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Mathematical models
| Models system for the description of processes existent on a watershed
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Purpose of model : Calculation of runoff and sediment load characteristics, and estimation of chemical substances transport from watersheds.
Mathematical structure :
Set of models of a different complexity and detailed elaboration (hydrodynamical, conceptual, empirical) for use on objects of a different complexity and with different level of investigation, and located in different geographical conditions.
Field of application :
Experimental watersheds of the Valdai branch of the State Hydrological Institute, Turkmen and Moldavian water-balance stations, Korzinskiy Meliorative Permanent Establishment (Karelia), Institute of a Hydrology Slovak АS (East Slovakia), irrigated fields of a cotton in Kazakhstan, objects of experimental investigations of Institute of Limnology RAS (experimental watersheds in Krasnodar territory, experimental watersheds in the Tver region, Lake Krasnoe and River Strannitsa watershed, inflows to Lake Bolshoe Rakovoe), watersheds of 20 small lakes located in Karelia, Latgallii (Latvia) and on Karelian Isthmus of Leningrad region, a watershed of St.- Petersburg Northern Station of Sewage Aeration.
Publications :
Кондратьев С.А. Математическое моделирование стока и выноса вещества с водосбора - Диссертация на соискание ученой степени доктора физико-математических наук, Спб, 1992. (in Russian)
Kondratyev S. Mathematical Model of Runoff and Pollution
Wash off from the Watershed of Northern Regions of St. Petersburg.-
Proc. of Int. Symp. "HYDROCOMP '92", Budapest, Hungary,
1992, pp. 349-355.
Rumyantsev V., Mendel O., Kondratyev S. Mathematical Modelling of Stream Water Quantity and Quality. - Proc. of Int. Symp."Man's Influence on Freshwater Ecosystems and Water Use", Boulder, USA, IAHS Publ. N 230, 1995.
Kondratyev S., Mendel O. Mathematical Modelling of Runoff and Material Transport from Drainage Areas into Recipient Water Bodies. - Hydrobiologia, 1996, v. 322, pp.237-240.
Purpose of model : One-dimensional non-stationary model TEMIX is intended for calculation of an annual thermal regime and the main characteristics of turbulent mixing in natural lakes and artificial reservoirs in dependent of meteorological conditions.
Mathematical structure :
The model is based on dimensionless phenomenological parameterizations of a vertical profile of water mass and bottom sediments temperatures. It allows avoiding the aprioristic assignment of coefficients of turbulent exchange by heat, mass and linear momentum. Parametrization, used in model, allows describing a thermal regime of a researched water body by means of a system of the ordinary differential equation having simple decisions.
The basic hydrothermodynamic parameters calculated in model:
- Streams of heat, mass and linear momentum on a border "water - atmosphere";
- Height of top quasi-homogeneous layer;
- A vertical structure of temperature and heat exchange in system "water mass – bottom sediments";
- Formation and destruction of an ice cover during winter-spring period.
The source information for model calculations :
Calculation of a hydrothermodynamic regime in model is made depending on meteorological conditions above a water body, described following parameters: windspeed, air temperature, nebulosity, humidity, total solar radiation, atmospheric pressure.
An initial vertical distribution of temperature in system "water weight – bottom sediments" is set as entry conditions for calculations.
Field of application :
The model was successfully applied for calculation of a hydrothermodynamic regime of polytypic lakes of various climatic zones. Among them there are shallow lakes of the Great Britain and Northwest of Russia, the central part of Lake Ladoga, Lake Sevan (Armenia), etc.
Purpose of model : One-dimensional non-stationary model REDLEX is intended for calculation of seasonal variability of the dissolved inorganic phosphorus contents taking into account its receipt from a watershed and mass transfer through the border "water – bottom sediments".
Mathematical structure :
The model will consist of several blocks, basic of which are:
- Hydrothermodynamic is intended for calculation of an annual thermal regime and the main
characteristics of turbulent exchange as in natural, as in artificial reservoirs in depending of meteorological conditions;
- Chemical and biological - describes the basic processes of carry and transformation of substance in a reservoir;
- The block of processes on a watershed, describing receipt of the inorganic dissolved phosphorus
in a reservoir taking into account an intensity of anthropogenous activity ont the watershed.
Main parameters considered in model:
- Streams of heat, mass and linear momentum on a border "water - atmosphere";
- Height of top quasi-homogeneous layer;
- A vertical structure of temperature and heat exchange in system "water mass – bottom sediments";
- Formation and destruction of an ice cover during winter-spring period;
- Mass transfer through the border "water – bottom sediments";
- Vertical distribution of the dissolved inorganic phosphorus concentration by water
weight and in pore solution of bottom sediments;
- Phosphorus intake from a watershed;
- A biomass of aggregated phytoplankton.
The source information for model calculations:
Calculations in model are made with use of the meteorological data and the data on receipt of biogenic elements from a watershed.
depending on meteorological conditions above a water body, described following
The main meteorologuical parameters are windspeed, air temperature, nebulosity (total), total solar radiation, atmospheric pressure.
Field of application :
The model was successfully applied for calculation of seasonal variability of inorganic dissolved phosphorus concentration and a phytoplankton biomass in shallow lakes of Karelian Isthmus (Northwest Russia) and in Lake Ladoga.
Purpose of model : Three-dimensional mathematical model for calculations of hydrodynamical processes in coastal zones of oceans and the seas.
Mathematical structure :
The system of hydrodynamic equations with Boissinesq approximation and hydrostatic equations with transformation of vertical coordinate. Energy flows and heat flows are set as a function of time.
