Correction to: Nature Astronomyhttps://doi.org/10.1038/s41550-024-02253-4, published online 29 April 2024 In the version of the article initially published, the affiliation of Wouter van der Wal was incorrect and has now been corrected to the Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands in the HTML and PDF versions of the article.

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Io exhibits widespread volcanism powered by tides raised by Jupiter. The distribution of volcanoes offers a window into the interior of the moon. The distribution shows more volcanism at the equator as well as peak volcanic output which is shifted by roughly 30-60 degrees to the east of the subjovian point [1]. Models of tidal dissipation that assume a spherically symmetric, solid Io cannot reproduce this shift [2]. More recently, it has been proposed that tidal dissipation in a magma ocean [3] or in a non-spherically symmetric, solid Io [4] can induce this lag. In this study, we explore the second option and show that solid-body dissipation can induce an eastward shift of the tidal dissipation pattern.@en

The Cassini mission has provided measurements of the gravity of several moons of Saturn as well as an estimate of the tidal response, which is expressed as the degree 2 Love number k₂ of its largest moon, Titan. The first estimates of Titan’s Love number were larger than pre-Cassini expectations. Interior modelling suggested it may be explained with a dense ocean, but the interpretation remains unclear. We analysed Cassini tracking data to determine Titan’s gravity field and its Love number. Our gravity results are consistent with earlier studies, but we find a lower Love number for Titan of k₂ = 0.375 ± 0.06. This lower value follows from an elaborate investigation of the tidal effects. We show that a dense ocean is not implied by the obtained Love number; instead, a water or ammonia ocean is more probable. A lower density ocean can increase the likeliness of contact between the silicate core and ocean, which can leach minerals into the ocean and could promote its habitability.

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Geodynamic processes in Antarctica such as glacial isostatic adjustment (GIA) and post-seismic deformation are measured by geo-detic observations such as global navigation satellite systems (GNSS) and satellite gravimetry. GNSS measurements have comprised both continuous measurements and episodic measurements since the mid-1990s. The estimated velocities typically reach an accuracy of 1 mm a⁻¹ for horizontal velocities and 2 mm a⁻¹ for vertical velocities. However, the elastic deformation due to present-day ice-load change needs to be considered accordingly. Space gravimetry derives mass changes from small variations in the inter-satellite distance of a pair of satellites, starting with the GRACE (Gravity Recovery and Climate Experiment) satellite mission in 2002 and continuing with the GRACE-FO (GRACE Follow-On) mission launched in 2018. The spatial resolution of the measurements is low (about 300 km) but the measurement error is homogeneous across Ant-arctica. The estimated trends contain signals from ice-mass change, and local and global GIA signals. To combine the strengths of the individual datasets, statistical combinations of GNSS, GRACE and satellite altimetry data have been developed. These combinations rely on realistic error estimates and assumptions of snow density. Nevertheless, they capture signals that are missing from geodynamic forward models such as the large uplift in the Amundsen Sea sector caused by a low-viscous response to century-scale ice-mass changes.

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The Antarctic mantle and lithosphere are known to have large lateral contrasts in seismic velocity and tectonic history. These contrasts suggest differences in the response timescale of mantle flow across the continent, similar to those documented between the northeastern and southwestern upper mantle of North America. Glacial isostatic adjustment and geodynamical modelling rely on independent estimates of lateral variability in effective viscosity. Recent improvements in imaging techniques and the distribution of seismic stations now allow reso-lution of both lateral and vertical variability of seismic velocity, making detailed inferences about lateral viscosity variations possible. Geodetic and palaeosea-level investigations of Antarctica provide quantitative ways of independently assessing the 3D mantle viscosity structure. While observational and causal connections between inferred lateral viscosity variability and seismic velocity changes are qualitatively reconciled, significant improvements in the quantitative relations between effective viscosity anomalies and those imaged by P-and S-wave tomography have remained elusive. Here we describe several methods for estimating effective viscosity from S-wave velocity. We then present and com-pare maps of the viscosity variability beneath Antarctica based on the recent S-wave velocity model ANT-20 using three different approaches.

