G. Spada

Creep, relaxation and viscosity properties for basic fractional models in rheology

Abstract.The purpose of this paper is twofold: from one side we provide a general survey to the viscoelastic models constructed via fractional calculus and from the other side we intend to analyze the basic fractional models as far as their creep, relaxation and viscosity properties are considered. The basic models are those that generalize via derivatives of fractional order the classical mechanical models characterized by two, three and four parameters, that we refer to as Kelvin–Voigt, Maxwell, Zener, anti–Zener and Burgers. For each fractional model we provide plots of the creep compliance, relaxation modulus and effective viscosity in non dimensional form in terms of a suitable time scale for different values of the order of fractional derivative. We also discuss the role of the order of fractional derivative in modifying the properties of the classical models.

Vertical GPS ground motion rates in the Euro‐Mediterranean region: New evidence of velocity gradients at different spatial scales along the Nubia‐Eurasia plate boundary

We use 2.5 to 14 years long position time series from >800 continuous Global Positioning System (GPS) stations to study vertical deformation rates in the Euro-Mediterranean region. We estimate and remove common mode errors in position time series using a principal component analysis, obtaining a significant gain in the signal-to-noise ratio of the displacements data. Following the results of a maximum likelihood estimation analysis, which gives a mean spectral index ~ −0.7, we adopt a power law + white noise stochastic model in estimating the final vertical rates and find 95% of the velocities within ±2 mm/yr, with uncertainties from filtered time series ~40% smaller than from the unfiltered ones. We highlight the presence of statistically significant velocity gradients where the stations density is higher. We find undulations of the vertical velocity field at different spatial scales both in tectonically active regions, like eastern Alps, Apennines, and eastern Mediterranean, and in regions characterized by a low or negligible tectonic activity, like central Iberia and western Alps. A correlation between smooth vertical velocities and topographic features is apparent in many sectors of the study area. Glacial isostatic adjustment and weathering processes do not completely explain the measured rates, and a combination of active tectonics and deep-seated geodynamic processes must be invoked. Excluding areas where localized processes are likely, or where subduction processes may be active, mantle dynamics is the most likely process, but regional mantle modeling is required for a better understanding.

Postglacial rebound in a non‐Newtonian spherical Earth

Using finite-elements we study the postglacial rebound in an axisymmetric Earth with a non-Newtonian rheology. Our approach allows for the first time a self-consistent study of the effect of a power-law creep on the long-wavelength deformations, which cannot be performed using flat models. We show that, for a non-Newtonian upper mantle, the time-variations of the low-degree Stokes coefficients become largely insensitive to the viscosity of the lower mantle. This is in contrast with previous evidence based on Newtonian models, which emphasized the importance of the low degree signatures for constraining the lower mantle viscosity.

Vertical ground displacement at Campi Flegrei (Italy) in the fifth century: Rapid subsidence driven by pore pressure drop

Campi Flegrei (Italy) caldera has experienced episodes of ground deformation throughout its geological history, alternating between uplift and subsidence phases. Although uplift periods are typically more alarming, here we focus on subsidence, looking for its driving mechanisms and its role in the caldera evolution. Historical and archaeological records constrain ground deformation over the last two millennia. Here we revise such records and combine them with published radiometric dating and with the simulation of sea level change. The resulting analysis highlights for the first time a rapid subsidence during the fifth century. We show that rate and magnitude of this subsidence are consistent with the compaction of porous material caused by a pressure drop of ~ 1 MPa within the hydrothermal system. We interpret this event as the decompression of the hydrothermal system following an unrecognized episode of unrest, during Roman times. These findings redefine the pattern of ground deformation and bear important implications for volcanic hazard assessment.

Large earthquakes and Earth rotation: The role of mantle relaxation

By means of a radially stratified spherical model and using the available moment tensor solutions, we evaluate the contribution of large earthquakes to polar drift in the last two decades. In contrast to previous analyses, which only considered the elastic response of the Earth to the release of seismic moment, we account for mantle rheology and thus for the time-dependent postseismic inertia changes driven by global seismicity. We confirm that during the last twenty years the global seismic activity has not significantly altered the rotational parameters of the Earth. However, we find that for values of asthenospheric viscosity suggested by recent studies of postseismic stress diffusion, the delayed relaxation of the mantle amplifies the average rate of polar drift by a factor of 1.2 to 1.7 (depending on the width of the asthenosphere) even on a decade time-scale.

