linear stability analysis that systematically unifies various boundary conditions, heating modes, and shell geometries characterized by inner‐out radius ratio. The analysis is eventually reduced to solving the critical Rayleigh number together with its corresponding spherical harmonic degree from a vanishing determinant of sixth order, and formulas for calculating the elements of the determinant are explicitly provided. The obtained high‐accuracy results clearly show that thermal instability depends strongly on the constraining strength posed by the boundary conditions, heating modes, and shell geometry. The critical Rayleigh number generally decreases with increasing internal heating fraction and inner‐out radius ratio, as well as with switches from fixed to free‐slip or from isothermal to constant heat flux in boundary conditions, whereas remarkable exceptional cases do exist. A thorough analysis is provided to fully explain the general trends and exceptional cases, which uncovers the complicated interplay between shell geometry, heating modes, and boundary conditions, and thus substantially improves our understanding of the underlying physics for thermal convections in spherical shells. Flow streamlines and perturbation temperature distributions for some typical cases of axisymmetric thermal convections are presented as well.

A point dislocation in a layered, transversely isotropic and self-gravitating Earth — Part II: accurate Green's functions

SUMMARYWe present an accurate approach for calculating the point-dislocation Green's functions (GFs) for a layered, spherical, transversely-isotropic and self-gravitating Earth. The formalism is based on the approach recently used to find analytical solutions for the dislocation Love numbers (DLNs). However, in order to make use of the DLNs, we first analyse their asymptotic behaviour, and then the behaviour of the GFs computed from the DLNs. We note that the summations used for different GF components evolve at different rates towards asymptotic convergence, requiring us to use two new and different truncation values for the harmonic degree (i.e. the index of summation). We exploit this knowledge to design a Kummer transformation that allows us to reduce the computation required to evaluate the GFs at the desired level of accuracy. Numerical examples are presented to clarify these issues and demonstrate the advantages of our approach. Even with the Kummer transformation, DLNs of high degree are still needed when the earth model contains very fine layers, so computational efficiency is important. The effect of anisotropy is assessed by comparing GFs for isotropic and transversely isotropic media. It is shown that this effect, though normally modest, can be significant in certain contexts, even in the far field.

Signature of solargmodes in first-orderp-mode frequency shifts

Context.Solar gravity modes (gmodes) are buoyancy waves that are trapped in the solar radiative zone and have been very difficult to detect at the surface. Solargmodes would complement solar pressure modes (pmodes) in probing the central regions of the Sun, for example the rotation rate of the core.Aims.A detection ofgmodes using changes in the large frequency separation ofpmodes has recently been reported. However, it is unclear howpandgmodes interact. The aim of this study is to evaluate to what extentgmodes can perturb the frequencies ofpmodes.Methods.We computed the first-order perturbation to globalp-mode frequencies due to a flow field and perturbations to solar structure (e.g. density and sound speed) caused by agmode. We focused on long-periodgmodes and assumed that theg-mode perturbations are constant in time. The surface amplitude ofgmodes is assumed to be 1 mm s−1, which is close to the observational limit set by Doppler observations.Results.Gravity modes do perturbp-mode frequencies to first order if the harmonic degree of thegmode is even and if its azimuthal order is zero. The effect is extremely small. For dipole and quadrupolepmodes, all frequency shifts are smaller than 0.1 nHz, or 2 × 10−8in relative numbers. This is because the relative perturbation to solar structure quantities caused by agmode of realistic amplitude is of the order of 10−6–10−5. Additionally, we find that structural changes dominate over advection. Surprisingly, the interaction ofgandpmodes takes place to a large part near the surface, wherepmodes spend most of their propagation times andgmodes generate the largest relative changes to solar structure. This is due to the steep density stratification, which compensates the evanescent behaviour ofgmodes in the convection zone.Conclusions.It appears to be impossible to detectgmodes solely through their signature inp-mode frequency shifts. Whethergmodes leave a detectable signature inp-mode travel times under a given observational setup remains an open question.

