Selected Papers of Academicians

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  • 1  Analysis of the process and mechanisms of genesis and development for 2014-2016 mega El Niño event
    DING Yihui
    2016, 39(6):722-734. DOI: 10.13878/j.cnki.dqkxxb.20161028003
    [Abstract](1872) [HTML](0) [PDF 5.22 M](11401)
    Abstract:
    The present paper has mainly analysed the process and mechanisms of genesis and development of the 2014-2016 mega El Niño event.It is shown that the entire lifecycle of the event is about 2 years(from April 2014 to May 2016),with four stages identified for its evolutive process:(1)Early and continuous westerly wind bursts(December 2013 to April 2014).The continuous three westerly wind burstsnot only changed the state of the easterly trade wind prevailing tropical central and easterly in the Pacific for long period of time,but also changed the cold water state in this region for the most recent 12 years,thus leading to SST rise and warming.Until early spring 2014,the SSTA exceeded 0.5℃,marking the possible occurrence of a new El Niño event.(2)Alternative weakening period(June 2014 to August 2015).Six westerly wind bursts continued to occur,thus maintaining and enhancing the warming of the equatorial central and eastern Pacific,while at the same time overcoming two periods of SST warming decrease or barrier,so that the initial development of El Niño was not aborted,and even changed into the stage of strong El Niño.Correspondingly,in the sub layer of the equatorial central and eastern Pacific,six warm Kelvin waves were observed to propagate eastward.The heat contents of these oceanic waves not only maintained the continuous warming in the equatorial central and eastern Pacific,but also caused El Niño to change from CP to EP type.(3)Peak period of development (September 2015 to February 2016).Two stronger westerly wind bursts were observed,which corresponded to very vigorous convective activity on the equatorial central and eastern Pacific.Rapid warming occurred in the Niño3.4 region,with 3℃observed in November 2015,classified as the mega-El Niño event.(4)Accelerating weakening stage(March to May 2016).The intensity of the El Niño rapidly weakened from 2 to 0.5℃ in the Niño3.4 region,then accelerated the transition to the cold water phase.In July to August 2016,the SSTA in the Niño3.4 region already approached -0.5℃.This rapid phase shift is a manifestation of the theory of delayed oscillation.From the above results,it is concluded that the development and shift of warm and cold phases is observationally consistent with the mechanism derived from the paradigm of the current theory of recharge oscillation and/or delayed oscillation theory.This clearly demonstrates that the results of the El Niño theory effectively underpin the development of related operational prediction.
    2  Interdecadal variability of largescale precipitation conditions over North China during rainy seasons
    Liu Haiwen Ding Yihui
    2011, 34(2):146-152.
    [Abstract](1252) [HTML](0) [PDF 1.79 M](2766)
    Abstract:
    Based on the daily precipitation data from 740 stations in China and NCEP/NCAR daily reanalysis datasets,the interdecadal variability of largescale precipitation conditions over North China during rainy seasons is analyzed by means of composite analysis methods.The results show that:with a turning point from flood period to dry period in 1978,anomalous moisture budget over North China changed from south boundary and west boundary to north boundary and east boundary and thus made water vapor become divergent and absent from convergent and abundant.Warm and moisture air mass from ocean could reach the north area of North China or even its northern border before 1978,but it could only reach the south bank after 1978.The interaction between the cold and warm moisture air mass along the Taihang Mountain became weaker.In addition,anomalous upward motion became anomalous downward motion over the North China;areal average vorticity changed from two layers structure(positive vorticity in the low levels and negative vorticity in the high levels) to three layers structure(negative vorticity in the middle and lower troposphere,positive vorticity in the middle and negative vorticity in up troposphere) and the value of total vorticity became smaller.Vertical configurations of areal average divergence and vertical velocity presented no conspicuous interdecadal variability,but their values all became smaller.
    3  Research Progress in the Effect of Vegetation Change on Regional Climate
    LI Qiao-ping DING Yi-hui
    2004, 27(1):131-140.
