Glaciological Observations Research Articles

Meteorological conditions of high-mountain part of the Altai Republic: present-day characteristics, estimates if variability, and effect on glacier regime

According to some meteorological and glaciological observations using statistical methods of cluster, harmony, and correlation analyses the assessment trends of the Altai mountainous region glaciological indicators and temperature regime have been obtained. The glaciological effect of the atmospheric circulation of the area under review by means of B.L. Dzerdzeevsky’s typing is studied. The temperature field is a good attribute spaces for classification of mountainous areas and identification of representative areas (classes) in the description of the dynamics of glaciers. Annual mass balance of the Malyi Aktru Glacier which is closely associated with amount of the accumulation decreases with the increase of zonal circulation. Warming which began in the mid 1970s has resulted in precipitation increase during the colder period and accumulation values intensity. Low-frequency harmonic with the period of 55–58 years (Quasi-half a century cycle) determined by the position of the planets of the Solar system and 3, 4, 6, 9, 11 years of periodicity (regularity) are revealed practically in all the ranks of indicators. Statistical characteristics of glaciological and climate indicators of the research can be used to predict seasonal, annual and current states of a climatic regime evaluation of the Altai glaciers and water resources in Western Siberia.

Low post-glacial rebound rates in the Weddell Sea due to Late Holocene ice-sheet readvance

Many ice-sheet reconstructions assume monotonic Holocene retreat for the West Antarctic Ice Sheet, but an increasing number of glaciological observations infer that some portions of the ice sheet may be readvancing, following retreat behind the present-day margin. A readvance in the Weddell Sea region can reconcile two outstanding problems: (i) the present-day widespread occurrence of seemingly stable ice streams grounded on beds that deepen inland; and (ii) the inability of models of glacial isostatic adjustment to match present-day uplift rates. By combining a suite of ice loading histories that include a readvance with a model of glacial isostatic adjustment we report substantial improvements to predictions of present-day uplift rates, including reconciling one problematic observation of land sinking. We suggest retreat behind present grounding lines occurred when the bed was lower, and isostatic recovery has since led to shallowing, ice sheet re-grounding and readvance. The paradoxical existence of grounding lines in apparently unstable configurations on reverse bed slopes may be resolved by invoking the process of unstable advance, in accordance with our load modelling.

Glaciological and meteorological observations at the SIGMA-D site, northwestern Greenland Ice Sheet（グリーンランド氷床北西部SIGMA-Dサイトにおける雪氷・気象観測）

Glaciological and meteorological observations at the SIGMA-D site, northwestern Greenland Ice Sheet（グリーンランド氷床北西部SIGMA-Dサイトにおける雪氷・気象観測）

‘For the sake of science and country’: the Ross Sea party 1914–1917

ABSTRACTIn December 1913 Sir Ernest Shackleton released a prospectus and announced The Imperial Trans-Antarctic Expedition. His goal was to undertake the first crossing of Antarctica from the Weddell Sea via the polar plateau to McMurdo Sound on the Ross Sea. The journey had already been attempted by Wilhelm Filchner whose shipDeutschland, had become beset in the Weddell Sea ice for nine months in 1912. Shackleton aimed ‘to make all possible scientific observations on [the Trans-Antarctic] journey; to carry on similar work by parties operating from the two bases on the Weddell and Ross Seas [and] to carry on scientific work, and travel unknown portions of the coastline, by the two ships of the expedition’(Shackleton 1913: 3). WithEndurancea continental crossing party of six led by Shackleton would begin from the Weddell Sea and a supporting depot laying party led byNimrodveteran Lieutenant Aeneas L.A. Mackintosh RNR, with the auxiliary barquentineAurorabased in McMurdo Sound. Unbeknown to each party, both experienced problems beyond their control.Endurancewas holed and sank in the Weddell Sea andAuroralocked in ice, although damaged, reached New Zealand. Here the ship was repaired and then undertook a relief expedition with Shackleton as a passenger, to McMurdo Sound. In spite of these major setbacks each party conducted valuable scientific observations.When Shackleton published his bookSouth(Shackleton 1919) on the expedition, compiled with New Zealand journalist and friend Edward Saunders, with exception of accounts on the Ross Sea party sledging and drift of the shipAurora, no recognition was given to work undertaken by the four Ross Sea party scientists and an assistant. Later publications have focused on the depot-laying, while books on Antarctic science have largely overlooked the science undertaken.The purpose of this paper is to make this better known, and to give credit to the four scientists involved. The science conducted although primarily concerned with meteorological observations, also covers limited glaciological observations including the ablation of lake ice, solution of glacier ice in salt water, tidal recordings, collection of zoological and other specimens, along with the use of improvised equipment to undertake observations. The science achieved was secondary to the field work. The Ross Sea party science was done however, under conditions not normally conducive for such field work with health issues a major contributing factor. A lack of funding, equipment, personality problems, concern forAuroraand crew, uncertainty of Shackleton's Antarctic crossing and their own relief, led to depression, sleeplessness and insomnia.