Polarization property of high harmonics generated from nitrogen molecule by bichromatic counter-rotating circularly polarized laser fields

The polarization property of high harmonics from N2 molecule is investigated in a bichromatic counter-rotating circularly polarized laser field by solving the two-dimensional time-dependent Schrödinger equation. The left- and right- circularly polarized harmonics are investigated and the results show that the helicity of the harmonic is reversed at particular harmonic orders. We also show that the electron ionized at one lobe of laser field will recombine with two different N nuclei at the same and the following laser lobes, which results in different polarization properties. The spatial distribution of the harmonic degree of circular polarization are presented to illustrate the effect on polarization properties of the electron recombined with different nuclei. The semi-classical analysis and the time-dependent probability density of electron wave packet are also demonstrated which provide a clear picture of the electron recombination process.

Cap integration in spectral gravity forward modelling up to the full gravity tensor

Cap-modified spectral gravity forward modelling is extended in this paper to the full gravity vector and tensor expressed in the local north-oriented reference frame. This is achieved by introducing three new groups of altitude-dependent Molodensky’s truncation coefficients. These are given by closed form and infinite spectral relations that are generalized for (i) an arbitrary harmonic degree, (ii) an arbitrary topography power, (iii) an arbitrary radial derivative, (iv) any radius larger than the radius of the reference sphere, and (v) for both near- and far-zone gravity effects. Thanks to the generalization for an arbitrary radial derivative, the cap-modified technique can efficiently be combined with the gradient approach for harmonic synthesis on irregular surfaces. In a numerical study, we exemplarily apply the new technique by forward modelling Earth’s degree-2159 topography up to degree 21,590, employing 30 topography powers. The experiment shows that near- and far-zone gravity effects can be synthesized on the topography with an accuracy (RMS) of 0.005–0.03 m $$^2$$ s $$^{-2}$$ (potential), 0.8–20 $$\upmu \mathrm {Gal}$$ (gravity vector), and 0.1 mE–1 E (gravity tensor). The numerical experiment also shows that the divergence effect of spherical harmonics comes into play around degree 10,795 when evaluating the series on the Earth’s surface. The difficult-to-compute truncation coefficients that are employed in the study are made freely available at http://edisk.cvt.stuba.sk/~xbuchab/ and are accompanied by MATLAB-based routines to evaluate them. Enclosed is also a MATLAB-based package to perform ultra-high-degree surface spherical harmonic analysis, a step of central importance in spectral gravity forward modelling techniques.

Core‐Mantle Boundary Dynamic Topography: Influence of Postperovskite Viscosity

AbstractCore‐mantle boundary (CMB) topography may provide useful hints on the deep mantle thermochemical structure, as clusters of thermal plumes and piles of chemically differentiated material, which are usually proposed as end‐member explanations for the large low shear‐wave velocity regions observed in the deep mantle, have different actions on this topography. CMB topography is further sensitive to several parameters, including mantle viscosity and its variations with thermal and compositional changes. Here we assess the influence of the postperovskite (pPv) phase viscosity on deep mantle dynamics and on CMB topography. We perform numerical simulations of thermal and thermochemical convection in spherical geometry, varying the ratio between pPv and bridgmanite viscosities, ΔηpPv, between 1 (regular pPv) and 10−3 (weak pPv). Thermochemical structures are dominated by smaller‐scale wavelengths (spherical harmonic degrees 3 to 6) and are more stable in weak than in regular pPv models. The amplitude of CMB topography is reduced by about a factor of 2 as ΔηpPv changes from 1 to 10−3, mostly due to a sharp drop in the depressions induced by downwellings reaching the CMB. By contrast, the topographies induced by plumes clusters and thermochemical piles are mostly unaffected. For all the values of ΔηpPv we tested, long‐wavelength CMB topography and reconstructed shear‐wave tomography are anticorrelated in purely thermal models, and correlated in thermochemical models with strong chemical density contrast (ΔρC = 140 kg/m3). In models with smaller density contrast (ΔρC = 90 kg/m3), topography and tomography are anticorrelated at ΔηpPv = 1, but correlated at ΔηpPv = 10−3.

