Spherical Harmonic Degree Research Articles

Relativistic Rayleigh–Taylor instability of a decelerating shell and its implications for gamma-ray bursts

Global linear stability analysis of a self-similar solution describing the interaction of a relativistic shell with an ambient medium is performed. The solution is shown to be unstable to convective Rayleigh–Taylor modes having angular scales smaller than the causality scale. Longer wavelength modes are stable and decay with time. For modes of sufficiently large spherical harmonic degree l the dimensionless growth rate scales as , where Γ is the Lorentz factor of the shell. The instability commences at the contact interface separating the shocked ejecta and shocked ambient gas and propagates to the shocks. The reverse shock front responds promptly to the instability and exhibits rapidly growing distortions at early times. Propagation to the forward shock is slower, and it is anticipated that the region near the contact will become fully turbulent before the instability is communicated to the forward shock. The non-universality of the Blandford–McKee blast wave solution suggests that turbulence generated by the instability in the shocked ambient medium may decay slowly with time and may be the origin of magnetic fields over a long portion of the blast wave evolution. It is also speculated that the instability may affect the emission from the shocked ejecta in the early post-prompt phase of gamma-ray bursts.

Coupled G-Mode Intersections and Solar-Cycle Variability

Wolff (Astrophys. J.193, 721, 1974) introduced the concept of g-mode coupling within the solar interior. Subsequently, Wolff developed a more quantitative model invoking a reciprocal interaction between coupled g modes and burning in the solar core. Coupling is proposed to occur for constant values of the spherical harmonic degree [l] creating rigidly rotating structures denoted as sets(l). Power would be concentrated near the core and the top of radiative zone [RZ] in narrow intervals of longitude on opposite sides of the Sun. Sets(l) would migrate retrograde in the RZ as function of l and their intersections would deposit extra energy at the top of the RZ. It is proposed that this enhances sunspot eruptions at particular longitudes and at regular time intervals. Juckett and Wolff (Solar Phys.252, 247, 2008) detected this enhancement by viewing selected spherical harmonics of sunspot patterns within stackplots twisted into the relative rotational frames of various sets(l). In subsequent work, the timings of the set(l) intersections were compared to the sub-decadal variability of the sunspot cycle. Seventeen sub-decadal intersection frequencies (0.63 – 7.0 year) were synchronous with 17 frequencies in the sunspot time-series with a mean correlation of 0.96. Six additional non-11-year frequencies (periods of 8.0 to 28.7 year) are now shown to be nearly synchronous between sunspot variability and the model. Two additional intersections have the same frequency as the solar cycle itself and peak during the rising phase of the solar cycle. This may be partly responsible for cycle asymmetry. These results are evidence that some of the solar-cycle variability may be attributable to deterministic components that are intermixed with a broad-spectrum stochastic and long-term chaotic background.

Deep versus shallow origin of gravity anomalies, topography and volcanism on Earth, Venus and Mars

The relation between gravity anomalies, topography and volcanism can yield important insights about the internal dynamics of planets. From the power spectra of gravity and topography on Earth, Venus and Mars we infer that gravity anomalies have likely predominantly sources below the lithosphere up to about spherical harmonic degree l = 30 for Earth, 40 for Venus and 5 for Mars. To interpret the low-degree part of the gravity spectrum in terms of possible sublithospheric density anomalies we derive radial mantle viscosity profiles consistent with mineral physics. For these viscosity profiles we then compute gravity and topography kernels, which indicate how much gravity anomaly and how much topography is caused by a density anomaly at a given depth. With these kernels, we firstly compute an expected gravity–topography ratio. Good agreement with the observed ratio indicates that for Venus, in contrast to Earth and Mars, long-wavelength topography is largely dynamically supported from the sublithospheric mantle. Secondly, we combine an empirical power spectrum of density anomalies inferred from seismic tomography in Earth’s mantle with gravity kernels to model the gravity power spectrum. We find a good match between modeled and observed gravity power spectrum for all three planets, except for 2 ⩽ l ⩽ 4 on Venus. Density anomalies in the Venusian mantle for these low degrees thus appear to be very small. We combine gravity kernels and the gravity field to derive radially averaged density anomaly models for the Martian and Venusian mantles. Gravity kernels for l ⩾ 5 are very small on Venus below ≈800 km depth. Thus our inferences on Venusian mantle density are basically restricted to the upper 800 km. On Mars, gravity anomalies for 2 ⩽ l ⩽ 5 may originate from density anomalies anywhere within its mantle. For Mars as for Earth, inferred density anomalies are dominated by l = 2 structure, but we cannot infer whether there are features in the lowermost mantle of Mars that correspond to Earth’s Large Low Shear Velocity Provinces (LLSVPs). We find that volcanism on Mars tends to occur primarily in regions above inferred low mantle density, but our model cannot distinguish whether or not there is a Martian analog for the finding that Earth’s Large Igneous Provinces mainly originate above the margins of LLSVPs.

