Horizontal Boundary Conditions Research Articles

Increasing accuracy of 3-D geomechanical-numerical models

SUMMARY The current crustal stress field is of key importance to understand geodynamic processes and to assess stability aspects during subsurface usage. To provide a 3-D continuous description of the stress state, linear elastic forward geomechanical-numerical models are used. These models solve the equilibrium of forces between gravitational volume forces and surfaces forces imposed mainly by plate tectonics. The latter are responsible for the horizontal stress anisotropy and impose the inverse problem to estimate horizontal displacement boundary conditions that provide a fit best to horizontal stress magnitude data within the model volume. However, horizontal stress magnitude data have high uncertainties and they are sparse, clustered and not necessarily representative for a larger rock volume. Even when Bayesian statistics are incorporated and additional stress information such as borehole failure observations or formation integrity test are used to further constrain the solution space, this approach may result in a low accuracy of the model results, that is the result is not correct. Here, we present an alternative approach that removes the dependence of the solution space based on stress magnitude data to avoid potential low accuracy. Initially, a solution space that contains all stress states that are physically reasonable is defined. Stress magnitude data and the additional stress information are then used in a Bayesian framework to evaluate which solutions are more likely than others. We first show and validate our approach with a generic truth model and then apply it to a case study of the Molasse foreland basin of the Alps in Southern Germany. The results show that the model's ability to predict a reliable stress state is increasing while the number of likely solutions may also increase, and that outlier of stress magnitude data can be identified. This alternative approach results in a substantial increase in computational speed as we perform most of the calculations analytically.

Retrieving 3D distributions of atmospheric particles using Atmospheric Tomography with 3D Radiative Transfer – Part 2: Local optimization

Abstract. Our global understanding of clouds and aerosols relies on the remote sensing of their optical, microphysical, and macrophysical properties using, in part, scattered solar radiation. Current retrievals assume clouds and aerosols form plane-parallel, homogeneous layers and utilize 1D radiative transfer (RT) models. These assumptions limit the detail that can be retrieved about the 3D variability in the cloud and aerosol fields and induce biases in the retrieved properties for highly heterogeneous structures such as cumulus clouds and smoke plumes. In Part 1 of this two-part study, we validated a tomographic method that utilizes multi-angle passive imagery to retrieve 3D distributions of species using 3D RT to overcome these issues. That validation characterized the uncertainty in the approximate Jacobian used in the tomographic retrieval over a wide range of atmospheric and surface conditions for several horizontal boundary conditions. Here, in Part 2, we test the algorithm's effectiveness on synthetic data to test whether the retrieval accuracy is limited by the use of the approximate Jacobian. We retrieve 3D distributions of a volume extinction coefficient (σ3D) at 40 m resolution from synthetic multi-angle, mono-spectral imagery at 35 m resolution derived from stochastically generated cumuliform-type clouds in (1 km)3 domains. The retrievals are idealized in that we neglect forward-modelling and instrumental errors, with the exception of radiometric noise; thus, reported retrieval errors are the lower bounds. σ3D is retrieved with, on average, a relative root mean square error (RRMSE) < 20 % and bias < 0.1 % for clouds with maximum optical depth (MOD) < 17, and the RRMSE of the radiances is < 0.5 %, indicating very high accuracy in shallow cumulus conditions. As the MOD of the clouds increases to 80, the RRMSE and biases in σ3D worsen to 60 % and −35 %, respectively, and the RRMSE of the radiances reaches 16 %, indicating incomplete convergence. This is expected from the increasing ill-conditioning of the inverse problem with the decreasing mean free path predicted by RT theory and discussed in detail in Part 1. We tested retrievals that use a forward model that is not only less ill-conditioned (in terms of condition number) but also less accurate, due to more aggressive delta-M scaling. This reduces the radiance RRMSE to 9 % and the bias in σ3D to −8 % in clouds with MOD ∼ 80, with no improvement in the RRMSE of σ3D. This illustrates a significant sensitivity of the retrieval to the numerical configuration of the RT model which, at least in our circumstances, improves the retrieval accuracy. All of these ensemble-averaged results are robust in response to the inclusion of radiometric noise during the retrieval. However, individual realizations can have large deviations of up to 18 % in the mean extinction in clouds with MOD ∼ 80, which indicates large uncertainties in the retrievals in the optically thick limit. Using less ill-conditioned forward model tomography can also accurately infer optical depths (ODs) in conditions spanning the majority of oceanic cumulus fields (MOD < 80), as the retrieval provides ODs with bias and RRMSE values better than −8 % and 36 %, respectively. This is a significant improvement over retrievals using 1D RT, which have OD biases between −30 % and −23 % and RRMSE between 29 % and 80 % for the clouds used here. Prior information or other sources of information will be required to improve the RRMSE of σ3D in the optically thick limit, where the RRMSE is shown to have a strong spatial structure that varies with the solar and viewing geometry.

