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Global existence and large-time behavior for primitive equations with the free boundary

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Science China Mathematics 2024年第10期67卷 2303-2330页

作者： Hai-Liang Li Chuangchuang Liang School of Mathematical Sciences Capital Normal UniversityBeijing 100048China Academy for Multidisciplinary Studies Capital Normal UniversityBeijing 100048China College of Mathematics and Statistics Chongqing UniversityChongqing 401331China

In the present paper, the primitive equations, which can be used to simulate the large-scale motion of the ocean and atmosphere, are considered in the three-dimensional domain bounded below by a fixed solid boundary and above by a free-moving boundary. The global existence and uniqueness of strong solutions are established, and the long-time convergence to the equilibrium state is shown to be either at an exponential rate for the horizontal periodic domain or at an algebraic rate for the horizontal whole space.

关键词： primitive equations the free boundary value problem global well-posedness large-time behavior

ON THE 3D VISCOUS primitive equations OF THE LARGE-SCALE ATMOSPHERE

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Acta Mathematica Scientia 2009年第4期29卷 846-866页

作者：郭柏灵黄代文 Institute of Applied Physics and Computational Mathematics Department of Mathematical Sciences Tsinghua University

This paper is devoted to considering the three-dimensional viscous primitive equations of the large-scale atmosphere. First, we prove the global well-posedness for the primitive equations with weaker initial data than that in [11]. Second, we obtain the existence of smooth solutions to the equations. Moreover, we obtain the compact global attractor in V for the dynamical system generated by the primitive equations of large-scale atmosphere, which improves the result of [11].

关键词： primitive equations Navier-Stokes equations global well-posedness smooth solutions

ON THE RIGOROUS MATHEMATICAL DERIVATION FOR THE VISCOUS primitive equations WITH DENSITY STRATIFICATION

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Acta Mathematica Scientia 2023年第3期43卷 1081-1104页

作者：蒲学科周文利 School of Mathematics and Information Science Guangzhou UniversityGuangzhou510006China

In this paper, we rigorously derive the governing equations describing the motion of a stable stratified fluid, from the mathematical point of view. In particular, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ratio goes to zero, and the rate of convergence is of the same order as the aspect ratio. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.

关键词： Boussinesq equations primitive equations density stratification hydrostatic approximation strong convergence

Quasi-hydrostatic primitive equations for Ocean Global Circulation Models

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Chinese Annals of Mathematics,Series B 2010年第6期31卷 939-952页

作者： Carine LUCAS Madalina PETCU Antoine ROUSSEAU Laboratoire MAPMO (UMR 6628) Fédération Denis Poisson (FDP-FR2964) Université d'OrléansBtiment de Mathématiques-Route de Chartres B. P. 6759 45067 Orléans Cedex 2 France Laboratoire de Mathématiques et Applications Téléport 2-B. P. 30179 Boulevard Marie et Pierre Curie 86962 Futuroscope Chasseneuil Cedex France INRIA Laboratoire Jean Kuntzmann 51 rue des Mathématiques B. P. 53 38041 Grenoble Cedex 9France

Global existence of weak and strong solutions to the quasi-hydrostatic primitive equations is studied in this paper. This model, that derives from the full non-hydrostatic model for geophysical fluid dynamics in the zero-limit of the aspect ratio, is more realistic than the classical hydrostatic model, since the traditional approximation that consists in neglecting a part of the Coriolis force is relaxed. After justifying the derivation of the model, the authors provide a rigorous proof of global existence of weak solutions, and well-posedness for strong solutions in dimension three.

关键词： Hydrostatic approximation Coriolis force Ocean global circulation models primitive equations Traditional approximation

A Second-Order Implicit-Explicit Scheme for the Baroclinic-Barotropic Split System of primitive equations

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Communications in Computational Physics 2023年第10期34卷 1306-1331页

作者： Rihui Lan Lili Ju Zhu Wang Max Gunzburger School of Mathematical Sciences Ocean University of ChinaQingdaoShandong 266100China Department of Mathematics University of South CarolinaColumbiaSC 29208USA Oden Institute for Computational Engineering and Sciences University of Texas at AustinAustinTexas 78712USA

The baroclinic-barotropic mode splitting technique is commonly employed in numerical solutions of the primitive equations for ocean modeling to deal with the multiple time scales of ocean *** this paper,a second-order implicit-explicit(IMEX)scheme is proposed to advance the baroclinic-barotropic split ***,the baroclinic mode and the layer thickness of fluid are evolved explicitly via the second-order strong stability preserving Runge-Kutta scheme,while the barotropic mode is advanced implicitly using the linearized Crank-Nicolson *** each time step,the baroclinic velocity is first computed using an intermediate barotropic *** barotropic velocity is then corrected by re-advancing the barotropic mode with an improved barotropic ***,the layer thickness is updated by coupling the baroclinic and barotropic velocities *** addition,numerical inconsistencies on the discretized sea surface height caused by the mode splitting are alleviated via a reconciliation process with carefully calculated flux *** truncation error is also analyzed to validate the second-order accuracy of the ***,two benchmark tests from the MPAS-Ocean platform are conducted to numerically demonstrate the performance of the proposed IMEX scheme.

关键词： primitive equations baroclinic-barotropic splitting implicit-explicit strong stability preserving RK SSH reconciliation

Algorithm for simulating ocean circulation on a quantum computer

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Science China Earth Sciences 2023年第10期66卷 2254-2264页

作者： Ruimin SHANG Zhimin WANG Shangshang SHI Jiaxin LI Yanan LI Yongjian GU College of Physics and Optoelectronic Engineering Ocean University of ChinaQingdao266100China

The accurate and efficient simulation of ocean circulation is a fundamental topic in marine science;however,it is also a well-known and dauntingly difficult problem that requires solving nonlinear partial differential equations with multiple *** this paper,we present for the first time an algorithm for simulating ocean circulation on a quantum computer to achieve a computational *** approach begins with using primitive equations describing the ocean dynamics and then discretizing these equations in time and *** results in several linear system of equations(LSE)with sparse coefficient *** solve these sparse LSE using the variational quantum linear solver that enables the present algorithm to run easily on near-term quantum ***,we develop a scheme for manipulating the data flow in the algorithm based on the quantum random access memory and l∞norm tomography *** efficiency of our algorithm is verified using multiple platforms,including MATLAB,a quantum virtual simulator,and a real quantum *** impact of the number of shots and the noise of quantum gates on the solution accuracy is also *** findings demonstrate that error mitigation techniques can efficiently improve the solution *** the rapid advancements in quantum computing,this work represents an important first step toward solving the challenging problem of simulating ocean circulation using quantum computers.

关键词： Ocean circulation primitive equations Linear system of equations Variational quantum linear solver Error mitigation technique

Partial and Spectral-Viscosity Models for Geophysical Flows

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Chinese Annals of Mathematics,Series B 2010年第5期31卷 579-606页

作者： Qingshan CHEN Max GUNZBURGER Xiaoming WANG Department of Scientific Computing Florida State University Tallahassee FL 32306 USA. Department of Mathematics. FloridaState University Tallahassee FL 32306 USA.

Two models based on the hydrostatic primitive equations are *** first model is the primitive equations with partial viscosity only,and is oriented towards large-scale wave structures in the ocean and *** second model is the viscous primitive equations with spectral eddy viscosity,and is oriented towards turbulent geophysical *** both models,the existence and uniqueness of global strong solutions are *** the second model,the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zero is also established.

关键词： primitive equations Partial viscosity Spectral-eddy viscosity