Recent Advances Concerning Certain Class of Geophysical Flows

Nonlinearity, 2001

The zero-Rossby limit for the primitive equations (PE) governing the atmospheric motions is analyzed. The limit is important in geophysics for large scale models (cf. Lions [13]) and is in the level of the zero relaxation limit for nonlinear partial differential equations (cf.

2002

Journal of Mathematical Fluid Mechanics, 2020

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced 3D primitive geostrophic adjustment model with weak dissipation. The term reduced model stems from the fact that the relevant physical quantities depends only on two spatial variables. The additional weak dissipation helps us overcome the ill-posedness of original model. We also prove the global well-posedness of the strong solution to the Voigt α-regularization of this model, and establish the convergence of the strong solution of the Voigt α-regularized model to the corresponding solution of original model. Furthermore, we derive a criterion for finite-time blow-up of reduced 3D primitive geostrophic adjustment model with weak dissipation based on Voigt α-regularization.

Applicable Analysis, 2006

A reformulation of the planetary geostrophic equations (PGEs) with inviscid balance equation is proposed and the existence of global weak solutions is established, provided that the mechanical forcing satisfies an integral constraint. There is only one prognostic equation for the temperature field and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition. Furthermore, the velocity profile can be accurately represented as a functional of the temperature gradient. In particular, the vertical velocity depends only on the first order derivative of the temperature. As a result, the bound for the L ∞ (0, t 1 ; L 2) ∩ L 2 (0, t 1 ; H 1) norm of the temperature field is sufficient to show the existence of the weak solution.

Communications in Mathematical Physics, 2012

The three-dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.

2006

Preface page xi 1 Barotropic geophysical flows and two-dimensional fluid flows: elementary introduction 1.1 Introduction 1.2 Some special exact solutions t 8 1.3 Conserved quantities 1.4 Barotropic geophysical flows in a channel domain-an important physical model 1.5 Variational derivatives and an optimization principle for elementary geophysical solutions 1.6 More equations for geophysical flows References 2 The response to large-scale forcing \ 2.1 Introduction 2.2 Non-linear stability with Kolmogorov forcing 2.3 Stability of flows with generalized Kolmogorov forcing References 3 The selective decay principle for basic geophysical flows 3.1 Introduction 3.2 Selective decay states and their invariance 3.3 Mathematical formulation of the selective decay principle 3.4 Energy-enstrophy decay 3.5 Bounds on the Dirichlet quotient, A(t) 3.6 Rigorous theory for selective decay 3.7 Numerical experiments demonstrating facets of selective decay References vi Contents A.l Stronger controls on A(?) A.2 The proof of the mathematical form of the selective decay principle in the presence of the beta-plane effect 4 Non-linear stability of steady geophysical flows 4.1 Introduction 4.2 Stability of simple steady states 4.3 Stability for more general steady states 4.4 Non-linear stability of zonal flows on the beta-plane 4.5 Variational characterization of the steady states References 5 Topographic mean flow interaction, non-linear instability, and chaotic dynamics 5.1 Introduction 5.2 Systems with layered topography 5.3 Integrable behavior 5.4 A limit regime with chaotic solutions 5.5 Numerical experiments • References Appendix 1 Appendix 2

SIAM Journal on Mathematical Analysis, 2001

Geophysical fluids all exhibit a common feature: their aspect ratio (depth to horizontal width) is very small. This leads to an asymptotic model widely used in meteorology, oceanography, and limnology, namely the hydrostatic approximation of the time-dependent incompressible Navier-Stokes equations. It relies on the hypothesis that pressure increases linearly in the vertical direction. In the following, we prove a convergence and existence theorem for this model by means of anisotropic estimates and a new time-compactness criterium.

Annali di Matematica Pura ed Applicata, 2005

In this article, we study the baroclinic flow in the primitive equations (PEs) of the ocean, which are known to be the fundamental equations of the ocean, [4]-[8]. We prove that the magnitude of the baroclinic flow in the L 2-norm is of order O(δ), where δ is the aspect ratio of the ocean. Some numerical simulations of the PEs of the ocean consistent with these estimates are also presented.

