Well-Balanced Central-Upwind Schemes For Thermal Rotating Shallow Water Equations And Coupled Moist-Convective Model

Thermal rotating shallow water equations are one of the most important hyperbolic partial differential equations,which are widely used in the theoretical analysis and numerical study of atmospheric motions.The main work of this thesis is to construct a well-balanced numerical method(exactly preserve the balanced states at the discrete level)for these equations and the established coupled moist-convective model.We expect that the proposed numerical methods will provide an efficient tool for atmospheric science study.The main difficulty of numerical simulation of atmospheric motion lies in the fact that the numerical method should be designed in such a way that it is consistent with the continuous atmosphere as far as possible in describing important physical laws and overall properties(for example,the(thermal)geostrophic equilibrium between the horizontal pressure gradient force and Coriolis parameters in the mid-latitude large-scale motion),besides accuracy and stability.Based on the Riemannian-problem-solver-free central-upwind numerical scheme,applying the flux globalization method and improving the computation of global variables,combining several important techniques,such as well-balanced reconstruction,well-balanced numerical flux correction,and draining time step setting,we develop a well-balanced and positive-preserving central-upwind numerical scheme.Furthermore,when designing the two dimensional scheme,we propose a numerical dissipation switch to improve the classical two dimensional central-upwind scheme.This thesis consists of the following three parts.In the first part,the theoretical analysis of the one-dimensional thermal rotating shallow water equations is given and a well-balanced central-upwind numerical scheme is developed.We use the Lagrangian variables to rewrite the studied equations and apply the linearization methods and Fourier transform to theoretical analysis.On the 1)-plane,we illustrate the existence and uniqueness of the decay solution at infinity(corresponding to the existence and uniqueness of the adjustment process),give the judgment conditions for the formation of breaking-waves,and prove that the frequency is super-inertial,that is,there is no trapped wave.On the equatorial -plane,we use the linearization method to explain the trapped waves will appear in the adjustment process and give the conditions for the occurrence of symmetric inertial instability.On the other hand,the equilibrium state of thermal rotating shallow water equations is very complicated(it is expressed as the equation generated by the interaction of the pressure gradient force,Coriolis force and the bottom function),the explicit solutions of each variable can not be obtained from this equation,which brings a lot of difficulties to the design well-balanced schemes.At the same time,the solution of the equation will form a shock wave in a finite time when it is out of equilibrium state,which requires the designed scheme to have the ability to capture the shock wave.In order to overcome these difficulties,a well-balanced centralupwind scheme was proposed and constructed based on the flux globalization method with a new global variable computation method.The source term is incorporated into the flux function and then one can obtain an equivalent hyperbolic system of conservation law under the equilibrium state and then combine the well-balanced reconstruction,wellbalanced numerical flux correction and draining time step setting several key technologies.We develop a well-balanced central-upwind scheme.In addition,two theorems are given to prove the well-balanced properties of the numerical scheme.Two numerical examples with non-flat bottom functions are given to verify the properties of positivity-preserving and well-balanced of the proposed scheme.Finally,we investigate the Rossby adjustment process,the wave-breaking and shock formation on the 1)-plane and the instability problem on the equatorial -plane to further verify the validity of the proposed scheme.In the second part,the classical two dimensional central-upwind scheme is improved and a well-balanced central-upwind scheme with a numerical dissipation switch is designed for the two dimensional thermal rotating shallow water equations.The centralupwind scheme is based on the average on the Riemann fans and takes into account the upwind information at the cell interface,which reduces the numerical dissipation of the staggered central scheme.However,the estimation of the one-sided local propagation speed at the cell interface is not the most accurate.Excessive absolute value estimates introduce additional numerical dissipation,especially for shear flow and contact discontinuity problems.In the numerical simulation of atmosphere,it is necessary to capture the information of various fronts,describe the complex solution structure and describe the formation and evolution of various instability,which requires a numerical scheme to have high resolution and low numerical dissipation.The estimation of local propagation speeds at the cell interface can be optimized,and the numerical dissipation of the classical central-upwind scheme is greatly reduced.The formation and development of convective instability of pure thermal vortices on the 1)-plane are numerically simulated,which is a good test of the effectiveness of the proposed optimized estimation and the improved central-upwind scheme.The numerical simulation of the propagation and evolution of an isolated vortex on the mid-latitude -plane and the fluctuation caused by the local pressure and temperature anomalies on the equatorial -plane shows that the proposed scheme can simulate the Rossby wave propagation and evolution process,is also able to accurately capture the characteristics of equatorial Kelvin waves propagation.In the third part,a moist-convective thermal rotating shallow water model is established.In the existing moist-convective rotating shallow water model,the latent heat generated by water phase transition is only related to the convection flux which causes the thickness change,but this is not consistent with the actual situation that latent heat will also cause the potential temperature increase in the atmospheric motion.Therefore,in this part,the coupled moist-convective model is established.The latent heat release caused by condensation was decomposed into two parts: the source term causing potential temperature variation and the sink term causing fluid thickness variation.A coupled moist convection model satisfying physical consistency was derived by parameterization method.Then,the evolution equation of humidity and precipitated water and the influence of surface evaporation,sea surface temperature and precipitation on fluid motion are further studied,and the established model is improved so that it can be better applied to real complex atmospheric motion.The high-resolution well-balanced center-upwind scheme with numerical dissipation switch developed in the second part is applied to the coupled moist-convective model,and the validity of the model is verified by numerical examples.