The convergence of harmonic reduction to an internal sphere

The boundary value problem of physical geodesy has been solved with the use of a harmonic reduction down to an internal sphere using a discrete procedure. (For gravity cf. Bjerhammar 1964 and for the potential cf. Bjerhammar 1968). This was a finite-dimensional approach mostly with one-to-one correspondence between observations and unknowns on the sphere. Earlier studies were made with the use of surface elements (on the sphere) with constantgravity. Integration over the surface elements was replaced by a discrete approach with the use of the distance to a point in the centre of the surface element. See Bjerhammar (1968) and (1969). This approach was later presented as a “reflexive prediction” technique for a weakly stationary stochastic process. Bjerhammar (1974, 1976). Krarup (1969) minimized the L²-norm of the potential on the internal sphere. It will here be proved that the two solutions are identical for a proper choice of the radii of the internal spheres. The proof is given for a spherical earth with selected choice of “carrier points”. The convergence problem is discussed. The L²-norm solution is found convergent for the fully harmonic case. Uniform convergence is obtained in the non-harmonic case with the use of the original procedure applied in accordance with the theorems of Keldych-Lavrentieff and Yamabe.     

On the geodetic boundary value problem for a fixed boundary surface—A satellite approach

Employing satellite-geometrical methods, the physical surface of the earth may be assumed to be known, while gravity measurements yield thelength of the gravity vector (including contributions from rotation). The problem then is to determine gravitational potential from such gravity observations. The corresponding linearized problem is an oblique derivative problem. The problem was discussed by Almqvist (1959), Koch (1970, 1971) and Koch and Pope (1972). Our presentation gives proofs for the existence (and uniqueness) of the solution in the non-linear case. The general implicit function theorem (in Banach spaces) is used to provewellposedness at least when data are close to given standard values (closeness is defined either in terms of Hölder or Sobolev norms). Iterative methods for solution by linear operators are given. The linearized problem is solved by harmonic reduction to an internal sphere in a generalization of the method by the first author for the Stokes problem. Also deflections of the vertical are treated.     

Solution of the altimetry-gravimetry problem

Unique solvability of the altimetry-gravimetry problem is proved for general ocean configurations and for models which are close to the classical Stokes model provided that a zero-degree component is removed from both gravity anomaly and disturbed potential. Data and solutions are to belong to suitable Sobolev spaces. The theory of pseudodifferential operators plays an important part in the proof. Finite element methods for numerical computations are suggested and convergence rates are estimated. Holota (1980, 1981a, b, 1982) has proved wellposedness of the problem without constraints for small continents by variational methods. The present paper yields wellposedness without any constraints when the diameter of the continental area is smaller than 62°.6. There are strong indications, very disturbing considering the amount of work spent on the method for the last two decades, that the direct least squares approach is unstable for mixed sets of data.     

The Greenland Ice Core Chronology 2005, 15–42 ka. Part 1: constructing the time scale

The Greenland Ice Core Chronology 2005, GICC05, is extended back to 42 ka b2k (before 2000 AD), i.e. to the end of Greenland Stadial 11. The chronology is based on independent multi-parameter counting of annual layers using comprehensive high-resolution measurements available from the North Greenland Ice Core Project, NGRIP. These are measurements of visual stratigraphy, conductivity of the solid ice, electrolytical melt water conductivity and the concentration of Na⁺, Ca²⁺, SO₄²⁻, NO₃⁻, NH₄⁺. An uncertainty estimate of the time scale is obtained from identification of ‘uncertain’ annual layers, which are counted as 0.5±0.5 years. The sum of the uncertain annual layers, the so-called maximum counting error of the presented chronology ranges from 4% in the warm interstadial periods to 7% in the cold stadials. The annual accumulation rates of the stadials and interstadials are on average one-third and half of the present day values, respectively, and the onset of the Greenland Interstadials 2, 3, and 8, based on 20 year averaged δ¹⁸O values, are determined as 23,340, 27,780, and 38,220 yr b2k in GICC05.     