Retrieving 3D distributions of atmospheric particles using Atmospheric Tomography with 3D Radiative Transfer – Part 1: Model description and Jacobian calculation

Abstract. Our global understanding of clouds and aerosols relies on the remote sensing of their optical, microphysical, and macrophysical properties using, in part, scattered solar radiation. These retrievals assume that clouds and aerosols form plane-parallel, homogeneous layers and utilize 1D radiative transfer (RT) models, limiting the detail that can be retrieved about the 3D variability in cloud and aerosol fields and inducing biases in the retrieved properties for highly heterogeneous structures such as cumulus clouds and smoke plumes. To overcome these limitations, we introduce and validate an algorithm for retrieving the 3D optical or microphysical properties of atmospheric particles using multi-angle, multi-pixel radiances and a 3D RT model. The retrieval software, which we have made publicly available, is called Atmospheric Tomography with 3D Radiative Transfer (AT3D). It uses an iterative, local optimization technique to solve a generalized least squares problem and thereby find a best-fitting atmospheric state. The iterative retrieval uses a fast, approximate Jacobian calculation, which we have extended from Levis et al. (2020) to accommodate open and periodic horizontal boundary conditions (BCs) and an improved treatment of non-black surfaces. We validated the accuracy of the approximate Jacobian calculation for derivatives with respect to both the 3D volume extinction coefficient and the parameters controlling the open horizontal boundary conditions across media with a range of optical depths and single-scattering properties and find that it is highly accurate for a majority of cloud and aerosol fields over oceanic surfaces. Relative root mean square errors in the approximate Jacobian for a 3D volume extinction coefficient in media with cloud-like single-scattering properties increase from 2 % to 12 % as the maximum optical depths (MODs) of the medium increase from 0.2 to 100.0 over surfaces with Lambertian albedos <0.2. Over surfaces with albedos of 0.7, these errors increase to 20 %. Errors in the approximate Jacobian for the optimization of open horizontal boundary conditions exceed 50 %, unless the plane-parallel media providing the boundary conditions are optically very thin (∼0.1). We use the theory of linear inverse RT to provide insight into the physical processes that control the cloud tomography problem and identify its limitations, supported by numerical experiments. We show that the Jacobian matrix becomes increasing ill-posed as the optical size of the medium increases and the forward-scattering peak of the phase function decreases. This suggests that tomographic retrievals of clouds will become increasingly difficult as clouds become optically thicker. Retrievals of asymptotically thick clouds will likely require other sources of information to be successful. In Loveridge et al. (2023a; hereafter Part 2), we examine how the accuracy of the retrieved 3D volume extinction coefficient varies as the optical size of the target medium increases using synthetic data. We do this to explore how the increasing error in the approximate Jacobian and the increasingly ill-posed nature of the inversion in the optically thick limit affect the retrieval. We also assess the accuracy of retrieved optical depths and compare them to retrievals using 1D radiative transfer.

Ground motion around a semi-cylindrical canyon in saturated soil: P wave incidence

Based on Biot wave theory in saturated two-phase medium, the analytical solution of plane P-wave scattering by semi-circular canyon in half space in saturated two-phase medium is given by using complex function method and Hankel function integral transformation method. Firstly, a semi-circular depression model in saturated half space is constructed, the wave field potential function in the medium is solved, and its general solution form is obtained. Secondly, the unknown coefficient value in the potential function is obtained by substituting the corresponding stress boundary conditions. With the help of Hankel function integral transformation method, the wave potential function in complex coordinates is transformed to meet the horizontal boundary conditions of half space in rectangular coordinates. Finally, the surface displacement amplitude at the semi-circular depression and the surrounding horizontal boundary is obtained through the total wave field potential function, and the variation law of surface displacement amplitude under the influence of different parameters is analyzed.

Collective Drifts in Vibrated Granular Packings: The Interplay of Friction and Structure.

