Gauss Coefficients Research Articles

On the determination and interpretation of the lithospheric induced magnetisation

We present a new technique for reducing the uncertainties inherent in the interpretation of lithospheric magnetic field observations over the Earth. This technique, without involving iterations, provides an improved estimate of the depth-integrated magnetic susceptibility of the crust as compared to previous approaches. Departing from the normal practice of using observations at specific locations, we model directly Gauss coefficients of the lithospheric magnetic field and use an a-priori initial lithospheric thickness model to circumvent magnetic annihilators (i.e. magnetisation distributions that cannot be determined from magnetic data since they have no impact on the lithospheric magnetic field). Of the several initial magnetic layer models tested, we prefer the model where magnetic thickness is based on the Moho or the regional estimate of the Curie depth when it is shallower than the Moho because it is physically reasonable and produces the fewest artefacts. The method is applied to a recent high-degree lithospheric magnetic field model called LCS-1 derived from CHAMP and Swarm magnetic satellite data. The technique is appropriate for regions where induced magnetisation dominates over remanent magnetisation. We show that high degrees of the final depth-integrated magnetic susceptibility variation are dependent only on the corresponding high degrees of the LCS-1 magnetic field model. Thus, the depth-integrated magnetic susceptibility variation is an important quantity derived in the study which enables readily qualitative interpretation of regional geology.

Transient core surface dynamics from ground and satellite geomagnetic data

SUMMARYWe present an update of the geomagnetic data assimilation tool pygeodyn, use it to analyse ground and satellite-based geomagnetic data sets, and report new findings on the dynamics of the Earth’s outer core on interannual to decadal timescales. Our results support the idea that quasi-geostrophic Magneto-Coriolis waves, recently discovered at a period of 7 yr, also operate on both shorter and longer timescales, specifically in period bands centred around 3.5 and 15 yr. We revisit the source of interannual variations in the length of day and argue that both geostrophic torsional Alfvén waves and quasi-geostrophic Magneto-Coriolis waves can possibly contribute to spectral lines that have been isolated around 8.5 and 6 yr. A significant improvement to our ensemble Kalman filter algorithm comes from accounting for cross-correlations between variables of the state vector forecast, using the ‘Graphical lasso’ method to help stabilize the correlation matrices. This allows us to avoid spurious shrinkage of the model uncertainties while (i) conserving important information contained in off-diagonal elements of the forecast covariance matrix, and (ii) considering a limited number of realizations, thus reducing the computational cost. Our updated scheme also permits us to use observations either in the form of Gauss coefficient data or more directly as ground-based and satellite-based virtual observatory series. It is thanks to these advances that we are able to place global constraints on core dynamics even at short periods.

Simulation for MSS-2 low-perigee elliptical orbit satellites: an example of lithospheric magnetic field modelling

A future constellation of at least four geomagnetic satellites (designated Macau Scientific Satellite-1 (MSS-1) and Macau Scientific Satellite-2 (MSS-2)) was recently proposed, to continue high-quality geomagnetic observations in the post-Swarm period, focusing especially on collecting data that will provide a global, three-dimensional survey of the geomagnetic field. In this paper, we present a simulation of two years of orbits (2020.01.01−2022.01.01) of two satellites (tentatively denoted as MSS-2) that are constellated in elliptical (200 × 5,300 km) low-perigee orbits. By comparing error variances of Gauss coefficients, we investigate the sensitivity of lithospheric magnetic field modelling to data collected from various satellite orbits, including a near circular reference orbit of 300 × 350 km, and elliptical orbit of 180 × 5,300 km, 220 × 5,300 km, 200 × 3,000 km and 200 × 1,500 km. We find that in two years the two MSS-2 satellites can collect 35,000 observations at altitude below 250 km, data that will be useful in advancing the quality of lithospheric magnetic field modelling; this number of observations reflects the fact that only 4.5% of the flight time of these satellites will be below 250 km (just 6.4% of their flight time below 300 km). By combining observations from the MSS-2 satellites’ elliptical orbits of 200 × 5,300 km with observations from a circular reference orbit, the variance of the geomagnetic model can be reduced by a factor of 285 at spherical harmonic degree <italic>n</italic> = 200 and by a factor of 1,300 at <italic>n</italic> = 250. The planned lower perigee of their orbits allows the new satellites to collect data at unprecedentedly lower altitudes, thus dramatically improving the spatial resolution of satellite-derived lithospheric field models, (up to 80% at <italic>n</italic>=150). In addition, lowering the apogee increases the time interval during which the satellites fly at near-Earth altitudes, thus improving the model predictions at all spherical harmonic degrees (around 52%−62% at <italic>n</italic> = 150). The upper limit of the expected improvement to the field model at the orbital apogee is not as good as at the perigee. However, data from the MSS-1 orbit can help fill the gap between data from the MSS-2 orbits and from the circular reference orbit for the low-degree part of the model. The feasibility of even lower-altitude flight requires further discussion with satellite engineers.