Non-axisymmetric low-frequency oscillations of rotating and magnetized neutron stars

We investigate non-axisymmetric low frequency modes of a rotating and magnetized neutron star, assuming that the star is threaded by a dipole magnetic field whose strength at the stellar surface, $B_0$, is less than $\sim 10^{12}$G, and whose magnetic axis is aligned with the rotation axis. For modal analysis, we use a neutron star model composed of a fluid ocean, a solid crust, and a fluid core, where we treat the core as being non-magnetic assuming that the magnetic pressure is much smaller than the gas pressure in the core. Here, we are interested in low frequency modes of a rotating and magnetized neutron star whose oscillation frequencies are similar to those of toroidal crust modes of low spherical harmonic degree and low radial order. For a magnetic field of $B_0\sim 10^7$G, we find Alfv\'en waves in the ocean have similar frequencies to the toroidal crust modes, and we find no $r$-modes confined in the ocean for this strength of the field. We calculate the toroidal crustal modes, the interfacial modes peaking at the crust/core interface, and the core inertial modes and $r$-modes, and all these modes are found to be insensitive to the magnetic field of strength $B_0\ltsim10^{12}$G. We find the displacement vector of the core $l^\prime=|m|$ $r$-modes have large amplitudes around the rotation axis at the stellar surface even in the presence of a surface magnetic field $B_0\sim10^{10}$G, where $l^\prime$ and $m$ are the spherical harmonic degree and the azimuthal wave number of the $r$-modes, respectively. We suggest that millisecond X-ray variations of accretion powered X-ray millisecond pulsars can be used as a probe into the core $r$-modes destabilized by gravitational wave radiation.

Prediction of vertical deflections from high-degree spherical harmonic synthesis and residual terrain model data

This study demonstrates that in mountainous areas the use of residual terrain model (RTM) data significantly improves the accuracy of vertical deflections obtained from high-degree spherical harmonic synthesis. The new Earth gravitational model EGM2008 is used to compute vertical deflections up to a spherical harmonic degree of 2,160. RTM data can be constructed as difference between high-resolution Shuttle Radar Topography Mission (SRTM) elevation data and the terrain model DTM2006.0 (a spherical harmonic terrain model that complements EGM2008) providing the long-wavelength reference surface. Because these RTM elevations imply most of the gravity field signal beyond spherical harmonic degree of 2,160, they can be used to augment EGM2008 vertical deflection predictions in the very high spherical harmonic degrees. In two mountainous test areas—the German and the Swiss Alps—the combined use of EGM2008 and RTM data was successfully tested at 223 stations with high-precision astrogeodetic vertical deflections from recent zenith camera observations (accuracy of about 0.1 arc seconds) available. The comparison of EGM2008 vertical deflections with the ground-truth astrogeodetic observations shows root mean square (RMS) values (from differences) of 3.5 arc seconds for ξ and 3.2 arc seconds for η, respectively. Using a combination of EGM2008 and RTM data for the prediction of vertical deflections considerably reduces the RMS values to the level of 0.8 arc seconds for both vertical deflection components, which is a significant improvement of about 75%. Density anomalies of the real topography with respect to the residual model topography are one factor limiting the accuracy of the approach. The proposed technique for vertical deflection predictions is based on three publicly available data sets: (1) EGM2008, (2) DTM2006.0 and (3) SRTM elevation data. This allows replication of the approach for improving the accuracy of EGM2008 vertical deflection predictions in regions with a rough topography or for improved validation of EGM2008 and future high-degree spherical harmonic models by means of independent ground truth data.

