High-order <Emphasis Type="Italic">h</Emphasis>-adaptive discontinuous Galerkin methods for ocean modelling

In this paper, we present an h-adaptive discontinuous Galerkin formulation of the shallow water equations. For a discontinuous Galerkin scheme using polynomials up to order , the spatial error of discretization of the method can be shown to be of the order of , where is the mesh spacing. It can be shown by rigorous error analysis that the discontinuous Galerkin method discretization error can be related to the amplitude of the inter-element jumps. Therefore, we use the information contained in jumps to build error metrics and size field. Results are presented for ocean modelling problems. A first experiment shows that the theoretical convergence rate is reached with the discontinuous Galerkin high-order h-adaptive method applied to the Stommel wind-driven gyre. A second experiment shows the propagation of an anticyclonic eddy in the Gulf of Mexico. An erratum to this article can be found at     

Six circumpolar currents—on the forcing of the Antarctic Circumpolar Current by wind and mixing

The transport of the Antarctic Circumpolar Current (ACC) is influenced by a variety of processes and parameters. A proper implementation of basin geometry, ocean topography and baroclinicity is known to be a fundamental requisite for a realistic simulation of the circulation and transport. Other, more subtle parameters are those of eddy-induced transports and diapycnal mixing of thermohaline tracers or buoyancy, either treated by eddy resolution or by a proper parameterization. Quite a number of realistic numerical simulations of the circulation in the Southern Ocean have recently been published. Many concepts on relations of the ACC transport to model parameters and forcing function are in discussion, however, without much generality and little success. We present a series of numerical simulations of circumpolar flow with a simplified numerical model, ranging from flat-bottom wind-driven flow to baroclinic flow with realistic topography and wind and buoyancy forcing. Analysis of the balances of momentum, vorticity, and baroclinic potential energy enables us to develop a new transport theory, which combines the most important mechanisms driving the circulation of the ACC and determining its zonal transport. The theory is based on the importance of the bottom vertical velocity in generating vorticity and shaping the baroclinic potential energy of the ACC. It explains the breaking of the -constraint by baroclinicity and brings together in one equation the wind and buoyancy forcing of the current. The theory emphasizes the role of Ekman pumping and eddy diffusion of buoyancy to determine the transport. It also demonstrates that eddy viscosity effects are irrelevant in the barotropic vorticity balance and that friction arises via eddy diffusion of density. In this regime, the classical Stommel model of vorticity balance is revived where the bottom friction coefficient is replaced by (with the Gent–McWilliams coefficient and the baroclinic Rossby radius ) and a modified wind curl forcing appears.     

Mapping nonlinear shallow-water tides: a look at the past and future

Overtides and compound tides are generated by nonlinear mechanisms operative primarily in shallow waters. Their presence complicates tidal analysis owing to the multitude of new constituents and their possible frequency overlap with astronomical tides. The science of nonlinear tides was greatly advanced by the pioneering researches of Christian Le Provost who employed analytical theory, physical modeling, and numerical modeling in many extensive studies, especially of the tides of the English Channel. Le Provost’s complementary work with satellite altimetry motivates our attempts to merge these two interests. After a brief review, we describe initial steps toward the assimilation of altimetry into models of nonlinear tides via generalized inverse methods. A series of barotropic inverse solutions is computed for the M tide over the northwest European Shelf. Future applications of altimetry to regions with fewer in situ measurements will require improved understanding of error covariance models because these control the tradeoffs between fitting hydrodynamics and data, a delicate issue in coastal regions. While M can now be robustly determined along the Topex/Poseidon satellite ground tracks, many other compound tides face serious aliasing problems. In memory of Christian Le Provost     

Empirische Untersuchung zur Frage der Beziehung zwischen durchschnittlicher und kennzeichnender Wellenperiode im Seegang

Zusammenfassung Als Zusammenhang zwischen der kennzeichnenden Wellenperiode und der durchschnittlichen Periode im Seegang wird die Formel angesetzt. Mit Hilfe empirischer Unterlagen wird nachgewiesen, daßc eine Funktion des von D. E. Cartwright und M. S. Longuet-Higgins [1956] eingeführten Spektralparameters ist. Es wird eine vorläufige quantitative Beziehung zwischenc und abgeleitet.