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Significant land uplift and horizontal motions have been recorded with Global Navigation Satellite Systems (GNSS) in areas such as Alaska, Iceland and the Northern Antarctic Peninsula (NAP) as a result of Glacial Isostatic Adjustment (GIA) due to ice melt after the Little Ice Age. Here, analysis of horizontal displacement rates can be of extra importance, as they are more sensitive to Earth properties in shallower layers than vertical displacement rates. Proper modelling of horizontal displacement rates with dedicated GIA models requires a spherical Earth with compressible rheology. However, in these small areas, the used GIA models are often incompressible using a Cartesian geometry to ease computation and in some cases allow for lateral viscosity changes or more complex rheology. We investigate the validity of modelled horizontal displacement rates using different approximations, that is using spherical or Cartesian Earth structures, and incompressible, material compressible or compressible rheology. Although the lack of self-gravity and sphericity compensate each other in the vertical, this is less the case for the horizontal. For a disc ice sheet with a radius just over 200 km and a thickness of 1000 m, differences due to sphericity are minimal and the modelled horizontal displacement rates of compressible Cartesian models differ from those simulated by a compressible spherical model by 0.63 mm a-1. Thus, compressible Cartesian GIA models can be applied for modelling horizontal displacement rates of small ice sheets like those in Alaska, Iceland and NAP. Unfortunately, the implementation of compressibility in Abaqus that we use here cannot be extended to spherical models as gravity can not be specified for a spherical body. Other modelling approaches are recommended in such cases.

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This paper presents a method that modifies commercial engineering-oriented finite element packages for the modelling of Glacial Isostatic Adjustment (GIA) on a self-gravitating, compressible and spherical Earth with 3-D structures. The approach, called the iterative finite element body and surface force (FEMIBSF) approach, solves the equilibrium equation for deformation using the ABAQUS finite element package and calculates potential perturbation consistently with finite element theory, avoiding the use of spherical harmonics. The key to this approach lies in computing the mean external body forces for each finite element within the Earth and pressure on Earth's surface and core-mantle boundary (CMB). These quantities, which drive the deformation and stress perturbation of GIA but are not included in the equation of motion of commercial finite element packages, are implemented therein. The method also demonstrates how to calculate degree-1 deformation directly in the spatial domain and Earth-load system for GIA models. To validate the FEMIBSF method, loading Love numbers (LLNs) for homogeneous and layered earth models are calculated and compared with three independent GIA methodologies: the normal-mode method, the iterative body force method and the spectral-finite element method. Results show that the FEMIBSF method can accurately reproduce the unstable modes for the homogeneous compressible model and agree reasonably well with the Love number results from other methods. It is found that the accuracy of the FEMIBSF method increases with higher resolution, but a non-conformal mesh should be avoided due to creating the so-called hanging nodes. The role of a potential force at the CMB is also studied and found to only affect the long-wavelength surface potential perturbation and deformation in the viscous time regime. In conclusion, the FEMIBSF method is ready for use in realistic GIA studies, with modelled vertical and horizontal displacement rates in a disc load case showing agreement with other two GIA methods within the uncertainty level of GNSS measurements.

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This chapter reviews glacial isostatic adjustment (GIA) and post-seismic deformation in Antarctica. It discusses numerical models and their inputs, and observations and inferences that have been made from them. Both processes are controlled by mantle viscosity but their forcings are different. Ongoing GIA induced by the loss of ice since the last glacial maximum (LGM) could have amounted to 5–15 m of global sea-level rise. However, mantle viscosity is so low in parts of West Antarctica (c. 10¹⁸ Pa s) that changes in ice thickness over the last centuries and decades have controlled the current uplift rates there. The uplift due to GIA has promoted ice-sheet stability since the LGM, and in West Antarctica GIA is a significant negative feedback on the current decline of the ice sheet. Post-seismic deformation following the 1998 earthquake near the Balleny Islands south of New Zealand has been detected in global navigation satellite system (GNSS) data and compared to model outputs. The best-fitting viscosity for this area is c. 10¹⁹ Pa s, similar to GIA-based estimates for the Antarctic Peninsula. Future work should focus on unifying descriptions of viscosity across geodynamic models, and integrating information from seismic, gravity, experimental and geological data.