We simulate vertically shaken dense granular packings with horizontal periodic boundary conditions. A coordinated translating motion of the whole medium emerges when the horizontal symmetry is broken by disorder or defects in the packing and the shaking is weak enough to conserve the structure. We argue that such a drift originates in the interplay between structural symmetry breaking and frictional forces transmitted by the vibrating plate. A nonlinear ratchet model with stick slips reproduces many faces of the phenomenon. The collective motion discussed here underlies phenomena observed recently with vibrofluidized granular materials, such as persistent rotations and anomalous diffusion.

Effects of Horizontal Resolution, Domain Size, Boundary Conditions, and Surface Heterogeneity on Coarse LES of a Convective Boundary Layer

Abstract Atmospheric properties in a convective boundary layer vary over a wide range of spatial scales and are commonly studied using large-eddy simulations (LES) in various configurations. We examine how the boundary layer depth and distribution of variability across scales are affected by LES grid spacing, domain size, inhomogeneity of surface properties, and external forcing. Two different setups of the Weather Research and Forecasting (WRF) model are analyzed. A semi-idealized configuration uses a periodic domain, flat surface, prescribed homogeneous surface heat fluxes, and horizontally uniform profiles of large-scale advective tendencies. A nested LES setup employs a larger domain and realistic initial and boundary conditions, including an interactive land surface model with representative topography and vegetation and soil types. Subdomains of identical size are analyzed for all simulations. Characteristic structure sizes are quantified using the variability scales L50 and L95, defined such that features smaller than that contain 50% and 95% of the total variance, respectively. Progressive increase in L50 from vertical velocity to temperature and moisture structures is systematically reproduced in all simulation configurations. This dependence of L50 on the considered variable complicates the development of scale-aware parameterizations for models with grid spacing in the “terra incognita”. In simulations using a larger domain with heterogeneous surface properties, the development of internal mesoscale patterns significantly affects variance distributions inside analyzed subdomains. Sizes of boundary layer structures also strongly depend on the LES grid spacing and, in case of heterogeneous surface and topography, on location of the subdomain inside a larger computational domain.

Estimation of groundwater recharge in a shallow sandy aquifer using unsaturated zone modeling and water table fluctuation method

Quantification of groundwater recharge is one of the most important issues in hydrogeology, especially in view of the ongoing changes in climate and land use. In this study, we use numerical models of 1D vertical flow in the vadose zone and the water table fluctuation (WTF) analysis to investigate local-scale recharge of a shallow sandy aquifer in the Brda outwash plain in northern Poland. We show that these two methods can be jointly used to improve confidence in recharge estimation. A set of preliminary numerical simulations based on soil water content measurements from 4 grassland and pine forest profiles provided a wide range of recharge estimates (263 mm to 839 mm for a 3-year period). Additional simulations were performed with the lower boundary condition specified as a functional relationship between the groundwater table elevation and the rate of groundwater outflow from the vertical profile (horizontal drains boundary condition). In this way, we could reproduce the water table fluctuations resulting from recharge and lateral discharge to nearby lakes. The agreement between simulated and observed groundwater levels differed depending on the specific set of parameters characterizing vadose zone flow, which allowed us to find the most representative parameter sets and refine the range of plausible recharge estimates (501 mm to 573 mm per 3 years). The recharge rates from WTF (410 mm to 606 mm per 3 years) were in good agreement with numerical simulations, providing that the effect of the natural recession of groundwater table due to lateral outflow was considered (master recession curve method). Our results show that: (i) the proposed approach combining 1D vadose zone modeling and WTF improves recharge estimation, (ii) multiple types of observations, including groundwater table positions, are needed to calibrate and validate vadose zone flow models, and (iii) extended periods of observations and simulations are necessary to capture year-to-year variability in the recharge rates.

Theoretical model for predicting the break-up of ice covers due to wave-ice interaction

An approximate solution for wave transmission and reflection between open water and a finite viscoelastic ice cover is developed in the present study, in which both the water part and the ice cover are treated as a continuum, each governed by its own equation of motion. The interface conditions include matching velocities and stresses between the two continua. In this study, only two modes of the dispersion relation are considered, and horizontal boundary conditions are approximated by matching mean values. The reflection and transmission coefficients are determined for simplified cases to compare with earlier theories. Behaviors of the resonance are obtained for pure elastic ice covers. The effect of viscosity is also investigated for viscoelastic ice covers. According to the above wave-ice interaction model, the stress and strain are calculated in the ice cover. By using a stress failure criterion, the break-up condition of ice covers with different thickness and length under the interaction of waves and sea ice is determined. Finally, a dynamic evolution model of the ice floe size distribution function in the marginal ice zone is established. The present study is useful for modeling the wave-in-ice climate on a geophysical scale.

