Some Mathematical Problems in Geophysical Fluid Dynamics

Chapter 6 Some mathematical problems in geophysical fluid dynamics

Boundary Value Problems for the Inviscid Primitive Equations in Limited Domains

<p>The matrix model for the barotropic vorticity equation on the torus and the 2-sphere, introduced by Zeitlin, remains a reference discretization, since it provides N conserved quantities with N degrees of freedom. Modin and Vivani recently also demonstrated its relevance for the numerical study of geophysical fluid dynamics. The origins of the discretization and its connection to the Moyal bracket of quantum mechanics are, however, somewhat mysterious, hampering the prospect of generalizing the ansatz to the shallow water and primitive equations. We show how the matrix model can be understood in the framework of variational, structure preserving discretizations of fluids introduced by Pavlov and co-workers, which has recently been extended to the finite element setting by Natale and Cotter as well as Gay-Balmaz and Gawlik. Pavlov et al.’s approach is to discretely mirror the continuous theory, where the dynamics take place in the space of (divergence free) vector fields, i.e. the Lie algebra of the (volume preserving) diffeomorphism group, and the reduced Euler-Poincaré variational principle yields the dynamical equations. Specifically, one considers the representation of the group and its Lie algebra on a finite dimensional function space, i.e. through their action on scalar functions, yielding an appropriate matrix group and Lie algebra as discrete configuration space. Because of the finite dimensional setting, one has to deviate at this point from the continuous theory and introduce a non-holonomic constraint, which amounts to restricting the finite dimensional Lie algebra to elements that correspond to vector fields. The Euler-Poincaré-d’Alembert principle has consequently also to be used to obtain semi-discrete time evolution equations. A modification of this methodology is to insist on the Euler-Poincaré theory from the continuous side and modify how the Lie algebra is discretized so that it remains applicable. Specifically, one can start with the action of a symmetry group on the configuration space, e.g. SO(3) on the 2-sphere, and consider the associated infinitesimal action of the Lie algebra on functions, which corresponds to vector fields, as in the approach by Pavlov et al. When the action admits a momentum map, it can equivalently be written using the Poisson bracket and Hamiltonians linear in the Lie algebra. Building on this and requiring that a generalization of the action on functions beyond linear Hamiltonians should be consistent with the group action, one is led to the iterated action of the Poisson algebra, which is equivalent to the Moyal bracket Lie algebra for the symmetry group (through the universal enveloping algebra of the original Lie algebra). When one then fixes a finite-dimensional spectral basis to discretize functions, this corresponds to a sub-algebra of gl(n). Finally, using Euler-Poincaré theory, as in the continuous case, on this Lie sub-algebra, one obtains the matrix model by Zeitlin that retains N conserved quantities for N degrees of freedom. We hope that our rationalization of the derivation of the matrix model opens up the possibility to generalize it to other equations for geophysical fluid dynamics, and we discuss possible directions for the shallow water and primitive equations.</p>

On the one hand, the primitive three-dimensional viscous equations for large-scale ocean and atmosphere dynamics are commonly used in weather and climate predictions. On the other hand, ever since the middle of the last century, it has been widely recognized that the climate variability exhibits long-time memory. In this paper, we first prove the global existence of weak solutions to the primitive equations of large-scale ocean and atmosphere dynamics with Caputo fractional time derivatives. Then we establish the existence of an absorbing set, which is positively invariant. Finally, an attractor (strictly speaking, the minimal attracting set containing all the limiting dynamics) is constructed for the time fractional primitive equations, which means that the present state of a system may have long-time influences on the states in far future. However, there was no work on the long-time behavior of the time fractional primitive equations and we fill this gap in this paper.

The purpose of this study is to present a new numerical treatment of the boundary conditions on the open boundary, which removes the non–physical reflection of the wave. In order to describe the motion of air in the earth's atmosphere, a numerical solution of the averaged primitive flow equations is considered in this paper. Appropriate initial and boundary conditions should be imposed on the equations for the solution to be uniquely determined. Our particular interest is the boundary conditions on an open boundary where the flow comes in or out. It is known that conventional treatment of the boundary conditions causes non–physical reflection. To determine suitable boundary conditions for primitive equations, the theory of characteristics provides guidance on the number and form of the boundary conditions. We determined the boundary conditions by the theory of characteristics. The primitive equations are discretised using the second–order Taylor–Galerkin method with linear finite elements, and boundary values are computed by the method of characteristics. We consider simple numerical experiments of one– and two–dimensional problems. As a result, it is concluded that spurious oscillation is eliminated.

In this article we consider the Primitive Equations without horizontal viscosity but with a mild vertical viscosity added in the hydrostatic equation, as in [13] and [16], which are the so-called $\delta-$Primitive Equations. We prove that the problem is well posed in the sense of Hadamard in certain types of spaces. This means that we prove the finite-in-time existence, uniqueness and continuous dependence on data for appropriate solutions. The results given in the 3D periodic space easily extend to dimension 2. We also consider a Boussinesq type of equation, meaning that the mild vertical viscosity present in the hydrostatic equation is replaced by the time derivative of the vertical velocity. We prove the same type of results as for the $\delta-$Primitive Equations; periodic boundary conditions are similarly considered.