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Ice losses from the Greenland and Antarctic ice sheets have accelerated since the 1990s, accounting for a significant increase in the global mean sea level. Here, we present a new 29-year record of ice sheet mass balance from 1992 to 2020 from the Ice Sheet Mass Balance Inter-comparison Exercise (IMBIE). We compare and combine 50 independent estimates of ice sheet mass balance derived from satellite observations of temporal changes in ice sheet flow, in ice sheet volume, and in Earth's gravity field. Between 1992 and 2020, the ice sheets contributed 21.0±1.9g€¯mm to global mean sea level, with the rate of mass loss rising from 105g€¯Gtg€¯yr-1 between 1992 and 1996 to 372g€¯Gtg€¯yr-1 between 2016 and 2020. In Greenland, the rate of mass loss is 169±9g€¯Gtg€¯yr-1 between 1992 and 2020, but there are large inter-annual variations in mass balance, with mass loss ranging from 86g€¯Gtg€¯yr-1 in 2017 to 444g€¯Gtg€¯yr-1 in 2019 due to large variability in surface mass balance. In Antarctica, ice losses continue to be dominated by mass loss from West Antarctica (82±9g€¯Gtg€¯yr-1) and, to a lesser extent, from the Antarctic Peninsula (13±5g€¯Gtg€¯yr-1). East Antarctica remains close to a state of balance, with a small gain of 3±15g€¯Gtg€¯yr-1, but is the most uncertain component of Antarctica's mass balance. The dataset is publicly available at 10.5285/77B64C55-7166-4A06-9DEF-2E400398E452 (IMBIE Team, 2021).

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The Antarctic mantle, bounded between the core and the Mohorovicǐćdiscontinuity, is one of the most difficult targets of study on Earth because of ice cover and rare outcrops. A multidisciplinary approach is adopted in this volume, using petrology, geochemistry, remote-sensed data and geodesy to characterize the Antarctic mantle. This characterization has application to rates of glacial isostatic adjustment, heat flow, sea-level rise and tectonics. It places the Antarctic mantle domain in a global framework on a scale not attempted before. In this chapter we review the historical development of mantle studies in Antarctica, outline current research directions, introduce the volume chapters and pro-vide a summary and outlook.

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Glacial isostatic adjustment (GIA) has a stabilizing effect on the evolution of the Antarctic ice sheet by reducing the grounding line migration following ice melt. The timescale and strength of this feedback depends on the spatially varying viscosity of the Earth's mantle. Most studies assume a relatively long and laterally homogenous response time of the bedrock. However, the mantle viscosity is spatially variable, with a high mantle viscosity beneath East Antarctica and a low mantle viscosity beneath West Antarctica. For this study, we have developed a new method to couple a 3D GIA model and an ice sheet model to study the interaction between the solid Earth and the Antarctic ice sheet during the last glacial cycle. With this method, the ice sheet model and GIA model exchange ice thickness and bedrock elevation during a fully coupled transient experiment. The feedback effect is taken into account with a high temporal resolution, where the coupling time steps between the ice sheet and GIA model are 5000 years over the glaciation phase and vary between 500 and 1000 years over the deglaciation phase of the last glacial cycle. During each coupling time step, the bedrock elevation is adjusted at every ice sheet model time step, and the deformation is computed for a linearly changing ice load. We applied the method using the ice sheet model ANICE and a 3D GIA finite element model. We used results from a regional seismic model for Antarctica embedded in the global seismic model SMEAN2 to determine the patterns in the mantle viscosity. The results of simulations over the last glacial cycle show that differences in mantle viscosity of an order of magnitude can lead to differences in the grounding line position up to 700gkm and to differences in ice thickness of the order of 2gkm for the present day near the Ross Embayment. These results underline and quantify the importance of including local GIA feedback effects in ice sheet models when simulating the Antarctic ice sheet evolution over the last glacial cycle.