Evaluation of the Surface Representation of the Greenland Ice Sheet in a General Circulation Model

Simulated surface conditions of the Goddard Earth Observing System model, version 5 (GEOS-5), atmospheric general circulation model (AGCM) are examined for the contemporary Greenland Ice Sheet (GrIS). A surface parameterization that explicitly models surface processes including snow compaction, meltwater percolation and refreezing, and surface albedo is found to remedy an erroneous deficit in the annual net surface energy flux and provide an adequate representation of surface mass balance (SMB) in an evaluation using simulations at two spatial resolutions. The simulated 1980–2008 GrIS SMB average is 24.7 ± 4.5 cm yr−1 water-equivalent (w.e.) at ½° model grid spacing, and 18.2 ± 3.3 cm yr−1 w.e. for 2° grid spacing. The spatial variability and seasonal cycle of the ½° simulation compare favorably to recent studies using regional climate models, while results from 2° integrations reproduce the primary features of the SMB field. In comparison to historical glaciological observations, the coarser-resolution model overestimates accumulation in the southern areas of the GrIS, while the overall SMB is underestimated. These changes relate to the sensitivity of accumulation and melt to the resolution of topography. The GEOS-5 SMB fields contrast with available corresponding atmospheric models simulations from phase 5 of the Coupled Model Intercomparison Project (CMIP5). It is found that only a few of the CMIP5 AGCMs examined provide significant summertime runoff, a dominant feature of the GrIS seasonal cycle. This is a condition that will need to be remedied if potential contributions to future eustatic change from polar ice sheets are to be examined with GCMs.

Glaciological observations in 2012 and 2013 at SIGMA-A site, Northwest Greenland（北西グリーンランド , SIGMA-Aサイトにおける 2012年, 2013年の雪氷調査）

Glaciological observations were conducted in 2012 and 2013 at the SIGMA‑A site on the northwest Greenland ice sheet (78°03�06�N 67°37�42�W 1490 m a.s.l. as part of the Snow Impurity and Glacial Microbe effects on abrupt warming in the Arctic (SIGMA project. The meteorological conditions during the two observations were quite different. The meteorological condition during the 2012 observation period was warm and heavy rainfall occurred during the observation period thus the snow was very wet. In contrast the meteorological condition during the observation period in 2013 was cold with a blowing snow event thus the snow was quite dry. The glaciological observations in 2012 consisted of 1 snow‑stake measurements 2 snow pit observations 3 grain size observations for validation of satellite‑derived snow products 4 snow specific surface area measurements using a near‑infrared camera 5 snow sampling for chemical analyses and 6 drilling of firn cores with a hand auger. The glaciological observations in 2013 consisted of 1 snow‑stake measurements 2 snow pit observations and 3 snow sampling for chemical analyses.