Estimation of Probable Maximum Precipitation for Dams in the Hongru River Catchment, China

Summary During the summer season, typhoons form in the western north Pacific Ocean and travel westward towards China. Some recurve northward off the coast, whereas others continue in over land. These typhoons bring heavy rainfall to the Huai river basin in eastern central China. In August 1975, the remnant of typhoon Nina caused exceptionally heavy rainfall in the Hongru river basin, in the mountainous upper reaches of the Huai river. The rainfall lasted five days from 4 to 8 August. This type of nearly stationary typhoon can cause rainfall of large intensity for a long duration, and is suitable for maximization to derive probable maximum precipitation (PMP) estimates. The PMP is transformed into a probable maximum flood hydrograph that is subsequently used to design spillways etc. In this study the PMP values have been estimated using a hydrometeorological method involving depth-area-duration analysis, moisture maximization, and altitude adjustment for typhoon Nina, for 1, 2, and 3 days duration. Areal PMP values were obtained for the entire Hongru river catchment, as well as for the subcatchments upstream the dams at Banqiao (762 km²), Shimantan (230 km²), Boshan (580 km²), and Suyahu (4 498 km²). For point values, the PMP was estimated to 1 200 mm/day, 1 460 mm/2 days, and 1 910 mm/3 days at altitudes about 100 m, which agrees well with previous studies. Received February 21, 1997 Revised May 27, 1997     

Occurrence,shape and movement of daily rainfall in the Huai Basin,China

The upper reaches of the Huai River in Central China are located in the East Asian monsoon region. Strong seasonality, as well as large interannual variability of rainfall, causes floods and an uneven supply of water. In order to conserve the water and mitigate the floods, dams and flood protection structures are constructed. Their design requires information about the rainfall. Daily observations from 1957 to 1986 from 78 rain gauges were used to study shape, orientation, movement and geographical and seasonal occurrence of storms in the 79 000 km² study area. The rainfall characteristics were described using graphical plots, cross‐ and autocorrelation. Storms larger than 50 mm/day were found to occur from February to November, whereas storms exceeding 350 mm/day were confined to the main rainfall season from late June to mid‐August. The southern part of the study area experienced a break in the rainfall season in late July, corresponding to the seasonal northward shift of the rain belt. A weekly periodicity of 7–8 days for rainfall was found during June–July, but not during August–September. During the whole period June–September, the spatial pattern of daily rainfall revealed an elongated shape, more pronounced during June–July than August–September. The rainfall area was orientated approximately from WSW to ENE during the whole period, and showed an anticlockwise rotation of about 16° per day during June–July. The cross‐correlation analysis revealed that the rainfall area moved about 100 km/day eastward. These results and an investigation of meteorological maps indicate that the spatial correlation pattern of daily rainfall is produced by cold fronts on the Mei‐Yu front. Suggestions are made as to how to use the results for the construction of design rainfalls in the study area. Copyright © 1999 John Wiley & Sons, Ltd.     

Prediction of British summer river flows using winter predictors

Summary Seasonal predictions of river flow provide managers with the opportunity to plan the use of water resources in advance. Summer (June–August) river flows for the period 1961–2002 in 20 catchments in Great Britain were grouped into two clusters, largely representing the southeast and the northwest of Britain. The cluster average summer river flows were predicted by linear regression using the previous winter’s (December–February) river flow, North Atlantic sea surface temperatures (SSTs), North American land air temperature and global airflow indices as potential predictors.Separate regression equations were derived for the northwest and southeast regions. Both equations contain eight predictors. Predictors for the southeast are shifted somewhat southward compared to the predictors for the northwest. The most influential predictors are the temperature difference between the North Atlantic Current and the Norwegian Sea for the northwest and the SST in the central part of the subtropical gyre in the North Atlantic (Central Gyre) for the southeast.The regression equations fitted to all data explain 55% and 61% of the variance for the northwest and southeast regions, respectively. The corresponding cross-validation correlations between the predicted and observed series are 0.54 and 0.62. The models are fairly skilful at avoiding the false positive or negative prediction of extremes, but less skilled at predicting medium sized river flows.     