Controls on Global Mantle Convective Structures and Their Comparison With Seismic Models

AbstractSeismic observations suggest (1) significant accumulation of subducted slabs above the 670‐km discontinuity in many subduction zones, (2) possible structure change at ~1,000‐km depth, and (3) the large low shear wave velocity provinces above the core‐mantle boundary in the African and Pacific lower mantle be associated with chemical heterogeneity. Global mantle convection models with realistic plate motion history reproduce most of these structures. However, it remains unclear how the convection models compare with seismic models at different spatial wavelengths and depths. By conducting quantitative analysis between mantle convection and seismic models, we found that mantle convective structures show significant correlations with seismic structures in the upper mantle and mantle transition zone for wavelengths up to spherical harmonic degree 20. However, the global correlation is weak at intermediate to short wavelengths (for degrees 4 and higher) in the lower mantle below ~1,000‐km depth. A weak layer beneath the spinel‐to‐postspinel phase change help consistently reproduce stagnant slabs in the western Pacific, while having insignificant effects elsewhere, that is, the large low shear wave velocity province structures. The cold slab structures and their correlations with the seismically fast anomalies are nearly identical for our convection models with and without the plumes, indicating that seismically fast anomalies in the mantle mainly result from the subducted slabs. Models with viscosity increase at 1,000‐km depth and the 670‐km depth phase change may reproduce seismic slab structures including the stagnant slabs in the mantle transition zone equally well as models with a thin weak layer below the 670‐km phase boundary.

Optimizing harmonics from Landsat time series data: the case of mapping rainfed and irrigated agriculture in Zimbabwe

ABSTRACTExplicit and large-scale information on farming systems is important for many applications such as crop production estimates, drought impact assessments or water footprint analysis. This contribution investigated the possibility of optimizing harmonic functions fitted to cloud-corrected Landsat NDVI (Normalized Difference Vegetation Index) time-series imagery to effectively map rainfed and irrigated croplands in Zimbabwe. The following harmonic optimizations were investigated: the effect of the linear trend, length of the time series data and the harmonics degree. The verification accuracy scores for mapping the two farming systems were used to ascertain the most optimal harmonics settings. The most accurate classification results (overall accuracy of 97%) were produced when the linear trend was excluded and the full time period (2013 to 2018) with a 7th degree harmonics fitting function was used. Optimized harmonics provide an effective way to compute vegetation seasonality signals while accounting for data gaps and residual noise inherent in Landsat time-series data.

The Mercury gravity field, orientation, love number, and ephemeris from the MESSENGER radiometric tracking data

The Mercury gravity field, spin-pole axis, rotation period, Love number, and ephemeris have been determined using the complete four years of MESSENGER tracking data from March 2011 to April 2015. The pole location and obliquity (1.99 ± 0.12 arcmin) is consistent with previous determinations. Since MESSENGER was in a highly elliptical orbit with periapsis in the far northern hemisphere, the gravity resolution over the surface of Mercury varies greatly from harmonic degree n = 12 at the south pole to n = 154 in a small region near the north pole which was covered with exceptionally low periapsis data near the end of mission. The gravity field MESS160A is determined to n = 160 and shows notable improvement in the correlation with topography. Three different constraint methods are used to generate the gravity field. The nominal method is a Kaula power law of 5 × 10−5/n2 to constraint the coefficients. One alternate constraint uses surface acceleration measurements, which only constrains the unobserved portion of the gravity field. The other constraint method sets the gravity uncertainties using the harmonic spectrum of the gravity derived from topography. The Mercury tidal Love number solution k2 = 0.53 ± 0.03 is larger than previous results but within suggested error bounds. In addition, the technique for estimating the Mercury ephemeris is discussed.

Apse-alignment in narrow-eccentric ringlets and its implications for the ϵ-ring of Uranus and the ring system of (10199) Chariklo