Where to find 1.5 million yr old ice for the IPICS "Oldest-Ice" ice core

Abstract. The recovery of a 1.5 million yr long ice core from Antarctica represents a keystone of our understanding of Quaternary climate, the progression of glaciation over this time period and the role of greenhouse gas cycles in this progression. Here we tackle the question of where such ice may still be found in the Antarctic ice sheet. We can show that such old ice is most likely to exist in the plateau area of the East Antarctic ice sheet (EAIS) without stratigraphic disturbance and should be able to be recovered after careful pre-site selection studies. Based on a simple ice and heat flow model and glaciological observations, we conclude that positions in the vicinity of major domes and saddle position on the East Antarctic Plateau will most likely have such old ice in store and represent the best study areas for dedicated reconnaissance studies in the near future. In contrast to previous ice core drill site selections, however, we strongly suggest significantly reduced ice thickness to avoid bottom melting. For example for the geothermal heat flux and accumulation conditions at Dome C, an ice thickness lower than but close to about 2500 m would be required to find 1.5 Myr old ice (i.e., more than 700 m less than at the current EPICA Dome C drill site). Within this constraint, the resolution of an Oldest-Ice record and the distance of such old ice to the bedrock should be maximized to avoid ice flow disturbances, for example, by finding locations with minimum geothermal heat flux. As the geothermal heat flux is largely unknown for the EAIS, this parameter has to be carefully determined beforehand. In addition, detailed bedrock topography and ice flow history has to be reconstructed for candidates of an Oldest-Ice ice coring site. Finally, we argue strongly for rapid access drilling before any full, deep ice coring activity commences to bring datable samples to the surface and to allow an age check of the oldest ice.

東シベリア・スンタルハヤタ地域氷河調査（2013年）

東シベリア・スンタルハヤタ地域氷河調査（2013年）

Hydrologic vulnerability to climate change of the Mandrone glacier (Adamello-Presanella group, Italian Alps)

In order to assess the annual mass balance of the Mandrone glacier in the Central Alps an energy-balance model was applied, supported by snowpack, meteorological and glaciological observations, together with satellite measurements of snow covered areas and albedo. The Physically based Distributed Snow Land and Ice Model (PDSLIM), a distributed multi-layer model for temperate glaciers, which was previously tested on both basin and point scales, was applied.Verification was performed with a network of ablation stakes over two summer periods. Satellite images processed within the Global Land Ice Measurements from Space (GLIMS) project were used to estimate the ice albedo and to verify the position of the simulated transient snowline on specific dates. The energy balance was estimated for the Mandrone and Presena glaciers in the Central Italian Alps. Their modeled balances (−1439 and −1503mmw.e.year−1, respectively), estimated over a 15year period, are in good agreement with those obtained with the glaciological method for the Caresèr glacier, a WGMS (World Glacier Monitoring Service) reference located in the nearby Ortles-Cevedale group.Projections according to the regional climate model COSMO-CLM (standing for COnsortium for Small-scale MOdeling model in CLimate Mode) indicate that the Mandrone glacier might not survive the current century and might be halved in size by 2050.

Spatial and temporal variations in Greenland glacial‐earthquake activity, 1993–2010

Glacial earthquakes are anomalous earthquakes associated with large ice‐loss events occurring at marine‐terminating glaciers, primarily in Greenland. They are detectable teleseismically, and a proper understanding of the source mechanism may provide a remote‐sensing tool to complement glaciological observations of these large outlet glaciers. We model teleseismic surface‐wave waveforms to obtain locations and centroid–single‐force source parameters for 121 glacial earthquakes occurring in Greenland during the period 2006–2010. We combine these results with those obtained by previous workers to analyze spatial and temporal trends in glacial‐earthquake occurrence over the 18‐year period from 1993–2010. We also examine earthquake occurrence at six individual glaciers, comparing the earthquake record to independently obtained observations of glacier change. Our findings confirm the inference that glacial‐earthquake seismogenesis occurs through the capsize of large, newly calved icebergs. We find a close correspondence between episodes of glacier retreat, thinning, and acceleration and the timing of glacial earthquakes, and document the northward progression of glacial earthquakes on Greenland's west coast over the 18‐year observing period. Our results also show that glacial earthquakes occur when the termini of the source glaciers are very close to the glacier grounding line, i.e., when the glaciers are grounded or nearly grounded.

Lichenometric ages of the little ice age moraines on king george island and of the last volcanic activity on penguin island (west antarctica)

The recently observed recession of glaciers on King George Island is associated with decades of climate warming in the Antarctic Peninsula region. However, with only 60 years of glaciological observations in the study area ages of the oldest moraines are still uncertain. The goal of the study was to estimate ages of lichen colonization on the oldest moraines of the Ecology and White Eagle Glaciers on King George Island and on the Principal Cone of Penguin Island volcano. The first lichenometric studies on these islands from the late 1970s used rates that had about four to five times slower Rhizocarpon growth rates. We re‐examined the sites and measured 996 thalli diameters to establish the surface ages. To estimate the age we used (1) long‐term Rhizocarpon lichen group growth rates established by authors using data from a previous lichenometric study on King George Island, and (2) previous data of lichen growth rates from other sub‐Antarctic islands. Our results suggest growth rates between 0.5 and 0.8 mmyr–1. According to these rates the ages of the oldest moraine ridges are of the Little Ice Age and were colonized at the beginning of the twentieth century. The mid‐twentieth century age of lichen colonization on the historically active Penguin Island volcano might support the date of the last eruption reported by whalers in the end of the nineteenth and the beginning of the twentieth century.