    [Abstract](1250) [HTML](0) [PDF 414.19 K](1982)
    Abstract:
    As one of the important influencing factors of global and regional climate change,vegetation variation has gradually attracted scientists in the recent decades.Many studies have been undertaken in terms of climate model to investigate the effects of various vegetation covers on climate change,and many remarkable advances in the field have been achieved.Results show that vegetation variation can influence the regional precipitation,circulation and temperature by changing the surface property,such as albedo,roughness length and soil moisture,and through radiation and moisture balance etc.This paper gives a summary of the recent researches in China and abroad,especially of the effects of vegetation change on regional climate of China.The evidence reveals that the large scale degradation of vegetation leads to the increase in surface temperature,the weakening of the east Asia monsoon,the drop of rainfall and the exacerbation of North China drought.Finally,problems in the past research and keystones in future research are also pointed out.
    4  Outline of the real-time seasonal haze pollution prediction in China in recent years
    YIN Zhicong WANG Huijun DUAN Mingkeng
    2019, 42(1):2-13. DOI: 10.13878/j.cnki.dqkxxb.20181224001
    [Abstract](1538) [HTML](0) [PDF 2.36 M](2907)
    Abstract:
    In recent years,severe haze pollution has been damaging human health,traffic security,the ecosystem and social economy in eastern China.In addition to the haze forecast within 1 week,seasonal haze prediction provides scientific support for longer periods to the decisions of emission reduction.In this study,taking the annual increment as the predictand,monthly prediction models were trained for the Beijing-Tianjin-Hebei and Yangtze Delta regions.The performances of the built models were similar,with 2 days of root-mean-square error and a>80% simulation rate of the anomalies' mathematical sign.In the real-time seasonal prediction for Beijing-Tianjin-Hebei haze days in the winter of 2016,the results with respect to the climate mean(the previous year) were completely (mostly) accurate.During the winter of 2017,the predicted biases for the December and January haze days in the Yangtze River Delta were very small,and the bias of February was nearly 2 days.
    5  Impact of Arctic warming and the super El Niño in winter 2015/2016 on the East Asian climate anomaly
    HE Shengping WANG Huijun XU Xinping LI Jingyi
    2016, 39(6):735-743. DOI: 10.13878/j.cnki.dqkxxb.20161008002
    [Abstract](1717) [HTML](0) [PDF 3.11 M](2850)
    Abstract:
    The influence of El Niño on the East Asian climate has been studied extensively.Generally,East Asia is warmer than normal in winter when El Niño occurs.During winter 2015/2016,an extreme El Niño event took place-one that was almost the strongest since records began.However,the winter of 2015/2016 was not much warmer than usual.By contrast,East Asia was even colder than normal in January 2016.At the same time,the Arctic experienced extreme warming in winter 2015/2016,which might have had an influence on the winter climate of East Asia.Therefore,in this study,the influences of Arctic warming and the super El Niño during winter 2015/2016 on the East Asian winter climate was investigated using the ENSO index of the Climate Prediction Center of NOAA,along with NCEP/NCAR reanalysis data,for the period 1980-2016.In winter 2015/2016,a super El Niño emerged in the central tropical Pacific.In January 2016,there was a dominant anomalous anticyclone over the western North Pacific and an anomalous cyclone over the Northeast Pacific.Meanwhile,a southerly anomaly was observed over East Asia,which was confined to the area south of 30°N.This meant that there was a particularly evident response of atmospheric circulation in January 2016 to the forcing of ENSO.However,significantly warmer conditions were not observed over East Asia as expected.On the contrary,it was much colder than normal in January.It is interesting to find that the northerly wind in the domain(30-50°N,110-125°E) was stronger in January 2016,which could explain the colder conditions over East Asia.The Rossby wave activity in January 2016 indicates a wave source over the Barents-Kara seas,suggesting that the colder conditions over East Asia might have been related to the Arctic warming that occurred in January 2016.To address this issue,an Arctic temperature index was defined as the area-weighted mean in the Barents-Kara seas(70-80°N,30-70°E).The statistical results from the reanalysis data during 1980-2016 revealed that the warming in the Barents-Kara seas was concurrent with a significant negative temperature anomaly over East Asia.The teleconnection between the Arctic and East Asian temperature was well supported by an obvious southeastward propagation of Rossby waves.Further analysis suggested that the warming in the Barents-Kara seas could induce anomalous ascending motion in situ.The anomalous ascending motion extended eastwards (around 70-90°E) and turned southwards due to the Coriolis force,and finally descended in Siberia(50-70°N,60-100°E).As the air mass accumulated over Siberia,the Siberian high was intensified,leading to anomalous northerly flow in its east,which favored the cold conditions over East Asia.To verify the combined effect of the Arctic warming and El Niño,a statistical prediction model was established via the multivariate linear regression of surface air temperature(SAT) onto the Arctic temperature index and Niño3.4 index.It was found that the hindcasted SAT anomaly over East Asia in January 2016 resembled the observed counterpart well,with a spatial correlation coefficient of 0.71.This implied that the Arctic warming and super El Niño during winter 2015/2016 can to some extent explain the colder than expected conditions in East Asia.