Evaluation of the Diurnal Cycle in the Atmospheric Boundary Layer Over Land as Represented by a Variety of Single-Column Models: The Second GABLS Experiment

We present the main results from the second model intercomparison within the GEWEX (Global Energy and Water cycle EXperiment) Atmospheric Boundary Layer Study (GABLS). The target is to examine the diurnal cycle over land in today??s numerical weather prediction and climate models for operational and research purposes. The set-up of the case is based on observations taken during the Cooperative Atmosphere-Surface Exchange Study-1999 (CASES-99), which was held in Kansas, USA in the early autumn with a strong diurnal cycle with no clouds present. The models are forced with a constant geostrophic wind, prescribed surface temperature and large-scale divergence. Results from 30 different model simulations and one large-eddy simulation (LES) are analyzed and compared with observations. Even though the surface temperature is prescribed, the models give variable near-surface air temperatures. This, in turn, gives rise to differences in low-level stability affecting the turbulence and the turbulent heat fluxes. The increase in modelled upward sensible heat flux during the morning transition is typically too weak and the growth of the convective boundary layer before noon is too slow. This is related to weak modelled near-surface winds during the morning hours. The agreement between the models, the LES and observations is the best during the late afternoon. From this intercomparison study, we find that modelling the diurnal cycle is still a big challenge. For the convective part of the diurnal cycle, some of the first-order schemes perform somewhat better while the turbulent kinetic energy (TKE) schemes tend to be slightly better during nighttime conditions. Finer vertical resolution tends to improve results to some extent, but is certainly not the solution to all the deficiencies identified.     

Arctic climate change in 21st century CMIP5 simulations with EC-Earth

The Arctic climate change is analyzed in an ensemble of future projection simulations performed with the global coupled climate model EC-Earth2.3. EC-Earth simulates the twentieth century Arctic climate relatively well but the Arctic is about 2 K too cold and the sea ice thickness and extent are overestimated. In the twenty-first century, the results show a continuation and strengthening of the Arctic trends observed over the recent decades, which leads to a dramatically changed Arctic climate, especially in the high emission scenario RCP8.5. The annually averaged Arctic mean near-surface temperature increases by 12 K in RCP8.5, with largest warming in the Barents Sea region. The warming is most pronounced in winter and autumn and in the lower atmosphere. The Arctic winter temperature inversion is reduced in all scenarios and disappears in RCP8.5. The Arctic becomes ice free in September in all RCP8.5 simulations after a rapid reduction event without recovery around year 2060. Taking into account the overestimation of ice in the twentieth century, our model results indicate a likely ice-free Arctic in September around 2040. Sea ice reductions are most pronounced in the Barents Sea in all RCPs, which lead to the most dramatic changes in this region. Here, surface heat fluxes are strongly enhanced and the cloudiness is substantially decreased. The meridional heat flux into the Arctic is reduced in the atmosphere but increases in the ocean. This oceanic increase is dominated by an enhanced heat flux into the Barents Sea, which strongly contributes to the large sea ice reduction and surface-air warming in this region. Increased precipitation and river runoff lead to more freshwater input into the Arctic Ocean. However, most of the additional freshwater is stored in the Arctic Ocean while the total Arctic freshwater export only slightly increases.     

Ice-wedge casts and relict polygonal patterns in Scandinavia

Ice-wedge casts in Scandinavia frequently occur in three regions: (a) southwestern Jutland (outside the limit of the Weichselian glaciation); (b) southwestern Sweden; and (c) northernmost Norway. The latter two areas are both early deglaciated regions. Characteristics of casts in vertical sections are discussed with reference to the host and infilling material. Aerial photographs provide an effective tool for detecting relict ice-wedge polygons (crop marks, micromorphology). Experience from present-day permafrost regions with active ice-wedge formation helps to explain the palaeoclimatic significance of the relict polygons. Attempts to establish the ages of formation are discussed from a methodological point of view.