A Numerical Study of The Outflow of Makassar Strait Using Regional Ocean Modelling System (ROMS)

In the Makassar Strait, upwelling occurred by wind-driven current and complexity of bottom topography. Analysis of upwelling phenomena could be determined by salinity, temperature and three-dimensional current value which happened in a year as a climatology change. Regional Ocean Modelling System (ROMS) is a terrain-following three-dimensional model to calculate a numerical value which needs stability for long-duration simulation. The model has been simulated for 1 year from 1 January 2017 to 31 December 2017 and verified by tidal data. Horizontal Boundary Condition which contains u-v velocities, salinity and temperature, is a climatology data from GLBa0.08 HYCOM. Atmospheric forcing was calculated from ERA-INTERIM ECMWF. The verification uses RMSE and RRE methods and showed good result between model and observation data with RMSE=0.1454 meter and RRE=6.5806%. The result was analysed from twelve sampling points, six points on the outflow of the Makassar Strait which generated upwelling and the six others on the canal that connect to the Flores Sea and the Java Sea. The model gave a good result on January, and indicate average w-velocity of 9.1183 m/day.

Singular limit of the nonisentropic compressible ideal MHD equations in a domain with boundary

ABSTRACT In this paper, we study the singular limit of three-dimensional nonisentropic compressible ideal MHD equations with horizontal boundary conditions. It is rigorously justified that, when the Mach number and Alfvén number are same and tend to zero, the solution of original equations converges to that of two-dimensional incompressible flow.

Performance of RegCM4 simulations over Croatia and adjacent climate regions

AbstractWe investigate the impact of horizontal resolution, convection parameterization and lateral boundary conditions on the ability of a regional climate model ‐ RegCM4 to simulate the observed temperature and precipitation climatology over Croatia and surrounding regions. An ensemble of RegCM4 simulations over Europe has been generated including two resolutions (50 and 12.5 km), two different deep convection schemes and different boundary conditions (a reanalysis and four global climate models). Using two observational data sets (with and without the undercatch correction) shows that the magnitude of RegCM4 errors in winter precipitation also depends on observational uncertainty. While the mean precipitation amounts in high‐resolution simulations are generally overestimated in the cold part of the year, low‐resolution RegCM4 simulations are comparable to most of other EURO‐CORDEX models. When forced by ERA‐Interim, RegCM4 performs relatively poorly in reproducing temporal correlations against observations, but performs well in terms of spatial correlation. Whereas precipitation amounts from high‐resolution simulations need further improvements, the RegCM4 skill in simulating near‐surface temperature is high and well suited for impact studies.

SINGV‐DA: A data assimilation system for convective‐scale numerical weather prediction over Singapore

AbstractSINGV‐DA is a convective‐scale numerical weather prediction system with regional data assimilation for Singapore and the surrounding region. This article documents SINGV‐DA's current operational configuration and the sensitivity studies that influenced its development. We show that background error covariances derived by bootstrapping (via the lagged National Meteorological Centre method) contain spurious vertical structures at higher model levels that may degrade forecast performance. We found that SINGV‐DA precipitation forecasts are sensitive to horizontal resolution and lateral boundary conditions. Our observing system experiments reveal that satellite radiance assimilation, while clearly beneficial for precipitation forecasts in this region, adversely affected model background temperatures and winds at higher altitudes. Benchmarked against the forecast model in isolation, the regional DA system adds significant value to precipitation forecasts in the nowcasting range, but not at longer lead times. Our findings point to the need for further research and development to improve the system.

A CLT for the total energy of the two-dimensional critical Ising model

Consider the Ising model on $([1,2N]\times[1,2M])\cap\mathbb{Z}^2$ at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let $E_{M,N}$ be its total energy (or Hamiltonian). Suppose $M$ is a function of $N$ satisfying $M\geq N/(\ln N)^{\alpha}$ for some $\alpha\in[0,1)$. In particular, one may take $M=N$. We prove that \begin{equation*} \frac{E_{M,N}+4\sqrt{2}M N-(4/\pi)N\ln N}{\sqrt{(32/\pi)MN\ln N}} \end{equation*} converges weakly to a standard Gaussian distribution as $N\rightarrow\infty$.