The discovery of ring systems around objects of the outer Solar System provides a strong motivation to apply theoretical models in order to better estimate their physical and orbital parameters, which can constrain scenarios for their origin.We review the criterion for maintaining apse-alignment across a ring and the balance between the energy input rate provided by a close by satellite and the internal dissipation rate occurring through ring particle collisions that is required to maintain ring eccentricity, as derived from the equations of motion governing the Lagrangian-displacements of the ring-particle orbits. We use the case of the ϵ-ring of Uranus, to calibrate our theoretical discussion and illustrate the basic dynamics governing these types of ring.In the case of the ring system of (10199) Chariklo, where the evidence that the rings are eccentric is not conclusive, we apply the theory of apse-alignment to derive information about the most plausible combination of values of the surface density and eccentricity-gradient, as well as the masses and locations of their postulated but - presently undetected - shepherd-satellites.When the balance conditions that we predict are applied to the ring system of (10199) Chariklo, we are able to estimate the minimum mass of a shepherd satellite required to prevent eccentricity decay, as a function of its orbital location, for two different models of dissipation. We conclude that the satellite mass required to maintain the m = 1 eccentric mode in the ring, would be similar or smaller than that needed to confine the rings radially.Our estimation of the most plausible combinations of eccentricity gradient and surface density consistent with apse-alignment are based on a standard model for the radial form of the surface density distribution, which approximately agrees with the optical depth profile derived by the stellar occultations. We find a diverse range of solutions, with combinations of eccentricity gradient and surface mass density that tend to minimize required enhanced collisional effects, having adopted estimated values of the form factor of the second degree harmonic of the gravitational potential.

Lithospheric and sublithospheric deformation under the Borborema Province of northeastern Brazil from receiver function harmonic stripping

Abstract. The depth-dependent anisotropic structure of the lithosphere under the Borborema Province in northeast Brazil has been investigated via harmonic stripping of receiver functions developed at 39 stations in the region. This method retrieves the first (k=1) and second (k=2) degree harmonics of a receiver function dataset, which characterize seismic anisotropy beneath a seismic station. Anisotropic fabrics are in turn directly related to the deformation of the lithosphere from past and current tectonic processes. Our results reveal the presence of anisotropy within the crust and the lithospheric mantle throughout the entire province. Most stations in the continental interior report consistent anisotropic orientations in the crust and lithospheric mantle, suggesting a dominant northeast–southwest pervasive deformation along lithospheric-scale shear zones developed during the Brasiliano–Pan-African orogeny. Several stations aligned along a northeast–southwest trend located above the (now aborted) Mesozoic Cariri–Potiguar rift display large uncertainties for the fast-axis direction. This non-azimuthal anisotropy may be related to a complex anisotropic fabric resulting from a combination of deformation along the ancient collision between Precambrian blocks, Mesozoic extension and thermomechanical erosion dragging by sublithospheric flow. Finally, several stations along the Atlantic coast reveal depth-dependent anisotropic orientations roughly (sub)perpendicular to the margin. These results suggest a more recent overprint, probably related to the presence of frozen anisotropy in the lithosphere due to stretching and rifting during the opening of the South Atlantic.

A new model of the crustal magnetic field of Mars using MGS and MAVEN.

While devoid of an active magnetic field today, Mars possesses a remanent magnetic field which may reach several thousand nT locally. The exact origin, and the events which have shaped the crustal magnetization remain largely enigmatic. Three magnetic field datasets from two spacecraft collected over 13 cumulative years have sampled the martian magnetic field over a range of altitudes from 90 km up to 6000 km: a- Mars Global Surveyor (MGS) magnetometer (1997-2006); b- MGS Electron Reflectometer (1999-2006); c- MAVEN magnetometer (2014-today). In this paper we combine these complementary datasets for the first time to build a new model of the martian internal magnetic field. This new model improves upon previous ones in several aspects: comprehensive data coverage; refined data selection scheme; modified modeling scheme; discrete-to-continuous transformation of the model; increased model resolution. The new model has a spatial resolution of ~ 160 km at the surface, corresponding to spherical harmonic degree 134. It shows small scales and well defined features, which can now be associated with geological signatures.

GRAIL-identified gravity anomalies in Oceanus Procellarum: Insight into subsurface impact and magmatic structures on the Moon

