冰雪热力过程中物理参数辨识及高分辨率计算
|
摘要
全球气候变暖及其伴随事件,尤其是海平面上升,极端天气频发对人类的生产生活产生一系列影响,准确进行气候变化的预测是制定适应气候变化对策的依据,是指导保障人们活动的关键。极区、亚极区海冰和淡水冰是全球气候系统的重要组成部分之一由于其对气候变化的敏感性和反馈作用,因此一直被认定为物候学指标。而目前研究表明海冰和淡水冰数值模式发展的制约因素已不再局限于模式结构本身,而是倾向于数值算法的改进和参数化方案的优化。本文以2007-2009年两个冬季芬兰淡水湖泊湖冰的生消过程观测,2008-2009年黑龙江省红旗泡水库冬季水库冰的现场观测数据,以及中国第22次南极科学考察度夏和越冬期间(2005年11月至2006年12月)普里兹湾中山站附近固定冰生消过程的观测数据为数据基础；以雪/冰热力学模式和湖热力学模式结合气候预报模式为研究手段,对雪/冰生消过程的数值模拟进行了分析和研究。建立了雪/冰表面非线性离散时滞动力系统和冰底能量平衡系统,利用非线性常微分方程、离散时滞微分方程理论和有界变差函数等证明了系统的解的存在与唯一性等。构造最优辨识模型,证明了模型最优解的存在性和最优性必要条件。构造优化算法,辨识得到雪/冰表面温度与气温变化之间时滞最优值、模拟初冰日和海洋热通量的时间序列,同时得到了雪/冰热力学过程中对雪/冰厚度影响较大的物理参数。本文所研究内容和取得的主要成果概括如下：
1.通过数值模拟验证了热红外遥感观测冰表面温度反演冰厚度技术；分别利用估值法和迭代法模拟雪/冰表面温度,分析两种不同计算方法与观测值间的偏差；首次将离散时滞应用到雪/冰表面温度的计算中,建立了计算雪/冰表面温度的非线性离散时滞动力系统,利用时滞微分方程基本理论证明了系统解的存在与唯一性等。构造了以雪/冰表面温度与气温变化之间时滞为参量,计算雪/冰表面温度和观测值差值为目标函数的最优辨识模型,证明了模型最优解的存在性及最优性必要条件。数值模拟结果得到芬兰湖冰的雪/冰表面温度与气温变化之间的最优时滞值为1小时。
2.利用简单的冰解析模式得到红旗泡水库冰冰厚上限和下限包络线,为水利工程设计提供数值参考。分析海冰和湖冰热力过程中物理参数差异,将一维高分辨雪/冰热力学数值模式应用到芬兰湖冰的数值模拟中。给出不同的气温变化趋势,根据数值模拟结果分析湖冰对气候变化的敏感性和反馈作用。气温升高1℃,湖冰冰期将缩短13天,年际最大冰厚增大6cm(占总冰厚的17%)。利用一维高分辨雪/冰热力学数值模式和湖热力学模式模拟芬兰雪/湖冰厚度,分析高分辨率计算方法在雪/冰厚度计算中的精确度,以及雪/冰厚度模拟作为湖热力学数值模式一部分的准确性。针对雪/冰热力学数值模式中初冰日模拟的缺乏,给出模拟初冰日的定义。以初冰日为辨识参数,以观测气温和数值模式计算冰厚为约束条件建立最优辨识模型,构造了优化算法,数值结果较好地逼近观测初冰日。
3.基于冰底薄层能量平衡方程,建立了冰底薄层能量平衡系统,证明了系统解的存在与唯一性等。以海洋热通量为辨识参量,以观测冰厚和计算冰厚差值为目标函数建立了最优辨识模型,利用有界变差函数理论证明了模型最优解的存在性和最优性必要条件,利用改进的遗传算法求解模型的最优解。此方法仅利用日常常规海冰生消过程的观测项-冰厚观测数据即可计算求得冰底海洋热通量,避免了已有的海洋热通量的计算方法中所需经验参数不确定性及温度的观测误差所带来的海洋热通量的计算偏差。依2006-2007年冰期内南极普里兹湾中山站附近固定冰厚观测数据,由本文给出的辨识模型求得了冰底海洋热通量及海洋热通量关于时间的拟合函数。并依据2005-2006年同一考察区域观测数据,检验了冰底薄层能量平衡系统的有效性。
Global warming and its resulting events, special for sea level rising, more extreme weather would make some directly or indirectly impacts on human's production and life. Reasonable forecast of climate change is crucial to put forward corresponding policy. Sea ice and fresh ice in polar or sub-polar regions are primary factors in global climate system, and serve as proxy climate records as their sensitivity to climate change. The optimization of parameterization and the arithmetic are more important than the model strategy to develop the ice thermodynamic model. Based on the data derived from the reservoir-ice measurements in Hongqi-pao reservoir in Heilongjiang province in the winter of2008-2009, lake-ice thermodynamic processes in two Finnish lakes in the winter of2007-2008and the winter of2008-2009, and the field campaigns of landfast sea-ice thermodynamic observations off Zhongshan Station in Prydz Bay, East Antarctica from November2005to December2006, the growth and decay of ice cover have been studied. The sonw or ice surface nonlinear discrete time delay system and the ice bottom energy balance system have been established. The theory of the nonlinear differential equation, the time delay differential equation and bounded variation are use to investigate the existence and uniqueness of the solution of the systems. The optimal identification models have been put forward, and the existence of the optimal solution of the optimal identification models have been proved. The optimal time delay between the snow or ice surface temperature and the air temperature, modelled freezing date and oceanic heat flux are calculated by optimization algorithm. The physical parameters in snow and ice thermodynamic processes which have more influence on the snow and ice growth or decay, have been indicated.The main contributions are as follows:
1The inversion of the surface temperature for ice thickness by thermal remote sensing has been proved by the model simulations. The deviations between the observed and calculated snow or ice surface temperature by assumption and iteration methods have been analyzed. The nonlinear discrete time delay system for calculating the time delay between the snow or ice surface temperature and the air temperature has been established. The existence and uniqueness of the solution of the system has been proved by the theory of the time delay differential equation. Taken the time dalay between the snow or ice surface temperature and the air temperature as the identified parameter, the temperature deviation of calculated snow or ice surface temperature and observations is defined as the performance criterion, and then the optimal identification model has been put forward. The existence of the optimal solution of the optimal identification model has been proved, and the optimality conditions of the optimal identification model are provided. Based on the field measurements of Finnish lake ice, the numerical simulations show the optimal time delay between the snow or ice surface temperature and the air temperature was one hour.
2The upper and lower envelope of the ice thickness in Hongqi-pao reservoir which can be cite as the reference values for the water project design, have been calculated by some simple analysis models. To notice the difference of the physical parameterization of sea ice and lake ice, snow and ice model has been applied in the numerical simulation of Finnish lake ice. The simulation results show that the sensibility of the snow or ice surface temperature to the variation of the air temperature. With the air temperature increasing1℃, the lake ice season will be shorten by13days, and the annual maximal ice thickness will be6cm(which is about17%of the total ice thickness) more than reference simulation. A high resolution thermodynamic snow and ice model and lake thermodynamic model are applied to investigate the snow and ice thickness in Finnish lakes. The model simulations show that the necessity of the high resolution method in the lake ice thickness calculation and the lake thermodynamic model can give more accurate simulation during the early ice growth period. The modelled freezing date has been definited. The establish of the optimal identification model, which contain the freezing date as identified parameter, the observed air temperature and modelled ice thickness as the constraints, to make the snow and ice model can calculate the freezing date. The freezing date identified by the the optimal identification model more close to the observation.
3According to the energy balance equation at the ice bottom, the ice bottom energy balance system has been given, and the existence and uniqueness of the solution of the system lias been proved. The oceanic heat flux is selected as identified parameter and the the ice thickness deviation as the performance criterion, so that the optimal identification model has been presented. The existence of the optimal solution and the optimality conditions of the optimal identification model have been considered by using the theory of bounded variation. This method to estimate the oceanic heat flux is only controlled by observed ice thickness, which can overcome the calculated bias caused by the technique error of observed temperature and the empirical parameters in other methods. Based on the the field campaigns of landfast sea-ice thermodynamic observation off Zhongshan Station in Prydz Bay, East Antarctica in March2006to November2006, the time series of the oceanic heat flux and the oceanic heat flux fitting function of time has been derived. The effectiveness of the ice bottom energy balance system has been provided by the sensitivity study on the measurements in the same observation area from May2005to November2005.
