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Balanced and Unbalanced Components of Moist Atmospheric Flows with Phase Changes

Chinese Annals of Mathematics,Series B 2019年第6期40卷 1005-1038页

作者： Alfredo N.WETZEL Leslie M.SMITH Samuel N.STECHMANN Jonathan E.MARTIN Department of Mathematics University of Wisconsin-MadisonMadisonWI 53706USA Department of Mathematics and Department of Engineering Physics University of Wisconsin-MadisonMadisonWI 53706USA Department of Mathematics and Department of Atmospheric and Oceanic Sciences University of Wisconsin-MadisonMadisonWI 53706USA Department of Atmospheric and Oceanic Sciences University of Wisconsin-MadisonMadisonWI 53706USA

Atmospheric variables(temperature, velocity, etc.) are often decomposed into balanced and unbalanced components that represent low-frequency and high-frequency waves, respectively. Such decompositions can be defined, for instance, in terms of eigenmodes of a linear operator. Traditionally these decompositions ignore phase changes of water since phase changes create a piecewise-linear operator that differs in different phases(cloudy versus non-cloudy). Here we investigate the following question: How can a balanced–unbalanced decomposition be performed in the presence of phase changes? A method is described here motivated by the case of small Froude and Rossby numbers,in which case the asymptotic limit yields precipitating quasi-geostrophic equations with phase changes. Facilitated by its zero-frequency eigenvalue, the balanced component can be found by potential vorticity(PV) inversion, by solving an elliptic partial differential equation(PDE), which includes Heaviside discontinuities due to phase changes. The method is also compared with two simpler methods: one which neglects phase changes, and one which simply treats the raw pressure data as a streamfunction. Tests are shown for both synthetic, idealized data and data from Weather Research and Forecasting(WRF) model simulations. In comparisons, the phase-change method and no-phase-change method produce substantial differences within cloudy regions, of approximately 5K in potential temperature, due to the presence of clouds and phase changes in the data. A theoretical justification is also derived in the form of a elliptic PDE for the differences in the two streamfunctions.

关键词： Potential vorticity inversion Moist atmospheric dynamics Slow-fast systems Balanced-unbalanced decomposition elliptic partial differential equations

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An exponential expanding meshes sequence and finite difference method adopted for two-dimensional elliptic equations

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International Journal of Modeling, Simulation, and Scientific Computing 2016年第2期7卷 109-125页

作者： Navnit Jha Neelesh Kumar Faculty of Mathematics and Computer Science Department of Mathematics South Asian University ChanakyapuriNew Delhi 110021India

We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative *** method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented,thus arriving at a compact *** general,a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular *** efficiency of using geometric mesh ratio parameter has been shown with the help of *** convergence of the scheme has been established using the matrix analysis,and irreducibility is proved with the help of strongly connected characteristics of the iteration *** difference scheme has been applied to test convection diffusion equation,steady state Burger’s equation,ocean model and a semi-linear elliptic *** computational results confirm the theoretical order and accuracy of the method.

关键词： Geometric mesh finite difference compact method elliptic partial differential equations convection diffusion equation Stommel ocean model

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APPROXIMATION　OF　GENERALIZED　BI－AXIALLY　SYMMETRIC　POTENTIALS　WITH　FAST　RATES　OF　GROWTH

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Acta Mathematica Scientia 1995年第4期15卷 458-467页

作者： H.S.Kasana(Dept.of Math., Birla Iust.of Tech. & Sci.,Pilani-333 031(Rajasthan), India.D.Kumar)(Dept.of Math., D. S. M.Degree College, Kanth-244 501 (Moradabad),U.P.,India.) Dept. of Math. Birla Inst. of Tech. & Sci. Pilani–333 031 Rajasthan India Dept. of Math. D. S. M. Degree College Kanth–244 501 (Moradabad) U. P. India

The paper deals with growth and approximation of solutions (not necessarily entire) of certain elliptic partial differential equations. These solutions are called Generalized Bi-axially Symmetric Potentials (GBSP's). The GBSP's are taken to be regular in a finite hyperball and influence of the growth of their maximum moduli on the rate of decay of their approximation errors in sup norm is studied. The authors obtain the characterizations of the q-type and lower q-type of a GBSP H ∈ HP,0 < R < ∞, in terms of rate of decay of approximation error E.(H,R0), 0 < R0

关键词： Generalized bi-axially symmetric potentials elliptic partial differential equations Index q Entire GBSP polynomials,Sup norm.

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GRID GENERATION OF COMPLEX PRACTICAL AIRCRAFT & ZONAL SOLUTION OF EULER equations FOR HIGH-INCIDENCE VORTICAL FLOWS(HIVF)

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Chinese Journal of Aeronautics 1997年第3期10卷 2-8页

作者： Zhang Zhengke, Zhu Zhiqiang, Zhuang Fenggan (Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China)

A complete process of grid generation for complex practical aircraft is described with a twin tail fighter as an example. Euler equations are discretized in the generated multiblock grid by a finite volume method and solved by a three stage explicit time stepping scheme in each block with some extra treatments of interface at each step. The predicted aerodynamic coefficients and vortical flow field are reasonable.

关键词： fighter aircraft grid generation (mathematics) elliptic partial differential equations Euler equations zonal method

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