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Empirical investigations of the relation between the mean and the significant wave period in the sea

Summary It is supposed that the formula represents the relation between the significant wave period and the mean period in the sea. With the aid of empirical data it is demonstrated thatc is a function of the spectral parameter introduced by D. E. Cartwright and M. S. Longuet-Higgins [1956]. A preliminary quantitative relation betweenc and is derived.

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Etudes empiriques de la relation entre la période moyenne et la période significative des vagues dans la houle

Résumé On suppose que la formule représente la relation entre la période significative des vagues et la période moyenne dans la houle. A l'aide des données empiriques on montre quec est une fonction du paramètre spectral , introduit par D. E. Cartwright et M. S. Longuet-Higgins [1956]. Une relation quantitative préliminaire entrec et est dérivée.

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Impacts of localisation in the EnKF and EnOI: experiments with a small model

The performance of an inexpensive, ensemble-based optimal interpolation (EnOI) scheme that uses a stationary ensemble of model anomalies to approximate forecast error covariances, is compared with that of an ensemble Kalman filter (EnKF). The model to which the methods are applied is a pair of “perfect”, one-dimensional, linear advection equations for two related variables. While EnOI is sub-optimal, it can give results that are comparable to those of the EnKF. The computational cost of EnOI is typically about times less than that of EnKF, where is the ensemble size. We suggest that EnOI may provide a practical and cost-effective alternative to the EnKF for some applications where computational cost is a limiting factor. We demonstrate that when the ensemble size is smaller than the dimension of the model’s sub-space, both the EnKF and EnOI may require localisation around each observation to eliminate effects of sampling error and to increase the effective number of independent ensemble members used to construct an analysis. However, localisation can degrade an analysis if the length-scales of the localising function are too short. We demonstrate that, as the length-scale of the localising function is decreased, localisation can significantly compromise the model’s dynamical balances. We also find that localisation artificially amplifies high frequencies for applications of the EnKF. Based on our experiments, for applications where localisation is necessary, the length-scales of the localisation should be larger than the decorrelation length-scales of the variables being updated.     

Das Spektrum der internen Bewegungsvorgänge der westlichen Ostsee im Periodenbereich von 0,3 bis 60 Minuten

Zusammenfassung Für ein Beobachtungsbeispiel (Stromgeschwindigkeit, Dichte) aus der Kieler Bucht wird eine Deutung ausgeprägter Maxima der zugehörigen Spektren zwischen 0,3 und 60 Minuten durch interne Wellen gegeben. Mit einer beobachteten Periode von zwei Minuten durchgeführte Rechnungen ergeben eine starke Abhängigkeit der Wellenlänge von der mittleren Strömung . Im Falle erhält man bei 23,5 m Wassertiefe eine Wellenlänge von etwa 70 m, im Falle von etwa 85 m. Die berechneten Schwingungen stellen uneigentliche interne Wellen dar (W. Krauß [1966]). Die Interpretation durch eine Grenzflächenwelle führt auf eine Wellenlänge von 86 m, die nur geringfügig von denen der internen Wellen 1. Ordnung in stetig sich ändernder Strömung abweicht.In einer theoretischen Untersuchung werden kleinräumige Anfangsstörungen (z. B. momentane Druckänderungen an der Meeresoberfläche) als mögliche Ursache für die Entstehung kurzperiodischer interner Wellen erkannt. Es zeigt sich, daß kurzzeitig wirksame Anfangsbeschleunigungen in ihrem Einwirkungsbereich stehende, allmählich abklingende interne Wellen erzeugen, während in der Umgebung gleichzeitig fortschreitende Wellen entstehen, deren Amplituden mit wachsender Entfernung vom Erregungsgebiet abnehmen. Die Perioden der Schwingungen haben größere Werte als die zu einer exponentiellen Schichtung gehörige Väisäläperiode und verändern sich in Abhängigkeit von der Größe des Anregungsgebietes wie die zellularer Stabilitätsschwingungen.

