Research On Several Mathematical Problems In Atmospheric Sciences

Three mathematical physics problems in atmospheric sciences are studied in this paper,mainly including the stability of the multilayer quasi-geostrophic flow,the application of Bayesian algorithm in Abel integral equation and the influence of topography on the airflow movement,the main results are as follows:1.The linear stability problem of multilayer quasi-geostrophic flow is studied by constructing infinite dimensional Hamiltonian system.The difficulty is how to transform the nonlinear partial differential equations into an infinite dimensional Hamiltonian system.In chapter two,by finding conserved quantity H(that is,the Hamilton function),establishing differential operator J and Poisson bracket,the multilayer quasi-geostrophic flow vorticity equations is transformed into infinite dimensional Hamiltonian system,and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms,physical meaning is clear,the form is concise and unified.2.Bayesian algorithm is used to solve Abel integral equation and compared with the inversion results of Tikhonov regularization in chapter three.According to the bending angle data that EGOPS(End-to-End Generic Occultation Performance Simulation and Processing System)simulated to inverse the refractive index.When bending angle data do not contain errors,Bayesian algorithm and Tikhonov regularization have good consistency,the root mean square error are both 2.5470e-08;however,in the process of actual observation,because of the influence of the ionosphere and the water vapor and so on,the bending angle data will inevitably contain errors,it may produce high frequency components,even exists discontinuous points,so the Gaussian errors are added to the bending angle.The results show that compared with Tikhonov regularization technology,Bayesian algorithm has higher inversion precision.3.By using the conformal mapping and complex potential,the airflow motion in two cases is discussed under the premise that the atmosphere is an ideal fluid.Firstly,the parallel flow movement through an arc-shaped mountain is studied,and the influence of horizontal velocity and of the angle between the mountain and the negative direction of horizontal ground on the airflow movement is analyzed,the results indicate that the fluctuation is more volatile with the increase of flow velocity and the decrease of the angle.Secondly,the motion of a parallel flow superimposed onto a point vortex is discussed when passing a linear boundary.Since the vortex core is an extreme point,its location cannot be changed no matter how fast the flow velocity is.That is,though the motion of the surrounding particles is changed,the vortex core remains motionless.4.On the assumption that the stream function is continuous in the vertical direction,the constant analytical solutions of the mesoscale motion under the influence of some typical terrains are obtained by using an analytical method and the movement of airflow is discussed correspondingly in this paper.The results show that the airflow does not present any fluctuation on the windward side of the terrain,and jet streams and rotors cannot form in the lower terrain.For the terrains with a certain height(depth),when the Froude number_rF(29)1/?,the atmospheric fluctuation only occurs near the terrain,the low-level streamlines are close to the terrain,and the strength of fluctuation decreases with the increase of the height;when 1/?(29)F_r(29)1/(7)2?(8),the fluctuation occurs not only near the terrain,but also on its leeward side.The wave length depends only on the background field,rather than the horizontal scale of the terrain,and the fluctuation decreases gradually with the increase of the height;when _rF<1/(7)2?(8),a jet stream forms in the middle atmosphere of the leeward side of terrain,then it divides into two branches,one locates in the upper atmosphere and the other locates in the lower atmosphere,and they finally converge in the middle atmosphere;rotors appear on one or both sides of the jet stream.