Ice‐avalanche impact landforms: the event in 2003 at the glacier nördliches bockkarkees, hohe tauern range, austria

Kellerer‐Pirklbauer, A., Slupetzky, H. and Avian, M., 2012. Ice‐avalanche impact landforms: the event in 2003 at the glacier Nördliches Bockkarkees, Hohe Tauern Range, Austria. Geografiska Annaler: Series A, Physical Geography, 94, 97–115. doi:10.1111/j.1468‐0459.2011.00446.xABSTRACTThe hanging glacier Nördliches Bockkarkees (47°07′21″N; 12°44′00″E) represents the only Austrian glacier that formed large and hazardous ice avalanches over the last decades. Only one major ice avalanche was released between August 1993 and July 2011 underlying the rareness of such large ice avalanches in the Austrian Alps. The ice avalanche happened on 26 June 2003 and was very unusual for two reasons: First, the ice avalanche occurred suddenly without major signs of glacier instabilities even a few days before the event. Second, the ice avalanche was released very early in the season. Ice and debris with a volume of 300-000m3 avalanched over a vertical distance of 1200 m. The main deposits with a volume of 250-000m3 and a debris content of 10% formed a flat avalanche cone in the Käfer Valley covering 82-000m2. This paper focuses on geomorphic and glaciological field measurements and observations at and below the ice avalanche deposit and its vicinity after the event. Studies were made (a) at the deposited material itself, (b) at the reshaped land surface below the deposit after ice melt, (c) at the front and margin of the deposit, and (d) outside the deposit. Our study reveals different types of ice‐avalanche impact landforms comparable to landforms caused by snow avalanches. It is shown that ice avalanches charged with debris at the Nördliches Bockkarkees are important in forming and reshaping erosional and depositional landforms along the track. However, the conservation potential of these landforms is low.

東シベリア・スンタルハヤタ地域氷河調査（2012年）

東シベリア・スンタルハヤタ地域氷河調査（2012年）

Favorable climatic regime for maintaining the present-day geometry of the Gregoriev Glacier, Inner Tien Shan

Abstract. We conducted 2 yr (2005–2007) of in situ meteorological and glaciological observations on the Gregoriev Glacier, a flat-top glacier within the Inner Tien Shan, Kyrgyzstan. Relative carrier-phase GPS surveys reveal a vertical lowering at the summit of the glacier. Based on snow density data and an energy-mass balance model, we estimate that the annual precipitation and summer mean temperature required to maintain the glacier in the current state are 289 mm and −3.8 °C at the glacier summit (4600 m a.s.l.), respectively. The good agreement between dynamically derived precipitation and the long-term observed precipitation at a nearby station in the Tien Shan (296 mm at 3614 m a.s.l. for the period 1930–2002) suggests that the glacier has been in a near steady-state in terms of mass supply. The glacier mass-balance, reconstructed based on meteorological data from the Tien Shan station for the past 80 yr, explains the observed fluctuations in glacier extent, particularly the negative mass balance in the 1990s.

A model for turbulent hydraulic fracture and application to crack propagation at glacier beds

Glaciological observations of under‐flooding suggest that fluid‐induced hydraulic fracture of an ice sheet from its bed sometimes occurs quickly, possibly driven by turbulently flowing water in a broad sheet flow. Taking the approximation of a fully turbulent flow into an elastic ice medium with small fracture toughness, we derive an approximate expression for the crack‐tip speed, opening displacement and pressure profile. We accomplish this by first showing that a Manning‐Strickler channel model for resistance to turbulent flow leads to a mathematical structure somewhat similar to that for resistance to laminar flow of a power law viscous fluid. We then adapt the plane‐strain asymptotic crack solution of Desroches et al. (1994) and the power law self‐similar solution of Adachi and Detournay (2002) for that case to calculate the desired quantities. The speed of crack growth is shown to scale as the overpressure (in excess of ice overburden) to the power 7/6, inversely as ice elastic modulus to the power 2/3, and as the ratio of crack length to wall roughness scale to the power 1/6. We tentatively apply our model by choosing parameter values thought appropriate for a basal crack driven by the rapid drainage of a surface meltwater lake near the margin of the Greenland Ice Sheet. Making various approximations perhaps relevant to this setting, we estimate fluid inflow rate to the basal fracture and vertical and horizontal surface displacements and find order‐of‐magnitude agreement with observations by Das et al. (2008) associated with lake drainage. Finally, we discuss how these preliminary estimates could be improved.