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The inferred density of Enceladus' core, together with evidence of hydrothermal activity within the moon, suggests that the core is porous. Tidal dissipation in an unconsolidated core has been proposed as the main source of Enceladus' geological activity. However, the tidal response of its core has generally been modeled assuming it behaves viscoelastically rather than poroviscoelastically. In this work, we analyze the poroviscoelastic response to better constrain the distribution of tidal dissipation within Enceladus. A poroviscoelastic body has a different tidal response than a viscoelastic one; pressure within the pores alters the stress field and induces a Darcian porous flow. This flow represents an additional pathway for energy dissipation. Using Biot's theory of poroviscoelasticity, we develop a new framework to obtain the tidal response of a spherically symmetric, self-gravitating moon with porous layers and apply it to Enceladus. We show that the boundary conditions at the interface of the core and overlying ocean play a key role in the tidal response. The ocean hinders the development of a large-amplitude Darcian flow, making negligible the Darcian contribution to the dissipation budget. We therefore infer that Enceladus' core can be the source of its geological activity only if it has a low rigidity and a very low viscosity. A future mission to Enceladus could test this hypothesis by measuring the phase lags of tidally induced changes of gravitational potential and surface displacements.

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ANET-POLENET (Antarctic Network of the Polar Earth Observing Network) bedrock GPS sites in the Ross Sea region of Antarctica are in close proximity to a major LGM load center in the Siple region, and therefore are thought to reflect motion due to GIA. For the simplest case, horizontal bedrock motion is expected in a radial pattern away from the former load, yet we instead observe three primary patterns of deformation; 1) motions are reversed towards the load in the southern region of the Transantarctic Mountains (TAM), 2) motions are radially away from the load in the Marie Byrd Land (MBL) region, and 3) an overall gradient in motion is present, with magnitudes progressively increasing from East to West Antarctica. We aim to understand these distinct patterns of horizontal bedrock motion and the causal sources of deformation by investigating alternative earth models and ice loading scenarios, with the goal of improving our understanding of GIA and ice mass change in Antarctica. We explore ice loading scenarios for the Wilkes Subglacial Basin (LGM time scale) and the Siple Coast (centennial and millennial time scales), using GIA models with 1D earth models. We find that the spatial extent of deformation resulting from Wilkes and Siple loading is significant, but that no 1D model, regardless of the earth model and ice loading scenario used, is able to reproduce all three distinct patterns of observed motion at the same time. For select loading scenarios, we also examine 3D GIA models by invoking a boundary in Earth properties beneath the Transantarctic Mountains. This approach accounts for the strong lateral gradient in earth properties across the continent by effectively separating East and West Antarctica into two different earth model profiles. Some of our GIA models utilizing 3D earth structure are able to reproduce predicted motions that directionally match all three observed patterns of deformation. Best fitting ice history and earth models are presented, including preferred upper mantle viscosity values.@en

Accurate glacial isostatic adjustment (GIA) models are required for correcting measurements of mass change in Antarctica and for improving knowledge of the sub-surface, especially in areas of large current ice loss such as the Amundsen Sea Embayment (ASE). Regionally, seismic and gravity data suggests lateral differences in viscosity (3D). Furthermore, mantle flow laws allow for a stress-dependent effective viscosity which changes over time (3D-s). In this study we investigate whether models with 3D/3D-s have significant effects on the uplift in the region. We use a finite element model with composite rheology consisting of diffusion and dislocation creep, forced by an ice deglaciation model starting in 1900. We use its uplift predictions as synthetic observations to test the performance of 1D model inversion in the presence of viscosity variations. Stress-dependent rheology results in lower viscosity beneath the load and a more localized uplift pattern. We demonstrate that the background stress from earlier ice load changes can both increase or decrease the influence of stress-induced effective viscosity changes. For the ASE, fitting 1D models to 3D model uplift results in a best fitting model with viscosity that represents the average of a large area, while for 3D-s rheology, local viscosity is more influential. 1D models are statistically indistinguishable from 3D/3D-s viscosity with current GPS stations. However, 3D and 3D-s models should be taken into account when accurate uplift and gravity rate patterns are needed, as uplift can differ up to 45% compared to 1D models in between existing GPS stations.