    6  A Preliminary Study of UV Radiation over Nanjing Area
    Zheng Youfei Shi Guangyu He Jinhai Ke Yaowen
    2000, 23(2):235-341.
    [Abstract](839) [HTML](0) [PDF 485.42 K](1881)
    Abstract:
    A simplified transmission model is used to estimate surface ultraviolet(UV) radiation over Nanjing area. Taking account of possible ozone variation in the future time, the value of UV radiation ampification factor(RAF) is obtained. Additionally, annual variations of UV, its relations to total solar radiation and weather conditions are investigated by virtue of one-year UV observations.
    7  Simulation and comparative analysis of stratosphere-troposphere exchange in the summers of 2009 and 2010 in China and surrounding areas
    CAO Zhiqiang LYU Daren
    2016, 39(3):300-308. DOI: 10.13878/j.cnki.dqkxxb.20140427001
    [Abstract](879) [HTML](0) [PDF 4.11 M](1915)
    Abstract:
    Stratosphere-troposphere exchange(STE) is a process not only involving transport from the troposphere to stratosphere,but also from the stratosphere to troposphere.STE is very important to the variation of water vapor and atmospheric chemical compositions.Correspondingly,it also plays an important role in the radiation balance and climate changes in the global climate system.Deep convection occurs frequently in summer.The strong upward flow in deep convective cloud can transport the atmosphere of the lower troposphere quickly to the upper troposphere,and even to the lower stratosphere.Thus,it is very important in the process of STE.To date,this conclusion has been reached largely with the help of aircraft observations.
    STE is a process that includes complex physical,chemical and dynamic processes.The time scale of STE can be climatological,synoptic,meso,and turbulent,while the spatial scale varies from local to global.Therefore,STE is a highly complex process,and to study such a complex process,using models to quantitatively simulate the transport processes is a good method.FLEXPART is a Lagrangian diffusion model.It is widely used in the transport of air pollution,the mesoscale transport of smoke from forest fires,as well as global STE.
    In this study,firstly,data on deep convection in China and surrounding areas for the period 2005-2012 were collected.The source of the data-collected 24 times a day and at a spatial resolution of 0.1°-was the FY-2 satellite.Through statistical analysis,the interannual variation of the deep convection was determined.One year of minimum convection and one year of maximum convection were selected from the overall study period.Then,the domain(15-30°N,75-135°E)was selected to represent China and its surrounding areas.In order to understand the bidirectional exchange characteristics of troposphere-to-stratosphere transport(TST)and stratosphere-to-troposphere transport(STT)in summer in this region,the atmospheric circulation status of the two selected years from June to August was simulated using FLEXPART.The statistical and model results were analyzed,revealing the following:
    In the summer season of 2009,deep convection occurred least frequently;while in the summer season of 2010,it occurred most frequently.The atmospheric circulation from June to August 2009 and 2010 was simulated using FLEXPART,and it was found that the area south of 30°N and the area north of 30°N possessed different characteristics of STE.In the area north of 30°N,TST and STT were active;whereas,in the area south of 30°N they were far less active than the area north of 30°N.However,the net transport (TST minus STT)was roughly equal.From June to August,in the area south of 30°N,the net transport was from the troposphere to stratosphere;while in June,in the area north of 30°N,the net transport was from the stratosphere to troposphere.In July and August,the net transport was from the troposphere to stratosphere.Comparing the TST and TST-minus-STT of summer 2009 and 2010,it was found that the total amount of TST and TST-minus-STT from June to August 2010 was more than that in 2009.This means that the transportation from the troposphere to stratosphere was larger in 2010 than in 2009.The TST and TST-minus-STT might be positively correlated with the deep convection activity,and this will be investigated in future work.
