THEORETICAL RESULTS ON THE EXISTENCE,REGULARITY AND ASYMPTOTIC STABILITY OF ENHANCED PULLBACK ATTRACTORS:APPLICATIONS TO 3D PRIMITIVE EQUATIONS

Several new concepts of enhanced pullback attractors for nonautonomous dynamical systems are introduced here by uniformly enhancing the compactness and attraction of the usual pullback attractors over an infinite forward time-interval under strong and weak topologies.Then we provide some theoretical results for the existence,regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which can be applied to a large class of PDEs.The existence of these enhanced attractors is harder to obtain than the backward case[33],since it is difficult to uniformly control the long-time pullback behavior of the systems over the forward time-interval.As applications of our theoretical results,we consider the famous 3D primitive equations modelling the large-scale ocean and atmosphere dynamics,and prove the existence,regularity and asymptotic stability of the enhanced pullback attractors in V×V and H^(2)×H^(2) for the time-dependent forces which satisfy some weak conditions.

ON THE RIGOROUS MATHEMATICAL DERIVATION FOR THE VISCOUS PRIMITIVE EQUATIONS WITH DENSITY STRATIFICATION

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.

ON THE 3D VISCOUS PRIMITIVE EQUATIONS OF THE LARGE-SCALE ATMOSPHERE

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].

Non-Acceleration Theorem in a Primitive Equation System: I. Acceleration of Zonal Mean Flow

Non-acceleration theorem in a primitive equation system is developed to investigate the influences of waves on the mean flow variation against external forcing. Numerical results show that mechanical forcing overwhelms thermal forcing in maintaining the mean flow in which the internal mechanical forcing associated with horizontal eddy flux of momentum plays the most important roles. Both internal forcing and external forcing are shown to be active and at the first place for the mean flow variations, whereas the forcing-induced mean meridional circulation is passive and secondary. It is also shown that the effects on mean flow of external mechanical forcing are concentrated in the lower troposphere, whereas those due to wave-mean flow interaction are more important in the upper troposphere. These act together and result in the vertically easterly shear in low latitudes and westerly shear in mid-latitudes. This vertical shear of mean flow is to some extent weakened by thermal forcing.

Elimination of Computational Systematic Errors and Improvements of Weather and Climate System Models in Relation to Baroclinic Primitive Equations

The design of a total energy conserving semi-implicit scheme for the multiple-level baroclinic primitive equation has remained an unsolved problem for a long time. In this work, however, we follow an energy perfect conserving semi-implicit scheme of a European Centre for Medium-Range Weather Forecasts (ECMWF) type sigma-coordinate primitive equation which has recently successfully formulated. Some real-data contrast tests between the model of the new conserving scheme and that of the ECMWF-type of global spectral semi-implicit scheme show that the RMS error of the averaged forecast Height at 850 hPa can be clearly improved after the first integral week. The reduction also reaches 50 percent by the 30th day. Further contrast tests demonstrate that the RMS error of the monthly mean height in the middle and lower troposphere also be largely reduced, and some well-known systematical defects can be greatly improved. More detailed analysis reveals that part of the positive contributions comes from improvements of the extra-long wave components. This indicates that a remarkable improvement of the model climate drift level can be achieved by the actual realizing of a conserving time-difference scheme, which thereby eliminates a corresponding computational systematic error source/sink found in the currently-used traditional type of weather and climate system models in relation to the baroclinic primitive equations.

LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS

In this article,we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions.The initial vacuum is allowed.

On two-dimensional large-scale primitive equations in oceanic dynamics (Ⅱ)

The initial boundary value problem for the two-dimensional primitive equations of large scale oceanic motion in geophysics is considered. It is assumed that the depth of the ocean is a positive constant. Firstly, if the initial data are square integrable, then by Fadeo-Galerkin method, the existence of the global weak solutions for the problem is obtained. Secondly, if the initial data and their vertical derivatives are all square integrable, then by Faedo-Galerkin method and anisotropic inequalities, the existerce and uniqueness of the global weakly strong solution for the above initial boundary problem are obtained.