Time-dependent residual deformations associated with the June 9, 1994 Bolivia Earthquake

By means of a spherical Earth model with Maxwell rheology we compute the coseismic and the postseismic displacements associated with the June 9, 1994 Bolivia earthquake. Our results are compared with the coseismic uplift detected by an array of broadband seismic stations in Southern Bolivia and with previous theoretical predictions. Our solution method has been recently employed to study the global deformations due to large lithospheric earthquakes. In contrast to purely elastic Earth models, our method includes the time-dependent effects associated with mantle relaxation. The results presented here show that the surface deformation due to a deep-focus earthquake are sensibly magnified by the delayed viscoelastic mantle relaxation. The time-scales which characterize mantle relaxation and the time evolution of surface observables depend on the rheological profile of the upper mantle, and particularly on the thickness of the viscoelastic asthenosphere.

A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus

Abstract We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter ν ∈ (0, 1], the logarithmic creep law known in rheology as Lomnitz law (obtained for ν = 1 ). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second kind. So doing we provide a full characterization of the new model both in creep and in relaxation representation, where the slow varying functions of logarithmic type play a fundamental role as required in processes of ultra slow kinetics.

A heuristic evaluation of long-term global sea-level acceleration

In view of the scientific and social implications, the global mean sea level rise (GMSLR) and its possible causes and future trend have been a challenge for so long. For the twentieth century, reconstructions generally indicate a rate of GMSLR in the range of 1.5 to 2.0 mm yr−1. However, the existence of nonlinear trends is still debated, and current estimates of the secular acceleration are subject to ample uncertainties. Here we use various GMSLR estimates published on scholarly journals since the 1940s for a heuristic assessment of global sea level acceleration. The approach, alternative to sea level reconstructions, is based on simple statistical methods and exploits the principles of meta-analysis. Our results point to a global sea level acceleration of 0.54 ± 0.27 mm/yr/century (1σ) between 1898 and 1975. This supports independent estimates and suggests that a sea level acceleration since the early 1900s is more likely than currently believed.

Evaluating model simulations of 20th century sea-level rise. Part 1: global mean sea-level change

AbstractSea level change is one of the major consequences of climate change and is projected to affect coastal communities around the world. Here, global mean sea level (GMSL) change estimated by 12 climate models from phase 5 of the World Climate Research Programme’s Climate Model Intercomparison Project (CMIP5) is compared to observational estimates for the period 1900–2015. Observed and simulated individual contributions to GMSL change (thermal expansion, glacier mass change, ice sheet mass change, landwater storage change) are analyzed and compared to observed GMSL change over the period 1900–2007 using tide gauge reconstructions, and over the period 1993–2015 using satellite altimetry estimates. The model-simulated contributions explain 50% ± 30% (uncertainties 1.65σ unless indicated otherwise) of the mean observed change from 1901–20 to 1988–2007. Based on attributable biases between observations and models, a number of corrections are proposed, which result in an improved explanation of 75% ± 38% o...

Regional sea level changes for the 20th and the 21st century induced by the regional variability in Greenland ice sheet surface mass loss

AbstractSurface mass balance (SMB) variations of the Greenland ice sheet (GrIS) have been identified as an important contributor to contemporary and projected global mean sea level variations, but their impact on the regional sea level change pattern is still poorly known. This study proposes estimates of GrIS SMB over 1900–2100 based on the output of 32 atmosphere–ocean general circulation models and Earth system models involved in phase 5 of the Climate Model Intercomparison Project (CMIP5). It is based on a downscaling technique calibrated against the Modele Atmospherique Regional (MAR) regional climate model and it provides an ensemble of 32 Greenland SMB estimates for each Greenland major drainage basin. Because the GrIS SMB does not respond uniformly to greenhouse gas (GHG) emissions, the southern part of the GrIS is more sensitive to climate warming. This study shows that this part should be in imbalance in the twenty-first century sooner than the northern part. This regional variability significan...