    8  A Study on Ground-Based Remote Sensing of Atmospheric Integrated Water Vapor and Cloud Liquid Water
    LIV Chao-shun LV Da-ren DU Bing-yu
    2006, 29(5):606-612.
    [Abstract](1247) [HTML](0) [PDF 1.76 M](1957)
    Abstract:
    The fundamental and method of the ground-based microwave radiometer remote sensing retrieval of integrated water vapor(V) and cloud liquid water(L) in the atmosphere are introduced herein.Climatologically representative retrieval equations for different seasons in Beijing are given,and the numerical simulation tests of the accuracy of retrieval results are also presented.The relative standard deviation of retrieved values of V is 3.1%,1.6%,2.2% and 2.4%,respectively,for spring,summer,autumn and winter.It is found from the retrieval results of the NASA microwave radiometer data taken at Xianghe that on average,the retrieved V is 0.21cm greater than that measured by radiosonde;the linear correlation coefficient between the two V is 0.988,and the rms error is 0.16cm;and the L value is always below 0.1mm except precipitation weather.
    9  A Primary Study on Three Dimensional Structure of Rainfall Rates of Typhoon Dan’s Precipitation Cloud Systems
    ZHONG Min Lü Da-ren DU Bing-yu
    2006, 29(1):41-47.
    [Abstract](812) [HTML](0) [PDF 931.62 K](1860)
    Abstract:
    Horizontal and vertical structure of rainfall rates of Typhoon Dan’s precipitation cloud systems at three different times during its life period is studied based on the data from TRMM Precipitation Radar(PR).Results show that stratiformis precipitation occupied more area and contributed more rainfall to the total than convective precipitation at the three times.The average rainfall rate of stratiformis precipitation at the three times varied little while that of convective precipitation had a considerable increase with the intensification of typhoon.In vertical,the profiles of convective and stratiformis precipitation were obviously different while the profiles of themself at the three times had small differences.Convective profiles can be devided into three sects according to the slope,the rainfall rate decrease with altitude and the rapidest decreased occurred within the altitude range from 5 to 6 km.Stratiformis profiles can be divided into four sects,and there was a distinct bright band structure near the altitude of 4.5km.
    10  Combinational anomalies of atmospheric circulation system and occurrences of extreme weather/climate events
    LI Chongyin YANG Hui ZHAO Jingjing
    2019, 42(3):321-333. DOI: 10.13878/j.cnki.dqkxxb.20190302001
    [Abstract](1174) [HTML](0) [PDF 6.67 M](1944)
    Abstract:
    Based on the analyses of lasting disaster weather of freezing rain and snow over South China during the January of 2008,the catastrophic flood over the Changjiang River basin during the summer of 1998 and the severe drought in Yunnan Province during the winter of 2009/2010,this paper emphasizes again that the combinational anomalies of atmospheric circulation system play important roles to the little probabilistic extreme weather and climate events.For the lasting disaster weather of freezing rain and snow in 2008,the combinational multiple circulation anomalies are clear,including Ural blocking high and a trough from Lake Baikal to Lake Balkhash that offer condition for continual cold air pouring southward from the west passage,and positive geopotential height anomalies over the eastern Asia and Japan that weaken the force of cold air from the north.Thus the front stays longer time over the South Mountain and its north area,which establishes background for continuous raining.Meanwhile,the Northwest Pacific subtropical high extents westward and its intensity also enhances,baring the cold air.The persistent strong Indian-Myanmar Trough and Northwest Pacific subtropical high transport warm and wet air to South China,and it is favorable to the occurrence of continuous rain.Persistent cold air activity and precipitation induce persistent low temperature,which creates conditions for the freezing weather.For the catastrophic flood over the Changjiang River basin in 1998,the combinational multiple circulation anomalies include the southwest monsoon surge,the Northwest Pacific subtropical high and cold air activities in summer,combined with the eastward propagation of convective system of the Tibetan Plateau.For the severe drought in Yunnan Province during the winter of 2009/2010,the combinational multiple circulation anomalies include that the Middle East subtropical jet at upper troposphere weakens,which influences Rossby wave and is unfavorable to the formation of the Tibetan Plateau-Bay of Bengal Trough;The Northwest Pacific subtropical high intensifies and shifts slightly southward,which inhibits the lower-level water vapor transmission into Yunnan;Teleconnection wave train anomaly caused by negative anomaly of NAO makes the cold air activity in the northeast of East Asia locate to the east and be difficult to reach the Yunnan region.It also makes the southern branch trough weak,and the warm and wet air could hardly reach to the Yunnan region.Generally the ENSO has a very important impact on weather and climate variations in China,but the influence is not same in every time and every region.The specific impact of ENSO must be analyzed case by case in order to determine its exact role in anomalous events,especially extreme weather and climate events.