Analysis-Prediction Experiments over Indian Region Using Primitive Equation Barotropic Model

The impact of initial guess and grid resolutions on the analysis and prediction has been investigated over the Indian region. For this purpose, an univariate objective analysis scheme and a primitive equation barotropic model have been used. The impact of initial guess and the resolutions on analysis and prediction is discussed.

Continuous Dependence for the 3D Primitive Equations of Large Scale Ocean Under Random Force

In this paper,we consider the initial-boundary value problem for the large scale three-dimensional(3D)viscous primitive equations under random force.Assuming that the random force and the heat source satisfy the some assumptions,we firstly establish rigorous a priori bounds with coefficients which depend only on boundary data,initial data and the geometry of the problem,and then with the aid of these a priori bounds,the continuous dependence of the solution on changes in the heat source is obtained.

A Regional Spectral Nested Multilevel Primitive Equation Model

By means of vertical normal modes a regional nested multilevel primitive equation model can be reduced to several sets of shallow water equations characterized by various equivalent depths. Therefore, time integration of the model in spectral form can be performed in the manner similar to those used in the spectral nested shallow water equation model case.

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

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.

A Second-Order Implicit-Explicit Scheme for the Baroclinic-Barotropic Split System of Primitive Equations

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 dynamics.In this paper,a second-order implicit-explicit(IMEX)scheme is proposed to advance the baroclinic-barotropic split system.Specifically,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 scheme.At each time step,the baroclinic velocity is first computed using an intermediate barotropic velocity.The barotropic velocity is then corrected by re-advancing the barotropic mode with an improved barotropic forcing.Finally,the layer thickness is updated by coupling the baroclinic and barotropic velocities together.In addition,numerical inconsistencies on the discretized sea surface height caused by the mode splitting are alleviated via a reconciliation process with carefully calculated flux deficits.Temporal truncation error is also analyzed to validate the second-order accuracy of the scheme.Finally,two benchmark tests from the MPAS-Ocean platform are conducted to numerically demonstrate the performance of the proposed IMEX scheme.

THE FORMULATION OF FIDELITY SCHEMES OF PHYSICAL CONSERVATION LAWS AND IMPROVEMENTS ON A TRADITIONAL SPECTRAL MODEL OF BAROCLINIC PRIMITIVE EQUATIONS FOR NUMERICAL PREDICTION

In this paper,two formulation theorems of time-difference fidelity schemes for general quadratic and cubic physical conservation laws are respectively constructed and proved,with earlier major conserving time-discretized schemes given as special cases.These two theorems can provide new mathematical basis for solving basic formulation problems of more types of conservative time- discrete fidelity schemes,and even for formulating conservative temporal-spatial discrete fidelity schemes by combining existing instantly conserving space-discretized schemes.Besides.the two theorems can also solve two large categories of problems about linear and nonlinear computational instability. The traditional global spectral-vertical finite-difference semi-implicit model for baroclinic primitive equations is currently used in many countries in the world for operational weather forecast and numerical simulations of general circulation.The present work,however,based on Theorem 2 formulated in this paper,develops and realizes a high-order total energy conserving semi-implicit time-difference fidelity scheme for global spectral-vertical finite-difference model of baroclinic primitive equations.Prior to this,such a basic formulation problem remains unsolved for long,whether in terms of theory or practice.The total energy conserving semi-implicit scheme formulated here is applicable to real data long-term numerical integration. The experiment of thirteen FGGE data 30-day numerical integration indicates that the new type of total energy conserving semi-implicit fidelity scheme can surely modify the systematic deviation of energy and mass conserving of the traditional scheme.It should be particularly noted that,under the experiment conditions of the present work,the systematic errors induced by the violation of physical laws of conservation in the time-discretized process regarding the traditional scheme designs(called type Z errors for short)can contribute up to one-third of the total systematic root-mean-square(RMS)error at the end of second week of the integration and exceed one half of the total amount four weeks afterwards.In contrast,by realizing a total energy conserving semi-implicit fidelity scheme and thereby eliminating corresponding type Z errors, roughly an average of one-fourth of the RMS errors in the traditional forecast cases can be reduced at the end of second week of the integration,and averagely more than one-third reduced at integral time of four weeks afterwards.In addition,experiment results also reveal that,in a sense,the effects of type Z errors are no less great than that of the real topographic forcing of the model.The prospects of the new type of total energy conserving fidelity schemes are very encouraging.