1.通过数值模拟验证了热红外遥感观测冰表面温度反演冰厚度技术；分别利用估值法和迭代法模拟雪/冰表面温度,分析两种不同计算方法与观测值间的偏差；首次将离散时滞应用到雪/冰表面温度的计算中,建立了计算雪/冰表面温度的非线性离散时滞动力系统,利用时滞微分方程基本理论证明了系统解的存在与唯一性等。构造了以雪/冰表面温度与气温变化之间时滞为参量,计算雪/冰表面温度和观测值差值为目标函数的最优辨识模型,证明了模型最优解的存在性及最优性必要条件。数值模拟结果得到芬兰湖冰的雪/冰表面温度与气温变化之间的最优时滞值为1小时。
2.利用简单的冰解析模式得到红旗泡水库冰冰厚上限和下限包络线,为水利工程设计提供数值参考。分析海冰和湖冰热力过程中物理参数差异,将一维高分辨雪/冰热力学数值模式应用到芬兰湖冰的数值模拟中。给出不同的气温变化趋势,根据数值模拟结果分析湖冰对气候变化的敏感性和反馈作用。气温升高1℃,湖冰冰期将缩短13天,年际最大冰厚增大6cm(占总冰厚的17%)。利用一维高分辨雪/冰热力学数值模式和湖热力学模式模拟芬兰雪/湖冰厚度,分析高分辨率计算方法在雪/冰厚度计算中的精确度,以及雪/冰厚度模拟作为湖热力学数值模式一部分的准确性。针对雪/冰热力学数值模式中初冰日模拟的缺乏,给出模拟初冰日的定义。以初冰日为辨识参数,以观测气温和数值模式计算冰厚为约束条件建立最优辨识模型,构造了优化算法,数值结果较好地逼近观测初冰日。
3.基于冰底薄层能量平衡方程,建立了冰底薄层能量平衡系统,证明了系统解的存在与唯一性等。以海洋热通量为辨识参量,以观测冰厚和计算冰厚差值为目标函数建立了最优辨识模型,利用有界变差函数理论证明了模型最优解的存在性和最优性必要条件,利用改进的遗传算法求解模型的最优解。此方法仅利用日常常规海冰生消过程的观测项-冰厚观测数据即可计算求得冰底海洋热通量,避免了已有的海洋热通量的计算方法中所需经验参数不确定性及温度的观测误差所带来的海洋热通量的计算偏差。依2006-2007年冰期内南极普里兹湾中山站附近固定冰厚观测数据,由本文给出的辨识模型求得了冰底海洋热通量及海洋热通量关于时间的拟合函数。并依据2005-2006年同一考察区域观测数据,检验了冰底薄层能量平衡系统的有效性。
Global warming and its resulting events, special for sea level rising, more extreme weather would make some directly or indirectly impacts on human's production and life. Reasonable forecast of climate change is crucial to put forward corresponding policy. Sea ice and fresh ice in polar or sub-polar regions are primary factors in global climate system, and serve as proxy climate records as their sensitivity to climate change. The optimization of parameterization and the arithmetic are more important than the model strategy to develop the ice thermodynamic model. Based on the data derived from the reservoir-ice measurements in Hongqi-pao reservoir in Heilongjiang province in the winter of2008-2009, lake-ice thermodynamic processes in two Finnish lakes in the winter of2007-2008and the winter of2008-2009, and the field campaigns of landfast sea-ice thermodynamic observations off Zhongshan Station in Prydz Bay, East Antarctica from November2005to December2006, the growth and decay of ice cover have been studied. The sonw or ice surface nonlinear discrete time delay system and the ice bottom energy balance system have been established. The theory of the nonlinear differential equation, the time delay differential equation and bounded variation are use to investigate the existence and uniqueness of the solution of the systems. The optimal identification models have been put forward, and the existence of the optimal solution of the optimal identification models have been proved. The optimal time delay between the snow or ice surface temperature and the air temperature, modelled freezing date and oceanic heat flux are calculated by optimization algorithm. The physical parameters in snow and ice thermodynamic processes which have more influence on the snow and ice growth or decay, have been indicated.The main contributions are as follows:
1The inversion of the surface temperature for ice thickness by thermal remote sensing has been proved by the model simulations. The deviations between the observed and calculated snow or ice surface temperature by assumption and iteration methods have been analyzed. The nonlinear discrete time delay system for calculating the time delay between the snow or ice surface temperature and the air temperature has been established. The existence and uniqueness of the solution of the system has been proved by the theory of the time delay differential equation. Taken the time dalay between the snow or ice surface temperature and the air temperature as the identified parameter, the temperature deviation of calculated snow or ice surface temperature and observations is defined as the performance criterion, and then the optimal identification model has been put forward. The existence of the optimal solution of the optimal identification model has been proved, and the optimality conditions of the optimal identification model are provided. Based on the field measurements of Finnish lake ice, the numerical simulations show the optimal time delay between the snow or ice surface temperature and the air temperature was one hour.
2The upper and lower envelope of the ice thickness in Hongqi-pao reservoir which can be cite as the reference values for the water project design, have been calculated by some simple analysis models. To notice the difference of the physical parameterization of sea ice and lake ice, snow and ice model has been applied in the numerical simulation of Finnish lake ice. The simulation results show that the sensibility of the snow or ice surface temperature to the variation of the air temperature. With the air temperature increasing1℃, the lake ice season will be shorten by13days, and the annual maximal ice thickness will be6cm(which is about17%of the total ice thickness) more than reference simulation. A high resolution thermodynamic snow and ice model and lake thermodynamic model are applied to investigate the snow and ice thickness in Finnish lakes. The model simulations show that the necessity of the high resolution method in the lake ice thickness calculation and the lake thermodynamic model can give more accurate simulation during the early ice growth period. The modelled freezing date has been definited. The establish of the optimal identification model, which contain the freezing date as identified parameter, the observed air temperature and modelled ice thickness as the constraints, to make the snow and ice model can calculate the freezing date. The freezing date identified by the the optimal identification model more close to the observation.
3According to the energy balance equation at the ice bottom, the ice bottom energy balance system has been given, and the existence and uniqueness of the solution of the system lias been proved. The oceanic heat flux is selected as identified parameter and the the ice thickness deviation as the performance criterion, so that the optimal identification model has been presented. The existence of the optimal solution and the optimality conditions of the optimal identification model have been considered by using the theory of bounded variation. This method to estimate the oceanic heat flux is only controlled by observed ice thickness, which can overcome the calculated bias caused by the technique error of observed temperature and the empirical parameters in other methods. Based on the the field campaigns of landfast sea-ice thermodynamic observation off Zhongshan Station in Prydz Bay, East Antarctica in March2006to November2006, the time series of the oceanic heat flux and the oceanic heat flux fitting function of time has been derived. The effectiveness of the ice bottom energy balance system has been provided by the sensitivity study on the measurements in the same observation area from May2005to November2005.
引文
[1]IPCC, WGI. The IPCC WGII Fourth Assessment Report:Summary for Policymakers [EB/OL]. http://www. ipcc. ch,2007.
[2]Stroeve J, Serreze M, Drobot S, et al. Arctic sea ice extent plummets in 2007 [J]. EOS, 2008,89(2):13-14.
[3]Rothrock D A, Yu Y, Maykut G A. Thinning of the Arctic sea-ice cover [J]. Geophysical Research Letters,1999,26(23):3469-3472.
[4]Comiso J C, Parkinson C L, Gersten R, et al. Accelerated decline in the Arctic sea ice cover [J]. Geophysical Research Letters,2008,35(L01703):doi:10.1029/2007GL031972.
[5]Stroeve J, Holland M M, Meier W, et al. Arctic sea ice decline:Faster than forecast [J]. Geophysical Research Letters,2007,34(9):1-5.
[6]Magnuson J J, Robertson D M, Benson B J, et al. Historical trends in lake and river ice cover in the Northern Hemisphere [J]. Science,2000,289(5485):1743-1746.
[7]Robinson D A. Hemispheric snow cover and surface albedo for model validation [J]. Annals of Glaciology,1997,25:241-245.
[8]刘钦政,黄嘉佑,白珊,等.渤海冬季海冰气候变异的成因分析[J].海洋学报(中文版),2004,26(2)：11-19.
[9]Turner J, Comiso J C, Marshall G J, et al. Non-annular atmospheric circulation change induced by stratospheric ozone depletion and its role in the recent increase of Antarctic sea ice extent [J]. Geophysical Research Letters,2009,36(8): doi:10.1029/2009GL037524.
[10]Zwally H J, Comiso J C, Parkinson C L, et al. Variability of Antarctic sea ice 1979-1998 [J]. Journal of Geophysical Research,2002,107(3041):doi:10.1029/2000JC000733.