    11  EvaluationoftheIPCCAR4ClimateModelsinSimulatingtheInterdecadalVariationsoftheEastChinaSummerPrecipitation,PDOandNAO
    GUWei1,2,LIChongyin2,3
    2010, 33(4):401-411.
    [Abstract](1987) [HTML](0) [PDF 1.30 M](3145)
    Abstract:
    利用1880—1999年中国东部35站的观测降水资料、英国Hadley中心的海温和海平面气压资料以及IPCC第4次评估报告(AR4)中20世纪气候模拟试验(20C3M)的模式输出结果,对IPCCAR4中22个耦合模式所模拟的我国东部夏季降水的年代际变化情况以及太平洋年代际涛动(PDO)和北大西洋涛动(NAO)的年代际变化情况进行了分析。结果显示,这些模式对20世纪我国东部夏季降水年代际变化的模拟结果并不理想,但对降水在20世纪70年代中期前后的突变具有一定的模拟能力。其中IAP_FGOALSL_0_G可以大致模拟出20世纪70年代中期前后降水型的突变特征,而BCCR_BCM2_0和UKMO_HadGEM1则可以模拟出华北地区降水在20世纪70年代中期之后减少的现象。对于引起我国东部夏季降水年代际变化的重要因子PDO和NAO,模式对它们年代际变化的模拟效果略好于降水。多数模式都可以模拟出PDO和NAO的空间模态,其中CNRM_CM3和UKMO_HadGEM1对PDO年代际变化(8a以上)的模拟与实际情况比较相似,并可以模拟出20世纪70年代中期之后PDO由负位相转变为正位相的情况,而模式UKMO_HadGEM1也对NAO的年代际变化以及1980年以来不断加强的趋势模拟较好。
    12  TENETS OF PRINCPAL OSCILLATION PATTERN ANALYSIS
    Zhang Jijia Ding Feng Wang Panxing
    1995, 18(1):93-99.
    [Abstract](1371) [HTML](0) [PDF 446.53 K](1994)
    Abstract:
    From a discrete time series derivation is done of two derived quantities for principal oscillation pattern analysis,i.e. ,the principal oscillation pattern(POP)and its associated correlation pattern(ACP).A pratical algorithm is presented from the examples of calculating the POP of a tropical Pacific SST anomaly time sequence and of ACP of the relatedarea 850 hPa wind.
    13  FEATURES OF 500hPa DIABATIC HEATING OVER THE NORTHERN HEMISPHERE DURING EL NINO AND ANTI-EL NINO YEARS
    Zhang Jijia Li Yueqing Lei Zhaochong Sun Zhaobo
    1991, 14(3):251-260.
    [Abstract](843) [HTML](0) [PDF 708.03 K](1832)
    Abstract:
    The monthly and seasonal diabatic heating anomaly fields in the atmosphere over the Northern Hemisphere during 6 E1 Nino and 6 anti-E1 Nino years (1967-1985) are calculated by using spectral representation. Results suggest that E1 Nino events have a significant influence upon the diabatic heating anomalies in the atmosphere. The atmosphere response is characterized by the well-organized positive and negative centers of diabatic heating in the lower and mid-high latitudes, with their phase and intensity having features of low frequency oscillation and difference centers showing quasi-bimonthly fluctuations. During anti-E1 Nino years the atmospheric distributions are the same as during E1 Nino years except for the anti-phase of the anomaly center values.As shown from the analysis of correlationship between sea surface temperature (SST) and atmospheric variables, the anomaly of SST is the dominant reason for the diabatic heating anomaly in the atmosphere.
    14  DYNAMICAL DIAGNOSIS OF A BLOCKING CASE
    Zhang Jijia Xu Hao
    1991, 14(2):135-142.