Rapidly rotating Rayleigh-Bénard convection with a tilted axis.

We numerically explore the dynamics of an incompressible fluid heated from below, bounded by free-slip horizontal plates and periodic lateral boundary conditions, subject to rapid rotation about a distant axis that is tilted with respect to the gravity vector. The angle ϕ between the rotation axis and the horizontal plane measures the tilting of the rotation axis; it can be taken as a proxy for latitude if we think of a local Cartesian representation of the convective dynamics in a rotating fluid shell. The results of the simulations indicate the existence of three different convective regimes, depending on the value of ϕ: (1) sheared, intermittent large-scale winds in the direction perpendicular to the plane defined by the gravity and rotation vectors, when rotation is "horizontal" (ϕ=0^{∘}); (2) a large-scale cyclonic vortex tilted along the rotation axis, when the angle between the rotation axis and the gravity vector is relatively small (ϕ between about 45^{∘} and 90^{∘}); and (3) a new intermediate regime characterized by vertically sheared large-scale winds perpendicular to both gravity and rotation. In this regime, the winds are organized in bands that are tilted along the rotation axis, with unit horizontal wave number in the plane defined by gravity and rotation at values of ϕ less than about 60^{∘}. This intermediate solution, studied for the first time in this work, is characterized by weaker vertical heat transport than the cases with large-scale vortices. For intermediate values of ϕ (between about 45^{∘} and 60^{∘}), the banded, sheared solution coexists with the large-scale vortex solution, with different initial conditions leading to one or the other dynamical behavior. A discussion of the possible implications of these results for the dynamics of rapidly rotating planetary atmospheres is provided.

Impact of the Diurnal Cycle of the Atmospheric Boundary Layer on Wind-Turbine Wakes: A Numerical Modelling Study

The wake characteristics of a wind turbine for different regimes occurring throughout the diurnal cycle are investigated systematically by means of large-eddy simulation. Idealized diurnal cycle simulations of the atmospheric boundary layer are performed with the geophysical flow solver EULAG over both homogeneous and heterogeneous terrain. Under homogeneous conditions, the diurnal cycle significantly affects the low-level wind shear and atmospheric turbulence. A strong vertical wind shear and veering with height occur in the nocturnal stable boundary layer and in the morning boundary layer, whereas atmospheric turbulence is much larger in the convective boundary layer and in the evening boundary layer. The increased shear under heterogeneous conditions changes these wind characteristics, counteracting the formation of the night-time Ekman spiral. The convective, stable, evening, and morning regimes of the atmospheric boundary layer over a homogeneous surface as well as the convective and stable regimes over a heterogeneous surface are used to study the flow in a wind-turbine wake. Synchronized turbulent inflow data from the idealized atmospheric boundary-layer simulations with periodic horizontal boundary conditions are applied to the wind-turbine simulations with open streamwise boundary conditions. The resulting wake is strongly influenced by the stability of the atmosphere. In both cases, the flow in the wake recovers more rapidly under convective conditions during the day than under stable conditions at night. The simulated wakes produced for the night-time situation completely differ between heterogeneous and homogeneous surface conditions. The wake characteristics of the transitional periods are influenced by the flow regime prior to the transition. Furthermore, there are different wake deflections over the height of the rotor, which reflect the incoming wind direction.

Localized traveling pulses in natural doubly diffusive convection

Two-dimensional natural doubly diffusive convection in a vertical slot driven by an imposed temperature difference in the horizontal is studied using numerical continuation and direct numerical simulation. Two cases are considered and compared. In the first a concentration difference that balances thermal buoyancy is imposed in the horizontal and stationary localized structures are found to be organized in a standard snakes-and-ladders bifurcation diagram. Disconnected branches of traveling pulses TPn consisting of n, n = 1,2, . . . , corotating cells are identified and shown to accumulate on a tertiary branch of traveling waves. With Robin or mixed concentration boundary conditions on one wall all localized states travel and the hitherto stationary localized states may connect up with the traveling pulses. The stability of the TPn states is determined and unstable TPn shown to evolve into spatio-temporal chaos. The calculations are done with no-slip boundary conditions in the horizontal and periodic boundary conditions in the vertical.