Study on the estimation of Euler angles for Macau Science Satellite-1

The Euler angle estimation is a calibration method for vector data measured by the magnetometer on a satellite. It is used to find the relative rotation between the coordinate system of the magnetometer and the satellite (usually determined by Star Imagers). Before launch of the low-orbit, low-inclination Macau Science Satellite-1 (known as MSS-1), we simulated the estimation of Euler angles by using the magnetic measurements of the in-orbit Swarm and China Seismo-Electromagnetic Satellite (noted as CSES), with various data combinations. In this study, 11 data sets were designed to analyze the estimation results for the MSS-1 orbit by using a joint estimation method of the geomagnetic field model parameters and Euler angles. For the model results, we found that all the spatial power spectral lines showed behavior consistent with that of the CHAOS-7.8 model at low degrees (corresponding to large-scale magnetic signals). The spectra of models without global data coverage deviated much more (by a maximum of ~10⁴ nT²) from those of the CHAOS-7.8 model at higher degrees. For models with global data coverage and with various data combinations, the spectral lines were distributed similarly. Moreover, the models with accordant power spectral distributions demonstrated different Euler angle estimations. As more vector data at higher latitudes were included, the estimated Euler angles varied monotonically in all three directions. The models with vector data in the same latitude range showed similar Euler angle results, regardless of whether the poleward scalar data were included. The largest value difference was found between the models using vector data within ±40° latitudes and those using vector data within ±60° latitudes, which reached to ~28″. Therefore, we concluded that the inversion of the spherical harmonic Gauss coefficients in our tests was mainly affected by the spatial coverage range of the data, whereas the estimation of Euler angles largely depended on the latitude range where the vector data could be obtained. These results can be used for future in-flight data testing. We expect the estimation of Euler angles to improve as other methods are adopted.

Modelling Earth’s lithospheric magnetic field using satellites in low-perigee elliptical orbits

SUMMARYThe sensitivity of magnetic measurements taken by satellites in elliptical orbits to the lithospheric magnetic field is studied by comparing the formal error variances of the lithospheric Gauss coefficients for various satellite orbital constellations. Analytical expressions are presented for the variances of the Gauss coefficients when either all three magnetic vector components or only the radial component are used. We compare the results obtained using a satellite in a near-polar circular orbit at 350 km altitude with those from a satellite in an elliptical orbit with perigee at 140 km (and apogee at 1500 km) and find that the latter leads to Gauss coefficient variances at spherical harmonic degree n = 180 (corresponding to a horizontal wavelength of λ = 220 km) that are 104 times smaller compared to those derived from a similar number of data measured at 350 km altitude. The improvements in variance ratio at degree n = 145 (λ = 275 km) and n = 110 (λ = 360 km) are 103 and 102, respectively. These findings are supported by an analysis of synthetic magnetic data along simulated satellite orbits from which the lithospheric Gauss coefficients are estimated and compared with the original ones used to generate the synthetic data. Coefficients at degree n are successfully determined if the power of the difference between retrieved and original coefficients at that degree is smaller than the power of the lithospheric field (i.e. of the input coefficients). Using 3 yr of simulated data we conclude that magnetic measurements from a satellite in an elliptical orbit with perigee at 140 km allow for a reliable determination of the lithospheric field up to spherical harmonic n ≈ 170 while a satellite in a circular orbit at 350 km height only enables lithospheric field modelling up to n ≈ 100. The analysis demonstrates that low-altitude magnetic data collected by satellites in low-perigee elliptical orbits—although only available for a fraction of each orbit—enable improved global lithospheric field modelling at spatial wavelengths well beyond what is currently possible with data from satellites in circular orbits that do not reach such low altitudes. We applied the approach to the orbital configuration proposed for the Daedalus satellite mission (140 km perigee); the method will however also help in the preparation for other satellite missions in near-polar low-perigee elliptical orbits like the Macau Science Satellite pair MSS-2A and MSS-2B (perigee of 200 km or lower).