Well-posedness for the stochastic 2D primitive equations with Lévy noise

The two-dimensional primitive equations with Lévy noise are studied in this paper.We prove the existence and uniqueness of the solutions in a fixed probability space which based on a priori estimates,weak convergence method and monotonicity arguments.

Quasi-hydrostatic Primitive Equations for Ocean Global Circulation Models

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.

Markov selection and W-strong Feller for 3D stochastic primitive equations

This paper studies some analytical properties of weak solutions of 3D stochastic primitive equations with periodic boundary conditions. The martingale problem associated to this model is shown to have a family of solutions satisfying the Markov property, which is achieved by means of an abstract selection principle. The Markov property is crucial to extend the regularity of the transition semigroup from small times to arbitrary times. Thus, under a regular additive noise, every Markov solution is shown to have a property of continuous dependence on initial conditions, which follows from employing the weak-strong uniqueness principle and the Bismut-Elworthy-Li formula.

Large Deviations for 2D Primitive Equations Driven by Multiplicative Lévy Noises

In this paper,we establish a large deviation principle for two-dimensional primitive equations driven by multiplicative Lévy noises.The proof is based on the weak convergence approach.

A Fourth Order Numerical Method for the Primitive Equations Formulated in Mean Vorticity

A fourth-order finite difference method is proposed and studied for the primitive equations(PEs)of large-scale atmospheric and oceanic flow based on mean vorticity formulation.Since the vertical average of the horizontal velocity field is divergence-free,we can introduce mean vorticity and mean stream function which are connected by a 2-D Poisson equation.As a result,the PEs can be reformulated such that the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of the horizontal velocity.The mean vorticity equation is approximated by a compact difference scheme due to the difficulty of the mean vorticity boundary condition,while fourth-order long-stencil approximations are utilized to deal with transport type equations for computational convenience.The numerical values for the total velocity field(both horizontal and vertical)are statically determined by a discrete realization of a differential equation at each fixed horizontal point.The method is highly efficient and is capable of producing highly resolved solutions at a reasonable computational cost.The full fourth-order accuracy is checked by an example of the reformulated PEs with force terms.Additionally,numerical results of a large-scale oceanic circulation are presented.

Algorithm for simulating ocean circulation on a quantum computer

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 variables.In this paper,we present for the first time an algorithm for simulating ocean circulation on a quantum computer to achieve a computational speedup.Our approach begins with using primitive equations describing the ocean dynamics and then discretizing these equations in time and space.It results in several linear system of equations(LSE)with sparse coefficient matrices.We solve these sparse LSE using the variational quantum linear solver that enables the present algorithm to run easily on near-term quantum computers.Additionally,we develop a scheme for manipulating the data flow in the algorithm based on the quantum random access memory and l∞norm tomography technique.The efficiency of our algorithm is verified using multiple platforms,including MATLAB,a quantum virtual simulator,and a real quantum computer.The impact of the number of shots and the noise of quantum gates on the solution accuracy is also discussed.Our findings demonstrate that error mitigation techniques can efficiently improve the solution accuracy.With 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.

ＲＥＧＵＬＡＲＩＴＹ ＲＥＳＵＬＴＳ ＦＯＲ ＬＩＮＥＡＲＥＬＬＩＰＴＩＣ ＰＲＯＢＬＥＭＳ ＲＥＬＡＴＥＤＴＯ ＴＨＥ ＰＲＩＭＩＴＩＶＥ ＥＱＵＡＴＩＯＮＳ

The authors study the regularity of soutions of the GFD-Stokes problem and of some second order linear elliptic partial differential equations related to the PrimitiveEquations of the ocean .The present work generallizes the regularity results in[18] by taking into consideraion the non- homogeneous boundary conditions and teh dependence of solutions on the thickness of the domain occupied by the ocean and its varying bottom topography. These regularity results are important tools in the study of the PEs(see e.g.[6]), and they seem also to possess their own interest.