This survey is concerned with the new developments on existence and uniqueness of solutions of some basic models in atmospheric dynamics, such as two- and three-dimensional quasi-geostrophic models and three-dimensional balanced model. The main aim of this paper is to introduce some results about the global and local (with respect to time) existence of solutions given by the authors in recent years, but others' important contributions and the literature on this subject are also quoted. We discuss briefly the relationships among the existence and uniqueness, physical instatibility and computational instability. In the appendixes, some key mathematical techniques in obtaining our results are presented, which are of vital importance to other problems in geophysical fluid dynamics as well.

Finite element solutions of the primitive equation (PE) form of the shallow water equations are notorious for the severe spurious 2Δx modes which appear. Wave equation (WE) solutions do not exhibit these numerical modes. In this paper we show that the severe spurious modes in PE solutions are strongly influenced by essential normal flow boundary conditions in the coupled continuity‐momentum system of equations. This is demonstrated through numerical examples that avoid the use of essential normal flow boundary conditions either by specifying elevation values over the entire boundary or by implementing natural flow boundary conditions in the weak weighted residual form of the continuity equation. Results from a series of convergence tests show that PE solutions are of nearly the same quality as WE solutions when spurious modes are suppressed by alternative specification of the boundary conditions. Network intercomparisons indicate that varying nodal support does not excite spurious modes in a solution, although it does enhance the spurious modes introduced when an essential normal flow boundary condition is used.Dispersion analysis of discrete equations for interior and boundary nodes offers an explanation of the observed solution behaviour. For certain PE algorithms a mixed situation can arise where the boundary nodes exhibit a monotonic (noise‐free) dispersion relationship and the interior nodes exhibit a folded (noisy) dispersion relationship. We have found that the mixed situation occurs when all boundary nodes are specified elevation nodes (which are enforced as essential conditions in the continuity equation) or when specified flow boundary nodes are treated as natural boundary conditions in the continuity equation. In either case the effect is to generate a solution that is essentially free of noise. Apparently, the monotonic dispersion behaviour at the boundaries suppresses the otherwise noisy behaviour caused by the folded dispersion relation on the interior.

The proofs of the existence, uniqueness, smoothness, and stability of solutions of problems in fluid dynamics are needed to give meaning to the equations and corresponding initial and boundary conditions that govern these problems. For any arbitrary reasonable choice of a class of admissible initial data, a problem in fluid dynamics must be well posed (in the Hadamard sense [1]). This means that (a) the problem has a solution for any initial data in this class, (b) this solution is unique for any initial conditions, (c) the solution depends continuously on the initial data. In this paper we give a survey of some aspects of problems on well-posedness from the point of view of fluid dynamics itself; these problems form a very difficult and at the same time important part of fluid mechanics.

The workshop “Geophysical Fluid Dynamics” addressed recent advances in analytical, stochastic, modeling and computational studies of geophysical rotating fluids models. Of particular interest on the analytical and stochastic sides were the contributions concerning dispersive mechanism, regularity verses finite-time formation of singularities of certain viscous and inviscid geostrophic models, the primitive equations, Boussinesq approximation, boundary layers and fast rotating fluids. Model reductions, based on asymptotic, scaling analysis and variational methods, were presented. In addition, computational investigations were provided in support of the claim that three-dimensional geophysical turbulent flows exhibit two-dimensional features, at small Rosby numbers.

The book explains the key notions and fundamental processes in the dynamics of the fluid envelopes of the Earth (transposable to other planets), and methods of their analysis, from the unifying viewpoint of rotating shallow-water model (RSW). The model, in its one- or two-layer versions, plays a distinguished role in geophysical fluid dynamics, having been used for around a century for conceptual understanding of various phenomena, for elaboration of approaches and methods, to be applied later in more complete models, for development and testing of numerical codes and schemes of data assimilations, and many other purposes. Principles of modelling of large-scale atmospheric and oceanic flows, and corresponding approximations, are explained and it is shown how single- and multi-layer versions of RSW arise from the primitive equations by vertical averaging, and how further time-averaging produces celebrated quasi-geostrophic reductions of the model. Key concepts of geophysical fluid dynamics are exposed and interpreted in RSW terms, and fundamentals of vortex and wave dynamics are explained in Part 1 of the book, which is supplied with exercises and can be used as a textbook. Solutions of the problems are available at Editorial Office by request. In-depth treatment of dynamical processes, with special accent on the primordial process of geostrophic adjustment, on instabilities in geophysical flows, vortex and wave turbulence and on nonlinear wave interactions follows in Part 2. Recently arisen new approaches in, and applications of RSW, including moist-convective processes constitute Part 3.

Simulations of the 2.5D inviscid primitive equations in a limited domain

The regularity of solutions of the primitive equations of the ocean in space dimension three

National audience

Global strong Lp well-posedness of the 3D primitive equations with heat and salinity diffusion