A new inclination-based method to evaluate the global geomagnetic configuration and axial dipole moment

The strength and configuration of the geomagnetic field control the average shape of the magnetosphere. The pure dipole assumption and the virtual dipole moment (VDM), determined by individual records, have been widely adopted to evaluate the strength of the geomagnetic field in geological time. However, such an assumption might not be valid during geomagnetic transitions, such as reversals and excursions. The traditional spherical harmonic modeling of the geomagnetic field could be difficult to implement because accurate global records are lacking. Here, we report that an empirical relationship exists between the ratio of the VDM to the true axial dipole moment (VDM/ADM) and the ratio of the power of the axial dipole to that of the non-axial dipole (AD/NAD) based on a new method utilizing globally distributed inclination records. The root mean square global deviation of inclination (RMSΔI) to the standard inclination distribution of the AD was fitted to the AD/NAD with a cubic polynomial by utilizing a large number of geodynamo simulations. Tests with geomagnetic field models showed that the AD/NAD derived from the RMSΔI agreed well with that calculated by using the Gauss coefficients, and the estimated ADM was consistent with the true value. Finally, the application of volcanic records during the Laschamp excursion showed the VDM might overestimate the ADM by a factor of 3. Our new method will be useful in future studies that characterize the configuration of the geomagnetic field and the strength of the axial dipole.

A Spherical Harmonic Model of Earth's Lithospheric Magnetic Field up to Degree 1050

AbstractDetailed mapping of the Earth's magnetic field brings key constraints on the composition, dynamics, and history of the crust. Satellite and near‐surface measurements detect different length scales and are complementary. Here, we build a model, selecting and processing magnetic field measurements from the German CHAMP and ESA Swarm satellites, which we merge with near‐surface scalar anomaly data. We follow a regional approach for modeling the magnetic field measurements and next transform the series of regional models into a unique set of spherical harmonic (SH) Gauss coefficients. This produces the first global model to SH degree 1050 derived by inversion of all available measurements. The new model agrees with previous satellite‐based models at large wavelengths and fits the CHAMP and Swarm satellite data down to expected noise levels. Further assessment in the geographical and spectral domains shows the model to be stable when downward continued to the Earth's surface.

A Spherical Harmonic Martian Crustal Magnetic Field Model Combining Data Sets of MAVEN and MGS

AbstractThis study presents a new spherical harmonic (SH) model of the crustal magnetic field of Mars, based on the magnetic field data set measured by the Mars Global Surveyor (MGS) and the Mars Atmosphere and Volatile EvolutioN (MAVEN) spacecraft. To minimize the influence of external fields due to solar wind interaction with Mars, we rejected data that were observed dayside and above an altitude of 500 km. The data points of MAVEN were reduced by using a proxy of solar wind activity that identified and rejected any data measured during magnetically disturbed intervals. We used a conventional least squares technique to estimate the Gauss coefficients fitted to the reduced data set and made a compromise between model misfit and model roughness by truncating the SH model at degree 110. This model is capable of representing crustal fields with a spatial resolution approaching ∼200 km at 120 km altitude and ∼260 km at the Martian surface. Since our model fits MAVEN's observational data better than previous models, especially the data obtained during MAVEN's low altitude periapsis passes, we conclude that it may more accurately approximate the low‐altitude crustal field. We calculate the crustal field power spectrum of various models and find that small‐scale fields at low altitudes were underestimated by most previous models. This new model could benefit future studies associated with the Martian crustal field and its interaction with the solar wind.