[11]Liu J, Curry J A. Accelerated warming of the Southern Ocean and its impacts on the hydrological cycle and sea ice [J]. Proceedings of the National Academy of Sciences, 2010,107(34):14987-14992.
[12]Meier W. Arctic sea ice breaks lowest extent on record [EB/OL]. http://nsidc. org/news/press/20120827_2012extentbreaks2007record.html,2012.
[13]中国新闻网.美国科学家语出惊人：南北两极海冰将完全消失[EB/OL].http://www. chinanews. com/gj/gj-sjkj/news/2009/05-21/1701176. shtml,2009.
[14]国家海洋局.2010年中国海平面公报[EB/0L].http://www. coi. gov. cn/gongbao/haipingmian/201107/t20110729_17734.html,2011.
[15]杨桂山.中国热带气旋灾害及全球变暖背景下的可能趋势分析[J].自然灾害学报,1996,5(2)：47-55.
[16]Fedotov V I, Cherepanov N V, Tyshko K P. Some features of the growth, structure and metamorphism of East Antarctic landfast sea ice [J]. Antarctic sea ice:physical processes, interactions and variability,1998:343-354.
[17]唐述林,秦大河,任贾文,等.极地海冰的研究及其在气候变化中的作用[J].冰川冻土,2006,28(001)：91-100.
[18]Heil P. Atmospheric conditions and fast ice at Davis, East Antarctica:A case study [J]. Journal of Geophysical Research,2006,111(C5):25741-25752.
[19]Dumas J, Carmack E, Melling H. Climate change impacts on the Beaufort shelf landfast ice [J]. Cold Regions Science and Technology,2005,42(1):41-51.
[20]Lepparanta M, Reinart A, Erm A, et al. Investigation of ice and water properties and under-ice light fields in fresh and brackish water bodies [J]. Nordic Hydrology,2003, 34(3):245-266.
[21]Livingstone D M. Lake oxygenation:application of a one-box model with ice cover [J]. Internationale Revue der gesamten Hydrobiologie und Hydrographie,1993,78(4): 465-480.
[22]Gerten D, Adrian R. Climate-driven changes in spring plankton dynamics and the sensitivity of shallow polyraictic lakes to the North Atlantic Oscillation [J]. Limnology and Oceanography,2000,45(5):1058-1066.
[23]Mironov D, Terzhevik A, Kirillin G, et al. Radiatively driven convection in ice-covered lakes:Observations, scaling, and a mixed layer model [J]. Journal of Geophysical Research,2002,107 (C4):doi:10.1029/2001JC000892.
[24]Vavrus S J, Wynne R H, Foley J A. Measuring the sensitivity of southern Wisconsin lake ice to climate variations and lake depth using a numerical model [J]. Limnology and Oceanography,1996,41 (5):822-831.
[25]Yen Y C. Review of thermal properties of snow, ice and sea ice [R]. Cold Regions Research and Engineering Laboratory Report 81-10, Hanover, New Hampshire,1981:1-27.
[26]Lepparanta M. A review of analytical models of sea-ice growth [J]. Atmosphere-Ocean, 1993,31(1):123-138.
[27]Ashton G D. Lake ice decay [J]. Cold Regions Science and Technology,1983,8(1):83-86.
[28]Bilello M A. Maximum thickness and subsequent decay if lake, river and fast sea ice in Canada and Alaska [R]. CREEL Report,1980.
[29]Untersteiner N. Calculations of temperature regime and heat budget of sea ice in the Central Arctic [J]. Journal of Geophysical Research,1964,69(22):4755-4766.
[30]Maykut G A, Untersteiner N. Some results from a time-dependent thermodynamic model of sea ice [J]. Journal of Geophysical Research,1971,76(6):1550-1575.
[31]Semtner Jr A J. A model for the thermodynamic growth of sea ice in numerical investigations of climate [J]. Journal of Physical Oceanography,1976,6(3):379-389.
[32]Hibler W D. A dynamic thermodynamic sea ice model [J]. Journal of Physical Oceanography, 1979,9:815-846.
[33]Lepparanta M. A growth model for black ice, snow ice and snow thickness in subarctic basins [J]. Nordic Hydrology,1983,14(2):59-70.
[34]程斌.一维海冰热力模式的守恒型差分格式和数值模拟[J].海洋通报,1996,15(4)：8-16.
[35]Lipscombz W H. An energy-conserving thermodynamic model of sea ice [J]. Journal of Geophysical Research,1999,104 (C7):15-669.
[36]Launiainen J, Cheng B. Modelling of ice thermodynamics in natural water bodies [J]. Cold Regions Science and Technology,1998,27(3):153-178.
[37]Cheng B, Vihma T, Launiainen J. Modelling of the superimposed ice formation and sub-surface melting in the Baltic Sea [J]. Geophysica,2003,39(1-2):31-50.
[38]Cheng B, Zhang Z, Vihma T, et al. Model experiments on snow and ice thermodynamics in the Arctic Ocean with CHINARE 2003 data [J]. Journal of Geophysical Research,2008, 113(C09020):doi:10.1029/2007JC004654.
[39]Wadhams P. Atmosphere-ice-ocean interactions in the Antarctic [J]. Antarctica and Global Climatic Change,1991:65-81.
[40]Heron R, Woo M. Decay of a High Arctic lake-ice cover:observations and modelling [J]. Journal of Glaciology,1994,40(135):283-292.
[41]Stefan H G, Fang X. Simulated climate change.effects on ice and snow covers on lakes in a temperate region [J]. Cold Regions Science and Technology,1997,25(2):137-152.
[42]Mironov D. Parameterization of lakes in numerical weather prediction. Description of a lake model [R]. Technical Report of COSMO, No.11.,2008:44pp.
[43]Mironov D, Heise E, et al. Implementation of the lake parameterisation scheme FLake into the numerical weather prediction model COSMO [J]. Boreal Environment Research, 2010,15:218-230.
[44]Kourzeneva E. External data for lake parameterization in Numerical Weather Prediction and climate modeling [J]. Boreal Environment Research,2010,15(2):165-177.
[45]Semmler T, Cheng B, Yang Y, et al. Snow and ice on Bear Lake (Alaska)-sensitivity experiments with two lake ice models [J]. Tellus A,2012,64:doi: 10.3402/tellusa. v64iO.17339.
[46]Flato G M, Brown R D. Variability and climate sensitivity of landfast Arctic sea ice [J]. Journal of Geophysical Research,1996, 101(C11):25725-25767.
[47]Duguay C R, Flato G M, Jeffries M 0, et al. Ice-cover variability on shallow lakes at high latitudes:model simulations and observations [J]. Hydrological Processes,2003, 17(17):3465-3483.
[48]Reid T, Crout N. A thermodynamic model of freshwater Antarctic lake ice [J]. Ecological Modelling,2008,210(3):231-241.
[49]Goldstine H H. A History of the Calculus of Variations from the 17th through the 19th Century [M]. Springer-Verlag,1980.
[50]Pontryagin, L S, Boltyanski V G, Gamkrelidze R V, Mischenko E E. The Mathematical Theory of Optimal Process [M]. Fizmatigiz, Interscience Publishers Ino., New York,1962.
[51]Bellman, R, Dreyfus S. Applied Dynamic Programming [M]. Izd. Inostr. Litri, Princeton University Press,1960.
[52]Kalman R E. Contributions to the theory of optimal control [J]. Bol. Soc. Mat. Mexicana, 1960,5(2):102-119.
[53]Krasovskii N N. On the theory of optimum control [J]. Journal of Applied Mathematics and Mechanics,1959,23(4):899-919.
[54]Filippov A F. On certain questions in the theory of optimal control [J]. Control Optimization,1962,1:76-84.
[55]Stoddart A W J. Existence of optimal controls [J]. Pacific Journal of Mathematics,1967, 20(1):167-177.
[56]Hartl R F, Sethi S P, Vickson R G. A survey of the maximum principles for optimal control problems with state constraints [J]. SIAM review,1995:181-218.
[57]Polak E. Optimization:algorithms and consistent approximations [M]. Springer-Verlag, 1997.
[58]Dontchev A L, Hager W W, Poore A B, et al. Optimality, stability, and convergence in nonlinear control [J]. Applied Mathematics & Optimization,1995,31(3):297-326.
[59]Schwartz A, Polak E. Consistent approximations for optimal control problems based on Runge-Kutta integration [J]. SIAM Journal on Control and Optimization,1996,34: 1235-1269.
[60]钱学森,宋健.工程控制论,下册[M].科学出版社,1981.