Four, quasi-circular, positive Bouguer gravity anomalies (PBGAs) that are similar in diameter (~90–190 km) and gravitational amplitude (>140 mGal contrast) are identified within the central Oceanus Procellarum region of the Moon. These spatially associated PBGAs are located south of Aristarchus Plateau, north of Flamsteed crater, and two are within the Marius Hills volcanic complex (north and south). Each is characterized by distinct surface geologic features suggestive of ancient impact craters and/or volcanic/plutonic activity. Here, we combine geologic analyses with forward modeling of high-resolution gravity data from the Gravity Recovery and Interior Laboratory (GRAIL) mission in order to constrain the subsurface structures that contribute to these four PBGAs. The GRAIL data presented here, at spherical harmonic degrees 6–660, permit higher resolution analyses of these anomalies than previously reported, and reveal new information about subsurface structures. Specifically, we find that the amplitudes of the four PBGAs cannot be explained solely by mare-flooded craters, as suggested in previous work; an additional density contrast is required to explain the high-amplitude of the PBGAs. For Northern Flamsteed (190 km diameter), the additional density contrast may be provided by impact-related mantle uplift. If the local crust has a density ~2800 kg/m3, then ~7 km of uplift is required for this anomaly, although less uplift is required if the local crust has a lower mean density of ~2500 kg/m3. For the Northern and Southern Marius Hills anomalies, the additional density contrast is consistent with the presence of a crustal complex of vertical dikes that occupies up to ~50% of the regionally thin crust. The structure of Southern Aristarchus Plateau (90 km diameter), an anomaly with crater-related topographic structures, remains ambiguous. Based on the relatively small size of the anomaly, we do not favor mantle uplift; however, understanding mantle response in a region of especially thin crust needs to be better resolved. It is more likely that this anomaly is due to subsurface magmatic material given the abundance of volcanic material in the surrounding region. Overall, the four PBGAs analyzed here are important in understanding the impact and volcanic/plutonic history of the Moon, specifically in a region of thin crust and elevated temperatures characteristic of the Procellarum KREEP Terrane.

Performance and Evaluation of Five-Phase Dual Random SVPWM Strategy With Optimized Probability Density Function

This paper presents a five-phase dual random space vector pulsewidth modulation (FPDR-SVPWM) strategy with an optimized probability density function (PDF) for a five-phase voltage-source inverter, and its performance and evaluation are also analyzed. Based on the nearest four vectors modulation method, the FPDR-SVPWM strategy, combined with random switching frequency and random zero vector, is investigated to significantly reduce the amplitude of high-order harmonics near the switching frequency and its integer multiples. For the nonsinusoidal power supply, third harmonic injection is introduced in this modulation strategy as well. Considering the effect of random numbers on the performance, an optimized Beta distribution is studied in detail. In order to analyze harmonic dispersion degree qualitatively and quantitatively, the power spectral density and the harmonic spread factor are both employed to assess the performance of the FPDR-SVPWM strategy. A series of results of simulation in MATLAB and experiments based on field programmable gate array demonstrate that the proposed modulation strategy is correct and valid, and it distributes high-order harmonics into a wide range without affecting the fundamental wave and third harmonic.

Lunar Cumulate Mantle Overturn: A Model Constrained by Ilmenite Rheology

AbstractLunar cumulate mantle overturn has been proposed to explain the abundances of TiO2 and heat‐producing elements (U, Th, and K) in the source region of lunar basalts. Ilmenite‐bearing cumulates (IBCs) that were formed near the end of lunar magma ocean solidification are the driving force for overturn. IBCs are enriched with dense TiO2 and FeO contents and have lower viscosity and solidus than those of the underlying lunar cumulate mantle. We investigate the effects of temperature‐ and ilmenite‐dependent mantle rheology on the dynamic process of lunar cumulate mantle overturn and conditions for long‐wavelength downwellings of an IBC layer in a 3‐D spherical geometry. Our results show that the ilmenite‐induced rheological weakening is necessary to decouple the IBC layer from the top stagnant lid and facilitate overturn. Models with IBC viscosity derived from the experimental scaling can only produce short‐wavelength downwellings (spherical harmonic degree >3) and show an overturn timescale more than 100 Ma. A viscosity of the IBC layer at least 10−4 lower than that of the ambient mantle can produce the long‐wavelength (spherical harmonic degree ≤3) downwellings in ~10 Ma and even a hemispheric downwelling. Such low IBC viscosity requires additional weakening mechanisms, such as remelting or/and water enrichment. During the overturn, the cold downwellings displace upward the materials from hot lower mantle and produce partial melting in upper mantle, which may serve as a viable mechanism for early lunar magmatisms. The settling of cold downwellings on the core‐mantle boundary stimulates a transient high heat flux, which may contribute to generating an early lunar dynamo event.