Glacier thickening and decay analysis from 50 years of glaciological observations performed on Glacier d’Argentière, Mont Blanc area, France

AbstractNumerous glaciological data have been obtained from measurements carried out on Glacier d’Argentière, Mont Blanc area, France, since the beginning of the 20th century. Moreover, data on annual mass balance, ice-flow velocity, thickness variation and length fluctuation have been obtained from yearly measurements performed since 1975. This dataset provides an excellent opportunity to analyze the relationships between surface mass balance and dynamic response over time periods during which net mass balance changed from positive to negative. Following a positive specific-net-balance period between 1960 and 1981, the ablation zone experienced a large increase in thickness and ice-flow velocities. Conversely, the highly negative specific-net-balance period since 1982 has led to strong thinning, deceleration and retreat of the tongue. The response of these observed dynamics to surface mass balance is analyzed from ice-flux calculations performed on three transverse cross-sections. Our results reveal that the ice fluxes are largely accommodated by ice-flow velocities. Velocity fluctuations are synchronous over the entire area studied. In the largest part of the glacier, no compressing/extending flow change has been observed over the last 30 years and thickness changes are solely driven by surface mass-balance changes. However, on the tongue of the glacier, thickness changes do not depend on surface mass balance but are mainly driven by changes in the longitudinal strain rate.

Evolución de los glaciares en los Andes venezolanos: picos Humboldt y Bonpland

Currently the Venezuelan glaciers are restricted to the northwest slopes of the Humboldt-Bonpland and Bolivar peaks of the Sierra Nevada de Merida. Only a few glaciological observations have been completed and so many parameters of their dynamics are still unknown. In 1952 the total glacial cover was calculated at 3 km2. However, their evolution until 2008, their rate of retreat and response to the climatic variability are not well understood. This work presents the first results using satellite observations made on the Siniguis glacier which is the largest in the region located on the Humboldt-Bonpland peaks. The covered area calculated from the multi-spectral analysis of the SPOT-5 images is 0.33 km2. This indicates that since 1952 the glacier has lost an area of approximately 1.7 km2 with and average retreat rate of 30.3 m/year which corresponds to 83.74% of its coverage area. lf this trend continues, the glaciers of the Venezuelan Andes could disappear in ten years.

Snow, runoff, and mass balance modeling for the entire Mittivakkat Glacier (1998–2006), Ammassalik Island, SE Greenland

Geografisk Tidsskrift, Danish Journal of Geography 108(1):121–136, 2008 SnowModel, a physically-based snow evolution modeling system that includes four submodels—MicroMet, EnBal, SnowPack, and SnowTran-3D—was used to simulate eight full-year (1998/99 through 2005/06) evolutions of snow accumulation, blowing snow sublimation, evaporation, snow and ice surface melt, runoff, and mass changes on the entire Mittivakkat Glacier (31 km2) in southeast Greenland. Meteorological observations from two meteorological stations inside the glacier catchment were used as model input, and glaciological mass balance observations were used for model calibration (1998/99 through 2001/02) and validation (2002/03 through 2005/06) of winter snow simulations. As confirmed by observations, the spatially modeled end-of-winter snow water equivalent (SWE) accumulation increased with elevation up to 700–800 m a.s.l. in response to elevation, topography, and dominating wind direction, and maximum snow deposition occurred on the lee side of the ridge east and south of the glacier. Simulated end-of-summer cumulative runoff decreased with elevation and minimum runoff occurred on the shadowed side of the ridge east and south of the glacier. The modeled test period averaged annual mass balance was 65 mm w. eq. y−1 or ∼8% more than the observed. For the simulation period, the glacier net mass balance varies from -199 to -1,834 mm w.eq. y−1, averaging -900 (±470) mm w.eq.y−1. The glacier averaged annual modeled precipitation ranged from 1,299 to 1,613 mm w.eq. y−1, evaporation and sublimation from 206 to 289 mm w.eq., and runoff from 1,531 to 2,869 mm w.eq. y−1. The model simulated Mittivakkat Glacier net loss of900 mm w.eq. y−1 contributes approximately 42% to the average simulated runoff of 2,140 mm w.eq. y−1, indicating a mean specific runoff of 67.8 l s−1 km−2.

