Primitive Equations

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Edriss S. Titi - One of the best experts on this subject based on the ideXlab platform.

  • Zero Mach Number Limit of the Compressible Primitive Equations: Well-Prepared Initial Data
    Archive for Rational Mechanics and Analysis, 2020
    Co-Authors: Edriss S. Titi
    Abstract:

    This work concerns the zero Mach number limit of the compressible Primitive Equations. The Primitive Equations with the incompressibility condition are identified as the limiting Equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order $$ \mathcal {O}(\varepsilon ) $$ O ( ε ) , as $$ \varepsilon \rightarrow 0^+ $$ ε → 0 + , where $$ \varepsilon $$ ε represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible Primitive Equations, which are close to the incompressible flows.

  • Zero Mach Number Limit of the Compressible Primitive Equations: Well-Prepared Initial Data
    Archive for Rational Mechanics and Analysis, 2020
    Co-Authors: Xin Liu, Edriss S. Titi
    Abstract:

    Author(s): Liu, Xin; Titi, Edriss S | Abstract: This work concerns the zero Mach number limit of the compressible Primitive Equations. The Primitive Equations with the incompressibility condition are identified as the limiting Equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order $ \mathcal O(\varepsilon) $, as $ \varepsilon \rightarrow 0^+ $, where $ \varepsilon $ represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible Primitive Equations, which are close to the incompressible flows.

  • Local Well-posedness of Strong Solutions to the Three-dimensional Compressible Primitive Equations
    arXiv: Analysis of PDEs, 2019
    Co-Authors: Xin Liu, Edriss S. Titi
    Abstract:

    Author(s): Liu, Xin; Titi, Edriss S | Abstract: This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible Primitive Equations of atmospheric dynamics. It is shown that strong solutions exist, unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity. We also introduce the free boundary problem for the compressible Primitive Equations.

  • Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics
    Communications in Mathematical Physics, 2015
    Co-Authors: Slim Ibrahim, Kenji Nakanishi, Edriss S. Titi
    Abstract:

    In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous Primitive Equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) Primitive Equations, if they exist, they blow up in finite time. Specifically, we consider the three-dimensional inviscid Primitive Equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom boundaries. For certain class of initial data we reduce this system into the two-dimensional system of Primitive Equations in an infinite horizontal strip with the same type of boundary conditions; and then we show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.

  • finite time blowup for the inviscid Primitive Equations of oceanic and atmospheric dynamics
    arXiv: Analysis of PDEs, 2012
    Co-Authors: Slim Ibrahim, Kenji Nakanishi, Edriss S. Titi
    Abstract:

    In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous Primitive Equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) Primitive Equations blow up in finite time. Specifically, we consider the three-dimensional inviscid Primitive Equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom, boundaries. For certain class of initial data we reduce this system into the two-dimensional system of Primitive Equations in an infinite horizontal strip with the same type of boundary conditions; and then show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.

Roger Temam - One of the best experts on this subject based on the ideXlab platform.

  • Numerical Simulations of the Two-Dimensional Inviscid Hydrostatic Primitive Equations with Humidity and Saturation
    Journal of Scientific Computing, 2020
    Co-Authors: Arthur Bousquet, Roger Temam, Youngjoon Hong, Joseph Tribbia
    Abstract:

    The two-dimensional inviscid hydrostatic Primitive Equations of the atmosphere with humidity and saturation are considered in the presence of topography. The model studied here describes the dynamics of the air or water in order to approximate global atmospheric flows. The heart of the paper is to derive a new set of transformed inviscid Primitive Equations using a version of the terrain-following coordinate systems and to develop an accurate numerical scheme to the Equations. In this regard, a fully discrete numerical algorithm based on a Godunov-type finite volume method is proposed and its convergence tested. We then use this algorithm to simulate the flows above a mountain using the terrain-following coordinate system with a dynamic bottom pressure.

  • Averaging method applied to the three-dimensional Primitive Equations.
    Discrete and Continuous Dynamical Systems, 2016
    Co-Authors: Madalina Petcu, Roger Temam, D. Wirosoetisno
    Abstract:

    In this article we study the small Rossby number asymptotics for the three-dimensional Primitive Equations of the oceans and of the atmosphere. The fast oscillations present in the exact solution are eliminated using an averaging method, the so-called renormalisation group method.