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The powerspectrum of internal motions in the western baltic between the periods 0.3 and 60 minutes. Part 1: Interpretation of the wavelike component of the internal unrest in the sea

Summary The powerspectra of the internal unrest in the sea show marked peaks in the range of periods between 0.3 and 60 minutes. An interpretation of these phenomena is given in terms of internal waves for a specific example obtained from short periodic current and density variations in the Kieler Bucht. The numerical calculations for an observed period of two minutes show an important influence of the vertical distribution of the current on the wavelength. In the case of the wavelength amounts to about 70 m, where as in the case of the length is about 85 m, assuming a depth of the sea of 23.5 m. The computed oscillations represent improper internal waves (W. Krauß [1966]). Interpretation by internal boundary waves yields a wavelength of 86 m, which is slightly different only from those of the first mode of internal waves in the case of continuously varying .By a theoretical investigation it is shown that short periodic internal waves may be caused by local initial perturbations (for instance by sudden variations of pressure at the surface). The solution of the problem describes slowly decreasing standing internal waves, which are generated within the area upon which short-dated initial accelerations have acted. At the same time a train of progressive waves is developed in the environment travelling away from the centre of the excitation. The amplitudes of these waves diminish with increasing distance from the origin. The periods of the computed oscillations yield higher values than the Väisäläperiod belonging to an exponential stratification. The variability in these periods is caused by variations in depth, by variations in stability, and by changes in the horizontal dimensions of the area of initial perturbation. This dependence is similar to that of cellular oscillations of stability.

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Spectre des oscillations internes de la mer Baltique Ouest pour des périodes comprises entre 0,3 et 60 minutes. 1ère Partie: Interprétation des éléments ondulatoires de mouvement

Résumé Pour un cas d'observation (vitesse de courant, densité) en baie de Kiel, des maximums bien marqués des spectres correspondants entre 0,3 et 60 minutes s'expliquent par des ondes internes. Des calculs effectués avec une période de deux minutes montrent que la longueur d'onde dépend beaucoup du courant moyen, . Pour , par 23,5 m de profondeur, on obtient une longueur d'onde environ 70 m; pour , une longueur d'environ 85 m. Les oscillations calculées représentent des ondes internes qui ne sont pas des ondes propres. L'interprétation par une onde de surface limite conduit à une longueur d'onde de 86 m très peu différente de celles des ondes internes du premier ordre dans un courant constamment variable.Une étude théorique montre que des perturbations initiales peu étendues (par exemple variations momentanées de la pression à la surface de la mer) peuvent être à l'origine d'ondes internes à courte période. Il apparaît que des accélérations initiales, agissant brièvement, font naître dans leur zone d'action des ondes internes stationnaires qui s'amortissent peu à peu, tandis qu'en même temps aux alentours se produisent des ondes progressives dont l'amplitude décroît à mesure qu'elles s'éloignent de la région où elles ont pris naissance. Les périodes des oscillations ont des valeurs plus grandes que celle de la période de Väisälä rapportée à une stratification exponentielle, et elles varient suivant la grandeur de la zone où elles ont pris naissance comme les oscillations de stabilité cellulaire.

     

A note on tidal current rotation

The rotational form of the vertically averaged equations of motion is applied to derive a formula, linear friction included, which establishes a direct connection between sense of rotation of tidal currents and features of tidal amphidromic systems. Two factors in the formula, called and , influence the sense of rotation of tidal currents; the factor involves the frequency of the tidal signal , the Coriolis parameter f, and the linear friction coefficient r. The sign of the cross-product of the logarithm of sea-surface elevation (), and phase () gradients determines whether the factor favors clockwise or anticlockwise sense of rotation. is a unit vector and is the angle between ln and . The limits ||0, ||0 and 0 lead to a clockwise sense of rotation in the Northern Hemisphere. 0 favors anticlockwise rotation in the Northern Hemisphere. Friction and low frequencies favor an anticlockwise sense of rotation. The theory works well in semi-enclosed regions like the North Sea. Although only linear friction and sea-surface elevation gradients were considered, there are ocean regions where the agreement between theory and observations is also good.Responsible Editor: Hans Burchard     

Internal and forced variability along a section between Greenland and Portugal in the CLIPPER Atlantic model

Numerical models are used to estimate the meridional overturning and transports along the paths of two hydrographic cruises, carried out in 1997 and 2002 from Greenland to Portugal. We have examined the influence of the different paths of the two cruises and found that it could explain 0.4 to 2 Sv of difference in overturning (the precise value is model-dependent). Models show a decrease in the overturning circulation between 1997 and 2002, with different amplitudes. The CLIPPER ATL6 model reproduces well the observed weakening of the overturning in density coordinates between the cruises; in the model, the change is due to the combination of interannual and high-frequency forcing and internal variability associated with eddies and meanders. Examination of the -coordinate overturning reveals model–data discrepancies: the vertical structure in the models does not change as much as the observed one. The East Greenland current variability is mainly wind-forced in the ATL6 model, while fluctuations due to eddies and instabilities explain a large part of the North Atlantic Current variability. The time-residual transport of dense water and heat due to eddy correlations between currents and properties is small across this section, which is normal to the direction of the main current.     