[61]Li W. Optimal control theory for infinite dimensional systems (Systems and control foundations and applications) [M]. Recherche,1994.
[62]高夯.半线性抛物方程支配系统的最优性条件[J].数学学报,1991,42(4)：705-714.
[63]Wang G W, Wang L. The Carlman inequality and its application to periodic optimal control governed by semilinear parabolic differential equations [J]. Journal of Optimization Theory and Applications,2003,118(2):429-461.
[64]宫召华,冯恩民.微生物批式流加发酵多阶段最优控制的最优性条件[J].运筹学学报,2009(4)：109-119.
[65]李晓红,冯恩民,修志龙.微生物连续培养非线性动力系统的性质及最优性条件[J].工程数学学报,2006,23(1)：7-12.
[66]江胜宗,韦艳华,钱伟懿,等.侧钻水平井轨道设计模型,算法及应用[J].运筹与管理,2002,11(2)：60-65.
[67]钱伟懿,江胜宗,冯恩民.三维水平井最优控制系统及应用[J].高校应用数学学报,2003,18(1)：8-14.
[68]冯恩民,赵海清,白乙拉,等.北极冰的热传递模型和数值模拟[J].工程数学学报,2007,24(3)：381-390.
[69]冯恩民,方海鹏,李志军.多区域耦合分布参数系统辨识与应用[J].应用基础与工程科学学报,2008,16(2)：153-160.
[70]张丹.三维北极海冰温度场的参数辨识与数值模拟[D].大连理工大学,2007.
[71]辛超.北极海冰温度场热力学参数辨识与数值模拟[D].大连理工大学,2007.
[72]孙丹丹.北极海冰雪的分布参数系统辨识与潮流的计算[D].大连理工大学,2007.
[73]白乙拉,李志军,冯恩民,等.静水条件下淡水冰有效导温系数的优化辨识[J].计算力学学报,2007,24(2)：187-191.
[74]Lv W, Feng E, Li Z. A coupled thermodynamic system of sea ice and its parameter identification [J]. Applied Mathematical Modelling,2008,32(7):1198-1207.
[75]徐鉴,陈予恕.时滞速度反馈对强迫自持系统动力学行为的影响[J].应用数学和力学,2004,25(5)：455-466.
[76]Xu J, Chung K W. Effects of time delayed position feedback on a van der Pol-Duffing oscillator [J]. Physica D:Nonlinear Phenomena,2003,180(1):17-39.
[77]Hong T, Hughes P C. Effect of time delay on the stability of flexible structures with rate feedback control [J]. Journal of Vibration and Control,2001,7(1):33-49.
[78]Zheng F, Frank P M. Robust control of uncertain distributed delay systems with application to the stabilization of combustion in rocket motor chambers [J]. Automatica, 2002,38(3):487-497.
[79]Qing Y W, Qi S L. Time delay-enhanced synchronization and regularization in two coupled chaotic neurons [J]. Chinese Physics Letters,2005,22(3):543-546.
[80]连涵生.带有时滞的微生物连续培养模型的分析与模拟[D].大连理工大学,2008.
[81]李晓芳,冯恩民,连涵生.连续时滞对微生物连续培养过程振荡行为影响[J].大连理工大学学报,2011,51(2)：309-312.
[82]Zimbidis A, Haberman S. The combined effect of delay and feedback on the insurance pricing process:a control theory approach [J]. Insurance:Mathematics and Economics, 2001,28(2):263-280.
[83]Grassia P S. Delay, feedback and quenching in financial markets [J]. The European Physical Journal B-Condensed Matter and Complex Systems,2000,17(2):347-362.
[84]Moore R M, Wallace D W R. A relationship between heat transfer to sea ice and temperature-salinity properties of Arctic Ocean waters [J]. Journal of Geophysical Research,1988,93(C1):565-571.
[85]Omstedt A, Svensson U. On the melt rate of drifting ice heated from below [J]. Cold Regions Science and Technology,1992,21(1):91-100.
[86]季顺迎.渤海海冰数值模拟及其工程应用[D].大连理工大学,2001.
[87]Lei R, Li Z, Cheng B, et al. Annual cycle of landfast sea ice in Prydz Bay, east Antarctica [J]. Journal of Geophysical Research,2010,115(C2):doi:10.1029/2008JC005223.
[88]Heil P, Allison I, Lytle V I. Seasonal and interannual variations of the oceanic heat flux under a landfast Antarctic sea ice cover [J]. Journal of Geophysical Research, 1996,101 (C11):25725-25741.
[89]Teo K L, Goh C J, Wong K H. A unified computational approach to optimal control problems [M]. The United States with John Wiley & Sons, Inc., New York,1990.
[90]马知恩,周义仓.常微分方程定性与稳定性方法[M].科学出版社,2001.
[91]秦元勋,刘永清,王联.带有时滞的动力系统的运动稳定性[M].科学出版社,1989.
[92]王康宁.最优控制的数学理论[M].国防工业出版社,1995.
[93]陈宝林.最优化理论与算法[M].清华大学出版社,2005.
[94]袁亚湘,孙文瑜.最优化理论与算法[M].北京：科学出版社,1997.
[95]程斌,刘春厚.海冰数值研究在海洋石油开发和工程设计中的释用[J].中国海上油气(工程),1996,8(1)：37-42.
[96]Yang Y, Lepparanta M, Cheng B, et al. Numerical modelling of snow and ice thicknesses in Lake Vanajavesi, Finland [J]. Tellus A,2012,64:doi:10.3402/tellusa. v64iO.17202.
[97]Cheng B, Vihma T, Pirazzini R, et al. Modelling of superimposed ice formation during the spring snowmelt period in the Baltic Sea [J]. Annals of Glaciology,2006,44(1): 139-146.
[98]Cheng B, Vihma T, Zhang Z, et al. Snow and sea ice thermodynamics in the Arctic:Model validation and sensitivity study against SHEBA data [J]. Chinese Journal of Polar Science,2008,19(2):108-122.
[99]Botzet M, Christensen J H, Claudius E, et al. A regional climate model of the Arctic atmosphere [J]. World Meteorological Organization-Publications-Wmo Td,1996: 169-173.
[100]Briegleb B P, Bitz C M, Hunke E C, et al. Scientific description of the sea ice component in the Community Climate System Model, version three [R]. Technical Note TN-463STR, NTIS\# PB2004-106574, NatNCAR Technical Report No. NCAR/TN-463+STR,2004:78pp.
[101]Curry J A, Schramm J L, Perovich D K, et al. Applications of SHEBA/FIRE data to evaluation of snow/ice albedo parameterizations [J]. Journal of Geophysical Research, 2001,106 (D14):15345-15355.
[102]杨清华,程斌,雷瑞波,等.北极夏季海冰反照率的观测和数值模拟试验[J].海洋学报,2011,33(2)：42-47.
[103]Pirazzini R, Vihma T, Granskog M A, et al. Surface albedo measurements over sea ice in the Baltic Sea during the spring snowmelt period [J]. Annals of Glaciology,2006, 44(1):7-14.
[104]Shine K P. Parametrization of the shortwave flux over high albedo surfaces as a function of cloud thickness and surface albedo [J]. Quarterly Journal of the Royal Meteorological Society,1984,110(465):747-764.
[105]Zillman J W. A study of some aspects of the radiation and heat budgets of the southern hemisphere oceans [M]. Australian Government Publishing Service,1972.
[106]Lumb F E. The influence of cloud on hourly amounts of total solar radiation at the sea surface [J]. Quarterly Journal of the Royal Meteorological Society,1964,90(383): 43-56.
[107]Bennett Jr T J. A coupled atmosphere-sea ice model study of the role of sea ice in climatic predictability [J]. Journal of Atmospheric Sciences,1982,39:1456-1465.
[108]Brunt D. Notes on radiation in the atmosphere. I [J]. Quarterly Journal of the Royal Meteorological Society,1932,58(247):389-420.
[109]Guest P S. Surface longwave radiation conditions in the eastern Weddell Sea during winter [J]. Journal of Geophysical Research,1998,103(C13):30761-30771.
[110]Berliand M E. Determining the net long-wave radiation of the Earth with consideration of the effect of cloudiness [J]. Izv. Akad. Nauk. SSSR Ser. Geofiz,1952,1:64-78.
[111]Efimova N A. On methods of calculating monthly values of net longwave radiation [J]. Meteorol. Gidrol,1961,10:28-33.
[112]Prata A J. A new long-wave formula for estimating downward clear-sky radiation at the surface [J]. Quarterly Journal of the Royal Meteorological Society,1996, 122(533):1127-1151.