Magnetometer-Based Orbit Determination via Fast Reconstruction of Three-Dimensional Decoupled Geomagnetic Field Model

In this paper, magnetometer-based orbit determination via the fast reconstruction of a 3-D decoupled geomagnetic field (TDGF) model is examined. For the TDGF model, the three components of the geomagnetic field vector at a certain point are calculated separately with three different independent sets of Gauss coefficients rather than one identical set. The fast reconstruction algorithm is based on a recursive least-squares algorithm with a low computational burden. According to the orbit determination accuracy requirements for near-Earth satellites with communication systems that must be operational to update the TDGF model, the on-orbit model update interval can be appropriately selected so that the orbit determination accuracy can be effectively maintained. The simulation results of real flight data from magnetic survey satellites and simulated data show that the robustness of the proposed algorithm for different values of the measurement noise covariance matrices makes the algorithm more practical and that the position estimation error achieved by the proposed algorithm can meet the requirements of low- and medium-precision missions for near-Earth micro- and nanosatellites. In this way, both the autonomy and the orbit determination accuracy of satellites are considered simultaneously at the expense of some unnecessary autonomous capabilities for satellite tasks.

Electrical conductivity and temperature of the Earth's mantle inferred from Bayesian inversion of Swarm vector magnetic data

Study of induced magnetic field is a powerful way to sound Earth internal structure. This work presents a full analysis and interpretation in terms of electrical conductivity and temperature of vector magnetic field measurements from Swarm Level1b data product from 26/11/2013 to 31/12/2019. Time series of the Gauss coefficients associated with the induced and inducing magnetic field are obtained from the data after removal of the core and lithospheric fields models and data selection. A Bayesian inversion of the induced field Gauss coefficients is then performed to obtain a new estimate of Earth's 1D mantle electrical conductivity down to 2000 km depth. This profile is fully compatible with the profiles derived from satellite and ground magnetic observatories data but does not present in the lower mantle the increase predicted by laboratory-based conductivity profile associated to classical mantle composition and temperature profile. Using the most recent database to model the electrical conductivity of all mineral mantle phases, two different methods are used to interpret Swarm data in terms of temperature for a given composition and water content. The first one is based on an interpretation of the conductivity estimates in terms of temperature by classical numerical root search. The second one consists in inferring a temperature probability density function from a Bayesian inversion of the Gauss coefficients associated to the induced magnetic field. Our results show that the later provide more reliable estimates of mantle temperatures, in relation to more physically grounded prior values. This second method provides also tighter constraints on the electrical conductivity estimates of the lower mantle.

The Mie representation for Mercury\u2019s magnetic field

The parameterization of the magnetospheric field contribution, generated by currents flowing in the magnetosphere is of major importance for the analysis of Mercury’s internal magnetic field. Using a combination of the Gauss and the Mie representation (toroidal–poloidal decomposition) for the parameterization of the magnetic field enables the analysis of magnetic field data measured in current carrying regions in the vicinity of Mercury. In view of the BepiColombo mission, the magnetic field resulting from the plasma interaction of Mercury with the solar wind is simulated with a hybrid simulation code and the internal Gauss coefficients for the dipole, quadrupole and octupole field are reconstructed from the data, evaluated along the prospective trajectories of the Mercury Planetary Orbiter (MPO) using Capon’s method. Especially, it turns out that a high-precision determination of Mercury’s octupole field is expectable from the future analysis of the magnetic field data measured by the magnetometer on board MPO. Furthermore, magnetic field data of the MESSENGER mission are analyzed and the reconstructed internal Gauss coefficients are in reasonable agreement with the results from more conventional methods such as the least-square fit.