Grundlagen für die Dimensionierung der Belüftung bei Belebtschlammanlagen

Summary The generally accepted theory for the calculation of the oxygen consumption caused by the microbes in an aeration basin assumes that the total of oxygen utilizedq _g is the sum of the endogenous respiration of the sludgeq _e plus a quantity for substrate respirationq _s . This theory—though physiologically probably not quite correct—, is very convenient because it allows a causal calculation of the necessary oxygen input into an activated sludge plant. Extensive data on endogenous respiration of activated sludge are presented (Table 1 and 5) and are correlated with sludge load and sludge age (Fig. 4 and 5). The term substrate respiration is discussed and a calculation method, based on the removal of organic C by the mixed liquor, is proposed. In view of the fact that nitrification is an inevitable reaction in activated sludge treatment (its extent being strictly a function of the operation parameters of a plant), oxygen consumption for the oxidation of ammonia has to be included into the term of substrate respiration. The amount of this oxygen fraction has to be predicted from the degree of nitrification to be expected at the desired operation conditions. The following formula for the calculation of the oxygen input into a completely mixed aeration basin is proposed (explanations of symbols see annex): or This same calculation method may be applied stepwise for aeration basins of the tubular reactor type (longitudinal flow basins) considering the concentration decrease of the substrates along the length profile of the basin (the term for endogenous respiration can be assumed constant for the entire aeration time).

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On the normal gravity formulae

The determination of the normal gravity according to the formula

Climatic significance of<Emphasis Type="Italic">δD</Emphasis> time series in tree rings from Tianmu Mountain

Based on cross-dating tree ring age from Tianmu Mountain, Zhejiang Province, theδD      

Der Brechungsindexgradient für elektromagnetische Wellen (Cm bis M-Band) in der maritimen Grenzschicht der Atmosphäre

Zusammenfassung Es werden Beziehungen abgeleitet zwischen dem vertikalen Gradienten des modifizierten BrechungsindexM (für Centimeter- bis Meterwellen) in der maritimen Grenzschicht der Atmosphäre und der Temperatur- und Wasserdampfdifferenz Luft-Wasser. Diese Beziehungen ergeben sich aus unseren Kenntnissen von dem Einflusse der Temperaturschichtung und der Windgeschwindigkeit auf die vertikale Wasserdampfschichtung über See und ermöglichen die Absehätzung der Refraktionsschichtung (Duct-Dicke,mittlerer Gradient u.a.) für elektromagnetische Wellen (Cm- bis M-Band) in den unteren Dekametern über dem Meer. Es werden Tabellen und Diagramme für eine praktische Anwendung des Verfahrens angegeben.Als Beispiel werden Häufigkeitsverteilungen der Duct-Dicke und des mittleren Brechungsindexgradienten für Sommer und Winter gezeigt, die sich aus vierjährigen Beobachtungen auf einem nordatlantischen Wetterschiff ergaben.

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On the gradient of the refractive index of electro-magnetic waves (centimetre- to metre-waves) in the maritime boundary layer of the atmosphere

Summary Correlations between the vertical gradient of the modified refractive indexM (of centimetre- to metre-waves) in the maritime boundary layer of the atmosphere and the difference air-water in temperature and water vapour are stated. The correlations are known from the influences exercised by the temperature stratification and the wind velocity on the vertical stratification of water vapour above the sea surface; they allow to estimate the refractive stratification (thickness of duct, medium gradient etc.) of the electro-magnetic waves (centimetre- to metre-waves) in the lower decametres above the ocean surface. Tables and diagrams are stated in the paper for a practical application of the method under discussion.As an example, the author included in this paper data of frequency distribution of the duct thickness and of the medium gradient of the refractive index for summer and winter which are derived from four years' observations on board of a weather ship in the North Atlantic.