[113]Jacobs J D. Radiation climate of Broughton Island [J]. Energy Budget Studies in Relation to Fast-Ice Breakup Processes in Davis Strait,1978,26:105-120.
[114]DrebsA, Nordlund A, KarlssonP, et al. Climatological statistics of Finland 1971-2000 [R]. Finnish Meteorological Institute:2002:99pp.
[115]Barry R G, Jacobs J D, Crane R G, et al. Energy budget studies in relation to fast-ice breakup processes in davis strait:climatological overview [M]. University of Colorado, Institute of Arctic and Alpine Research,1978.
[116]Lepparanta M, Lewis J E. Observations of ice surface temperature and thickness in the Baltic Sea [J]. International Journal of Remote Sensing,2007,28(17):3963-3977.
[117]Wang X, Key J R, Liu Y. A thermodynamic model for estimating sea and lake ice thickness with optical satellite data [J]. Journal of Geophysical Research,2010,115(12):1-14.
[118]Marko M, Hsieh C, Schalek R, et al. Focused-ion-beam thinning of frozen-hydrated biological specimens for cryo-electron microscopy [J]. Nature Methods,2007,4(3): 215-217.
[119]王索成,孟凡静,卢永超,等.红旗泡水库水质监测分析[M].黑龙江水利科技,2008,36(3)：134-134.
[120]肖建民,金龙海,谢永刚,等.寒区水库冰盖形成与消融机理分析[J].水利学报,2004,6：80-85.
[121]倪汉根,徐福生,金生.水库冰盖厚度及弧门自开启原因的研究[J].水利学报,1998(12)：1-6.
[122]Shen H T, Yapa P D. A unified degree-day method for river ice cover thickness simulation [J]. Canadian Journal of Civil Engineering,1985,12(1):54-62.
[123]Eicken H. From the microscopic, to the macroscopic, to the regional scale:growth, microstructure and properties of sea ice [J]. Sea Ice:An Introduction to its Physics, Chemistry, Biology and Geology,2003:22-81.
[124]Korhonen J. Suomen vesistojen jaaolot [M]. Suomen ymparistokeskus,2005.
[125]Lepparanta M, Uusikivi J. The annual cycle of the Lake Paajarvi ice [J]. Lammi Notes, 2002,29:4-9.
[126]Saloranta T M. Modeling the evolution of snow, snow ice and ice in the Baltic Sea [J]. Tellus A,2000,52(1):93-108.
[127]Kovacs A. Sea ice. part 1. bulk salinity versus ice floe thickness [R]. DTIC Document: 1996.
[128]Karkas E. The ice season of Lake Paajarvi in southern Finland [J]. Geophysica,2000, 36(1-2):69-84.
[129]Lei R, Lepparanta M, Erm A, et al. Field investigations of apparent optical properties of ice cover in Finnish and Estonian lakes in winter 2009 [J]. Estonian Journal of Earth Sciences,2011,60:50-64.
[130]Shirasawa K, Lepparanta M, Kawamura T, et al. Measurements and modelling of the water: ice heat flux in natural waters [C]//Proceedings of the 18th IAHR International Symposium on Ice. Sapporo, Japan:2006,1:85-91.
[131]Jakkila J, Lepparanta M, Kawamura T, et al. Radiation transfer and heat budget during the ice season in Lake Paajarvi, Finland [J]. Aquatic Ecology,2009,43(3):681-692.
[132]Burda N Y. Computer simulation for Ladoga Lake ice dynamics based on remotely sensed data [C]//IGARSS'99 Proceedings.1999,2:937-939.
[133]Wake A R R. Great lakes ice dynamics simulation [J]. Journal of Waterway, Port, Coastal and Ocean Engineering,1983,109:86-102.
[134]Lepparanta M. Modelling the formation and decay of lake ice [J]. The Impact of Climate Change on European Lakes,2010,4:63-83.
[135]Palecki M A, Barry R G. Freeze-up and break-up of lakes as an index of temperature changes during the transition seasons:A case study for Finland [J]. Journal of Applied Meteorology and Climatology Advanced Search,1986,25(7):893-902.
[136]Nonaka T, Matsunaga T, Hoyano A. Estimating ice breakup dates on Eurasian lakes using water temperature trends and threshold surface temperatures derived from MODIS data [J]. International Journal of Remote Sensing,2007,28(10):2163-2179.
[137]Anderson E A. A point energy and mass balance model of a snow cover [M]. United States. Office of Hydrology,1976.
[138]Zhou X, Li S, Stamnes K. Effects of vertical inhomogeneity on snow spectral albedo and its implication for optical remote sensing of snow [J]. Journal of Geophysical Research,2003,108(4738):doi:10.1029/2003JD003859.
[139]Sturm M, Holmgren J, Konig M, et al. The thermal conductivity of seasonal snow [J]. Journal of Glaciology,1997,43(143):26-41.
[140]Patterson J C, Hamblin P F. Thermal simulation of a lake with winter ice cover [J]. Limnology and Oceanography,1988,33(3):323-338.
[141]Brown L C, Duguay C R. The response and role of ice cover in lake-climate interactions[J]. Progress in Physical Geography,2010,34(5):671-704.
[142]Parkinson C L, Washington W M. A large-scale numerical model of sea ice [J]. Journal of Geophysical Research,1979,84(C1):311-337.
[143]Wettlaufer J S. Heat flux at the ice-ocean interface [J]. Journal of Geophysical Research,1991,96(C4):7215-7236.
[144]McPhee M G, Maykut G A, Morison J H. Dynamics and thermodynamics of the ice/upper ocean system in the marginal ice zone of the Greenland Sea [J]. Journal of Geophysical Research,1987,92(C7):7017-7031.
[145]Harvey L D D. Testing alternative parameterizations of lateral melting and upward basal heat flux in a thermodynamic sea ice model [J]. Journal of Geophysical Research,1990, 95(C5):7359-7365.
[146]Omstedt A, Wettlaufer J S. Ice growth and oceanic heat flux:Models and measurements [J]. Journal of Geophysical Research,1992,97(C6):9383-9390.
[147]Shirasawa K, Grant Ingram R, Hudier E J J. Oceanic heat fluxes under thin sea ice in Saroma-ko Lagoon, Hokkaido, Japan [J]. Journal of marine systems,1997,11(1):9-19.
[148]McPhee M G, Untersteiner N. Using sea ice to measure vertical heat flux in the ocean [J]. Journal of Geophysical Research,1982,87(C3):2071-2074.
[149]季顺迎,岳前进.工程海冰数值模型及应用[M].北京：科学出版社,2011.
[150]白乙拉.非光滑分布参数系统参数辨识及其应用[D].大连理工大学,2006.
[151]周明,孙树栋.遗传算法原理及其应用[M].北京：国防工业出版社,1999.
[152]Lei R, Li Z, Cheng Y, et al. A new apparatus for monitoring sea ice thickness based on the Magnetostrictive-Delay-Line principle [J]. Journal of Atmospheric and Oceanic Technology,2009,26(4):818-827.
[153]陆龙骅,卞林根,逯昌贵,等.近20年中国南极科学考察的气象业务进展[J].气象,2005,31(001)：3-8.
[154]Perovich D K. The optical properties of sea ice [R]. DTIC Document,1996:23pp.
[155]Yang Y, Li Z, Lepparanta M, et al. Estimation of oceanic heat flux under landfast sea ice in Prydz Bay, East Antarctica[C]//20th IAHR International Symposium on Ice,2010, No.47.
[156]Nazintsev Y L. Nekotorye dannye k raschetu teplovykh Svoistv morskogo 1'da (Some data on the calculation of thermal properties of sea ice) [J]. Tr. Arkt. Antarkt. Nauchlo Issled Inst,1964,267:31-47.
[157]Makshtas A P. Thermodynamics of sea ice [J]. Physics of Ice-covered Seas,1998,1: 289-303.
[2]Stroeve J, Serreze M, Drobot S, et al. Arctic sea ice extent plummets in 2007 [J]. EOS, 2008,89(2):13-14.
[3]Rothrock D A, Yu Y, Maykut G A. Thinning of the Arctic sea-ice cover [J]. Geophysical Research Letters,1999,26(23):3469-3472.
[4]Comiso J C, Parkinson C L, Gersten R, et al. Accelerated decline in the Arctic sea ice cover [J]. Geophysical Research Letters,2008,35(L01703):doi:10.1029/2007GL031972.
[5]Stroeve J, Holland M M, Meier W, et al. Arctic sea ice decline:Faster than forecast [J]. Geophysical Research Letters,2007,34(9):1-5.
[6]Magnuson J J, Robertson D M, Benson B J, et al. Historical trends in lake and river ice cover in the Northern Hemisphere [J]. Science,2000,289(5485):1743-1746.