  • Finite Dimensions of the Global Attractor for 3D Primitive Equations with Viscosity
    Journal of Nonlinear Science, 2014
    Co-Authors: Roger Temam
    Abstract:

    For a set of non-periodic boundary conditions, we prove the uniform boundedness of the \(H^2\) norms of the solutions of the 3D Primitive Equations with viscosity. An absorbing set of the solutions in \(H^2\) is also obtained. As an application of this result, we prove also the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solutions of the 3D Primitive Equations with viscosity. Our results also improve the existing results for the case with periodic boundary conditions.

  • Asymptotic analysis for the 3D Primitive Equations in a channel
    Discrete & Continuous Dynamical Systems - S, 2013
    Co-Authors: Makram Hamouda, Chang-yeol Jung, Roger Temam
    Abstract:

    In this article, we give an asymptotic expansion, with respect to the viscosity which is considered here to be small, of the solutions of the $3D$ linearized Primitive Equations (EPs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice of the non-local boundary conditions for the non-viscous PEs.

  • Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise
    Nonlinearity, 2012
    Co-Authors: Arnaud Debussche, Roger Temam, Nathan Glatt-holtz, Mohammed Ziane
    Abstract:

    The Primitive Equations are a basic model in the study of large scale Oceanic and Atmospheric dynamics. These systems form the analytical core of the most advanced General Circulation Models. For this reason and due to their challenging nonlinear and anisotropic structure the Primitive Equations have recently received considerable attention from the mathematical community. In view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential Equations in connection with the Primitive Equations and more generally. In this work we study a stochastic version of the Primitive Equations. We establish the global existence of strong, pathwise solutions for these Equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, $L^{p}_{t}L^{q}_{x}$ estimates on the pressure and stopping time arguments.

Boling Guo - One of the best experts on this subject based on the ideXlab platform.

  • The exponential behavior of 3D stochastic Primitive Equations driven by fractional noise
    2020
    Co-Authors: Lidan Wang, Guoli Zhou, Boling Guo
    Abstract:

    In this article, we study the exponential behavior of 3D stochastic Primitive Equations driven by fractional noise. Since fractional Brownian motion is essentially different from Brownian motion, the standard method via classic stochastic analysis tools is not available. Here, we develop a method which is close to the method from dynamic system to show that the weak solutions to 3D stochastic Primitive Equations driven by fractional noise converge exponentially to the unique stationary solution of Primitive Equations. This method may be applied to other stochastic hydrodynamic Equations and other noises including Brownian motion and Levy noise.

  • Stochastic 2D Primitive Equations: Central limit theorem and moderate deviation principle
    Computers & Mathematics with Applications, 2019
    Co-Authors: Rangrang Zhang, Guoli Zhou, Boling Guo
    Abstract:

    Abstract In this paper, we establish a central limit theorem and a moderate deviation for two-dimensional stochastic Primitive Equations driven by multiplicative noise. This is the first result about the limit theorem and the moderate deviations for stochastic Primitive Equations. The proof of the results relies on the weak convergence method and some delicate and careful a p r i o r i estimates.

  • On the backward uniqueness of the stochastic Primitive Equations with additive noise
    Discrete & Continuous Dynamical Systems - B, 2019
    Co-Authors: Boling Guo, Guoli Zhou
    Abstract:

    The previous works focus on the uniqueness for the initial-value problems of stochastic Primitive Equations. Uniqueness for the initial-value problems means that if the two initial conditions are the same, then the two solutions coincide with each other. However there is no work to answer what will happen to the solutions if the two initial conditions are different. This problem for the stochastic three dimensional Primitive Equations is addressed by the backward uniqueness established in this article. The backward uniqueness means that if two solutions intersect at time $t>0, $ then they are equal everywhere on the interval $(0, t).$ In other words, given two different initial-value conditions, the corresponding two solutions will never cross in the future. Hence this article can be viewed as a further study of the dependence of the solutions on the initial data.