CHAOS-2���a geomagnetic field model derived from one decade of continuous satellite data

SUMMARY We have derived a model of the near-Earth's magnetic field using more than 10 yr of high-precision geomagnetic measurements from the three satellites Orsted, CHAMP and SAC-C. This model is an update of the two previous models, CHAOS (Olsen et al. 2006) and xCHAOS (Olsen & Mandea 2008). Data selection and model parameterization follow closely those chosen for deriving these models. The main difference concerns the maximum spherical harmonic degree of the static field (n= 60 compared to n= 50 for CHAOS and xCHAOS), and of the core field time changes, for which spherical harmonic expansion coefficients up to n= 20 are described by order 5 splines (with 6-month knot spacing) spanning the years from 1997.0 to 2009.5. Compared to its predecessors, the temporal regularization of the CHAOS-2 model is also modified. Indeed, second and higher order time derivatives of the core field are damped by minimizing the second time derivative of the squared magnetic field intensity at the core–mantle boundary. The CHAOS-2 model describes rapid time changes, as monitored by the ground magnetic observatories, much better than its predecessors.

Constraints on opacities from complex asteroseismology of B-type pulsators: the β Cephei star θ Ophiuchi

We present results of a comprehensive asteroseismic modelling of the β Cephei variable θ Ophiuchi. We call these studies complex asteroseismology because our goal is to reproduce both pulsational frequencies and corresponding values of a complex, non-adiabatic parameter, f, defined by the radiative flux perturbation. To this end, we apply the method of simultaneous determination of the spherical harmonic degree, ℓ, of excited pulsational mode and the corresponding non-adiabatic f parameter from combined multicolour photometry and radial velocity data. Using both the OP and OPAL opacity data, we find a family of seismic models which reproduce the radial and dipole centroid mode frequencies, as well as the f parameter associated with the radial mode. Adding the non-adiabatic parameter to seismic modelling of the B-type main-sequence pulsators yields very strong constraints on stellar opacities. In particular, only with one source of opacities it is possible to agree the empirical values of f with their theoretical counterparts. Our results for θ Oph point substantially to preference for the OPAL data.

Generation of plate‐like behavior and mantle heterogeneity from a spherical, viscoplastic convection model

How plate tectonics arises from mantle convection is a question that has only very recently become feasible to address with spherical, viscoplastic computations. We present mainly internally heated convection results with temperature‐dependent viscosity and explore parts of the Rayleigh number (Ra)–yield stress (σy) phase space, as well as the effects of depth‐dependent σy, bottom heating, and a low‐viscosity asthenosphere. Convective planform and toroidal‐poloidal velocity field ratio (TPR) are affected by near‐surface viscosity variations, and TPR values are close to observed values for our most plate‐like models. At the relatively low convective vigor that is accessible at present, most models favor spherical harmonic degree one convection, though models with a weaker surface viscosity form degree two patterns and reproduce tomographically observed power spectra. An asthenospheric viscosity reduction improves plate‐like nature, as expected. For our incompressible computations, pure bottom heating produces strong plumes that tend to destroy plates at the surface. This implies that significant internal heating may be required, both to reduce the role of active upwellings and to form a low‐viscosity zone beneath the upper boundary layer.

Low-degree earth deformation from reprocessed GPS observations

Surface mass variations of low spherical harmonic degree are derived from residual displacements of continuously tracking global positioning system (GPS) sites. Reprocessed GPS observations of 14 years are adjusted to obtain surface load coefficients up to degree nmax = 6 together with station positions and velocities from a rigorous parameter combination. Amplitude and phase estimates of the degree-1 annual variations are partly in good agreement with previously published results, but also show interannual differences of up to 2 mm and about 30 days, respectively. The results of this paper reveal significant impacts from different GPS observation modeling approaches on estimated degree-1 coefficients. We obtain displacements of the center of figure (CF) relative to the center of mass (CM), ΔrCF–CM, that differ by about 10 mm in maximum when compared to those of the commonly used coordinate residual approach. Neglected higher-order ionospheric terms are found to induce artificial seasonal and long-term variations especially for the z-component of ΔrCF–CM. Daily degree-1 estimates are examined in the frequency domain to assess alias contributions from model deficiencies with regard to satellite orbits. Finally, we directly compare our estimated low-degree surface load coefficients with recent results that involve data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission.