A secular variation candidate model for IGRF-13 based on Swarm data and ensemble inverse geodynamo modelling

This paper describes the design of a candidate secular variation model for the 13th generation of the International Geomagnetic Reference Field. This candidate is based upon the integration of an ensemble of 100 numerical models of the geodynamo between epochs 2019.0 and 2025.0. The only difference between each ensemble member lies in the initial condition that is used for the numerical integration, all other control parameters being fixed. An initial condition is defined as follows: an estimate of the magnetic field and its rate-of-change at the core surface for 2019.0 is obtained from a year (2018.5–2019.5) of vector Swarm data. This estimate (common to all ensemble members) is subject to prior constraints: the statistical properties of the numerical dynamo model for the main geomagnetic field and its secular variation, and prescribed covariances for the other sources. One next considers 100 three-dimensional core states (in terms of flow, buoyancy and magnetic fields) extracted at different discrete times from a dynamo simulation that is not constrained by observations, with the time distance between each state exceeding the dynamo decorrelation time. Each state is adjusted (in three dimensions) in order to take the estimate of the geomagnetic field and its rate-of-change for 2019.0 into account. This methodology provides 100 different initial conditions for subsequent numerical integration of the dynamo model up to epoch 2025.0. Focussing on the 2020.0–2025.0 time window, we use the median average rate-of-change of each Gauss coefficient of the ensemble and its statistics to define the geomagnetic secular variation over that time frame and its uncertainties.

A general study of charged particles dynamics near magnetic planets

A general study of the Störmer problem is carried taking into account the combined effect of the dipolar and quadrupolar magnetic terms as well as the gauss coefficients for Earth and Saturn. The resulted trajectories as well as the critical states for various kinds of charged dust grains are studied.

Bootstrapping Swarm and observatory data to generate candidates for the DGRF and IGRF-13

As posted by the Working Group V of the International Association of Geomagnetism and Aeronomy (IAGA), the 13th generation of the International Geomagnetic Reference Field (IGRF) has been released at the end of 2019. Following IAGA recommendations, in this work we present a candidate model for the IGRF-13, for which we have used the available Swarm satellite and geomagnetic observatory ground data for the last year. In order to provide the IGRF-13 candidate, we have extrapolated the Gauss coefficients of the main field and its secular variation to January 1st, 2020. In addition, we have generated a Definitive Geomagnetic Reference Field model for 2015.0 using the same modelling approach, but focussed on a 1-year time window of data centred on 2015.0. To jointly model both satellite and ground data, we have followed the classical protocols and data filters applied in geomagnetic field modelling. Novelty arrives from the application of bootstrap analysis to solve issues related to the inhomogeneity of the spatial and temporal data distributions. This new approach allows the estimation of not only the Gauss coefficients, but also their uncertainties.

Covariant Giant Gaussian Process Models With Improved Reproduction of Palaeosecular Variation

AbstractA commonly used family of statistical magnetic field models is based on a giant Gaussian process (GGP), which assumes each Gauss coefficient can be realized from an independent normal distribution. GGP models are capable of generating suites of plausible Gauss coefficients, allowing for palaeomagnetic data to be tested against the expected distribution arising from a time‐averaged geomagnetic field. However, existing GGP models do not simultaneously reproduce the distribution of field strength and palaeosecular variation estimates reported for the past 10 million years and tend to underpredict virtual geomagnetic pole (VGP) dispersion at high latitudes unless trade‐offs are made to the fit at lower latitudes. Here we introduce a new family of GGP models, BB18 and BB18.Z3 (the latter includes non‐zero‐mean zonal terms for spherical harmonic degrees 2 and 3). Our models are distinct from prior GGP models by simultaneously treating the axial dipole variance separately from higher degree terms, applying an odd‐even variance structure, and incorporating a covariance between certain Gauss coefficients. Covariance between Gauss coefficients, a property both expected from dynamo theory and observed in numerical dynamo simulations, has not previously been included in GGP models. Introducing covariance between certain Gauss coefficients inferred from an ensemble of “Earth‐like” dynamo simulations and predicted by theory yields a reduced misfit to VGP dispersion, allowing for GGP models which generate improved reproductions of the distribution of field strengths and palaeosecular variation observed for the last 10 million years.