  • Diffusion limit of 3D Primitive Equations of the large-scale ocean under fast oscillating random force
    Journal of Differential Equations, 2015
    Co-Authors: Boling Guo, Daiwen Huang, Wei Wang
    Abstract:

    Abstract The three-dimensional (3D) viscous Primitive Equations describing the large-scale oceanic motions under fast oscillating random perturbation are studied. Under some assumptions on the random force, the solution to the initial boundary value problem (IBVP) of the 3D random Primitive Equations converges in distribution to that of IBVP of the limiting Equations, which are the 3D stochastic Primitive Equations describing the large-scale oceanic motions under a white in time noise forcing. This also implies the convergence of the stationary solution of the 3D random Primitive Equations.

  • existence of the universal attractor for the 3 d viscous Primitive Equations of large scale moist atmosphere
    Journal of Differential Equations, 2011
    Co-Authors: Boling Guo, Daiwen Huang
    Abstract:

    Abstract In this paper, we consider the initial–boundary value problem for the three-dimensional viscous Primitive Equations of large-scale moist atmosphere which are used to describe the turbulent behavior of long-term weather prediction and climate changes. By obtaining the existence and uniqueness of global strong solutions for the problem and studying the long-time behavior of strong solutions, we prove the existence of the universal attractor for the dynamical system generated by the Primitive Equations of large-scale moist atmosphere.

Rangrang Zhang - One of the best experts on this subject based on the ideXlab platform.

D. Wirosoetisno - One of the best experts on this subject based on the ideXlab platform.

  • Averaging method applied to the three-dimensional Primitive Equations.
    Discrete and Continuous Dynamical Systems, 2016
    Co-Authors: Madalina Petcu, Roger Temam, D. Wirosoetisno
    Abstract:

    In this article we study the small Rossby number asymptotics for the three-dimensional Primitive Equations of the oceans and of the atmosphere. The fast oscillations present in the exact solution are eliminated using an averaging method, the so-called renormalisation group method.

  • Slow Manifolds and Invariant Sets of the Primitive Equations
    Journal of the Atmospheric Sciences, 2011
    Co-Authors: Roger Temam, D. Wirosoetisno
    Abstract:

    Abstract The authors review, in a geophysical setting, several recent mathematical results on the forced–dissipative hydrostatic Primitive Equations with a linear equation of state in the limit of strong rotation and stratification, starting with existence and regularity (smoothness) results and describing their implications for the long-time behavior of the solution. These results are used to show how the solution of the Primitive Equations in a periodic box comes close to geostrophic balance as t → ∞. Then a review follows of how geostrophic balance could be extended to higher orders in the Rossby number, and it is shown that the solution of the Primitive Equations also satisfies a higher-order balance up to an exponentially small error. Finally, the connection between balance dynamics in the Primitive Equations and its global attractor, which is the only known invariant set (for a sufficiently general forcing), is discussed.

  • Stability of the slow manifold in the Primitive Equations
    arXiv: Analysis of PDEs, 2008
    Co-Authors: Roger Temam, D. Wirosoetisno
    Abstract:

    We show that, under reasonably mild hypotheses, the solution of the forced--dissipative rotating Primitive Equations of the ocean loses most of its fast, inertia--gravity, component in the small Rossby number limit as $t\to\infty$. At leading order, the solution approaches what is known as "geostrophic balance" even under ageostrophic, slowly time-dependent forcing. Higher-order results can be obtained if one further assumes that the forcing is time-independent and sufficiently smooth. If the forcing lies in some Gevrey space, the solution will be exponentially close to a finite-dimensional "slow manifold" after some time.

  • Exponential approximations for the Primitive Equations of the ocean
    arXiv: Analysis of PDEs, 2006
    Co-Authors: Roger Temam, D. Wirosoetisno
    Abstract:

    We show that in the limit of small Rossby number $\eps$, the Primitive Equations of the ocean (OPEs) can be approximated by ``higher-order quasi-geostrophic Equations'' up to an exponential accuracy in $\eps$. This approximation assumes well-prepared initial data and is valid for a timescale of order one (independent of $\eps$). Our construction uses Gevrey regularity of the OPEs and a classical method to bound errors in higher-order perturbation theory.

  • Renormalization group method applied to the Primitive Equations
    Journal of Differential Equations, 2005
    Co-Authors: Madalina Petcu, Roger Temam, D. Wirosoetisno
    Abstract:

    Abstract In this article we study the limit, as the Rossby number e goes to zero, of the Primitive Equations of the atmosphere and the ocean. From the mathematical viewpoint we study the averaging of a penalization problem displaying oscillations generated by an antisymmetric operator and by the presence of two time scales.