Ensemble inversion of time‐dependent core flow models

Quasi‐geostrophic core flow models are built from two secular variation models spanning the periods 1960–2002 and 1997–2008. We rely on an ensemble method to account for the contributions of the unresolved small‐scale magnetic field interacting with core surface flows to the observed magnetic field changes. The different core flow members of the ensemble solution agree up to spherical harmonic degree ℓ ≃ 10, and this resolved component varies only weakly with regularization. Taking into account the finite correlation time of the small‐scale concealed magnetic field, we find that the time variations of the magnetic field occurring over short time scales, such as the geomagnetic jerks, can be accounted for by the resolved (large‐scale) part of the flow to a large extent. Residuals from our flow models are 30% smaller for recent epochs, after 1995. This result is attributed to an improvement in the quality of geomagnetic data. The magnetic field models show little frozen flux violation for the most recent epochs, within our estimate of the apparent magnetic flux changes at the core‐mantle boundary arising from spatial resolution errors. We associate the more important flux changes detected at earlier epochs with uncertainties in the field models at large harmonic degrees. Our core flow models show, at all epochs, an eccentric and planetary‐scale anticyclonic gyre circling around the cylindrical surface tangent to the inner core, at approximately 30 and 60 latitude under the Indian and Pacific oceans, respectively. They account well for the changes in core angular momentum for the most recent epochs.

Improved nearside gravity field of the Moon by localizing the power law constraint

The problem associated with the large data gap on the farside of the Moon is addressed for constructing a high‐resolution global gravity model. By localizing the power law constraint and making it effective only within the farside and limb regions, we mitigate the undesired power‐limiting effect on the nearside. Compared to the solution estimated from Lunar Prospector and other satellite tracking data with the globally‐applied power law, the locally‐constrained solution shows significant improvement of the nearside gravity estimates. Around the areas dominated by craters in the southern hemisphere of the nearside, the correlation with topography approaches nearly 0.95 and the admittance converges to 100–110 mGal/km up to spherical harmonic degree 130, while the globally‐constrained solutions distort starting at degree 90. The proposed analysis can benefit the science and operation of other existing and future planetary missions and enhance the geophysical interpretation of the gravity field.

Crustal concealing of small-scale core-field secular variation

SUMMARY The Earth’s magnetic field is mainly produced within the Earth’s liquid and electrically conducting core, as a result of a process known as the geodynamo. Many other sources also contribute to the magnetic signal accessible to observation at the Earth’s surface, partly obscuring the main core magnetic field signal. Thanks to a series of very successful satellites and to advances in magnetic field modelling techniques, considerable progress has, however, been made in the recent years toward better identifying the signal of each of these sources. In particular, temporal changes in the field of internal origin happen to be detectable now in spherical harmonic degrees up to, perhaps, 16. All of these changes are usually attributed to changes in the core field itself, the secular variation, on the ground that the lithospheric magnetization cannot produce such signals. It has, however, been pointed out, on empirical grounds, that temporal changes in the field of internal origin produced by the induced part of the lithospheric magnetization could dominate the core field signal beyond degree 22. This short note revisits this issue by taking advantage of our improved knowledge of the small-scale field changes and of the likely sources of the lithospheric field. We rely on a simple extrapolation of the observed spatial spectrum of the field changes beyond degree 16 and use a forward approach based on a recent geological model of lithospheric magnetization. This leads us to confirm that the main cause of the observed changes in the field of internal origin up to some critical degree, N C, is indeed likely to be the secular variation of the core field, but that the signal produced by the time-varying lithospheric field is bound to dominate and conceal the time-varying core signal beyond that critical degree, in very much the same way the permanent component of the lithospheric field dominates and conceals the permanent component of the core field beyond degree 14. All uncertainties taken into account, we estimate N C to lie between 22 and 24. We, however, also note that in practice, the main limitation to the observation of the core field small-scale secular variation is not so much its concealing by the field of lithospheric origin but its fast changing nature and small magnitude. This leads us to conclude that whereas cumulative small-scale lithospheric field changes might be detected some day, detection of core-field secular variation beyond degree 18 is likely to remain a severe challenge for some time.

