scholarly journals Modelling the Evolution of the East Antarctic Ice Sheet: Problems and Perspectives (Abstract)

1988 ◽  
Vol 11 ◽  
pp. 217
Author(s):  
S. S. Grigoryan ◽  
I. Yu. Ignat’yeva ◽  
M. S. Krass ◽  
P. A. Shumskiy
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Surface Temperature ◽  
Ice Sheet ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Antarctic Ice Sheet ◽  
The Antarctic ◽  
Geothermal Flux ◽  
East Antarctic Ice Sheet

Modelling the thermodynamics of a large ice sheet is in essence a problem of the deformation of a non-isothermal and relatively thin viscous layer. Reasons for forecasting the behaviour of a large ice sheet are: (a) to specify the relationship between the existence and development of the ice sheet and regional climatic and tectonic conditions; (b) to study the influence of variations in surface temperature, mass balance, and geothermal flux (as functions of coordinates and time) on the evolution of the thickness of the ice sheet. Hence the principal requirements for modelling are: 1. The model must be evolutionary; in particular, various stationary stages should be derived from it (rather than postulated). 2. The model must be non-isothermal because temperature variations amount to some tens of degrees. 3. Two-dimensionality of the model is essential if we are to use it for making realistic forecasts. 4. Conditions at the boundary between land ice and the sea, which are the basic factors controlling the location of the margin of the ice sheet, should be taken into account. 5. Finally, the model must be acceptable for digital analysis by computer. Contemporary mathematical models of the dynamics of the Antarctic and Greenland ice sheets are too simplified and do not fulfil these requirements. Our mathematical model of the evolution of the Antarctic ice sheet is based on temperature parameterization. The parameterization correctly describes the distribution of temperature and its dependence on surface temperature, advection of ice, geothermal flux, and heat losses. The problem reduces to Cauchy’s problem for a non-linear parabolic partial differential equation with three-step dependence of ice temperature on depth. A linear viscous flow law is assumed. Tests treat the whole problem of the thermo-hydromechanics of the ice sheet. The ice sheet is assumed to be bounded by a grounding line or ice wall where the ice is in hydrostatic equilibrium with the sea. In this case, the problem reduces to a non-linear parabolic partial differential equation with a complicated boundary condition at the moving edge. A digital representation of the model is presented first. Tests demonstrated the possibility of using an explicit three-layered scheme. Calculations for the East Antarctic ice sheet indicated that the ice margin should retreat slightly under contemporary climatic conditions (which corresponds to the field evidence). In future the boundary should tend towards a stationary state. A forecast of the behaviour of the ice sheet is derived. The East Antarctic ice sheet is found to be stable during various changes in climatic factors. The calculated hydromechanical characteristics of the ice sheet agree with the observed field data.

scholarly journals Modelling the Evolution of the East Antarctic Ice Sheet: Problems and Perspectives (Abstract)

1988 ◽  
Vol 11 ◽  
pp. 217-217
Author(s):  
S. S. Grigoryan ◽  
I. Yu. Ignat’yeva ◽  
M. S. Krass ◽  
P. A. Shumskiy
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Surface Temperature ◽  
Ice Sheet ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Antarctic Ice Sheet ◽  
The Antarctic ◽  
Geothermal Flux ◽  
East Antarctic Ice Sheet

Modelling the thermodynamics of a large ice sheet is in essence a problem of the deformation of a non-isothermal and relatively thin viscous layer. Reasons for forecasting the behaviour of a large ice sheet are: (a) to specify the relationship between the existence and development of the ice sheet and regional climatic and tectonic conditions; (b) to study the influence of variations in surface temperature, mass balance, and geothermal flux (as functions of coordinates and time) on the evolution of the thickness of the ice sheet. Hence the principal requirements for modelling are: 1.The model must be evolutionary; in particular, various stationary stages should be derived from it (rather than postulated).2.The model must be non-isothermal because temperature variations amount to some tens of degrees.3.Two-dimensionality of the model is essential if we are to use it for making realistic forecasts.4.Conditions at the boundary between land ice and the sea, which are the basic factors controlling the location of the margin of the ice sheet, should be taken into account.5.Finally, the model must be acceptable for digital analysis by computer.Contemporary mathematical models of the dynamics of the Antarctic and Greenland ice sheets are too simplified and do not fulfil these requirements. Our mathematical model of the evolution of the Antarctic ice sheet is based on temperature parameterization. The parameterization correctly describes the distribution of temperature and its dependence on surface temperature, advection of ice, geothermal flux, and heat losses. The problem reduces to Cauchy’s problem for a non-linear parabolic partial differential equation with three-step dependence of ice temperature on depth. A linear viscous flow law is assumed. Tests treat the whole problem of the thermo-hydromechanics of the ice sheet. The ice sheet is assumed to be bounded by a grounding line or ice wall where the ice is in hydrostatic equilibrium with the sea. In this case, the problem reduces to a non-linear parabolic partial differential equation with a complicated boundary condition at the moving edge. A digital representation of the model is presented first. Tests demonstrated the possibility of using an explicit three-layered scheme. Calculations for the East Antarctic ice sheet indicated that the ice margin should retreat slightly under contemporary climatic conditions (which corresponds to the field evidence). In future the boundary should tend towards a stationary state. A forecast of the behaviour of the ice sheet is derived. The East Antarctic ice sheet is found to be stable during various changes in climatic factors. The calculated hydromechanical characteristics of the ice sheet agree with the observed field data.

scholarly journals Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Guanglu Zhou ◽  
Boying Wu ◽  
Wen Ji ◽  
Seungmin Rho
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Sensor Array ◽  
Variational Iteration Method ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Variational Iteration

This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems.

scholarly journals USING GENERALIZED STOCHASTIC METHOD TO EVALUATE PROBABILITY OF CONFLICT IN CONTROLLED AIR TRAFFIC

Aviation ◽  
2007 ◽  
Vol 11 (2) ◽  
pp. 31-36 ◽  
Author(s):  
Vitaly Babak ◽  
Volodymyr Kharchenko ◽  
Volodymyr Vasylyev
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Traffic Safety ◽  
Evaluation Method ◽  
Air Traffic ◽  
Collision Probability ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Potential Conflict ◽  
Infinitesimal Operator

The introduction of the new concepts of air traffic management (ATM) and transition from centralized to decentralized air traffic control (ATC) with the change of traditional ATM to Cooperative ATM sets new tasks and opens new capabilities for air traffic safety systems. This paper is devoted to the problem of evaluating the probability of aircraft collision under the condition of Cooperative ATM, when the necessary information is available to the subjects involved in the decision‐making process. The generalized stochastic conflict probability evaluation method is developed. This method is based on the generalized conflict probability equation for evaluation of potential conflict probability and aircraft collision probability that is derived by taking into account stochastic nature and time correlation of deviation from planned flight trajectory in controlled air traffic. This equation is described as a multi‐dimensional parabolic partial differential equation using a differential (infinitesimal) operator of the multi‐dimensional stochastic process of relative aircraft movement. The common procedure for the prediction of conflict probability is given, and the practical application of the generalized method presented is shown. All equational coefficients of a differential operator for a practical solution of a parabolic partial differential equation are derived. For some conditions, the numerical solution of the conflict probability equation is obtained and illustrated graphically.

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