    [Abstract](863) [HTML](0) [PDF 596.16 K](1740)
    Abstract:
    On the basis of the spherical grid-point data from the ECMWF,a preliminary dynamical diagnosis is made of the blocking high whichpersisted over the Urals during October 4-27, 1981.Results show thatlike the decaying eddies on the track of the Atlantic and Pacificstorms at their last stage, the upstream eddies in this case have propertiesof upgradient transfer, and similar eddy forcing processes also exist inthe persistence of the Ural blocking. Furthermore, the orographic forcingof the Urals should not be neglected. In conclusion, the eddy forcingand the orographic effect of the Urals play a concurrent role in thepersistence of the blocking.
    15  STATISTICAL ANALYSIS OF THE BLOCKING PATTERNS IN THE NORTHERN HEMISPHERE
    Zhang Jijia Xu Hao
    1991, 14(1):1-9.
    [Abstract](793) [HTML](0) [PDF 2.64 M](1817)
    Abstract:
    Using the monthly NH 500hPa mean height field data from 1951 through 1988,the blocking patterns in the northern hemisphere areanalyzed statistically from the viewpoint of monthly means. Power and cross spectrum analyses show that the results from Scheme Care similarto those from Scheme D; periods of two and a half years (about 30months), one and a half years (18-20 months) and 2-5 months are all of considerable importance for the time series of area index in the regions;there is a good negative in-phase correlation between Region Ⅰ and Region Ⅱ: and periods with remarked coagulation exist in both the longand short time periods.
    16  MAJOR CLIMATIC DISASTERS IN CHINA AND THEIR IMPACT ON AGRICULTURAL PRODUCTION
    章基嘉 周曙光
    1990, 13(3):259-265.
    [Abstract](731) [HTML](0) [PDF 532.92 K](1827)
    Abstract:
    Preliminary studies are made of the time and spatial distribution of the major climatic disasters, such as drought, flooding and cool injury, in China and their impact on agricultural production. And the causes of their formation are analyzed in brief.
    17  INFLUENCE OF THE OCEAN HEATING SCALE ON THE STRUCTURE OF THE VERTICAL CIRCULATION CELL IN THE TROPICAL ATMOSPHERE
    Chao Jiping Wang Zhanggui
    1991, 14(1):10-17.
    [Abstract](818) [HTML](0) [PDF 515.66 K](1784)
    Abstract:
    A steady, shallow water atmospheric model is used to studythe influence of the ocean heating scale on the structure of Walker and Had1ey circulation in the tropical atmosphere. It is shown that whenthe heating scale is smaller than the critical scale, the results are similarto Gill’s (1980), that is,a hadley cell is located in the region of Rossby waves and an anti-Hadley cell in the area of Kelvin waves. When theheating scale is larger than the critical scale. a Hadley cell exists inthe entire heating region. In the latter case, a large scale Hadley cellcan transfer a great amount of heat and energy from the ocean to the extratropical region. This causes the atmospheric circulation there tobe anomalous. The results are consistent with statistical facts and sup-port the oretically the teleconnection which Bjerknes put forward inthe 1960s.
    18  VARIATIONAL PRINCIPLE OF INSTABILITY OF ATMOSPHERIC MOTIONS
    Zeng Qingcun
    1990, 13(2):123-157.
    [Abstract](873) [HTML](0) [PDF 1.87 M](1832)
    Abstract:
    Problems of instability of rotating atmospheric motions are investigated by using nonlinear governing equations and the variational principle. The method suggested in this palter is universal for obtaining criteria of instability in all models with all possible basic flows. For example, the model can be barotropic or baroelinic,layer or continuous, quasi-geostrophic or primitive equations, the basic flow can be zonal or nonzonal, steady or unsteady. Although the basic flows possess a great deal of variety,they all are the stationary points in the functional space determined by an appropriate invariant functional. The basic flow is an unsteady one if the conservation of angular momentum is included in the associated functional. The second variation, linear or nonlinear, gives the criteria of instability. Especially, the general criteria of instability for unsteady basic flow, orographically disturbed flow as well as nongeostrophic flow are first obtained by the method described in this paper. It is also shown that the difference between the criteria of instability obtained by the linear theory and our variational principle clearly indicates the importance of using nonlinear governing equations. In the appendix the theory is extended to cases such as in a 13-plane where the fluid does not possess finite total energy,hence the variational principle can not be directly applied. However, a generalized Liapounoff norm can still be obtained on the basis of variational consideration.

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