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Sur le gradient de l'indice de réfraction des ondes électromagnétiques (de 0,01 m á 1m) présentes dans la couche limite maritime de l'atmosphère

Résumé On révèle des corrélations entre le gradient vertical de l'indice modifié de réfractionM (par rapport aux ondes de 0,01 à 1 m) et la différence: air eau de la température et du vapeur d'eau. Ces corrélations, nous le savons, résultent de l'effect que la stratification de la température a sur la stratification verticale du vapeur d'eau au-dessus de la surface de la mer. La connaissance de ces corrélations nous permet d'estimer la stratification de réfraction (l'épaisseur du duct, gradient moyen etc.) des ondes électromagnétiques présentes dans les bas décamètres voisins de la surface de l'océan. Des tableaux et des diagrammes permettant d'appliquer au travail pratique la méthode discutée ici, sont indiqués.Pour offrir un exemple, l'auteur décrit dans cet article la distribution en fréquence de l'épaisseur du duct et du gradient moyen de l'indice de réfraction pour l'hiver et pour l'été en se basant sur les observations faites à bord d'un bateau météorologique en Océan Atlantique Nord pendant une période de quatre ans.

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Herrn o. ö. Professor Dr. Paul Raethjen, Direktor des Geophysikalischen Instituts der Universität Hamburg, zum 60. Geburtstag gewidmet.     

On a certain doubt L. Euler had with respect to determining the rotation of the earth and of celestial bodies

Summary The application of Euler's postulate , which determines rotation (here denotes the derivative of the angular momentum with respect to time, L the resultant torque of exterior forces relative to the centre of inertia), to a celestial body, which is not spherically symmetric, had aroused doubts in Euler himself because of the displacement of the resultant attractive force of other celestial bodies from the said centre. As Minding had noted, it is more logical to compute the resultant torque relative to a point at which the torque has a minimum. This point should be named the attrahentis centrum after Euler. Newton was aware of this displacement of the resultant force and had mentioned it in the Principia. A review of Euler's works connected with this subject is given.     

Puhimau Thermal Area: A Window into the Upper East Rift Zone of Volcano, Hawaii?

We report the results of two soil CO₂ efflux surveys by the closed chamber circulation method at the Puhimau thermal area in the upper East Rift Zone (ERZ) of volcano, Hawaii. The surveys were undertaken in 1996 and 1998 to constrain how much CO₂ might be reaching the ERZ after degassing beneath the summit caldera and whether the Puhimau thermal area might be a significant contributor to the overall CO₂ budget of . The area was revisited in 2001 to determine the effects of surface disturbance on efflux values by the collar emplacement technique utilized in the earlier surveys. Utilizing a cutoff value of 50 g m⁻² d⁻¹ for the surrounding forest background efflux, the CO₂ emission rates for the anomaly at Puhimau thermal area were 27 t d⁻¹ in 1996 and 17 t d⁻¹ in 1998. Water vapor was removed before analysis in all cases in order to obtain CO₂ values on a dry air basis and mitigate the effect of water vapor dilution on the measurements. It is clear that Puhimau thermal area is not a significant contributor to CO₂ output and that most of CO₂ (8500 t d⁻¹) is degassed at the summit, leaving only magma with its remaining stored volatiles, such as SO₂, for injection down the ERZ. Because of the low CO₂ emission rate and the presence of a shallow water table in the upper ERZ that effectively scrubs SO₂ and other acid gases, Puhimau thermal area currently does not appear to be generally well suited for observing temporal changes in degassing at .     

Theoretical topocentric distance of a lunar station

Summary Drawing on [1], the theoretical distance between two general points on the surface of the Earth and Moon is derived using ultraspherical polynomials, retaining terms of the order of 10^–11 .     

Strato-mesospheric ozone and IMF sector boundary

Summary We find no effect of the interplanetary magnetic field(IMF) sector boundary crossing in ozone at altitudes ranging from the middle stratosphere(10 hPa) to the lower mesosphere(0.4 hPa) at middle latitudes(40°–60° N), based on winter data over the period of December 1979–December 1982.