Incorporating self‐consistently calculated mineral physics into thermochemical mantle convection simulations in a 3‐D spherical shell and its influence on seismic anomalies in Earth's mantle

Phase assemblages of mantle rocks calculated from the ratios of five oxides (CaO‐FeO‐MgO‐Al2O3‐SiO2) by free energy minimization were used to calculate the material properties density, thermal expansivity, specific heat capacity, and seismic velocity as a function of temperature, pressure, and composition, which were incorporated into a numerical thermochemical mantle convection model in a 3‐D spherical shell. The advantage of using such an approach is that thermodynamic parameters are included implicitly and self‐consistently, obviating the need for ad hoc parameterizations of phase transitions which can be complex in regions such as the transition zone particularly if compositional variations are taken into account. Convective planforms for isochemical and thermochemical cases are, however, not much different from those computed using our previous, simple parameterized reference state, which means that our previous results are robust in this respect. The spectrum and amplitude of seismic velocity anomalies obtained using the self‐consistently calculated material properties are more “realistic” than those obtained when seismic velocity is linearly dependent on temperature and composition because elastic properties are dependent on phase relationship of mantle minerals, in other words, pressure and temperature. In all cases, the spectra are dominated by long wavelengths (spherical harmonic degree 1 to 2), similar or even longer wavelength than seismic tomographic models of Earth, which is probably due to self‐consistent plate tectonics and depth‐dependent viscosity. In conclusion, this combined approach of mantle convection and self‐consistently calculated mineral physics is a powerful and useful technique for predicting thermal‐chemical‐phase structures in Earth's mantle. However, because of uncertainties in various parameters, there are still some shortcomings in the treatment of the postperovskite phase transition. Additionally, transport properties such as thermal conductivity and viscosity are not calculated by this treatment and are thus subject to the usual uncertainties.

Synthetic tomography of plume clusters and thermochemical piles

Seismic tomography elucidates broad, low shear-wave velocity structures in the lower mantle beneath Africa and the central Pacific with uncertain physical and compositional origins. One hypothesis suggests that these anomalies are caused by relatively hot and intrinsically dense material that has been swept into large thermochemical piles by mantle flow. An alternative hypothesis suggests that they are instead poorly imaged clusters of narrow thermal plumes. Geodynamical calculations predict fundamentally different characters of the temperature fields of plume clusters and thermochemical piles. However the heterogeneous resolution of tomographic models makes direct comparison between geodynamical temperature fields and tomographic shear-wave anomalies tenuous at best. Here, we compute synthetic tomographic images for 3D spherical mantle convection models and evaluate how well thermal plumes and thermochemical piles can be reconciled with actual seismic tomography images. Geodynamical temperature fields are converted to shear-wave velocity using experimental and theoretical mineral physics constraints. The resultant shear-wave velocity fields are subsequently convolved with the resolution operator from seismic model S20RTS to mimic the damping and distortion associated with heterogeneous seismic sampling of the mantle. We demonstrate that plume clusters are tomographically blurred into two broad, antipodal velocity anomalies in agreement with S20RTS and other global seismic models. Large, thermochemical piles are weakly distorted by the tomographic filter. The power spectrum of velocity heterogeneity peaks at spherical harmonic degree 3, unlike the degree-2 maximum in S20RTS, but decays rapidly similar to S20RTS and many other seismic models. Predicted tomography from thermochemical pile and plume cluster models correlate equally well with S20RTS given the uncertainties in the numerical modeling parameters. However, thermochemical piles match tomography better in visual comparison and in the overall character of the harmonic spectrum.