[7]Robinson D A. Hemispheric snow cover and surface albedo for model validation [J]. Annals of Glaciology,1997,25:241-245.
[8]刘钦政,黄嘉佑,白珊,等.渤海冬季海冰气候变异的成因分析[J].海洋学报(中文版),2004,26(2)：11-19.
[9]Turner J, Comiso J C, Marshall G J, et al. Non-annular atmospheric circulation change induced by stratospheric ozone depletion and its role in the recent increase of Antarctic sea ice extent [J]. Geophysical Research Letters,2009,36(8): doi:10.1029/2009GL037524.
[10]Zwally H J, Comiso J C, Parkinson C L, et al. Variability of Antarctic sea ice 1979-1998 [J]. Journal of Geophysical Research,2002,107(3041):doi:10.1029/2000JC000733.
[11]Liu J, Curry J A. Accelerated warming of the Southern Ocean and its impacts on the hydrological cycle and sea ice [J]. Proceedings of the National Academy of Sciences, 2010,107(34):14987-14992.
[12]Meier W. Arctic sea ice breaks lowest extent on record [EB/OL]. http://nsidc. org/news/press/20120827_2012extentbreaks2007record.html,2012.
[13]中国新闻网.美国科学家语出惊人：南北两极海冰将完全消失[EB/OL].http://www. chinanews. com/gj/gj-sjkj/news/2009/05-21/1701176. shtml,2009.
[14]国家海洋局.2010年中国海平面公报[EB/0L].http://www. coi. gov. cn/gongbao/haipingmian/201107/t20110729_17734.html,2011.
[15]杨桂山.中国热带气旋灾害及全球变暖背景下的可能趋势分析[J].自然灾害学报,1996,5(2)：47-55.
[16]Fedotov V I, Cherepanov N V, Tyshko K P. Some features of the growth, structure and metamorphism of East Antarctic landfast sea ice [J]. Antarctic sea ice:physical processes, interactions and variability,1998:343-354.
[17]唐述林,秦大河,任贾文,等.极地海冰的研究及其在气候变化中的作用[J].冰川冻土,2006,28(001)：91-100.
[18]Heil P. Atmospheric conditions and fast ice at Davis, East Antarctica:A case study [J]. Journal of Geophysical Research,2006,111(C5):25741-25752.
[19]Dumas J, Carmack E, Melling H. Climate change impacts on the Beaufort shelf landfast ice [J]. Cold Regions Science and Technology,2005,42(1):41-51.
[20]Lepparanta M, Reinart A, Erm A, et al. Investigation of ice and water properties and under-ice light fields in fresh and brackish water bodies [J]. Nordic Hydrology,2003, 34(3):245-266.
[21]Livingstone D M. Lake oxygenation:application of a one-box model with ice cover [J]. Internationale Revue der gesamten Hydrobiologie und Hydrographie,1993,78(4): 465-480.
[22]Gerten D, Adrian R. Climate-driven changes in spring plankton dynamics and the sensitivity of shallow polyraictic lakes to the North Atlantic Oscillation [J]. Limnology and Oceanography,2000,45(5):1058-1066.
[23]Mironov D, Terzhevik A, Kirillin G, et al. Radiatively driven convection in ice-covered lakes:Observations, scaling, and a mixed layer model [J]. Journal of Geophysical Research,2002,107 (C4):doi:10.1029/2001JC000892.
[24]Vavrus S J, Wynne R H, Foley J A. Measuring the sensitivity of southern Wisconsin lake ice to climate variations and lake depth using a numerical model [J]. Limnology and Oceanography,1996,41 (5):822-831.
[25]Yen Y C. Review of thermal properties of snow, ice and sea ice [R]. Cold Regions Research and Engineering Laboratory Report 81-10, Hanover, New Hampshire,1981:1-27.
[26]Lepparanta M. A review of analytical models of sea-ice growth [J]. Atmosphere-Ocean, 1993,31(1):123-138.
[27]Ashton G D. Lake ice decay [J]. Cold Regions Science and Technology,1983,8(1):83-86.
[28]Bilello M A. Maximum thickness and subsequent decay if lake, river and fast sea ice in Canada and Alaska [R]. CREEL Report,1980.
[29]Untersteiner N. Calculations of temperature regime and heat budget of sea ice in the Central Arctic [J]. Journal of Geophysical Research,1964,69(22):4755-4766.
[30]Maykut G A, Untersteiner N. Some results from a time-dependent thermodynamic model of sea ice [J]. Journal of Geophysical Research,1971,76(6):1550-1575.
[31]Semtner Jr A J. A model for the thermodynamic growth of sea ice in numerical investigations of climate [J]. Journal of Physical Oceanography,1976,6(3):379-389.
[32]Hibler W D. A dynamic thermodynamic sea ice model [J]. Journal of Physical Oceanography, 1979,9:815-846.
[33]Lepparanta M. A growth model for black ice, snow ice and snow thickness in subarctic basins [J]. Nordic Hydrology,1983,14(2):59-70.
[34]程斌.一维海冰热力模式的守恒型差分格式和数值模拟[J].海洋通报,1996,15(4)：8-16.
[35]Lipscombz W H. An energy-conserving thermodynamic model of sea ice [J]. Journal of Geophysical Research,1999,104 (C7):15-669.
[36]Launiainen J, Cheng B. Modelling of ice thermodynamics in natural water bodies [J]. Cold Regions Science and Technology,1998,27(3):153-178.
[37]Cheng B, Vihma T, Launiainen J. Modelling of the superimposed ice formation and sub-surface melting in the Baltic Sea [J]. Geophysica,2003,39(1-2):31-50.
[38]Cheng B, Zhang Z, Vihma T, et al. Model experiments on snow and ice thermodynamics in the Arctic Ocean with CHINARE 2003 data [J]. Journal of Geophysical Research,2008, 113(C09020):doi:10.1029/2007JC004654.
[39]Wadhams P. Atmosphere-ice-ocean interactions in the Antarctic [J]. Antarctica and Global Climatic Change,1991:65-81.
[40]Heron R, Woo M. Decay of a High Arctic lake-ice cover:observations and modelling [J]. Journal of Glaciology,1994,40(135):283-292.
[41]Stefan H G, Fang X. Simulated climate change.effects on ice and snow covers on lakes in a temperate region [J]. Cold Regions Science and Technology,1997,25(2):137-152.
[42]Mironov D. Parameterization of lakes in numerical weather prediction. Description of a lake model [R]. Technical Report of COSMO, No.11.,2008:44pp.
[43]Mironov D, Heise E, et al. Implementation of the lake parameterisation scheme FLake into the numerical weather prediction model COSMO [J]. Boreal Environment Research, 2010,15:218-230.
[44]Kourzeneva E. External data for lake parameterization in Numerical Weather Prediction and climate modeling [J]. Boreal Environment Research,2010,15(2):165-177.
[45]Semmler T, Cheng B, Yang Y, et al. Snow and ice on Bear Lake (Alaska)-sensitivity experiments with two lake ice models [J]. Tellus A,2012,64:doi: 10.3402/tellusa. v64iO.17339.
[46]Flato G M, Brown R D. Variability and climate sensitivity of landfast Arctic sea ice [J]. Journal of Geophysical Research,1996, 101(C11):25725-25767.
[47]Duguay C R, Flato G M, Jeffries M 0, et al. Ice-cover variability on shallow lakes at high latitudes:model simulations and observations [J]. Hydrological Processes,2003, 17(17):3465-3483.
[48]Reid T, Crout N. A thermodynamic model of freshwater Antarctic lake ice [J]. Ecological Modelling,2008,210(3):231-241.
[49]Goldstine H H. A History of the Calculus of Variations from the 17th through the 19th Century [M]. Springer-Verlag,1980.
[50]Pontryagin, L S, Boltyanski V G, Gamkrelidze R V, Mischenko E E. The Mathematical Theory of Optimal Process [M]. Fizmatigiz, Interscience Publishers Ino., New York,1962.
[51]Bellman, R, Dreyfus S. Applied Dynamic Programming [M]. Izd. Inostr. Litri, Princeton University Press,1960.
[52]Kalman R E. Contributions to the theory of optimal control [J]. Bol. Soc. Mat. Mexicana, 1960,5(2):102-119.
[53]Krasovskii N N. On the theory of optimum control [J]. Journal of Applied Mathematics and Mechanics,1959,23(4):899-919.
[54]Filippov A F. On certain questions in the theory of optimal control [J]. Control Optimization,1962,1:76-84.