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Резюме Мы не нaшлu нuкaко о эффекma nересеченuя секmорноŭ рaнuцы межnлaнеmно о мa нumно о nоля в озоне в облaсmu высоm меж?rt;у сре?rt;неŭ сmрamосфероŭ(10 Пa) u нuжнеŭ мезосфероŭ(0,4 Пa) нa сре?rt;нuх шuроmaх(40°–60° с.ш.), nрuменяя зuмнuе ?rt;aнные зa nерuо?rt; ?rt;екaбрь 1979 – ?rt;екaбрь 1982 .

     

Background aspects of lake restoration: water balance,heavy metal content,phosphorus homeostasis

Perturbations of the lake water balance, inputs of heavy metals to lakes, and intensifying fertilization of lakes through input and accumulation of phosphorus—these three classes of phenomena are among the more important background processes in lake restoration. Lake restoration consists of a series of measures animed at producing a homeostatic response of a lake system to external perturbations. The success of its implementation is affected by the morphometric and edaphic parameters of different types of lakes. The relationship between the volume (V) and mean-depth of fresh-water lakes indicates a trend of . Glacial lakes occuring on or near crystalline shields have relatively shallow depths, whereas volcanic lakes, rift valley and deep valley lakes have relatively greater depths for the same volume. For saline lakes (21 lakes, V>1 km³) that undergo cycles of expansion and shrinkage, the V to relationship is closer to power 1. Water residence times (τ) of small and big fresh-water lakes show a trend of τ approximately linear in or τ∝V^0.3. Volcanic lakes and Maare have longer residence times in comparison to other lakes of similar volumes. For the major inorganic chemical species and heavy metals, the regulatory upper-limit concentrations in drinking water in the USA and EEC are from several times to more than 100 times higher than their concentrations in a global mean river water. Only three elements (Fe, P, and Al) occur in river water at concentrations approaching such upper-limit recommendations. Rates of accumulation of phosphorus in lake water and sediments, computed as the difference between input and ouflow removal rates for 23 fresh-water lakes, are generally lower for lakes of longer water residence time. The rate of accumulation is a measure of homeostatic response of the lake system to input load: it is equivalent to the rate of all the removal processes needed to maintain phosphorus concentration in lake water at a steady state.     

Weak Earthquake of Aug. 10, 1987 connected with the Jílovice fault (Northern Bohemia)

Summary The paper describes the seismic activity of the Jílovice fault and the last earthquake of Aug. 10, 1987.

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Резюме Оnuсывaеmся сеŭсмuческaя aкmuвносmь рaзломa йuловuце в северовосmочноŭ Чехuu u землеmрясенuе с 10-о о aв усma 1987 , очa коmоро о был связaн с ?rt;aнным разломом.

     

An algorithm for the optimum distribution of a regional seismic network — II. An analysis of the accuracy of location of local earthquakes depending on the number of seismic stations

Summary Seven optimal networks consisting of 4 to 10 stations are compared for a given region, where velocity-depth profiles and the distribution of seismic intensity are known. Assuming that the standard error of arrival time is _t =0.05 s and the standard errors of the parameters of velocity-depth profiles are equal to 5% of their values, the average standard errors of the origin time and focus coordinates are estimated. The application of optimum methods to the planning of seismic networks in the Lublin Coal Basin is presented, and maps of standard errors of origin time , depth and epicenter ( _xy ) for the case of an optimum network of 6 seismic stations are given.     

Attenuation of P,S, and coda waves in Koyna region,India

The attenuation properties of the crust in the Koyna region of the Indian shield have been investigated using 164 seismograms from 37 local earthquakes that occurred in the region. The extended coda normalization method has been used to estimate the quality factors for P waves and S waves , and the single back-scattering model has been used to determine the quality factor for coda waves (Q _c). The earthquakes used in the present study have the focal depth in the range of 1–9 km, and the epicentral distance vary from 11 to 55 km. The values of and Q _c show a dependence on frequency in the Koyna region. The average frequency dependent relationships (Q = Q ₀ f ⁿ) estimated for the region are , and . The ratio is found to be greater than one for the frequency range considered here (1.5–18 Hz). This ratio, along with the frequency dependence of quality factors, indicates that scattering is an important factor contributing to the attenuation of body waves in the region. A comparison of Q _c and in the present study shows that for frequencies below 4 Hz and for the frequencies greater than 4 Hz. This may be due to the multiple scattering effect of the medium. The outcome of this study is expected to be useful for the estimation of source parameters and near-source simulation of earthquake ground motion, which in turn are required in the seismic hazard assessment of a region.