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Thousands of exoplanets have been discovered; however, the detection of exomoons remains elusive. Tidally heated exomoons have been proposed as candidate targets for observation; vigorous tidal dissipation can raise the moon's surface temperature, making direct imaging possible, and cause widespread volcanism that can have a signature in transits. We assess whether the required amounts of tidal dissipation can be attained and how long it can be sustained. In a first step, we look at the thermal state of a super-Io for different orbital configurations. We show that close-in exomoons with moderate (e ≈ eIo) to high (e ≈ 0.1) orbital eccentricities can feature surface heat fluxes 1-3 orders of magnitude higher than that of Io if heat transfer is dominated by heat piping or the moon has a magma ocean. In a second step, we investigate the longevity of a super-Io. The free eccentricity of an isolated close-in exomoon is quickly dampened due to tides; high orbital eccentricities can be maintained if the moon is in a mean-motion resonance with another moon and the planet is highly dissipative. However, this scenario leads to fast orbital migration. For a Mars-sized exomoon, we find that tides alone can raise the surface temperatures to more than 400 K for 10 million yr, and surface heat fluxes higher than that of Io can be maintained for billions of years. Such tidally active bodies are expected to feature more vigorous volcanic activity than Io. The material outgassed via volcanism might be detected in transits.

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In Southeast Alaska, extreme uplift rates are primarily caused by glacial isostatic adjustment (GIA), as a result of ice thickness changes from the Little Ice Age to the present combined with a low-viscosity asthenosphere. Previous GIA models adopted a 1-D Earth structure. However, the actual Earth structure is likely more complex due to the long history of subduction and tectonism and the transition from a continental to an oceanic plate. Seismic evidence indeed shows a laterally heterogenous Earth structure. In this study, a numeral model is constructed for Southeast Alaska, which allows for the inclusion of lateral viscosity variations. The viscosity follows from scaling relationships between seismic velocity anomalies and viscosity variations. We use this scaling relationship to constrain the thermal effect on seismic variations and investigate the importance of lateral viscosity variations. We find that a thermal contribution to seismic anomalies of 10% is required to explain the GIA observations. This implies that non-thermal effects control seismic anomaly variations in the shallow upper mantle. Due to the regional geologic history, it is likely that hydration of the mantle impact both viscosity and seismic velocity. The best-fit model has a background viscosity of 5.0 × 10¹⁹ Pa-s, and viscosities at ∼80 km depth range from 1.8 × 10¹⁹ to 4.5 × 10¹⁹ Pa-s. A 1-D averaged version of the 3-D model performed slightly better, however, the two models were statistically equivalent within a 2σ measurement uncertainty. Thus, lateral viscosity variations do not contribute significantly to the uplift rates measured with the current accuracy and distribution of sites.