The Capon Method for Mercury's Magnetic Field Analysis

Characterization of Mercury's internal and external magnetic field is one of the primary goals of the magnetometer experiment on board the BepiColombo MPO (Mercury Planetary Orbiter) spacecraft. A novel data analysis tool is developed to determine the Gauss coefficients in the multipole expansion using Capon's minimum variance projection method. The construction of the estimator is presented along with a test against the numerical simulation data of Mercury's magnetosphere and a comparison with the least square fitting method shows, that Capon's estimator is in better agreement with the coefficients, implemented in the simulation, than the least square fit estimator.

Correlation based snapshot models of the archeomagnetic field

SUMMARY For the time stationary global geomagnetic field, a new modelling concept is presented. A Bayesian non-parametric approach provides realistic location dependent uncertainty estimates. Modelling related variabilities are dealt with systematically by making little subjective a priori assumptions. Rather than parametrizing the model by Gauss coefficients, a functional analytic approach is applied. The geomagnetic potential is assumed a Gaussian process to describe a distribution over functions. A priori correlations are given by an explicit kernel function with non-informative dipole contribution. A refined modelling strategy is proposed that accommodates non-linearities of archeomagnetic observables: First, a rough field estimate is obtained considering only sites that provide full field vector records. Subsequently, this estimate supports the linearization that incorporates the remaining incomplete records. The comparison of results for the archeomagnetic field over the past 1000 yr is in general agreement with previous models while improved model uncertainty estimates are provided.

Giant Gaussian process models of geomagnetic palaeosecular variation: a directional outlook

SUMMARY Assessment of long-term palaeosecular variation (PSV) of the geomagnetic field is frequently based on simplified versions of a class of statistical models known as giant Gaussian processes (GGP) used to represent temporal variations in spherical harmonic descriptions of the field. Here we propose a new type of analysis to assess the shape and dispersion of the directional distributions caused by PSV. The quantities analysed in this study are equal-area coordinates of rotated distributions of palaeomagnetic directions, ${x_E}$ (east−west) and ${x_N}\ $(north−south) and their standard deviations (${\sigma _E}$ and ${\sigma _N}$). These are easy to determine, and can readily be numerically predicted for any GGP model, avoiding the need for the numerous simulations generally used to determine the scatter and/or elongation of directional distributions. Mean predictions of $\overline {{x_N}} $ for a simplified GGP model are different from the expected geocentric axial dipole (GAD) directions, in agreement with inclination differences noted in previous studies. The best estimates for palaeomagnetic inclination are the expected directions from the mean of virtual geomagnetic poles (VGPs) calculated using an iterative angular cut-off process. Predictions of ${\sigma _{\rm E}}$ and ${\sigma _{\rm N}}$ vary with latitude and are symmetric about the Equator. The N–S direction (${\sigma _{\rm N}}$) is always larger than E–W (${\sigma _{\rm E}}$), but the difference decreases from a maximum at the Equator to the poles, where ${\sigma _{\rm N}} = \ {\sigma _{\rm E}}$. A simplified GGP model is used to show that the parameter α (affecting variances in all Gauss coefficients) is positively correlated with ${\sigma _{\rm E}}$ and ${\sigma _{\rm N}}\ $ while the β parameter, the ratio of dipole to quadrupole family standard deviations, modifies the latitudinal dependence of ${\sigma _{\rm E}}$ and ${\sigma _{\rm N}}$. Experimental error in ${\sigma _{\rm E}}$ and ${\sigma _{\rm N}}$ can be accommodated using the common statistical parameters found in palaeomagnetic data sets, as ${\alpha _{95}}$ from site-mean directions. Predictions of simplified GGP models are compared with both numerical simulations and real data spanning the last 10 Ma. The latitudinal dependence of the proposed measures of PSV (${\sigma _{\rm E}}$ and ${\sigma _N}$) provide useful diagnostics for testing the validity of a GGP model. For the past 10 Ma the best-fitting GGP model with a mean GAD field set to $g_1^0 = \ - 18\ \mu T$ has α = 6.7 µT and β = 4.2. These new directional diagnostics will be used to investigate changes in overall geomagnetic field behaviour over other geological time intervals.