Insights into convective momentum transport and its parametrization from idealized simulations of organized convection

Deep convection is a multiscale process that influences the budgets of heat, moisture and momentum significantly. In global climate models, the thermodynamic effects of convection are normally treated by parametrization schemes, with a separate formulation for convective momentum transport (CMT). The schemes for current thermodynamic and momentum parametrizations are based on upright entraining plume models that do not account for vertically tilted mesoscale circulations, which characterize organized convection in sheared environments. The associated countergradient vertical transport of horizontal momentum fundamentally affects dynamical interactions between the convection and the mean flow. This study examines the CMT properties of simulated idealized mesoscale convective systems, including their sensitivity to horizontal resolution, domain size and lateral boundary conditions. It is found that, even for large domains, the horizontal gradient terms are important, especially the mesoscale pressure gradients, which are neglected in CMT parametrizations. A nonlinear analytic model provides a dynamical foundation for the effects of convective organization, including the role of the horizontal pressure gradient. It is found that a small computational domain affects the convective organization adversely by generating artificially large compensating subsidence and an unrealistic evolution of CMT. Finally, analyses of the cross‐updraught/downdraught pressure gradients expose significant uncertainties in their representation in contemporary CMT parametrization schemes.

Deformation of an elastic substrate due to a resting sessile droplet

On a sufficiently soft substrate, a resting fluid droplet will cause significant deformation of the substrate. This deformation is driven by a combination of capillary forces at the contact line and the fluid pressure at the solid surface. These forces are balanced at the surface by the solid traction stress induced by the substrate deformation. Young's Law, which predicts the equilibrium contact angle of the droplet, also indicates an a priori radial force balance for rigid substrates, but not necessarily for soft substrates that deform under loading. It remains an open question whether the contact line transmits a non-zero force tangent to the substrate surface in addition to the conventional normal (vertical) force. We present an analytic Fourier transform solution technique that includes general interfacial energy conditions, which govern the contact angle of a 2D droplet. This includes evaluating the effect of gravity on the droplet shape in order to determine the correct fluid pressure at the substrate surface for larger droplets. Importantly, we find that in order to avoid a strain singularity at the contact line under a non-zero tangential contact line force, it is necessary to include a previously neglected horizontal traction boundary condition. To quantify the effects of the contact line and identify key quantities that will be experimentally accessible for testing the model, we evaluate solutions for the substrate surface displacement field as a function of Poisson's ratio and zero/non-zero tangential contact line forces.

Numerical simulations of oscillation-driven regolith motion: Brazil-nut effect

Many if not most small asteroids are rubble piles covered by regolith, and small perturbations may be enough to disturb their surfaces in complex ways due to microgravity. Experiments to study low-gravity regolith dynamics are challenging, and properly validated numerical simulations can provide valuable insights. In this paper, we investigate numerically size segregation among regolith grains, which is likely to occur after repeated shaking events. In particular, we are interested in the so-called Brazil-nut effect (BNE), i.e. the migration of a large intruder towards the top of a vertically shaken granular system. We go a step forward in simulating this effect by implementing horizontal periodic boundary conditions (PBC) in the N-body code pkdgrav, with the aim of making the simulations more representative of the expected asteroid environment. We study the influence of PBC on the BNE in Earth gravity and compare them with a walled case. We also investigate the influence of static and rolling friction on the BNE. With walls, we observe the well-known convection mechanism driving the BNE. However, we find that a different mechanism, consisting of void filling, is responsible for the BNE with PBC, and we discuss its relevance in light of previous studies. By running simulations in 10−4g, we show that this void-filling mechanism remains relevant in a low-gravity environment. However, we find that depending on the gravity level, the void-filling mechanism is differently influenced by the friction properties of particles. We speculate that this is likely due to a change in the granular flow time-scales.

Present-day uplift of the western Alps

Collisional mountain belts grow as a consequence of continental plate convergence and eventually disappear under the combined effects of gravitational collapse and erosion. Using a decade of GPS data, we show that the western Alps are currently characterized by zero horizontal velocity boundary conditions, offering the opportunity to investigate orogen evolution at the time of cessation of plate convergence. We find no significant horizontal motion within the belt, but GPS and levelling measurements independently show a regional pattern of uplift reaching ~2.5 mm/yr in the northwestern Alps. Unless a low viscosity crustal root under the northwestern Alps locally enhances the vertical response to surface unloading, the summed effects of isostatic responses to erosion and glaciation explain at most 60% of the observed uplift rates. Rock-uplift rates corrected from transient glacial isostatic adjustment contributions likely exceed erosion rates in the northwestern Alps. In the absence of active convergence, the observed surface uplift must result from deep-seated processes.