Subglacial drainage by groundwater-channel coupling, and the origin of esker systems: Part 1—glaciological observations

Overall This work is presented in two parts. Part I presents observations on the coupling between subglacial channel flow and groundwater flow in determining subglacial hydraulic regime and creating eskers from an Icelandic glacier that is suggested as an analogue for many parts of Pleistocene ice sheets. Part II develops a theory of perennial subglacial stream flow and the origin of esker systems, and models the evolution of the subglacial stream system and associated groundwater flow in a glacier of the type described in Part I. It is suggested that groundwater flow may be the predominant mechanism whereby meltwater at the glacier bed finds its way to the major subglacial streams that discharge water to glacier margins. Part I Boreholes drilled through an Icelandic glacier into an underlying till and aquifer system have been used to measure variations in head in the vicinity of a perennial subglacial stream tunnel during late summer and early winter. They reveal a subglacial groundwater catchment that is drained by a subglacial stream along its axis. The stream tunnel is characterised by low water pressures, and acts as a drain for the groundwater catchment, so that groundwater flow is predominantly transverse to ice flow, towards the channel. These perennial streams flow both in summer and winter. Their portals have lain along the same axes for the 5 km of retreat that has occurred since the end of the Little Ice Age, 100 years ago, suggesting that the groundwater catchments have been relatively stable for at least this period. In the winter season, stream discharges are largely derived from basal melting, but during summer, water derived from the glacier surface finds its way, via fractures and moulins, to the glacier bed, where it dominates the meltwater flux. Additional subglacial streams are created in summer to help drain this greater flux from beneath the glacier, through poorly integrated and unstable networks. Summer streams cease to flow during winter and tend not to form in the same places in the following summer. Perennial streams are the stable component of the system and are the main sources of extensive esker systems. Strong flow of groundwater towards low-pressure areas along channels and the ice margin is a source of major upwelling that can produce sediment liquefaction and instability. A theory is developed to show how this could have a major effect on subglacial sedimentary processes.

Snow Distribution and Melt Modeling for Mittivakkat Glacier, Ammassalik Island, Southeast Greenland

Abstract A physically based snow-evolution modeling system (SnowModel) that includes four submodels—the Micrometeorological Model (MicroMet), EnBal, SnowPack, and SnowTran-3D—was used to simulate five full-year evolutions of snow accumulation, distribution, sublimation, and surface melt on the Mittivakkat Glacier, in southeast Greenland. Model modifications were implemented and used 1) to adjust underestimated observed meteorological station solid precipitation until the model matched the observed Mittivakkat Glacier winter mass balance, and 2) to simulate glacier-ice melt after the winter snow accumulation had ablated. Meteorological observations from two meteorological stations were used as model inputs, and glaciological mass balance observations were used for model calibration and testing of solid precipitation observations. The modeled end-of-winter snow-water equivalent (w.eq.) accumulation increased with elevation from 200 to 700 m above sea level (ASL) in response to both elevation and topographic influences, and the simulated end-of-summer location of the glacier equilibrium line altitude was confirmed by glaciological observations and digital images. The modeled test-period-averaged annual mass balance was 150 mm w.eq. yr−1, or ∼15%, less than the observed. Approximately 12% of the precipitation was returned to the atmosphere by sublimation. Glacier-averaged mean annual modeled surface melt ranged from 1272 to 2221 mm w.eq. yr−1, of which snowmelt contributed from 610 to 1040 mm w.eq. yr−1. The surface-melt period started between mid-May and the beginning of June, and lasted until mid-September; there were as many as 120 melt days at the glacier terminus. The model simulated a Mittivakkat Glacier recession averaging −616 mm w.eq. yr−1, almost equal to the observed −600 mm w.eq. yr−1.