Journal of Mathematical Physics, 2012

We study a viscous two-layer quasi-geostrophic beta-plane model that is forced by imposition of a spatially uniform vertical shear in the eastward (zonal) component of the layer flows, or equivalently a spatially uniform north-south temperature gradient. We prove that the model is linearly unstable, but that non-linear solutions are bounded in time by a bound which is independent of the initial data and is determined only by the physical parameters of the model. We further prove, using arguments first presented in the study of the Kuramoto-Sivashinsky equation, the existence of an absorbing ball in appropriate function spaces, and in fact the existence of a compact finitedimensional attractor, and provide upper bounds for the fractal and Hausdorff dimensions of the attractor. Finally, we show the existence of an inertial manifold for the dynamical system generated by the model's solution operator. Our results provide rigorous justification for observations made by Panetta based on long-time numerical integrations of the model equations.

Communications in Mathematical Physics, 2015

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 threedimensional 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.

Journal of Mathematical Analysis and Applications, 1999

In this paper we study the long time behavior of the energy of solutions to the Boussinesq, planetary geostrophic, and primitive equations. The equations are considered in the whole space R 3 . The asymptotic behavior will depend on the type of data and how many damping constants are nonzero in the equations. In several cases we are able to establish an algebraic rate of decay of the same order as the solutions of the underlying linear equations. In the case with less damping our results establish that either the energy of the solutions decays with no rate to an equilibrium or it will be oscillating.

Journal of the Atmospheric Sciences, 2011

Dynamical influence of moist convection upon development of the barotropic instability is studied in the rotating shallow-water model. First, an exhaustive linear ''dry'' stability analysis of the Bickley jet is performed, and the most unstable mode identified in this way is used to initialize simulations to compare the development and the saturation of the instability in dry and moist configurations. High-resolution numerical simulations with a well-balanced finite-volume scheme reveal substantial qualitative and quantitative differences in the evolution of dry and moist-convective instabilities. The moist effects affect both balanced and unbalanced components of the flow. The most important differences between dry and moist evolution are 1) the enhanced efficiency of the moist-convective instability, which manifests itself by the increase of the growth rate at the onset of precipitation, and by a stronger deviation of the end state from the initial one, measured with a number of different norms; 2) a pronounced cyclone-anticyclone asymmetry during the nonlinear evolution of the moist-convective instability, which leads to an additional, with respect to the dry case, geostrophic adjustment, and the modification of the end state; and 3) an enhanced ageostrophic activity in the precipitation zones but also in the nonprecipitating areas because of the secondary geostrophic adjustment.

2003

Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. D. BRESCH B. DESJARDINS DMA-02-31 Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model.

Communications on Pure and Applied Mathematics, 2003

In this paper we consider a three-dimensional planetary geostrophic viscous model of the gyre-scale mid-latitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite-dimensional global attractor to this dissipative evolution system.

Dynamics of Atmospheres and Oceans, 1990

Buffoni, G. and Griffa, A., 1990. The finite-difference barotropic vorticity equation in ocean circulation modeling: basic properties of the solutions. Dyn. Atmos. Oceans, 15: 1-33. We study a discrete form of the barotropic vorticity equation used in the numerical studies of the ocean general circulation. Our goal is to contribute to the understanding of the numerical solutions, introducing some new tools of investigation. We begin by examining the basic properties of the discrete solutions for time-independent forcing. All the solutions are shown to be ultimately bounded in a finite region in the phase space and an estimate of the energy and enstrophy bounds is made. Steady-state solutions are shown always to exist for arbitrary forcing and non-zero dissipation. A sufficient condition of asymptotic stability and uniqueness for the steady-state solutions is derived. The limit solutions for zero dissipation and infinite forcing are considered. For the infinite forcing case, a class of asymptotic steady-state solutions ~k is recovered, proportional to the eigenvectors of the Laplacian operator. In this limit, the discrete problem is shown to be different from the continuous problem that allows for a wider class of asymptotic solutions. The analytical results have been used to study a number of numerical solutions computed for the single-gyre .and double-gyre wind stresses and for various values of the forcing and dissipation parameters. We focus on solutions at high non-linearity, obtained either for decreasing dissipation or for increasing forcing. When the dissipation is decreased, the energy of the solutions increases at a slower rate than the upper bound. For sufficiently small values of the dissipation parameters, the steady-state solutions become unstable and the computed solutions are time dependent. The transition to instability is characterized by the rapid increase of the non-linear contribution to the Jacobian matrix. When the forcing is increased, the energy of the solutions increases and tends to the theoretical bound. These solutions are always stationary and asymptotically stable. The non-linear contribution to the Jacobian matrix increases linearly with the forcing. The steady-state solutions in this case converge to the asymptotic solutions ~1 and ~z for the single-gyre and the double-gyre respectively.