The landscape of Saturn’s internal magnetic field from the Cassini Grand Finale

The Cassini mission entered the Grand Finale phase in April 2017 and executed 22.5 highly inclined, close-in orbits around Saturn before diving into the planet on September 15th 2017. Here we present our analysis of the Cassini Grand Finale magnetometer (MAG) dataset, focusing on Saturn’s internal magnetic field. These measurements demonstrate that Saturn’s internal magnetic field is exceptionally axisymmetric, with a dipole tilt less than 0.007 degrees (25.2 arcsecs). Saturn’s magnetic equator was directly measured to be shifted northward by ∼ 0.0468 ± 0.00043 (1σ) RS, 2820±26 km, at cylindrical radial distances between 1.034 and 1.069 RS from the spin-axis. Although almost perfectly axisymmetric, Saturn’s internal magnetic field exhibits features on many characteristic length scales in the latitudinal direction. Examining Br at the a=0.75RS, c=0.6993RS isobaric surface, the degree 4 to 11 contributions correspond to latitudinally banded magnetic perturbations with characteristic width ∼15∘, similar to that of the off-equatorial zonal jets observed in the atmosphere of Saturn. Saturn’s internal magnetic field beyond 60∘, in particular the small-scale features, are less well constrained by the available measurements, mainly due to incomplete spatial coverage in the polar region. Magnetic fields associated with the ionospheric Hall currents were estimated and found to contribute less than 2.5 nT to Gauss coefficients beyond degree 3. The magneto-disk field features orbit-to-orbit variations between 12 nT and 15.4 nT along the close-in part of Grand Finale orbits, offering an opportunity to measure the electromagnetic induction response from the interior of Saturn. A stably stratified layer thicker than 2500 km likely exists above Saturn’s deep dynamo to filter out the non-axisymmetric internal magnetic field. A heat transport mechanism other than pure conduction, e.g. double diffusive convection, must be operating within this layer to be compatible with Saturn’s observed luminosity. The latitudinally banded magnetic perturbations likely arise from a shallow secondary dynamo action with latitudinally banded differential rotation in the semi-conducting layer.

Spatial power spectrum of the photospheric magnetic field during solar minimum

Context. During solar minima the spatial power spectrum of the photospheric magnetic field is dominated by the low-degree zonal (axisymmetric; m = 0) harmonic components, reflecting the large polar coronal holes of unipolar magnetic field. However, measuring polar fields is difficult because of the unequal visibility of the two poles during most of the year and the small line-of-sight component of the roughly radial field at high solar latitudes. Aims. In this paper we derive the spatial power spectrum of the photospheric magnetic field in terms of the harmonic coefficients of the radial component (Br) as well as in terms of the harmonic coefficients of the internal potential (known as Gauss coefficients). We calculate the zonal spatial power spectrum using Mount Wilson Observatory synoptic maps from 1995–1996, during the solar minimum between solar cycles 22 and 23, and investigate how filling or not filling the polar data gaps affects the zonal harmonic coefficients. Methods. We eliminated the vantage point effect by removing the highest 5° of the measured magnetic field and calculating the latitudinal profile of the zonal median field over the two years, which ensured equal latitudinal data coverage of both solar hemispheres. We then derived the zonal harmonic coefficients using this latitudinal profile of Br. Results. We find that when the polar data gaps are left unfilled, a strong artificial power above l = 8 is produced. Only the first five zonal harmonic coefficients can be considered reliable in this case. Therefore polar filling is essential to obtain a realistic spatial power spectrum. Filling the polar gap with a constant (non-zero) value yields zonal harmonics that are reliable up to l = 9. We find that the zonal octupole component contributes most to the total spatial power, more than the zonal dipole, even during the solar minimum conditions. This difference is seen more clearly in the case of polar filling. We also prove that the asymmetry of the polar fields during this solar minimum is statistically significant. Conclusions. Our results emphasize the importance of filling the polar data gaps in order to obtain a correct estimate of the spatial power spectrum of the photospheric field. This helps in estimating the reliability of polar fields and the large-scale structure in synoptic maps of different origin. Our results also verify the asymmetric nature of the polar fields, which is important for the heliospheric magnetic field and for solar dynamo modeling.