On the geographical distribution of induced time-varying crustal magnetic fields

[1] A long standing question in geomagnetism is whether the time variation of the induced crustal field is a detectable quantity and, if so, at which spatial wavelengths. We tackle this problem with the help of a forward modeling approach using a vertically integrated susceptibility (VIS) grid of the Earth's crust. For spherical harmonic degrees 15–90, we estimate the root mean square of the crustal magnetic field secular variation to amount 0.06–0.12 nT/yr at the terrestrial surface between epochs 1960–2002.5. The geographical distribution of the signal shows absolute values reaching 0.65–1.30 nT/yr over South America. Unfortunately, most of the world magnetic observatories currently lie on quasi-stationary features where the crustal field signal variations are expected to be very low. However, this long sought signal could be detected over well chosen regions, provided that satellite, observatory, and repeat station measurements are available over several decades.

Global maps of the step-wise topography corrected and crustal components stripped geoids using the CRUST 2.0 model

Global maps of the step-wise topography corrected and crustal components stripped geoids using the CRUST 2.0 modelWe compile global maps of the step-wise topography corrected and crustal components stripped geoids based on the geopotential model EGM'08 complete to spherical harmonic degree 180 and the CRUST 2.0 global crustal model. The spectral resolution complete to degree 180 is used to compute the primary indirect bathymetric stripping and topographic effects on the geoid, while degree 90 for the primary indirect ice stripping effect. The primary indirect stripping effects of the soft and hard sediments, and the upper, middle and lower consolidated crust components are forward modeled in spatial form using the 2 × 2 arc-deg discrete data of the CRUST 2.0 model. The ocean, ice, sediment and consolidated crust density contrasts are defined relative to the adopted reference crustal density of 2670 kg/m3. Finally we compute and apply the primary indirect stripping effect of the density contrast (relative to the mantle) of the reference crust. The constant value of -520 kg/m3is adopted for this density contrast relative to the mantle. All data are evaluated on a 1 × 1 arc-deg geographical grid. The complete crust-stripped geoidal undulations, globally having a range of approximately 1.5 km, contain the gravitational signal coming from the global mantle lithosphere (upper mantle) morphology and density composition, and from the sub-lithospheric density heterogeneities. Large errors in the complete crust-stripped geoid are expected due to uncertainties of the CRUST 2.0 model, i.e., due to deviations of the CRUST 2.0 model density from the real earth's crustal density and due to the Moho-boundary uncertainties.

On the applicability of the frozen flux approximation in core flow modelling as a function of temporal frequency and spatial degree

SUMMARY The secular variation of the geomagnetic field is due to flow of molten iron in the Earth’s core. One can invert for the flow at the core-surface, assuming that the magnetic field is frozen into the core fluid. The validity of the frozen flux approximation (FFA) depends on the scale length of the magnetic field and the period of core motions. A first-order quantitative assessment can be made by solving the induction equation for a spherical conductor oscillating in an ambient magnetic field. The period, at which the FFA holds, strongly decreases with increasing spherical harmonic degree of the ambient field. For degrees smaller than five, the flux of the ambient magnetic field can be considered as frozen-in for periods of more than a thousand years. For ambient magnetic fields of spherical harmonic degree 10 and higher, the FFA still holds up to periods of about a hundred years. However, the real situation in the core could be more complex if the field cannot be assumed as ambient but is generated by the flow itself.

A benchmark study on mantle convection in a 3‐D spherical shell using CitcomS

As high‐performance computing facilities and sophisticated modeling software become available, modeling mantle convection in a three‐dimensional (3‐D) spherical shell geometry with realistic physical parameters and processes becomes increasingly feasible. However, there is still a lack of comprehensive benchmark studies for 3‐D spherical mantle convection. Here we present benchmark and test calculations using a finite element code CitcomS for 3‐D spherical convection. Two classes of model calculations are presented: the Stokes' flow and thermal and thermochemical convection. For Stokes' flow, response functions of characteristic flow velocity, topography, and geoid at the surface and core‐mantle boundary (CMB) at different spherical harmonic degrees are computed using CitcomS and are compared with those from analytic solutions using a propagator matrix method. For thermal and thermochemical convection, 24 cases are computed with different model parameters including Rayleigh number (7 × 103 or 105) and viscosity contrast due to temperature dependence (1 to 107). For each case, time‐averaged quantities at the steady state are computed, including surface and CMB Nussult numbers, RMS velocity, averaged temperature, and maximum and minimum flow velocity, and temperature at the midmantle depth and their standard deviations. For thermochemical convection cases, in addition to outputs for thermal convection, we also quantified entrainment of an initially dense component of the convection and the relative errors in conserving its volume. For nine thermal convection cases that have small viscosity variations and where previously published results were available, we find that the CitcomS results are mostly consistent with these previously published with less than 1% relative differences in globally averaged quantities including Nussult numbers and RMS velocities. For other 15 cases with either strongly temperature‐dependent viscosity or thermochemical convection, no previous calculations are available for comparison, but these 15 test calculations from CitcomS are useful for future code developments and comparisons. We also presented results for parallel efficiency for CitcomS, showing that the code achieves 57% efficiency with 3072 cores on Texas Advanced Computing Center's parallel supercomputer Ranger.