[55]Stoddart A W J. Existence of optimal controls [J]. Pacific Journal of Mathematics,1967, 20(1):167-177.
[56]Hartl R F, Sethi S P, Vickson R G. A survey of the maximum principles for optimal control problems with state constraints [J]. SIAM review,1995:181-218.
[57]Polak E. Optimization:algorithms and consistent approximations [M]. Springer-Verlag, 1997.
[58]Dontchev A L, Hager W W, Poore A B, et al. Optimality, stability, and convergence in nonlinear control [J]. Applied Mathematics & Optimization,1995,31(3):297-326.
[59]Schwartz A, Polak E. Consistent approximations for optimal control problems based on Runge-Kutta integration [J]. SIAM Journal on Control and Optimization,1996,34: 1235-1269.
[60]钱学森,宋健.工程控制论,下册[M].科学出版社,1981.
[61]Li W. Optimal control theory for infinite dimensional systems (Systems and control foundations and applications) [M]. Recherche,1994.
[62]高夯.半线性抛物方程支配系统的最优性条件[J].数学学报,1991,42(4)：705-714.
[63]Wang G W, Wang L. The Carlman inequality and its application to periodic optimal control governed by semilinear parabolic differential equations [J]. Journal of Optimization Theory and Applications,2003,118(2):429-461.
[64]宫召华,冯恩民.微生物批式流加发酵多阶段最优控制的最优性条件[J].运筹学学报,2009(4)：109-119.
[65]李晓红,冯恩民,修志龙.微生物连续培养非线性动力系统的性质及最优性条件[J].工程数学学报,2006,23(1)：7-12.
[66]江胜宗,韦艳华,钱伟懿,等.侧钻水平井轨道设计模型,算法及应用[J].运筹与管理,2002,11(2)：60-65.
[67]钱伟懿,江胜宗,冯恩民.三维水平井最优控制系统及应用[J].高校应用数学学报,2003,18(1)：8-14.
[68]冯恩民,赵海清,白乙拉,等.北极冰的热传递模型和数值模拟[J].工程数学学报,2007,24(3)：381-390.
[69]冯恩民,方海鹏,李志军.多区域耦合分布参数系统辨识与应用[J].应用基础与工程科学学报,2008,16(2)：153-160.
[70]张丹.三维北极海冰温度场的参数辨识与数值模拟[D].大连理工大学,2007.
[71]辛超.北极海冰温度场热力学参数辨识与数值模拟[D].大连理工大学,2007.
[72]孙丹丹.北极海冰雪的分布参数系统辨识与潮流的计算[D].大连理工大学,2007.
[73]白乙拉,李志军,冯恩民,等.静水条件下淡水冰有效导温系数的优化辨识[J].计算力学学报,2007,24(2)：187-191.
[74]Lv W, Feng E, Li Z. A coupled thermodynamic system of sea ice and its parameter identification [J]. Applied Mathematical Modelling,2008,32(7):1198-1207.
[75]徐鉴,陈予恕.时滞速度反馈对强迫自持系统动力学行为的影响[J].应用数学和力学,2004,25(5)：455-466.
[76]Xu J, Chung K W. Effects of time delayed position feedback on a van der Pol-Duffing oscillator [J]. Physica D:Nonlinear Phenomena,2003,180(1):17-39.
[77]Hong T, Hughes P C. Effect of time delay on the stability of flexible structures with rate feedback control [J]. Journal of Vibration and Control,2001,7(1):33-49.
[78]Zheng F, Frank P M. Robust control of uncertain distributed delay systems with application to the stabilization of combustion in rocket motor chambers [J]. Automatica, 2002,38(3):487-497.
[79]Qing Y W, Qi S L. Time delay-enhanced synchronization and regularization in two coupled chaotic neurons [J]. Chinese Physics Letters,2005,22(3):543-546.
[80]连涵生.带有时滞的微生物连续培养模型的分析与模拟[D].大连理工大学,2008.
[81]李晓芳,冯恩民,连涵生.连续时滞对微生物连续培养过程振荡行为影响[J].大连理工大学学报,2011,51(2)：309-312.
[82]Zimbidis A, Haberman S. The combined effect of delay and feedback on the insurance pricing process:a control theory approach [J]. Insurance:Mathematics and Economics, 2001,28(2):263-280.
[83]Grassia P S. Delay, feedback and quenching in financial markets [J]. The European Physical Journal B-Condensed Matter and Complex Systems,2000,17(2):347-362.
[84]Moore R M, Wallace D W R. A relationship between heat transfer to sea ice and temperature-salinity properties of Arctic Ocean waters [J]. Journal of Geophysical Research,1988,93(C1):565-571.
[85]Omstedt A, Svensson U. On the melt rate of drifting ice heated from below [J]. Cold Regions Science and Technology,1992,21(1):91-100.
[86]季顺迎.渤海海冰数值模拟及其工程应用[D].大连理工大学,2001.
[87]Lei R, Li Z, Cheng B, et al. Annual cycle of landfast sea ice in Prydz Bay, east Antarctica [J]. Journal of Geophysical Research,2010,115(C2):doi:10.1029/2008JC005223.
[88]Heil P, Allison I, Lytle V I. Seasonal and interannual variations of the oceanic heat flux under a landfast Antarctic sea ice cover [J]. Journal of Geophysical Research, 1996,101 (C11):25725-25741.
[89]Teo K L, Goh C J, Wong K H. A unified computational approach to optimal control problems [M]. The United States with John Wiley & Sons, Inc., New York,1990.
[90]马知恩,周义仓.常微分方程定性与稳定性方法[M].科学出版社,2001.
[91]秦元勋,刘永清,王联.带有时滞的动力系统的运动稳定性[M].科学出版社,1989.
[92]王康宁.最优控制的数学理论[M].国防工业出版社,1995.
[93]陈宝林.最优化理论与算法[M].清华大学出版社,2005.
[94]袁亚湘,孙文瑜.最优化理论与算法[M].北京：科学出版社,1997.
[95]程斌,刘春厚.海冰数值研究在海洋石油开发和工程设计中的释用[J].中国海上油气(工程),1996,8(1)：37-42.
[96]Yang Y, Lepparanta M, Cheng B, et al. Numerical modelling of snow and ice thicknesses in Lake Vanajavesi, Finland [J]. Tellus A,2012,64:doi:10.3402/tellusa. v64iO.17202.
[97]Cheng B, Vihma T, Pirazzini R, et al. Modelling of superimposed ice formation during the spring snowmelt period in the Baltic Sea [J]. Annals of Glaciology,2006,44(1): 139-146.
[98]Cheng B, Vihma T, Zhang Z, et al. Snow and sea ice thermodynamics in the Arctic:Model validation and sensitivity study against SHEBA data [J]. Chinese Journal of Polar Science,2008,19(2):108-122.
[99]Botzet M, Christensen J H, Claudius E, et al. A regional climate model of the Arctic atmosphere [J]. World Meteorological Organization-Publications-Wmo Td,1996: 169-173.
[100]Briegleb B P, Bitz C M, Hunke E C, et al. Scientific description of the sea ice component in the Community Climate System Model, version three [R]. Technical Note TN-463STR, NTIS\# PB2004-106574, NatNCAR Technical Report No. NCAR/TN-463+STR,2004:78pp.
[101]Curry J A, Schramm J L, Perovich D K, et al. Applications of SHEBA/FIRE data to evaluation of snow/ice albedo parameterizations [J]. Journal of Geophysical Research, 2001,106 (D14):15345-15355.
[102]杨清华,程斌,雷瑞波,等.北极夏季海冰反照率的观测和数值模拟试验[J].海洋学报,2011,33(2)：42-47.
[103]Pirazzini R, Vihma T, Granskog M A, et al. Surface albedo measurements over sea ice in the Baltic Sea during the spring snowmelt period [J]. Annals of Glaciology,2006, 44(1):7-14.
[104]Shine K P. Parametrization of the shortwave flux over high albedo surfaces as a function of cloud thickness and surface albedo [J]. Quarterly Journal of the Royal Meteorological Society,1984,110(465):747-764.
[105]Zillman J W. A study of some aspects of the radiation and heat budgets of the southern hemisphere oceans [M]. Australian Government Publishing Service,1972.
[106]Lumb F E. The influence of cloud on hourly amounts of total solar radiation at the sea surface [J]. Quarterly Journal of the Royal Meteorological Society,1964,90(383): 43-56.
[107]Bennett Jr T J. A coupled atmosphere-sea ice model study of the role of sea ice in climatic predictability [J]. Journal of Atmospheric Sciences,1982,39:1456-1465.
[108]Brunt D. Notes on radiation in the atmosphere. I [J]. Quarterly Journal of the Royal Meteorological Society,1932,58(247):389-420.