Field of application :
Coastal zones of oceans and the seas; the big deep lakes [the equation for calculation of fresh water density (Chen and Millero, 1986) implemented by АЮТ].
Publications :
Blumberg, A. F. and G. L. Mellor, 1987, A description of a three-dimensional
coastal ocean circulation model, Three-Dimensional Coastal ocean
Models, edited by N. Heaps, 208 pp., American Geophysical Union.
Mellor, G. L. and A. F. Blumberg, 1985, Modeling
vertical and horizontal diffusivities with the sigma coordinate
system, Mon. Wea. Rev., 113, 1380-1383.
Mellor, G. L., 1992, User's guide for a three-dimensional,
primitive equation, numerical ocean model, 35 pp., Prog. in Atmos.
and Ocean. Sci, Princeton University.
O'Connor, W.P., and Schwab, D.J., 1994, Sensitivity
of Great Lakes Forecasting System Nowcasts to Meteorological Fields
and Model Parameters, in: M.L. Spaulding, K. Bedford, A. Blumberg,
R. Cheng, and C. Swanson (eds.), Estaurine and Coastal Modeling
III, Proceedings of the 3rd International Conference, ASCE, Sep.
8-10, 1993, Oak Brook, IL, 149-157.
Purpose of model : Stochastic model of a synthesis of meteorological parameters necessary for calculation of a total heat stream on a water surface (temperature and humidity of air, nebulosity, windspeed).
Mathematical structure :
The principle of a work is based on use of monthly average values of parameters from accounts of their variability (long-term average, strong - weak, warm - cold, etc.) and generation of random numbers as multipliers for reception of daily average values of parameters.
Field of application :
Reservoirs or their parts, where a distribution of meteorological parameters can be considered as a quasi-homogeneous.
Publications :
Тержевик А.Ю., 1992, Простейшая имитационная модель погодных условий
над Ладожским озером. В кн. Ладожское озеро - критерии состояния
экосистемы. СПб., 41-47 (in Russian).
Тержевик А.Ю.1992, Моделирование различных сезонов.
В кн. Ладожское озеро - критерии состояния экосистемы. СПб., 48-53 (in Russian).
Purpose of model : Modelling of the natural processes having property of periodic nonstationarity (annual rhythmics, a daily course).
Mathematical structure :
Periodically non-stationary process Xt of discrete argument t
is represented as m- measured vector of a set of the stationary and permanently connected casual processes:
{Xi,j}, i=1,...,m, j=0,1,...,
Vector process {Xi,j}
is described by multivariate process of autoregression. Identification of model is based on the decision of the multivariate equations of Jula-Walker.
Field of application :
Imitating reproduction of the natural processes having annual (or daily) rhythmics.
Publications :
Рожков В.А., Трапезников Ю.А. Вероятностные модели океанологических
процессов. - Л.: Гидрометеоиздат, 1990, 270 с. (in Russian)
Purpose of model : Modelling of a set of casual processes having property of periodic nonstationarity (annual rhythmics, a daily course).
Mathematical structure :
Set of n periodically correlated casual processes Xk,t , k=1,...,n
is represented as nm -measured vector of stationary and permanently connected casual processes Xt .
Stages of model construction:
- Transformation of casual vector Xt with marginal functions of distribution P in Gaussian vector
Zt which components are distributed under the standard normal law;
- Approximation of vector process Zt by multivariate model of autoregress with identification of parameters of model on the basis of the decision of the multivariate equation of Jula-Walker;
- Transformation a component of vector Zt in process with functions of distribution P .
Field of application :
Imitating reproduction of set of the natural processes having annual (or daily) rhythmics.
Publications :
Григорьев А.С. Модель годового хода и межгодовой изменчивости совокупности
гидрометеорологических процессов. - International Scientific Conference
"STOHASTIC MODELS of HYDROLOGICAL PROCESSES and THEIR APPLICATIONS to
PROBLEMS of ENVIRONMENTAL PRESERVATION" -23-27 November, 1998.
Purpose of model : The model is intended for imitation of casual processes with infinity divisibility distribution law and arbitrary spectrum.
Mathematical structure :
The model generalizes linear Gaussian processes АРСС to a class of linear processes with infinity divisibility distribution law.
Field of application :
Imitating reproduction of the natural processes having arbitrary spectrum and infinity divisibility distribution law.
Publications :
Рожков В.А., Трапезников Ю.А. Вероятностные модели океанологических
процессов. - Л.: Гидрометеоиздат, 1990, 270 с. (in Russian)
Purpose of model : Imitating modelling of casual processes having arbitrary spectrum and infinity divisibility distribution law.
Mathematical structure :
The model generalizes linear processes АРСС to a class of linear processes with infinity divisibility distribution law.
Stages of model construction:
- Transformation of initial process
Xt with the purpose of reception of intermediate process yt
with normal or infinitely decomposable distribution law;
- Approximation of intermediate process
yt by linear (or known nonlinear) process АРСС;
- Return transformation, converted process
yt in process with given distribution law.
Field of application :
Imitating reproduction of the natural processes having arbitrary spectrum and arbitrary distribution function.
Publications :
Рожков В.А., Трапезников Ю.А. Вероятностные модели океанологических
процессов. - Л.: Гидрометеоиздат, 1990, 270 с. (in Russian)
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The institute participated in the organization of conference «the Ecological condition of continental reservoirs of the Arctic zone in connection with industrial development of northern territories »,which took place in Arkhangelsk in the period 20-25 June 2005.