Assessment of systematic errors in the surface gravity anomalies over North America using the GRACE gravity model

SUMMARY The surface gravity data collected via traditional techniques such as ground-based, shipboard and airborne gravimetry describe precisely the local gravity field, but they are often biased by systematic errors. On the other hand, the spherical harmonic gravity models determined from satellite missions, in particular, recent models from CHAMP and GRACE, homogenously and accurately describe the low-degree components of the Earth’s gravity field. However, they are subject to large omission errors. The surface and satellite gravity data are therefore complementary in terms of spectral composition. In this paper, we aim to assess the systematic errors of low spherical harmonic degrees in the surface gravity anomalies over North America using a GRACE gravity model. A prerequisite is the extraction of the low-degree components from the surface data to make them compatible with GRACE data. Three types of methods are tested using synthetic data: low-pass filtering, the inverse Stokes integral, and spherical harmonic analysis. The results demonstrate that the spherical harmonic analysis works best. Eighty-five per cent of difference between the synthetic gravity anomalies generated from EGM96 and GGM02S from degrees 2 to 90 can be modelled for a region covering North America and neighbouring areas. Assuming EGM96 is developed solely from the surface gravity data with the same accuracy and GGM02S errorless, one way to understand the 85 per cent difference is that it represents the systematic error from the region of study, while the remaining 15 per cent originates from the data outside of the region. To estimate systematic errors in the surface gravity data, Helmert gravity anomalies are generated from both surface and GRACE data on the geoid. Their differences are expanded into surface spherical harmonics. The results show that the systematic errors for degrees 2 to 90 range from about −6 to 13 mGal with a RMS value of 1.4 mGal over North America. A few significant data gaps can be identified from the resulting error map. The errors over oceans appear to be related to the sea surface topography. These systematic errors must be taken into consideration when the surface gravity data are used to validate future satellite gravity missions.

Can core‐surface flow models be used to improve the forecast of the Earth's main magnetic field?

Geomagnetic main field models used for navigation are updated every 5 years and contain a forecast of the geomagnetic secular variation for the upcoming epoch. Forecasting secular variation is a difficult task. The change of the main magnetic field is thought to be principally due to advection of the field by flow at the surface of the outer core on short timescales and when large length scales are considered. With accurate secular variation (SV) and secular acceleration (SA) models now available from new satellite missions, inverting for the flow and advecting it forward could lead to a more accurate prediction of the main field. However, this scheme faces two significant challenges. The first arises from the truncation of the observable main field at spherical harmonic degree 13. This can however be handled if the true core flow is large scale and has a rapidly decaying energy spectrum. The second is that even at a given single epoch the instantaneous SV and SA cannot simultaneously be explained by a steady flow. Nevertheless, we find that it may be feasible to use flow models for an improved temporal extrapolation of the main field. A medium‐term (≈10 years) hindcast of the field using a steady flow model outperforms the usual extrapolation using the presently observed SV and SA. On the other hand, our accelerated, toroidal flow model, which explains a larger portion of the observed average SA over the 2000–2005 period, fails to improve both the short‐term and medium‐term hindcasts of the field. This somewhat paradoxical result is related to the occurrence of so‐called geomagnetic jerks, the still poorly known dynamical nature of which remains the main obstacle to improved geomagnetic field forecasts.