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The Earth’s surface and interior deform due to a changing load of the Antarctic Ice Sheet (AIS) during the last glacial cycle, called Glacial Isostatic Adjustment (GIA). This deformation changes the surface height of the ice sheet and indirectly the groundling line position. These changes in surface height and grounding line position influence the evolution of the AIS and consequently, again the load on the Earth’s surface. As a result, GIA operates as a negative feedback loop and could stabilize the evolution of the AIS. This feedback maybe particularly relevant for relatively low viscosities of the mantle in West Antarctica which lead to a relatively fast response time of the bedrock due to changes in the West Antarctic Ice Sheet loading. Most studies capture this process by ignoring lateral variations in the viscosity of the mantle and the stabilizing GIA feedback loop. Here we present a new method to couple an ice sheet model to a GIA model at a variable timestep in the order of a thousand years. Several experiments have been done using different radial and lateral varying rheologies for simulations of the last glacial cycle. It is shown that the effect of including lateral variations and accounting for the stabilizing GIA feedback is up to 80 kilometers for the grounding line position and 400 meters for the ice thickness. The largest differences are observed close to the grounding line of the Ronne ice shelf and at several locations in East Antarctica. The total ice volume of the AIS increases by 0.5 percent over 5000 years when including the 3D GIA feedback loops in the coupled model. These results quantify the local importance of including GIA feedback effects in ice dynamic models when simulating the Antarctic Ice Sheet evolution over the full glacial cycle. @en

Long-term monitoring of global mass transport within the Earth system improves our ability to mitigate natural hazards and better understand their relations to climate change. Satellite gravity is widely used to monitor surface mass variations for its unprecedented spatial and temporal coverage. However, the gravity data contain signals from visco-elastic deformation in response to past ice sheet melting, preventing us from extracting signals of present-day surface mass trend (PDMT) directly. Here we present a global inversion scheme that separates PDMT and visco-elastic glacial isostatic adjustment (GIA) signatures by combining satellite gravimetry with satellite altimetry and ground observations. Our inversion provides global dual data coverage that enables a robust separation of PDMT and GIA spherical harmonic coefficients. It has the advantage of providing estimates of Earth's long wavelength deformation signatures and their uncertainties. Our GIA result, along with its uncertainty estimates, can be used in future GRACE processing to better assess the impact of GIA on surface mass change. Our GIA estimates include a rapid GIA uplift in the Southeast Alaska and the Amundsen Sea Embayment, due to the visco-elastic response to recent glacial unloading. We estimate the average surface mass change rate from 2002–2010 to be −203 ± 3 GT·a⁻¹ in Greenland, −126 ± 18 GT·a⁻¹ in Antarctica and, −62 ± 5 GT·a⁻¹ in Alaska. The GIA low degree spherical harmonic coefficients are sensitive to rheological properties in Earth's deep interior. Our low-degree GIA estimates include geocenter motion and (Formula presented.) which provide unique constraints to understand Earth's lower mantle and ice history.

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Tidal heating can play an important role in the formation and evolution of subsurface oceans of outer-planet moons. Up until now tidal heating has only been studied in subsurface oceans of spatially uniform thickness. We develop a numerical model to consider oceans of spatially variable thickness. We use the Laplace Tidal Equations for the ocean and model the ice shell using membrane theory. The problem is solved using the commercial Finite Element software Comsol Multiphysics®. We use this new model to study the tidal response of Enceladus' ocean with a twofold objective: to understand how ocean thickness variations modify the tidal response of a subsurface ocean and to assess if tidal dissipation in an Enceladan ocean with varying ocean thickness can explain the high heat flux emanating from Enceladus' South Polar Terrain and the perdurance of a subsurface ocean. We consider the effect of meridional ocean thickness changes of spherical harmonic degree two and three as suggested by topography and gravimetry data. We observe that an ocean with degree two topography responds with the same eigenmodes as an ocean of constant thickness but resonances occur for thicker oceans. However, resonant ocean thicknesses are still thin compared to current estimates for Enceladus ocean thickness. Rossby-Haurwitz waves, excited by the obliquity tide for thick oceans of constant thickness, are not excited at the tidal frequency when oceans of variable thickness are considered. This result implies that the role of the obliquity tide in ocean tidal-dissipation might have been overestimated for Enceladus and other icy worlds. An antisymmetric, degree-three ocean thickness variation mixes the ocean modes excited in a constant thickness ocean by the eccentricity and obliquity tide.

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The movement of sediment across Earth’s surface affects sea level by deforming the solid Earth, modifying the gravity field, and displacing and absorbing water. Recent studies show that accounting for sediment redistribution is important for interpreting and predicting sea-level change@en