Study On The Well-posedness Of The Primitive Equations Of The Wet Atmosphere And Continuous Data Assimilatio

Numerical weather forecasting is an important weather forecasting method,which plays an extremely important role in daily weather forecasting and natural disaster prediction.In the study of numerical weather forecasting,determining the equations that describe the laws of weather change and data assimilation are two key research topics.Based on the principles of aerodynamics,thermodynamics,and cloud microphysical processes,this thesis establishes a moist atmospheric primitive equations that describes large-scale air motion,temperature changes,and water vapor conversion,and studies its well-posedness and data assimilation issues.In the first part,a mathematical model is constructed to describe air motion,temperature,and humidity changes in multi-phase saturated humid atmosphere,where the air humidity equation consists of three equations that describe the changes in cloud water,rain,and moist air.The Heaviside function is introduced into the source terms to describe the phase transition between water vapor,and it is assumed that supersaturation does not occur.Air motion and temperature changes are described by a fully viscous primitive equations.We obtain the global existence of the strong solution of the model by using the penalty function method.In physical practice,the large-scale atmosphere is subjected to strong horizontal turbulence,which makes the horizontal viscosity of air motion much greater than the vertical viscosity.Therefore,in the second part,we construct a partially viscous large-scale moist atmospheric primitive equations,in which there is only a horizontal viscosity operator and no vertical viscosity in the air motion equation.This model allows for the occurrence of supersaturation,which is more physically realistic.Using the viscosity elimination method and the z-weak solution method,the local existence of a strong solution for the system is obtained,and then improve the regularity of initial data to get the global existence of strong solutions.Using the elimination relationship between source terms,two new variables can be constructed to obtain the uniqueness of the strong solution.In the third part,based on the results of the H1regularity of the solution to the moist atmosphere primitive equations given in the previous two parts,we consider the higher regularity and the existence of attractor.We first improve the regularity of the H1strong solution in the vertical direction,and then improve the regularity of the solution in the horizontal direction,thereby obtaining the well-posedness of the H2strong solution.Furthermore,using the uniform Gronwall inequality,we obtain the uniform boundedness of the H2 strong solution,and use the compactness theorem to obtain the existence of the H2 attractor.In the fourth part,the problem of time continuous data assimilation for the primitive equations of moist atmosphere is studied.When the viscosity coefficient of the system is known,we prove that the strong solution of the assimilation system will converge to the strong solution of the reference system exponentially in the sense of L2.When the viscosity coefficient of the system is unknown,we determine the large time error between the reference solution and the assimilation solution caused by the difference between the approximate viscosity coefficient and the real viscosity coefficient.On this basis,we propose an algorithm to repair the viscosity coefficient of the assimilation system.Finally,by proving that the strong solutions of a series of difference quotient equations converge to the solutions of the sensitivity equation,we prove the sensitivity of the assimilation system respect to the viscosity coefficient.Starting from physical reality,this paper constructs mathematical models to describe the large-scale moist atmosphere,and systematically considers continuous data assimilation issues,thereby providing a theoretical basis for more accurate numerical weather forecasting.