[109]Guest P S. Surface longwave radiation conditions in the eastern Weddell Sea during winter [J]. Journal of Geophysical Research,1998,103(C13):30761-30771.
[110]Berliand M E. Determining the net long-wave radiation of the Earth with consideration of the effect of cloudiness [J]. Izv. Akad. Nauk. SSSR Ser. Geofiz,1952,1:64-78.
[111]Efimova N A. On methods of calculating monthly values of net longwave radiation [J]. Meteorol. Gidrol,1961,10:28-33.
[112]Prata A J. A new long-wave formula for estimating downward clear-sky radiation at the surface [J]. Quarterly Journal of the Royal Meteorological Society,1996, 122(533):1127-1151.
[113]Jacobs J D. Radiation climate of Broughton Island [J]. Energy Budget Studies in Relation to Fast-Ice Breakup Processes in Davis Strait,1978,26:105-120.
[114]DrebsA, Nordlund A, KarlssonP, et al. Climatological statistics of Finland 1971-2000 [R]. Finnish Meteorological Institute:2002:99pp.
[115]Barry R G, Jacobs J D, Crane R G, et al. Energy budget studies in relation to fast-ice breakup processes in davis strait:climatological overview [M]. University of Colorado, Institute of Arctic and Alpine Research,1978.
[116]Lepparanta M, Lewis J E. Observations of ice surface temperature and thickness in the Baltic Sea [J]. International Journal of Remote Sensing,2007,28(17):3963-3977.
[117]Wang X, Key J R, Liu Y. A thermodynamic model for estimating sea and lake ice thickness with optical satellite data [J]. Journal of Geophysical Research,2010,115(12):1-14.
[118]Marko M, Hsieh C, Schalek R, et al. Focused-ion-beam thinning of frozen-hydrated biological specimens for cryo-electron microscopy [J]. Nature Methods,2007,4(3): 215-217.
[119]王索成,孟凡静,卢永超,等.红旗泡水库水质监测分析[M].黑龙江水利科技,2008,36(3)：134-134.
[120]肖建民,金龙海,谢永刚,等.寒区水库冰盖形成与消融机理分析[J].水利学报,2004,6：80-85.
[121]倪汉根,徐福生,金生.水库冰盖厚度及弧门自开启原因的研究[J].水利学报,1998(12)：1-6.
[122]Shen H T, Yapa P D. A unified degree-day method for river ice cover thickness simulation [J]. Canadian Journal of Civil Engineering,1985,12(1):54-62.
[123]Eicken H. From the microscopic, to the macroscopic, to the regional scale:growth, microstructure and properties of sea ice [J]. Sea Ice:An Introduction to its Physics, Chemistry, Biology and Geology,2003:22-81.
[124]Korhonen J. Suomen vesistojen jaaolot [M]. Suomen ymparistokeskus,2005.
[125]Lepparanta M, Uusikivi J. The annual cycle of the Lake Paajarvi ice [J]. Lammi Notes, 2002,29:4-9.
[126]Saloranta T M. Modeling the evolution of snow, snow ice and ice in the Baltic Sea [J]. Tellus A,2000,52(1):93-108.
[127]Kovacs A. Sea ice. part 1. bulk salinity versus ice floe thickness [R]. DTIC Document: 1996.
[128]Karkas E. The ice season of Lake Paajarvi in southern Finland [J]. Geophysica,2000, 36(1-2):69-84.
[129]Lei R, Lepparanta M, Erm A, et al. Field investigations of apparent optical properties of ice cover in Finnish and Estonian lakes in winter 2009 [J]. Estonian Journal of Earth Sciences,2011,60:50-64.
[130]Shirasawa K, Lepparanta M, Kawamura T, et al. Measurements and modelling of the water: ice heat flux in natural waters [C]//Proceedings of the 18th IAHR International Symposium on Ice. Sapporo, Japan:2006,1:85-91.
[131]Jakkila J, Lepparanta M, Kawamura T, et al. Radiation transfer and heat budget during the ice season in Lake Paajarvi, Finland [J]. Aquatic Ecology,2009,43(3):681-692.
[132]Burda N Y. Computer simulation for Ladoga Lake ice dynamics based on remotely sensed data [C]//IGARSS'99 Proceedings.1999,2:937-939.
[133]Wake A R R. Great lakes ice dynamics simulation [J]. Journal of Waterway, Port, Coastal and Ocean Engineering,1983,109:86-102.
[134]Lepparanta M. Modelling the formation and decay of lake ice [J]. The Impact of Climate Change on European Lakes,2010,4:63-83.
[135]Palecki M A, Barry R G. Freeze-up and break-up of lakes as an index of temperature changes during the transition seasons:A case study for Finland [J]. Journal of Applied Meteorology and Climatology Advanced Search,1986,25(7):893-902.
[136]Nonaka T, Matsunaga T, Hoyano A. Estimating ice breakup dates on Eurasian lakes using water temperature trends and threshold surface temperatures derived from MODIS data [J]. International Journal of Remote Sensing,2007,28(10):2163-2179.
[137]Anderson E A. A point energy and mass balance model of a snow cover [M]. United States. Office of Hydrology,1976.
[138]Zhou X, Li S, Stamnes K. Effects of vertical inhomogeneity on snow spectral albedo and its implication for optical remote sensing of snow [J]. Journal of Geophysical Research,2003,108(4738):doi:10.1029/2003JD003859.
[139]Sturm M, Holmgren J, Konig M, et al. The thermal conductivity of seasonal snow [J]. Journal of Glaciology,1997,43(143):26-41.
[140]Patterson J C, Hamblin P F. Thermal simulation of a lake with winter ice cover [J]. Limnology and Oceanography,1988,33(3):323-338.
[141]Brown L C, Duguay C R. The response and role of ice cover in lake-climate interactions[J]. Progress in Physical Geography,2010,34(5):671-704.
[142]Parkinson C L, Washington W M. A large-scale numerical model of sea ice [J]. Journal of Geophysical Research,1979,84(C1):311-337.
[143]Wettlaufer J S. Heat flux at the ice-ocean interface [J]. Journal of Geophysical Research,1991,96(C4):7215-7236.
[144]McPhee M G, Maykut G A, Morison J H. Dynamics and thermodynamics of the ice/upper ocean system in the marginal ice zone of the Greenland Sea [J]. Journal of Geophysical Research,1987,92(C7):7017-7031.
[145]Harvey L D D. Testing alternative parameterizations of lateral melting and upward basal heat flux in a thermodynamic sea ice model [J]. Journal of Geophysical Research,1990, 95(C5):7359-7365.
[146]Omstedt A, Wettlaufer J S. Ice growth and oceanic heat flux:Models and measurements [J]. Journal of Geophysical Research,1992,97(C6):9383-9390.
[147]Shirasawa K, Grant Ingram R, Hudier E J J. Oceanic heat fluxes under thin sea ice in Saroma-ko Lagoon, Hokkaido, Japan [J]. Journal of marine systems,1997,11(1):9-19.
[148]McPhee M G, Untersteiner N. Using sea ice to measure vertical heat flux in the ocean [J]. Journal of Geophysical Research,1982,87(C3):2071-2074.
[149]季顺迎,岳前进.工程海冰数值模型及应用[M].北京：科学出版社,2011.
[150]白乙拉.非光滑分布参数系统参数辨识及其应用[D].大连理工大学,2006.
[151]周明,孙树栋.遗传算法原理及其应用[M].北京：国防工业出版社,1999.
[152]Lei R, Li Z, Cheng Y, et al. A new apparatus for monitoring sea ice thickness based on the Magnetostrictive-Delay-Line principle [J]. Journal of Atmospheric and Oceanic Technology,2009,26(4):818-827.
[153]陆龙骅,卞林根,逯昌贵,等.近20年中国南极科学考察的气象业务进展[J].气象,2005,31(001)：3-8.
[154]Perovich D K. The optical properties of sea ice [R]. DTIC Document,1996:23pp.
[155]Yang Y, Li Z, Lepparanta M, et al. Estimation of oceanic heat flux under landfast sea ice in Prydz Bay, East Antarctica[C]//20th IAHR International Symposium on Ice,2010, No.47.
[156]Nazintsev Y L. Nekotorye dannye k raschetu teplovykh Svoistv morskogo 1'da (Some data on the calculation of thermal properties of sea ice) [J]. Tr. Arkt. Antarkt. Nauchlo Issled Inst,1964,267:31-47.
[157]Makshtas A P. Thermodynamics of sea ice [J]. Physics of Ice-covered Seas,1998,1: 289-303.