The innermost Galilean moon, Io, exhibits widespread tidally-driven volcanism. Monitoring of its volcanoes has revealed that they are not homogeneously distributed across its surface: volcanic activity is higher at low latitudes and peaks east of the sub- and anti-Jovian points. Dissipation in a radially symmetric solid body cannot explain the observed longitudinal shift but dissipation in a magma ocean can. However, recent observations show that Io does not have one. Here, we demonstrate that a longitudinal shift in the heating pattern naturally arises from the feedback between tidal heating and melt production. The feedback between tidal dissipation and interior properties that results in interiors that deviate from radial symmetry is expected to drive the interior evolution of other tidally-active worlds, including icy moons such as Europa and Enceladus and exo-planets/moons with high eccentricity or obliquity.

Rates of relative sea-level rise during the final stage of the last deglaciation, the early Holocene, are key to understanding future ice melt and sea-level change under a warming climate¹. Data about these rates are scarce², and this limits insight into the relative contributions of the North American and Antarctic ice sheets to global sea-level rise during the early Holocene. Here we present an early Holocene sea-level curve based on 88 sea-level data points (13.7–6.2 thousand years ago (ka)) from the North Sea (Doggerland^3,4). After removing the pattern of regional glacial isostatic adjustment caused by the melting of the Eurasian Ice Sheet, the residual sea-level signal highlights two phases of accelerated sea-level rise. Meltwater sourced from the North American and Antarctic ice sheets drove these two phases, peaking around 10.3 ka and 8.3 ka with rates between 8 mm yr⁻¹ and 9 mm yr⁻¹. Our results also show that global mean sea-level rise between 11 ka and 3 ka amounted to 37.7 m (2σ range, 29.3–42.2 m), reconciling the mismatch that existed between estimates of global mean sea-level rise based on ice-sheet reconstructions and previously limited early Holocene sea-level data. With its broad spatiotemporal coverage, the North Sea dataset provides critical constraints on the patterns and rates of the late-stage deglaciation of the North American and Antarctic ice sheets, improving our understanding of the Earth-system response to climate change.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.